Comparative Analysis of Deterministic and Nondeterministic Decision Trees (Intelligent Systems Reference Library, 179) 3030417271, 9783030417277

Table of contents :
Preface
Acknowledgements
Contents
1 Introduction
1.1 Main Directions of Study
1.1.1 Decision Tables with Many-Valued Decisions
1.1.2 Local Approach to Study of Decision Trees
1.1.3 Global Approach to Study of Decision Trees
1.2 Contents of Book
1.2.1 Part I. Decision Trees for Decision Tables
1.2.2 Part II. Decision Trees for Problems. Local Approach
1.2.3 Part III. Decision Trees for Problems. Global Approach
1.2.4 Index and Notation
1.3 Use of Book
References
Part I Decision Trees for Decision Tables
2 Basic Definitions and Notation
2.1 Common Notions
2.2 Decision Tables
2.3 Schemes of Decision Trees
2.4 Different Types of Decision Trees for Decision Tables
2.5 Complexity Functions
2.6 Enumerated Signatures
References
3 Lower Bounds on Complexity of Deterministic Decision Trees for Decision Tables
3.1 Definitions and Notation
3.2 Auxiliary Statements
3.3 Lower Bounds
3.4 Approach to Proof of Lower Bounds
References
4 Upper Bounds on Complexity and Algorithms for Construction of Deterministic Decision Trees for Decision Tables. First Approach
4.1 Difference-Bounded Uncertainty Measures for Decision Tables
4.2 Process Uρ of Schema Construction
4.3 Auxiliary Statements
4.4 Upper Bounds on ψ(Uρ(T,γ,ψ))
4.5 Corollaries
4.6 Algorithm Uρ,,ψ
4.7 Algorithms Uρ,GHρ,h, Uρ,RHρm+1,h, and Uρ,Hρm+1,h
4.8 Algorithms Wρ,GHρ,h, Wρ,RHρ2,h, and Wρ,Hρ2,h
References
5 Upper Bounds and Algorithms for Construction of Deterministic Decision Trees for Decision Tables. Second Approach
5.1 Reduced Deterministic Decision Trees for Decision Tables
5.2 Additive-Bounded Uncertainty Measures for Decision Tables
5.3 Process Yρ,mathcalN of Schema Construction
5.4 Upper Bounds on ψρd(T)
5.5 Upper Bounds on hρd(T)
5.6 Algorithm Yρ,η,,ψ
References
6 Bounds on Complexity and Algorithms for Construction of Nondeterministic and Strongly Nondeterministic Decision Trees for Decision Tables
6.1 Bounds on ψρa(T) and ψρs(T)
6.2 Approach to Proof of Lower Bounds on ψρs(T) and ψρa(T)
6.3 Algorithms Vρ,,ψa and Vρ,,ψs
Reference
7 Closed Classes of Boolean Functions
7.1 Main Definitions and Notation
7.2 Auxiliary Statements
7.3 Bounds for Individual Closed Classes
7.4 Bounds for Arbitrary Closed Classes
References
8 Algorithmic Problems
8.1 Relationships Among Algorithmic Problems
8.2 Examples of Decidable and Undecidable Problems
8.3 Possible Variants of Algorithmic Problem Behavior
8.4 Examples of Problems with Polynomial Complexity
8.5 Examples of NP-Hard Problems
References
Part II Decision Trees for Problems. Local Approach
9 Basic Definitions and Notation
9.1 Information Systems
9.2 Decision Trees
9.3 Problem Schemes and Problems
9.4 Decision Trees Solving Problems
9.5 Decision Tree Schemes Corresponding to Problem Schemes
9.6 Complexity Functions
9.7 Matrices of Upper and Lower Local Bounds for Sccf-Triple
9.8 Types of Functions and Sccf-Triples
References
10 Main Reductions
10.1 Decision Trees and Decision Tree Schemes
10.2 Arbitrary Classes of Information Systems and One-Element Classes of Information Systems
10.3 Upper and Lower Bounds
References
11 Functions on Main Diagonal and Below
11.1 Function τii
11.2 Function τdi
11.3 Function τdd
11.4 Functions τai, τad, τaa, τsi, τsd, τsa, and τss
References
12 Functions Over Main Diagonal
12.1 Functions τid, τia, and τis
12.2 Function τas
12.3 Function τds
12.4 Function τda
References
13 Local Upper Types of Restricted Sccf-Triples
13.1 Possible Local Upper Types
13.2 Examples of Restricted Sccf-Triples
13.3 Main Statements
Reference
14 Bounds Inside Types
14.1 Set ρ(1)
14.2 Set ρ(2)
14.3 Set ρ(3)
14.4 Set ρ(4)
14.5 Set ρ(5)
14.6 Set ρ(6)
Reference
15 Matrices of Lower Local Bounds
15.1 Possible Local Lower Types
15.2 Auxiliary Statements
15.3 Bounds Inside Types
Reference
16 Algorithmic Problems. Local Approach
16.1 Relationships Among Algorithmic Problems
16.2 Algorithm for Table Tρ(z,K) Construction
16.3 Algorithms for Construction of Decision Tree Schemes
References
Part III Decision Trees for Problems. Global Approach
17 Basic Definitions and Notation
17.1 Complexity Functions
17.2 Matrices of Upper and Lower Global Bounds for Sccf-Triple
17.3 Types of Sccf-Triples
References
18 Some Reductions
18.1 Problem Schemes and Decision Tables
18.2 Arbitrary Classes of Information Systems and One-Element Classes of Information Systems
18.3 Set mathbbtildeR
18.4 Primitive Composition of Disjoint Simple Sccf-Triples
18.5 Composition of Simple Sccf-Triples
18.6 Operation rotimes
18.7 Relationships Between Ψτ and τ
18.8 Transition from One Signature to Another
References
19 Functions on Main Diagonal and Below
19.1 Preliminary Lemmas
19.2 Function Ψτii
19.3 Function Ψτdi
19.4 Function Ψτdd
19.5 Functions Ψτai, Ψτad, Ψτaa, Ψτsi, Ψτsd, Ψτsa, and Ψτss
References
20 Functions over Main Diagonal
20.1 Functions Ψτid, Ψτia, and Ψτis
20.2 Function Ψτas
20.3 Function Ψτds
20.4 Function Ψτda
Reference
21 Possible Global Upper Types of Sccf-Triples
21.1 Possible Global Upper Types
21.2 Criteria for Equalities Typ Ψτ=Tpj
21.3 Global Upper Type of Composition of Simple Sccf-Triples
References
22 Realizable Global Upper Types of Sccf-Triples
22.1 Sccf-Triples from Ws(1)
22.2 Sccf-Triples from Ws(7)
22.3 Sccf-Triples from Ws(2) and Ws(9)
22.4 Sccf-Triples from Ws(8)
22.5 Sccf-Triples from Ws(4)
22.6 Sccf-Triples from Ws(6)
22.7 Sccf-Triples from Ws(3)
22.8 Sccf-Triples from Ws(5)
22.9 Sccf-Triples from Ws(10)
22.10 Sccf-Triples τ with TypΨτ=Tp11
22.11 Main Statements
Reference
23 Bounds Inside Types
23.1 Auxiliary Statements
23.2 Set Wρ(1)
23.3 Set Wρ(2)
23.4 Set Wρ(3)
23.5 Set Wρ(4)
23.6 Set Wρ(5)
23.7 Set Wρ(6)
23.8 Set Wρ(7)
23.9 Set Wρ(8)
23.10 Set Wρ(9)
23.11 Set Wρ(10)
Reference
24 Matrices of Lower Global Bounds
24.1 Possible Global Lower Types
24.2 Auxiliary Statements
24.3 Bounds Inside Types
Reference
25 Algorithmic Problems. Global Approach
25.1 Some Relationships Among Algorithmic Problems
25.2 Proper Weighted Depth
References
Appendix Final Remarks
Appendix Notation
Symbol Index
Appendix Index
Index

Citation preview

Intelligent Systems Reference Library 179

Mikhail Moshkov

Comparative Analysis of Deterministic and Nondeterministic Decision Trees

Intelligent Systems Reference Library Volume 179

Series Editors Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland Lakhmi C. Jain, Faculty of Engineering and Information Technology, Centre for Artificial Intelligence, University of Technology, Sydney, NSW, Australia; KES International, Shoreham-by-Sea, UK; Liverpool Hope University, Liverpool, UK

The aim of this series is to publish a Reference Library, including novel advances and developments in all aspects of Intelligent Systems in an easily accessible and well structured form. The series includes reference works, handbooks, compendia, textbooks, well-structured monographs, dictionaries, and encyclopedias. It contains well integrated knowledge and current information in the field of Intelligent Systems. The series covers the theory, applications, and design methods of Intelligent Systems. Virtually all disciplines such as engineering, computer science, avionics, business, e-commerce, environment, healthcare, physics and life science are included. The list of topics spans all the areas of modern intelligent systems such as: Ambient intelligence, Computational intelligence, Social intelligence, Computational neuroscience, Artificial life, Virtual society, Cognitive systems, DNA and immunity-based systems, e-Learning and teaching, Human-centred computing and Machine ethics, Intelligent control, Intelligent data analysis, Knowledge-based paradigms, Knowledge management, Intelligent agents, Intelligent decision making, Intelligent network security, Interactive entertainment, Learning paradigms, Recommender systems, Robotics and Mechatronics including human-machine teaming, Self-organizing and adaptive systems, Soft computing including Neural systems, Fuzzy systems, Evolutionary computing and the Fusion of these paradigms, Perception and Vision, Web intelligence and Multimedia. ** Indexing: The books of this series are submitted to ISI Web of Science, SCOPUS, DBLP and Springerlink.

More information about this series at http://www.springer.com/series/8578

Mikhail Moshkov

Comparative Analysis of Deterministic and Nondeterministic Decision Trees

123

Mikhail Moshkov Computer, Electrical and Mathematical Science and Engineering Division King Abdullah University of Science and Technology Thuwal, Saudi Arabia

ISSN 1868-4394 ISSN 1868-4408 (electronic) Intelligent Systems Reference Library ISBN 978-3-030-41727-7 ISBN 978-3-030-41728-4 (eBook) https://doi.org/10.1007/978-3-030-41728-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my family

Preface

This book is devoted to the comparative analysis of deterministic, nondeterministic, and strongly nondeterministic decision trees for problems over information systems. An information system consists of a universe and a set of attributes defined on the universe. Various problems of fault diagnosis, computational geometry, combinatorial optimization, etc., can be represented as problems over appropriate finite or infinite information systems. Each decision table can be interpreted as a problem over corresponding finite information system. Deterministic decision trees are widely used as classifiers, as a means of knowledge representation, and as algorithms. Nondeterministic decision trees are less known. A nondeterministic decision tree can be interpreted as a system of decision rules for a given problem that covers all inputs. Strongly nondeterministic decision trees can be considered for problems that have two decisions: 0 and 1. A strongly nondeterministic decision tree accepts only inputs with the decision 1 and can be interpreted as a system of decision rules which covers all inputs with the decision 1 and only such inputs. We design tools for study of problems over information systems: lower and upper bounds on complexity and algorithms for construction of deterministic, nondeterministic, and strongly nondeterministic decision trees for decision tables with many-valued decisions. We consider two approaches to the study of decision trees for problems over information systems: local, when we can use in decision trees only attributes from the problem representation, and global, when we can use arbitrary attributes from the information system. In the frameworks of each approach, we compare the complexity of problem representation and minimum complexities of deterministic, nondeterministic, and strongly nondeterministic decision trees solving problem. For the local and global approaches, we describe all possible types of relationships among these four parameters. We also study relationships among the following algorithmic problems: problems of computation of the minimum complexity of deterministic, nondeterministic, and strongly nondeterministic decision

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trees; problems of construction of decision trees with the minimum complexity; and the problem of solvability of systems of equations over information systems. The results presented in this book can be useful for researchers who use decision trees and rules in design and analysis of algorithms, and in data analysis, especially those who work in rough set theory, test theory, and logical analysis of data. The book can be used for the creation of courses for graduate students. Thuwal, Saudi Arabia November 2019

Mikhail Moshkov

Acknowledgements

The author wishes to express his deep gratitude to the late Al. A. Markov, O. B. Lupanov, Z. Pawlak, and S. V. Yablonskii, and also to thank A. Skowron and Ju. I. Zhuravlev who influenced greatly the author’s views on the subject of the present investigation. The author is greatly indebted to Lobachevsky State University of Nizhni Novgorod, University of Warsaw, University of Silesia in Katowice, Stanford University, and King Abdullah University of Science and Technology for their hospitality and support during the preparation of the book. The author extend an expression of gratitude to Prof. J. Kacprzyk, Dr. T. Ditzinger, and the Series Intelligent Systems Reference Library staff at Springer for their support in making this book possible.

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Lower Bounds on Complexity of Deterministic Decision Trees for Decision Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Auxiliary Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Main Directions of Study . . . . . . . . . . . . . . . . . . . . 1.1.1 Decision Tables with Many-Valued Decisions 1.1.2 Local Approach to Study of Decision Trees . . 1.1.3 Global Approach to Study of Decision Trees . 1.2 Contents of Book . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Part I. Decision Trees for Decision Tables . . . 1.2.2 Part II. Decision Trees for Problems. Local Approach . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Part III. Decision Trees for Problems. Global Approach . . . . . . . . . . . . . . . . . . . . . 1.2.4 Index and Notation . . . . . . . . . . . . . . . . . . . . 1.3 Use of Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Decision Trees for Decision Tables

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3.3 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Approach to Proof of Lower Bounds . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

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Upper Bounds on Complexity and Algorithms for Construction of Deterministic Decision Trees for Decision Tables. First Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Difference-Bounded Uncertainty Measures for Decision Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Process Uq of Schema Construction . . . . . . . . . . . . . . . . . . 4.3 Auxiliary Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Upper Bounds on wðUq ðT; c; wÞÞ . . . . . . . . . . . . . . . . . . . . 4.5 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Algorithm Uq;u;w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Algorithms Uq;GHq ;h , Uq;RHqm þ 1 ;h , and Uq;Hqm þ 1 ;h . . . . . . . . . . 4.8 Algorithms Wq;GHq ;h , Wq;RHq2 ;h , and Wq;Hq2 ;h . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Upper Bounds and Algorithms for Construction of Deterministic Decision Trees for Decision Tables. Second Approach . . . . . . . . 5.1 Reduced Deterministic Decision Trees for Decision Tables . 5.2 Additive-Bounded Uncertainty Measures for Decision Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Process Yq;N of Schema Construction . . . . . . . . . . . . . . . . 5.4 Upper Bounds on wdq ðTÞ . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Upper Bounds on hdq ðTÞ . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Algorithm Yq;g;u;w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Bounds on Complexity and Algorithms for Construction of Nondeterministic and Strongly Nondeterministic Decision Trees for Decision Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Bounds on waq ðTÞ and wsq ðTÞ . . . . . . . . . . . . . . . . . . . . . . . 6.2 Approach to Proof of Lower Bounds on wsq ðTÞ and waq ðTÞ . a s 6.3 Algorithms Vq;u;w and Vq;u;w ....................... Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Closed Classes of Boolean Functions . . . . . . 7.1 Main Definitions and Notation . . . . . . 7.2 Auxiliary Statements . . . . . . . . . . . . . 7.3 Bounds for Individual Closed Classes 7.4 Bounds for Arbitrary Closed Classes . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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Algorithmic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Relationships Among Algorithmic Problems . . . . . . 8.2 Examples of Decidable and Undecidable Problems . 8.3 Possible Variants of Algorithmic Problem Behavior 8.4 Examples of Problems with Polynomial Complexity 8.5 Examples of NP-Hard Problems . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part II 9

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Decision Trees for Problems. Local Approach

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Definitions and Notation . . . . . . . . . . . . . . . . . . . Information Systems . . . . . . . . . . . . . . . . . . . . . . Decision Trees . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Schemes and Problems . . . . . . . . . . . . . . Decision Trees Solving Problems . . . . . . . . . . . . . Decision Tree Schemes Corresponding to Problem Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Complexity Functions . . . . . . . . . . . . . . . . . . . . . 9.7 Matrices of Upper and Lower Local Bounds for Sccf-Triple . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Types of Functions and Sccf-Triples . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Main Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Decision Trees and Decision Tree Schemes . . . . . 10.2 Arbitrary Classes of Information Systems and One-Element Classes of Information Systems . 10.3 Upper and Lower Bounds . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Functions on Main Diagonal and Below . . . . . . . . . . . ^ ii . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Function W s ^ di . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Function W s ^ dd . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Function W s ^ ai , W ^ ad , W ^ aa , W ^ si , W ^ sd , W ^ sa , and W ^ ss 11.4 Functions W s s s s s s s References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Functions Over Main Diagonal . . . ^ id , W ^ ia , and W ^ is . 12.1 Functions W s s s ^ as . . . . . . . . . . . 12.2 Function W s ^ ds . . . . . . . . . . . 12.3 Function W s ^ da . . . . . . . . . . . 12.4 Function W s References . . . . . . . . . . . . . . . . . . .

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13 Local Upper Types of Restricted Sccf-Triples . 13.1 Possible Local Upper Types . . . . . . . . . 13.2 Examples of Restricted Sccf-Triples . . . . 13.3 Main Statements . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Bounds Inside Types . ^ q ð1Þ . . . . 14.1 Set W ^ q ð2Þ . . . . 14.2 Set W ^ q ð3Þ . . . . 14.3 Set W ^ q ð4Þ . . . . 14.4 Set W ^ q ð5Þ . . . . 14.5 Set W ^ q ð6Þ . . . . 14.6 Set W Reference . . . . . . . . . .

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187 187 188 191 193

16 Algorithmic Problems. Local Approach . . . . . . . . . . . . . . . . 16.1 Relationships Among Algorithmic Problems . . . . . . . . . 16.2 Algorithm for Table Tq ðz; KÞ Construction . . . . . . . . . . 16.3 Algorithms for Construction of Decision Tree Schemes . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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195 195 198 199 202

15 Matrices of Lower Local Bounds . 15.1 Possible Local Lower Types 15.2 Auxiliary Statements . . . . . . 15.3 Bounds Inside Types . . . . . . Reference . . . . . . . . . . . . . . . . . . . .

Part III

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Decision Trees for Problems. Global Approach

17 Basic Definitions and Notation . . . . . . . . . . . . . . . . 17.1 Complexity Functions . . . . . . . . . . . . . . . . . . 17.2 Matrices of Upper and Lower Global Bounds for Sccf-Triple . . . . . . . . . . . . . . . . . . . . . . . 17.3 Types of Sccf-Triples . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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18 Some Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Problem Schemes and Decision Tables . . . . . . . . . . . . 18.2 Arbitrary Classes of Information Systems and One-Element Classes of Information Systems . . . . ~ .................................... 18.3 Set R 18.4 Primitive Composition of Disjoint Simple Sccf-Triples 18.5 Composition of Simple Sccf-Triples . . . . . . . . . . . . . . 18.6 Operation r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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210 211 219 221

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xv

^ s . . . . . . . . . . . . . . . . . . . . . 222 18.7 Relationships Between Ws and W 18.8 Transition from One Signature to Another . . . . . . . . . . . . . . . 223 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 19 Functions on Main Diagonal and Below . . . . . . . . . . . 19.1 Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . . 19.2 Function Wiis . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Function Wdi s . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Function Wdd s . . . . . . . . . . . . . . . . . . . . . . . . . . ad aa si sd sa ss 19.5 Functions Wai s , Ws , Ws , Ws , Ws , Ws , and Ws References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Functions over Main Diagonal . . . ia is 20.1 Functions Wid s , Ws , and Ws as 20.2 Function Ws . . . . . . . . . . . 20.3 Function Wds s . . . . . . . . . . . 20.4 Function Wda s . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . 225 . . . . . . . . . . 225 . . . . . . . . . . 226 . . . . . . . . . . 227 . . . . . . . . . . 229 . . . . . . . . . . 229 . . . . . . . . . . 232

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21 Possible Global Upper Types of Sccf-Triples . . . . . . 21.1 Possible Global Upper Types . . . . . . . . . . . . . 21.2 Criteria for Equalities Typ Ws ¼ Tp j . . . . . . . 21.3 Global Upper Type of Composition of Simple Sccf-Triples . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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22 Realizable Global Upper Types of Sccf-Triples . 22.1 Sccf-Triples from W s ð1Þ . . . . . . . . . . . . . 22.2 Sccf-Triples from W s ð7Þ . . . . . . . . . . . . . 22.3 Sccf-Triples from W s ð2Þ and W s ð9Þ . . . . . 22.4 Sccf-Triples from W s ð8Þ . . . . . . . . . . . . . 22.5 Sccf-Triples from W s ð4Þ . . . . . . . . . . . . . 22.6 Sccf-Triples from W s ð6Þ . . . . . . . . . . . . . 22.7 Sccf-Triples from W s ð3Þ . . . . . . . . . . . . . 22.8 Sccf-Triples from W s ð5Þ . . . . . . . . . . . . . 22.9 Sccf-Triples from W s ð10Þ . . . . . . . . . . . . 22.10 Sccf-Triples s with Typ Ws ¼ Tp11 . . . . 22.11 Main Statements . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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249 249 250 253 256 256 259 260 261 262 263 263 264

23 Bounds Inside Types . . . . . 23.1 Auxiliary Statements 23.2 Set Wq ð1Þ . . . . . . . . 23.3 Set Wq ð2Þ . . . . . . . .

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265 265 266 267

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xvi

Contents

23.4 Set 23.5 Set 23.6 Set 23.7 Set 23.8 Set 23.9 Set 23.10 Set 23.11 Set Reference .

Wq ð3Þ . Wq ð4Þ . Wq ð5Þ . Wq ð6Þ . Wq ð7Þ . Wq ð8Þ . Wq ð9Þ . Wq ð10Þ

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267 268 268 269 269 270 271 271 272

24 Matrices of Lower Global Bounds . 24.1 Possible Global Lower Types 24.2 Auxiliary Statements . . . . . . . 24.3 Bounds Inside Types . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . .

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273 273 274 277 280

25 Algorithmic Problems. Global Approach . . . . . . . . . . . 25.1 Some Relationships Among Algorithmic Problems 25.2 Proper Weighted Depth . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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281 281 283 285

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Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Chapter 1

Introduction

The aim of this book is to compare deterministic, nondeterministic, and strongly nondeterministic decision trees. Conventional (deterministic) decision trees are widely used as classifiers, as a means of knowledge representation, and as algorithms [7, 20, 27]. Nondeterministic and especially strongly nondeterministic decision trees [12] are less known. They are closely related to systems of decision rules. The investigations of relationships between deterministic and nondeterministic universal algorithms (in particular, relationships between classes P and N P) attract attention for many years. The investigations of relationships between deterministic and nondeterministic decision trees are mainly related to decision trees for Boolean functions [1, 4]. Note, that for a Boolean function, the minimum depth of a nondeterministic (strongly nondeterministic) decision tree is equal to the certificate complexity (1-certificate complexity) of this function [8]. In this book, we compare different kinds of decision trees for problems over arbitrary (finite and infinite) information systems. An information system [23] consists of a universe (a set of objects) and a set of k-valued attributes (functions) defined on this universe. We study problems with many-valued decisions over the considered information system. Any problem is specified by a number of attributes that divide the universe into domains in which these attributes have fixed values. A nonempty finite set of decisions is attached to each domain. For a given object from the universe, it is required to find a decision from the set attached to the domain containing this object. Various problems of fault diagnosis, computational geometry, combinatorial optimization, etc., can be represented as problems over appropriate finite or infinite information systems. In particular, each decision table can be interpreted as a problem over corresponding finite information system. Examples of decision problems with many-valued decisions can be found in the book [2]. As algorithms for problem solving we study deterministic, nondeterministic, and strongly nondeterministic decision trees. One can interpret nondeterministic decision trees solving a problem as a way for representation of arbitrary complete (covering all objects) decision rule systems for the problem. Strongly nondeterministic decision trees are applicable to problems that have only two decisions: 0 and 1. A strongly © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_1

1

2

1 Introduction

nondeterministic decision tree accepts objects with the decision 1 and can be interpreted as a system of decision rules which covers all objects with the decision 1 and only such objects. We consider various complexity measures which characterize time complexity of decision trees. One of the most known among them is the depth. The depth of a decision tree is the maximum number of nodes labeled with attributes in a path from the root to a terminal node of the tree. We develop two approaches to the decision tree investigation: the local approach, where, for a problem, the decision trees are considered that use only attributes from the problem representation, and the global one, where, for problem solving, all attributes from the considered information system can be used. This book deals with comparative analysis of the four parameters of problems over arbitrary information system: the complexity of a problem representation (in the case of the depth, for example, this is the number of attributes in the problem representation), and the minimum complexities of deterministic, nondeterministic, and strongly nondeterministic decision trees solving this problem. We design tools for the study of problems over information systems that includes bounds on complexity and algorithms for construction of deterministic, nondeterministic, and strongly nondeterministic decision trees for decision tables with manyvalued decisions. After that, we investigate a rough classification of relations among the four parameters of problems over information systems: we enumerate and study all possible six types of these relations for the local approach and all possible 11 types for the global approach. We also discuss various algorithmic problems related to the optimization of decision trees.

1.1 Main Directions of Study There are three main directions of study in this book.

1.1.1 Decision Tables with Many-Valued Decisions The first direction is devoted to the study of decision trees for decision tables with many-valued decisions in which each row is labeled with a nonempty finite set of decisions. Such decision tables are known also as multi-label decision tables. They are studied intensively last decades [3, 26, 30, 31]. For a row of a decision table with many-valued decisions given by values of attributes, we should return a decision from the set of decisions attached to this row. To this end, we can use deterministic and nondeterministic decision trees—see decision table T1 and deterministic and nondeterministic decision trees for this table depicted in Fig. 1.1. The considered nondeterministic decision tree can be interpreted

1.1 Main Directions of Study

3

Fig. 1.1 Decision table with many-valued decisions T1 and two its decision trees: deterministic and nondeterministic

Fig. 1.2 Decision table with many-valued decisions T2 and three its decision trees: deterministic, nondeterministic, and strongly nondeterministic

as a system of true for T1 decision rules that cover all rows of T1 : {( f 1 = 1) → 1, ( f 2 = 1) → 2, ( f 3 = 1) → 3} . Each such system of decision riles can be represented as a nondeterministic decision tree. If in a decision table each row is labeled with the set of decisions {0} or {1} and at least one row is labeled with {1}, then we can use deterministic, nondeterministic, and strongly nondeterministic decision trees—see decision table T2 and deterministic, nondeterministic, and strongly nondeterministic decision trees for this table depicted in Fig. 1.2. The considered strongly nondeterministic decision tree can be interpreted as a system of true for T2 decision rules that cover all rows of T2 labeled with the set of decisions {1} and only such rows: {( f 1 = 1) → 1, ( f 2 = 1) → 1} .

4

1 Introduction

Each such system of decision riles can be represented as a strongly nondeterministic decision tree. We study decision tables of a signature ρ = (F, k), where F is a set of attribute names and k ≥ 2. In such decision tables, rows are filled with numbers from the set E k = {0, 1, . . . , k − 1}, columns are labeled with names of attributes from the set F, and each row is labeled with a set of decisions—a nonempty finite subset of the set ω of nonnegative integers. A complexity function ψ of the signature ρ is defined on the words over the alphabet F including the empty word λ and has values from the set ω. We consider mainly so-called restricted complexity functions satisfying the following properties: 1. ψ(α1 α2 ) ≤ ψ(α1 ) + ψ(α2 ), 2. ψ(α1 α2 α3 ) ≥ ψ(α1 α3 ), 3. ψ(α1 ) ≥ |α1 |, 4. ψ(λ) = 0, where α1 , α2 , α3 are words over F and |α1 | is the length of the word α1 . We study decision trees of the signature ρ in which nonterminal nodes with the exception of the root are labeled with names of attributes from F, terminal nodes are labeled with numbers from ω, and edges are labeled with numbers from E k . For a decision tree , let αξ be a word over F corresponding to a path ξ in  from the root to a terminal node. Letters of this word are names of attributes attached to nodes of the path ξ . The complexity ψ() of the tree  is the maximum value of ψ(αξ ) among all the paths ξ from the root of  to a terminal node. The most known examples of the restricted complexity functions are the depth and the weighted depth. We created lower and upper bounds on complexity and algorithms for construction of deterministic, nondeterministic, and strongly nondeterministic decision trees for decision tables with many-valued decisions, study the depth of decision trees computing Boolean functions from closed classes, and consider algorithmic problems related to the optimization of decision trees for the decision tables. An essential part of the obtained results is used for the investigation of decision trees for problems over information systems. Many results related to the first direction of study were published in [11, 12, 17, 20, 21].

1.1.2 Local Approach to Study of Decision Trees The second direction is devoted to the development of the local approach to the study of decision trees for problems, where decision trees can use only attributes from problem representation. An information system of the signature ρ = (F, k) is a nonempty set of objects A (universe) and an interpretation of the set F which corresponds to each attribute name from F an attribute—a function from A to E k = {0, 1, . . . , k − 1}. A problem schema of the signature ρ is a tuple z = (ν, f 1 , . . . , f n ), where f 1 , . . . , f n ∈ F and ν is a mapping that corresponds to each n-tuple from E kn a finite

1.1 Main Directions of Study

5

set of decisions from ω = {0, 1, 2, . . .}. For a given information system of the signature ρ, the problem schema z becomes a problem: attributes with names f 1 , . . . , f n divide the universe into nonempty domains in each of which these attributes have fixed values, and the mapping ν corresponds to each domain a set of decisions. For a given object from the universe, we should return a decision from the set attached to the domain to which this object belongs. To this end, we use deterministic, nondeterministic, and strongly nondeterministic decision trees. Really, we have schemes of decision trees of the signature ρ (see, for example, Figs. 1.1 and 1.2) which become decision trees when we fix an information system. Let ρ be a signature, K be a nonempty class of information systems of the signature ρ, and ψ be a complexity function of the signature ρ. The triple τ = (ρ, K , ψ) is called a sccf-triple (sccf are the first letters of the words signature, class, complexity function). The triple τ is called restricted if the complexity function ψ is restricted (has the properties 1, 2, 3, and 4). A problem schema z = (ν, f 1 , . . . , f n ) of the signature ρ is called a τ -0-1-problem schema, if the mapping ν has only two possible values {0} and {1} and, for at least one information system from K for at least one object from the universe, the problem corresponding to z has the solution 1. i (z) = ψ( f 1 · · · f n ) is For a problem schema z = (ν, f 1 , . . . , f n ), the value ψˆ ρ,K the complexity of representation of z (or the complexity of trivial deterministic decision tree solving a problem corresponding to z by sequential computation of d (z) the minimum values of attributes with names f 1 , . . . , f n ). We denote by ψˆ ρ,K complexity of a schema of deterministic decision tree  of the signature ρ which uses only names of attributes f 1 , . . . , f n and, for each information system from the class K , the decision tree corresponding to  solves the problem corresponding to z. a s (z) and ψˆ ρ,K (z) are defined in a similar way for nondeterministic The parameters ψˆ ρ,K decision trees (index a) and for strongly nondeterministic decision trees (index s). In the latter case, we assume z is a τ -0-1-problem schema. Let b, c ∈ {i, d, a, s}. If b = s and c = s, then we denote by τbc the set of all schemes of problems of the signature ρ. If b = s or c = s, then τbc is the set of τ -0-1-problem schemes of the signature ρ. For b, c ∈ {i, d, a, s}, we define partial functions Ψˆ τbc : ω → ω and Φˆ τb,c : ω → ω b c (z) and ψˆ ρ,K (z): that describe relationships between parameters ψˆ ρ,K b c (z) : z ∈ τbc , ψˆ ρ,K (z) ≤ n} , Ψˆ τbc (n) = max{ψˆ ρ,K b c Φˆ τbc (n) = min{ψˆ ρ,K (z) : z ∈ τbc , ψˆ ρ,K (z) ≥ n} . c b If m, t ∈ ω and m ≤ ψˆ ρ,K (z) ≤ t, then Φˆ τbc (m) ≤ ψˆ ρ,K (z) ≤ Ψˆ τbc (t) and the last bounds are unimprovable if defined. We denote by Ψˆ τ the matrix with four rows and four columns in which rows from the top to the bottom and columns from the left to the right are labeled with indices i, d, a, s, and at the intersection of the row with index b ∈ {i, d, a, s} and the column

6

1 Introduction

with index c ∈ {i, d, a, s} the function Ψˆ τbc is placed. The matrix Ψˆ τ is called the matrix of upper local bounds for the sccf-triple τ . It is very difficult to study the matrix Ψˆ τ directly. Instead of this, we study the matrix Typ Ψˆ τ obtained from the matrix Ψˆ τ in the following: for any b, c ∈ {i, d, a, s}, instead of the function Ψˆ τbc , the matrix Typ Ψˆ τ contains its upper type Typ Ψˆ τbc which is defined later. The matrix Typ Ψˆ τ is called the local upper type of the sccf-triple τ . Let f be a partial function from ω to ω. We denote by Arg f the domain of f . We now define the value Typ f ∈ {ε, λ, χ , ω} which is called the upper type of the function f . • If Arg f is an infinite set and there exists c ∈ ω such that f (n) ≤ c for any n ∈ Arg f , then Typ f = ε. • If Arg f is an infinite set, there is no c ∈ ω such that f (n) ≤ c for any n ∈ Arg f , and the set {n : n ∈ Arg f, f (n) ≥ n} is finite, then Typ f = λ. • If the set {n : n ∈ Arg f, f (n) ≥ n} is infinite, then Typ f = χ . • If the set Arg f is finite, then Typ f = ω. For example, if Typ Ψˆ τdi = λ, then, for problems with large enough complexity of representation, optimal deterministic decision trees have less complexity than trivial deterministic decision trees corresponding to problem representations. If Typ Ψˆ τdi = χ , then there exist problems with arbitrarily large complexity of representation for which optimal deterministic decision trees have the same complexity as trivial deterministic decision trees corresponding to problem representations. We describe all possible six local upper types of restricted sccf-triples Tpi, i ∈ {1, . . . , 6}:

i Tp1 = d a s

i χ ε ε ε

d ω ε ε ε

a ω ε ε ε

s ω ε ε ε

i Tp2 = d a s

i χ λ ε ε

d ω χ ε ε

a ω ω ε ε

s ω ω ε ε

i Tp3 = d a s

i χ χ χ χ

d ω χ χ χ

a ω ω χ χ

s ω ω χ χ

i Tp4 = d a s

i χ χ χ χ

d ω χ χ χ

a ω χ χ χ

s ω χ χ χ

i Tp5 = d a s

i χ χ χ χ

d ω χ χ χ

a ω ω χ χ

s ω ω ω χ

i Tp6 = d a s

i χ χ χ χ

d ω χ χ χ

a ω χ χ χ

s ω ω ω χ

For each of these six types, we consider the criterion of its implementation and give an example of a restricted sccf-triple with this type. For a given signature ρ and each possible local upper type of restricted sccftriples Tpi, i ∈ {1, . . . , 6}, we consider the set Wˆ ρ (i) of restricted sccf-triples τ with Typ Ψˆ τ = Tpi. For each pair (b, c) ∈ {i, d, a, s}2 such that in the matrix Tpi at the intersection of the row with index b and the column with index c either λ or χ

1.1 Main Directions of Study

7

stays, we study upper and lower bounds on the function Ψˆ τbc true for any sccf-triple τ ∈ Wˆ ρ (i). We define the matrix Φˆ τ of lower local bounds for the sccf-triple τ in a similar way as the matrix Ψˆ τ of upper local bounds for the sccf-triple τ . For a partial function f : ω → ω we define the lower type typ f ∈ {ε, γ , μ, ω} of the function f . We also define the matrix typ Φˆ τ called the local lower type of the sccf-triple τ . In this matrix, instead of the functions Φˆ τbc we have their lower types. We show that the matrix of upper local bounds for a sccf-triple completely defines its matrix of lower local bounds. In particular, the local upper type of a sccf-triple completely defines its local lower type. We describe and study all possible six local lower types tp1, . . . , tp6 of restricted sccf-triples which correspond to the local upper types Tp1, . . . , Tp6, respectively. We also discuss algorithmic problems related to the optimization of decision trees in the frameworks of the local approach. Upper types of functions considered in this book were introduced in [18]. Types of functions slightly different from the upper and lower types were investigated in [13, 14, 16, 19]. Some results similar to considered in the second direction were published in [2, 11, 13, 14, 16, 19, 20, 22].

1.1.3 Global Approach to Study of Decision Trees The third direction is devoted to the development of the global approach to the study of decision trees for problems, where decision trees can use arbitrary attributes from the considered information system. Let τ = (ρ, K , ψ) be a sccf-triple and ρ = (F, k). For a problem schema z = i (z) = ψ( f 1 · · · f n ) is the complexity (ν, f 1 , . . . , f n ) of the signature ρ, the value ψρ,K of representation of z (or the complexity of trivial deterministic decision tree solving a problem corresponding to z by sequential computation of values of attributes with d (z) the minimum complexity of a schema names f 1 , . . . , f n ). We denote by ψρ,K of deterministic decision tree  of the signature ρ such that, for each information system from the class K , the decision tree corresponding to  solves the problem a s (z) and ψρ,K (z) are defined in a similar corresponding to z. The parameters ψρ,K way for nondeterministic decision trees (index a) and for strongly nondeterministic decision trees (index s). In the latter case, we assume that z is a τ -0-1-problem schema. For b, c ∈ {i, d, a, s}, we define partial functions Ψτbc : ω → ω and Φτb,c : ω → ω b c (z) and ψρ,K (z): that describe relationships between parameters ψρ,K b c Ψτbc (n) = max{ψρ,K (z) : z ∈ τbc , ψρ,K (z) ≤ n} , b c Φτbc (n) = min{ψρ,K (z) : z ∈ τbc , ψρ,K (z) ≥ n} .

8

1 Introduction

c b If m, t ∈ ω and m ≤ ψρ,K (z) ≤ t, then Φτbc (m) ≤ ψρ,K (z) ≤ Ψτbc (t) and the last bounds are unimprovable if defined. We denote by Ψτ the matrix with four rows and four columns in which rows from the top to the bottom and columns from the left to the right are labeled with indices i, d, a, s, and at the intersection of the row with index b ∈ {i, d, a, s} and the column with index c ∈ {i, d, a, s} the function Ψτbc is placed. The matrix Ψτ is called the matrix of upper global bounds for the sccf-triple τ . It is too difficult to study the matrix Ψτ directly. Instead of this, we study the matrix Typ Ψτ obtained from the matrix Ψτ in the following: for any b, c ∈ {i, d, a, s}, instead of the function Ψτbc , the matrix Typ Ψτ contains its upper type Typ Ψτbc . The matrix Typ Ψτ is called the global upper type of the sccf-triple τ . We describe all possible 11 global upper types Tp1, . . . , Tp11 of sccf-triples, where

i Tp7 = d a s

i χ λ ε ε

d ω χ ε ε

a ω ω ε ε

s ω ε ε ε

i Tp10 = d a s

i Tp8 = d a s i χ χ χ χ

d ω χ χ χ

a ω ω χ χ

s ω χ χ χ

i χ χ ε ε

d ω χ ε ε

a ω ω ε ε

s ω ε ε ε

i Tp11 = d a s

i Tp9 = d a s i ε ε ε ε

d ε ε ε ε

a ε ε ε ε

i χ χ ε ε

d ω χ ε ε

a ω ω ε ε

s ω ω ε ε

s ε ε ε ε

and all possible 10 global upper types Tp1, . . . , Tp10 of restricted sccf-triples. For each of the last 10 global upper types, we consider the criterion of its implementation for restricted sccf-triples. For each i ∈ {1, . . . , 11}, we prove that there exists a sccftriple τ such that Typ Ψτ = Tpi. We also prove that, for each i ∈ {1, . . . , 10}, there exists a restricted sccf-triple τ such that Typ Ψτ = Tpi. For a given signature ρ and each possible global upper type of restricted sccftriples Tpi, i ∈ {1, . . . , 10}, we consider the set Wρ (i) of restricted sccf-triples τ with Typ Ψτ = Tpi. For each pair (b, c) ∈ {i, d, a, s}2 such that in the matrix Tpi at the intersection of the row with index b and the column with index c either λ or χ stays, we study upper and lower bounds on the function Ψτbc true for any sccf-triple τ ∈ Wρ (i). We define the matrix Φτ of lower global bounds for the sccf-triple τ in a similar way as the matrix Ψτ of upper global bounds for the sccf-triple τ . We also define the matrix typ Φτ called the global lower type of the triple τ . In this matrix, instead of the functions Φτbc we have their lower types. We show that the global upper type of a sccf-triple completely defines its global lower type. We describe all possible 11 global lower types tp1, . . . , tp11 of sccftriples which correspond to the global upper types Tp1, . . . , Tp11, respectively. We

1.1 Main Directions of Study

9

also describe and study all possible 10 global lower types tp1, . . . , tp10 of restricted sccf-triples which correspond to the global upper types Tp1, . . . , Tp10. We study algorithmic problems related to the global approach to the investigation of decision trees: problems of computation of the minimum complexity of decision trees, problems of construction of decision trees with the minimum complexity, and the problem of solvability of systems of equations over information systems. Some results similar to considered in the third direction were published in [2, 11, 14–16, 20, 22].

1.2 Contents of Book This book includes Introduction and three parts consisting of 24 chapters.

1.2.1 Part I. Decision Trees for Decision Tables In this part, lower and upper bounds on complexity and algorithms for construction of deterministic, nondeterministic, and strongly nondeterministic decision trees for decision tables are considered. This part consists of seven chapters. Chapter 2 contains definitions of the notions of decision table with many-valued decisions, decision tree schema, deterministic, nondeterministic, and strongly nondeterministic decision trees for decision tables, and complexity function. In Chap. 3, lower bounds on complexity of deterministic decision trees for decision tables are studied that are based on the notions of super-cover, super-partition, test, and system of representatives for decision tables. An approach to the proof of lower bounds is also considered which is based on the use of so-called proof-trees. In Chap. 4, for complexity functions having the properties 1, 2, and 3, upper bounds on the minimum complexity and algorithms for construction of deterministic decision trees for decision tables are considered. These bounds and algorithms are based on the use of so-called difference-bounded uncertainty measures for decision tables. In Chap. 5, upper bounds on the minimum complexity and algorithms for construction of deterministic decision trees for decision tables are considered. These bounds and algorithms are based on the use of so-called additive-bounded uncertainty measures for decision tables. The bounds are true for any complexity function having the property 1. When developing algorithms, we assume that the complexity functions have properties 1, 2, and 3. In Chap. 6, we consider bounds on the minimum complexity, an approach to proof of lower bounds, and algorithms for construction of nondeterministic and strongly nondeterministic decision trees. The bounds on complexity are true for arbitrary complexity functions. The approach to proof of lower bounds assumes that the complexity functions have the property 2. The considered algorithms require complexity functions having properties 1, 2, and 3.

10

1 Introduction

In Chap. 7, for decision tables corresponding to functions from an arbitrary closed class of Boolean functions, the depth of deterministic, nondeterministic, and strongly nondeterministic decision trees is studied. The obtained results have some independent interest. Proofs of these results illustrate mainly methods for the proof of lower bounds on complexity of decision trees. In Chaps. 2–7, we study, mainly, approximate bounds on the minimum complexity and approximate algorithms for optimization of decision trees. In Chap. 8, we consider algorithmic problems of computation of the minimum complexity of deterministic, nondeterministic, and strongly nondeterministic decision trees, and of construction of decision trees with the minimum complexity.

1.2.2 Part II. Decision Trees for Problems. Local Approach In this part, we develop local approach to the study of decision trees for problems, where decision trees can use only attributes from problem representation. This part consists of eight chapters. In Chap. 9, we discuss main notions and notation for the local approach to the study of decision trees for problems. We consider information systems, decision trees over information systems, problems over information systems, classes of information systems, relationships among different parameters of problems, and upper and lower types of these relationships. In Chap. 10, we consider some reductions which will be used later in the investigations of decision trees for problems. We show that, in the frameworks of the local approach, the study of decision trees for problems can be reduced to the study of decision trees for decision tables. We prove that, instead of arbitrary classes of information systems, we can consider classes containing only one information system. We also show that the matrix of upper local bounds for a sccf-triple completely defines its matrix of lower local bounds and vice versa. In particular, the local upper type of a sccf-triple completely defines its local lower type and vice versa. In Chap. 11, for restricted sccf-triples τ , we study functions Ψˆ τbc , b, c ∈ {i, d, a, s}, located in the matrix of upper local bounds for the triple τ on the main diagonal and below. For each of these functions, we list all possible upper types and consider criterion for each such type. In several cases, we give upper and lower bounds for the considered functions. In Chap. 12, for restricted sccf-triples τ , we study functions Ψˆ τbc , b, c ∈ {i, d, a, s}, located in the matrix of upper local bounds for the triple τ over the main diagonal. For each of these functions, we list all possible upper types and consider criterion for each such type. In several cases, we give upper and lower bounds for the considered functions. In Chap. 13, we describe all possible six local upper types of restricted sccf-triples. For each of these six types, we consider the criterion of its implementation and give an example of a restricted sccf-triple with this type.

1.2 Contents of Book

11

In Chap. 14, for a given signature ρ and each possible local upper type of restricted sccf-triples Tpi, i ∈ {1, . . . , 6}, we consider the set Wˆ ρ (i) of restricted sccf-triples τ with Typ Ψˆ τ = Tpi. For each pair (b, c) ∈ {i, d, a, s}2 such that in the matrix Tpi at the intersection of the row with index b and the column with index c either λ or χ stays, we study upper and lower bounds on the function Ψˆ τbc true for any sccf-triple τ ∈ Wˆ ρ (i). In Chap. 15, we describe all possible six local lower types tp1, . . . , tp6 of restricted sccf-triples which correspond to the local upper types Tp1, . . . , Tp6, respectively. For a given signature ρ, each local lower type tpi, i ∈ {1, . . . , 6}, and each pair (b, c) ∈ {i, d, a, s}2 such that in the matrix tpi at the intersection of the row with index b and the column with index c either μ or γ stays, we study upper and lower bounds on the function Φˆ τbc true for any sccf-triple τ ∈ Wˆ ρ (i). In Chap. 16, we study algorithmic problems related to the local approach to the investigation of decision trees: problems of computation of the minimum complexity of deterministic, nondeterministic, and strongly nondeterministic decision trees, problems of construction of decision trees with the minimum complexity, and the problem of solvability of systems of equations over information systems. We study relationships among these problems and describe all variants of algorithmic problem behavior. We also discuss an algorithm for decision table construction and two algorithms that construct deterministic decision trees.

1.2.3 Part III. Decision Trees for Problems. Global Approach In this part, we develop global approach to the study of decision trees for problems, where decision trees can use arbitrary attributes from the considered information system. This part consists of nine chapters. In Chap. 17, we discuss main notions and notation for the global approach to the study of decision trees for problems. We consider different parameters of problems related to complexity of decision trees, relationships among these parameters, and upper and lower types of these relationships. In Chap. 18, we consider some reductions which will be used later in the investigations of decision trees in the framework of the global approach. We consider relationships among decision trees for problems and decision trees for decision tables. We prove that, instead of arbitrary classes of information systems, we can consider classes containing only one information system. We discuss some operations on sccf-triples, relationships between matrices of upper local and global bounds for sccf-triples, and possibilities to transfer results from one signature to another. In Chap. 19, for arbitrary and for the restricted sccf-triples τ , we study functions Ψτbc , b, c ∈ {i, d, a, s}, located in the matrix of upper global bounds for the triple τ on the main diagonal and below. For each of these functions, we list all possible upper types and consider criterion for each such type (with the exception of the function Ψτdi ). In several cases, we give bounds for the considered functions.

12

1 Introduction

In Chap. 20, for arbitrary and for the restricted sccf-triples τ , we study functions Ψτbc , b, c ∈ {i, d, a, s}, located in the matrix of upper global bounds for the triple τ over the main diagonal. For each of these functions, we list all possible upper types and consider criterion for each such type. In several cases, we give upper and lower bounds for the considered functions. In Chap. 21, we describe all possible 11 global upper types Tp1, . . . , Tp11 of sccf-triples and all possible 10 global upper types Tp1, . . . , Tp10 of restricted sccftriples. For each of the last 10 global upper types, we consider the criterion of its implementation for restricted sccf-triples. In Chap. 22, for each i ∈ {1, . . . , 11}, we prove that there exists a sccf-triple τ such that Typ Ψτ = Tpi. We also prove that, for each i ∈ {1, . . . , 10}, there exists a restricted sccf-triple τ such that Typ Ψτ = Tpi. In Chap. 23, for a given signature ρ and each possible global upper type of restricted sccf-triples Tpi, i ∈ {1, . . . , 10}, we consider the set Wρ (i) of restricted sccf-triples τ with Typ Ψτ = Tpi. For each pair (b, c) ∈ {i, d, a, s}2 such that in the matrix Tpi at the intersection of the row with index b and the column with index c either λ or χ stays, we study upper and lower bounds on the function Ψτbc true for any sccf-triple τ ∈ Wρ (i). In Chap. 24, we describe all possible 11 global lower types tp1, . . . , tp11 of sccftriples which correspond to the global upper types Tp1, . . . , Tp11, respectively. We also describe all possible 10 global lower types tp1, . . . , tp10 of restricted sccf-triples which correspond to the global upper types Tp1, . . . , Tp10. For a given signature ρ, each global lower type tpi, i ∈ {1, . . . , 10}, and each pair (b, c) ∈ {i, d, a, s}2 such that in the matrix tpi at the intersection of the row with index b and the column with index c either μ or γ stays, we study upper and lower bounds on the function Φτbc true for any sccf-triple τ ∈ Wρ (i). In Chap. 25, we study algorithmic problems related to the global approach to the investigation of decision trees: problems of computation of the minimum complexity of deterministic, nondeterministic, and strongly nondeterministic decision trees, problems of construction of decision trees with the minimum complexity, and the problem of solvability of systems of equations over information systems. We study relationships among these problems. We also discuss the notion of a proper weighted depth for which the problems of computation of the minimum complexity of decision trees and problems of construction of decision trees with the minimum complexity are decidable if the problem of solvability of systems of equations over information systems is decidable.

1.2.4 Index and Notation This book contains both Index and Notation. At the beginning of Notation we mention special symbols not related to Latin and Greek letters. After that, we group the notation according to Latin letters such that a Greek letter is located jointly with the first Latin letter in its name. For example, α (alpha) is located jointly with A, χ (chi) is located jointly with C, etc.

1.3 Use of Book

13

1.3 Use of Book The obtained theoretical results and tools designed for decision tables with manyvalued decisions can be useful for researchers who use decision trees and rules in design and analysis of algorithms, and in data analysis. In particular, the results presented in this book can be useful in rough set theory and its applications [24, 25, 28], where decision rules and decision trees are widely used, in logical analysis of data [5, 6, 10] which is based mainly on the use of patterns (decision rules), and in test theory [9, 29, 32] which study both decision trees and decision rules. This book can also be used for the creation of courses for graduate students.

References 1. AbouEisha, H., Amin, T., Chikalov, I., Hussain, S., Moshkov, M.: Extensions of Dynamic Programming for Combinatorial Optimization and Data Mining. Intelligent Systems Reference Library, vol. 146. Springer, Cham (2019) 2. Alsolami, F., Azad, M., Chikalov, I., Moshkov, M.: Decision and Inhibitory Trees and Rules for Decision Tables with Many-valued Decisions. Intelligent Systems Reference Library, vol. 156. Springer, Cham (2020) 3. Blockeel, H., Schietgat, L., Struyf, J., Dzeroski, S., Clare, A.: Decision trees for hierarchical multilabel classification: A case study in functional genomics. In: Fürnkranz, J., Scheffer, T., Spiliopoulou, M. (eds.) Knowledge Discovery in Databases: 10th European Conference on Principles and Practice of Knowledge Discovery in Databases, PKDD 2006, Berlin, Germany, September 18–22, 2006, Lecture Notes in Computer Science, vol. 4213, pp. 18–29. Springer, Berlin (2006) 4. Blum, M., Impagliazzo, R.: Generic oracles and oracle classes (extended abstract). In: 28th Annual Symposium on Foundations of Computer Science, Los Angeles, California, USA, October 27–29, 1987, pp. 118–126. IEEE Computer Society (1987) 5. Boros, E., Hammer, P.L., Ibaraki, T., Kogan, A.: Logical analysis of numerical data. Math. Program. 79, 163–190 (1997) 6. Boros, E., Hammer, P.L., Ibaraki, T., Kogan, A., Mayoraz, E., Muchnik, I.: An implementation of logical analysis of data. IEEE Trans. Knowl. Data Eng. 12, 292–306 (2000) 7. Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J.: Classification and Regression Trees. Wadsworth and Brooks, Monterey, CA (1984) 8. Buhrman, H., de Wolf, R.: Complexity measures and decision tree complexity: a survey. Theor. Comput. Sci. 288(1), 21–43 (2002) 9. Chegis, I.A., Yablonskii, S.V.: Logical methods of control of work of electric schemes. Trudy Mat. Inst. Steklov (in Russian) 51, 270–360 (1958) 10. Crama, Y., Hammer, P.L., Ibaraki, T.: Cause-effect relationships and partially defined Boolean functions. Ann. Oper. Res. 16, 299–326 (1988) 11. Moshkov, M.: Decision Trees. Theory and Applications (in Russian). Nizhny Novgorod University Publishers, Nizhny Novgorod (1994) 12. Moshkov, M.: About the depth of decision trees computing Boolean functions. Fundam. Inform. 22(3), 203–215 (1995) 13. Moshkov, M.: Comparative analysis of complexity of deterministic and nondeterministic decision trees. Local approach. In: Actual Problems of Modern Mathematics 1 (in Russian), pp. 109–113. NII MIOO NGU Publishers, Novosibirsk (1995)

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14. Moshkov, M.: Two approaches to investigation of deterministic and nondeterministic decision trees complexity. In: Glas, M.D., Pawlak Z. (eds.) Second World Conference on the Fundamentals of Artificial Intelligence, WOCFAI 1995, Paris, France, 3–7 July 1995, pp. 275–280 (1995) 15. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity. Global approach. Fundam. Inform. 25(2), 201–214 (1996) 16. Moshkov, M.: Local and global approaches to comparative analysis of complexity of deterministic and nondeterministic decision trees. In: Actual Problems of Modern Mathematics 2 (in Russian), pp. 110–118. NII MIOO NGU Publishers, Novosibirsk (1996) 17. Moshkov, M.: Lower bounds for the time complexity of deterministic conditional tests. Diskret. Mat. (in Russian) 8(3), 98–110 (1996). https://doi.org/10.4213/dm538 18. Moshkov, M.: Unimprovable upper bounds on time complexity of decision trees. Fundam. Inform. 31(2), 157–184 (1997) 19. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity. Local approach. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets IV, Lecture Notes in Computer Science, vol. 3700, pp. 125–143. Springer, Berlin (2005) 20. Moshkov, M.: Time complexity of decision trees. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III, Lecture Notes in Computer Science, vol. 3400, pp. 244–459. Springer, Berlin (2005) 21. Moshkov, M.: Bounds on complexity and algorithms for construction of deterministic conditional tests. In: Mat. Vopr. Kibern. (in Russian), vol. 16, pp. 79–124. Fizmatlit, Moscow (2007). http://library.keldysh.ru/mvk.asp?id=2007-79 22. Moshkov, M., Zielosko, B.: Combinatorial Machine Learning - A Rough Set Approach, Studies in Computational Intelligence, vol. 360. Springer, Berlin (2011) 23. Pawlak, Z.: Information systems theoretical foundations. Inf. Syst. 6(3), 205–218 (1981) 24. Pawlak, Z.: Rough Sets - Theoretical Aspect of Reasoning About Data. Kluwer Academic Publishers, Dordrecht (1991) 25. Pawlak, Z., Skowron, A.: Rudiments of rough sets. Inf. Sci. 177(1), 3–27 (2007) 26. Read, J., Martino, L., Olmos, P.M., Luengo, D.: Scalable multi-output label prediction: from classifier chains to classifier trellises. Pattern Recognit. 48(6), 2096–2109 (2015) 27. Rokach, L., Maimon, O.: Data Mining with Decision Trees: Theory and Applications. World Scientific Publishing, River Edge, NJ (2008) 28. Skowron, A., Rauszer, C.: The discernibility matrices and functions in information systems. In: Słowi´nski, R. (ed.) Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory, pp. 331–362. Kluwer Academic Publishers, Dordrecht (1992) 29. Soloviev, N.A.: Tests (Theory, Construction, Applications). Nauka, Novosibirsk (1978). (in Russian) 30. Wieczorkowska, A., Synak, P., Lewis, R.A., Ras, Z.W.: Extracting emotions from music data. In: Hacid, M., Murray, N.V., Ras, Z.W., Tsumoto, S. (eds.) Foundations of Intelligent Systems, 15th International Symposium, ISMIS 2005, Saratoga Springs, NY, USA, May 25–28, 2005. Lecture Notes in Computer Science, vol. 3488, pp. 456–465. Springer, Berlin (2005) 31. Zhou, Z.H., Jiang, K., Li, M.: Multi-instance learning based web mining. Appl. Intell. 22(2), 135–147 (2005) 32. Zhuravlev, J.I.: On a class of partial Boolean functions. Diskret. Analiz (in Russian) 2, 23–27 (1964)

Part I

Decision Trees for Decision Tables

In this part, lower and upper bounds on complexity and algorithms for construction of deterministic, nondeterministic, and strongly nondeterministic decision trees for decision tables are considered. This part consists of seven chapters. Chapter 2 contains definitions of the notions of decision table with many-valued decisions, decision tree schema, deterministic, nondeterministic, and strongly nondeterministic decision trees for decision tables, and complexity function. In Chap. 3, lower bounds on complexity of deterministic decision trees for decision tables are studied that are based on the notions of super-cover, super-partition, test, and system of representatives for decision tables. An approach to the proof of lower bounds is also considered which is based on the use of so-called proof-trees. In Chap. 4, for complexity functions having the properties Λ1, Λ2, and Λ3, upper bounds on the minimum complexity and algorithms for construction of deterministic decision trees for decision tables are considered. These bounds and algorithms are based on the use of so-called difference-bounded uncertainty measures for decision tables. In Chap. 5, upper bounds on the minimum complexity and algorithms for construction of deterministic decision trees for decision tables are considered. These bounds and algorithms are based on the use of so-called additive-bounded uncertainty measures for decision tables. The bounds are true for any complexity function having the property Λ1 . When developing algorithms, we assume that the complexity functions have properties Λ1, Λ2, and Λ3. In Chap. 6, we consider bounds on the minimum complexity, an approach to proof of lower bounds, and algorithms for construction of nondeterministic and strongly nondeterministic decision trees. The bounds on complexity are true for arbitrary complexity functions. The approach to proof of lower bounds assumes that the complexity functions have the property Λ2. The considered algorithms require complexity functions having properties Λ1, Λ2, and Λ3. In Chap. 7, for decision tables corresponding to functions from an arbitrary closed class of Boolean functions, the depth of deterministic, nondeterministic, and strongly nondeterministic decision trees is studied. The obtained results have some independent interest. Proofs of these results illustrate mainly methods for the proof of lower bounds on complexity of decision trees.

16

Decision Trees for Decision Tables

In Chaps. 2–7, we study, mainly, approximate bounds on the minimum complexity and approximate algorithms for optimization of decision trees. In Chap. 8, we consider algorithmic problems of computation of the minimum complexity of deterministic, nondeterministic, and strongly nondeterministic decision trees, and of construction of decision trees with the minimum complexity.

Chapter 2

Basic Definitions and Notation

This chapter contains definitions of the notions of decision table with many-valued decisions, decision tree schema, deterministic, nondeterministic, and strongly nondeterministic decision trees for decision tables, and complexity function. We study decision tables with many-valued decisions in which each row is labeled with a set of decisions. For a given row, we should find a decision from the set attached to this row. To solve this problem, we use deterministic and nondeterministic decision trees. For decision tables in which each row is labeled with one of the sets of decisions {0} and {1}, we use also strongly nondeterministic decision trees. Each terminal node of such a tree is labeled with the decision 1. Paths from the root to terminal nodes in this tree accept all rows labeled with {1} and only such rows. Examples of decision tables with many-valued decisions were considered by the author in 1981 [4]. Such tables were investigated systematically in [6, 8]. Decision tables with many-valued decisions (known also as multi-label decision tables) are studied intensively last decades in different applications—see, for example, [1]. Deterministic decision trees are well known as classifiers, as means of knowledge representations, and as algorithms [2, 7, 9]. Nondeterministic decision trees are mainly used in the study of complexity of Boolean functions—see, for example [5]. The minimum depth of a nondeterministic decision tree for a Boolean function f is equal to the certificate complexity of f [3]. The notion of a strongly nondeterministic decision tree for a Boolean function was introduced in [5]. The minimum depth of a strongly nondeterministic decision tree for a Boolean function f is equal to the 1-certificate complexity of f [3].

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_2

17

18

2 Basic Definitions and Notation

2.1 Common Notions Let ω = {0, 1, 2, . . .}. We denote by P(ω) the set of all nonempty finite subsets of the set ω. For k ∈ ω \ {0, 1}, set E k = {0, 1, . . . , k − 1}. A pair ρ = (F, k) will be called a signature if F is a nonempty set and k ∈ ω \ {0, 1}. Elements from F will be considered as names of attributes with values from the set E k . For an arbitrary nonempty set A we denote by A∗ the set of all finite words over the alphabet A including the empty word λ. Denote Ωρ = {( f, δ) : f ∈ F, δ ∈ E k }∗ . We now define on the set 2ω two partial functions min and max with values from ω. Let B ⊆ ω. If B = ∅, then the value of min B is undefined, and if B = ∅, then min B is the minimum number from B. If B is an empty or infinite set, then the value of max B is undefined, and if B is a nonempty finite set, then max B is the maximum number from B.

2.2 Decision Tables A triple T = (Δ, ν, μ) will be called a decision table of the signature ρ if there exists n ∈ ω \ {0} such that Δ ⊆ E kn , ν : Δ → P(ω), μ : {1, . . . , n} → F, and, for any i, j ∈ {1, . . . , n}, if i = j, then μ(i) = μ( j). Later the set Δ and functions ν and μ, defining the table T , will be denoted by Δ(T ), νT and μT , respectively. The number n will be called the dimension of the table T and will be denoted by dim T . The decision table T can be represented as a rectangular table with n columns labeled with pairwise different names of attributes μT (1), . . . , μT (n). The set Δ(T ) is the set of rows of T . Rows are interpreted as n-tuples of values of attributes ¯ μT (1), . . . , μT (n). Each row δ¯ ∈ Δ(T ) is labeled with the set of decisions νT (δ). We correspond to the table T the following problem. For a given row of T , we should find a decision from the set of decisions attached to this row using values of attributes from the row. To solve this problem, we use different types of decision trees. Denote P(T ) = {μT (i)  : i = 1, . . . , n} the set of attribute names attached to ¯ columns of T , Π (T ) = δ∈Δ(T ¯ ) νT (δ) the set of common decisions for T which belong to each set of decisions attached to rows of T , and Ωρ (T ) = {( f, δ) : f ∈ P(T ), δ ∈ E k }∗ . A pair ( f, δ) is interpreted as the condition f = δ, and a word from Ωρ (T ) is interpreted as the conjunction of conditions that are letters of this word. Let α ∈ Ωρ (T ). We correspond to the table T and to the word α a decision table T α of the signature ρ. If α = λ, then Tα = T . Let α = λ, α = ( f 1 , δ1 ) · · · ( f m , δm ) and i 1 , . . . , i m be numbers from {1, . . . , n} such that μ(i 1 ) = f 1 , . . . , μ(i m ) = f m . Then Δ(Tα ) = {(σ1 , . . . , σn ) : (σ1 , . . . , σn ) ∈ Δ(T ), σi1 = δ1 , . . . , σim = δm }, νTα is the restriction of the mapping νT to the set Δ(Tα ), and μTα is the mapping coinciding with the mapping μT . In the other words, Tα is a subtable of the table T which

2.2 Decision Tables

19

contains only rows that at the intersection with columns f 1 , . . . , f m have numbers δ1 , . . . , δm , respectively. Denote Mρ the set of all decision tables of the signature ρ and Mρ C = {T : T ∈ Mρ , Π (T ) = ∅} ∪ {T : T ∈ Mρ , Δ(T ) = ∅} the set of all decision tables of the signature ρ that are either empty (have no rows) or have common decisions. ¯ = {0} or By Mρ0−1 we denote the set of all tables T from Mρ such that νT (δ) ¯ ¯ ¯ ¯ νT (δ) = {1} for any δ ∈ Δ(T ), and there exists δ ∈ Δ(T ) for which νT (δ)  = {1}.  For ¯ ≤m m ∈ ω \ {0}, denote Mρ (m) the set of all tables T from Mρ such that νT (δ) for any δ¯ ∈ Δ(T ). We denote by Mρ F the set of all tables T from Mρ (1) such that Δ(T ) = E kdim T . Such tables describe functions from E kn to ω. A table T ∈ Mρ will be called diagnostic if it satisfies the following conditions: • Δ(T ) = ∅. • For any δ¯1 , δ¯2 ∈ Δ(T ), if δ¯1 = δ¯2 , then νT (δ¯1 ) ∩ νT (δ¯2 ) = ∅. Diagnostic decision tables arise, in particular, in the problems of fault diagnosis.

2.3 Schemes of Decision Trees First, we consider the notion of a schema of decision tree of the signature ρ. Later, we will use this notion to define the notions of decision trees for decision tables and decision trees for problems. A finite directed tree with the root is a finite directed tree in which exactly one node has no entering edges. This node is called the root. Nodes of the tree which have no leaving edges are called terminal nodes. A complete path in a finite directed tree with the root G is a sequence ξ = w0 , d0 , . . . , wm , dm , wm+1 of nodes and edges of the tree G in which w0 is the root of G, wm+1 is a terminal node of G, and, for i = 0, . . . , m, the edge di leaves the node wi and enters the node wi+1 . A schema of decision tree of the signature ρ (schema of the signature ρ) is a finite directed tree with the root with at least two nodes in which: • The root and edges leaving the root are not labeled. • Each node, which is not the root nor terminal node, is labeled with an element from the set F. • Each edge leaving a node, which is not the root, is labeled with a number from E k . • Each terminal node is labeled with a number from ω. We denote by Cρ the set of all schemes of decision trees of the signature ρ. A schema from Cρ is called deterministic if it satisfies the following conditions: • Exactly one edge leaves the root. • For any node, which is not the root nor terminal node, the edges leaving this node are labeled with pairwise different numbers.

20

2 Basic Definitions and Notation

Let Γ ∈ Cρ . Denote by P(Γ ) the set of elements from F attached to nodes of Γ which are not the root nor terminal nodes. Set Ωρ (Γ ) = {( f, δ) : f ∈ P(Γ ), δ ∈ E k }∗ . Let us correspond to each node w of the schema Γ a word πΓ (w) ∈ Ωρ (Γ ). We correspond the empty word λ to the root of Γ and to each node connected with the root by an edge. Let an edge labeled with a number δ leave a node w1 and enter a node w2 , and the node w1 be labeled with an element f . Then πΓ (w2 ) = πΓ (w1 )( f, δ). Later, instead of πΓ (w), we will write often π(w). Let ξ = w0 , d0 , . . . , wm , dm , wm+1 be a complete path in Γ . Denote π(ξ ) = πΓ (wm+1 ). If m = 0, then π(ξ ) = λ. Let m > 0 and, for i = 1, . . . , m, the node wi be labeled with an element f i , and the edge di be labeled with a number δi . Then π(ξ ) = ( f 1 , δ1 ) · · · ( f m , δm ). We denote by (Γ ) the set of all complete paths in the schema Γ .

2.4 Different Types of Decision Trees for Decision Tables Let T ∈ Mρ . A nondeterministic decision tree for the table T is a schema Γ ∈ Cρ which satisfies the following conditions: •  P(Γ ) ⊆ P(T ). • ξ ∈(Γ ) Δ(T π(ξ )) = Δ(T ). • For any complete path ξ ∈ (Γ ), the relation T π(ξ ) ∈ Mρ C holds, and if Δ(T π(ξ )) = ∅, then the terminal node of the path ξ is labeled with a number from the set Π (T π(ξ )). Consider now an equivalent but less formal definition of the notion of a nondeterministic decision tree. Let δ¯ be a row of T and ξ be a complete path in Γ . We say that ξ accepts δ¯ if δ¯ is a row of the table T π(ξ ). The schema Γ is a nondeterministic decision tree for the table T if Γ uses only attributes (names of attributes) from T , for each row of T , there is a complete path in Γ which accepts this row and, for each ¯ the number (decision) attached to the row δ¯ and each complete path ξ accepting δ, ¯ attached to the row δ. ¯ terminal node of ξ belongs to the set of decisions νT (δ) A deterministic decision tree for the table T is a deterministic schema from Cρ which is a nondeterministic decision tree for the table T . Let T ∈ Mρ0−1 . A strongly nondeterministic decision tree for the table T is a schema Γ ∈ Cρ which satisfies the following conditions: •  P(Γ ) ⊆ P(T ). ¯ = {1}}. • ξ ∈(Γ ) Δ(T π(ξ )) = {δ¯ : δ¯ ∈ Δ(T ), νT (δ) • Each terminal node of Γ is labeled with the number 1. Consider now an equivalent but less formal definition. The schema Γ is a strongly nondeterministic decision tree for the table T if Γ uses only attributes (names of attributes) from T , each terminal node of Γ is labeled with the number (decision) 1, and the set of rows of T accepted by complete paths in Γ is equal to the set of rows of T labeled with the set of decisions {1}.

2.5 Complexity Functions

21

2.5 Complexity Functions A complexity function of the signature ρ is an arbitrary mapping ψ : F ∗ → ω. Let us extend the complexity function ψ to the sets Ωρ and Cρ . Values of ψ on empty words from Ωρ and F ∗ coincide. Let α ∈ Ωρ , α = λ and α = ( f 1 , δ1 ) · · · ( f m , δm ). Then ψ(α) = ψ( f 1 · · · f m ). Let Γ ∈ Cρ . Then ψ(Γ ) = max{ψ(π(ξ )) : ξ ∈ (Γ )}. We now define relations Rρd ⊆ Cρ × Mρ , Rρa ⊆ Cρ × Mρ , and Rρs ⊆ Cρ × Mρ0−1 . Let (Γ, T ) ∈ Cρ × Mρ . Then (Γ, T ) ∈ Rρd if and only if Γ is a deterministic decision tree for the table T , and (Γ, T ) ∈ Rρa if and only if Γ is a nondeterministic decision tree for the table T . Let (Γ, T ) ∈ Cρ × Mρ0−1 . Then (Γ, T ) ∈ Rρs if and only if Γ is a strongly nondeterministic decision tree for the table T . Let us correspond functions ψρi : Mρ → ω, ψρd : Mρ → ω, ψρa : Mρ → ω, and s ψρ : Mρ0−1 → ω to the complexity function ψ. Let T be a table from Mρ , and the dimension of T be equal to n. Then ψρi (T ) = ψ(μT (1) · · · μT (n)) , ψρd (T ) = min{ψ(Γ ) : (Γ, T ) ∈ Rρd } , ψρa (T ) = min{ψ(Γ ) : (Γ, T ) ∈ Rρa } . Let T ∈ Mρ0−1 . Then ψρs (T ) = min{ψ(Γ ) : (Γ, T ) ∈ Rρs } . Later we will require sometimes that the complexity function ψ has some of the following properties: 1. 2. 3. 4.

Λ1. Λ2. Λ3. Λ4.

ψ(α1 α2 ) ≤ ψ(α1 ) + ψ(α2 ) for any α1 , α2 ∈ F ∗ . ψ(α1 α2 α3 ) ≥ ψ(α1 α3 ) for any α1 , α2 , α3 ∈ F ∗ . ψ(α) ≥ |α|, for any α ∈ F ∗ , where |α| is the length of the word α. ψ(λ) = 0.

We now consider examples of complexity functions. • A complexity function ψ : F ∗ → ω will be called a weighted depth if it satisfies the following conditions: ψ(λ) = 0, ψ( f ) > 0 for any element f from F, and ψ( f 1 · · · f m ) =

m 

ψ( f i )

i=1

for any nonempty word f 1 · · · f m from F ∗ . A weighted depth ψ is called the depth and is denoted by h if ψ( f ) = 1 for any f ∈ F. One can show that weighted depth has the properties Λ1, Λ2, Λ3, and Λ4.

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2 Basic Definitions and Notation

• A complexity function ψ : F ∗ → ω is called extremal if it satisfies the following conditions: ψ(λ) = 0, and ψ( f 1 · · · f m ) = max{ψ( f 1 ), . . . , ψ( f 1 )} for any nonempty word f 1 · · · f m from F ∗ . One can show that extremal complexity function has the properties Λ1, Λ2, and Λ4. Let ψ1 , ψ2 be complexity functions of the signature ρ and i ∈ {1, 2, 4}. It is not difficult to show that if the functions ψ1 and ψ2 have the property Λi, then the functions ψ3 and ψ4 have the property Λi, where ψ3 (α) = ψ1 (α) + ψ2 (α) and ψ4 (α) = max{ψ1 (α), ψ2 (α)} for any α ∈ F ∗ . If the function ψ1 has the property Λ3, then the functions ψ3 and ψ4 have the property Λ3. Complexity functions having the properties Λ1, Λ2, Λ3, and Λ4 will be called restricted complexity functions. In particular, the sum of the extremal complexity function and the depth is a restricted complexity function.

2.6 Enumerated Signatures We denote by [i]2 the binary representation of the number i ∈ ω. Let ρ = (F, k) be a signature, and there exist a letter f such that f ∈ / {0, 1}∗ and F = { f [i]2 : i ∈ ω}. In this case the signature ρ will be called enumerated. An element f [i]2 ∈ F will be denoted often by f i . Let f i1 , . . . , f im ∈ F and δ1 , . . . , δm ∈ E k . The word ( f [i 1 ]2 , [δ1 ]2 ) · · · ( f [i m ]2 , [δm ]2 ) will be called the binary form of the word ( f i1 , δ1 ) · · · ( f im , δm ). ¯ 2 = ([δ1 ]2 , . . . , [δn ]2 ). For an arbiFor a tuple δ¯ = (δ1 , . . . , δn ) ∈ E kn , denote [δ] trary finite nonempty set D = {d1 , . . . , dr } ⊂ ω, denote [D]2 = ([d1 ]2 , . . . , [dr ]2 ). Let T ∈ Mρ , dim T = n, and Δ(T ) = {δ¯1 , . . . , δ¯m }. The word (([δ¯1 ]2 , [νT (δ¯1 )]2 ), . . . , ([δ¯m ]2 , [νT (δ¯m )]2 ), (μT (1), . . . , μT (n))) in the alphabet {(, ),, , f, 0, 1} will be called the binary form of the table T . Later, when we will study algorithms over decision tables, we assume that the considered signature ρ is enumerated, any table T ∈ Mρ is represented in the binary form, and any word from Ωρ (T ) is represented in the binary form.

References 1. Boutell, M.R., Luo, J., Shen, X., Brown, C.M.: Learning multi-label scene classification. Pattern Recognit. 37(9), 1757–1771 (2004) 2. Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J.: Classification and Regression Trees. Wadsworth and Brooks, Monterey, CA (1984)

References

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3. Buhrman, H., de Wolf, R.: Complexity measures and decision tree complexity: a survey. Theor. Comput. Sci. 288(1), 21–43 (2002) 4. Moshkov, M.: On the uniqueness of dead-end tests for recognition problems with linear decision rules. In: Markov, A.A. (ed.) Combinatorial-Algebraic Methods in Applied Mathematics (in Russian), pp. 97–109. Gorky University Press, Gorky (1981) 5. Moshkov, M.: About the depth of decision trees computing Boolean functions. Fundam. Inform. 22(3), 203–215 (1995) 6. Moshkov, M.: Lower bounds for the time complexity of deterministic conditional tests. Disk. Mat. (in Russian) 8(3), 98–110 (1996). https://doi.org/10.4213/dm538 7. Moshkov, M.: Time complexity of decision trees. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III, Lecture Notes in Computer Science, vol. 3400, pp. 244–459. Springer, Berlin (2005) 8. Moshkov, M.: Bounds on complexity and algorithms for construction of deterministic conditional tests. In: Mat. Vopr. Kibern. (in Russian), vol. 16, pp. 79–124. Fizmatlit, Moscow (2007). http:// library.keldysh.ru/mvk.asp?id=2007-79 9. Rokach, L., Maimon, O.: Data Mining with Decision Trees: Theory and Applications. World Scientific Publishing, River Edge, NJ (2008)

Chapter 3

Lower Bounds on Complexity of Deterministic Decision Trees for Decision Tables

In this chapter, lower bounds on complexity of deterministic decision trees for decision tables are studied that are based on the notions of super-cover, super-partition, test, and system of representatives for decision tables. An approach to the proof of lower bounds is also considered which is based on the use of a proof-tree for the bound ψρd (T ) ≥ r . This is a marked tree of a special kind which exists if and only if the inequality ψρd (T ) ≥ r holds. The parameter Mρ,h (T ) related to the notion of super-cover and corresponding lower bound were introduced in [2]. After publication of the paper [1] this parameter is known as the generalized teaching dimension. The paper [3] contains generalizations of this parameter to various complexity functions and the proof of corresponding lower bound for decision tables with single-valued decisions. Others lower bounds on the depth of deterministic decision trees for such tables can be found in [4] (see also [7, 8]). These bounds are based on the number of different decisions and the minimum cardinality of a test for the decision table. The notion of a proof-tree was introduced in [4] and was used in [5] to evaluate the depth of decision trees computing Boolean functions (see also [7, 8]). The considered approach to the proof of lower bounds is similar to the methods of analysis of search problems developed in [9]. The most part of results presented in this chapter were published in [6], where we studied decision tables with many-valued decisions.

3.1 Definitions and Notation Let ρ = (F, k) be a signature, and α ∈ Ωρ . We denote by χ (α) the set of letters from the alphabet {( f, δ) : f ∈ F, δ ∈ E k } contained in α. We will say that the word α is inconsistent if there exists an element f ∈ F and numbers σ, δ ∈ E k such that ( f, σ ) ∈ χ (α), ( f, δ) ∈ χ (α) and σ = δ. If the word α is not inconsistent, then we will say that α is consistent. Let α, β ∈ Ωρ . If the word αβ is consistent, then we © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_3

25

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3 Lower Bounds on Complexity of Deterministic Decision Trees for Decision Tables

will say that the words α and β are compatible. Otherwise, we will say that the words α and β are incompatible. Let ψ be a complexity function of the signature ρ, and Ω be a nonempty finite subset of the set Ωρ . Denote ψ(Ω) = max{ψ(α) : α ∈ Ω}. Let T be a table of dimension n from Mρ and μT (1) = f 1 , . . . , μT (n) = f n . We denote by Ωˆ ρ (T ) the set of consistent words from Ωρ (T ). A finite set Ω ⊆ Ωˆ ρ (T ) will be called a super-cover for the table T if the following conditions hold: • For any n-tuple (δ1 , . . . , δn ) ∈ E kn , there exists a word α ∈ Ω such that χ (α) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}. • T α ∈ Mρ C for any α ∈ Ω. We denote by SCρ (T ) the set of super-covers for the table T . We now define functions Mρ,ψ : Mρ → ω and lρ : Mρ → ω. Let T ∈ Mρ . Then Mρ,ψ (T ) = min{ψ(Ω) : Ω ∈ SCρ (T )} and lρ (T ) = min{|Ω| : Ω ∈ SCρ (T )} . Let us consider one more definition of the function Mρ,ψ . For δ¯ = (δ1 , . . . , δn ) ∈ let

E kn ,

¯ = min{ψ(α) : α ∈ Ωρ (T ), χ (α) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}, Mρ,ψ (T, δ) T α ∈ Mρ C } . Later (see Lemma 3.5) we will show that ¯ : δ¯ ∈ E kn } . Mρ,ψ (T ) = max{Mρ,ψ (T, δ) A finite set Ω ⊆ Ωˆ ρ (T ) will be called a super-partition for the table T if the following conditions hold: • The set Ω is a super-cover for the table T . • For any α, β ∈ Ω, if α = β, then the words α and β are incompatible. We denote by S Pρ (T ) the set of super-partitions for the table T . We now define functions Mˆ ρ,ψ : Mρ → ω and lˆρ : Mρ → ω. Let T ∈ Mρ . Then Mˆ ρ,ψ (T ) = min{ψ(Ω) : Ω ∈ S Pρ (T )} and

lˆρ (T ) = min{|Ω| : Ω ∈ S Pρ (T )} .

3.1 Definitions and Notation

27

A subset B of the set P(T ) will be called a test for the table T if it satisfies the following conditions: • If B = ∅, then T ∈ Mρ C . • If B = ∅ and B = { f i1 , . . . , f im }, then T ( f i1 , δ1 ), . . . , ( f im , δm ) ∈ Mρ C for any δ1 , . . . , δm ∈ E k . A word α ∈ P(T )∗ will be called an unconditional test for the table T if the set of letters from the alphabet F contained in α is a test for the table T . We will denote by U Cρ (T ) the set of unconditional tests for the table T . We now define the function Θρ,ψ : Mρ → ω. Let T ∈ Mρ . Then Θρ,ψ (T ) = min{ψ(α) : α ∈ U Cρ (T )} . One can show that Θρ,h (T ) is the minimum cardinality of a test for the table T . A finite nonempty subset D of the set ω will be called a system of representatives ¯ = ∅ for any δ¯ ∈ Δ(T ). By definition, the empty set is for the table T if D ∩ νT (δ) a system of representatives for the table T if and only if Δ(T ) = ∅. We denote by l r (T ) the minimum cardinality of a system of representatives for the table T . Let α ∈ F ∗ . We now define a deterministic schema G ρ (α) from Cρ . If α = λ, then G ρ (α) contains exactly two nodes: the root and the terminal node which is labeled with the number 0. Let α = λ and α = f i1 · · · f im . Then G ρ (α) is a schema in which the set of nodes is divided into m + 2 layers. The layer number 0 consists of the root of schema. The edge leaving the root enters the node of the layer number 1. For j = 1, . . . , m + 1, the layer number j contains k j−1 nodes. For j = 1, . . . , m, each node of the layer number j is labeled with the element f i j , k edges labeling with numbers 0, . . . , k − 1 leave this node and enter nodes of the layer number j + 1. Each node of the layer number m + 1 is labeled with the number 0.

3.2 Auxiliary Statements First, we prove six lemmas. Let us fix a signature ρ = (F, k). Lemma 3.1 For any complexity function ψ of the signature ρ and any table T ∈ Mρ , the values of ψρd (T ) and Θρ,ψ (T ) are definite, and the following inequalities hold: ψρd (T ) ≤ Θρ,ψ (T ) ≤ ψρi (T ) . Proof Let T be a table from Mρ which dimension is equal to n. It is clear, that the word μT (1) · · · μT (n) is an unconditional test for the table T . Therefore the value of Θρ,ψ (T ) is definite, Θρ,ψ (T ) ≤ ψρi (T ), and there exists an unconditional test α for the table T such that ψ(α) = Θρ,ψ (T ). Let us define a deterministic schema Γα from Cρ . The schema Γα coincides with the schema G ρ (α) (defined at the end of Sect. 3.1) in which, for each terminal node w, the number attached to this node

28

3 Lower Bounds on Complexity of Deterministic Decision Trees for Decision Tables

is modified in the following way. Let ξ be the complete path in the schema G ρ (α) ending in the node w. If Δ(T π(ξ )) = ∅, then the node w is labeled with the number 0. If Δ(T π(ξ )) = ∅, then the node w is labeled with the minimum number from the set (T π(ξ )). One can show that the schema Γα is a deterministic decision tree for the table T . Hence the value of ψρd (T ) is definite. It is clear that ψρd (T ) ≤ ψ(Γα ) and ψ(Γα ) = ψ(α). Using the equality ψ(α) = Θρ,ψ (T ) we obtain ψρd (T )  ≤ Θρ,ψ (T ). Lemma 3.2 (a) Let T ∈ Mρ and Γ be a nondeterministic decision tree for the table T . Then the set P(Γ ) is a test for the table T . (b) Let T ∈ Mρ0−1 and Γ be a strongly nondeterministic decision tree for the table T . Then the set P(Γ ) is a test for the table T . Proof Let T ∈ Mρ and Γ be a nondeterministic decision tree for the table T . If / Mρ C . T ∈ Mρ C , then, evidently, the set P(T ) is a test for the table T . Let T ∈ One can show that in this case P(T ) = ∅. Let P(T ) = { f 1 , . . . , f m }. Assume the set P(T ) is not a test for the table T . Then there exist numbers σ1 , . . . , σm ∈ E k such that / Mρ C . Denote β = ( f 1 , σ1 ) · · · ( f m , σm ). Let us choose a T ( f 1 , σ1 ) · · · ( f m , σm ) ∈ tuple δ¯ ∈ Δ(Tβ). Since Γ is a nondeterministic decision tree for the table T , there exists a complete path ξ in the schema Γ such that δ¯ ∈ Δ(T π(ξ )) and T π(ξ ) ∈ Mρ C . One can show that χ (π(ξ )) ⊆ χ (β). Therefore Δ(Tβ) ⊆ Δ(T π(ξ )). Taking into account that T π(ξ ) ∈ Mρ C we obtain Tβ ∈ Mρ C , but this contradicts the assumption. Hence P(Γ ) is a test for the table T . Thus, the part (a) of the lemma statement is proved. The part (b) of the lemma statement can be proved in the same way. Note only ¯ = {1}.  that the tuple δ¯ from the set Δ(Tβ) must be chosen such that νT (δ) Let Γ ∈ Cρ . We denote by L 1 (Γ ) the number of nodes in the schema Γ which are not neither the root nor a terminal node. Lemma 3.3 Let Γ be a deterministic schema of the signature ρ = (F, k) such that h(Γ ) ≥ 1. Then h(Γ )−1  ki . L 1 (Γ ) ≤ i=1

Proof Let Γ  be a deterministic schema of the signature ρ in which exactly k edges leave each node that is not neither the root nor a terminal node, and each complete path contains exactly h(Γ ) + 1 nodes. One can show that L 1 (Γ ) ≤ L 1 (Γ  ) =  h(Γ )−1 i k.  i=1 Lemma 3.4 For any complexity function ψ of the signature ρ and any table T ∈ Mρ , the values of Mρ,ψ (T ), Mˆ ρ,ψ (T ), lρ (T ), lˆρ (T ), and l r (T ) are definite. Proof Let the dimension of the table T be equal to n. One can show that the set {(μT (1), δ1 ), . . . , (μT (n), δn ) : (δ1 , . . . , δn ) ∈ E kn }

3.2 Auxiliary Statements

29

is a super-cover and a super-partition for the table T . Evidently, the set 

¯ vT (δ)

¯ δ∈Δ(T )

is a system of representatives for the table T . Therefore the values of Mρ,ψ (T ),  Mˆ ρ,ψ (T ), lρ (T ), lˆρ (T ), and l r (T ) are definite. Lemma 3.5 Let ψ be a complexity function of the signature ρ = (F, k), and T be a table of dimension n from Mρ . Then the following statements hold: ¯ is definite. (a) For any δ¯ ∈ E kn , the value Mρ,ψ (T, δ) ¯ : δ¯ ∈ E kn }. (b) Mρ,ψ (T ) = max{Mρ,ψ (T, δ) Proof Let μT (1) = f 1 , . . . , μT (n) = f n , and δ¯ = (δ1 , . . . , δn ) ∈ E kn . Denote β = ( f 1 , δ1 ) · · · ( f n , δn ). It is clear that χ (β) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )} and Tβ ∈ Mρ C . ¯ is definite. Therefore, for any tuple δ¯ = (δ1 , . . . , δn ) ∈ Hence the value of Mρ,ψ (T, δ) ¯ such that χ (α(δ)) ¯ ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}, T α(δ) ¯ ∈ E kn , there exists a word α(δ) ¯ = Mρ,ψ (T, δ). ¯ Mρ C , and ψ(α(δ)) ¯ : δ¯ ∈ E kn }. Evidently, the set Ω0 is a super-cover for the table Denote Ω0 = {α(δ) ¯ : δ¯ ∈ E kn }. Therefore T , and ψ(Ω0 ) = max{Mρ,ψ (T, δ) ¯ : δ¯ ∈ E kn } . Mρ,ψ (T ) ≤ max{Mρ,ψ (T, δ) From Lemma 3.4 it follows that there exists a super-cover Ω for the table T such that ψ(Ω) = Mρ,ψ (T ). Let δ¯ = (δ1 , . . . , δn ) ∈ E kn . Then there exists a word α ∈ Ω for which χ (α) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )} and T α ∈ Mρ C . Evidently, ψ(α) ≤ ¯ ≤ Mρ,ψ (T ). Taking into account that δ¯ is an arbitrary Mρ,ψ (T ). Hence Mρ,ψ (T, δ) ¯ : δ¯ ∈ E kn } ≤ Mρ,ψ (T ).  tuple from E kn we obtain max{Mρ,ψ (T, δ) Lemma 3.6 Let ψ be a complexity function of the signature ρ = (F, k) having the property Λ3, T ∈ Mρ , and Ω be a super-partition for the table T . Then |Ω| ≤ k ψ(Ω) , and |α| ≤ ψ(Ω) for any α ∈ Ω. Proof Let T be a table of dimension n. Denote m = ψ(Ω). Since the function ψ has the property Λ3, the following inequality holds for any α ∈ Ω: |α| ≤ m .

(3.1)

We now define a table T  of dimension n from Mρ : Δ(T  ) = E kn , νT  ≡ {0}, and μT  ≡ μT . Let α1 , α2 ∈ Ω and α1 = α2 . Since the words α1 and α2 are incompatible, we have (3.2) Δ(T  α1 ) ∩ Δ(T  α2 ) = ∅ . Taking into account that Ω is a super-partition for the table T one can show that   Δ(T α) = E kn . From this equality and from (3.2) it follows that α∈Ω

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3 Lower Bounds on Complexity of Deterministic Decision Trees for Decision Tables

  Δ(T  α) = k n .

(3.3)

α∈Ω

From (3.1), from the equality Δ(T  ) = E kn , and from the relation Ω ⊆ Ωˆ ρ (T ) it  follows that Δ(T  α) ≥ k n−m for any α ∈ Ω. Using this inequality and (3.3) we  obtain |Ω| k n−m ≤ k n . Therefore |Ω| ≤ k m .

3.3 Lower Bounds We now consider several lower bounds on the minimum complexity of deterministic decision trees for decision tables. Theorem 3.1 For any complexity function ψ of the signature ρ = (F, k) and any table T ∈ Mρ , the following inequalities hold: ψρd (T ) ≥ Mˆ ρ,ψ (T ) ≥ Mρ,ψ (T ) . Proof Let T be a table of dimension n and μT (1) = f 1 , . . . , μT (n) = f n . Using Lemma 3.1 we conclude that there exists a deterministic decision tree Γ for the table T such that ψ(Γ ) = ψρd (T ). Denote Q(Γ ) the set of nodes of Γ satisfying the following conditions: the node is not neither the root nor a terminal node, and at most k − 1 edges leave this node. Let w ∈ Q(Γ ). We denote by E(w) the set of numbers attached to edges leaving the node w. Let us choose a path τ from the node w to a terminal node of Γ . We now define a word α ∈ F ∗ . If the path τ contains exactly two nodes, then α = λ. Let τ = w0 , d0 , w1 , d1 , . . . , wm , dm , wm+1 , where m ≥ 1, w0 = w, for j = 0, . . . , m, the edge d j leaves the node w j and enters the node w j+1 , and, for j = 1, . . . , m, the node w j is labeled with an element f i j ∈ F. Then α = f i1 · · · f im . We denote by G a finite directed tree with the root which is obtained from the schema G ρ (α) (defined at the end of Sect. 3.1) by removal of the root and the edge leaving the root. For each δ ∈ E k \ E(w), we add to the schema Γ the tree G and the edge leaving the node w and entering the root of the tree G. We mark this edge by the number δ. Let us transform in the same way all nodes from the set Q(Γ ). As a result, we obtain a schema from Cρ . We denote this schema by Γ  . Set Ω = {π(ξ ) : ξ ∈ Ξ (Γ  )} ∩ Ωˆ ρ (T ). One can show that ψ(Γ  ) = ψ(Γ ) and ψ(Ω) ≤ ψ(Γ  ). Taking into account that ψ(Γ ) = ψρd (T ) we obtain ψ(Ω) ≤ ψρd (T ) .

(3.4)

Let us show that the set Ω is a super-partition for the table T . For each node of the schema Γ  , which is not neither the root nor terminal node, for any δ ∈ E k , an edge labeled with the number δ leaves this node. Moreover, P(Γ  ) = P(Γ ) and P(Γ ) ⊆ P(T ). Using these facts it is not difficult to show that, for any tuple (δ1 , . . . , δn ) ∈ E kn , there exists a path ξ ∈ Ξ (Γ  ) such that χ (π(ξ )) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}. From

3.3 Lower Bounds

31

the last relation it follows that π(ξ ) ∈ Ωˆ ρ (T ). Thus, for any tuple (δ1 , . . . , δn ) ∈ E kn , there exists a word α from Ω such that α ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}. Evidently, the schema Γ  is a deterministic schema. Using this fact it is not difficult to show that, for any ξ1 , ξ2 ∈ Ξ (Γ  ), if ξ1 = ξ2 , then the words π(ξ1 ) and π(ξ2 ) are incompatible. Therefore any two different words from the set Ω are incompatible. Let ξ1 , ξ2 ∈ Ξ (Γ  ) and ξ1 = ξ2 . Since the words π(ξ1 ) and π(ξ2 ) are incompatible, we have (3.5) Δ(T π(ξ1 )) ∩ Δ(T π(ξ2 )) = ∅ . Evidently, Ξ (Γ ) ⊆ Ξ (Γ  ). Let ξ ∈ Ξ (Γ ). Taking into account that Γ is a deterministic decision tree for the table T , we obtain T π(ξ ) ∈ Mρ C . Let ξ ∈ Assume the contrary. Then, Ξ (Γ  ) \ Ξ (Γ ). We now show that Δ(T π(ξ )) = ∅.  using (3.5) and the relation Ξ (Γ ) ⊆ Ξ (Γ  ), we obtain ξ ∈Ξ (Γ ) Δ(T π(ξ )) = Δ(T ) which is impossible since Γ is a deterministic decision tree for the table T . Hence T α ∈ Mρ C for any word α ∈ Ω. Thus, Ω is a super-partition for the table T . Evidently, Mˆ ρ,ψ (T ) ≤ ψ(Ω). Using (3.4) we obtain Mˆ ρ,ψ (T ) ≤ ψρd (T ). Since any super-partition for the table T is a super-cover for the table T , we have  Mρ,ψ (T ) ≤ Mˆ ρ,ψ (T ).  Let us consider a corollary of Theorem 3.1. Let T ∈ Mρ . Denote νT (T ) = ¯ ¯ δ∈Δ(T ) νT (δ). For an arbitrary m ∈ νT (T ), set ¯ . Δ(T, m) = {δ¯ : δ¯ ∈ Δ(T ), m ∈ νT (δ)} We denote by dρ (T, m) the maximum r ∈ ω such that k r is a divisor of the number |Δ(T, m)|. Set dρ (T ) = min{dρ (T, m) : m ∈ νT (T )} . Proposition 3.1 Let ψ be a complexity function of the signature ρ = (F, k) having the property Λ3, and T ∈ Mρ F . Then ψρd (T ) ≥ dim T − dρ (T ) . Proof From Lemma 3.4 it follows that there exists a super-partition Ω for the table T (T, m) = dρ (T ). such that ψ(Ω) = Mˆ ρ,ψ (T ). Let us choose m ∈ νT (T ) such that dρ  Since T ∈ Mρ F , there exists a subset Ω  of the set Ω for which α∈Ω  Δ(T α) = Δ(T, m). Taking into account that any two different words from Ω are incompatible,  we obtain α∈Ω  |Δ(T α)| = |Δ(T, m)|. Since Δ(T ) = E kdim T , for any α ∈ Ω, the equality |Δ(T α)| = k dim T −m(α) holds, where m(α) ∈ ω and m(α) ≤ |α|. Therefore k dim T − p is a divisor of the number |Δ(T, m)|, where p = max{|α| : α ∈ Ω  }. Hence dim T − p ≤ dρ (T, m). Taking into account that dρ (T, m) = dρ (T ) we obtain p ≥ dim T − dρ (T ). Since the function ψ has the property Λ3, we have ψ(Ω) ≥ p. Taking into account that ψ(Ω) = Mˆ ρ,ψ (T ) we obtain Mˆ ρ,ψ (T ) ≥ dim T − dρ (T ). From Theorem 3.1 it follows that ψρd (T ) ≥ Mˆ ρ,ψ (T ). Therefore ψρd (T ) ≥ dim T − dρ (T ). 

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Remark 3.1 If ρ = (F, 2), T ∈ Mρ F , and there exists m ∈ νT (T ) such that |Δ(T, m)| is an odd number, then from Lemma 3.1 and Proposition 3.1 it follows that h dρ (T ) = dim T . Theorem 3.2 Let ψ be a complexity function of the signature ρ = (F, k) having the property Λ3, and T be a table from Mρ . Then ψρd (T ) ≥ logk lˆρ (T ) ≥ logk lρ (T ) , and if Δ(T ) = ∅, then

logk lρ (T ) ≥ logk l r (T ) .

Proof Let dim T = n and μT (1) = f 1 , . . . , μT (n) = f n . From Lemma 3.4 it follows that there exists a super-partition Ω for the table T such that ψ(Ω) = Mˆ ρ,ψ (T ). ˆ Using Lemma 3.6 we obtain |Ω| ≤ k Mρ,ψ (T ) . Evidently, lˆρ (T ) ≤ |Ω|. Hence ˆ

k Mρ,ψ (T ) ≥ lˆρ (T ) .

(3.6)

Since any super-partition for the table T is a super-cover for T , we have lˆρ (T ) ≥ lρ (T ) .

(3.7)

From the definition of a super-cover it follows that any super-cover for the table T is a nonempty set. Therefore lρ (T ) ≥ 1. From this inequality and from (3.6) and (3.7) it follows that Mˆ ρ,ψ (T ) ≥ logk lˆρ (T ) ≥ logk lρ (T ). By Theorem 3.1, ψρd (T ) ≥ Mˆ ρ,ψ (T ). Hence ψρd (T ) ≥ logk lˆρ (T ) ≥ logk lρ (T ). Let Δ(T ) = ∅. From Lemma 3.4 it follows that there exists a super-cover Ω1 for the table T such that |Ω1 | = lρ (T ). Let us correspond a number n(α) ∈ ω to any word α ∈ Ω1 . If Δ(T α) = ∅, then n(α) = 0. If Δ(T α) = ∅, then n(α) is the minimum number from the set (T α). Denote D = {n(α) : α ∈ Ω1 }. Let δ¯ = (δ1 , . . . , δn ) ∈ Δ(T ). Then there exists a word α ∈ Ω1 such that χ (α) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}. ¯ Since Δ(T α) = ∅, we have Evidently, δ¯ ∈ Δ(T α). Therefore (T α) ⊆ νT (δ). ¯ and D ∩ νT (δ) ¯ = ∅. Taking into account that δ¯ n(α) ∈ (T α). Hence n(α) ∈ νT (δ) is an arbitrary tuple from Δ(T ) we conclude that the set D is a system of representatives for the table T . Therefore l r (T ) ≤ |D|. Evidently, |D| ≤ |Ω1 |. Taking into account that |Ω1 | = lρ (T ), we obtain lρ (T ) ≥ l r (T ). Since Δ(T ) = ∅, we obtain  l r (T ) ≥ 1. Therefore logk lρ (T ) ≥ logk l r (T ). Let us consider a statement which is a simple corollary of Theorem 3.2. We now define a function Nρ : Mρ → ω. Let T ∈ Mρ . Then Nρ (T ) = |Δ(T )|. Proposition 3.2 For any complexity function ψ of the signature ρ = (F, k) having the property Λ3 and for any diagnostic table T ∈ Mρ , the following inequality holds: ψρd (T ) ≥ logk Nρ (T ) .

3.3 Lower Bounds

33

Proof Evidently, Δ(T ) = ∅. By Theorem 3.2, ψρd (T ) ≥ logk l r (T ). One can show  that l r (T ) = Nρ (T ). Hence ψρd (T ) ≥ logk Nρ (T ). Theorem 3.3 For any complexity function ψ of the signature ρ = (F, k) having the property Λ3 and for any table T ∈ Mρ , the following inequality holds: ψρd (T ) ≥ logk (Θρ,h (T ) + 1) . Proof If Θρ,h (T ) = 0, then the statement of theorem holds. Let Θρ,h (T ) ≥ 1. From Lemma 3.1 it follows that there exists a deterministic decision tree Γ for the table T such that ψ(Γ ) = ψρd (T ). Taking into account that any deterministic decision tree for the table T is a nondeterministic decision tree for the table T and using Lemma 3.2 we conclude that the set P(Γ ) is a test for the table T . One can show that the value Θρ,h (T ) is equal to the minimum cardinality of a test for the table T . Hence Θρ,h (T ) ≤ |P(Γ )|. It is clear that |P(Γ )| ≤ L 1 (Γ ). Therefore Θρ,h (T ) ≤ L 1 (Γ ). Taking into account that Θρ,h (T ) ≥ 1 we obtain h(Γ ) ≥ 1. Lemma 3.3 shows h(Γ )−1 i h(Γ ) k = k k−1−1 ≤ k h(Γ ) − 1. Therefore Θρ,h (T ) ≤ k h(Γ ) − 1. that L 1 (Γ ) ≤ i=0 Hence h(Γ ) ≥ logk (Θρ,h (T ) + 1). Since the function ψ has the property Λ3, the inequality ψ(Γ ) ≥ h(Γ ) holds. Taking into account that ψ(Γ ) = ψρd (T ) we obtain  ψρd (T ) ≥ logk (Θρ,h (T ) + 1).

3.4 Approach to Proof of Lower Bounds Let ρ = (F, k) be a signature, A be a nonempty finite subset of the set F, and A = { f 1 , . . . , f n }. A marked finite directed tree with the root will be called an (A, ρ)tree if it satisfies the following conditions. The nodes of the tree are not labeled. Each edge of the tree is labeled with a pair from the set {( f i , δ) : f i ∈ A, δ ∈ E k }. Either the root is a terminal node, or n edges leave the root and these edges are labeled with pairs of the form ( f 1 , δ1 ), . . . , ( f n , δn ) respectively, where δ1 , . . . , δn ∈ E k . Let w be a node of the tree which is not the root, and B(w) be the set of elements from A which are not contained in pairs attached to edges of the path from the root to the node w. If B(w) = ∅, then w is a terminal node. Let B(w) = ∅ and B(w) = { f i1 , . . . , f im }. Then either w is a terminal node, or m edges leave w, and these edges are labeled with pairs of the form ( f i1 , σ1 ), . . . , ( f im , σm ) respectively, where σ1 , . . . , σm ∈ E k . Let G be an (A, ρ)-tree. We now correspond to each node w of the tree G a word ζ (w) ∈ {( f i , δ) : f i ∈ A, δ ∈ E k }∗ . We correspond the empty word λ to the root of G . Let an edge d leave a node w1 , enter a node w2 , and be labeled with a pair ( f i , δ). Then ζ (w2 ) = ζ (w1 )( f i , δ). Let ψ be a complexity function of the signature ρ, T ∈ Mρ , r ∈ ω, and G be a (P(T ), ρ)-tree. The tree G will be called a proof-tree for the bound ψρd (T ) ≥ r if Δ(T ζ (w)) = ∅ and ψ(ζ (w)) ≥ r for any terminal node w of the tree G, and T ζ (u) ∈ / Mρ C for any node u of the tree G which is not terminal.

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3 Lower Bounds on Complexity of Deterministic Decision Trees for Decision Tables

Let T ∈ Mρ , dim T = n > 1, f ∈ P(T ), and δ ∈ E k . We now define a table T [ f, δ] ∈ Mρ . Let i ∈ {1, . . . , n} and μT (i) = f . Then dim T [ f, δ] = n − 1, the set Δ(T [ f, δ]) is obtained from the set Δ(T ( f, δ)) by removal of i-th component from each tuple belonging to Δ(T ( f, δ)), the tuple (μT [ f,δ] (1), . . . , μT [ f,δ] (n − 1)) is obtained from the tuple (μT (1), . . . , μT (n)) by removal of i-th component. Let δ¯ ∈ Δ(T [ f, δ]) and the tuple δ¯ is obtained from the tuple σ¯ ∈ Δ(T ( f, δ)) by removal ¯ = νT (σ¯ ). of i-th component. Then νT [ f,δ] (δ) Theorem 3.4 Let ψ be a complexity function of the signature ρ having the property Λ2, T ∈ Mρ , Δ(T ) = ∅, and r ∈ ω. Then a proof-tree for the bound ψρd (T ) ≥ r exists if and only if the inequality ψρd (T ) ≥ r holds. Proof Let G be a proof-tree for the bound ψρd (T ) ≥ r . From Lemma 3.1 it follows that there exists a deterministic decision tree Γ for the table T such that ψ(Γ ) = ψρd (T ). Let us correspond to each node w of the schema Γ a word π(w) = πΓ (w) ∈ Ωρ (Γ ) as it was done in Sect. 2.3. We now describe a process of construction of a directed path ξ = w0 , d0 , w1 , . . . in the schema Γ starting in the root of Γ , and a complete path τ = v1 , q1 , v2 , . . . in the tree G. During the first step, we construct the fragment w0 , d0 , w1 of the path ξ and the node v1 of the path τ , where w0 is the root of Γ , d0 is the edge leaving w0 , w1 is the node which d0 enters, and v1 is the root of G. In what follows, during each step with the exception of the last one, we add one edge and one node to the path ξ and at most one edge and one node to the path τ . Let i steps have been made. Let us describe the step number (i + 1). Let wi and v j be the last nodes of paths ξ and τ , respectively, constructed during the first i steps. If v j is a terminal node, then the process of path construction is completed. Let the node v j be not terminal. Then, as it will be shown later, the node wi is not terminal. Let the node wi be labeled with an element f t . From the description of the tree G it follows that, among the edges in the path from the root of G to the node v j and the edges leaving the node v j , there is exactly one edge l labeled with a pair of the form ( f t , δ), where δ ∈ E k . In the capacity of the edge di , we add to the path ξ an edge which leaves the node wi and is labeled with the number δ (the existence of such edge will be proved later). In the capacity of the node wi+1 , we add the node which the edge di enters. If the edge l leaves the node v j , then we add the edge l to the path τ as the edge q j and, in the capacity of the node v j+1 , we add to the path τ the node which the edge l enters. If the edge l leaves a node different from v j , then we do not add anything to the path τ . We now consider two statements relating to the description of the step number (i + 1). Let the node v j be not terminal. We show that the node wi is not terminal too. From the description of the process of paths ξ and τ construction it follows that χ (π(wi )) = χ (ζ (v j )). Since the node v j is not a terminal node of the tree G, we / Mρ C . Hence T π(wi ) ∈ / Mρ C . Assume that the node wi is a terminal have T ζ (v j ) ∈

3.4 Approach to Proof of Lower Bounds

35

node. Let γ be the complete path in Γ ending in wi . Since Γ is a deterministic decision tree for the table T , we have T π(γ ) ∈ Mρ C . Evidently, π(wi ) = π(γ ) and T π(wi ) ∈ Mρ C which is impossible. Hence the node wi is not terminal. Let us show that there exists an edge which leaves the node wi and which is labeled with the number δ. Assume the contrary. One can show that there exists a node v of the tree G such that χ (π(wi )( f t , δ)) = χ (ζ (v)). Since Δ(T ζ (v)) = ∅, we have Δ(T π(wi )( f t , δ)) = ∅. Taking into account that Γ is a deterministic schema, one can show that Δ(T π(ξ )) ∩ Δ(T π(wi )( f t , δ)) = ∅ for any complete path ξ in the schema Γ . Hence ξ ∈Ξ (Γ ) Δ(T π(ξ )) = Δ(T ) which is impossible since Γ is a deterministic decision tree for the table T . Therefore the considered edge exists. Since there are no infinite directed paths in Γ , the process of construction of the directed path ξ and the complete path τ will end after a finite number of steps. Let wm and v p be the last nodes of the paths ξ and τ , respectively. It is not difficult to show that the word ζ (v p ) can be obtained from the word π(wm ) by removal of some letters. Evidently, there exists a complete path γ in the schema Γ which passes through the node wm . It is clear that the word π(wm ) is a prefix of the word π(γ ). Since the function ψ has the property Λ2, the inequalities ψ(π(γ )) ≥ ψ(π(wm )) ≥ ψ(ζ (v p )) hold. Taking into account that v p is a terminal node of the tree G we conclude that ψ(ζ (v p )) ≥ r . Hence ψ(Γ ) ≥ r . Since ψ(Γ ) = ψρd (T ), we obtain ψρd (T ) ≥ r . We now prove by induction on n that, for any table T ∈ Mρ with dim T = n and Δ(T ) = ∅, any complexity function ψ having the property Λ2, and for any r ∈ ω, if the inequality ψρd (T ) ≥ r holds, then there exists a proof-tree for the bound ψρd (T ) ≥ r . Let us show that the considered statement holds for n = 1. Let dim T = 1. Let T ∈ Mρ C and m ∈ (T ). Then a schema of the signature ρ, which has exactly two nodes and in which the terminal node is labeled with m, is a deterministic decision tree for the table T . Hence ψρd (T ) ≤ ψ(λ). Since the function ψ has the property Λ2, the inequality ψ(α) ≥ ψ(λ) holds for any α ∈ F ∗ . Therefore ψρd (T ) = ψ(λ). One can show that the tree G  consisting of a single node, which is not marked, is a proof-tree for the bound ψρd (T ) ≥ ψ(λ). Let the inequality ψρd (T ) ≥ r hold. Then ψ(λ) ≥ r . Hence G  is a proof-tree for the bound ψρd (T ) ≥ r . Let T ∈ / Mρ C , Γ be an arbitrary deterministic decision tree for the table T , and ξ ∈ Ξ (Γ ). Since T ∈ / Mρ C , we have π(ξ ) = λ. Taking into account that the function ψ has the property Λ2, we conclude that ψ(π(ζ )) ≥ ψ( f ), where f is the unique element of the set P(T ). Hence ψρd (T ) ≥ ψ( f ). One can show that there exists a deterministic decision tree Γ for the table T such that ψ(Γ ) = ψ( f ). Therefore / Mρ C , there exists δ ∈ E k such that Δ(T ( f, δ)) = ∅. ψρd (T ) = ψ( f ). Since T ∈ Hence the tree G  , consisting of two nodes which are not marked and the edge connecting them and labeled with the pair ( f, δ), is a proof-tree for the bound ψρd (T ) ≥ ψ( f ). Let the inequality ψρd (T ) ≥ r hold. Then ψ( f ) ≥ r . Therefore G  is a proof-tree for the bound ψρd (T ) ≥ r . Let the considered statement hold for some n ≥ 1. We now prove that this statement holds for n + 1 too. Let dim T = n + 1, and the inequality ψρd (T ) ≥ r hold. Let T ∈ Mρ C . As in the case of dim T = 1, we conclude that the tree G  is a proof-

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3 Lower Bounds on Complexity of Deterministic Decision Trees for Decision Tables

tree for the bound ψρd (T ) ≥ r . Let T ∈ / Mρ C and P(T ) = { f 1 , . . . , f n+1 }. For any f i ∈ P(T ), we define a complexity function f i ψ of the signature ρ as follows: f i ψ(α) = ψ( f i α) for any α ∈ F ∗ . Since the function ψ has the property Λ2, the following relations hold for any α1 , α2 , α3 ∈ F ∗ : f i ψ(α1 α2 α3 ) = ψ( f i α1 α2 α3 ) ≥ ψ( f i α1 α3 ) ≥ f i ψ(α1 α3 ) . Therefore the function f i ψ has the property Λ2. One can show that, for any f i ∈ P(T ), the following inequality holds: ψρd (T ) ≤ max{ f i ψρd (T [ f i , δ]) : δ ∈ E k , Δ(T [ f i , δ]) = ∅} . Hence, for each f i ∈ P(T ), there exists δi ∈ E k such that Δ(T [ f i , δi ]) = ∅ and f i ψρd (T [ f i , δi ]) ≥ ψρd (T ) ≥ r . Evidently, dim T [ f i , δi ] = n for any f i ∈ P(T ). Using the inductive hypothesis we conclude that, for any f i ∈ P(T ), there exists a proof-tree G i for the bound f i ψρd (T [ f i , δi ]) ≥ r . We denote by G a marked finite directed tree with the root of the following type: for any f i ∈ P(T ), an edge labeled with the pair ( f i , δi ) leaves the root of G and enters the root of the tree G i . One can  show that G is a proof-tree for the bound ψρd (T ) ≥ r .

References 1. Hegedüs, T.: Generalized teaching dimensions and the query complexity of learning. In: Maass, W. (ed.) Proceedings of the Eigth Annual Conference on Computational Learning Theory, COLT 1995, Santa Cruz, California, USA, July 5–8, 1995, pp. 108–117. ACM (1995) 2. Moshkov, M.: On conditional tests. Sov. Phys. Dokl. 27, 528–530 (1982) 3. Moshkov, M.: Conditional tests. In: Yablonskii, S.V. (ed.) Problemy Kibernetiki (in Russian), vol. 40, pp. 131–170. Nauka Publishers, Moscow (1983) 4. Moshkov, M.: Decision Trees. Theory and Applications (in Russian). Nizhny Novgorod University Publishers, Nizhny Novgorod (1994) 5. Moshkov, M.: About the depth of decision trees computing Boolean functions. Fundam. Inform. 22(3), 203–215 (1995) 6. Moshkov, M.: Lower bounds for the time complexity of deterministic conditional tests. Diskret. Mat. (in Russian) 8(3), 98–110 (1996). https://doi.org/10.4213/dm538 7. Moshkov, M.: Time complexity of decision trees. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III, Lecture Notes in Computer Science, vol. 3400, pp. 244–459. Springer, Berlin (2005) 8. Moshkov, M., Zielosko, B.: Combinatorial Machine Learning - A Rough Set Approach. Studies in Computational Intelligence, vol. 360. Springer, Heidelberg (2011) 9. Tarasova, V.P.: Opponent Strategy Method in Optimal Search Problems (in Russian). Moscow University Publishers, Moscow (1988)

Chapter 4

Upper Bounds on Complexity and Algorithms for Construction of Deterministic Decision Trees for Decision Tables. First Approach

In this chapter, for complexity functions having the properties Λ1, Λ2, and Λ3, upper bounds on the minimum complexity and algorithms for construction of deterministic decision trees for decision tables are considered. These bounds and algorithms are based on the use of so-called difference-bounded uncertainty measures for decision tables. The first step in this direction was done in [1], where a simple greedy algorithm for construction of deterministic decision trees was considered and a bound on the depth of the constructed tree was discussed. This algorithm was based on the uncertainty measure which is the number of unordered pairs of rows of the decision table with different decisions. Apparently, the first publication which proposed a similar algorithm for decision tree construction was [4]. In [2], the considered greedy algorithm for decision tables with single-valued decisions was generalized to various uncertainty measures and cost functions. To construct a decision tree, this algorithm chooses an attribute which reduces the uncertainty to the greatest extent under certain constraints on the complexity of the attribute. The most part of results presented in this chapter were published in [3], where we studied decision tables with many-valued decisions.

4.1 Difference-Bounded Uncertainty Measures for Decision Tables Let ρ = (F, k) be a signature, T ∈ Mρ , dim T = n, and μT (1) = f 1 , . . . , μT (n) = f n . A word α ∈ Ωρ (T ) will be called complete for the table T if {( f 1 , δ1 ), . . . , ( f n , δn )} ⊆ χ (α) for some δ1 , . . . , δn ∈ E k . A function γ : Ωρ (T ) → ω will be called a difference-bounded uncertainty measure for the decision table T if it has the following properties:

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_4

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• Difference-boundedness: for any α, β, ( f i , δ) ∈ Ωρ (T ), if β( f i , δ) ∈ Ωˆ ρ (T ) (i.e., if β( f i , δ) is a consistent word from Ωρ (T )), then γ (α) − γ (α( f i , δ)) ≥ γ (αβ) − γ (αβ( f i , δ)). • Partial coordination: for any α ∈ Ωρ (T ), if γ (α) = 0, then T α ∈ Mρ C . • Completeness: γ (α) = 0 for any word α ∈ Ωρ (T ) which is complete for the table T. • Monotonicity: for any α, β ∈ Ωρ (T ), if γ (β) = 0, then γ (αβ) = 0. A difference-bounded uncertainty measure γ for the table T will be called coordinated if it has the following property: • Coordination: for any α ∈ Ωρ (T ), the equality γ (α) = 0 holds if and only if T α ∈ Mρ C . Let p, q ∈ ω and p + q ≥ 1. One can show that if functions γ1 and γ2 are difference-bounded uncertainty measures for the decision table T , then the function γ3 = pγ1 + qγ2 is a difference-bounded uncertainty measure for the decision table T too. If the functions γ1 and γ2 have the property of coordination, then the function γ3 has this property too. Remark 4.1 Let T ∈ Mρ and γ : Ωρ (T ) → ω. One can show that if the function γ has the property of coordination, then it has the properties of partial coordination, completeness, and monotonicity. We now consider examples of difference-bounded uncertainty measures. Let T ∈ Mρ . A table T  will be called a separable subtable of the table T if there exists a word α ∈ Ωρ (T ) such that T  = T α. The table T will be called a boundC for any f i ∈ P(T ) ary table if T ∈ / Mρ C and T ( f i , δ) = T or T ( f i , δ) ∈ Mρ ¯ Let I = νT (δ). and δ ∈ E k . Let A ⊆ Δ(T ) and A = ∅. Denote Π (A, T ) = δ∈A ¯ {( f i , δ) : ( f i , δ) ∈ Ωρ (T ), A ⊆ Δ(T ( f i , δ))}. Let us define a word α(A) ∈ Ωρ (T ). If I = ∅, then α(A) = λ. If I = ∅ and I = {( f i1 , δ1 ), . . . , ( f im , δm )}, then α(A) = ( f i1 , δ1 ) · · · ( f im , δm ). We will say that the set A separates the subtable T α(A) of the table T . Let m ∈ ω \ {0, 1}. We now describe functions G Hρ , R Hρm , and Hρm from Mρ to ω. Let T ∈ Mρ . Then • G Hρ (T ) is the number of different separable subtables of the table T which are boundary subtables. • R Hρm (T ) is the number of different nonempty subsets A of the set Δ(T ) such that |A| ≤ m and Π (A, T ) = ∅. • Hρm (T ) is the number of different nonempty subsets A of the set Δ(T ) such that |A| ≤ m, Π (A, T ) = ∅, and the subtable T α(A) of the table T separated by the set A is a boundary table. Let f : Mρ → ω and T ∈ Mρ . We now define a function T ∗ f : Ωρ (T ) → ω. Let α ∈ Ωρ (T ). Then T ∗ f (α) = f (T α).

4.1 Difference-Bounded Uncertainty Measures for Decision Tables

39

Proposition 4.1 For any table T ∈ Mρ , the function T ∗ G Hρ is a coordinated difference-bounded uncertainty measure for the decision table T . Let m ∈ ω \ {0}. Then, for any table T ∈ Mρ (m), the functions T ∗ R Hρm+1 and T ∗ Hρm+1 are coordinated difference-bounded uncertainty measures for the decision table T . Proof Let T ∈ Mρ , α, β, ( f i , δ) ∈ Ωρ (T ), and β( f i , δ) ∈ Ωˆ ρ (T ). (a) We denote by G the function T ∗ G Hρ . By B (Bβ respectively) we denote the set of boundary tables which are separable subtables of the table T α (T αβ respectively) but are not separable subtables of the table T α( f i , δ) (T αβ( f i , δ) respectively). One can show that T  ∈ B (T  ∈ Bβ respectively) if and only if the table T  is a boundary table, T  is a separable subtable of the table T α (T αβ respectively), and T  ( f i , δ) ∈ Mρ C . One can also show that if T  is a separable subtable of the table T αβ, then T  is a separable subtable of the table T α too. Therefore Bβ ⊆ B. It is not difficult to show that |B| = G(α) − G(α( f i , δ)) and |Bβ| = G(αβ) − G(αβ( f i , δ)). Hence G(α) − G(α( f i , δ)) ≥ G(αβ) − G(αβ( f i , δ)). Thus, the function G has the property of difference-boundedness. / Mρ C . Then either T α is a Evidently, if T α ∈ Mρ C , then G(α) = 0. Let T α ∈ boundary table and, hence, G(α) ≥ 1, or there exist f j ∈ P(T ) and σ ∈ E k such / Mρ C and T α( f j , σ ) = T α. In the last case, we consider the table that T α( f j , σ ) ∈ T α( f j , σ ) instead of the table T α, and will continue this process until obtain a boundary table which is a separable subtable of the table T α. The considered process is finite and leads to a boundary table since after each step the obtained table does not belong to the set Mρ C , and the value of the function Nρ for this table is less than the value of the function Nρ for the previous table. Hence G(α) ≥ 1. Thus, the function G has the property of coordination. Taking into account Remark 4.1 we conclude that G is a coordinated difference-bounded uncertainty measure for the table T . (b) We denote by R the function T ∗ R Hρm+1 . Let T ∈ Mρ (m). By D (Dβ respectively) we denote the set of nonempty subsets A of the set Δ(T α) (Δ(T αβ) respectively) having the following properties: |A| ≤ m + 1, Π (A, T ) = ∅, and A  Δ(T ( f i , δ)). One can show that Dβ ⊆ D, |D| = R(α) − R(α( f i , δ)), and |Dβ| = R(αβ) − R(αβ( f i , δ)). Therefore R(α) − R(α( f i , δ)) ≥ R(αβ) − R(αβ( f i , δ)). Hence the function R has the property of difference-boundedness. / Mρ C and δ¯ ∈ Δ(T α). Then, If T α ∈ Mρ C , then, evidently, R(α) = 0. Let T α ∈ ¯ there exists δ¯ j ∈ Δ(T α) such that j ∈ ¯ Denote A = {δ} ¯ ∪ / νT (δ). for any j ∈ νT (δ), ¯ Since T ∈ Mρ (m), we have |A| ≤ m + 1. Evidently, Π (A, T α) = {δ¯ j : j ∈ νT (δ)}. ∅. Hence R(α) ≥ 1. Thus, the function R has the property of coordination. Using Remark 4.1 we conclude that R is a coordinated difference-bounded uncertainty measure for the table T . (c) We now consider the function T ∗ Hρm+1 . Let Q ∈ Mρ , A be a nonempty subset of the set Δ(Q), Π (A, Q) = ∅, and Q  be a separable subtable of the table Q which is a boundary table. Let us show that the set A separates the subtable Q  if and only if A ⊆ Δ(Q  ).

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4 Upper Bounds on Complexity and Algorithms …

Let the set A separate the subtable Q  . Then, evidently, A ⊆ Δ(Q  ). Let A ⊆ Δ(Q  ). Denote I = {( f j , σ ) : ( f j , σ ) ∈ Ωρ (Q), A ⊆ Δ(Q( f j , σ ))} . Let I = ∅ and I = {( f i1 , σ1 ), . . . , ( f i p , σ p )}. Set α = ( f i1 , σ1 ) · · · ( f i p , σ p ). Let us show that Q  = Qα. Since Q  is a separable subtable of the table Q, the equality Q  = Qβ holds for some word β ∈ Ωρ (Q). From the relation A ⊆ Δ(Qβ) it follows that χ (β) ⊆ χ (α). Therefore Δ(Qα) ⊆ Δ(Qβ). Let us show that Δ(Qβ) ⊆ Δ(Qα). Assume the contrary. Then there exists a pair ( fi j , σ j ) ∈ χ (α) such that Q  ( f i j , σ j ) = / Mρ C which is impossible since Q  is a boundary table. Hence Q  and Q  ( f i j , σ j ) ∈  Q = Qα. In the case I = ∅, the proof of the equality Q  = Qα is similar to the proof for the case I = ∅. Thus, A separates the subtable Q  if and only if A ⊆ Δ(Q  ). From this statement it follows that, for any table Q ∈ Mρ , the following equality holds: Hρm+1 (Q) =



R Hρm+1 (Q  ) ,

(4.1)

Q  ∈gρ (Q)

where gρ (Q) is the set of separable subtables of the table Q which are boundary tables. Let T ∈ Mρ (m). We define the sets B and Bβ in the same way as in the part (a) of the proof. Then using (4.1) we obtain T ∗ Hρm+1 (α) − T ∗ Hρm+1 (α( f i , δ)) =



R Hρm+1 (T  )

T  ∈B



and T ∗ Hρm+1 (αβ) − T ∗ Hρm+1 (αβ( f i , δ)) =

R Hρm+1 (T  ) .

T  ∈Bβ

From these equalities and from the relation Bβ ⊆ B it follows that the function T ∗ Hρm+1 has the property of difference-boundedness. If T α ∈ Mρ C , then T ∗ Hρm+1 (α) = 0. Let T ∗ Hρm+1 (α) = 0. From this equality, from (4.1), and from the fact that, for any table T  ∈ Mρ (m), the function T  ∗ R Hρm+1 has the property of coordination it follows that gρ (T α) = ∅. Therefore T ∗ G Hρ (α) = 0. Since the function T ∗ G Hρ has the property of coordination, we obtain T α ∈ Mρ C . Therefore the function T ∗ Hρm+1 has the property of coordination. Using Remark 4.1 we conclude that T ∗ Hρm+1 is a coordinated differencebounded uncertainty measure for the table T .  Remark 4.2 Let m ∈ ω \ {0} and T ∈ Mρ (m). Then G Hρ (T ) ≤ Hρm+1 (T ) ≤ R Hρm+1 (T ) . The inequality Hρm+1 (T ) ≤ R Hρm+1 (T ) is obvious. The inequality

4.1 Difference-Bounded Uncertainty Measures for Decision Tables

41

G Hρ (T ) ≤ Hρm+1 (T ) follows from (4.1) and from the fact that, for any table T  ∈ Mρ (m), the function T  ∗ R Hρm+1 has the property of coordination. Remark 4.3 Let ρ be an enumerated signature and m ∈ ω \ {0}. One can show that there exist algorithms which compute values of functions Hρm+1 and R Hρm+1 on tables from Mρ (m), and which time complexities are bounded from above by some polynomials on the length of binary representations of input tables. Simple analysis of Proposition 4.1 proof shows that each boundary separable subtable of a table T ∈ Mρ (m) is separated by a nonempty subset A of the set Δ(T ) such that |A| ≤ m + 1 and Π (A, T ) = ∅. Using this fact one can show that there exists an algorithm which computes values of the function G Hρ on tables from Mρ (m), and which time complexity is bounded from above by some polynomial on the length of binary representations of input tables. Let T ∈ Mρ . We denote by P(Ωˆ ρ (T )) the set of nonempty finite subsets of the set Ωˆ ρ (T ). A nonempty finite subset D of the set P(Ωˆ ρ (T )) will be called a proper cover for the table T if it has the following properties:  ¯ ¯ (a) For  exists a word α ∈ d∈D d such that δ ∈ Δ(T α).  any δ ∈ Δ(T ), there (b) α∈d Π (T α) = ∅ or α∈d Δ(T α) = ∅ for any d ∈ D. (c) For any d1 , d2 ∈ D and α1 ∈ d1 , α2 ∈ d2 , if d1 = d2 , then the words α1 and α2 are incompatible. Let D be a proper cover for the table T and α ∈ Ωρ (T ). For an arbitrary d ∈ D, we denote by d[α] the set of words from d which are compatible with the word α. Denote η(D, α) = {(α1 , α2 ) : α1 ∈ d1 [α], α2 ∈ d2 [α], d1 , d2 ∈ D, d1 = d2 } . We now define a function γ D : Ωρ (T ) → ω. For an arbitrary α ∈ Ωρ (T ), let γ D (α) = |η(D, α)| .

Proposition 4.2 Let T ∈ Mρ and D be a proper cover for the table T . Then the function γ D is a difference-bounded uncertainty measure for the table T . Proof Let T ∈ Mρ , dim T = n, and μT (1) = f 1 , . . . , μT (n) = f n . We denote by γ the function γ D . Let a, b ∈ Ωρ (T ). One can show that η(D, ab) ⊆ η(D, a). Therefore, for any a, b ∈ Ωρ (T ), γ (a) ≥ γ (ab) . (4.2)

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Let α, β, ( f i , δ) ∈ Ωρ (T ) and β( f i , δ) ∈ Ωˆ ρ (T ). Let us show that γ (α) − γ (α( f i , δ)) ≥ γ (αβ) − γ (αβ( f i , δ)) .

(4.3)

/ Ωˆ ρ (T ). Since β( f i , δ) ∈ Ωˆ ρ (T ), we have αβ ∈ / Ωˆ ρ (T ) or α( f i , δ) ∈ / Let αβ( f i , δ) ∈ Ωˆ ρ (T ). Let αβ ∈ / Ωˆ ρ (T ). Then γ (αβ) = γ (αβ( f i , δ)) = 0. From (4.2) it follows / Ωˆ ρ (T ). Then γ (α( f i , that γ (α) ≥ γ (α( f i , δ)). Therefore (4.3) holds. Let α( f i , δ) ∈ δ)) = γ (αβ( f i , δ)) = 0. From (4.2) it follows that γ (α) ≥ γ (αβ). Therefore (4.3) holds. Let αβ( f i , δ) ∈ Ωˆ ρ (T ). We denote by Bα (Bαβ respectively) the set of pairs of words from η(D, α) (η(D, αβ) respectively) in each of which at least one word contains a letter ( f i , σ ), where σ ∈ E k and σ = δ. One can show that η(D, αβ) ⊆ η(D, α). Therefore Bαβ ⊆ Bα. Taking into account that αβ( f i , δ) ∈ Ωˆ ρ (T ) it is easy to show that γ (α) − γ (α( f i , δ)) = |Bα| and γ (αβ) − γ (αβ( f i , δ)) = |Bαβ|. The inequality (4.3) follows from these equalities and from the relation Bαβ ⊆ Bα. Thus, the function γ has the propertyof difference-boundedness. Let α ∈ Ωρ (T ). Denote D[α] = d∈D d[α]. Let us show that Δ(T α) ⊆



Δ(Tβ) .

(4.4)

β∈D[α]

If Δ(T ) = ∅, then (4.4), evidently, holds. Let Δ(T ) = ∅ and δ¯ = (δ1 , . . . , δn ) ∈ Δ(T α). Using  the property (a) of the proper cover D we conclude that there exists a word τ ∈ d∈D d such that δ¯ ∈ Δ(T τ ). It is clear that χ (α) ⊆ {( f 1 , δ1 ), . . . , , ( f n , δn )}. Therefore the words α and τ are com( f n , δn )} and χ (τ ) ⊆ {( f 1 , δ1 ), . . . patible. Hence τ ∈ D[α] and δ¯ ∈ β∈D[α] Δ(Tβ). Taking into account that δ¯ is an arbitrary tuple from Δ(T α), we conclude that (4.4) holds. Let γ (α) = d[α] for some d ∈ D. Using (4.4) we obtain  = 0. Then D[α] Δ(T α) ⊆ β∈d[α] Δ(Tβ) ⊆ β∈d Δ(Tβ). Using the property (b) of the proper cover D we conclude that T α ∈ Mρ C . Thus, the function γ has the property of partial coordination. Let α be a complete word for the table T . If α is an inconsistent word, then, evidently, γ (α) = 0. Let α be a consistent word. One can show that in this case any two words from the set D[α] are compatible. Using the property (c) of the proper cover D we obtain γ (α) = 0. Thus, the function γ has the property of completeness. Let α, β ∈ Ωρ (T ) and γ (β) = 0. Using the obvious relation η(D, αβ) ⊆ η(D, β) we obtain γ (αβ) = 0. Thus, the function γ has the property of monotonicity.  The following statement gives us examples of proper covers for decision tables. Proposition 4.3 Let T ∈ Mρ and Ω be a super-partition for the table T . Then the set D = {{α} : α ∈ Ω} is a proper cover for the table T . Proof Let dim T = n and μT (1) = f 1 , . . . , μT (n) = f n . It is clear that D is a nonempty finite subset of the set P(Ωˆ ρ (T )). Let δ¯ = (δ1 , . . . , δn ) ∈ Δ(T ). Then

4.1 Difference-Bounded Uncertainty Measures for Decision Tables

43

there exists a word α ∈ Ω such that χ (α) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}. Evidently, δ¯ ∈ Δ(T α). Hence the set D has the property (a) of a proper cover for the table T . Since T α ∈ Mρ C for any α ∈ Ω, the set D has the property (b) of a proper cover. Since, for any α, β ∈ Ω, if α = β, then the words α and β are incompatible, the set D has the property (c) of a proper cover. Thus, the set D is a proper cover for the table T . 

4.2 Process Uρ of Schema Construction We now describe a process Uρ which, for a table T ∈ Mρ , a difference-bounded uncertainty measure γ for the table T , and a complexity function ψ of the signature ρ = (F, k), constructs a schema Uρ (T, γ , ψ) that is a deterministic decision tree for the table T . Since the table T and the functions γ , ψ can be nonconstructive objects, the process Uρ in general case is not an algorithm but only a way for description of the schema Uρ (T, γ , ψ). Process Uρ . Step 1. Construct a tree containing nodes w1 , w2 and an edge which leaves w1 and enters w2 . Let T ∈ Mρ C . If Δ(T ) = ∅, then the node w2 is labeled with the number 0. If Δ(T ) = ∅, then the node w2 is labeled with the minimum number from the set Π (T ). We denote the obtained tree by Uρ (T, γ , ψ). The process Uρ is completed. Let T ∈ / Mρ C . Mark the node w2 by the word λ and proceed to the second step. Assume that t steps were made. We denote by D the tree obtained in the step t. Step (t + 1). If in the tree D there are no nodes labeled with words from Ωρ (T ), then denote by Uρ (T, γ , ψ) the tree D. The process Uρ is completed. Otherwise, choose in the tree D a node w which is labeled with a word from Ωρ (T ). Let the node w be labeled with the word α. If T α ∈ Mρ C , then, instead of the word α, we mark the node w by the minimum number from the set Π (T α), and proceed to the step (t + 2). Let T α ∈ / Mρ C . For any f i ∈ P(T ), let σi be the minimum number from E k such that γ (α( f i , σi )) = max{γ (α( f i , σ )) : σ ∈ E k } . Set I α = { f i : f i ∈ P(T ), γ (α) > γ (α( f i , σi ))} . For each f i ∈ I α, set  d fi = max ψ( f i ),

γ (α) γ (α) − γ (α( f i , σi ))

 .

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4 Upper Bounds on Complexity and Algorithms …

Let j be the minimum number from the set {1, . . . , dim T } such that μT ( j) ∈ I α and dμT ( j) = min{d fi : f i ∈ I α} . Let μT ( j) = f i0 . Instead of the word α, we mark the node w by the element f i0 . For each δ ∈ E k such that Δ(T α( f i0 , δ)) = ∅, we add to the tree D a node w(δ) and an edge leaving w and entering w(δ). We mark this edge by the number δ, and mark the node w(δ) by the word α( f i0 , δ). Proceed to the step (t + 2). Note that, for the depth h, in the capacity of the value d fi we can consider the value γ (α( f i , σi )). The constructed schema coincides with the schema Uρ (T, γ , h). Let us show that the set I α (see the description of the step (t + 1) of the process Uρ ) is a nonempty set. Lemma 4.1 Let ρ = (F, k), T ∈ Mρ , dim T = n, μT (1) = f 1 , . . . , μT (n) = f n , γ be a difference-bounded uncertainty measure for the table T , α ∈ Ωρ (T ), and T ∈ / Mρ C . Then there exists an element f i ∈ P(T ) such that γ (α) > max{γ (α( f i , δ)) : δ ∈ E k } . Proof For each f i ∈ P(T ), we denote by σi the minimum number from E k having the following property: γ (α( f i , σi )) = max{γ (α( f i , σ )) : σ ∈ E k } . Denote β0 = λ and, for j = 1, . . . , n, denote β j = ( f 1 , σ1 ) · · · ( f j , σ j ). Using the property of partial coordination of the function γ and the relation T ∈ / Mρ C we obtain γ (αβ0 ) > 0. Evidently, αβn is a complete for the table T word. Using the property of completeness of the function γ we obtain that γ (αβn ) = 0. Hence there exists j ∈ {0, . . . , n − 1} such that γ (αβ j ) > 0 and γ (αβ j+1 ) = 0. Using the property of difference-boundedness of the function γ and obvious relation β j+1 ∈ Ωˆ ρ (T )  we obtain γ (α) − γ (α( f j+1 , σ j+1 )) ≥ γ (αβ j ) − γ (αβ j+1 ) > 0. Using the description of the process Uρ , properties of the function γ , Lemma 4.1 and the definition of deterministic decision tree, it is not difficult to prove the following statement. Proposition 4.4 For any table T ∈ Mρ , any difference-bounded uncertainty measure γ for the table T , and any complexity function ψ of the signature ρ, the process Uρ ends after a finite number of steps. The schema Uρ (T, γ , ψ) constructed by the process Uρ is a deterministic decision tree for the table T . Corollary 4.1 For any table T ∈ Mρ , any difference-bounded uncertainty measure γ for the table T , and any complexity function ψ of the signature ρ, the inequality ψρd (T ) ≤ ψ(Uρ (T, γ , ψ)) holds.

4.3 Auxiliary Statements

45

4.3 Auxiliary Statements We now prove four lemmas. Let ρ = (F, k), T ∈ Mρ , dim T = n, μT (1) = f 1 , . . . , μT (n) = f n , γ be a difference-bounded uncertainty measure for the table T , α ∈ Ωρ (T ), δ¯ = (δ1 , . . . , δn ) ∈ E kn , and ψ be a complexity function of the signature ρ. Denote ¯ = min{ψ(β) : χ (β) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}, γ (αβ) = 0} M˜ ρ,ψ (γ , α, δ) and

¯ : δ¯ ∈ E kn } . M˜ ρ,ψ (γ , α) = max{ M˜ ρ,ψ (γ , α, δ)

Lemma 4.2 Let ρ = (F, k), T ∈ Mρ , dim T = n, α ∈ Ωρ (T ), δ¯ ∈ E kn , γ be a difference-bounded uncertainty measure for the table T , and ψ be a complexity function of the signature ρ. Then ¯ is definite. (a) The value of M˜ ρ,ψ (γ , α, δ) (b) If the function γ has the property of coordination, then M˜ ρ,ψ (γ , α) = Mρ,ψ (T α) . Proof Let μT (1) = f 1 , . . . , μT (n) = f n . Set β = ( f 1 , δ1 ) · · · ( f n , δn ). Evidently, χ (β) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )} , and αβ is a complete word for the table T . Since the function γ has the property of ¯ is definite. completeness, γ (αβ) = 0. Hence the value of M˜ ρ,ψ (γ , α, δ) Let the function γ have the property of coordination. One can show that in this ¯ = M˜ ρ,ψ (γ , α, δ). ¯ Taking into account that δ¯ is an arbitrary tuple case Mρ,ψ (T α, δ) n from E k and using Lemma 3.5 we conclude that M˜ ρ,ψ (γ , α) = Mρ,ψ (T α).  Lemma 4.3 Let ρ = (F, k), T ∈ Mρ , α ∈ Ωρ (T ), γ be a difference-bounded uncertainty measure for the table T , and ψ be a complexity function of the signature ρ. Then M˜ ρ,ψ (γ , λ) ≥ M˜ ρ,ψ (γ , α). Proof Let dim T = n, μT (1) = f 1 , . . . , μT (n) = f n , and δ¯ = (δ1 , . . . , δn ) ∈ E kn . From Lemma 4.2 it follows that there exists a word β ∈ Ωρ (T ) such that χ (β) ⊆ ¯ Using the property {( f 1 , δ1 ), . . . , ( f n , δn )}, γ (β) = 0 and ψ(β) = M˜ ρ,ψ (γ , λ, δ). ¯ ≤ of monotonicity of the function γ we obtain γ (αβ) = 0. Hence M˜ ρ,ψ (γ , α, δ) n ˜ ¯ Taking into account that δ¯ is an arbitrary tuple from E k we conclude Mρ,ψ (γ , λ, δ).  that M˜ ρ,ψ (γ , α) ≤ M˜ ρ,ψ (γ , λ). Lemma 4.4 Let ρ = (F, k), T ∈ Mρ \ Mρ C , γ be a difference-bounded uncertainty measure for the table T , ψ be a complexity function of the signature ρ having

46

4 Upper Bounds on Complexity and Algorithms …

the properties Λ2 and Λ3, and ξ = w0 , d0 , . . . , wm , dm , wm+1 be an arbitrary complete path in the schema Uρ (T, γ , ψ). Let, for j = 1, . . . , m, the node w j be labeled with an element f t j , and the edge d j be labeled with a number δ j . Let α0 = λ and α j = ( f t1 δ1 ) · · · ( f t j δ j ) for j = 1, . . . , m. Then, for j = 0, . . . , m − 1, ψ( f t j+1 ) ≤ M˜ ρ,ψ (γ , α j ) and γ (α j+1 ) ≤ γ (α j )

M˜ ρ,ψ (γ , α j ) − 1 . M˜ ρ,ψ (γ , α j )

Proof For j = 0, . . . , m, denote M j = M˜ ρ,ψ (γ , α j ). Let us fix a number j ∈ {0, . . . , m − 1} . Let dim T = n and μT (1) = f 1 , . . . , μT (n) = f n . For each f i ∈ P(T ), we denote by σi the minimum number from E kn having the following property: γ (α j ( f i , σi )) = max{γ (α j ( f i , σ )) : σ ∈ E k } . From Lemma 4.2 it follows that there exists a word β ∈ Ωρ (T ) such that χ (β) ⊆ {( f 1 , σ1 ), . . . , ( f n , σn )}, γ (α j β) = 0 and ψ(β) = M˜ ρ,ψ (γ , α j , (σ1 , . . . , σn )). Let β = ( fl1 , σl1 ) · · · ( flr , σlr ). From the description of the schema Uρ (T, γ , ψ) it fol/ Mρ C . Using the property of partial coordination of the function lows that T α j ∈ γ we obtain γ (α j ) > 0. From this inequality and from the equality γ (α j β) = 0 it follows that 1≤r . (4.5) Using the property Λ3 of the function ψ we obtain r ≤ ψ(β) .

(4.6)

From the choice of the word β and from the definition of the value M j it follows that ψ(β) ≤ M j .

(4.7)

Since γ (α j β) = 0, the following equality holds: γ (α j ) − (γ (α j ) − γ (α j ( fl1 , σl1 ))) −(γ (α j ( fl1 , σl1 )) − γ (α j ( fl1 , σl1 )( fl2 , σl2 ))) − · · · −(γ (α j ( fl1 , σl1 ) · · · ( flr −1 , σlr −1 )) − γ (α j β)) = 0 .

(4.8)

4.3 Auxiliary Statements

47

Choose q ∈ {l1 , . . . , lr } such that γ (α j ( f q , σq )) = min{γ (α j ( f p , σ p )) : p ∈ {l1 , . . . , lr }} . It is clear that ( fl1 , σl1 ) · · · ( fls , σls ) ∈ Ωˆ ρ (T ) for s = 2, . . . , r . Using the property of difference-boundedness of the function γ we obtain γ (α j ( fl1 , σl1 ) · · · ( fls−1 , σls−1 )) − γ (α j ( fl1 , σl1 ) · · · ( fls , σls )) ≤ γ (α j ) − γ (α j ( fls , σls )) for s = 2, . . . , r . From the choice of q it follows that γ (α j ) − γ (α j ( fls , σls )) ≤ γ (α j ) − γ (α j ( f q , σq )) for s = 1, . . . , r . Using these inequalities and (4.8) we obtain γ (α j ) − r (γ (α j ) − γ (α j ( f q , σq ))) ≤ 0 . From this inequality and from (4.5) it follows that γ (α j ( f q , σq )) ≤ γ (α j )

r −1 . r

From this inequality, from (4.6) and from (4.7) it follows that γ (α j ( f q , σq )) ≤ γ (α j )

Mj − 1 . Mj

(4.9)

Using the relation q ∈ {l1 , . . . , lr } and the property Λ2 of the function ψ we conclude that ψ( f q ) ≤ ψ(β). From this inequality and from (4.7) it follows that ψ( f q ) ≤ M j .

(4.10)

By the description of the schema Uρ (T, γ , ψ), the element f t j+1 is defined in the following way. Let I α j = { f i : f i ∈ P(T ), γ (α j ) > γ (α j ( f i , σi ))} . For each f i ∈ I α j , set  d fi = max ψ( f i ),

γ (α j ) γ (α j ) − γ (α j ( f i , σi ))

 .

Let i 0 be the minimum number from {1, . . . , n} such that f i0 ∈ I α j and d fi0 = min{d fi : f i ∈ I α j }. Then f t j+1 = f i0 .

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4 Upper Bounds on Complexity and Algorithms …

From (4.5)–(4.7), (4.9) and the inequality γ (α j ) > 0 it follows that f q ∈ I α j . One can show that d fi > 0 for any f i ∈ I α j , and d fi is the minimum number . From among all numbers d such that ψ( f i ) ≤ d and γ (α j ( f i , σi )) ≤ γ (α j ) d−1 d here, from (4.9), and from (4.10) it follows that d fq ≤ M j . Therefore d ft j+1 ≤ M j . From this inequality and from the definition of the value d ft j+1 it follows that M −1

ψ( f t j+1 ) ≤ M j and γ (α j ( f t j+1 , σt j+1 )) ≤ γ (α j ) Mj j . The statement of the lemma follows from these inequalities and from the choice of σt j+1 according to which  γ (α j+1 ) ≤ γ (α j ( f t j+1 , σt j+1 )). Lemma 4.5 Let ρ = (F, k), T ∈ Mρ and r ∈ {0, 1}. Then (a) Mρ,h (T ) = r if and only if h dρ (T ) = r . (b) If Mρ,h (T ) = 1, then there exists an element f i ∈ P(T ) such that T ( f i , δ) ∈ Mρ C for any δ ∈ E k . Proof Let Mρ,h (T ) = 0. Then, evidently, T ∈ Mρ C . One can show that in this case h dρ (T ) = 0. Let h dρ (T ) = 0. Using Theorem 3.1 we obtain Mρ,h (T ) = 0. Let Mρ,h (T ) = 1. We now show that there exists an element f i ∈ P(T ) such that T ( f i , δ) ∈ Mρ C for any δ ∈ E k . Assume the contrary. Let dim T = n, μT (1) = / f 1 , . . . , μT (n) = f n and, for any f i ∈ P(T ), there exists δi ∈ E k such that T ( f i , δi ) ∈ ¯ ≥ 2 which is imposMρ C . Denote δ¯ = (δ1 , . . . , δn ). One can show that Mρ,h (T, δ) ¯ ≤ Mρ,h (T ) = 1. Hence there exists f i ∈ P(T ) such that sible since Mρ,h (T, δ) T ( f i , δ) ∈ Mρ C for any δ ∈ E k . Using this fact one can show that there exists a deterministic decision tree Γ for the table T such that h(Γ ) = 1. Hence h dρ (T ) ≤ 1. Using the equality Mρ,h (T ) = 1 and Theorem 3.1 we obtain h dρ (T ) = 1. Let h dρ (T ) = 1. Using Theorem 3.1 we obtain Mρ,h (T ) ≤ 1. Assume that Mρ,h (T ) = 0. Then, by  proved above, h dρ (T ) = 0 which is impossible. Hence Mρ,h (T ) = 1.

4.4 Upper Bounds on ψ(Uρ (T, γ , ψ)) In this section, upper bounds on ψ(Uρ (T, γ , ψ)) are considered. From Corollary 4.1 it follows that these bounds are also upper bounds on ψρd (T ). The case, when γ is a coordinated difference-bounded uncertainty measure and ψ = h, is studied more thoroughly. Theorem 4.1 Let T ∈ Mρ , γ be a difference-bounded uncertainty measure for the table T , and ψ be a complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3. Then  ψ(Uρ (T, γ , ψ)) ≤

ψ(λ), if T ∈ Mρ C , ( M˜ ρ,ψ (γ , λ))2 ln γ (λ) + M˜ ρ,ψ (γ , λ), if T ∈ / Mρ C ,

and ψρd (T ) ≤ ψ(Uρ (T, γ , ψ)) .

4.4 Upper Bounds on ψ(Uρ (T, γ , ψ))

49

Proof Let T ∈ Mρ C . One can show that in this case ψ(Uρ (T, γ , ψ)) = ψ(λ). Let T ∈ / Mρ C , dim T = n, and μT (1) = f 1 , . . . , μT (n) = f n . Let us consider an arbitrary complete path ξ = w0 , d0 , . . . , wm , dm , wm+1 in the schema Uρ (T, γ , ψ). Using Proposition 4.4 we conclude that the schema Uρ (T, γ , ψ) is a deterministic decision tree for the table T . Since T ∈ / Mρ C , we have m ≥ 1. Let, for j = 1, . . . , m, the node w j be labeled with an element f t j , and the edge d j be labeled with a number δ j . Set α0 = λ and, for j = 1, . . . , m, set α j = ( f t1 , δ1 ) · · · ( f t j , δ j ). We now show that (4.11) m ≤ M˜ ρ,ψ (γ , λ) ln γ (λ) + 1 . From the condition T ∈ / Mρ C and from the property of partial coordination of the function γ it follows that γ (λ) ≥ 1. Evidently, M˜ ρ,ψ (γ , λ) ≥ 0. Therefore if m = 1, then (4.11) holds. Let m ≥ 2. From Lemma 4.4 it follows that γ (αm−1 ) ≤ γ (λ)

m−2  j=0

M˜ ρ,ψ (γ , α j ) − 1 . M˜ ρ,ψ (γ , α j )

(4.12)

By the description of the schema Uρ (T, γ , ψ), we have T αm−1 ∈ / Mρ C . From this relation and from the property of partial coordination of the function γ it follows that γ (αm−1 ) ≥ 1. Using this inequality and (4.12) we obtain m−2  j=0

M˜ ρ,ψ (γ , α j )

M˜ ρ,ψ (γ , α j ) − 1

≤ γ (λ) .

(4.13)

From Lemma 4.3 it follows that M˜ ρ,ψ (γ , α j ) ≤ M˜ ρ,ψ (γ , λ)

(4.14)

for j = 0, . . . , m. By (4.13) and (4.14), (m − 1) ln 1 +

1 M˜ ρ,ψ (γ , λ) − 1

≤ ln γ (λ) .

(4.15)

Let us show that M˜ ρ,ψ (γ , λ) ≥ 2. Using (4.5)–(4.7) we obtain M˜ ρ,ψ (γ , λ) ≥ 1. From the inequality m ≥ 2 and from Lemma 4.4 it follows that γ (α1 ) ≤ γ (λ)

M˜ ρ,ψ (γ , λ) − 1 . M˜ ρ,ψ (γ , λ)

Since m ≥ 2, we have T α1 ∈ / Mρ C . Using the property of partial coordination of the function γ we obtain γ (α1 ) ≥ 1. Hence M˜ ρ,ψ (γ , λ) = 1. Therefore M˜ ρ,ψ (γ , λ) ≥ 2. Taking into account this inequality and also known inequality

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4 Upper Bounds on Complexity and Algorithms …

 1 1 > ln 1 + n n+1 which is true for any natural n we obtain ln 1 +

1 M˜ ρ,ψ (γ , λ) − 1

>

1 . ˜ Mρ,ψ (γ , λ)

From this inequality and from (4.15) it follows (4.11). From Lemma 4.4, from (4.11), (4.14), and from the property Λ1 of the function ψ it follows that ψ(π(ξ )) = ψ( f t1 . . . f tm ) ≤ ( M˜ ρ,ψ (γ , λ))2 ln γ (λ) + M˜ ρ,ψ (γ , λ) . Taking into account that ξ is an arbitrary complete path in the schema Uρ (T, γ , ψ) we obtain ψ(Uρ (T, γ , ψ)) ≤ ( M˜ ρ,ψ (γ , λ))2 ln γ (λ) + M˜ ρ,ψ (γ , λ) . By Corollary 4.1, ψρd (T ) ≤ ψ(Uρ (T, γ , ψ)).



Corollary 4.2 Let T ∈ Mρ , γ be a coordinated difference-bounded uncertainty measure for the table T , and ψ be a complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3. Then  ψρd (T ) ≤ ψ(Uρ (T, γ , ψ)) ≤

ψ(λ), if T ∈ Mρ C , / Mρ C . (Mρ,ψ (T ))2 ln γ (λ) + Mρ,ψ (T ), if T ∈

Proof The considered statement follows from Theorem 4.1 and Lemma 4.2.



Corollary 4.3 Let T ∈ Mρ and γ be a difference-bounded uncertainty measure for the table T . Then  0, if T ∈ Mρ C , d h ρ (T ) ≤ h(Uρ (T, γ , h)) ≤ ˜ Mρ,h (γ , λ) ln γ (λ) + 1, if T ∈ / Mρ C . Proof Evidently, the function h has the properties Λ1, Λ2, and Λ3. Let T ∈ Mρ C . From Theorem 4.1 it follows that h(Uρ (T, γ , h)) ≤ h(λ) = 0. Let T ∈ / Mρ C and ξ = w0 , d0 , . . . , wm , dm , wm+1 be an arbitrary complete path in the schema Uρ (T, γ , h). By (4.11), h(π(ξ )) = m ≤ M˜ ρ,ψ (γ , λ) ln γ (λ) + 1. Taking into account that ξ is an arbitrary complete path in the schema Uρ (T, γ , h) we obtain h(Uρ (T, γ , h)) ≤ M˜ ρ,ψ (γ , λ) ln γ (λ) + 1. The inequality h dρ (T ) ≤ h(Uρ (T, γ , h)) follows from Theorem 4.1.  Theorem 4.2 Let ρ = (F, k), T ∈ Mρ and γ be a coordinated difference-bounded uncertainty measure for the table T . Then

4.4 Upper Bounds on ψ(Uρ (T, γ , ψ))

 h dρ (T ) ≤ h(Uρ (T, γ , h)) ≤

51

if Mρ,h (T ) ≤ 1 , Mρ,h (T ), Mρ,h (T )(ln γ (λ) − ln Mρ,h (T ) + 1), if Mρ,h (T ) ≥ 2 .

Proof Let Mρ,h (T ) = 0. Then, evidently, T ∈ Mρ C . Using Corollary 4.3 we obtain h(Uρ (T, γ , h)) ≤ 0. Let Mρ,h (T ) = 1. Using Lemma 4.5 we conclude that there exists an element f i ∈ P(T ) such that T ( f i , δ) ∈ Mρ C for any δ ∈ E k . Since the function γ has the property of coordination, 

γ (λ) max h( f i ), γ (λ) − max{γ (( f i , δ)) : δ ∈ E k }

 =1.

Let w be the node of the schema Uρ (T, γ , h) which is connected with the root by an edge leaving the root and entering w. From the last equality and from the description of the process Uρ it follows that the node w is labeled with an element f j ∈ P(T ) such that  max h( f j ),

γ (λ) γ (λ) − max{γ (( f j , δ)) : δ ∈ E k }

 =1.

Taking into account that the function γ has the property of coordination we obtain T ( f j , δ) ∈ Mρ C for any δ ∈ E k . From here and from the description of the process Uρ it follows that, with the exception of the root and the node w, all nodes of the schema Uρ (T, γ , h) are terminal nodes. Therefore h(Uρ (T, γ , h)) = 1 . Let Mρ,h (T ) ≥ 2. Let ξ = w0 , d0 , . . . , wm , dm , wm+1 be the longest complete path in the schema Uρ (T, γ , h). Evidently, h(Uρ (T, γ , h)) = m. Since the schema Uρ (T, γ , h) is a deterministic decision tree for the table T , from the inequality Mρ,h (T ) ≥ 2 and from Theorem 3.1 it follows that m ≥ 2. Let, for j = 1, . . . , m, the node w j be labeled with an element f t j and the edge d j be labeled with a number δ j . Set α0 = λ and, for j = 1, . . . , m, set α j = ( f t1 , δ1 ) · · · ( f t j , δ j ). For j = 0, . . . , m, set M j = M˜ ρ,h (γ , α j ). Let us show that for, i = 0, . . . , m, Mm−i ≤ i .

(4.16)

We denote by Γi the subtree of the tree Uρ (T, γ , h) which root is the node wm−i+1 . Add to Γi a new node and an edge leaving this node and entering the node wm−i+1 . We denote the obtained schema by Γi . One can show that Γi is a deterministic decision tree for the table T αm−i . Taking into account that ξ is the longest complete path of the schema Uρ (T, γ , h) we conclude that h(Γi ) = i. Since Γi is a deterministic decision tree for the table T αm−i , we have h dρ (T αm−i ) ≤ h(Γi ) = i. By Theorem 3.1, Mρ,h (T αm−i ) ≤ h dρ (T αm−i ). Therefore Mρ,h (T αm−i ) ≤ i. The function γ has the

52

4 Upper Bounds on Complexity and Algorithms …

property of coordination. Using this fact and Lemma 4.2 we conclude that Mm−i = Mρ,h (T αm−i ). Therefore Mm−i ≤ i, and (4.16) holds. Evidently, the function h has the properties Λ2 and Λ3. From the inequality m ≥ 2 and from Lemma 4.4 it follows that γ (αm−1 ) ≤ γ (λ)

m−2 

Mj − 1 . Mj

j=0

(4.17)

Using the description of the schema Uρ (T, γ , h) we obtain that T αm−1 ∈ / Mρ C . From here and from the property of partial coordination of the function γ it follows that γ (αm−1 ) ≥ 1. Using this inequality and (4.17) we obtain m−2  j=0

Mj ≤ γ (λ) . Mj − 1

(4.18)

M j ≤ M0 .

(4.19)

By Lemma 4.3, for j = 1, . . . , m,

From (4.16), (4.18), and (4.19) it follows that 

M0 M0 − 1

m−M0 M 0 −2 j=0

M0 − j ≤ γ (λ) . M0 − j − 1

Taking natural logarithm of both sides of this equation we obtain  (m − M0 ) ln 1 + Using known inequality

1 M0 − 1

≤ ln γ (λ) − ln M0 .

(4.20)

 1 1 > , ln 1 + n n+1

which holds for any natural n, and the relations M0 = Mρ,h (T ) ≥ 2 we obtain  ln 1 +

1 M0 − 1

>

1 . M0

From this inequality and from (4.20) it follows that m < M0 (ln γ (λ) − ln M0 + 1). Taking into account that m = h(Uρ (T, γ , h)) and M0 = Mρ,h (T ) we obtain h(Uρ (T, γ , h)) < Mρ,h (T )(ln γ (λ) − ln Mρ,h (T ) + 1). From Corollary 4.3 it fol lows that h dρ (T ) ≤ h(Uρ (T, γ , h)).

4.5 Corollaries

53

4.5 Corollaries In this section, we consider upper and lower bounds on the value ψρd (T ) depending on the value Mˆ ρ,ψ (T ). For tables from Mρ F , we discuss upper and lower bounds on the value ψρd (T ) depending on the values Mρ,ψ (T ) and Θρ,ψ (T ). These bounds are corollaries of results from Chaps. 3 and 4. Theorem 4.3 Let ρ = (F, k), T ∈ Mρ and ψ be a complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3. Then Mˆ ρ,ψ (T ) ≤ ψρd (T ) ≤



ψ(λ), if T ∈ Mρ C , 2( Mˆ ρ,ψ (T ))3 ln k + Mˆ ρ,ψ (T ), if T ∈ / Mρ C .

Proof Let dim T = n and μT (1) = f 1 , . . . , μT (n) = f n . From Lemma 3.4 it follows that there exists a super-partition Ω for the table T such that ψ(Ω) = Mˆ ρ,ψ (T ). Set D = {{α} : α ∈ Ω}. Using Propositions 4.2 and 4.3 we conclude that the function γ D is a difference-bounded uncertainty measure for the table T . Let us show that M˜ ρ,ψ (γ D , λ) ≤ Mˆ ρ,ψ (T ) .

(4.21)

Let δ¯ = (δ1 , . . . , δn ) ∈ E kn . Then there exists a word β ∈ Ω such that χ (β) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}. Since any two different words from Ω are incompati¯ ≤ ble, γ D (β) = 0. Evidently, ψ(β) ≤ ψ(Ω) = Mˆ ρ,ψ (T ). Hence M˜ ρ,ψ (γ D , λ, δ) n Mˆ ρ,ψ (T ). Taking into account that δ¯ is an arbitrary tuple from E k we conclude that ˆ (4.21) holds. From Lemma 3.6 it follows that |Ω| ≤ k Mρ,ψ (T ) . Therefore γ D (λ) ≤ 2 Mˆ ρ,ψ (T ) k . From this inequality, from (4.21), and from Theorem 4.1 it follows that  ψρd (T )

≤

ψ(λ), if T ∈ Mρ C , 2( Mˆ ρ,ψ (T ))3 ln k + Mˆ ρ,ψ (T ), if T ∈ / Mρ C .

The inequality Mˆ ρ,ψ (T ) ≤ ψρd (T ) follows from Theorem 3.1.



Corollary 4.4 Let ρ = (F, k) and T ∈ Mρ . Then Mˆ ρ,h (T ) ≤ h dρ (T ) ≤



0, if T ∈ Mρ C , 2( Mˆ ρ,h (T ))2 ln k + 1, if T ∈ / Mρ C .

Proof Evidently, h is a complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3. Let T ∈ Mρ C . From Theorem 4.3 it follows that h dρ (T ) ≤ h(λ) = 0. Let T ∈ / Mρ C . From Lemma 3.4 it follows that there exists a superpartition Ω for the table T such that h(Ω) = Mˆ ρ,ψ (T ). Set D = {{α} : α ∈ Ω}. By (4.21), M˜ ρ,h (γ D , λ) ≤ Mˆ ρ,h (T ). Using Lemma 3.6 we obtain γ D (λ) ≤ (|Ω|)2 ≤ ˆ k 2 Mρ,h (T ) . From these inequalities and from Corollary 4.3 it follows that h dρ (T ) ≤

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4 Upper Bounds on Complexity and Algorithms …

2( Mˆ ρ,ψ (T ))2 ln k + 1. Theorem 3.1.

The

inequality

Mˆ ρ,h (T ) ≤ h dρ (T )

follows

from 

Proposition 4.5 Let ρ = (F, k), T ∈ Mρ F , and ψ be a complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3. Then  ≤

max{Mρ,ψ (T ), logk (Θρ,h (T ) + 1)} ≤ ψρd (T ) ψ(λ), if T ∈ Mρ C , / Mρ C . 2(Mρ,ψ (T ))3 (ln Θρ,h (T ) + ln k) + Mρ,ψ (T ), if T ∈

Proof The lower bound on the value ψρd (T ) follows from Theorems 3.1 and 3.3. Let T ∈ Mρ C . Then from Theorem 4.1 it follows that ψρd (T ) ≤ ψ(λ). Let T ∈ / Mρ C , dim T = n, and μT (1) = f 1 , . . . , μT (n) = f n . For δ¯ = (δ1 , . . . , δn ) ∈ Δ(T ), set ¯ . ¯ = {β : χ (β) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}, Tβ ∈ Mρ C , ψ(β) ≤ Mρ,ψ (T, δ)} B(δ) ¯ = ∅. For each δ¯ ∈ Δ(T ), choose in the set From Lemma 3.5 it follows that B(δ) ¯ a word with the minimum length. We denote this word by α(δ). ¯ For each B(δ)  ¯ ¯ ¯ ¯ ν ( δ), set d = {α( δ) : δ ∈ Δ(T ), ν ( δ) = {i}}. Denote D= i ∈ νT (T ) = δ∈Δ(T ¯ i T ) T {di : i ∈ νT (T )}. Let us show that the set D is a proper cover for the table T . Evidently, ¯ D is a nonempty finite subset of the set P(Ωˆ ρ (T )). It is clear that δ¯ ∈ Δ(T α(δ)) for any δ¯ ∈ Δ(T ). Therefore the set D has the property (a) of proper cover for the ¯ table T . Let di ∈ D and α ∈ di . Evidently, there exists δ¯ ∈ Δ(T ) such that α = α(δ) ¯ = {i}. Taking into account that T ∈ Mρ F , T α ∈ Mρ C and δ¯ ∈ Δ(T α), and νT (δ)  we obtain Π (T α) = {i}. Hence α∈di Π (T α) = {i}. Therefore the set D has the property (b) of proper cover for the table T . Let i, j ∈ νT (T ), i = j, α ∈ di , and β ∈ d j . Assume that the words α and β are compatible. Taking into account that ¯ = {i} T ∈ Mρ F , we obtain Δ(T α) ∩ Δ(Tβ) = ∅ which is impossible since νT (δ) for any δ¯ ∈ Δ(T α), and νT (σ¯ ) = { j} for any σ¯ ∈ Δ(Tβ). Hence the set D has the property (c) of proper cover for the table T . Thus, the set D is a proper cover for the table T . Using Proposition 4.2 we conclude that the function γ D is a differencebounded uncertainty measure for the table T . Let us show that the function γ D has the property of coordination. Let α ∈ Ωρ (T ). If γ D (α) = 0, then using the property of partial coordination of the function γ D we obtain T α ∈ Mρ C . Let T α ∈ Mρ C . If α ∈ / Ωˆ ρ (T ), then, evidently, di [α] = ∅ for any di ∈ D (di [α] is the set of words from di which are compatible with the word α). Hence γ D (α) = 0. Let α ∈ Ωˆ ρ (T ). Since T ∈ Mρ F , we have Δ(T α) = ∅. Hence ¯ = {i} for any δ¯ ∈ Δ(T α). Let j ∈ νT (T ) there exists i ∈ νT (T ) such that νT (δ) and j = i. Let us show that d j [α] = ∅. Assume the contrary. Let β ∈ d j [α]. Since T ∈ Mρ F , we have Δ(T α) ∩ Δ(Tβ) = ∅ which is, evidently, impossible. Hence γ D (α) = 0. Thus, γ D is a coordinated difference-bounded uncertainty measure for the table T .

4.5 Corollaries

55

Let f i ∈ P(T ) and there exist two tuples δ¯ and σ¯ from Δ(T ) which are different ¯ = νT (σ¯ ). Then the element f i will only in the ith element, and for which νT (δ) be called essential for the table T . Let B be an arbitrary test for the table T . Let / Mρ C , we have B = ∅. Let us show that f i ∈ B. Assume the contrary. Since T ∈ / B, the tuples δ¯ and σ¯ belong to B = { f i1 , . . . , f im } and δ¯ = (δ1 , . . . , δn ). Since f i ∈ / Mρ C which is the set Δ(T ( f i1 , δi1 ) · · · ( f im , δim )). Hence T ( f i1 , δi1 ) · · · ( f im , δim ) ∈ impossible since B is a test for the table T . Therefore f i ∈ B. We denote by P0 (T ) the set of all elements from P(T ) which are essential for the table T . By proved above, |P0 (T )| ≤ Θρ,h (T ) . (4.22) ¯ ( f j , δ j ) ∈ χ (α) and f j ∈ / P0 (T ). We Let δ¯ = (δ1 , . . . , δn ) ∈ Δ(T ), α ∈ B(δ), denote by β the word obtained from α by removal of all occurrences of the let¯ Evidently, ter ( f j , δ j ). Let us show that β ∈ B(δ). χ (β) ⊆ χ (α) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )} , ¯ Let Since the function ψ has the property Λ2, we have ψ(β) ≤ ψ(α) ≤ Mρ,ψ (T, δ). us show that Tβ ∈ Mρ C . Assume the contrary. Then there exists a tuple σ¯ ∈ Δ(Tβ) ¯ We denote by τ¯ the tuple obtained from the tuple σ¯ by subsuch that νT (σ¯ ) = νT (δ). stitution of δ j instead of the jth element. Since T ∈ Mρ F , we have τ¯ ∈ Δ(T α). ¯ Hence f j ∈ P0 (T ) which is impossible. So Tβ ∈ Mρ C . Therefore νT (τ¯ ) = νT (δ). ¯ ¯ Since α(δ) ¯ is a word with the minimum Thus, β ∈ B(δ). Consider the word α(δ). ¯ ¯ length from B(δ), we have χ (α(δ)) ⊆ {( f i , δi ) : f i ∈ P0 (T )}. Since the function ψ

¯ ¯ ¯ has  the property Λ3, we have α(δ) ≤ ψ(α(δ)) ≤ Mρ,ψ (T, δ) ≤∗ Mρ,ψ (T ). Set A = Then A ⊆ {α : α ∈ {( f i , δ) : f i ∈ P0 (T ), δ ∈ E k } , |α| ≤ Mρ,ψ (T )}. di ∈D di . Using (4.22) we obtain |A| ≤ (|P0 (T )| k) Mρ,ψ (T ) ≤ (Θρ,h (T )k) Mρ,ψ (T ) . Evidently, γ D (λ) ≤ |A|2 . Therefore γ D (λ) ≤ (Θρ,h (T )k)2Mρ,ψ (T ) . Using Corollary 4.2 and the fact, that γ D is a coordinated difference-bounded uncertainty measure for the table  T , we conclude that ψρd (T ) ≤ 2(Mρ,ψ (T ))3 (ln Θρ,h (T ) + ln k) + Mρ,ψ (T ). Corollary 4.5 Let T ∈ Mρ F , where ρ = (F, k). Then  ≤

max{Mρ,h (T ), logk (Θρ,h (T ) + 1)} ≤ h dρ (T ) if Mρ,h (T ) ≤ 1 , Mρ,h (T ), Mρ,h (T )(2Mρ,h (T )(ln Θρ,h (T ) + ln k) − ln Mρ,h (T ) + 1), if Mρ,h (T ) ≥ 2 .

Proof Evidently, h is a complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3. The lower bound on the value h dρ (T ) follows immediately from Proposition 4.5. The upper bound on the value h dρ (T ) in the case Mρ,h (T ) ≤ 1 follows from Theorem 4.2. / Mρ C . Let us define the function γ D Let Mρ,h (T ) ≥ 2. Then, evidently, T ∈ in the same way as in the proof of Proposition 4.5. In the capacity of the function ψ, we take the function h. Then, as it was shown in the proof of Proposi-

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tion 4.5, γ D is a coordinated difference-bounded uncertainty measure for the table T such that γ D (λ) ≤ (Θρ,h (T )k)2Mρ,h (T ) . Using Theorem 4.2 we obtain h dρ (T ) ≤  Mρ,h (T )(2Mρ,h (T )(ln Θρ,h (T ) + ln k) − ln Mρ,h (T ) + 1).

4.6 Algorithm Uρ,ϕ,ψ Let ρ = (F, k) be an enumerated signature, F = { f [i]2 : i ∈ ω}, ψ be a computable complexity function of the signature ρ, ϕ : Mρ → ω be a computable function, and Mρ,ϕ be the set of tables T from Mρ such that the function T ∗ ϕ is a differencebounded uncertainty measure of the signature ρ for the table T . Let Mρ,ϕ = ∅. We now describe an algorithm Uρ,ϕ,ψ which, for a given table T ∈ Mρ,ϕ , constructs a schema Uρ,ϕ,ψ (T ) that is a deterministic decision tree for the table T . Algorithm Uρ,ϕ,ψ . For the table T , the algorithm Uρ,ϕ,ψ makes the same steps as the process Uρ , when it constructs the schema Uρ (T, T ∗ ϕ, ψ). Using Proposition 4.4 we conclude that the algorithm Uρ,ϕ,ψ constructs the schema Uρ,ϕ,ψ (T ) = Uρ (T, T ∗ ϕ, ψ) which is a deterministic decision tree for the table T , and makes finite number of steps. Evidently, ψ(Uρ,ϕ,ψ (T )) = ψ(Uρ (T, T ∗ ϕ, ψ)) for any table T ∈ Mρ,ϕ . Therefore if the complexity function ψ has the properties Λ1, Λ2, and Λ3, then, in the capacity of bounds on ψ(Uρ,ϕ,ψ (T )), we can use bounds on the value ψ(Uρ (T, T ∗ ϕ, ψ)) from Sect. 4.4. Let us estimate from the above the number of steps which algorithm Uρ,ϕ,ψ makes when it constructs the schema Uρ,ϕ,ψ (T ). Let Γ be a finite directed tree with the root. We denote by L(Γ ) the number of nodes in Γ , and by L 0 (Γ ) we denote the number of terminal nodes in the tree Γ . Lemma 4.6 Let Γ be a finite directed tree with the root such that only one edge leaves the root of Γ , and at least two edges leave each node which is not neither the root nor a terminal node. Then L(Γ ) ≤ 2L 0 (Γ ). Proof We prove the statement of the lemma by induction on L(Γ ). Evidently, if L(Γ ) = 2, then L(Γ ) ≤ 2L 0 (Γ ). One can show that there is no tree Γ such that L(Γ ) = 3 and Γ satisfies the conditions of the lemma. Let for some n ∈ ω, n ≥ 3, for any tree Γ , which satisfies the conditions of the lemma and for which L(Γ ) ≤ n, the considered inequality hold. Let Γ be a tree which satisfies the conditions of the lemma and for which L(Γ ) = n + 1. Let us show that L(Γ ) ≤ 2L 0 (Γ ). One can prove that there exists a node w of the tree Γ which is not neither the root nor a terminal node and for which each edge leaving w enters a terminal node. We denote by Γ  the tree obtained from Γ by removal of all edges leaving w and all nodes which these edges enter. Assume that the node w had m leaving edges. It is clear that m ≥ 2. One can show that L(Γ  ) = L(Γ ) − m,

4.6 Algorithm Uρ,ϕ,ψ

57

L 0 (Γ  ) = L 0 (Γ ) − m + 1 and the schema Γ  satisfies the conditions of the lemma. By the inductive hypothesis, L(Γ  ) ≤ 2L 0 (Γ  ). Therefore L(Γ ) − m ≤ 2(L 0 (Γ ) −  m + 1) = 2L 0 (Γ ) − 2m + 2. Hence L(Γ ) ≤ 2L 0 (Γ ) − m + 2 ≤ 2L 0 (Γ ). Theorem 4.4 Let ρ = (F, k) be an enumerated signature, ψ be a computable complexity function of the signature ρ, ϕ : Mρ → ω be a computable function, and T ∈ Mρ,ϕ . Then the algorithm Uρ,ϕ,ψ makes at most 2Nρ (T ) + 1 steps, when it constructs the schema Uρ,ϕ,ψ (T ). Proof Let T ∈ Mρ C . From the description of the algorithm Uρ,ϕ,ψ it follows that this algorithm makes exactly one step, when constructs the schema Uρ,ϕ,ψ (T ). Therefore the statement of the theorem holds in the considered case. Let T ∈ / Mρ C . We denote by Γ the schema Uρ,ϕ,ψ (T ). Let us correspond to each node w of the schema Γ the word π(w) = πΓ (w) ∈ Ωρ (T ) as it was described in Sect. 2.3. Let w be an arbitrary node of the schema Γ which is not neither the root nor a terminal node. Let the node w be labeled with an element f i . From the description of the algorithm Uρ,ϕ,ψ it follows that Δ(T π(w)) = ∅ and ϕ(T π(w)) > max{ϕ(T π(w)( f i , δ)) : δ ∈ E k }. Therefore the cardinality of the set {δ : δ ∈ E k , Δ(T π(w)( f i , δ)) = ∅} is at least 2. Hence there are at least two edges leaving the node w. Evidently, Γ is a deterministic schema. Therefore only one edge leaves the root of Γ . From the description of the algorithm Uρ,ϕ,ψ it follows that Δ(T π(ξ )) = ∅ for any complete path ξ in the schema Γ . Evidently, Δ(T π(ξ1 )) ∩ Δ(T π(ξ2 )) = ∅ for any two different complete paths ξ1 and ξ2 in the schema Γ . Therefore the number of complete paths in the schema Γ is at most Nρ (T ). Hence L 0 (Γ ) ≤ Nρ (T ). Using Lemma 4.6 we obtain L(Γ ) ≤ 2Nρ (T ). Finally, one can show that the number of steps making by the algorithm Uρ,ϕ,ψ during the construction of the schema Γ is at most L(Γ ) + 1.  Remark 4.4 Let ρ = (F, k) be an enumerated signature, ψ be a computable complexity function of the signature ρ, ϕ : Mρ → ω be a computable function, A ⊆ Mρ,ϕ , and A = ∅. Using Theorem 4.4 and the description of the algorithm Uρ,ϕ,ψ one can show that the algorithm Uρ,ϕ,ψ has polynomial time complexity on the set of tables A if there exists a polynomial algorithm which, for a given f i ∈ F, computes the value ψ( f i ), and there exists a polynomial algorithm which, for given T ∈ A and α ∈ Ωρ (T ), computes the value ϕ(T α).

4.7 Algorithms Uρ,G Hρ ,h , Uρ,R H m+1 ,h , and Uρ,H m+1 ,h ρ

ρ

Let ρ be an enumerated signature, m ∈ ω \ {0}, and ϕ be a function from the set {G Hρ , R Hρm+1 , Hρm+1 }. From Proposition 4.1 it follows that Mρ (m) ⊆ Mρ,ϕ . In this section, we consider upper bounds on the depth of schemes constructed by the algorithm Uρ,ϕ,h for tables from Mρ (m), and also some considerations about the complexity of this algorithm.

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Theorem 4.5 Let ρ be an enumerated signature, m ∈ ω \ {0}, ϕ be a function from the set {G Hρ , R Hρm+1 , Hρm+1 }, and T ∈ Mρ (m). Then the schema Uρ,ϕ,h (T ) is a deterministic decision tree for the table T such that  h(Uρ,ϕ,h (T )) ≤

if Mρ,h (T ) ≤ 1 , Mρ,h (T ), Mρ,h (T )(ln ϕ(T ) − ln Mρ,h (T ) + 1), if Mρ,h (T ) ≥ 2 .

Proof The statement of the theorem follows from Proposition 4.1, Proposition 4.4, Theorem 4.2, and obvious equality Uρ,ϕ,h (T ) = Uρ (T, T ∗ ϕ, h).  Let ϕ ∈ {G Hρ , R Hρ2 , Hρ2 }. We now consider in details upper bounds on the value h(Uρ,ϕ,h (T )) for tables T ∈ Mρ (1). Let ρ be a signature, A be a nonempty subset of the set Mρ , and f, g be functions from A to ω. Denote B( f, g, A) = {( f (T ), g(T )) : T ∈ A}. Using Theorem 3.3 from [2] one can prove the following statement. Theorem 4.6 Let ρ be an enumerated signature and ϕ be a function from the set {G Hρ , R Hρ2 , Hρ2 }. Then B(Mρ,h , ϕ, Mρ (1)) = B(h dρ , ϕ, Mρ (1)) = {(0, 0)} ∪ {(n, r ) : n ∈ ω \ {0}, r ∈ ω \ {0}, n ≤ r } , and, for any pair (n, r ) ∈ B(Mρ,h , ϕ, Mρ (1)), there exists a table T (n, r ) ∈ Mρ (1) such that Mρ,h (T (n, r )) = h dρ (T (n, r )) = n, ϕ(T (n, r )) = r and  h(Uρ,ϕ,h (T (n, r ))) ≥

n, if n < 2 or r < 3n , (n − 1)(ln r − ln(3n) + 1), if n ≥ 2 and r ≥ 3n .

Remark 4.5 From Theorem 4.6 it follows that if m = 1, then the bound from Theorem 4.5 for decision tables from Mρ (1) is close to unimprovable. Remark 4.6 Let ϕ be a function from the set {G Hρ , R Hρ2 , Hρ2 }. From Lemma 4.5 and from Theorems 3.1, 4.5 and 4.6 it follows that, for any table T ∈ Mρ (1),  h(Uρ,ϕ,h (T )) ≤

h dρ (T ), d h ρ (T )(ln ϕ(T ) − ln h dρ (T )

if h dρ (T ) ≤ 1 , + 1), if h dρ (T ) ≥ 2

and, for tables from Mρ (1), this bound is close to unimprovable. Remark 4.7 Let ϕ ∈ {G Hρ , R Hρ2 , Hρ2 }. From Theorem 4.6 it follows that there is no function q : ω → ω such that h(Uρ,ϕ,h (T )) ≤ q(h dρ (T )) for any table T ∈ Mρ (1). For diagnostic tables from Mρ (1), we have another situation. Proposition 4.6 Let ρ = (F, k) be an enumerated signature, ϕ be a function from the set {G Hρ , R Hρ2 , Hρ2 }, and T be a diagnostic table from Mρ (1). Then

4.7 Algorithms Uρ,G Hρ ,h , Uρ,R Hρm+1 ,h , and Uρ,Hρm+1 ,h

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h(Uρ,ϕ,h (T )) ≤ 2 ln k(h dρ (T ))2 . Proof Let Mρ,h (T ) ≤ 1. Then, using Theorems 3.1 and 4.5, we obtain h(Uρ,ϕ,h (T )) ≤ Mρ,h (T ) ≤ h dρ (T ) . It is clear that h dρ (T ) ≤ 2 ln k(h dρ (T ))2 . Let Mρ,h (T ) ≥ 2. Then, using Theorem 4.5, we obtain h(Uρ,ϕ,h (T )) ≤ Mρ,h (T )(ln ϕ(T ) − ln Mρ,h (T ) + 1) . N (T )2

Using Remark 4.2 we obtain ϕ(T ) ≤ R Hρ2 (T ). One can show that R Hρ2 (T ) ≤ ρ 2 . Using these inequalities and the inequality Mρ,h (T ) ≥ 2 we obtain h(Uρ,ϕ,h (T )) ≤ 2Mρ,h (T ) ln Nρ (T ) = 2Mρ,h (T ) ln k logk Nρ (T ). From Theorem 3.1 it follows that Mρ,h (T ) ≤ h dρ (T ). Using Proposition 3.2 we obtain logk Nρ (T ) ≤ h dρ (T ). Therefore  h(Uρ,ϕ,h (T )) ≤ 2 ln k(h dρ (T ))2 . Let m ∈ ω \ {0}. We now consider in more detail upper bounds on the value h(Uρ,G Hρ ,h (T )) for tables T ∈ Mρ (m). Lemma 4.7 Let ρ = (F, k) be a signature and T ∈ Mρ . Then h dρ (T ) ≤ G Hρ (T ). Proof We denote by G the function G Hρ . Let us prove the statement of the lemma by induction on the value G(T ). Let G(T ) = 0. By Proposition 4.1, the function T ∗ G has the property of coordination. Therefore T ∈ Mρ C . Using Corollary 4.3 we obtain that h dρ (T ) = 0. Hence in the case G(T ) = 0 the statement of the lemma holds. Assume that n ∈ ω \ {0} and, for any table T ∈ Mρ with G(T ) < n, the statement of the lemma holds. Consider a table T ∈ Mρ with G(T ) = n. Let T  be a separable subtable of the table T which is a boundary table. Choose an element f i ∈ P(T ) such that the cardinality of the set {δ : δ ∈ E k , Δ(T  ( f i , δ)) = ∅} is at / Mρ C . From least two. The existence of such element follows from the relation T  ∈ the definition of a boundary table and from the choice of the element f i it follows that T  ( f i , δ) ∈ Mρ C for any δ ∈ E k . Let δ ∈ E k . One can show that any separable subtable of the table T ( f i , δ) is a separable subtable of the table T . By the choice of the element f i , the table T  is not a separable subtable of the table T ( f i , δ). Hence G(T ( f i , δ)) ≤ n − 1. From this inequality, from the inductive hypothesis and from Lemma 3.1 it follows that there exists a deterministic decision tree Γδ for the table T ( f i , δ) such that h(Γδ ) ≤ n − 1. We denote by Γδ a labeled finite directed tree with the root which is obtained from schema Γδ by removal of the root and the edge leaving the root. Let us consider a schema Γ obtained from the trees Γδ in the following way. Denote by w the node to which the unique edge enters that leaves the root of Γ . The node w is labeled with the element f i . For each δ ∈ E k , an edge leaves the node w which is labeled with the number δ. This edge enters the root of the tree Γδ . One can show that Γ is a deterministic decision tree for the table T , and h(Γ ) ≤ n.  Hence h dρ (T ) ≤ n.

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Lemma 4.8 Let ρ be an enumerated signature and m ∈ ω \ {0}. Then B(Mρ,h , G Hρ , Mρ (m)) = B(h dρ , G Hρ , Mρ (m)) = {(0, 0)} ∪ {(n, r ) : n ∈ ω \ {0}, r ∈ ω \ {0}, n ≤ r } . Proof Denote D = {(0, 0)} ∪ {(n, r ) : n ∈ ω \ {0}, r ∈ ω \ {0}, n ≤ r }. Theorem 4.6 and from the relation Mρ (1) ⊆ Mρ (m) it follows that

From

D ⊆ B(Mρ,h , G Hρ , Mρ (m)) and D ⊆ B(h dρ , G Hρ , Mρ (m)). By Proposition 4.1, for any table T ∈ Mρ (m), the function T ∗ G Hρ has the property of coordination. Using this fact, Lemma 4.7 and Theorem 3.1 one can show that B(Mρ,h , G Hρ , Mρ (m)) ⊆ D and B(h dρ , G Hρ , Mρ (m)) ⊆ D .  Remark 4.8 Let m ∈ ω \ {0}. From Lemma 4.8, from the relation Mρ (1) ⊆ Mρ (m) and from Theorem 4.6 it follows that the bound from Theorem 4.5 is close to unimprovable for ϕ = G Hρ and for tables from Mρ (m). Remark 4.9 Let m ∈ ω \ {0}. From the relation Mρ (1) ⊆ Mρ (m), from Lemmas 4.5, 4.7, 4.8 and from Theorems 3.1, 4.5 and 4.6 it follows that, for any table T ∈ Mρ (m),  h(Uρ,G Hρ ,h (T )) ≤

h dρ (T ), d h ρ (T )(ln G Hρ (T ) − ln h dρ (T )

if h dρ (T ) ≤ 1 , + 1), if h dρ (T ) ≥ 2

and this bound is close to unimprovable. Remark 4.10 Let m ∈ ω \ {0}. Evidently, if T ∈ Mρ (m) and α ∈ Ωρ (T ), then T α ∈ Mρ (m). Using Remarks 4.3 and 4.4 we conclude that the algorithms Uρ,G Hρ ,h , Uρ,Hρm+1 ,h and Uρ,R Hρm+1 ,h have polynomial time complexity on the set Mρ (m).

4.8 Algorithms Wρ,G Hρ ,h , Wρ,R Hρ2 ,h , and Wρ,Hρ2 ,h In this section, we consider three algorithms that will be used in the next chapter. Let ρ = (F, k) be a signature, T ∈ Mρ , dim T = n, δ¯ = (δ1 , . . . , δn ) ∈ E kn , and ¯ i)) = {δ} ¯ ∪ ¯ i) ∈ Mρ (1) in the following way: Δ(T (δ, i ∈ ω. We define a table T (δ, ¯ ≡ μT , νT (δ,i) (δ) = {1} and νT (δ,i) ¯ ) = {0} for {σ¯ : σ¯ ∈ Δ(T ), i ∈ / νT (σ¯ )}, μT (δ,i) ¯ ¯ (σ ¯ ¯ ¯ i)) \ {δ}. ¯ Denote νT (T ) = δ∈Δ(T ν ( δ). any σ¯ ∈ Δ(T (δ, ¯ ) T

4.8 Algorithms Wρ,G Hρ ,h , Wρ,R Hρ2 ,h , and Wρ,Hρ2 ,h

61

Proposition 4.7 Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ, T ∈ Mρ , dim T = n, μT (1) = f 1 , . . . , μT (n) = f n , δ¯ = (δ1 , . . . , δn ) ∈ E kn , t be the minimum number from the set ω \ νT (T ), and  B=

/ Δ(T ) , {t} ∪ νT (T ), if δ¯ ∈ ¯ if δ¯ ∈ Δ(T ) . νT (δ),

Then ¯ i), and ξ is a (a) If i ∈ B, Γ is a deterministic decision tree for the table T (δ, ¯ i)π(ξ )), then T π(ξ ) ∈ Mρ C complete path in the schema Γ such that δ¯ ∈ Δ(T (δ, and χ (π(ξ )) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}. ¯ i)) : i ∈ B} = Mρ,ψ (T, δ) ¯ holds. (b) The equality min{ψρd (T (δ, ¯ i), ξ be a Proof (a) Let i ∈ B, Γ be a deterministic decision tree for the table T (δ, ¯ ¯ ¯ ¯ complete path in Γ , and δ ∈ Δ(T (δ, i)π(ξ )). Then Δ(T (δ, i)π(ξ )) = {δ}. Therefore if i = t, then Δ(T π(ξ )) = ∅, and if i = t, then i ∈ νT (σ¯ ) for any σ¯ ∈ Δ(T π(ξ )). Hence T π(ξ ) ∈ Mρ C . The relation χ (π(ξ )) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )} is obvious. ¯ i)) : i ∈ B} ≥ (b) From the part (a) of the statement it follows that min{ψρd (T (δ, d ¯ Let us show that there exists i ∈ B such that ψρ (T (δ, ¯ i)) ≤ Mρ,ψ (T, δ). ¯ Mρ,ψ (T, δ). From Lemma 3.5 it follows that there exists a word β having the following properties: ¯ If Δ(Tβ) = χ (β) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}, Tβ ∈ Mρ C , and ψ(β) = Mρ,ψ (T, δ). ∅, then set i equal to t. If Δ(Tβ) = ∅, then set i equal to the minimum number from the set Π (Tβ). One can show that i ∈ B. Let us correspond a word α ∈ F ∗ to the word β. If β = λ, then α = λ. If β = λ and β = ( f i1 , δi1 ) · · · ( f im , δim ), then α = f i1 · · · f im . We denote by G the schema obtained from the schema G ρ (α) (see definition in Sect. 3.1) in the following way. Let ξ be a complete path in the schema G ρ (α) such that π(ξ ) = β. Then we label the terminal node of the path ξ with the number 1 instead of the number 0. One can show that the schema G is a deterministic decision ¯ Hence ψρd (T (δ, ¯ i)) ≤ ¯ i) and ψ(G) = ψ(β) = Mρ,ψ (T, δ). tree for the table T (δ, ¯ Mρ,ψ (T, δ).  Let ρ = (F, k) be an enumerated signature, ϕ be a function from the set {G Hρ , R Hρ2 , Hρ2 } , and ψ be a computable complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3. We now describe an algorithm Wρ,ϕ,ψ which, for a given table T ∈ Mρ and an arbitrary tuple δ¯ = (δ1 , . . . , δdim T ) ∈ E kdim T , constructs a word ¯ having the following properties: Wρ,ϕ,ψ (T, δ) ¯ ⊆ {(μT (1), δ1 ), . . . , (μT (dim T ), δdim T )} χ (Wρ,ϕ,ψ (T, δ)) ¯ ∈ Mρ C . and T Wρ,ϕ,ψ (T, δ)

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Algorithm Wρ,ϕ,ψ . ¯ = λ. Let T ∈ If T ∈ Mρ C , then Wρ,ϕ,ψ (T, δ) / Mρ . Set t equal to the minimum / Δ(T ), then B = {t} ∪ νT (T ). number from the set ω \ νT (T ). Define a set B. If δ¯ ∈ If δ¯ ∈ Δ(T ), then B = νT (T ). For any i ∈ B, construct by the algorithm Uρ,ϕ,ψ ¯ i)). Find the minimum number i 0 ∈ B such that the schema Γi = Uρ,ϕ,ψ (T (δ, ψ(Γi0 ) = min{ψ(Γi ) : i ∈ B}. In the schema Γi0 , find the complete path ξ such ¯ = π(ξ ). ¯ i 0 )π(ξ )). Set Wρ,ϕ,ψ (T, δ) that δ¯ ∈ Δ(T (δ, Theorem 4.7 Let ρ = (F, k) be an enumerated signature, ϕ be a function from the set {G Hρ , R Hρ2 , Hρ2 }, ψ be a computable complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3, T ∈ Mρ , dim T = n, μT (1) = f i1 , . . . , μT (n) = f in and δ¯ = (δ1 , . . . , δn ) ∈ E kn . Then ¯ ⊆ {( f i1 , δ1 ), . . . , ( f in , δn )} , χ (Wρ,ϕ,ψ (T, δ)) ¯ ∈ Mρ C and T Wρ,ϕ,ψ (T, δ) ¯ ≤ ψ(Wρ,ϕ,ψ (T, δ))



ψ(λ), if T ∈ Mρ C , ¯ 2 ln Nρ (T ) + Mρ,ψ (T, δ), ¯ if T ∈ / Mρ C . (Mρ,ψ (T, δ))

If ψ = h, then ¯ ≤ h(Wρ,ϕ,h (T, δ))



0, if T ∈ Mρ C , ¯ ln Nρ (T ) + 1, if T ∈ / Mρ C . Mρ,h (T, δ)

Proof In the case T ∈ Mρ C , the statement of the theorem follows immediately from the description of the algorithm Wρ,ϕ,ψ . Let T ∈ / Mρ C . We denote by t the minimum number from the set ω \ νT (T ). Let us define a set B. If δ¯ ∈ / Δ(T ), then B = {t} ∪ νT (T ). If δ¯ ∈ Δ(T ), then ¯ i)) B = νT (T ). By Propositions 4.1 and 4.4, for any i ∈ B, the schema Uρ,ϕ,ψ (T (δ, ¯ i). Using the description of the is a deterministic decision tree for the table T (δ, algorithm Wρ,ϕ,ψ and the part (a) of the statement of Proposition 4.7 we con¯ ⊆ {( f i1 , δ1 ), . . . , ( f in , δn )}, T Wρ,ϕ,ψ (T, δ) ¯ ∈ Mρ C and clude that χ (Wρ,ϕ,ψ (T, δ)) ¯ ≤ min{ψ(Uρ,ϕ,ψ (T (δ, ¯ i))) : i ∈ B}. ψ(Wρ,ϕ,ψ (T, δ)) Using the part (b) of the statement of Proposition 4.7 we conclude that there exists ¯ i)) = Mρ,ψ (T, δ). ¯ From this equality and from Theorem i ∈ B such that ψρd (T (δ, ¯ ¯ One can show that T (δ, ¯ i) ∈ 3.1 it follows that Mρ,ψ (T (δ, i)) ≤ Mρ,ψ (T, δ). / Mρ C 2 ¯ and R Hρ (T (δ, i)) ≤ Nρ (T ). From the last inequality and from Remark 4.2 it follows ¯ i)) ≤ Nρ (T ). Using Proposition 4.1 and Corollary 4.2 we obtain that ϕ(T (δ, ¯ ≤ ψ(Uρ,ϕ,ψ (T (δ, ¯ i))) ψ(Wρ,ϕ,ψ (T, δ)) 2 ¯ ¯ ¯ i)) ≤ (Mρ,ψ (T (δ, i))) ln ϕ(T (δ, i)) + Mρ,ψ (T (δ, ¯ 2 ln Nρ (T ) + Mρ,ψ (T, δ) ¯ . ≤ (Mρ,ψ (T, δ))

4.8 Algorithms Wρ,G Hρ ,h , Wρ,R Hρ2 ,h , and Wρ,Hρ2 ,h

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Let ψ = h. Then, using Proposition 4.1, Corollary 4.3, and Lemma 4.2, we obtain ¯ ≤ Mρ,h (T, δ) ¯ ln Nρ (T ) + 1.  h(Wρ,ϕ,h (T, δ)) Remark 4.11 Let ρ be an enumerated signature and ϕ be a function from the set {G Hρ , R Hρ2 , Hρ2 }. Let there exists a polynomial time algorithm for the computation of the function ψ on the set F ∗ . It is clear that if T ∈ Mρ (1) and α ∈ Ωρ (T ), then T α ∈ Mρ (1). Using Remarks 4.3 and 4.4 we conclude that the algorithm Uρ,ϕ,ψ has polynomial time complexity on the set Mρ (1). Using this fact one can show that the algorithm Wρ,ϕ,ψ has polynomial time complexity too.

References 1. Moshkov, M.: On conditional tests. Sov. Phys. Dokl. 27, 528–530 (1982) 2. Moshkov, M.: Conditional tests. In: Yablonskii, S.V. (ed.) Problemy Kibernetiki (in Russian), vol. 40, pp. 131–170. Nauka Publishers, Moscow (1983) 3. Moshkov, M.: Bounds on complexity and algorithms for construction of deterministic conditional tests. In: Mat. Vopr. Kibern. (in Russian), vol. 16, pp. 79–124. Fizmatlit, Moscow (2007). http:// library.keldysh.ru/mvk.asp?id=2007-79 4. Quinlan, J.R.: Discovering rules by induction from large collections of examples. In: D. Michie (ed.) Experts Systems in the Microelectronic Age, pp. 168–201. Edinburg University Press (1979)

Chapter 5

Upper Bounds and Algorithms for Construction of Deterministic Decision Trees for Decision Tables. Second Approach

In this chapter, upper bounds on the minimum complexity and algorithms for construction of deterministic decision trees for decision tables are considered. These bounds and algorithms are based on the use of so-called additive-bounded uncertainty measures for decision tables. The bounds are true for any complexity function having the property Λ1. When developing algorithms, we assume that the complexity functions have properties Λ1, Λ2, and Λ3. In [1], an upper bound on the minimum depth of a deterministic decision tree was considered. This is a bound on the depth of a decision tree constructed by an algorithm which uses the uncertainty measure equal to the number of rows in a decision table. In [2], this algorithm for decision tables with single-valued decisions was generalized to various uncertainty measures and cost functions. To construct a decision tree, the algorithm minimizes the complexity of a sequence of attributes which either reduces the uncertainty by half or gives the solution of the problem (a decision for the considered row). The most part of results presented in this chapter were published in [3], where we studied decision tables with many-valued decisions.

5.1 Reduced Deterministic Decision Trees for Decision Tables Let ρ = (F, k) be a signature, T ∈ Mρ , and Γ be a deterministic decision tree for the table T . Let us correspond to each node w of Γ the word π(w) = πΓ (w) ∈ Ωρ (T ) in the same way as in Sect. 2.3. The deterministic decision tree Γ for the table T will be called reduced if 1. Δ(T π(w)) = ∅ for any terminal node w of the schema Γ . 2. At least two edges leave any node of the schema Γ which is not neither the root nor a terminal node. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_5

65

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3. T π(w) ∈ / Mρ C for any node w of the schema Γ which is not neither the root nor a terminal node. T.

We denote by Cρ0 (T ) the set of reduced deterministic decision trees for the table

A node w2 of the schema Γ will be called a successor of a node w1 of the schema Γ if there exists an edge leaving w1 and entering w2 . If this edge is labeled with a number δ, then the node w2 will be called the δ-successor of the node w1 . A node of the schema Γ will be called preterminal if it is not neither the root nor a terminal node, and each edge leaving this node enters a terminal node. Lemma 5.1 Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ having the property Λ2, T ∈ Mρ , and Δ(T ) = ∅. Then there exists a reduced deterministic decision tree Γ for the table T such that ψ(Γ ) = ψρd (T ). Proof By Lemma 3.1, there exists a deterministic decision tree Γ1 for the table T such that ψ(Γ1 ) = ψρd (T ). For an arbitrary node w of the schema Γ1 , we denote by π(w) the word πΓ1 (w). / Let w be an arbitrary node of the schema Γ1 which is not the root. If T π(w) ∈ Mρ C , then we leave the node w untouched. If Δ(T π(w)) = ∅, then we remove from the schema Γ1 the node w and the edge entering the node w. Let T π(w) ∈ Mρ C and Δ(T π(w)) = ∅. If w is the successor of the root, then, instead of the previous label, we mark the node w by the minimum number from the set Π (T π(w)). Let the node w be a successor of a node w1 which is not the root of the schema Γ1 . If T π(w1 ) ∈ Mρ C , then we remove from the schema Γ1 the node w and the edge / Mρ C , then, instead of the previous label, we mark entering the node w. If T π(w1 ) ∈ the node w by the minimum number from the set Π (T π(w)). Let us treat in this way all nodes of the schema Γ which are not the root. We begin with terminal nodes, then treat nodes all successors of which are treated, etc. As a result, we obtain a schema of the signature ρ. We denote this schema by Γ2 . Let w be an arbitrary node of the schema Γ2 which is not neither the root nor a terminal node. If the node w has at least two successors, then we leave the node w untouched. Let the node w have exactly one successor w1 . Remove from the schema Γ2 the node w and the edge leaving w. The edge entering the node w will enter the node w1 . Let us treat in this way all nodes of the schema Γ2 which are not neither the root nor a terminal node. As a result, we obtain a schema of the signature ρ. We denote this schema by Γ . One can show that Γ is a reduced deterministic decision tree for the table T . Since ψ has the property Λ2, we have ψ(Γ ) ≤ ψ(Γ1 ). By the  choice of the schema Γ1 , we have ψ(Γ ) = ψρd (T ).

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5.2 Additive-Bounded Uncertainty Measures for Decision Tables Let T ∈ Mρ . A function γ : Ωρ (T ) → ω will be called an additive-bounded uncertainty measure for the decision table T if it has the following properties: • Additive-boundedness: for any α, ( f i , δ1 ), ( f i , δ2 ) ∈ Ωρ (T ), if δ1 = δ2 , then γ (α) ≥ γ (α( f i , δ1 )) + γ (α( f i , δ2 )). • Partial coordination: for any α ∈ Ωρ (T ), if γ (α) = 0, then T ∈ Mρ C . • Nondecrease: for any α, β, ε ∈ Ωρ (T ), γ (αεβ) ≤ γ (αβ). Let p, q ∈ ω and p + q ≥ 1. One can show that if functions γ1 and γ2 are additivebounded uncertainty measures for the decision table T ∈ Mρ , then the functions pγ1 + qγ2 , min{γ1 , γ2 } and γ1 γ2 are additive-bounded uncertainty measures for the decision table T too. Let γ : Ωρ (T ) → ω. We will say that the function γ has the property of commutativity if γ (αβ) = γ (βα) for any α, β ∈ Ωρ (T ). Remark 5.1 Let T ∈ Mρ and γ : Ωρ (T ) → ω. If the function γ has the properties of commutativity and additive-boundedness, then the function γ has the property of nondecrease. Really, let α, β, ε ∈ Ωρ (T ). Then γ (αεβ) = γ (βαε) since γ has the property of commutativity, γ (βαε) ≤ γ (βα) since γ has the property of additiveboundedness, and γ (βα) = γ (αβ) since γ has the property of commutativity. We now consider examples of additive-bounded uncertainty measures. Let T ∈ Mρ and n ∈ ω. Denote   ¯  . σ (T, n) = {δ¯ : δ¯ ∈ Δ(T ), n ∈ νT (δ)} We describe functions Nρ , Jρ , L K ρ , and K ρ from Mρ to ω. Let T ∈ Mρ . Then • Nρ (T ) = |Δ(T )|. • Jρ (T ) = Nρ (T ) − max{σ (T, n) : n ∈ ω}. • If Δ(T ) = ∅, then L K ρ (T ) = 0. Let Δ(T ) = ∅ and Γ be a schema of the signature ρ. We denote by L K ρ (Γ ) the number of preterminal nodes of the schema Γ . Then L K ρ (T ) = max{L K ρ (Γ ) : Γ ∈ Cρ0 (T )}. • If Δ(T ) = ∅, then K ρ (T ) = 0. Let Δ(T ) = ∅ and Γ ∈ Cρ0 (T ). We correspond a number K ρ (w) to each node w of the schema Γ which is not the root. If w is a terminal node, then K ρ (w) = 0. If w is a preterminal node, then K ρ (w) = 1. Let w be a node of the schema Γ which is not neither the root, nor a terminal node, nor a preterminal node. Then K ρ (w) = max{K ρ (w1 ) + K ρ (w2 ) : w1 and w2 are successors of w and w1 = w2 }. Set K ρ (Γ ) = K ρ (w0 ), where w0 is the successor of the root of the schema Γ . Then K ρ (T ) = max{K ρ (Γ ) : Γ ∈ Cρ0 (T )}. Proposition 5.1 Let ρ = (F, k) be a signature and T ∈ Mρ . Then the functions T ∗ Nρ , T ∗ Jρ , T ∗ L K ρ , and T ∗ K ρ are additive-bounded uncertainty measures for the table T .

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Proof Let α, β, ε ∈ Ωρ (T ), f i ∈ P(T ), δ1 , δ2 ∈ E k , and δ1 = δ2 .  (a) The function T ∗ Nρ . We denote by N the function T ∗ Nρ . Evidently, N (α) = δ∈E k N (α( f i , δ)). From this equality it follows that the function N has the property of additive-boundedness. If N (α) = 0, then, evidently, T α ∈ Mρ C . Therefore the function N has the property of partial coordination. It is clear that the function N has the property of commutativity. By Remark 5.1, the function N has the property of nondecrease. (b) The function T ∗ Jρ . We denote by J the function T ∗ Jρ . Let m ∈ ω and σ (T α, m) = max{σ (T α, n) : n ∈ ω}. Then J (α) = Nρ (T α) − σ (T α, m) = ≥





(Nρ (T α( f i , δ)) − σ (T α( f i , δ), m))

δ∈E k

(Nρ (T α( f i , δ)) − max{σ (T α( f i , δ), n) : n ∈ ω}) =

δ∈E k



J (α( f i , δ)) .

δ∈E k

From these relations it follows that the function J has the property of additiveboundedness. Let J (α) = 0. Then Δ(T α) = ∅ or Π (T α) = ∅. Hence T α ∈ Mρ C . Therefore the function J has the property of partial coordination. Evidently, the function J has the property of commutativity. By Remark 5.1, the function J has the property of nondecrease. (c) The function T ∗ L K ρ . Let us show that the value of L K ρ (T ) is definite, and the following inequality holds: L K ρ (T ) ≤

Nρ (T ) . 2

(5.1)

Let Δ(T ) = ∅. Then L K ρ (T ) = 0 and, evidently, (5.1) holds. Let Δ(T ) = ∅. Using Lemma 5.1 we obtain Cρ0 (T ) = ∅. Let Γ ∈ Cρ0 (T ). Then Δ(T π(ξ )) = ∅ for any complete path ξ in the schema Γ . Evidently, Δ(T π(ξ1 )) ∩ Δ(T π(ξ2 )) = ∅ for any different complete paths ξ1 and ξ2 in the schema Γ . Therefore the number of terminal nodes in the schema Γ is at most Nρ (T ). Taking into account N (T ) that at least two edges leave each preterminal node of Γ we obtain L K ρ (Γ ) ≤ ρ2 . Hence the value of L K ρ (T ) is definite and the inequality (5.1) holds. We denote by L the function T ∗ L K ρ . Let Δ(T ) = ∅. Then L(α) = 0 for any α ∈ Ωρ (T ). One can show that in this case the function L has the properties of additive-boundedness, partial coordination, and nondecrease. We now assume that Δ(T ) = ∅. Let T α ∈ Mρ C . Then, as it is not difficult to verify, L(α) = L(α( f i , δ1 )) = L(α( f i , δ2 )) = 0. Therefore L(α) ≥ L(α( f i , δ1 )) + / Mρ C . Set E = {δ : δ ∈ E k , Δ(T α( f i , δ)) = ∅}. Let |E| = L(α( f i , δ2 )). Let T α ∈ 1 and E = {δ}. Then L(α( f i , δ)) = L(α) and L(α( f i , σ )) = 0 for any σ ∈ E k \ {δ}. Therefore L(α) ≥ L(α( f i , δ2 )) + L(α( f i , δ2 )). Let |E| ≥ 2. We choose for each

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δ ∈ E a schema Γδ ∈ Cρ0 (T α( f i , δ)) such that L K ρ (Γδ ) = L(α( f i , δ)). For each δ ∈ E, we denote by Γδ a labeled finite directed tree with the root which is obtained from Γδ by removal of the root and the edge leaving the root. Let us consider a schema Γ which is obtained from the trees Γδ in the following way. We denote by w the successor of the schema Γ root. The node w is labeled with the element f i . For any δ ∈ E, the root of the tree Γδ is the δ-successor of the node w. There are no any other successors of the node w. One can show that Γ ∈ Cρ0 (T α) and L K ρ (Γ ) =

 δ∈E

L K ρ (Γδ ) =



L(α( f i , δ)) .

δ∈E

Taking into account that L(α) ≥ L K ρ (Γ ) and L(α( f i , δ)) = 0 for any δ ∈ E k \ E we obtain L(α) ≥ δ∈Ek L(α( f i , δ)). Hence the function L has the property of additive-boundedness. Let L(α) = 0. If Δ(T α) = ∅, then T α ∈ Mρ C . Let Δ(T α) = ∅. Using Lemma 5.1 we obtain Cρ0 (T α) = ∅. Let Γ ∈ Cρ0 (T α). Then L K ρ (Γ ) = 0. One can show that in this case the schema Γ has the only complete path ξ and π(ξ ) = λ for this path. Hence T α ∈ Mρ C . Therefore the function L has the property of partial coordination. Evidently, the function L has the property of commutativity. By Remark 5.1, the function L has the property of nondecrease. (d) The function T ∗ K ρ . Let us show that the value of K ρ (T ) is definite. If Δ(T ) = ∅, then K ρ (T ) = 0. Let Δ(T ) = ∅. Using Lemma 5.1 we obtain Cρ0 (T ) = ∅. Let Γ ∈ Cρ0 (T ). One can show that K ρ (Γ ) ≤ L K ρ (Γ ). Taking into account that the value of L K ρ (T ) is definite we conclude that the value of K ρ (T ) is definite too. We denote by K the function T ∗ K ρ . Let Δ(T ) = ∅. Then K (α) = 0 for any α ∈ Ωρ (T ). One can show that in this case the function K has the properties of additive-boundedness, partial coordination, and nondecrease. We now assume that Δ(T ) = ∅. Let T α ∈ Mρ C . Then, as it is not difficult to verify, K (α) = K (α( f i , δ1 )) = K (α( f i , δ2 )) = 0. Therefore K (α) ≥ K (α( f i , δ1 )) + / Mρ C . Set E = {δ : δ ∈ E k , Δ(T α( f i , δ)) = ∅}. Let |E| = K (α( f i , δ2 )). Let T α ∈ 1 and E = {δ}. Then K (α( f i , δ)) = K (α) and K (α( f i , σ )) = 0 for any σ ∈ E k \ {δ}. Therefore K (α) ≥ K (α( f i , δ1 )) + K (α( f i , δ2 )). Let |E| ≥ 2. Let us choose for each δ ∈ E a schema Γδ ∈ Cρ0 (T α( f i , δ)) such that K ρ (Γδ ) = K (α( f i , δ)). We denote by Γ the schema which is obtained from the schemes Γδ in the same way as in the previous part of the proof. It is not difficult to verify that Γ ∈ Cρ0 (T α). One can show that K ρ (Γ ) = K ρ (Γσ1 ) + K ρ (Γσ2 ) = K (α( f i , σ1 )) + K (α( f i , σ2 )), where σ1 , σ2 ∈ E, σ1 = σ2 and if |E| ≥ 3, then K (α( f i , σ1 )) ≥ K (α( f i , σ )) and K (α( f i , σ2 )) ≥ K (α( f i , σ )) for any σ ∈ E \ {σ1 , σ2 }. Taking into account that K (T α( f i , δ)) = 0 for any δ ∈ E k \ E, and K (α) ≥ K ρ (Γ ) we conclude that K (α) ≥ K (α( f i , δ1 )) + K (α( f i , δ2 )). Hence the function K has the property of additiveboundedness. Let K (α) = 0. If Δ(T α) = ∅, then T α ∈ Mρ C . Let Δ(T α) = ∅. Using Lemma 5.1 we obtain Cρ0 (T α) = ∅. Let Γ ∈ Cρ0 (T α). Then K ρ (Γ ) = 0. Hence there are no

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preterminal nodes in the schema Γ . One can show that in this case the schema Γ has the only complete path ξ and π(ξ ) = λ for this path. Hence T α ∈ Mρ C . Therefore the function K has the property of partial coordination. Evidently, the function K has the property of commutativity. By Remark 5.1, the function K has the property of nondecrease.  The following statement characterizes the class of additive-bounded uncertainty measures for an arbitrary decision table. Proposition 5.2 Let T ∈ Mρ and γ be an additive-bounded uncertainty measure for the table T . Then γ (λ) ≥ K ρ (T ) . Proof Let Δ(T ) = ∅. Then K ρ (T ) = 0 and, evidently, the considered inequality holds. Let Δ(T ) = ∅. By the above (see the proof of Proposition 5.1), the value of K ρ (T ) is definite. Hence there exists a schema Γ ∈ Cρ0 (T ) such that K ρ (Γ ) = K ρ (T ). For each node w of the schema Γ , we denote by π(w) the word πΓ (w). By the definition of the value K ρ (Γ ), a number K ρ (w) is associated with each node w of the schema Γ which is not the root. Let w be an arbitrary node of the schema Γ which is not the root. We now show that γ (π(w)) ≥ K ρ (w). Let w be a terminal node of the schema Γ . Then K ρ (w) = 0 and γ (π(w)) ≥ 0. Therefore γ (π(w)) ≥ K ρ (w). Let ω be a preterminal node of the schema Γ . Since Γ ∈ Cρ0 (T ), we have T π(w) ∈ / Mρ C . Using the property of partial coordination of the function γ we obtain γ (π(w)) ≥ 1. Taking into account that K ρ (w) = 1 we conclude that γ (π(w)) ≥ K ρ (w). Let w be a node of the schema Γ which is not neither the root nor a preterminal node, nor a terminal node. Assume that for all successors of the node w the considered statement holds. Let us show that the considered statement holds for the node w too. By definition, K ρ (w) = K ρ (w1 ) + K ρ (w2 ) for some different successors w1 and w2 of the node w. Using the property of additive-boundedness of the function γ we obtain γ (π(w)) ≥ γ (π(w1 )) + γ (π(w2 )). By the assumption, γ (π(w1 )) ≥ K ρ (w1 ) and γ (π(w2 )) ≥ K ρ (w2 ). Therefore γ (π(w)) ≥ K ρ (w). Thus, the considered statement holds. Let w0 be the successor of the root of Γ . Then γ (π(w0 )) ≥ K ρ (w0 ). By definition, π(w0 ) = λ and K ρ (w0 ) = K ρ (Γ ). By the choice of Γ , the equality K ρ (Γ ) = K ρ (T )  holds. Hence γ (λ) ≥ K ρ (T ).

5.3 Process Yρ,N of Schema Construction Let ρ = (F, k) be a signature. Denote Dρ = {(T, γ ) : T ∈ Mρ , γ is an additivebounded measure for the table T }. Let D be a nonempty subset of the set Dρ . Denote Z˜ ρ (D) = {(T, α, γ ) : (T, γ ) ∈ D, α ∈ Ωρ (T ), T α ∈ / Mρ C } and Cρd the set

5.3 Process Yρ,N of Schema Construction

71

of deterministic schemes of the signature ρ. A mapping N : Z˜ ρ (D) → Cρd will be called admissible if, for any tuple (T, α, γ ) ∈ Z˜ ρ (D), the following conditions hold: 1. P(N (T, α, γ )) ⊆ P(T ). 2. Δ(T α) = ξ ∈Ξ (N (T,α,γ )) Δ(T απ(ξ )). 3. T απ(ξ ) ∈ Mρ C or γ (α) > γ (απ(ξ )) for any ξ ∈ Ξ (N (T, α, γ )). Let N : Z˜ ρ (D) → Cρd be an admissible mapping. We now describe a process Yρ,N which, for a given pair (T, γ ) ∈ D, constructs a schema Yρ,N (T, γ ) that is a deterministic decision tree for the table T . In general case, the process Yρ,N is not an algorithm but only a way for description of the schema Yρ,N (T, γ ). Process Yρ,N . Step 1. Construct a tree containing nodes w1 , w2 and the edge which leaves w1 and enters w2 . Let T ∈ Mρ C . If Δ(T ) = ∅, then label the node w2 with the number 0. If Δ(T ) = ∅, then label the node w2 with the minimum number from the set Π (T ). We denote the obtained tree by Yρ,N (T, γ ). The process Yρ,N is completed. Let T ∈ / Mρ C . Label the node w2 with the word λ and pass to the second step. (We will say that the node w1 was constructed in step 1.) Assume that t steps were made. We denote by G the tree obtained in step t. Step (t + 1). If in the tree G there are no nodes labeled with words from Ωρ (T ), then we denote by Yρ,N (T, γ ) the tree G. The process Yρ,N is completed. Otherwise, choose in the tree G a node w which is labeled with a word from Ωρ (T ). Let the node w be labeled with the word α. Let T α ∈ Mρ C . If Δ(T α) = ∅, then instead of the word α, we label the node w with the number 0. If Δ(T α) = ∅, then instead of the word α, we label the node w with the minimum number from the set Π (T α). (We will say that the node w was constructed in step (t + 1).) Pass to step (t + 2). Let T α ∈ / Mρ C . For each complete path ξ in the schema N (T, α, γ ), instead of a number, we label the terminal node of this path with the word απ(ξ ). Remove the root of the schema N (T, α, γ ) and the edge leaving the root. We denote by Γ the obtained labeled finite directed tree with the root. Remove from the tree G the node w. The edge entering the node w will enter the root of the tree Γ . (We will say that the nodes of the tree Γ , which are not terminal, were constructed in step (t + 1).) Pass to step (t + 2). Using the description of the process Yρ,N , properties of the mapping N , and the property of partial coordination of the function γ one can prove the following statement. Proposition 5.3 Let ρ be a signature, D be a nonempty subset of the set Dρ , and N : Z˜ ρ (D) → Cρd be an admissible mapping. Then, for any pair (T, γ ) ∈ D, the process Yρ,N ends after a finite number of steps. The schema Yρ,N (T, γ ) constructed by the process Yρ,N is a deterministic decision tree for the table T .

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5.4 Upper Bounds on ψρd (T ) Let ρ = (F, k) be a signature and κρ : {(T, α) : T ∈ Mρ , α ∈ Ωρ (T )} → ω be a function having the following property: if T ∈ Mρ , α, β ∈ Ωρ (T ) and κρ (T, α) = κρ (T, β), then α = β. Let T ∈ Mρ and α ∈ Ωρ (T ). The number κρ (T, α) will be called the T -number of the word α. Let ψ be a complexity function of the signature ρ. We now define a mapping Nψ : Z˜ ρ (Dρ ) → Cρd . Mapping Nψ . Let (T, α, γ ) ∈ Z˜ ρ (Dρ ), dim T = n, μT (1) = f 1 , . . . , μT (n) = f n . For each f i ∈ P(T ), we denote by σi the minimum number from E k such that γ (α( f i , σi )) = max{γ (α( f i , σ )) : σ ∈ E k } . Set σ¯ = (σ1 , . . . , σn ). Let β be the word from Ωρ (T ) with the minimum T number having the following properties: χ (β) ⊆ {( f 1 , σ1 ), . . . , ( f n , σn )}, T αβ ∈ / Mρ C , we have β = λ. Let β = Mρ C , and ψ(β) = Mρ,ψ (T, σ¯ ). Since T α ∈ ( f i1 , σi1 ) · · · ( f im , σim ). Set ε = f i1 · · · f im . Then Nψ (T, α, γ ) = G ρ (ε) (the definition of the schema G ρ (ε) can be found in Sect. 3.1). Lemma 5.2 Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ, and (T, α, γ ) ∈ Z˜ ρ (Dρ ). Then, for any complete path ξ in the schema Nψ (T, α, γ ), the following statements hold: (a) ψ(π(ξ )) ≤ Mρ,ψ (T ). . (b) If T απ(ξ ) ∈ / Mρ C , then γ (απ(ξ )) ≤ γ (α) 2 Proof Let dim T = n and μT (1) = f 1 , . . . , μT (n) = f n . For each f i ∈ P(T ), we denote by σi the minimum number from E k such that γ (α( f i , σi )) = max{γ (α( f i , σ )) : σ ∈ E k }. Let σ¯ = (σ1 , . . . , σn ) and β be the word from Ωρ (T ) with the minimum T -number which has the following properties: χ (β) ⊆ {( f 1 , σ1 ), . . . , ( f n , σn )}, T αβ ∈ Mρ C , and ψ(β) = Mρ,ψ (T α, σ¯ ). Let ξ be an arbitrary complete path in the schema Nψ (T, α, γ ). Then, evidently, ψ(π(ξ )) = ψ(β) = Mρ,ψ (T α, σ¯ ). One can show that Mρ,ψ (T α, σ¯ ) ≤ Mρ,ψ (T, σ¯ ). From Lemma 3.5 it follows that Mρ,ψ (T, σ¯ ) ≤ Mρ,ψ (T ). Therefore ψ(π(ξ )) ≤ Mρ,ψ (T ). Let T απ(ξ ) ∈ / Mρ C . One can show that in this case π(ξ ) = β, and there exist f i ∈ P(T ) and δ ∈ E k such that δ = σi and ( f i , δ) ∈ χ (π(ξ )). Using the property of nondecrease of the function γ we obtain γ (απ(ξ )) ≤ γ (α( f i , δ)). Using the property of additive-boundedness of the function γ and taking into account the choice of the number σi we obtain 2γ (α( f i , δ)) ≤ γ (α( f i , δ)) + γ (α( f i , σi )) ≤ γ (α). Hence .  γ (απ(ξ )) ≤ γ (α) 2 Corollary 5.1 Let ρ be a signature, ψ be a complexity function of the signature ρ, T ∈ Mρ , and γ be an additive-bounded uncertainty measure for the table T . Then the schema Yρ,N ψ (T, γ ) is a deterministic decision tree for the table T .

5.4 Upper Bounds on ψρd (T )

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Proof Using the definition of the mapping Nψ and Lemma 5.2 one can show that Nψ is an admissible mapping. By Proposition 5.3, the schema Yρ,N ψ (T, γ ) is a deterministic decision tree for the table T .  Theorem 5.1 Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ having the property Λ1, T ∈ Mρ , and γ be an additive-bounded uncertainty measure for the table T . Then  ψρd (T )

≤

ψ(λ), if T ∈ Mρ C , / Mρ C . Mρ,ψ (T )(1 + log2 γ (λ)), if T ∈

Proof Let T ∈ Mρ C . From the description of the process Yρ,N ψ it follows that ψ(Yρ,N ψ (T, γ )) = ψ(λ). Let T ∈ / Mρ C . We now consider an arbitrary complete path ξ in the schema Yρ,N ψ (T, γ ). From the description of the process Yρ,N ψ (T, γ ) and from the relation T ∈ / Mρ C it follows that, for some m ∈ ω \ {0}, the equality π(ξ ) = π(ξ1 ) · · · π(ξm ) holds, where ξ1 is a complete path in the schema Nψ (T, λ, γ ) and if m ≥ 2, then ξi is a complete path in the schema Nψ (T, π(ξ1 ) · · · π(ξi−1 ), γ ) for i = 2, . . . , m. By the assumption, T λ ∈ / Mρ C . From the description of the process Yρ,N ψ it follows that if / Mρ C for i = 2, . . . , m. Using the part (a) of the m ≥ 2, then T π(ξ1 ) · · · π(ξi−1 ) ∈ statement of Lemma 5.2 we conclude that ψ(π(ξi )) ≤ Mρ,ψ (T ) for i = 1, . . . , m. Using the property Λ1 of the function ψ we obtain ψ(π(ξ )) ≤

m 

ψ(π(ξi )) ≤ m Mρ,ψ (T ) .

(5.2)

i=1

Let us show that m ≤ 1 + log2 γ (λ). Since T ∈ / Mρ C and the function γ has the property of partial coordination, we have γ (λ) ≥ 1. Therefore if m = 1, then the considered inequality holds. Let m ≥ 2. Using the part (b) of the statement of Lemma 5.2 we obtain γ (λ) γ (π(ξ1 ) · · · π(ξm−1 )) ≤ m−1 . 2 Taking into account that T π(ξ1 ) · · · π(ξm−1 ) ∈ / Mρ C and using the property of partial coordination of the function γ we obtain γ (π(ξ1 ) · · · π(ξm−1 )) ≥ 1. Hence 2m−1 ≤ γ (λ) and m ≤ 1 + log2 γ (λ). Using the inequality (5.2) we obtain ψ(π(ξ )) ≤ Mρ,ψ (T )(1 + log2 γ (λ)). Taking into account that ξ is an arbitrary complete path in the schema Yρ,N ψ (T, γ ) we obtain ψ(Yρ,N ψ (T, γ )) ≤ Mρ,ψ (T )(1 + log2 γ (λ)). Finally, using Corollary 5.1 we conclude that ψρd (T ) ≤ Mρ,ψ (T )(1 + log2 γ (λ)).  Corollary 5.2 Let ρ be a signature, ψ be a complexity function of the signature ρ having the property Λ1, and T ∈ Mρ . Then  ψρd (T ) ≤

ψ(λ), if T ∈ Mρ C , / Mρ C . Mρ,ψ (T ) log2 Nρ (T ), if T ∈

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Proof By Proposition 5.1, the function T ∗ L K ρ is an additive-bounded uncertainty N (T ) measure for the table T . From (5.1) it follows that T ∗ L K ρ (λ) ≤ ρ2 . Using Theorem 5.1 we obtain the considered inequality.  Corollary 5.3 Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ having the properties Λ1 and Λ2, and T be a diagnostic table from Mρ \ Mρ C . Then max{Mρ,ψ (T ), logk Nρ (T )} ≤ ψρd (T ) ≤ Mρ,ψ (T ) log2 Nρ (T ) . Proof The lower bounds on the value ψρd (T ) follow from Theorem 3.1 and Proposition 3.2. The upper bound follows from Corollary 5.2. 

5.5 Upper Bounds on hρd (T ) Let ρ = (F, k) be a signature. We define a mapping Nρh : Z˜ ρ (Dρ ) → Cρd . Mapping Nρh . Let (T, α, γ ) ∈ Z˜ ρ (Dρ ), dim T = n, and μT (1) = f 1 , . . . , μT (n) = f n . For each f i ∈ P(T ), we denote by σi the minimum number from E k such that γ (α( f i , σi )) = max{γ (α( f i , σ )) : σ ∈ E k } . Set σ¯ = (σ1 , . . . , σn ). Let β be the word from Ωρ (T ) with the minimum T -number having the following properties: χ (β) ⊆ {( f 1 , σ1 ), . . . , ( f n , σn )}, T αβ ∈ Mρ C , and h(β) = Mρ,h (T, σ¯ ). The existence of the word β follows from Lemma 3.5. Since Tα ∈ / Mρ C , we have β = λ. Let β = ( f i1 , σi1 ) · · · ( f im , σim ). We now describe the process of construction of the schema Nρh (T, α, γ ). Step 1. Construct a tree containing nodes w1 , w2 and the edge which leaves w1 and enters w2 . Label the node w2 with the word α. We denote the obtained tree by G 1 . Set I1 = { f i1 , . . . , f im }. Pass to the second step. Assume that t steps were made, and the tree G t and the set It were constructed. Step (t + 1). Find in the tree G t the unique node w which is labeled with a word from the set Ωρ (T ). Let the node w be labeled with the word ε. If It = ∅, then instead of the word ε, we label the node w with the number 0. We denote the obtained tree by Nρh (T, α, γ ). The process of the schema Nρh (T, α, γ ) construction is completed. Let It = ∅ and j be the minimum number from the set {1, . . . , n} having the following properties: f j ∈ It and, for any fl ∈ It , max{γ (ε( f j , σ )) : σ ∈ E k \ {σ j }} ≥ max{γ (ε( fl , σ )) : σ ∈ E k \ {σl }} . Instead of the word ε, we label the node w with the element f j . For each σ ∈ E k , we add to the tree G t the node wσ and the edge leaving w and entering wσ . We label this

5.5 Upper Bounds on h dρ (T )

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edge with the number σ . If σ = σ j , then we label the node wσ with the number 0. If σ = σ j , then we label the node wσ with the word ε( f j , σ j ). We denote the obtained tree by G t+1 and denote by It+1 the set It \ { f j }. Pass to step (t + 2). Lemma 5.3 Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ, and (T, α, γ ) ∈ Z˜ ρ (Dρ ). Then, for any complete path ξ in the schema Nρh (T, α, γ ), the following statements hold: (a) h(π(ξ )) ≤ Mρ,h (T ). γ (α) . (b) If T απ(ξ ) ∈ / Mρ C , then γ (απ(ξ )) ≤ max{2,h(π(ξ ))} Proof Let dim T = n and μT (1) = f 1 , . . . , μT (n) = f n . For each f i ∈ P(T ), we denote by σi the minimum number from E k such that γ (α( f i , σi )) = max{γ (α( f i , σ )) : σ ∈ E k }. Let σ¯ = (σ1 , . . . , σn ) and β be the word from Ωρ (T ) with the minimum T -number which has the following properties: χ (β) ⊆ {( f 1 , σ1 ), . . . , ( f n , σn )}, T αβ ∈ Mρ C , and h(β) = Mρ,h (T α, σ¯ ). One can show that in the schema Nρh (T, α, γ ) there exists a complete path ξ0 such that χ (π(ξ0 )) = χ (β). Let π(ξ0 ) = ( f j1 , σ j1 ) · · · ( f jm , σ jm ). Set α0 = λ and, for i = 1, . . . , m, set αi = ( f j1 , σ j1 ) · · · ( f ji , σ ji ). For i = 1, . . . , m, we denote by δ ji the minimum number from the set E k \ {σ ji } such that γ (ααi−1 ( f ji , δ ji )) = max{γ (ααi−1 ( f ji , σ )) : σ ∈ E k \ {σ ji }} . Let ξ be an arbitrary complete path in the schema Nρh (T, α, γ ). Then, evidently, h(π(ξ )) ≤ h(β) = Mρ,h (T α, σ¯ ). One can show that Mρ,h (T α, σ¯ ) ≤ Mρ,h (T, σ¯ ). By Lemma 3.5, Mρ,h (T, σ¯ ) ≤ Mρ,h (T ). Therefore h(π(ξ )) ≤ Mρ,h (T ). Let T απ(ξ ) ∈ / Mρ C . Then ξ = ξ0 . One can show that in this case, for some r ∈ {1, . . . , m}, the equality π(ξ ) = αr −1 ( f jr , δ) holds, where δ = σ jr . Evidently, . Using the property of nondecrease r = h(π(ξ )). Let us show that γ (απ(ξ )) ≤ γ (α) 2 of the function γ we obtain γ (απ(ξ )) ≤ γ (α( f jr , δ)). Using the property of additiveboundedness of the function γ and taking into account the choice of the number σ jr we obtain 2γ (α( f jr , δ)) ≤ γ (α( f jr , δ)) + γ (α( f jr , σ jr )) ≤ γ (α). Hence γ (απ(ξ )) ≤ γ (α) . 2 Let r ≥ 2. We now show that γ (απ(ξ )) ≤ γ (α) . From the property of additiver boundedness of the function γ it follows that, for i = 0, . . . , r − 2, the inequality γ (ααi+1 ) + γ (ααi ( f ji+1 , δ ji+1 )) ≤ γ (ααi ) holds. Summing these inequalities over i from 0 to r − 2 we obtain γ (ααr −1 ) +

r −2 

γ (ααi ( f ji+1 , δ ji+1 )) ≤ γ (α) .

(5.3)

i=0

Let us show that, for any i ∈ {0, . . . , r − 2}, γ (απ(ξ )) ≤ γ (ααi ( f ji+1 , δ ji+1 )) .

(5.4)

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The inequality γ (ααi ( f jr , δ)) ≤ γ (ααi ( f ji+1 , δ ji+1 )) follows from the choice of the element f ji+1 (see the description of the schema Nρh (T, α, γ )) and the definition of the number δ ji+1 . The inequality γ (απ(ξ )) ≤ γ (ααi ( f jr , δ)) is a corollary of the property of nondecrease of the function γ . The inequality (5.4) follows from these two inequalities. The inequality γ (απ(ξ )) ≤ γ (ααr −1 ) is a corollary of the property of nondecrease of the function γ . From this inequality, from (5.3), and (5.4) the inequality r γ (απ(ξ )) ≤ γ (α) follows. Taking into account that r ≥ 2 we obtain .  γ (απ(ξ )) ≤ γ (α) r Corollary 5.4 Let ρ be a signature, T ∈ Mρ , and γ be an additive-bounded uncertainty measure for the table T . Then the schema Yρ,N ρh (T, γ ) is a deterministic decision tree for the table T . Proof Using the definition of the mapping Nρh and Lemma 5.3 one can show that Nρh is an admissible mapping. By Proposition 5.3, the schema Yρ,N ρh (T, γ ) is a deterministic decision tree for the table T .  Theorem 5.2 Let ρ = (F, k) be a signature, T ∈ Mρ , and γ be an additivebounded uncertainty measure for the table T . Then

h dρ (T )

⎧ ⎨

Mρ,h (T ), if Mρ,h (T ) ≤ 1 , γ (λ) + M (T ), if 2 ≤ Mρ,h (T ) ≤ 3 , 2 log ρ,h ≤ 2 ⎩ Mρ,h (T ) log2 γ (λ) + Mρ,h (T ), if Mρ,h (T ) ≥ 4 . log Mρ,h (T ) 2

Proof In the case Mρ,h (T )≤1, the statement of the theorem follows from Lemma 4.5. Let Mρ,h (T ) ≥ 2. From this inequality it follows that T ∈ / Mρ C . Let us consider an arbitrary complete path ξ in the schema Yρ,N ρh (T, γ ). From the description of / Mρ C it follows that, for some m ∈ the process Yρ,N ρh and from the relation T ∈ ω \ {0}, the equality π(ξ ) = π(ξ1 ) · · · π(ξm ) holds, where ξ1 is a complete path in the schema Nρh (T, λ, γ ) and if m ≥ 2, then ξi is a complete path in the schema 2, . . . , m. For i = 1, . . . , m, set ri = h(π(ξi )). Nρh (T, π(ξ1 ) · · · π(ξi−1 ), γ ) for i =  m ri . We now show that Let us estimate the value h(π(ξ )) = i=1 m  i=1

ri ≤

2 log2 γ (λ) + Mρ,h (T ), Mρ,h (T ) log2 γ (λ) + Mρ,h (T ), log2 Mρ,h (T )

if 2 ≤ Mρ,h (T ) ≤ 4 , if Mρ,h (T ) ≥ 4 .

(5.5)

/ Mρ C and from the Let m = 1. By Lemma 5.3, r1 ≤ Mρ,h (T ). From the relation T ∈ property of partial coordination of the function γ it follows that γ (λ) ≥ 1. Therefore if m = 1, then the inequality (5.5) holds. Let m ≥ 2. For i = 1, . . . , m, set z i = / Mρ C . From the description of the process Yρ,N ρh max{2, ri }. By assumption, T λ ∈ it follows that T π(ξ1 ) · · · π(ξi ) ∈ / Mρ C for i = 1, . . . , m − 1. Using Lemma 5.3 and the inequality m ≥ 2 we obtain γ (λ) γ (π(ξ1 ) · · · π(ξm−1 )) ≤ m−1 . i=1 z i

5.5 Upper Bounds on h dρ (T )

77

Using the property of partial coordination of the function γ and the relation / Mρ C T π(ξ1 ) · · · π(ξm−1 ) ∈ m−1 we obtain γ (π(ξ1 ) · · · π(ξm−1 )) ≥ 1. Hence i=1 z i ≤ γ (λ). Let us take the binary m−1 logarithm of both sides of this inequality. As a result, we have i=1 log2 z i ≤ log2 γ (λ). From this inequality it follows that m 

≤ rm +

m−1  i=1

i=1



log2 z i max 

ri log2 z i

m−1 

ri log2 z i i=1  : i ∈ {1, . . . , m − 1}

ri = rm +

log2 z i

ri : i ∈ {1, . . . , m − 1} ≤ rm + log2 γ (λ) max log2 z i

(5.6)

 .

x Let us consider the function q(x) = log (max{2,x}) defined on the set of real num2 bers. One can show that q(1) = 1, q(2) = 2, q(3) < 2, q(4) = 2 and, for x ≥ 3, the function q is a monotone increasing function. Therefore, for any n ∈ ω \ {0},

⎧ if n = 1 , ⎨ 1, 2, if 2 ≤ n ≤ 4 , max{q(i) : i ∈ {1, . . . , n}} = ⎩ n , if n ≥ 4 . log n

(5.7)

2

Using Lemma 5.3 we conclude that, for i = 1, . . . , m, ri ≤ Mρ,h (T ) .

(5.8)

From (5.7), (5.8) and the inequality Mρ,h (T ) ≥ 2 it follows that  max

 ri : i ∈ {1, . . . , m − 1} ≤ max{q(i) : i ∈ {1, . . . , Mρ,h (T )}} (5.9) log2 z i

2, if Mρ,h (T ) ≤ 4 , = Mρ,h (T ) , if Mρ,h (T ) ≥ 4 . log Mρ,h (T ) 2

The inequality (5.5) follows from (5.6), (5.8), and (5.9). By Corollary 5.4, the schema Yρ,N ρh (T, γ ) is a deterministic decision tree for the table T . Taking into account that ξ is an arbitrary complete path in the schema Yρ,N ρh (T, γ ) and using (5.5) we conclude that the statement of the theorem also holds if Mρ,h (T ) ≥ 2.  We now look in more detail at the bounds on the value h dρ (T ) depending on Mρ,h (T ) and K ρ (T ), and depending on Mρ,h (T ) and Nρ (T ).

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Lemma 5.4 Let ρ = (F, k) be a signature, T ∈ Mρ , and Δ(T ) = ∅. Then h dρ (T ) ≤ Nρ (T ) − 1. Proof We prove the statement of the lemma by induction on the value Nρ (T ). If Nρ (T ) = 1, then, evidently, h dρ (T ) = 0. Hence if Nρ (T ) = 1, then the statement of the lemma holds. Let n ∈ ω, n ≥ 2 and, for any table T ∈ Mρ such that Δ(T ) = ∅ and Nρ (T ) < n, the statement of the lemma hold. Assume now that T ∈ Mρ and Nρ (T ) = n. Let us choose an element f i ∈ P(T ) such that the cardinality of the set E( f i ) = {δ : δ ∈ E k , Δ(T ( f i , δ)) = ∅} is at least 2. The existence of such element follows from the inequality Nρ (T ) ≥ 2. Let δ ∈ E( f i ). Then, evidently, Nρ (T ( f i , δ)) < n. From this inequality, from the inductive hypothesis, and from Lemma 3.1 it follows that there exists a deterministic decision tree Γδ for the table T ( f i , δ) such that h(Γδ ) ≤ n − 2. We denote by Γδ the labeled finite directed tree with the root which is obtained from the schema Γδ by removal of the root and the edge leaving the root. Let us consider a schema Γ which is obtained from the trees Γδ , δ ∈ E( f i ), in the following way. We denote by w the node which the edge enters leaving the root of Γ . The node w is labeled with the element f i and, for each δ ∈ E( f i ), an edge leaves w which is labeled with the number δ and which enters the root of the tree Γδ . There are no any other edges leaving the node w. One can show that Γ is a deterministic decision tree for the table T and h(Γ ) ≤ n − 1. Thus,  h dρ (T ) ≤ n − 1. Corollary 5.5 Let ρ be a signature, T ∈ Mρ , and Δ(T ) = ∅. Then Mρ,h (T ) ≤ Nρ (T ) − 1. Proof The considered statement follows immediately from Lemma 5.4 and Theorem 3.1.  Lemma 5.5 Let ρ be a signature and T ∈ Mρ . Then K ρ (T ) = 0 if and only if Mρ,h (T ) = 0. Proof Let K ρ (T ) = 0. Using Proposition 5.1 we conclude that the function T ∗ K ρ has the property of partial coordination. Therefore T ∈ Mρ C . Hence Mρ,h (T ) = 0. Let Mρ,h (T ) = 0. Then, as it is not difficult to show, T ∈ Mρ C . If Δ(T ) = ∅, then K ρ (T ) = 0 by definition of the function K ρ . Let Δ(T ) = ∅. Then any schema  from Cρ0 (T ) contains exactly two nodes. Therefore K ρ (T ) = 0. Let ρ be a signature, A be a nonempty subset of the set Mρ and f, g be functions from A to ω. Then B( f, g, A) = {( f (T ), g(T )) : T ∈ A}. For a real number a, we denote by a the maximum integer which is at most a. Simple analysis of Theorem 2.4 from [2] and its proof, and also the use of Corollary 5.5 and Lemma 5.5 allow us to prove the following statement. Theorem 5.3 Let ρ = (F, k) be a signature such that the set F is infinite and A ∈ {Mρ } ∪ {Mρ (m) : m ∈ ω \ {0}}. Then (a) B(Mρ,h , K ρ , A) = {(0, 0)} ∪ {(n, r ) : n ∈ ω \ {0}, r ∈ ω \ {0}} and, for any pair (n, r ) ∈ B(Mρ,h , K ρ , A), there exist tables T1 (n, r ) and T2 (n, r ) from A such

5.5 Upper Bounds on h dρ (T )

79

that Mρ,h (T1 (n, r )) = Mρ,h (T2 (n, r )) = n, h dρ (T1 (n, r )) = n and

h dρ (T2 (n, r )) ≥

K ρ (T1 (n, r )) = K ρ (T2 (n, r )) = r ,

⎧ ⎪ ⎨

n, if 0 ≤ n ≤ 1 , 1 + log r, if n = 2 , 2   ⎪ log r 2 ⎩ (n − 2) + n − 1 , if n ≥ 3. log (n−1) 2

(b) B(Mρ,h , Nρ , A) = {(0, 0)} ∪ {(n, r ) : n ∈ ω, r ∈ ω \ {0}, r ≥ n + 1} and, for any pair (n, r ) ∈ B(Mρ,h , Nρ , A), there exist tables T3 (n, r ) and T4 (n, r ) from A such that Mρ,h (T3 (n, r )) = Mρ,h (T4 (n, r )) = n, Nρ (T3 (n, r )) = r , Nρ (T4 (n, r )) = r , h dρ (T3 (n, r )) = n and

h dρ (T4 (n, r ))

≥

⎧ ⎪ ⎨ ⎪ ⎩ (n − 2)



n, log2 r, 

log2 r −log2 n log2 (n−1)

if 0 ≤ n ≤ 1 , if n = 2 , + n − 1,

if n ≥ 3 .

In the following three remarks we assume that ρ = (F, k) is a signature with an infinite set F. Remark 5.2 Let A ∈ {Mρ } ∪ {Mρ (m) : m ∈ ω \ {0}} and ϕ ∈ {Nρ , K ρ }. From Proposition 5.1, Theorems 5.2 and 5.3 it follows that, for any table T ∈ A, h dρ (T )

⎧ ⎨

Mρ,h (T ), if Mρ,h (T ) ≤ 1 , ϕ(T ) + M (T ), if 2 ≤ Mρ,h (T ) ≤ 3 , 2 log ρ,h ≤ 2 ⎩ Mρ,h (T ) log2 ϕ(T ) + Mρ,h (T ), if Mρ,h (T ) ≥ 4 log Mρ,h (T ) 2

and, for tables from A, this bound is close to unimprovable. Remark 5.3 Let A ∈ {Mρ } ∪ {Mρ (m) : m ∈ ω \ {0}} and ϕ ∈ {Nρ , K ρ }. From Theorems 3.1 and 5.3 it follows that h dρ (T ) ≥ Mρ,h (T ) for any table T ∈ A and, for tables from A, this bound is the unimprovable lower bound on the value h dρ (T ) depending on values Mρ,h (T ) and ϕ(T ). Remark 5.4 Let A ∈ {Mρ } ∪ {Mρ (m) : m ∈ ω \ {0}}. From Theorem 5.3 it follows that there is no function g : ω → ω such that h dρ (T ) ≤ g(Mρ,h (T )) for any table T ∈ A. Comparison of this result with Corollary 4.4 shows that the functions Mˆ ρ,h (T ) and Mρ,h (T ) are very different.

5.6 Algorithm Yρ,η,ϕ,ψ Let ρ = (F, k) be an enumerated signature, η : Mρ → ω be a computable function such that, for any table T ∈ Mρ , the function η ∗ T is an additive-bounded uncertainty measure for the table T , ϕ ∈ {G Hρ , Hρ2 , R Hρ2 }, and ψ be a computable complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3.

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Denote Dρ (η) = {(T, T ∗ η) : T ∈ Mρ }. We now define a mapping Nη,ϕ,ψ : Z˜ ρ (Dρ (η)) → Cρd . To this end, we describe an algorithm which, for an arbitrary / Mρ C , constructs the schema pair (T, α), where T ∈ Mρ , α ∈ Ωρ (T ) and T α ∈ Nη,ϕ,ψ (T, α, T ∗ η). Mapping Nη,ϕ,ψ . Let dim T = n and μT (1) = fl1 , . . . , μT (n) = fln . For each f i ∈ P(T ), we denote by σi the minimum number from E k such that T ∗ η(α( f i , σi )) = max{T ∗ η(α( f i , σ )) : σ ∈ E k }. Set σ¯ = (σ1 , . . . , σn ). Let us apply the algorithm Wρ,ϕ,ψ to the pair (T α, σ¯ ). Set β = Wρ,ϕ,ψ (T α, σ¯ ). Evidently, β = λ. Let β = ( f i1 , σi1 ) · · · ( f im , σim ). We now construct a labeled finite directed tree with the root Γ of the following type. There is a path w0 , d0 , w1 , . . . , wm−1 , dm−1 , wm in Γ in which w0 is the root of Γ and, for j = 0, . . . , m − 1, the edge d j leaves the node w j and enters the node w j+1 . The node w0 and the edge d0 are not labeled. For j = 1, . . . , m, the node w j is labeled with the element f i j . For j = 1, . . . , m − 1, the edge d j is labeled with the number σi j . Set α0 = λ and, for j = 1, . . . , m − 1, set α j = ( f i1 , σi1 ) · · · ( f i j , σi j ). For j = 1, . . . , m − 1 for each δ ∈ E k \ {σi j } such that Δ(T αα j−1 ( f i j , δ)) = ∅, an edge leaves the node w j which is labeled with the number δ. This edge enters a terminal node of Γ which is labeled with the number 0. For each δ ∈ E k such that Δ(T ααm−1 ( f im , δ)) = ∅, an edge leaves the node wm which is labeled with the number δ. This edge enters a terminal node of Γ which is labeled with the number 0. There are no any other nodes end edges in Γ . From the description of the algorithm Wρ,ϕ,ψ it follows that there exists δ ∈ E k such that Δ(T ααm−1 ( f im , δ)) = ∅. Therefore Γ ∈ Cρd . We denote by Nη,ϕ,ψ (T, α, T ∗ η) the schema Γ . Lemma 5.6 Let ρ = (F, k) be an enumerated signature, η : Mρ → ω be a computable function such that, for any table T ∈ Mρ , the function η ∗ T is an additivebounded uncertainty measure for the table T , ϕ ∈ {G Hρ , Hρ2 , R Hρ2 }, ψ be a computable complexity function of the signature ρ having the properties Λ1, Λ2, and / Mρ C . Then the schema Nη,ϕ,ψ (T, α, T ∗ η) Λ3, T ∈ Mρ , α ∈ Ωρ (T ), and T α ∈ has at lest two complete paths, and, for any complete path ξ in this schema, the following statements hold: (a) Δ(T απ(ξ )) = ∅. (b) ψ(π(ξ )) ≤ (Mρ,ψ (T ))2 ln Nρ (T ) + Mρ,ψ (T ) and if ψ = h, then h(π(ξ )) ≤ Mρ,h (T ) ln Nρ (T ) + 1. (c) If T απ(ξ ) ∈ / Mρ C , then T ∗ η(απ(ξ )) ≤ T ∗η(α) . 2 Proof Let dim T = n and μT (1) = fl1 , . . . , μT (n) = fln . For each f i ∈ P(T ), we denote by σi the minimum number from E k such that T ∗ η(α( f i , σi )) = max{T ∗ η(α( f i , σ )) : σ ∈ E k }. Set σ¯ = (σ1 , . . . , σn ) and β = Wρ,ϕ,ψ (T α, σ¯ ). Evidently, β = λ. Let β = ( f i1 , σi1 ) · · · ( f im , σim ). Set α0 = λ, and, for j = 1, . . . , m − 1, set α j = ( f i1 , σi1 ) · · · ( f i j , σi j ). Let ξ be a complete path in the schema Nη,ϕ,ψ (T, α, T ∗ η). From the description of the algorithm constructing this schema it follows that Δ(T απ(ξ )) = ∅. One can show that π(ξ ) = β or, for some j ∈ {1, . . . , m}, the equality π(ξ ) = α j−1 ( f i j , δ)

5.6 Algorithm Yρ,η,ϕ,ψ

81

holds, where δ = σi j . Taking into account that the function ψ has the property Λ2 we obtain ψ(π(ξ )) ≤ ψ(β). Using Theorem 4.7 and the relation T α ∈ / Mρ C we conclude that ψ(β) ≤ (Mρ,ψ (T α, σ¯ ))2 ln Nρ (T α) + Mρ,ψ (T α, σ¯ ) and if ψ = h, then h(β) ≤ Mρ,h (T α, σ¯ ) ln Nρ (T α) + 1. Evidently, Nρ (T α) ≤ Nρ (T ). One can show that Mρ,ψ (T α, σ¯ ) ≤ Mρ,ψ (T, σ¯ ). Using Lemma 3.5 we obtain Mρ,ψ (T, σ¯ ) ≤ Mρ,ψ (T ). Hence ψ(π(ξ )) ≤ (Mρ,ψ (T ))2 ln Nρ (T ) + Mρ,ψ (T ) and if ψ = h, then h(π(ξ )) ≤ Mρ,h (T ) ln Nρ (T ) + 1. Let T απ(ξ ) ∈ / Mρ C . Then π(ξ ) = β and, hence, for some j ∈ {1, . . . , m}, the equality π(ξ ) = α j−1 ( f i j , δ) holds, where δ = σi j . Using the property of nondecrease of the function T ∗ η we obtain T ∗ η(απ(ξ )) ≤ T ∗ η(α( f i j , δ)). Using the property of additive-boundedness of the function T ∗ η and taking into account the choice of the number σi j we obtain 2T ∗ η(α( f i j , δ)) ≤ T ∗ η(α( f i j , δ)) + T ∗ η(α( f i j , σi j )) ≤ T ∗ η(α) . . Thus, T ∗ η(απ(ξ )) ≤ T ∗η(α) 2 Let ξ be a complete path in the schema Nη,ϕ,ψ (T, α, T ∗ η). By the above, T απ(ξ ) ∈ Mρ C or η(T απ(ξ )) ≤ η(T2 α) . Since T α ∈ / Mρ C , we obtain T απ(ξ ) = T α. From the description of the algorithm for construction of the schema Nη,ϕ,ψ (T, α, T ∗ η) it follows that, for any δ¯ ∈ Δ(T α), there is a complete path τ in the considered schema such that δ¯ ∈ Δ(T απ(τ )). Hence the schema Nη,ϕ,ψ (T, α, T ∗ η) has at least two complete paths.  Corollary 5.6 Let ρ = (F, k) be an enumerated signature, η : Mρ → ω be a computable function such that, for any table T ∈ Mρ , the function T ∗ η is an additivebounded uncertainty measure for the table T , ϕ ∈ {G Hρ , Hρ2 , R Hρ2 }, and ψ be a computable complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3. Then, for any table T ∈ Mρ for the pair (T, T ∗ η), the process Yρ,N η,ϕ,ψ ends after a finite number of steps. The schema Yρ,N η,ϕ,ψ (T, T ∗ η) constructed by the process Yρ,N η,ϕ,ψ is a deterministic decision tree for the table T . Proof Using the definition of the mapping Nη,ϕ,ψ and Lemma 5.6 one can show that Nη,ϕ,ψ is an admissible mapping. Using Proposition 5.3 we obtain the considered statement.  Let us describe an algorithm Yρ,η,ϕ,ψ which, for an arbitrary table T ∈ Mρ , constructs a schema Yρ,η,ϕ,ψ (T ) that is a deterministic decision tree for the table T . Algorithm Yρ,η,ϕ,ψ . For the table T , the algorithm Yρ,η,ϕ,ψ makes the same steps as the process Yρ,N η,ϕ,ψ for the pair (T, T ∗ η). Using Corollary 5.6 we conclude that the algorithm Yρ,η,ϕ,ψ ends after a finite number of steps and constructs a schema Yρ,η,ϕ,ψ (T ) = Yρ,N η,ϕ,ψ (T, T ∗ η) which

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is a deterministic decision tree for the table T . Let us consider upper bounds on complexity of the schema Yρ,η,ϕ,ψ (T ). Theorem 5.4 Let ρ = (F, k) be an enumerated signature, η : Mρ → ω be a computable function such that, for any table T ∈ Mρ , the function T ∗ η is an additivebounded uncertainty measure for the table T , ϕ be an arbitrary function from the set {G Hρ , Hρ2 , R Hρ2 }, ψ be a computable complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3, and T ∈ Mρ . Then ψ(Yρ,η,ϕ,ψ (T ))  ≤

ψ(λ), if T ∈ Mρ C , / Mρ C . (Mρ,ψ (T ))2 ln Nρ (T ) + Mρ,ψ (T ))(1 + log2 η(T )), if T ∈

Let ψ = h. Then  h(Yρ,η,ϕ,h (T )) ≤

0, if T ∈ Mρ C , / Mρ C . (Mρ,h (T ) ln Nρ (T ) + 1)(1 + log2 η(T )), if T ∈

Proof The upper bounds for the case T ∈ Mρ C follow immediately from the description of the algorithm Yρ,η,ϕ,ψ . Let T ∈ / Mρ C . We denote by Γ the schema Yρ,η,ϕ,ψ (T ). Let us consider an arbitrary complete path ξ in the schema Γ . From the description of the algorithm / Mρ C it follows that, for some m ∈ ω \ {0}, Yρ,η,ϕ,ψ and from the relation T ∈ the equality π(ξ ) = π(ξ1 ) · · · π(ξm ) holds, where ξ1 is a complete path in the schema Nη,ϕ,ψ (T, λ, T ∗ η) and if m ≥ 2, then ξi is a complete path in the schema / Mρ C . Nη,ϕ,ψ (T, π(ξ1 ) · · · π(ξi−1 ), T ∗ η) for i = 2, . . . , m. By assumption, T λ ∈ / From the description of the algorithm Yρ,η,ϕ,ψ it follows that T π(ξ1 ) · · · π(ξi−1 ) ∈ Mρ C for i = 2, . . . , m. Using the part (c) of the statement of Lemma 5.6 we obtain T ∗ η(π(ξ1 ) · · · π(ξm−1 )) ≤ T2∗η(λ) / m−1 . Taking into account that T π(ξ1 ) · · · π(ξm−1 ) ∈ Mρ C and using the property of partial coordination of the function T ∗ η we obtain T ∗ η(π(ξ1 ) · · · π(ξm−1 )) ≥ 1. Hence 2m−1 ≤ T ∗ η(λ). Therefore m ≤ 1 + log2 η(T ) .

(5.10)

Using the property Λ1 of the function ψ we obtain ψ(π(ξ )) ≤

m 

ψ(π(ξi )) .

(5.11)

i=1

Using the part (b) of the statement of Lemma 5.6 we conclude that, for i = 1, . . . , m, ψ(π(ξi )) ≤ (Mρ,ψ (T ))2 ln Nρ (T ) + Mρ,ψ (T ) , and if ψ = h, then

h(π(ξi )) ≤ Mρ,h (T ) ln Nρ (T ) + 1 .

5.6 Algorithm Yρ,η,ϕ,ψ

83

From these inequalities, from (5.10) and from (5.11) it follows that ψ(π(ξ )) ≤ ((Mρ,ψ (T ))2 ln Nρ (T ) + Mρ,ψ (T ))(1 + log2 η(T )) , and if ψ = h, then h(π(ξ )) ≤ (Mρ,h (T ) ln Nρ (T ) + 1)(1 + log2 η(T )) . Taking into account that ξ is an arbitrary complete path in the schema Γ we conclude  that the statement of the theorem holds in the case T ∈ / Mρ C too. Let us estimate the number of steps which the algorithm Yρ,η,ϕ,ψ makes during the schema Yρ,η,ϕ,ψ (T ) construction. Theorem 5.5 Let ρ = (F, k) be an enumerated signature, η : Mρ → ω be a computable function such that, for any table T ∈ Mρ , the function T ∗ η is an additivebounded uncertainty measure for the table T , ϕ be an arbitrary function from the set {G Hρ , Hρ2 , R Hρ2 }, ψ be a computable complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3, and T ∈ Mρ . Then during the schema Yρ,η,ϕ,ψ (T ) construction the algorithm Yρ,η,ϕ,ψ makes at most 2Nρ (T ) + 1 steps. Proof Let T ∈ Mρ C . From the description of the algorithm Yρ,η,ϕ,ψ it follows that during the construction of the schema Yρ,η,ϕ,ψ (T ) it makes exactly one step. Therefore the statement of the theorem holds if T ∈ Mρ C . Let T ∈ / Mρ C . We denote by Γ the schema Yρ,η,ϕ,ψ (T ). From the description of the algorithm Yρ,η,ϕ,ψ and from the part (a) of the statement of Lemma 5.6 it follows that Δ(T π(ξ )) = ∅ for any complete path ξ in the schema Γ . It is clear that Δ(T π(ξ1 )) ∩ Δ(T π(ξ2 )) = ∅ for any two different complete paths ξ1 and ξ2 in the schema Γ . Therefore the number of complete paths in the schema Γ is at most Nρ (T ). Let the algorithm Yρ,η,ϕ,ψ make n steps during the schema Yρ,η,ϕ,ψ (T ) construction. Since T ∈ / Mρ C , we have n ≥ 2. For each t ∈ {1, . . . , n − 1}, we remove from the schema Γ all edges which both leave and enter nodes constructed in the step t, and merge all nodes constructed in the step t. We denote the obtained finite directed tree with the root by G. One can show that exactly one edge leaves the root of G. Using Lemma 5.6 we conclude that at least two edges leave each node of G, which is neither the root nor a terminal node. One can show that L 0 (Γ ) = L 0 (G). Therefore L 0 (G) ≤ Nρ (T ). Using Lemma 4.6 we conclude that L(G) ≤ 2Nρ (T ). Finally, one can show that the number of steps which the algorithm Yρ,η,ϕ,ψ makes during the  construction of the schema Yρ,η,ϕ,ψ (T ) coincides with the number L(G) + 1. Remark 5.5 Let ρ = (F, k) be an enumerated signature, η : Mρ → ω be a computable function such that, for any table T ∈ Mρ , the function T ∗ η is an additivebounded uncertainty measure for the table T , ϕ be an arbitrary function from the set {G Hρ , Hρ2 , R Hρ2 }, and ψ be a computable complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3. Let there exist a polynomial time algorithm for the computation of the function ψ on the set F ∗ , and there exist a polynomial

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time algorithm for the computation of the function η on the set Mρ . By Remark 4.11, the algorithm Wρ,ϕ,ψ has polynomial time complexity. Using this fact and Theorem 5.5 one can show that the algorithm Yρ,η,ϕ,ψ has polynomial time complexity. In particular, if η ∈ {Nρ , Jρ } and ψ = h, then the algorithm Yρ,η,ϕ,ψ has polynomial time complexity.

References 1. Moshkov, M.: On conditional tests. Sov. Phys. Dokl. 27, 528–530 (1982) 2. Moshkov, M.: Conditional tests. In: Yablonskii, S.V. (ed.) Problemy Kibernetiki (in Russian), vol. 40, pp. 131–170. Nauka Publishers, Moscow (1983) 3. Moshkov, M.: Bounds on complexity and algorithms for construction of deterministic conditional tests. In: Mat. Vopr. Kibern. (in Russian), vol. 16, pp. 79–124. Fizmatlit, Moscow (2007). http:// library.keldysh.ru/mvk.asp?id=2007-79

Chapter 6

Bounds on Complexity and Algorithms for Construction of Nondeterministic and Strongly Nondeterministic Decision Trees for Decision Tables

In this chapter, we consider bounds on the minimum complexity, an approach to proof of lower bounds, and algorithms for construction of nondeterministic and strongly nondeterministic decision trees. The bounds on complexity are true for arbitrary complexity functions. The approach to proof of lower bounds assumes that the complexity functions have the property Λ2. The considered algorithms require complexity functions having properties Λ1, Λ2, and Λ3. Some similar results were published in [1] for decision tables that represent Boolean functions.

6.1 Bounds on ψρa (T ) and ψρs (T ) Let ρ = (F, k) be a signature, T ∈ Mρ , α ∈ Ωρ (T ), and T α ∈ Mρ C . We now define a schema G(α, T ) of the signature ρ. Let α = λ. The schema G(λ, T ) consists of two nodes w0 and w1 and the edge which leaves w0 and enters w1 . If Δ(T ) = ∅, then the node w1 is labeled with the number 0. If Δ(T ) = ∅, then the node w1 is labeled with the minimum number from the set Π (T ). Let α = λ and α = ( f 1 , δ1 ) · · · ( f m , δm ). Then the schema G(α, T ) consists of nodes w0 , w1 , . . . , wm+1 and edges d0 , . . . , dm . For i = 0, . . . , m, the edge di leaves the node wi and enters the node wi+1 . For i = 1, . . . , m, the node wi is labeled with the element f i , and the edge di is labeled with the number δi . If Δ(T ) = ∅, then the node wm+1 is labeled with the number 0. If Δ(T α) = ∅, then the node wm+1 is labeled with the minimum number from the set Π (T α). Theorem 6.1 Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ, and T ∈ Mρ . Then the value ψρa (T ) is definite and if Δ(T ) = ∅, then ¯ : δ¯ ∈ Δ(T )} . ψρa (T ) = max{Mρ,ψ (T, δ) © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_6

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Proof Let dim T = n and μT (1) = f 1 , . . . , μT (n) = f n . If Δ(T ) = ∅, then the schema G(λ, T ) is a nondeterministic decision tree for the table T . Therefore the value ψρa (T ) is definite. Let Δ(T ) = ∅. For each δ¯ = (δ1 , . . . , δn ) ∈ Δ(T ), ¯ from Ωρ (T ) having the following properties: χ (α(δ)) ¯ ⊆ we choose a word α(δ) ¯ ¯ ¯ {( f 1 , δ1 ), . . . , ( f n , δn )}, T α(δ) ∈ Mρ C , and ψ(α(δ)) = Mρ,ψ (T, δ). The existence of such word follows from Lemma 3.5. We denote by G the schema of the signature ¯ T ) : δ¯ ∈ Δ(T )}. ρ obtained by merging of roots of all schemes from the set {G(α(δ), One can show that G is a nondeterministic decision tree for the table T such that ¯ : δ¯ ∈ Δ(T )}. Hence the value ψρa (T ) is definite, and the ψ(G) = max{Mρ,ψ (T, δ) following inequality holds: ¯ : δ¯ ∈ Δ(T )} . ψρa (T ) ≤ max{Mρ,ψ (T, δ)

(6.1)

Since the value ψρa (T ) is definite, there exists a nondeterministic decision tree Γ for the table T such that ψ(Γ ) = ψρa (T ). Let δ¯ = (δ1 , . . . , δn ) ∈ Δ(T ). Then there exists a complete path ξ in the schema Γ such that δ¯ ∈ Δ(T π(ξ )) and T π(ξ ) ∈ Mρ C . Evidently, the word π(ξ ) has the following properties: χ (π(ξ )) ⊆ ¯ and {( f 1 , δ1 ), . . . , ( f n , δn )} and T π(ξ ) ∈ Mρ C . Therefore ψ(π(ξ )) ≥ Mρ,ψ (T, δ) ¯ Taking into account that δ¯ is an arbitrary tuple from Δ(T ) ψ(Γ ) ≥ Mρ,ψ (T, δ). ¯ : δ¯ ∈ Δ(T )}. From this and ψ(Γ ) = ψρa (T ) we obtain ψρa (T ) ≥ max{Mρ,ψ (T, δ) a ¯ : δ¯ ∈ Δ(T )}.  inequality and from (6.1) it follows that ψρ (T ) = max{Mρ,ψ (T, δ) Corollary 6.1 Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3, and T ∈ Mρ F . Then  ≤

max{ψρa (T ), logk (Θρ,h (T ) + 1)} ≤ ψρd (T ) ψ(λ), if T ∈ Mρ C , / Mρ C . 2(ψρa (T ))3 (ln Θρ,h (T ) + ln k) + ψρa (T ), if T ∈

and  ≤

max{h aρ (T ), logk (Θρ,h (T ) + 1)} ≤ h dρ (T ) h aρ (T ), a a h ρ (T )(2h ρ (T )(ln Θρ,h (T ) + ln k)

−

ln h aρ (T )

if h aρ (T ) ≤ 1 , + 1), if h aρ (T ) ≥ 2 .

Proof Since T ∈ Mρ F , we have Δ(T ) = E kdim T . From this equality, from Lemma 3.5 and from Theorem 6.1 it follows that ψρa (T ) = Mρ,ψ (T ) and, in particular, h aρ (T ) = Mρ,h (T ). Using these equalities, Proposition 4.5 and Corollary 4.5 we obtain the considered statement.  ¯ = {1}}. For an arbitrary table T ∈ Mρ0−1 , set Δ(T, 1) = {δ¯ : δ¯ ∈ Δ(T ), νT (δ) Theorem 6.2 Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ and T ∈ Mρ0−1 . Then the value ψρs (T ) is definite, and the following equality holds:

6.1 Bounds on ψρa (T ) and ψρs (T )

87

¯ : δ¯ ∈ Δ(T, 1)} . ψρs (T ) = max{Mρ,ψ (T, δ) Proof Let dim T = n and μT (1) = f 1 , . . . , μT (n) = f n . For each δ¯ = (δ1 , . . . , δn ) ¯ from Ωρ (T ) having the following properties: ∈ Δ(T, 1), choose a word α(δ) ¯ ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )} , χ (α(δ)) ¯ = Mρ,ψ (T, δ). ¯ The existence of such word follows ¯ ∈ Mρ C , and ψ(α(δ)) T α(δ) from Lemma 3.5. We denote by G the schema of the signature ρ obtained by merging ¯ T ) : δ¯ ∈ Δ(T, 1)}. One can show that of roots of all schemes from the set {G(α(δ), G is a strongly nondeterministic decision tree for the table T such that ψ(G) = ¯ : δ¯ ∈ Δ(T, 1)}. Hence the value ψρs (T ) is definite, and the following max{Mρ,ψ (T, δ) inequality holds: ¯ : δ¯ ∈ Δ(T, 1)} . ψρs (T ) ≤ max{Mρ,ψ (T, δ)

(6.2)

Since the value ψρs (T ) is definite, there exists a strongly nondeterministic decision tree Γ for the table T such that ψ(Γ ) = ψρs (T ). Let δ¯ = (δ1 , . . . , δn ) ∈ Δ(T, 1). Then there exists a complete path ξ in the schema Γ such that δ¯ ∈ Δ(T π(ξ )). Evidently, the word π(ξ ) has the following properties: χ (π(ξ )) ⊆ ¯ and {( f 1 , δ1 ), . . . , ( f n , δn )} and T π(ξ ) ∈ Mρ C . Therefore ψ(π(ξ )) ≥ Mρ,ψ (T, δ) ¯ Taking into account that δ¯ is an arbitrary tuple from Δ(T, 1) ψ(Γ ) ≥ Mρ,ψ (T, δ). ¯ : δ¯ ∈ Δ(T, 1)}. From this and ψ(Γ ) = ψρs (T ) we obtain ψρs (T ) ≥ max{Mρ,ψ (T, δ) s ¯ : δ¯ ∈ inequality and from (6.2) it follows that ψρ (T ) = max{Mρ,ψ (T, δ) Δ(T, 1)}.  Proposition 6.1 Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ, and T ∈ Mρ . Then the following statements hold: (a) ψρa (T ) ≤ ψρd (T ). (b) If T ∈ Mρ0−1 , then ψρs (T ) ≤ ψρa (T ) ≤ ψρd (T ). Proof Evidently, any deterministic decision tree for the table T is a nondeterministic decision tree for the table T . Therefore ψρa (T ) ≤ ψρd (T ). Let T ∈ Mρ0−1 . From  Theorems 6.1 and 6.2 it follows that ψρs (T ) ≤ ψρa (T ).

6.2 Approach to Proof of Lower Bounds on ψρs (T ) and ψρa (T ) Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ having the property Λ2, T ∈ Mρ , dim T = n, μT (1) = f 1 , . . . , μT (n) = f n , and δ¯ = (δ1 , . . . , δn ) ∈ E kn . Section 3.4 contains the definition of a (P(T ), ρ)-tree. Let G be a (P(T ), ρ)-tree. We associate a word ζ (w) ∈ Ωρ (T ) with each node w of the tree G in the same way as it was described in Sect. 3.4. Let r ∈ ω. The tree G

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¯ ≥ r if it satisfies the following will be called a proof-tree for the bound Mρ,ψ (T, δ) conditions: (a) Each edge of the tree G is labeled with a pair from the set {( f 1 , δ1 ), . . . , ( f n , δn )}. (b) For any node w of the tree G, which is not terminal, T ζ (w) ∈ / Mρ C . (c) For any terminal node w of the tree G, the inequality ψ(ζ (w)) ≥ r holds. Proposition 6.2 Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ having the property Λ2, T ∈ Mρ , dim T = n, μT (1) = f 1 , . . . , μT (n) = ¯ ≥ f n , δ¯ = (δ1 , . . . , δn ) ∈ E kn , and r ∈ ω. Then a proof-tree for the bound Mρ,ψ (T, δ) ¯ ≥ r holds. r exists if and only if the inequality Mρ,ψ (T, δ) ¯ ≥ r . We associate the word Proof Let G be a proof-tree for the bound Mρ,ψ (T, δ) ζ (w) with each node w of the tree G as it was described in Sect. 3.4. Set ¯ = {β : β ∈ Ωρ (T ), χ (β) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}, B(δ) ¯ . Tβ ∈ Mρ C , ψ(β) = Mρ,ψ (T, δ)} ¯ = ∅. Let α be a word with the minimum length from the set By Lemma 3.5, B(δ) ¯ Let α = λ. Then T ∈ Mρ C . In this case the tree G consists of one node w B(δ). ¯ ≥ r . Let such that ζ (w) = λ and ψ(ζ (w)) ≥ r . Therefore ψ(α) ≥ r and Mρ,ψ (T, δ) α = λ and α = ( f i1 , δi1 ) · · · ( f im , δim ). Taking into account the choice of the word α and the property Λ2 of the function ψ we conclude that T α ∈ Mρ C and, for any l, j ∈ {1, . . . , m}, if l = j, then f il = f i j . Using the listed properties of the word α one can show that there exists a terminal node w of the tree G such that ζ (w) = ( f i1 , δi1 ) · · · ( f it , δit ) for some t ∈ {0, 1, . . . , m} (ζ (w) = λ if t = 0). Using the property Λ2 of the function ψ we obtain ψ(α) ≥ ψ(ζ (w)). Since w is a terminal ¯ ≥ r. node of the tree G, we have ψ(ζ (w)) ≥ r . Hence ψ(α) ≥ r and Mρ,ψ (T, δ) ¯ ≥ r hold. We denote by G a (P(T ), ρ)-tree in Let the inequality Mρ,ψ (T, δ) which each edge is labeled with a pair from the set {( f 1 , δ1 ), . . . , ( f n , δn )}, and each complete path contains exactly n + 1 nodes. We associate a word ζ (w) ∈ Ωρ (T ) with each node w of the tree G in the same way as it was described in Sect. 3.4. Let w be a node of the tree G. If T ζ (w) ∈ / Mρ C , then we leave the node w untouched. Let T ζ (w) ∈ Mρ C . If w is the root of G, then we leave the node w untouched. Let w be / Mρ C , not the root of G, and an edge enter w which leaves a node w1 . If T ζ (w1 ) ∈ then we leave the node w untouched. If T ζ (w1 ) ∈ Mρ C , then we remove the node w and the edge entering the node w. Let us treat in the same way all nodes of the tree G: first, terminal nodes, then the nodes for which all successors are treated, etc., right up to the root. As a result, we obtain a tree. We denote this tree by Γ . One can show that Γ is a (P(T ), ρ)-tree in which each edge is labeled with a pair from the set {( f 1 , δ1 ), . . . , ( f n , δn )}. We associate the word ζ (w) ∈ Ωρ (T ) with each node of Γ in the same way as it was described in Sect. 3.4. Let w be a node of the tree Γ which is not terminal. From the description of the process of the tree Γ construction it follows that T ζ (w) ∈ / Mρ C . Evidently, T ζ (w) ∈ Mρ C for any terminal node w of the tree G. Therefore T ζ (w) ∈ Mρ C for any terminal node w of

6.2 Approach to Proof of Lower Bounds on ψρs (T ) and ψρa (T )

89

the tree Γ . Let w be an arbitrary terminal node of the tree Γ . Then T ζ (w) ∈ Mρ C ¯ Taking and χ (ζ (w)) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}. Therefore ψ(ζ (w)) ≥ Mρ,ψ (T, δ). ¯ ≥ r we obtain ψ(ζ (w)) ≥ r . Hence the tree Γ is a into account that Mρ,ψ (T, δ) ¯ ≥ r. proof-tree for the bound Mρ,ψ (T, δ)  Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ having the property Λ2, T ∈ Mρ , Δ(T ) = ∅, G be a (P(T ), ρ)-tree, and r ∈ ω. The tree G will be called a proof-tree for the bound ψρa (T ) ≥ r if, for some δ¯ ∈ Δ(T ), the ¯ ≥ r. tree G is a proof-tree for the bound Mρ,ψ (T, δ) Theorem 6.3 Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ having the property Λ2, T ∈ Mρ , Δ(T ) = ∅, and r ∈ ω. A proof-tree for the bound ψρa (T ) ≥ r exists if and only if the inequality ψρa (T ) ≥ r holds. Proof Let G be a proof-tree for the bound ψρa (T ) ≥ r . Then, for some δ¯ ∈ Δ(T ), ¯ ≥ r . By Proposition 6.2, the tree G is a proof-tree for the bound Mρ,ψ (T, δ) a ¯ Mρ,ψ (T, δ) ≥ r . Using Theorem 6.1 we obtain ψρ (T ) ≥ r . Let the inequality ψρa (T ) ≥ r hold. Using Theorem 6.1 we obtain that there exists ¯ ≥ r . Using Proposition 6.2 we conclude that there δ¯ ∈ Δ(T ) such that Mρ,ψ (T, δ) ¯ ≥ r . It is clear that G exists a tree G which is a proof-tree for the bound Mρ,ψ (T, δ)  is a proof-tree for the bound ψρa (T ) ≥ r . Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ having the property Λ2, T ∈ Mρ0−1 , G be a (P(T ), ρ)-tree, and r ∈ ω. The tree G will be called a proof-tree for the bound ψρs (T ) ≥ r if, for some δ¯ ∈ Δ(T, 1), the ¯ ≥ r. tree G is a proof-tree for the bound Mρ,ψ (T, δ) Theorem 6.4 Let ρ = (F, k) be a signature, ψ be a complexity function of the signature ρ having the property Λ2, T ∈ Mρ0−1 , and r ∈ ω. A proof-tree for the bound ψρs (T ) ≥ r exists if and only if the inequality ψρs (T ) ≥ r holds. Proof Let G be a proof-tree for the bound ψρs (T ) ≥ r . Then, for some δ¯ ∈ Δ(T, 1), ¯ ≥ r . By Proposition 6.2, the tree G is a proof-tree for the bound Mρ,ψ (T, δ) s ¯ Mρ,ψ (T, δ) ≥ r . Using Theorem 6.2 we obtain ψρ (T ) ≥ r . Let the inequality ψρs (T ) ≥ r hold. Using Theorem 6.2 we obtain that there exists ¯ ≥ r . By Proposition 6.2, there exists a tree G δ¯ ∈ Δ(T, 1) such that Mρ,ψ (T, δ) ¯ ≥ r . It is clear that G is a proof-tree which is a proof-tree for the bound Mρ,ψ (T, δ)  for the bound ψρs (T ) ≥ r .

a s 6.3 Algorithms Vρ,ϕ,ψ and Vρ,ϕ,ψ

Let ρ = (F, k) be an enumerated signature, ϕ be an arbitrary function from the set {G Hρ , Hρ2 , R Hρ2 }, and ψ be a complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3.

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a We now describe an algorithm Vρ,ϕ,ψ which, for a given table T ∈ Mρ , constructs a a schema Vρ,ϕ,ψ (T ) that is a nondeterministic decision tree for the table T . a . Algorithm Vρ,ϕ,ψ a (T ) consists of two nodes w1 , w2 and the edge which Let T ∈ Mρ C . Then Vρ,ϕ,ψ leaves w1 and enters w2 . If Δ(T ) = ∅, then the node w2 is labeled with the number 0. If Δ(T ) = ∅, then the node w2 is labeled with the minimum number from the set Π (T ). Let T ∈ / Mρ C . For each δ¯ ∈ Δ(T ), with the help of the algorithm Wρ,ϕ,ψ we ¯ For each δ¯ ∈ Δ(T ), we construct the schema ¯ = Wρ,ϕ,ψ (T, δ). construct the word α(δ) ¯ T ). Let us merge the roots of all schemes from the set {G(α(δ), ¯ T ) : δ¯ ∈ G(α(δ), a (T ). Δ(T )}. We denote the obtained schema by Vρ,ϕ,ψ s We now describe an algorithm Vρ,ϕ,ψ which, for a given table T ∈ Mρ0−1 , cons structs a schema Vρ,ϕ,ψ (T ) that is a strongly nondeterministic decision tree for the table T . s . Algorithm Vρ,ϕ,ψ s (T ) consists of two nodes w1 , w2 and the edge which Let T ∈ Mρ C . Then Vρ,ϕ,ψ leaves w1 and enters w2 . The node w2 is labeled with the number 1. Let T ∈ / Mρ C . For each δ¯ ∈ Δ(T, 1), with the help of the algorithm Wρ,ϕ,ψ ¯ = Wρ,ϕ,ψ (T, δ). ¯ For each δ¯ ∈ Δ(T, 1), we construct the we construct the word α(δ) ¯ T ). Let us merge the roots of all schemes from the set {G(α(δ), ¯ T) : schema G(α(δ), s (T ). δ¯ ∈ Δ(T, 1)}. We denote the obtained schema by Vρ,ϕ,ψ

Theorem 6.5 Let ρ = (F, k) be an enumerated signature, ϕ be an arbitrary function from the set {G Hρ , Hρ2 , R Hρ2 }, ψ be a computable complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3, T ∈ Mρ , and Δ(T ) = ∅. Then a (T ) is a nondeterministic decision tree for the table T and the schema Vρ,ϕ,ψ  a ψ(Vρ,ϕ,ψ (T )) ≤

ψ(λ), if T ∈ Mρ C , / Mρ C . (ψρa (T ))2 ln Nρ (T ) + ψρa (T ), if T ∈

If ψ = h, then  a h(Vρ,ϕ,h (T ))

≤

0, if T ∈ Mρ C , / Mρ C . h aρ (T ) ln Nρ (T ) + 1, if T ∈

a (T ) is a nonProof Using Theorem 4.7 one can show that the schema Vρ,ϕ,ψ deterministic decision tree for the table T . In the case T ∈ Mρ C , the consida . Let ered bounds follow immediately from the description of the algorithm Vρ,ϕ,ψ a (T ). T ∈ / Mρ C . We now consider an arbitrary complete path ξ in the schema Vρ,ϕ,ψ ¯ ¯ Evidently, π(ξ ) = Wρ,ϕ,ψ (T, δ) for some δ ∈ Δ(T ). Using Theorem 4.7 we con¯ 2 ln Nρ (T ) + Mρ,ψ (T, δ) ¯ and if ψ = h, then clude that ψ(π(ξ )) ≤ (Mρ,ψ (T, δ)) ¯ ¯ ≤ h(π(ξ )) ≤ Mρ,h (T, δ) ln Nρ (T ) + 1. Using Theorem 6.1 we obtain Mρ,ψ (T, δ)

a s 6.3 Algorithms Vρ,ϕ,ψ and Vρ,ϕ,ψ

91

¯ ≤ h aρ (T ). Using these inequalities and taking ψρa (T ) and, in particular, Mρ,h (T, δ) into account that ξ is an arbitrary complete path in the considered schema we obtain a a (T )) ≤ (ψρa (T ))2 ln Nρ (T ) + ψρa (T ) and if ψ = h, then h(Vρ,ϕ,h (T )) ≤ ψ(Vρ,ϕ,ψ a h ρ (T ) ln Nρ (T ) + 1.  Theorem 6.6 Let ρ = (F, k) be an enumerated signature, ϕ be an arbitrary function from the set {G Hρ , Hρ2 , R Hρ2 }, ψ be a computable complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3, and T ∈ Mρ0−1 . Then the schema s (T ) is a strongly nondeterministic decision tree for the table T and Vρ,ϕ,ψ  s ψ(Vρ,ϕ,ψ (T ))

≤

ψ(λ), if T ∈ Mρ C , / Mρ C . (ψρs (T ))2 ln Nρ (T ) + ψρs (T ), if T ∈

Let ψ = h. Then  s h(Vρ,ϕ,h (T )) ≤

0, if T ∈ Mρ C , / Mρ C . h sρ (T ) ln Nρ (T ) + 1, if T ∈

s (T ) is a strongly nondeterministic decision Proof By Theorem 4.7, the schema Vρ,ϕ,ψ tree for the table T . In the case T ∈ Mρ C , the considered bounds follow immediately s . from the description of the algorithm Vρ,ϕ,ψ Let T ∈ / Mρ C . We now consider an arbitrary complete path ξ in the schema s ¯ for some δ¯ ∈ Δ(T, 1). Using Theo(T ). Evidently, π(ξ ) = Wρ,ϕ,ψ (T, δ) Vρ,ϕ,ψ ¯ 2 ln Nρ (T ) + Mρ,ψ (T, δ) ¯ and rem 4.7 we conclude that ψ(π(ξ )) ≤ (Mρ,ψ (T, δ)) ¯ ln Nρ (T ) + 1. Using Theorem 6.2 we obtain if ψ = h, then h(π(ξ )) ≤ Mρ,h (T, δ) ¯ ≤ ψρs (T ) and, in particular, Mρ,h (T, δ) ¯ ≤ h sρ (T ). Using these inequalMρ,ψ (T, δ) ities and taking into account that ξ is an arbitrary complete path in the considered s (T )) ≤ (ψρs (T ))2 ln Nρ (T ) + ψρs (T ) and if ψ = h, then schema we obtain ψ(Vρ,ϕ,ψ s s  h(Vρ,ϕ,h (T )) ≤ h ρ (T ) ln Nρ (T ) + 1.

Remark 6.1 Let ρ = (F, k) be an enumerated signature, ϕ be an arbitrary function from the set {G Hρ , Hρ2 , R Hρ2 }, and ψ be a computable complexity function of the signature ρ having the properties Λ1, Λ2, and Λ3. Let there exist a polynomial time algorithm for the computation of the function ψ on the set F ∗ . By Remark 4.11, the algorithm Wρ,ϕ,ψ has polynomial time complexity. Using this fact one can show that a s and Vρ,ϕ,ψ have polynomial time complexity. In particular, the the algorithms Vρ,ϕ,ψ a s algorithms Vρ,ϕ,h and Vρ,ϕ,h have polynomial time complexity.

Reference 1. Moshkov, M.: About the depth of decision trees computing Boolean functions. Fundam. Inform. 22(3), 203–215 (1995)

Chapter 7

Closed Classes of Boolean Functions

In this chapter, for decision tables corresponding to functions from an arbitrary closed class of Boolean functions, the depth of deterministic, nondeterministic, and strongly nondeterministic decision trees is studied. The obtained results have some independent interest. Proofs of these results illustrate mainly methods for the proof of lower bounds on complexity of decision trees. The structure of all classes of Boolean functions closed relatively the operation of substitution was described by Post in [3, 4]. In [5], S.V. Yablonskii, G.P. Gavrilov, and V.B. Kudriavtzev considered the structure of all classes of Boolean functions closed relatively the operation of substitution and the operations of insertion and deletion of unessential variables. We study closed classes from this structure which is slightly different from the Post’s structure. The structure of closed classes considered in [5] contains, in Post’s notation, the following classes: Ci , Ai (Mi in [5]), i ∈ {1, 2, 3, 4}; D j , j ∈ {1, 2, 3}; L k , k ∈ {1, 2, 3, 4, 5}; St , Pt , t ∈ {1, 3, 5, 6}; Om , m ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9}; Flr , l ∈ {1, 2, 3, 4, 5, 6, 7, 8}, r ∈ {∞} ∪ {2, 3, 4, . . .}. Results considered in this chapter were published in [1]. Part of these results related to the deterministic decision trees was included into the monograph [2]. This monograph contains complete description of the structure of closed classes from [5]. In this chapter, we use some definitions, notation, and results from [5] (see also [2]).

7.1 Main Definitions and Notation Let V = {vi : i ∈ ω} be the set of variables. We will denote by ε2 the signature (V, 2). Let P2 be the set of Boolean functions with variables from V including constants 0 and 1. Let n ∈ ω \ {0}. We denote by P2 (n) the set of functions from P2 depending on n variables. Let f (vi1 , . . . , vin ) ∈ P2 (n). We correspond to the function f a table T ( f ) from Mε2 in the following way: dim T ( f ) = n, Δ(T ( f )) = E 2n , © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_7

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¯ = { f (δ)} ¯ for any δ¯ ∈ Δ(T ( f )). μT ( f ) ( j) = vi j for any j ∈ {1, . . . , n}, and νT ( f ) (δ) Let b ∈ {d, a, s}. We now define a function h b : P2 → ω. Let f ∈ P2 . If f ≡ const, then h b ( f ) = 0. If f ≡ const, then h b ( f ) = h bε2 (T ( f )). Let n ∈ ω \ {0}, f (vi1 , . . . , vin ) ∈ P2 (n), and j ∈ {1, . . . , n}. A variable vi j is called an essential variable of the function f if there exist two tuples δ¯ and σ¯ from ¯ = f (σ¯ ). If a E kn which are different only in jth component and for which f (δ) variable of the function f is not essential, then it is called an unessential variable of the function f . As in [5], we assume that together with a function f ∈ P2 all functions are given which can be obtained from f by insertion or deletion of some unessential variables. Let U be a nonempty set of Boolean functions. We denote by [U ] the closure of the set U relative to the operation of substitution. The set U will be called a closed class if U = [U ]. Let U be a closed class of Boolean functions. For n ∈ ω \ {0}, set U (n) = U ∩ P2 (n). Let b ∈ {d, a, s}. We now define a function h Ub : ω \ {0} → ω. Let n ∈ ω \ {0}. Then h Ub (n) = max{h b ( f ) : f ∈ U (n)}. We denote the negation by the symbol ¬: ¬v1 = 0 if v1 = 1, and ¬v1 = 1 if v1 = 0. Let n ∈ ω \ {0}. For t ∈ E 2 , we denote by t˜n the tuple (t, . . . , t) from E 2n . Let f ∈ P2 (n). The function f will be called an α-function if f (t˜n ) = t for any t ∈ E 2 , β-function if f (t˜n ) = 1 for any t ∈ E 2 , and γ -function if f (t˜n ) = 0 for any t ∈ E 2 . Let m ∈ ω \ {0, 1}. We will say that the function f satisfies the condition (a m ) if, for any m tuples from E 2n on which f has the value 0, there exists j ∈ {1, . . . , n} such that, in each of the considered m tuples, jth component is equal to 0. We will say that the function f satisfies the condition (a ∞ ) if there exists j ∈ {1, . . . , n} such that, in any tuple from E 2n on which f has the value 0, jth component is equal to 0. By definition, the constant 1 satisfies the condition (a ∞ ). By definition, the constant 0 does not satisfy the condition (a 2 ). A function f will be called a linear function if f = c0 + c1 x1 + · · · + cn xn , where ci ∈ E 2 , 0 ≤ i ≤ n, and + is the sum modulo 2. A function f will be called a self-dual function if f (x1 , . . . , xn ) = ¬ f (¬x1 , . . . , ¬xn ). A function f will be called a monotone function if, for any tuples δ¯ = (δ1 , . . . , δn ) and σ¯ = (σ1 , . . . , σn ) ¯ ≤ f (σ¯ ) holds. from E 2n such that δi ≤ σi , 1 ≤ i ≤ n, the inequality f (δ) Let us define some closed classes which will be used later: O1 = [{v0 }], O2 = [{1}], O3 = [{0}], O4 = [{¬v0 }], O5 = [{v0 , 1}], O6 = [{v0 , 0}], O7 = [{0, 1}], O8 = [{v0 , 0, 1}], O9 = [{¬v0 , 0}], S1 = [{v0 ∨ v1 }], S3 = [{v0 ∨ v1 , 1}], S5 = [{v0 ∨ v1 , 0}], S6 = [{v0 ∨ v1 , 0, 1}], P1 = [{v0 ∧ v1 }], L 2 is the set of all linear α- and β-functions, L 3 is the set of all linear α- and γ -functions, L 4 is the set of all linear α-functions, L 5 is the set of all linear self-dual functions, D1 is the set of all self-dual α-functions, D2 is the set of all self-dual monotone functions, D3 is the set of all self-dual functions. Let r ∈ {∞} ∪ {m : m ∈ ω \ {0, 1}}. Then F1r is the set of all α-functions satisfying the condition (a r ), F2r is the set of all monotone α-functions satisfying the condition (a r ), F3r is the set of all monotone functions satisfying the condition (a r ), and F4r is the set of all functions satisfying the condition (a r ).

7.1 Main Definitions and Notation

95

We now define some Boolean functions which will be used later. For n = 1, 2, . . ., let kn = kn (v1 , . . . , vn ) = v1 ∧ . . . ∧ vn , dn = dn (v1 , . . . , vn ) = v1 ∨ · · · ∨ vn , ln = ln (v1 , . . . , vn ) = v1 + · · · + vn , ¬ln = ¬ln (v1 , . . . , vn ) = v1 + · · · + vn + 1, and πn = πn (v1 , . . . , vn ) = v1 . For n = 2, 3, . . ., let m n = m n (v1 , . . . , vn ) be the function obtained from the function ln−1 by insertion of the unessential variable vn , and qn = qn (v1 , . . . , vn ) = v1 ∨ v2 ∧ · · · ∧ vn . For n = 3, 4, . . ., let rn = rn (v1 , . . . , vn ) = v1 ∧ (v2 ∨ · · · ∨ vn ) ∨ v2 ∧ · · · ∧ vn .

7.2 Auxiliary Statements Let us define a function E V : P2 → ω. Let f ∈ P2 . If f ≡ const, then E V ( f ) = 0. If f ≡ const, then E V ( f ) is the number of essential variables of the function f . Lemma 7.1 Let n ∈ ω \ {0} and f ∈ P2 (n). Then h s ( f ) ≤ h a ( f ) ≤ h d ( f ) ≤ E V ( f ) ≤ n. Proof If f ≡ const, then, evidently, the statement of the lemma holds. Let f ≡ const. Then T ( f ) ∈ Mε0−1 . Using Proposition 6.1 we obtain 2 h sε2 (T ( f )) ≤ h aε2 (T ( f )) ≤ h dε2 (T ( f )) . Therefore h s ( f ) ≤ h a ( f ) ≤ h d ( f ). It is clear that E V ( f ) ≤ n. We now show that h d ( f ) ≤ E V ( f ). ¯ σ¯ ∈ E 2n such that Let f = f (vi1 , . . . , vin ). Let us show that, for any tuples δ, ¯ = f (σ¯ ), there exists a variable vit having the following properties: f (δ) (a) vit is an essential variable of the function f . (b) The tuples δ¯ and σ¯ are different in the component with number t. Let A be the set of numbers of components in which the tuples δ¯ and σ¯ are different. One can show that, for some m ∈ ω \ {0, 1}, there exists a sequence δ¯1 , . . . , δ¯m of ¯ δ¯m = σ¯ , for j = 1, . . . , m − 1, the tuples δ¯ j and δ¯ j+1 tuples from E 2n such that δ¯1 = δ, are different only in one component, and the number of this component belongs to the set A. Since f (δ¯1 ) = f (δ¯m ), there exists j ∈ {1, . . . , m − 1} such that f (δ¯ j ) = f (δ¯ j+1 ). Let t be the number of component in which the tuples δ¯ j and δ¯ j+1 are different. It is clear that t ∈ A and vit is an essential variable of the function f . Thus, the considered statement holds. Using this statement it is not difficult to prove that the set of essential variables of the function f is a test for the table T ( f ). Therefore Θε2 ,h (T ( f )) ≤ E V ( f ). From Lemma 3.1 it follows that h dε2 (T ( f )) ≤ Θε2 ,h (T ( f )).  Therefore h dε2 (T ( f )) ≤ E V ( f ). Hence h d ( f ) ≤ E V ( f ). Corollary 7.1 Let U be a closed class of Boolean functions and n ∈ ω \ {0}. Then the values h Us (n), h Ua (n), and h Ud (n) are definite, and the inequalities h Us (n) ≤ h Ua (n) ≤ h Ud (n) ≤ n hold.

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Proof Evidently, U (n) = ∅. By Lemma 7.1, the values h Us (n), h Ua (n), and h Ud (n) are definite, and the considered inequalities hold.  Let n ∈ ω \ {0} and δ¯ = (δ1 , . . . , δn ) ∈ E 2n . Denote ¯ = {(¬δ1 , δ2 , . . . , δn ), (δ1 , ¬δ2 , . . . , δn ), . . . , (δ1 , δ2 , . . . , ¬δn )} . O(δ)   ¯ f (δ) ¯ = f (σ¯ )}. ¯ = {σ¯ : σ¯ ∈ O(δ), Lets f ∈ P2 (n) and δ¯ ∈ E 2n . Denote o( f, δ) Lemma 7.2 Let n ∈ ω \ {0}, f ∈ P2 , and δ¯ ∈ E 2n . Then the following statements hold: ¯ ≥ o( f, δ). ¯ (a) Mε2 ,h (T ( f ), δ) ¯ = n if and only if o( f, δ) ¯ = n. (b) Mε2 ,h (T ( f ), δ) Proof Let f = f (vi1 , . . . , vin ) and δ¯ = (δ1 , . . . , δn ). We denote by G a (P(T ( f )), ε2 )-tree in which each edge is labeled with a pair from the set {(vi1 , δ1 ), . . . , (vin , δn )}, ¯ + 1 nodes. We correspond to each and each complete path contains exactly o( f, δ) node w of the tree G the word ζ (w) ∈ Ωε2 (T ( f )) in the same way as it was described ¯ for any terminal node w of the tree G. in Sect. 3.4. It is clear that h(ζ (w)) = o( f, δ) Let w be a node of the tree G which is not terminal. Then, evidently, the length of the ¯ − 1. Therefore there exists a tuple σ¯ ∈ O(δ) ¯ such that word ζ (w) is at most o( f, δ) ¯ and σ¯ ∈ Δ(T ( f )ζ (w)). Hence T ( f )ζ (w) ∈ f (σ¯ ) = f (δ) / Mρ C . Thus, G is a proof¯ ≥ o( f, δ). ¯ By Proposition 6.2, Mε2 ,h (T ( f ), δ) ¯ ≥ tree for the bound Mε2 ,h (T ( f ), δ) ¯ o( f, δ). ¯ ≥ n. Using Lemma 3.5, Theorem ¯ = n. By the above, Mε2 ,h (T ( f ), δ) Let o( f, δ) ¯ = n. Let o( f, δ) ¯ < n. Then there exists 3.1, and Lemma 3.1 we obtain Mε2 ,h (T ( f ), δ) ¯ such that f (σ¯ ) = f (δ). ¯ Let σ¯ = (δ1 , . . . , δ j−1 , ¬δ j , δ j+1 , . . . , δn ). a tuple σ¯ ∈ O(δ) Set α = (vi1 , δ1 ) · · · (vi j−1 δ j−1 )(vi j+1 , δ j+1 ) · · · (vin , δn ) . Evidently, χ (α) ⊆ {(vi1 , δ1 ), . . . , (vin , δn )} and h(α) = n − 1. One can show that ¯ ≤ n − 1.  ¯ σ¯ }. Therefore T ( f )α ∈ Mε2 C . Hence Mε2 ,h (T ( f ), δ) Δ(T ( f )α) = {δ, Corollary 7.2 Let n ∈ ω \ {0}, f ∈ P2 (n), and δ¯ ∈ E 2n . Then the following statements hold: ¯ (a) h a ( f ) ≥ o( f, δ). ¯ ¯ (b) If f (δ) = 1, then h s ( f ) ≥ o( f, δ). Proof If f ≡ const, then, evidently, the statements (a) and (b) hold. Let f ≡ const. The statement (a) follows from Lemma 7.2 and Theorem 6.1. The statement (b) follows from Lemma 7.2 and Theorem 6.2.  Lemma 7.3 Let n ∈ ω \ {0} and f (vi1 , . . . , vin ) be a monotone function from P2 (n). Then the following statements hold: (a) h a ( f ) = n if and only if f (vi1 , . . . , vin ) = vi1 ∨ . . . ∨ vin or f (vi1 , . . . , vin ) = vi1 ∧ . . . ∧ vin . (b) h s ( f ) = n if and only if f (vi1 , . . . , vin ) = vi1 ∧ . . . ∧ vin .

7.2 Auxiliary Statements

97

Proof Let f (vi1 , . . . , vin ) = vi1 ∨ · · · ∨ vin . Then o( f, 0˜ n ) = n. By Corollary 7.2, h a ( f ) ≥ n. From this inequality and from Lemma 7.1 it follows that h a ( f ) = n. Let f (vi1 , . . . , vin ) = vi1 ∧ · · · ∧ vin . Then o( f, 1˜ n ) = n. It is clear that f (1˜ n ) = 1. Using Corollary 7.2 we obtain h a ( f ) ≥ n and h s ( f ) ≥ n. From these inequalities and from Lemma 7.1 it follows that h a ( f ) = n and h s ( f ) = n. Let h a ( f ) = n. By Theorem 6.1, there exists a tuple δ¯ ∈ E 2n such that ¯ =n. Mε2 ,h (T ( f ), δ) ¯ = n. Taking into From this equality and from Lemma 7.2 it follows that o( f, δ) account that f is a monotone function we conclude that either δ¯ = 1˜ n , f (1˜ n ) = 1, and f (σ¯ ) = 0 for any σ¯ ∈ E 2n \ {1˜ n }, or δ¯ = 0˜ n , f (0˜ n ) = 0 and f (σ¯ ) = 1 for any σ¯ ∈ E 2n \ {0˜ n }. Therefore f (vi1 , . . . , vin ) = vi1 ∨ . . . ∨ vin or f (vi1 , . . . , vin ) = vi1 ∧ · · · ∧ vin . ¯ =1 Let h s ( f ) = n. By Theorem 6.2, there exists a tuple δ¯ ∈ E 2n such that f (δ) ¯ = n. From the last equality and from Lemma 7.2 it follows and Mε2 ,h (T ( f ), δ) ¯ = n. Taking into account that f (δ) ¯ = 1 and f is a monotone functhat o( f, δ) tion we conclude that δ¯ = 1˜ n and f (σ¯ ) = 0 for any σ¯ ∈ E 2n \ {1˜ n }. Therefore  f (vi1 , . . . , vin ) = vi1 ∧ · · · ∧ vin . Lemma 7.4 For any n ∈ ω \ {0}, the following inequalities hold: h s (kn ) ≥ n, h a (dn ) ≥ n, h s (ln ) ≥ n, h s (¬ln ) ≥ n, and h s (πn ) ≥ 1. For any n ∈ ω \ {0, 1}, the following inequalities hold: h s (m n ) ≥ n − 1 and h s (qn ) ≥ n − 1. For any n ∈ ω \ {0, 1, 2}, the inequality h s (rn ) ≥ n − 1 holds. Proof Let n ∈ ω \ {0}. Then kn (1˜ n ) = 1 and o(kn , 1˜ n ) = n, dn (0˜ n ) = 0 and o(dn , 0˜ n ) = n , ¬ln (0˜ n ) = 1 and o(¬ln , 0˜ n ) = n, πn (1˜ n ) = 1 and o(πn , 1˜ n ) = 1. Let δ¯ ∈ E 2n and ¯ = 1 and o(ln , δ) ¯ = n. Using Corollary 7.2 we obtain δ¯ = (1, 0, . . . , 0). Then ln (δ) h s (kn ) ≥ n, h a (dn ) ≥ n, h s (ln ) ≥ n, h s (¬ln ) ≥ n, and h s (πn ) ≥ 1. ¯ σ¯ ∈ E 2n , δ¯ = (1, 0, . . . , 0) and σ¯ = (0, 1, . . . , 1). Then Let n ∈ ω \ {0, 1}, δ, ¯ = 1 and o(m n , δ) ¯ = n − 1, qn (σ¯ ) = 1 and o(qn , σ¯ ) = n − 1. Using Corolm n (δ) lary 7.2 we obtain h s (m n ) ≥ n − 1 and h s (qn ) ≥ n − 1. Let n ∈ ω \ {0, 1, 2}, σ¯ ∈ E 2n and σ¯ = (0, 1, . . . , 1). Then rn (σ¯ ) = 1 and o(rn , σ¯ )  = n − 1. Using Corollary 7.2 we obtain h s (rn ) ≥ n − 1. Lemma 7.5 Let n ∈ ω \ {0, 1, 2}. Then h d (rn ) ≥ n. Proof Let us define a ({v1 , . . . , vn }, ε2 )-tree Hn . In this tree, each complete path contains exactly n + 1 nodes. Let ξ = w1 , d1 , . . . , wn , dn , wn+1 be an arbitrary complete path in the tree Hn , i ∈ {1, . . . , n}, and the edge di be labeled with a pair (v j , σ ). If j = 1, then σ = 0. If j = 1 and i = n − 1, then σ = 1. Let j = 1 and i = n − 1. If among the pairs attached to edges in the path from w1 to wn−1 there is no the pair (v1 , 0), then σ = 0. Otherwise, σ = 1.

98

7 Closed Classes of Boolean Functions

Let us show that the tree Hn is a proof-tree for the bound h dε2 (T (rn )) ≥ n. Let us correspond to each node w of the tree Hn the word ζ (w) ∈ Ωε2 (T (rn )) in the same way as it was described in Sect. 3.4. Let w be an arbitrary terminal node of the tree Hn . One can show that h(ζ (w)) = n and Δ(T (rn )ζ (w)) = ∅. Let us show that, for / Mε2 C . To this end, it any node w of the tree Hn which is not terminal, T (rn )ζ (w) ∈ is sufficient to consider all nodes which have the following property: the node is not terminal, and all edges leaving this node enter terminal nodes of the tree Hn . Let w be one of such nodes. Then there exists i ∈ {2, . . . , n} such that χ (ζ (w)) = {(v1 , 0)} ∪ {(v j , 1) : j ∈ {2, . . . , n} \ {i}}

(7.1)

χ (ζ (w)) = {(vi , 0)} ∪ {(v j , 1) : j ∈ {2, . . . , n} \ {i}} .

(7.2)

or We denote by γ¯ the tuple from E 2n in which the first component is equal to 0 and all other components are equal to 1. We denote by δ¯ the tuple from E 2n in which the first and the i-th components are equal to 0 and all other components are equal to 1. We denote by σ¯ the tuple from E 2n in which the i-th component is equal to 0 and all other components are equal to 1. Let (7.1) hold. Then the tuples γ¯ and δ¯ belong to the set ¯ = {0}. Let (7.2) hold. Then the tuples δ¯ Δ(T (rn )ζ (w)), νT (rn ) (γ¯ ) = {1} and νT (rn ) (δ) ¯ = {0} and νT (rn ) (σ¯ ) = {1}. Therefore and σ¯ belong to the set Δ(T (rn )ζ (w)), νT (rn ) (δ) / Mε2 C . Hence the tree Hn is a proof-tree for the bound h dε2 (T (rn )) ≥ n. T (rn )ζ (w) ∈  Using Theorem 3.4 we obtain h dε2 (T (rn )) ≥ n. Thus, h d (rn ) ≥ n. ¯ = 1, o( f, δ) ¯ = n, t be the Lemma 7.6 Let n ∈ ω \ {0, 1}, f ∈ P2 (n), δ¯ ∈ E 2n , f (δ) ¯ number of 0 components in the tuple δ, m ∈ ω \ {0, 1}, and m ≥ t. Then the function f does not satisfy the condition (a m ). Proof Evidently, if t = 0 or t = 1, then the function f does not satisfy the condition (a 2 ) and, hence, does not satisfy the condition (a m ). Let t ≥ 2 and i 1 , . . . , i t be ¯ For j = 1, . . . , t, we denote by σ¯ j the tuple numbers of 0 components in the tuple δ. ¯ obtained from the tuple δ by replacing 0 in the i j th component with 1. Then f (σ¯ j ) = 0 for j = 1, . . . , t and, for any i ∈ {1, . . . , n}, in the set {σ¯ j : j ∈ {1, . . . , t}} there exists a tuple in which ith component is equal to 1. Therefore the function f does not satisfy the condition (a t ). Since m ≥ t, the function f does not satisfy the con dition (a m ).

7.3 Bounds for Individual Closed Classes Lemma 7.7 Let U ∈ {O2 , O3 , O7 }, b ∈ {s, a, d}, and n ∈ ω \ {0}. Then h Ub (n) = 0. Proof Evidently, f ≡ const for any f ∈ U (n). Therefore h Ub (n) = 0.



7.3 Bounds for Individual Closed Classes

99

Lemma 7.8 Let U ∈ {O1 , O4 , O5 , O6 , O8 , O9 }, b ∈ {s, a, d}, and n ∈ ω \ {0}. Then h Ub (n) = 1. Proof One can show that πn ∈ U (n). Using Lemmas 7.4 and 7.1 we obtain h Ub (n) ≥ 1. It is clear that E V ( f ) ≤ 1 for any function f ∈ U (n). By Lemma 7.1,  h Ub (n) ≤ 1. Lemma 7.9 Let U ∈ {L 4 , L 5 }, b ∈ {s, a, d}, and n ∈ ω \ {0}. Then  h Ub (n)

=

n, if n is odd , n − 1, if n is even .

Proof Let n be odd. Then ln ∈ U (n). Using Lemmas 7.4 and 7.1 we obtain h Ub (n) = n. Let n be even. Then m n ∈ U (n). Using Lemmas 7.4 and 7.1 we obtain h Ub (n) ≥ n − 1. One can show that E V ( f ) ≤ n − 1 for any function f ∈ U (n). By Lemma  7.1, h Ub (n) ≤ n − 1. Lemma 7.10 Let U ∈ {D1 , D3 }, b ∈ {s, a, d}, and n ∈ ω \ {0}. Then  h Ub (n) =

n, if n ≥ 3 , 1, if n ≤ 2 .

Proof Let n ≤ 2. One can show that πn ∈ U (n). Using Lemmas 7.4 and 7.1 we obtain h Ub (n) ≥ 1. One can show that E V ( f ) ≤ 1 for any function f ∈ U (n). By Lemma 7.1, h Ub (n) ≤ 1. Let n ≥ 3. One can show that there exists a self-dual α-function f ∈ P2 (n) such that f (1˜ n ) = 1 and f (σ¯ ) = 0 for any σ¯ ∈ O(1˜ n ). Evidently, f ∈ U (n), f (1˜ n ) = 1, and o( f, 1˜ n ) = n. Using Corollary 7.2 we obtain h s ( f ) ≥ n. By Lemma 7.1,  h Ub (n) = n. Lemma 7.11 Let n ∈ ω \ {0}. Then  h sD2 (n) = h aD2 (n) = 

and h dD2 (n) =

n − 1, if n ≥ 3 , 1, if n ≤ 2

n, if n ≥ 3 , 1, if n ≤ 2 .

Proof Let n ≤ 2. Evidently, πn ∈ D2 (n). Using Lemma 7.4 we obtain h sD2 (n) ≥ 1. One can show that E V ( f ) ≤ 1 for any function f ∈ D2 (n). By Lemma 7.1, 1 ≤ h sD2 (n) ≤ h aD2 (n) ≤ h dD2 (n) ≤ 1. Let n ≥ 3. One can show that rn ∈ D2 (n). From Lemma 7.5 it follows that h d (rn ) ≥ n. By Lemma 7.1, h dD2 (n) = n. From Lemma 7.4 it follows that h s (rn ) ≥ n − 1. Using Lemma 7.1 we obtain n − 1 ≤ h sD2 (n) ≤ h aD2 (n). Let f (vi1 , . . . , vin ) ∈

100

7 Closed Classes of Boolean Functions

D2 (n). Assume that h a ( f ) = n. Then, using Lemma 7.3, we obtain f (vi1 , . . . , vin ) = vi1 ∧ · · · ∧ vin or f (vi1 , . . . , vin ) = vi1 ∨ · · · ∨ vin which is impossible since these functions are not self-dual. Therefore h a ( f ) ≤ n − 1. Hence h aD2 (n) ≤ n − 1.  Lemma 7.12 Let U ∈ {S1 , S3 , S5 , S6 } and n ∈ ω \ {0}. Then h Us (n) = 1. Proof Evidently, πn ∈ U (n). By Lemma 7.4, h Us (n) ≥ 1. Let f (vi1 , . . . , vin ) ∈ U (n). If f ≡ const, then h s ( f ) = 0. Let f ≡ const. Then, for a nonempty subset {vi j (1) , . . . , vi j (m) } of the set {vi1 , . . . , vin }, the equality f (vi1 , . . . , vin ) = vi j (1) ∨ ¯ = 1. Then, evidently, δt = 1 · · · ∨ vi j (m) holds. Let δ¯ = (δ1 , . . . , δn ) ∈ E 2n and f (δ) for some t ∈ { j (1), . . . , j (m)}. One can show that (vit , 1) ∈ {(vi1 , δ1 ), . . . , (vin , δn )} ¯ ≤ 1. Taking into account that and T ( f )(vit , 1) ∈ Mε2 C . Therefore Mε2 ,h (T ( f ), δ) ¯δ is an arbitrary tuple from E 2n such that f (δ) ¯ = 1 and using Theorem 6.2 we obtain  h s ( f ) ≤ 1. Since f is an arbitrary function from U (n), we have h Us (n) ≤ 1. Lemma 7.13 Let U ∈ {F1∞ , F2∞ , F3∞ , F4∞ } ∪ {F2m : m ∈ ω \ {0, 1}} ∪ {F3m : m ∈ ω \ {0, 1}} and n ∈ ω \ {0}. Then  h Us (n) =

n − 1, if n ≥ 2 , 1, if n = 1 .

Proof Let n = 1. Then π1 ∈ U (1). By Lemma 7.4, h Us (1) ≥ 1. Using Lemma 7.1 we obtain h Us (1) = 1. Let n ≥ 2. One can show that qn ∈ U (n). By Lemma 7.4, h Us (n) ≥ n − 1. ¯ = 1, and A = {σ¯ : σ¯ ∈ Let U ∈ {F1∞ , F2∞ , F3∞ , F4∞ }, f ∈ U (n), δ¯ ∈ E 2n , f (δ) ¯ O(δ), f (σ¯ ) = 0}. Since the function f satisfies the condition (a ∞ ), there exists i ∈ {1, . . . , n} such that, in each tuple from A, the component with number i is equal ¯ ≤ ¯ ≤ n − 1. By Lemma 7.2, Mε2 ,h (T ( f ), δ) to 0. Therefore |A| ≤ n − 1 and o( f, δ) n ¯ ¯ n − 1. Taking into account that δ is an arbitrary tuple from E 2 such that f (δ) = 1 and using Theorem 6.2 we obtain h s ( f ) ≤ n − 1. Since f is an arbitrary function from U (n) we have h Us (n) ≤ n − 1. Let U ∈ {F2m : m ∈ ω \ {0, 1}} ∪ {F3m : m ∈ ω \ {0, 1}} and f (vi1 , . . . , vin ) ∈ U (n). Then f is a monotone function satisfying the condition (a 2 ). Assume that h s ( f ) = n. Then, using Lemma 7.3, we obtain f (vi1 , . . . , vin ) = vi1 ∧ · · · ∧ vin which is impossible since the function vi1 ∧ · · · ∧ vin does not satisfy the condition (a 2 ). Hence h s ( f ) ≤ n − 1. Taking into account that f is an arbitrary function  from U (n) we have h Us (n) ≤ n − 1. Lemma 7.14 Let m ∈ ω \ {0, 1} and n ∈ ω \ {0}. Then h sF1m (n) =

⎧ ⎨

n, if n ≥ m + 2 , n − 1, if 2 ≤ n ≤ m + 1 , ⎩ 1, if n = 1 .

Proof Evidently, F1∞ ⊆ F1m . Using Lemma 7.13 we conclude that h sF m (1) ≥ 1 and 1 if 2 ≤ n ≤ m + 1, then h sF m (n) ≥ n − 1. By Lemma 7.1, h sF m (1) = 1. Let 2 ≤ n ≤ 1

1

7.3 Bounds for Individual Closed Classes

101

m + 1 and f ∈ F1m . Assume that h s ( f ) = n. Then, using Theorem 6.2 and Lemma ¯ = 1 and o( f, δ) ¯ = n. Since 7.2, we conclude that there exists δ¯ ∈ E 2n such that f (δ) f ∈ F1m , f is an α-function. Therefore δ¯ = 0˜ n . We denote by t the number of 0 ¯ Then t ≤ n − 1 ≤ m. By Lemma 7.6, f does not satisfy components in the tuple δ. the condition (a m ) which is impossible. Hence h sF m (n) ≤ n − 1. 1 Let n ≥ m + 2. Taking into account that m ≥ 2 we obtain n ≥ 4. We now define n a function f ∈ P2 (n). Let δ¯ ∈ E 2 and δ¯ = (1, 0, . . . , 0). Then f (σ¯ ) = 0 for any ¯ Since n ≥ 4, f is an α-function. ¯ and f (γ¯ ) = 1 for any γ¯ ∈ E 2n \ O(δ). σ¯ ∈ O(δ), Taking into account that n ≥ m + 2 one can show that the function f satisfies the ¯ = 1 and o( f, δ) ¯ = n. Using condition (a m ). Therefore f ∈ F1m (n). Evidently, f (δ) Corollary 7.2 we obtain h s ( f ) ≥ n. Hence h sF m (n) ≥ n. From Lemma 7.1 it follows 1  that h sF m (n) ≤ n. 1

Lemma 7.15 Let m ∈ ω \ {0, 1} and n ∈ ω \ {0}. Then h sF4m (n) =

⎧ ⎨

n, if n ≥ m + 1 , n − 1, if 2 ≤ n ≤ m , ⎩ 1, if n = 1 .

Proof Let n ≤ m. Taking into account that F1m ⊆ F4m and using Lemma 7.14 we obtain  n − 1, if 2 ≤ n ≤ m , h sF4m (n) ≥ 1, if n = 1 . Let n = 1. By Lemma 7.1, h sF m (1) ≤ 1. Let 2 ≤ n ≤ m and f ∈ F4m (n). Assume that 4 h s ( f ) = n. Then, using Theorem 6.2 and Lemma 7.2, we conclude that there exists ¯ = 1 and o( f, δ) ¯ = n. Evidently, the tuple δ¯ contains a tuple δ¯ ∈ E 2n such that f (δ) at most n components which are equal to 0. Taking into account that 2 ≤ n ≤ m and using Lemma 7.6 we conclude that the function f does not satisfy the condition (a m ) which is impossible. Hence h s ( f ) ≤ n − 1. Taking into account that f is an arbitrary function from F4m (n) we obtain h sF m (n) ≤ n − 1. 4 Let n ≥ m + 1. We define a function f ∈ P2 (n) as follows: f (σ¯ ) = 0 for any σ¯ ∈ n O(0˜ n ), and f (γ¯ ) = 1 for any γ¯ ∈ E 2 \ O(0˜ n ). Taking into account that n ≥ m + 1 one can show that the function f satisfies the condition (a m ). Therefore f ∈ F4m . Evidently, f (0˜ n ) = 1 and o( f, 0˜ n ) = n. Using Corollary 7.2 we obtain h s ( f ) ≥ n.  Hence h sF m (n) ≥ n. By Corollary 7.1, h sF m (n) ≤ n. 4

4

Lemma 7.16 Let U ∈ {L 2 , L 3 , P1 } and n ∈ ω \ {0}. Then h Us (n) = n. Proof One can show that at least one of the functions ln , ¬ln , kn belongs to the set U (n). By Lemma 7.4, h Us (n) ≥ n. Using Corollary 7.1 we conclude that h Us (n) ≤ n.  Lemma 7.17 Let U ∈ {S1 , L 2 , L 3 , P1 } and n ∈ ω \ {0}. Then h Ua (n) = h Ud (n) = n. Proof One can show that at least one of the functions dn , ln , ¬ln , kn belongs to the set U (n). Using Lemmas 7.4 and 7.1 we obtain n ≤ h Ua (n) ≤ h Ud (n). By Corollary  7.1, h Ud (n) ≤ n.

102

7 Closed Classes of Boolean Functions

7.4 Bounds for Arbitrary Closed Classes Theorem 7.1 Let U be a closed class of Boolean functions and n ∈ ω \ {0}. Then (a) If U ∈ {O2 , O3 , O7 }, then h Us (n) = 0. (b) If U ∈ {O1 , O4 , O5 , O6 , O8 , O9, S1 , S3 , S5 , S6 }, then h Us (n) = 1. n, if n is odd, (c) If U ∈ {L 4 , L 5 }, then h Us (n) = n − 1, if n is even. (d) If U ∈ {D2 , F1∞ , F2∞ , F3∞ , F4∞ } ∪ {F2m : m ∈ ω \ {0, 1}} ∪ {F3m : m ∈ ω \ n − 1, if n ≥ 2, {0, 1}}, then h Us (n) = 1, if n = 1. (e) If m ∈ ω \ {0, 1} and U ∈ {F1m , F4m+1 }, then h Us (n) =

⎧ ⎨

n, if n ≥ m + 2, n − 1, if 2 ≤ n ≤ m + 1, ⎩ 1, if n = 1. 

n, if n ≥ 3, 1, if n ≤ 2. (g) If the class U coincides with neither of above-mentioned classes, then h Us (n) = n. (f) If U ∈ {D1 , D3 , F42 }, then h Us (n) =

Proof The part (a) of the theorem statement follows from Lemma 7.7, the part (b) follows from Lemmas 7.8 and 7.12, the part (c) follows from Lemma 7.9, the part (d) follows from Lemmas 7.11 and 7.13, the part (e) follows from Lemmas 7.14 and 7.15, and the part (f) follows from Lemmas 7.10 and 7.15. Let U coincide with neither of classes mentioned in the parts (a)–(f) of the theorem statement. Using the description of all closed classes of Boolean functions [5] (see also [2]) one can show that in this case at least one of the following relations holds: L 2 ⊆ U , L 3 ⊆ U ,  P1 ⊆ U . Using Lemma 7.16 and Corollary 7.1 we obtain h Us (n) = n. Theorem 7.2 Let U be a closed class of Boolean functions and n ∈ ω \ {0}. Then (a) If U ∈ {O2 , O3 , O7 }, then h Ua (n) = 0. (b) If U ∈ {O1 , O4 , O5 , O6 , O8 , O9}, then h Ua (n) = 1. n, if n is odd, (c) If U ∈ {L 4 , L 5 }, then h Ua (n) = n − 1, if n is even.  n − 1, if n ≥ 2, a (d) If U = D2 , then h U (n) = 1, if n = 1.  n, if n ≥ 3, a (e) If U ∈ {D1 , D3 }, then h U (n) = 1, if n ≤ 2. (f) If the class U coincides with neither of above-mentioned classes, then h Ua (n) = n. Proof The part (a) of the theorem statement follows from Lemma 7.7, the part (b) follows from Lemma 7.8, the part (c) follows from Lemma 7.9, the part (d) follows from Lemma 7.11, and the part (e) follows from Lemma 7.10. Let U coincide with

7.4 Bounds for Arbitrary Closed Classes

103

neither of classes mentioned in the parts (a)–(e) of the theorem statement. Using the description of all closed classes of Boolean functions [5] (see also [2]) one can show that in this case at least one of the following relations holds: S1 ⊆ U , L 2 ⊆ U , L 3 ⊆ U , P1 ⊆ U . Using Lemma 7.17 and Corollary 7.1 we obtain h Ua (n) = n.  Theorem 7.3 Let U be a closed class of Boolean functions and n ∈ ω \ {0}. Then (a) If U ∈ {O2 , O3 , O7 }, then h Ud (n) = 0. (b) If U ∈ {O1 , O4 , O5 , O6 , O8 , O9}, then h Ud (n) = 1. n, if n is odd, (c) If U ∈ {L 4 , L 5 }, then h Ud (n) = n − 1, if n is even.  n, if n ≥ 3, (d) If U ∈ {D1 , D2 , D3 }, then h Ud (n) = 1, if n ≤ 2. (e) If the class U coincides with neither of above-mentioned classes, then h Ud (n) = n. Proof The part (a) of the theorem statement follows from Lemma 7.7, the part (b) follows from Lemma 7.8, the part (c) follows from Lemma 7.9, and the part (d) follows from Lemmas 7.10 and 7.11. Let U coincide with neither of classes mentioned in the parts (a)–(d) of the theorem statement. Using the description of all closed classes of Boolean functions [5] (see also [2]) one can show that in this case at least one of the following relations holds: S1 ⊆ U , L 2 ⊆ U , L 3 ⊆ U , P1 ⊆ U .  Using Lemma 7.17 and Corollary 7.1 we obtain h Ud (n) = n.

References 1. Moshkov, M.: About the depth of decision trees computing Boolean functions. Fundam. Inform. 22(3), 203–215 (1995) 2. Moshkov, M.: Time complexity of decision trees. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III, Lecture Notes in Computer Science, vol. 3400, pp. 244–459. Springer, Berlin (2005) 3. Post, E.: Introduction to a general theory of elementary propositions. Am. J. Math. 43, 163–185 (1921) 4. Post, E.: Two-Valued Iterative Systems of Mathematical Logic. Annals of Mathematics Studies, vol. 5. Princeton University Press, Princeton-London (1941) 5. Yablonskii, S.V., Gavrilov, G.P., Kudriavtzev, V.B.: Functions of the Algebra of Logic and Post Classes. Nauka Publishers, Moscow (1966). (in Russian)

Chapter 8

Algorithmic Problems

In the previous chapters, we considered, mainly, approximate bounds on the minimum complexity and approximate algorithms for optimization of decision trees. In this chapter, we consider algorithmic problems of computation of the minimum complexity of deterministic, nondeterministic, and strongly nondeterministic decision trees, and of construction of decision trees with the minimum complexity. We study relationships among these algorithmic problems, consider examples of decidable and undecidable problems, describe all variants of algorithmic problem behavior, and discuss examples of problems with polynomial time complexity and examples of problems that are N P-hard. Some results for deterministic decision trees considered in this chapter were published in [2, 3].

8.1 Relationships Among Algorithmic Problems Let ρ be an enumerated signature, ψ be a computable complexity function of the signature ρ, and b ∈ {s, a, d}. We define the set of tables Nρb in the following way: Nρs = Mρ0−1 and Nρa = Nρd = Mρ . We now define two algorithmic problems: Com b (ρ, ψ) and Des b (ρ, ψ). Problem Com b (ρ, ψ): for a given table T ∈ Nρb , it is required to compute the value ψρb (T ). Problem Des b (ρ, ψ): for a given table T ∈ Nρb , it is required to construct a schema Γ ∈ Cρ such that (Γ, T ) ∈ Rρb and ψ(Γ ) = ψρb (T ) (the definition of the set Rρb can be found in Sect. 2.5). Proposition 8.1 Let ρ be an enumerated signature, ψ be a computable complexity function of the signature ρ, and b ∈ {s, a, d}. Then the problem Com b (ρ, ψ) is decidable if and only if the problem Des b (ρ, ψ) is decidable. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_8

105

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Proof Let the problem Des b (ρ, ψ) be decidable and T ∈ Nρb . Using the decidability of the problem Des b (ρ, ψ) we construct a schema Γ ∈ Cρ such that (Γ, T ) ∈ Rρb and ψ(Γ ) = ψρb (T ). Using the computability of the function ψ we compute the value ψ(Γ ). It coincides with the value ψρb (T ). Hence the problem Com b (ρ, ψ) is decidable. Let the problem Com b (ρ, ψ) be decidable. One can show that the relation Rρb is decidable, the function ψ is computable on the set Cρ , and there is an algorithm which enumerates all elements of the set Cρ . Let T ∈ Nρb . Using the decidability of the problem Com b (ρ, ψ) we compute the value ψρb (T ). Among listed schemes from the set Cρ , we choose a schema Γ such that (Γ, T ) ∈ Rρb and ψ(Γ ) = ψρb (T ).  Hence the problem Des b (ρ, ψ) is decidable. Proposition 8.2 Let ρ be an enumerated signature and ψ be a computable complexity function of the signature ρ. If the problem Com d (ρ, ψ) is decidable, then the problems Com s (ρ, ψ) and Com a (ρ, ψ) are decidable. Proof Let the problem Com d (ρ, ψ) be decidable. Let T ∈ Mρ . If Δ(T ) = ∅, then, using the decidability of the problem Com d (ρ, ψ), we compute the value ψρd (T ). Using the equality Δ(T ) = ∅ one can show that ψρa (T ) = ψρd (T ). Let Δ(T ) = ∅. Using the decidability of the problem Com d (ρ, ψ), for each δ¯ ∈ Δ(T ) and each ¯ we compute the value ψρd (T (δ, ¯ i)) (the definition of the table T (δ, ¯ i) can i ∈ νT (δ), ¯ To this end, be found in Sect. 4.8). For each δ¯ ∈ Δ(T ), we find the value Mρ,ψ (T, δ). ¯ = min{ψρd (T (δ, ¯ i)) : i ∈ νT (δ)} ¯ which follows from we use the equality Mρ,ψ (T, δ) a ¯ : δ¯ ∈ Proposition 4.7. From Theorem 6.1 it follows that ψρ (T ) = max{Mρ,ψ (T, δ) a a Δ(T )}. Using this equality we find the value ψρ (T ). Thus, the problem Com (ρ, ψ) is decidable. ¯ = {1}}. Using the decidLet T ∈ Mρ0−1 . Denote Δ(T, 1) = {δ¯ : δ¯ ∈ Δ(T ), νT (δ) d ability of the problem Com (ρ, ψ), for each δ¯ ∈ Δ(T, 1), we compute the value ¯ 1)). By Proposition 4.7, Mρ,ψ (T, δ) ¯ = ψρd (T (δ, ¯ 1)). From Theorem 6.2 it ψρd (T (δ, s ¯ ¯ follows that ψρ (T ) = max{Mρ,ψ (T, δ) : δ ∈ Δ(T, 1)}. Using this equality we find  the value ψρs (T ). Thus, the problem Com s (ρ, ψ) is decidable. Lemma 8.1 Let ρ be a signature, ψ be a complexity function of the signature ρ, ¯ Then ψρd (T (δ, ¯ i)) = ψρa (T (δ, ¯ i)) = T ∈ Mρ , Δ(T ) = ∅, δ¯ ∈ Δ(T ), and i ∈ νT (δ). s ¯ ψρ (T (δ, i)). ¯ = {1}. ¯ i). Evidently, Q ∈ Mρ0−1 , δ¯ ∈ Δ(Q), and ν Q (δ) Proof Denote Q = T (δ, d ¯ 1)) = Mρ,ψ (Q, δ). ¯ One can show From Proposition 4.7 it follows that ψρ (Q(δ, ¯ 1) = Q. Therefore ψρd (Q) = Mρ,ψ (Q, δ). ¯ By Theorem 6.2, Mρ,ψ (Q, δ) ¯ = that Q(δ, s d s s a ψρ (Q). Hence ψρ (Q) = ψρ (Q). Using Proposition 6.1 we obtain ψρ (Q) ≤ ψρ (Q)  ≤ ψρd (Q). Therefore ψρd (Q) = ψρa (Q) = ψρs (Q). Proposition 8.3 Let ρ be an enumerated signature and ψ be a computable complexity function of the signature ρ. Then the problem Com s (ρ, ψ) is decidable if and only if the problem Com a (ρ, ψ) is decidable.

8.1 Relationships Among Algorithmic Problems

107

Proof Let the problem Com s (ρ, ψ) be decidable. Let T ∈ Mρ . Assume that Δ(T ) = ∅. We define a table T  ∈ Mρ0−1 in the following way: dim T  = dim T , ¯ where δ¯ = (1, . . . , 1), μT  ≡ μT , and νT  (δ) ¯ = {1}. Using the decidΔ(T  ) = {δ}, s ability of the problem Com (ρ, ψ) we compute the value ψρs (T  ). Using the equality Δ(T ) = ∅ one can show that ψρa (T ) = ψρs (T  ). Assume now that Δ(T ) = ∅. Using the decidability of the problem Com s (ρ, ψ), ¯ we compute the value ψρs (T (δ, ¯ i)). From for each δ¯ ∈ Δ(T ) and each i ∈ νT (δ), s ¯ ¯ i)) : i ∈ Proposition 4.7 and Lemma 8.1 it follows that Mρ,ψ (T, δ) = min{ψρ (T (δ, ¯ for any δ¯ ∈ Δ(T ). Using this equality we compute the value Mρ,ψ (T, δ) ¯ for νT (δ)} ¯ : δ¯ ∈ Δ(T )}. Using this each δ¯ ∈ Δ(T ). By Theorem 6.1, ψρa (T ) = max{Mρ,ψ (T, δ) equality we find the value ψρa (T ). Thus, if the problem Com s (ρ, ψ) is decidable, then the problem Com a (ρ, ψ) is decidable too. Let the problem Com a (ρ, ψ) be decidable. Let T ∈ Mρ0−1 . Denote Δ(T, 1) = ¯ = {1}}. Using the decidability of the problem Com a (ρ, ψ) {δ¯ : δ¯ ∈ Δ(T ), νT (δ) ¯ 1)) for each δ¯ ∈ Δ(T, 1). By Proposition 4.7 and we compute the value ψρa (T (δ, a ¯ = ψρ (T (δ, ¯ 1)). From Theorem 6.2 it follows that ψρs (T ) = Lemma 8.1, Mρ,ψ (T, δ) ¯ : δ¯ ∈ Δ(T, 1)}. Using this equality we find the value ψρs (T ). Thus, max{Mρ,ψ (T, δ) if the problem Com a (ρ, ψ) is decidable, then the problem Com s (ρ, ψ) is decidable too. 

8.2 Examples of Decidable and Undecidable Problems Let a be a letter and n ∈ ω \ {0}. We define the word a n in the following way: a 1 = a and if n ≥ 2, then a n = a n−1 a. Proposition 8.4 Let ρ = (F, k) be an enumerated signature. Then there exists a computable complexity function ψ of the signature ρ such that the problems Com d (ρ, ψ), Des d (ρ, ψ), Com a (ρ, ψ), Des a (ρ, ψ), Com s (ρ, ψ), and Des s (ρ, ψ) are undecidable. Proof Let ϕ(x) be a total recursive function for which the range Range(ϕ) (the set of values of ϕ) is not recursive [4]. We now define a function ψ : F ∗ → ω. Let α ∈ F ∗ . If α = λ, then ψ(α) = 1. Let α = λ. If α contains different letters from F, then ψ(α) = 1. Let α = f in , where f i ∈ F and n ∈ ω \ {0}. Then  ψ(α) =

0, if ϕ(n − 1) = i , 1, if ϕ(n − 1) = i .

Evidently, ψ is a computable complexity function of the signature ρ. For i ∈ ω, we define a table T i ∈ Mρ0−1 as follows: dim T i = 1, Δ(T i ) = {(1), (0)}, μT i (1) = f i , νT i ((1)) = {1}, and νT i ((0)) = {0}. Using Theorem 6.2 we obtain ψρs (T i ) = Mρ,ψ (T i , (1)). It is clear that Mρ,ψ (T i , (1)) = min{ψ(α) :

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χ (α) ⊆ {( f i , 1)}, T i α ∈ Mρ C }. Let χ (α) ⊆ {( f i , 1)}. One can show that T i α ∈ Mρ C if and only if α = λ. Therefore  Mρ,ψ (T , (1)) = i

0, if i ∈ Range(ϕ) , 1, if i ∈ / Range(ϕ) .

Hence, for any i ∈ ω, the equality ψρs (T i ) = 0 holds if and only if i ∈ Range(ϕ). Assume that the problem Com s (ρ, ψ) is decidable. Then the set Range(ϕ) is recursive which is impossible. Hence the problem Com s (ρ, ψ) is undecidable. Using Propositions 8.2 and 8.3 we conclude that the problems Com d (ρ, ψ) and Com a (ρ, ψ) are undecidable. By Proposition 8.1, the problems Des s (ρ, ψ), Des a (ρ, ψ), and Des d (ρ, ψ) are undecidable.  Proposition 8.5 Let ρ = (F, k) be an enumerated signature. Then there exists a computable complexity function ψ of signature ρ such that problems Com d (ρ, ψ), Des d (ρ, ψ) are undecidable and problems Com a (ρ, ψ), Des a (ρ, ψ), Com s (ρ, ψ), and Des s (ρ, ψ) are decidable. Proof Let ϕ(x) be a total recursive function for which the range Range(ϕ) is not recursive [4]. Let F = { f [i]2 : i ∈ ω}. For i ∈ ω, set ai = f [3i]2 , bi = f [3i + 1]2 , and ci = f [3i + 2]2 . We now define a function ψ : F ∗ → ω. Let α ∈ F ∗ . We denote by χ (α) the set of letters from F contained in α. The word α will be called homogeneous if χ (α) ⊆ {ai , bi , ci } for some i ∈ ω. If the word α is not homogeneous, then ψ(α) = 1. Let the word α be homogeneous. If |χ (α)| ≤ 1, then ψ(α) = 1. If |χ (α)| = 3, then ψ(α) = 2. Let |χ (α)| = 2. If α ∈ {ai bi , bi ci , ci ai } ∪ {ai cin+1 : n ∈ ω, ϕ(n) = i}, then ψ(α) = 1. Otherwise, ψ(α) = 2. Evidently, ψ is a computable complexity function of the signature ρ. We now show that the problem Com s (ρ, ψ) is decidable. Let T ∈ Mρ0−1 . If dim T ≤ 2 or dim T ≥ 4, then, evidently, ψρs (T ) = 1. If dim T = 3 and there is no i ∈ ω such that P(T ) = {ai , bi , ci }, then, evidently, ψρs (T ) = 1. Assume that there exists i ∈ ω such that P(T ) = {ai , bi , ci }. Let, for the definiteness, μT (1) = ¯ = {1}. If ai , μT (2) = bi , and μT (3) = ci . Let δ¯ = (δ1 , δ2 , δ3 ) ∈ Δ(T ) and νT (δ) there exists a word β ∈ {(ai , δ1 )(bi , δ2 ), (bi , δ2 )(ci , δ3 ), (ci , δ3 )(ai , δ1 )} such that ¯ = 1. Otherwise, Mρ,ψ (T, δ) ¯ = 2. By Theorem 6.2, the Tβ ∈ Mρ C , then Mρ,ψ (T, δ) s s ¯ : δ¯ ∈ value ψρ (T ) can be found according to the equality ψρ (T ) = max{Mρ,ψ (T, δ) s ¯ = {1}}. Thus, the problem Com (ρ, ψ) is decidable. Using Δ(T ), νT (δ) Propositions 8.1 and 8.3 we conclude that the problems Des s (ρ, ψ), Com a (ρ, ψ), and Des a (ρ, ψ) are decidable. We now show that the problem Com d (ρ, ψ) is undecidable. Let i ∈ ω. We define a table T i ∈ Mρ as follows: dim T i = 3, Δ(T i ) = {(0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 0, 1)} , μT i (1) = ai , μT i (2) = bi , μT i (3) = ci , νT i ((0, 0, 0)) = {1}, νT i ((0, 1, 0)) = {2}, νT i ((1, 0, 0)) = {3}, and νT i ((1, 0, 1)) = {4}. Let us show that ψρd (T i ) = 1 if and

8.2 Examples of Decidable and Undecidable Problems

109

only if i ∈ Range(ϕ). Let i ∈ Range(ϕ). Then ϕ(n) = i for some n ∈ ω. Set α = cin+1 . For t ∈ E 2 , we denote by ξt the complete path in the schema G ρ (α) (the definition of this schema can be found in Sect. 3.1) in which each edge is labeled with the number t. Instead of the number 0, we label the terminal node of the path ξ0 with the number 3. Instead of the number 0, we label the terminal node of the path ξ1 with the number 4. Remove from the schema G ρ (α) the root and the edge leaving the root. As a result, we obtain some labeled finite directed tree with the root. We denote this tree by G. We now define a schema Γ ∈ Cρ . Exactly one edge leaves the root of Γ . This edge enters a node w1 which is labeled with the element ai . 1-Successor of the node w1 is the root of the tree G. 0-Successor of the node w1 is a node w2 . The node w1 has no any other successors. The node w2 is labeled with the element bi . 0-Successor of the node w2 is a terminal node of the schema Γ labeled with the number 1. 1-Successor of the node w2 is a terminal node of the schema Γ labeled with the number 2. The node w2 has no any other successors. One can show that the schema Γ is a deterministic decision tree for the table T i , and ψ(Γ ) = 1. Hence ψρd (T i ) = 1. Let i ∈ / Range(ϕ). We now show that ψρd (T i ) = 1. Assume the contrary. Let ψρd (T i ) = 1. By Lemma 3.1, there exists a deterministic decision tree Γ for the table T i such that ψ(Γ ) = ψρd (T i ). Let w be the node which is the successor of the root / Mρ C , the node w is not terminal. of schema Γ . Since T i ∈ Let w be labeled with the element ai . Since (1, 0, 0) ∈ Δ(T i ) and Γ is a deterministic decision tree for the table T i , there exists a complete path ξ in the schema Γ such that χ (π(ξ )) ⊆ {(ai , 1), (bi , 0), (ci , 0)} and T i π(ξ ) ∈ Mρ C . Since ψ(Γ ) = 1 and i ∈ / Range(ϕ), we have π(ξ ) ∈ {(ai , 1)n : n ∈ ω \ {0}} ∪ {(ai , 1)(bi , 0)}. One / Mρ C for any n ∈ ω \ {0}, and T i (ai , 1)(bi , 0) ∈ / Mρ C . can show that T i (ai , 1)n ∈ Hence the node w is not labeled with the element ai . Let w be labeled with the element bi . Since (0, 0, 0) ∈ Δ(T i ) and Γ is a deterministic decision tree for the table T i , there exists a complete path ξ in the schema Γ such that χ (π(ξ )) ⊆ {(ai , 0), (bi , 0), (ci , 0)} and T i π(ξ ) ∈ Mρ C . Since ψ(Γ ) = 1 and i ∈ / Range(ϕ), we have π(ξ ) ∈ {(bi , 0)n : n ∈ ω \ {0}} ∪ {(bi , 0)(ci , 0)}. One / Mρ C for any n ∈ ω \ {0}, and T i (bi , 0)(ci , 0) ∈ / Mρ C . can show that T i (bi , 0)n ∈ Hence the node w is not labeled with the element bi . Let w be labeled with the element ci . Since (0, 0, 0) ∈ Δ(T i ) and Γ is a deterministic decision tree for the table T i , there exists a complete path ξ in the schema Γ such that χ (π(ξ )) ⊆ {(ai , 0), (bi , 0), (ci , 0)} and T i π(ξ ) ∈ Mρ C . Since ψ(Γ ) = 1 and i ∈ / Range(ϕ), we have π(ξ ) ∈ {(ci , 0)n : n ∈ ω \ {0}} ∪ {(ci , 0)(ai , 0)}. One / Mρ C for any n ∈ ω \ {0}, and T i (ci , 0)(ai , 0) ∈ / Mρ C . can show that T i (ci , 0)n ∈ Hence the node w is not labeled with the element ci . Thus, we obtain a contradiction. Hence ψρd (T i ) = 1. Assume that the problem Com d (ρ, ψ) is decidable. Then the set Range(ϕ) is recursive which is impossible. Hence the problem Com d (ρ, ψ) is undecidable. Using  Proposition 8.1 we conclude that the problem Des d (ρ, ψ) is undecidable. Proposition 8.6 Let ρ be an enumerated signature and ψ be a computable complexity function of the signature ρ having the property 2. Then the problems

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Com d (ρ, ψ), Des d (ρ, ψ), Com a (ρ, ψ), Des a (ρ, ψ), Com s (ρ, ψ), and Des s (ρ, ψ) are decidable. Proof Let T ∈ Mρ \ Mρ C . We denote by Cρ2 (T ) the set of deterministic schemes Γ of the signature ρ having the following properties: (a) P(Γ ) ⊆ P(T ). (b) Terminal nodes of Γ are labeled with numbers from the set νT (T ) =



¯ . νT (δ)

¯ δ∈Δ(T )

(c) In each complete path in the schema Γ , nodes which are neither the root nor a terminal node are labeled with pairwise different elements. One can show that Cρ2 (T ) is a finite set, and there exists an algorithm which, for a given table T ∈ Mρ \ Mρ C , enumerates all schemes from Cρ2 (T ). In Sect. 2.5, a relation Rρd ⊆ Cρ × Mρ was defined. One can show that this relation is decidable. Let us describe an algorithm for the problem Des d (ρ, ψ) solving. Let T ∈ Mρ . Assume that T ∈ Mρ C . We now define a schema Γ ∈ Cρ . The schema Γ consists of two nodes w1 , w2 and an edge leaving w1 and entering w2 . If Δ(T ) = ∅, then the node w2 is labeled with the number 0. If Δ(T ) = ∅, then the node w2 is labeled with the minimum number from the set Π (T ). Evidently, the schema Γ is a deterministic decision tree for the table T . Taking into account that the function ψ has the property 2 it / Mρ C . Searching through the set is not difficult to show that ψ(Γ ) = ψρd (T ). Let T ∈ Cρ2 (T ) we find a schema Γ which satisfies the following conditions: (i) Γ ∈ Cρ2 (T ), (ii) (Γ, T ) ∈ Rρd , and (iii) ψ(Γ ) = min{ψ(G) : G ∈ Cρ2 (T ), (G, T ) ∈ Rρd }. One can show that each reduced deterministic decision tree for the table T belongs to the set Cρ2 (T ). By Lemma 5.1, there exists a reduced deterministic decision tree Γ0 for the table T such that ψ(Γ0 ) = ψρd (T ). Hence a schema Γ satisfying the conditions (i)– (iii) exists and ψ(Γ ) = ψρd (T ). Thus, the problem Des d (ρ, ψ) is decidable. Using Propositions 8.1 and 8.2 we conclude that the problems Com d (ρ, ψ), Com a (ρ, ψ),  Des a (ρ, ψ), Com s (ρ, ψ), and Des s (ρ, ψ) are decidable. Proposition 8.7 Let ρ be an enumerated signature and ψ be a computable complexity function of the signature ρ having the property 3. Then the problems Com d (ρ, ψ), Des d (ρ, ψ), Com a (ρ, ψ), Des a (ρ, ψ), Com s (ρ, ψ), and Des s (ρ, ψ) are decidable. Proof Let T ∈ Mρ . We now define a set e(T ). If Δ(T ) = ∅, then e(T ) = {0}. If Δ(T ) = ∅, then e(T ) = νT (T ). We denote by Cρ3 (T ) the set of deterministic schemes Γ of the signature ρ having the following properties: (a) P(Γ ) ⊆ P(T ). (b) Terminal nodes of Γ are labeled with numbers from e(T ). (c) h(Γ ) ≤ ψρi (T ) = ψ(μT (1) · · · μT (dim T )). One can show that Cρ3 (T ) is a finite set, and there exists an algorithm which, for a given table T ∈ Mρ , enumerates all schemes from the set Cρ3 (T ). Evidently, the

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111

relation Rρd is decidable. We know also that the function ψ is computable on Cρ . Let us describe an algorithm for the problem Des d (ρ, ψ) solving. Let T ∈ Mρ . Searching through the set Cρ3 (T ) we find a schema Γ which satisfies the following conditions: (i) Γ ∈ Cρ3 (T ), (ii) (Γ, T ) ∈ Rρd , and (iii) ψ(Γ ) = min{ψ(G) : G ∈ Cρ3 (T ), (G, T ) ∈ Rρd }. By Lemma 3.1, there exists a deterministic decision tree Γ0 for that table T such that ψ(Γ0 ) = ψρd (T ). From Lemma 3.1 it follows that ψρd (T ) ≤ ψρi (T ). Taking into account that the function ψ has property 3 we obtain h(Γ0 ) ≤ ψ(Γ0 ). Therefore h(Γ0 ) ≤ ψρi (T ). For each terminal node of the schema Γ0 labeled with a number which does not belong to the set e(T ), we remove this number and label the considered node by the minimum number from e(T ). We denote the obtained schema by Γ1 . One can show that Γ1 ∈ Cρ3 (T ), ψ(Γ1 ) = ψρd (T ), and Γ1 is a deterministic decision tree for the table T . Therefore a schema Γ satisfying the conditions (i)–(iii) exists and ψ(Γ ) = ψρd (T ). Thus, the problem Des d (ρ, ψ) is decidable. Using Propositions 8.1 and 8.2 we conclude that the problems Com d (ρ, ψ), Com a (ρ, ψ), Des a (ρ, ψ),  Com s (ρ, ψ), and Des s (ρ, ψ) are decidable.

8.3 Possible Variants of Algorithmic Problem Behavior Theorem 8.1 (I) Let ρ be an enumerated signature and ψ be a computable complexity function of the signature ρ. Then one of the following three statements holds: (a) Problems Com s (ρ, ψ), Des s (ρ, ψ), Com a (ρ, ψ), Des a (ρ, ψ), Com d (ρ, ψ), and Des d (ρ, ψ) are undecidable. (b) Problems Com s (ρ, ψ), Des s (ρ, ψ), Com a (ρ, ψ), and Des a (ρ, ψ) are decidable, and problems Com d (ρ, ψ) and Des d (ρ, ψ) are undecidable. (c) Problems Com s (ρ, ψ), Des s (ρ, ψ), Com a (ρ, ψ), Des a (ρ, ψ), Com d (ρ, ψ), and Des d (ρ, ψ) are decidable. (II) For each of the statements (a), (b) and (c), there exist an enumerated signature ρ and a computable complexity function ψ of the signature ρ for which the considered statement holds. Proof The part (I) of the theorem statement follows form Propositions 8.1–8.3. The part (II) of the theorem statement follows from Propositions 8.4–8.7. 

8.4 Examples of Problems with Polynomial Complexity Let ρ = (F, k) be a signature and ψ be a complexity function of the signature ρ. The function ψ is called an extremal complexity function if it satisfies the following conditions: ψ(λ) = 0 and ψ( f i1 · · · f im ) = max{ψ( f i j ) : j = 1, . . . , m} for any nonempty word f i1 · · · f im ∈ F ∗ . Let ψ be an extremal complexity function, and A be a finite subset of the set F. We now define the value ψ(A). If A = ∅, then ψ(A) = 0.

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If A = ∅, then ψ(A) = max{ψ( f i ) : f i ∈ A}. Let T ∈ Mρ . We denote by Nˆ Cρ (T ) the set of all tests for the table T , and Θˆ ρ,ψ (T ) = min{ψ(A) : A ∈ Nˆ Cρ (T )}. By Lemma 3.1, Nˆ Cρ (T ) = ∅. Therefore the value Θˆ ρ,ψ (T ) is definite. Lemma 8.2 Let ρ = (F, k) be a signature and ψ be an extremal complexity function of the signature ρ. Then the following statements hold: (a) If T ∈ Mρ , then ψρa (T ) = ψρd (T ) = Θˆ ρ,ψ (T ). (b) If T ∈ Mρ0−1 , then ψρs (T ) = ψρa (T ) = ψρd (T ) = Θˆ ρ,ψ (T ). Proof Let Γ ∈ Cρ . One can show that ψ(Γ ) = ψ(P(Γ )) .

(8.1)

Let T ∈ Mρ . It is not difficult to prove that Θˆ ρ,ψ (T ) = Θρ,ψ (T ) .

(8.2)

By Lemma 3.1, ψρd (T ) ≤ Θρ,ψ (T ). Using Proposition 6.1 we obtain ψρa (T ) ≤ ψρd (T ). By Theorem 6.1, there exists a nondeterministic decision tree Γ for the table T such that ψ(Γ ) = ψρa (T ). Using (8.1) we obtain ψ(P(Γ )) = ψ(Γ ). From Lemma 3.2 it follows that the set P(Γ ) is a test for the table T . Therefore Θˆ ρ,ψ (T ) ≤ ψ(P(Γ )). Hence Θˆ ρ,ψ (T ) ≤ ψρa (T ) ≤ ψρd (T ) ≤ Θρ,ψ (T ) .

(8.3)

The part (a) of the lemma statement follows from (8.3) and (8.2). Let T ∈ Mρ0−1 . Using Proposition 6.1 we obtain ψρs (T ) ≤ ψρa (T ). By Theorem 6.2, there exists a strongly nondeterministic decision tree Γ for the table T such that ψ(Γ ) = ψρs (T ). From (8.1) it follows that ψ(P(Γ )) = ψ(Γ ). Using Lemma 3.2 we conclude that the set P(Γ ) is a test for the table T . Hence Θˆ ρ,ψ (T ) ≤ ψ(P(Γ )). Thus, Θˆ ρ,ψ (T ) ≤ ψρs (T ) ≤ ψρa (T ). The part (b) of the lemma statement follows from these inequalities, from (8.3), and from (8.2).  Let ρ = (F, k) be an enumerated signature and ψ be a computable extremal complexity function of the signature ρ. We will say that ψ has a polynomial complexity if there exists a polynomial time algorithm for the computation of ψ on the set F ∗ . We will say that an algorithmic problem defined on a subset of the set Mρ has a polynomial complexity if there exists a polynomial time algorithm which solves the considered problem on the considered subset of Mρ . We define a problem Des(ρ, ψ) as follows: for a given table T ∈ Mρ , it is required to construct a test A for the table T such that ψ(A) = Θˆ ρ,ψ (T ). Lemma 8.3 Let ρ = (F, k) be an enumerated signature and ψ be a computable extremal complexity function of the signature ρ which has a polynomial complexity. Then the problem Des(ρ, ψ) has a polynomial complexity.

8.4 Examples of Problems with Polynomial Complexity

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Proof We define a problem T es(ρ) as follows: for a given table T ∈ Mρ and a given set A ⊆ P(T ), it is required to recognize if A is a test for the table T . Let us show that T es(ρ) has a polynomial complexity. We now define a set B(T, A) ⊆ Ωρ (T ). If A = ∅, then B(T, A) = {λ}. Let A = ∅ and A = { f i1 , . . . , f im }. Then B(T, A) = {( f i1 , δ1 ) · · · ( f im , δm ) : δ1 , . . . , δm ∈ E k , Δ(T ( f i1 , δ1 ) · · · ( f im , δm )) = ∅}. One can show that the set A is a test for the table T if and only if T α ∈ Mρ C for any α ∈ B(T, A). One can also show that there exists a polynomial time algorithm which, for a given table T ∈ Mρ and a given set A ⊆ P(T ), constructs the set B(T, A) and, for each α ∈ B(T, A), verifies the condition T α ∈ Mρ C . Therefore the problem T es(ρ) has a polynomial complexity. Let T ∈ Mρ . Denote D(T, ψ) = {0} ∪ {ψ( f i ) : f i ∈ P(T )}. For each j ∈ D(T, ψ), set P j (T, ψ) = { f i : f i ∈ P(T ), ψ( f i ) ≤ j}. Let j0 be the minimum number from D(T, ψ) such that the set P j0 (T, ψ) is a test for the table T . Let us show that j0 = ψ(P j0 (T, ψ)) = Θˆ ρ,ψ (T ). Assume the contrary. Then there exists a test A for the table T such that ψ(A) < j0 . Let ψ(A) = j1 . One can show that j1 ∈ D(T, ψ), and the set P j1 (T, ψ) is a test for the table T which is impossible. Taking into account that the problem T es(ρ) has a polynomial complexity, one can show that there exists a polynomial time algorithm which, for a given table T ∈ Mρ , constructs the set D(T, ψ), finds the minimum number j0 ∈ D(T, ψ) such that P j0 (T, ψ) is a test for the table T , and constructs the set P j0 (T, ψ). Hence the problem Des(ρ, ψ) has a polynomial complexity.  Proposition 8.8 Let ρ = (F, k) be an enumerated signature and ψ be a computable extremal complexity function of the signature ρ which has a polynomial complexity. Then each of the problems Com s (ρ, ψ), Des s (ρ, ψ), Com a (ρ, ψ), Des a (ρ, ψ), Com d (ρ, ψ), and Des d (ρ, ψ) has a polynomial complexity. Proof By Lemmas 8.2 and 8.3, problems Com s (ρ, ψ), Com a (ρ, ψ), Com d (ρ, ψ) have a polynomial complexity. Let as describe an algorithm which solves the problems Des s (ρ, ψ), Des a (ρ, ψ), and Des d (ρ, ψ). Let T ∈ Mρ . By Lemma 8.3, there exists a polynomial time algorithm which solves the problem Des(ρ, ψ). Using this algorithm we construct a test A for the table T such that ψ(A) = Θˆ ρ,ψ (T ). We now construct a schema Γ (A, T ) ∈ Cρ . Let α ∈ Ωρ (T ). We now define a set χ(α) ˆ ⊂ F. If α = λ, then χˆ (α) = ∅. If α = λ and α = ( f i1 , δ1 ) · · · ( f im , δm ), then χ(α) ˆ = { f i1 , . . . , f im }. Step 1. We construct a tree consisting of the nodes w1 , w2 and the edge leaving w1 and entering w2 . If Δ(T ) = ∅, then we label the node w2 with the number 0 and denote the obtained schema by Γ (A, T ). Let Δ(T ) = ∅. Then we label the node w2 with the word λ and pass to step 2. Let t steps have been made. We now describe step (t + 1). Step (t + 1). If in the tree G, constructed during step t, there are no nodes labeled with words from Ωρ (T ), then we denote this tree by Γ (A, T ). Otherwise, choose a node w of the tree G which is labeled with a word from Ωρ (T ). Let the node w be labeled with the word α. If T α ∈ Mρ C , then we remove the word α, label the

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node w with the minimum number from the set Π (T α), and pass to step (t + 2). Let Tα ∈ / Mρ C , and f i be the element from the set A\ χˆ (α) with the minimum number i. Remove the word α and label the node w with the element f i . For each δ ∈ E k such that Δ(T α( f i , δ)) = ∅, add to the tree G a node w(δ) and an edge which leaves the node w and enters the node w(δ). This edge is labeled with the number δ. Label the node w(δ) with the word α( f i , δ). Pass to step (t + 2). Let T ∈ Mρ0−1 . We now construct a schema Γ s (A, T ). This schema is obtained from the schema Γ (A, T ) by removal of all nodes end edges that do not belong to complete paths with terminal nodes labeled with the number 1. One can show that the schema Γ (A, T ) is a deterministic decision tree for the table T , and P(Γ (A, T )) ⊆ A. Therefore ψ(Γ (A, T )) ≤ Θˆ ρ,ψ (T ). Using Lemma 8.2 we conclude that the schema Γ (A, T ) is a solution of the problems Des d (ρ, ψ) and Des a (ρ, ψ). One can show that the schema Γ s (A, T ) is a strongly nondeterministic decision tree for the table T , and P(Γ s (A, T )) ⊆ A. Therefore ψ(Γ s (A, T )) ≤ Θˆ ρ,ψ (T ). Using Lemma 8.2 we conclude that the schema Γ s (A, T ) is a solution of the problem Des s (ρ, ψ) for the table T . Let T ∈ Mρ and Δ(T ) = ∅. One can show that Δ(T π(ξ )) = ∅ for each complete path ξ in the schema Γ (A, T ). Therefore the number of complete paths in the schema Γ (A, T ) is at most Nρ (T ). Evidently, in each complete path in the schema Γ (A, T ), there are at most dim T + 2 nodes. Therefore the number of nodes in the schema Γ (A, T ) is at most Nρ (T )(dim T + 2), and the number of steps under the construction of the schema Γ (A, T ) is at most Nρ (T )(dim T + 2) + 1. Evidently, the obtained bound on the number of steps holds for the case Δ(T ) = ∅ too. Using this bound one can show that the considered algorithm for solving of the problems Des s (ρ, ψ), Des a (ρ, ψ), and Des d (ρ, ψ) has a polynomial time complexity. 

8.5 Examples of NP-Hard Problems An undirected graph without multiple edges and loops is a pair G = (W, R), where W is a nonempty finite set and R is a set of two-element subsets of the set W . The elements of the set W are called vertices and the elements of the set R are called edges of the graph G. Let V = {vi : i ∈ ω}. We denote by H V the set of all undirected graphs without multiple edges and loops G = (W, R) such that W ⊂ V and if |W | = n, then W = {v1 , . . . , vn }. Let G = (W, R) ∈ H V , and U ⊆ W . The set U will be called a vertex cover for the graph G if it satisfies the following conditions: (a) If U = ∅, then R = ∅. (b) If R = ∅, then, for each edge {vi , v j } ∈ R, at least one of the relations vi ∈ U and v j ∈ U holds. We denote by PC(G) the minimum cardinality of a vertex cover for the graph G. Let ρ = (F, k) be an enumerated signature. We now describe a table Tρ (G) ∈ Mρ0−1 corresponding to the graph G. Let W = {v1 , . . . , vn }, r ∈ R, and r = {vi1 , vi2 }. We

8.5 Examples of NP-Hard Problems

115

¯ ) the tuple from E 2n in which the components with numbers i 1 and i 2 are denote by δ(r equal to 1 and all other components are equal to 0. Then dim Tρ (G) = n, Δ(Tρ (G)) = ¯ ) : r ∈ R}, μTρ (G) (1) = f 1 , . . . , μTρ (G) (n) = f n , νTρ (G) (0˜ n ) = {1}, and {0˜ n } ∪ {δ(r ¯ νTρ (G) (δ(r )) = {0} for any r ∈ R. Lemma 8.4 Let ρ = (F, k) be an enumerated signature and G ∈ H V . Then h dρ (Tρ (G)) = h aρ (Tρ (G)) = h sρ (Tρ (G)) = PC(G) . Proof Let G = (W, R) and W = {v1 , . . . , vn }. Denote Q = Tρ (G). By Lemma 8.1, h dρ (Q(0˜ n , 1)) = h aρ (Q(0˜ n , 1)) = h sρ (Q(0˜ n , 1)). One can show that Q(0˜ n , 1) = Q. Therefore h dρ (Q) = h aρ (Q) = h sρ (Q). Using Theorem 6.2 we obtain h sρ (Q) = Mρ,h (Q, 0˜ n ). Evidently, Mρ,h (Q, 0˜ n ) = 0 if and only if R = ∅. Let R = ∅ and j1 , . . . , jm ∈ {1, . . . , n}. One can show that Q( f j1 , 0) · · · ( f jm , 0) ∈ Mρ C if and only if the set {v j1 , . . . , v jm } is a vertex cover for the graph G. Hence Mρ,h (Q, 0˜ n ) = PC(G).  Proposition 8.9 Let ρ = (F, k) be en enumerated signature. Then each of problems Com s (ρ, h), Des s (ρ, h), Com a (ρ, h), Des a (ρ, h), Com d (ρ, h), and Des d (ρ, h) is N P-hard. Proof Let us consider the vertex cover problem: for a given graph G = (W, R) ∈ H V and given m ∈ ω, m ≤ |W |, it is required to verify the inequality PC(G) ≤ m. From results of [1] it follows that this problem is N P-hard. By Lemma 8.4, there exists a polynomial time reduction of the vertex cover problem to each of the following problems: Com s (ρ, h), Des s (ρ, h), Com a (ρ, h), Des a (ρ, h), Com d (ρ, h),  and Des d (ρ, h). Therefore the considered problems are N P-hard.

References 1. Garey, M.R., Jonson, D.S.: Computers and Intractability. A Guide to the Theory of NPCompleteness. W.H. Freeman and Co., San Francisco (1979) 2. Moshkov, M.: Decision Trees. Theory and Applications (in Russian). Nizhny Novgorod University Publishers, Nizhny Novgorod (1994) 3. Moshkov, M.: Time complexity of decision trees. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III, Lecture Notes in Computer Science, vol. 3400, pp. 244–459. Springer, Berlin (2005) 4. Rogers Jr., H.: Theory of Recursive Functions and Effective Computability. MIT Press, Cambridge, MA, USA (1987)

Part II

Decision Trees for Problems. Local Approach

In this part, we develop local approach to the study of decision trees for problems where decision trees can use only attributes from problem representation. This part consists of eight chapters. In Chap. 9, we discuss main notions and notation for the local approach to the study of decision trees for problems. We consider information systems, decision trees over information systems, problems over information systems, classes of information systems, relationships among different parameters of problems, and upper and lower types of these relationships. In Chap. 10, we consider some reductions which will be used later in the investigations of decision trees for problems. We show that, in the frameworks of the local approach, the study of decision trees for problems can be reduced to the study of decision trees for decision tables. We prove that, instead of arbitrary classes of information systems, we can consider classes containing only one information system. We also show that the matrix of upper local bounds for a sccf-triple completely defines its matrix of lower local bounds and vice versa. In particular, the local upper type of a sccf-triple completely defines its local lower type and vice versa. ˆ τbc , b, c ∈ {i, d, a, s}, In Chap. 11, for restricted sccf-triples τ , we study functions  located in the matrix of upper local bounds for the triple τ on the main diagonal and below. For each of these functions, we list all possible upper types and consider criterion for each such type. In several cases, we give upper and lower bounds for the considered functions. ˆ τbc , b, c ∈ {i, d, a, s}, In Chap. 12, for restricted sccf-triples τ , we study functions  located in the matrix of upper local bounds for the triple τ over the main diagonal. For each of these functions, we list all possible upper types and consider criterion for each such type. In several cases, we give upper and lower bounds for the considered functions. In Chap. 13, we describe all possible six local upper types of restricted sccf-triples. For each of these six types, we consider the criterion of its implementation and give an example of a restricted sccf-triple with this type. In Chap. 14, for a given signature ρ and each possible local upper type of restricted sccf-triples Tpi, i ∈ {1, . . . , 6}, we consider the set Wˆ ρ (i) of restricted sccf-triples ˆ τ = Tpi. For each pair (b, c) ∈ {i, d, a, s}2 such that in the matrix τ with Typ 

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Tpi at the intersection of the row with index b and the column with index c either ˆ τbc true for any λ or χ stays, we study upper and lower bounds on the function  sccf-triple τ ∈ Wˆ ρ (i). In Chap. 15, we describe all possible six local lower types tp1, . . . , tp6 of restricted sccf-triples which correspond to the local upper types Tp1, . . . , Tp6, respectively. For a given signature ρ, each local lower type tpi, i ∈ {1, . . . , 6}, and each pair (b, c) ∈ {i, d, a, s}2 such that in the matrix tpi at the intersection of the row with index b and the column with index c either μ or γ stays, we study upper and lower ˆ ˆ bc bounds on the function  τ true for any sccf-triple τ ∈ Wρ (i). In Chap. 16, we study algorithmic problems related to the local approach to the investigation of decision trees: problems of computation of the minimum complexity of deterministic, nondeterministic, and strongly nondeterministic decision trees, problems of construction of decision trees with the minimum complexity, and the problem of solvability of systems of equations over information systems. We study relationships among these problems and describe all variants of algorithmic problem behavior. We also discuss an algorithm for decision table construction and two algorithms that construct deterministic decision trees.

Chapter 9

Basic Definitions and Notation

In this chapter, we discuss main notions and notation for the local approach to the study of decision trees for problems. We consider information systems, decision trees over information systems, problems over information systems, classes of information systems, relationships among different parameters of problems, and upper and lower types of these relationships. The notion of information system considered in this chapter is close to the notion introduced by Pawlak [4]. Upper types of functions were introduced in [2]. Types of functions slightly different from upper and lower types were investigated in [1, 3].

9.1 Information Systems Let ρ = (F, k) be a signature. A pair U = (A, γ ) will be called an information system of the signature ρ if the following conditions hold: (a) A is a nonempty set. (b) γ is a mapping from F to E kA , where E kA is the set of mappings from A to E k . The set A will be called the universe of the information system U , and the mapping γ will be called the interpretation of the signature ρ in A. The mappings γ ( f ), f ∈ F, will be called attributes of the information system U . Further, instead of γ ( f ), we will often write f U or even f if it is clear which information system being considered. The set A and the mapping γ defining the information system U will be denoted by AU and γU . Let U be an information system of the signature ρ. We correspond to each word α ∈ Ωρ a subset AU (α) of the set AU . If α = λ, then AU (α) = AU . Let α = λ and α = ( f 1 , δ1 ) · · · ( f m , δm ). Then AU (α) is the set of solutions over A of the equation system { f 1U (x) = δ1 , . . . , f mU (x) = δm } .

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_9

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9.2 Decision Trees Let U = (A, γ ) be an information system of the signature ρ. A pair D = (Γ, U ) will be called a decision tree of the signature ρ over U (a decision tree over U ) if Γ is a schema of decision tree of the signature ρ. The schema Γ will be called the schema of the decision tree D.

9.3 Problem Schemes and Problems A tuple z = (v, f 1 , . . . , f n ) will be called a problem schema of the signature ρ if n ∈ ω \ {0}, f 1 , . . . , f n ∈ F and ν : E kn → P(ω). Set P(z) = { f 1 , . . . , f n }. We denote by ρ the set of all problem schemes of the signature ρ, and by ρ0−1 we denote the set of all problem schemes z = (ν, f 1 , . . . , f n ) of the signature ρ such ¯ = {0} or ν(δ) ¯ = {1} for any δ¯ ∈ E kn . that ν(δ) Let U be an information system of the signature ρ. A pair t = (z, U ), where z ∈ ρ , will be called a problem of the signature ρ over U (a problem over U ). The tuple z will be called the schema of the problem t. Let us correspond a mapping ϕt : AU → P(ω) to the problem t. Let z = (v, f 1 , . . . , f n ) and a ∈ AU . Then ϕt (a) = v( f 1U (a), . . . , f nU (a)). For a given a ∈ AU , we should find a number (decision) from the set ϕt (a). Let z ∈ ρ0−1 . The problem t = (z, U ) will be called nondegenerate if there exists an element a ∈ AU such that ϕt (a) = {1}.

9.4 Decision Trees Solving Problems Let U be an information system of the signature ρ, t = (z, U ) be a problem of the signature ρ over U , and D = (Γ, U ) be a decision tree of the signature ρ over U . Let ξ be a complete path in Γ and a ∈ AU . We say that ξ accepts a if a ∈ AU (π(ξ )). We will say that the decision tree D solves the problem t nondeterministically if the following conditions hold:  (a) ξ ∈Ξ (Γ ) AU (π(ξ )) = AU (each element from AU is accepted by at least one complete path in Γ ). (b) For any complete path ξ ∈ Ξ (Γ ), either AU (π(ξ )) = ∅ or the number d attached to the terminal node of the path ξ belongs to the set a∈AU (π(ξ )) ϕt (a) (the number d belongs to the set ϕt (a) for each element a accepted by ξ ). We will say that the decision tree D solves the problem t deterministically if the schema of the tree D is deterministic, and the decision tree D solves the problem t nondeterministically. Let z ∈ ρ0−1 and t = (z, U ) be a nondegenerate problem. We will say that the decision tree D solves the problem t strongly nondeterministically if the following conditions hold:

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 (a) ξ ∈Ξ (Γ ) AU (π(ξ )) = {a : a ∈ AU , ϕt (a) = {1}} (an element a ∈ AU is accepted by at least one complete path in Γ if and only if ϕt (a) = {1}). (b) All terminal nodes of the schema of the decision tree D are labeled with the number 1.

9.5 Decision Tree Schemes Corresponding to Problem Schemes We study not only individual information systems by also classes of information systems. Let K be a nonempty class (set) of information systems of the signature ρ. We denote by ρ0−1 (K ) the set of problem schemes z ∈ ρ0−1 having the following property: there exists an information system U ∈ K such that the problem (z, U ) is nondegenerate. Let Γ ∈ Cρ and z ∈ ρ . We will say that the schema Γ corresponds nondeterministically to the problem schema z relative to the class K if, for any information system U ∈ K , the decision tree (Γ, U ) solves the problem (z, U ) nondeterministically. We will say that the schema Γ corresponds deterministically to the problem schema z relative to the class K if, for any information system U ∈ K , the decision tree (Γ, U ) solves the problem (z, U ) deterministically. Let z ∈ ρ0−1 (K ). We will say that the schema Γ corresponds strongly nondeterministically to the problem schema z relative to the class K if, for any information system U ∈ K , the decision tree (Γ, U ) solves the problem (z, U ) strongly nondeterministically.

9.6 Complexity Functions Let ψ be a complexity function of the signature ρ, and K be a nonempty class of information systems of the signature ρ. d a s ⊆ Cρ × ρ , Rˆ ρ,K ⊆ Cρ × ρ , and Rˆ ρ,K ⊆ Cρ × We now define relations Rˆ ρ,K d 0−1 ˆ ρ (K ). Let (Γ, z) ∈ Cρ × ρ . Then (Γ, z) ∈ Rρ,K if and only if P(Γ ) ⊆ P(z) and the schema Γ corresponds deterministically to the problem schema z relative to a if and only if P(Γ ) ⊆ P(z) and the class K . The pair (Γ, z) belongs to the set Rˆ ρ,K the schema Γ corresponds nondeterministically to the problem schema z relative to s if and only if P(Γ ) ⊆ the class K . Let (Γ, z) ∈ Cρ × ρ0−1 (K ). Then (Γ, z) ∈ Rˆ ρ,K P(z) and the schema Γ corresponds strongly nondeterministically to the problem schema z relative to the class K .

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i d a We now define functions ψˆ ρ,K : ρ → ω, ψˆ ρ,K : ρ → ω, ψˆ ρ,K : ρ → ω, s i 0−1 ˆ (z) = and ψρ,K : ρ (K ) → ω. Let z ∈ ρ and z = (ν, f 1 , . . . , f n ). Then ψˆ ρ,K d d a ˆ ˆ ˆ ψ( f 1 · · · f n ), ψρ,K (z) = {min ψ(Γ ) : (Γ, z) ∈ Rρ,K }, and ψρ,K (z) = {min ψ(Γ ) : a s s (Γ, z) ∈ Rˆ ρ,K }. Let z ∈ ρ0−1 (K ). Then ψˆ ρ,K (z) = {min ψ(Γ ) : (Γ, z) ∈ Rˆ ρ,K }.

9.7 Matrices of Upper and Lower Local Bounds for Sccf-Triple Let ρ be a signature, K be a nonempty class of information systems of the signature ρ, and ψ be a complexity function of the signature ρ. Denote τ = (ρ, K , ψ). Such triples will be called later sccf-triples. Here sccf are the first letters of the words signature, class, complexity function. The triple τ will be called restricted if the complexity function ψ is restricted (has the properties Λ1, Λ2, Λ3, and Λ4). Let b, c ∈ {i, d, a, s}. We now define partial functions Ψˆ τbc : ω → ω and Φˆ τb,c : b c (z) and ψˆ ρ,K (z). Let ω → ω that describe relationships between parameters ψˆ ρ,K n ∈ ω. If b = s and c = s, then b c Ψˆ τbc (n) = max{ψˆ ρ,K (z) : z ∈ ρ , ψˆ ρ,K (z) ≤ n} , b c Φˆ τbc (n) = min{ψˆ ρ,K (z) : z ∈ ρ , ψˆ ρ,K (z) ≥ n} .

If b = s or c = s, then b c Ψˆ τbc (n) = max{ψˆ ρ,K (z) : z ∈ ρ0−1 (K ), ψˆ ρ,K (z) ≤ n} , b c Φˆ τbc (n) = min{ψˆ ρ,K (z) : z ∈ ρ0−1 (K ), ψˆ ρ,K (z) ≥ n} . c If the value Ψˆ τbc (n) is defined, then from the inequality ψˆ ρ,K (z) ≤ n it follows b bc the unimprovable inequality ψˆ ρ,K (z) ≤ Ψˆ τ (n). If the value Φˆ τbc (n) is defined, then c b (z) ≥ n it follows the unimprovable inequality ψˆ ρ,K (z) ≥ from the inequality ψˆ ρ,K bc ˆ Φτ (n). We denote by Ψˆ τ the matrix with four rows and four columns in which rows from the top to the bottom and columns from the left to the right are labeled with indices i, d, a, s, and at the intersection of the row with index b ∈ {i, d, a, s} and the column with index c ∈ {i, d, a, s} the function Ψˆ τbc is placed. The matrix Ψˆ τ will be called the matrix of upper local bounds for the triple τ . We denote by Φˆ τ the matrix with four rows and four columns in which rows from the top to the bottom and columns from the left to the right are labeled with indices

9.7 Matrices of Upper and Lower Local Bounds for Sccf-Triple

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i, d, a, s, and at the intersection of the row with the index b ∈ {i, d, a, s} and the column with the index c ∈ {i, d, a, s} the function Φˆ τbc is placed. The matrix Φˆ τ will be called the matrix of lower local bounds for the triple τ .

9.8 Types of Functions and Sccf-Triples Let f be a partial function from ω to ω. We denote by Arg f the domain of f . We now define the value Typ f ∈ {ε, λ, χ , ω} which will be called the upper type of the function f . • If Arg f is an infinite set and there exists c ∈ ω such that f (n) ≤ c for any n ∈ Arg f , then Typ f = ε. • If Arg f is an infinite set, there is no c ∈ ω such that f (n) ≤ c for any n ∈ Arg f , and the set {n : n ∈ Arg f, f (n) ≥ n} is finite, then Typ f = λ. • If the set {n : n ∈ Arg f, f (n) ≥ n} is infinite, then Typ f = χ . • If the set Arg f is finite, then Typ f = ω. Define the value typ f ∈ {ε, γ , μ, ω} which will be called the lower type of the function f . • If Arg f is an infinite set and there exists c ∈ ω such that f (n) ≤ c for any n ∈ Arg f , then typ f = ε. • If the set {n : n ∈ Arg f, f (n) ≤ n} is an infinite set and there is no c ∈ ω such that f (n) ≤ c for any n ∈ Arg f , then typ f = γ . • If Arg f is an infinite set and the set {n : n ∈ Arg f, f (n) ≤ n} is finite, then Typ f = μ. • If the set Arg f is finite, then typ f = ω. Let ρ be a signature, K be a nonempty class of information systems of the signature ρ, and ψ be a complexity function of the signature ρ. Denote τ = (ρ, K , ψ). We denote by Typ Ψˆ τ the matrix with four rows and four columns in which rows from the top to the bottom and columns from the left to the right are labeled with indices i, d, a, s, and at the intersection of the row with index b ∈ {i, d, a, s} and the column with index c ∈ {i, d, a, s} the value Typ Ψˆ τbc is placed. The matrix Typ Ψˆ τ will be called the local upper type of the triple τ . We denote by typ Φˆ τ the matrix with four rows and four columns in which rows from the top to the bottom and columns from the left to the right are labeled with indices i, d, a, s, and at the intersection of the row with index b ∈ {i, d, a, s} and the column with index c ∈ {i, d, a, s} the value typ Φˆ τbc is placed. The matrix typ Φˆ τ will be called the local lower type of the triple τ .

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References 1. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity, Global approach. Fundam. Inform. 25(2), 201–214 (1996) 2. Moshkov, M.: Unimprovable upper bounds on time complexity of decision trees. Fundam. Inform. 31(2), 157–184 (1997) 3. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity, Local approach. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets IV, Lecture Notes in Computer Science, vol. 3700, pp. 125–143. Springer, Berlin (2005) 4. Pawlak, Z.: Information systems theoretical foundations. Inf. Syst. 6(3), 205–218 (1981)

Chapter 10

Main Reductions

In this chapter, we consider some reductions which will be used later in the investigations of decision trees for problems. We show that, in the frameworks of the local approach, the study of decision trees for problems can be reduced to the study of decision trees for decision tables. We prove that, instead of arbitrary classes of information systems, we can consider classes containing only one information system. We also show that the matrix of upper local bounds for a sccf-triple completely defines its matrix of lower local bounds and vice versa. In particular, the local upper type of a sccf-triple completely defines its local lower type and vice versa. Similar result was obtained in [1, 2] for types of functions slightly different from upper and lower types considered here.

10.1 Decision Trees and Decision Tree Schemes Let ρ = (F, k) be a signature and z = (ν, f 1 , . . . , f n ) be a problem schema of the signature ρ. Let ( f i1 , . . . , f im ) be a tuple satisfying the following conditions: (a) { f i1 , . . . , f im } = { f 1 , . . . , f n }. (b) For any l, t ∈ {1, . . . , m}, if l = t, then f il = f it . (c) i 1 < i 2 < · · · < i m . (d) For any l ∈ {1, . . . , m} and j ∈ {1, . . . , n}, if j < il , then f j = f il . We now define a mapping γ : E km → P(ω). Let (δ1 , . . . , δm ) ∈ E km . Let us define a tuple (σ1 , . . . , σn ) ∈ E kn . Let j ∈ {1, . . . , n} and f j = f il for l ∈ {1, . . . , m}. Then σ j = δl and γ (δ1 , . . . , δm ) = ν(σ1 , . . . , σn ). Denote υ(z) = (γ , f i1 , . . . , f im ). The tuple υ(z) will be called the normal form of the problem schema z. Let K be a nonempty class of information systems of the signature ρ and z be a problem schema of the signature ρ. We now define a decision table Tρ (z, K ) of the signature ρ. Let υ(z) = (ν, f 1 , . . . , f n ). Then dim Tρ (z, K ) = n and μTρ (z,K ) (i) = f i for any i ∈ {1, . . . , n}. The set Δ(Tρ (z, K )) coincides with the set of tuples © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_10

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(δ1 , . . . , δn ) ∈ E kn having the following property: there exists a system U ∈ K such that the system of equations { f 1U (x) = δ1 , . . . , f nU (x) = δn } has a solution over the set AU . The mapping νTρ (z,K ) is the restriction of the mapping ν to the set Δ(Tρ (z, K )). Theorem 10.1 Let ρ = (F, k) be a signature, K be a nonempty class of information systems of the signature ρ, z be a problem schema of the signature ρ, and Γ be a decision tree schema of the signature ρ such that P(Γ ) ⊆ P(z). Then the following statements hold: (a) The schema Γ corresponds nondeterministically to the problem schema z relative to the class K if and only if the schema Γ is a nondeterministic decision tree for the table Tρ (z, K ). (b) The schema Γ corresponds deterministically to the problem schema z relative to the class K if and only if the schema Γ is a deterministic decision tree for the table Tρ (z, K ). (c) If z ∈ ρ0−1 (K ), then the schema Γ corresponds strongly nondeterministically to the problem schema z relative to the class K if and only if the schema Γ is a strongly nondeterministic decision tree for the table Tρ (z, K ). Proof Let υ(z) = (ν, f 1 , . . . , f n ). One can show that ϕ(z,U ) (a) = ϕ(υ(z),U ) (a) for any information system U ∈ K and any a ∈ AU . Set T = Tρ (z, K ). Let U ∈ K and a ∈ AU . Denote δ¯U (a) = ( f 1U (a), . . . , f nU (a)). Let ξ ∈ Ξ (Γ ). One can show that a ∈ AU (π(ξ )) if and only if δ¯U (a) ∈ Δ(T π(ξ )). (a) Let us consider the following four statements:  (a.1) There exists a system U ∈ K such that ξ ∈Ξ (Γ ) AU (π(ξ )) = AU . (a.2) There exists a system U ∈ K and a path ξ ∈ Ξ (Γ ) such that AU (π(ξ )) = ∅ and  the number attached to the terminal node of the path ξ does not belong to the set a∈AU (π(ξ )) ϕ(υ(z),U ) (a). (a.3) ξ ∈Ξ (Γ ) Δ(T π(ξ )) = Δ(T ). (a.4) There exists a path ξ ∈ Ξ (Γ ) such that Δ(T π(ξ )) = ∅ and the number attached to the terminal node of the path ξ does not belong to the set Π (T π(ξ )). Let us show that the statement (a.1) holds if and only if the statement (a.3) holds. Let the statement (a.1) hold. Then there exists an element a ∈ AU such that / Δ(T π(ξ )) for any ξ ∈ Ξ (Γ ). a∈ / AU (π(ξ )) for any ξ ∈ Ξ (Γ ). Therefore δ¯U (a) ∈ Evidently, δ¯U (a) ∈ Δ(T ). Hence the statement (a.3) holds. Let the statement (a.3) / hold. Then there exists a system U ∈ K and an element a ∈ AU such that δ¯U (a) ∈ Δ(T π(ξ )) for any ξ ∈ Ξ (Γ ). Therefore a ∈ / AU (π(ξ )) for any ξ ∈ Ξ (Γ ). Hence the statement (a.1) holds. Let us show that the statement (a.2) holds if and only if the statement (a.4) holds. Let the statement (a.2) hold and the terminal node of the path ξ be labeled / with a number m. Then there exists an element a ∈ AU (π(ξ )) such that m ∈ ϕ(υ(z),U ) (a). Evidently, δ¯U (a) ∈ Δ(T π(ξ )) and νT (δ¯U (a)) = ϕ(υ(z),U ) (a). Therefore

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m∈ / Π (T π(ξ )). Hence the statement (a.4) holds. Let the statement (a.4) hold and the terminal node of the path ξ be labeled with a number m. Then there exist an information system U ∈ K and an element α ∈ AU such that δ¯U (a) ∈ Δ(T π(ξ )) and m∈ / νT (δ¯U(a)). It is clear that a ∈ AU (π(ξ )) and ϕ(υ(z),U ) (a) = νT (δ¯U (a)). Therefore m ∈ / a∈AU (π(ξ )) ϕ(υ(z),U ) (a). Hence the statement (a.2) holds. a Let us show that (Γ, υ(z)) ∈ Rˆ ρ,K if and only if (Γ, T ) ∈ Rρa . Let (Γ, υ(z)) ∈ / a ˆ Rρ,K . Then at least one of the statements (a.1) and (a.2) holds. Hence at least one / Rρa . of the statements (a.3) and (a.4) holds. Therefore (Γ, T ) ∈ / Rρa . Let (Γ, T ) ∈ Then at least one of the statements (a.3) and (a.4) holds. Hence at least one of the a . statements (a.1) and (a.2) holds. Therefore (Γ, υ(z)) ∈ / Rˆ ρ,K a a One can show that (Γ, υ(z)) ∈ Rˆ ρ,K if and only if (Γ, z) ∈ Rˆ ρ,K . Thus, the statement (a) of the theorem holds. d a . Then Γ is a deterministic schema and (Γ, z) ∈ Rˆ ρ,K . (b). Let (Γ, z) ∈ Rˆ ρ,K By the above, (Γ, T ) ∈ Rρa . Hence (Γ, T ) ∈ Rρd . Let (Γ, T ) ∈ Rρd . Then Γ is a a . Hence (Γ, z) ∈ deterministic schema and (Γ, T ) ∈ Rρa . By the above, (Γ, z) ∈ Rˆ ρ,K d ˆ Rρ,K . Thus, the statement (b) of the theorem holds. (c) Let us consider the following three statements: ¯ = {1}}. (c.1) ξ ∈Ξ (Γ ) Δ(T π(ξ )) = {δ¯ : δ¯ ∈ Δ(T ), νT (δ) (c.2) There exists a terminal node of the schema Γ which is labeled with a number that does not equal to 1.  (c.3) There exists a system U ∈ K such that ξ ∈Ξ (Γ ) AU (π(ξ )) = {a : a ∈ AU , ϕ(υ(z),U ) (a) = {1}}. Let z ∈ ρ0−1 (K ). In this case T ∈ Mρ0−1 . Let us show that the statement (c.1) holds if and only if the statement (c.3) holds. One can show that the statement (c.1) holds if and only if at least one of the following statements holds: (c.1.1) There exists a system U ∈ K and an element a ∈ AU such that νT (δ¯U (a)) = / Δ(T π(ξ )) for any ξ ∈ Ξ (Γ ). {1} and δ¯U (a) ∈ (c.1.2) There exists a system U ∈ K , an element a ∈ AU , and a path ξ ∈ Ξ (Γ ) such that νT (δ¯U (a)) = {0} and δ¯U (a) ∈ Δ(T π(ξ )). One can show that the statement (c.3) holds if and only if at least one of the following statements holds: (c.3.1) There exists a system U ∈ K and an element a ∈ AU such that ϕ(υ(z),U ) (a) = {1} and a ∈ / AU (π(ξ )) for any ξ ∈ Ξ (Γ ). (c.3.2) There exists a system U ∈ K , an element a ∈ AU , and a path ξ ∈ Ξ (Γ ) such that ϕ(υ(z),U ) (a) = {0} and a ∈ AU (π(ξ )). Finally, one can show that the statement (c.1.1) holds if and only if the statement (c.3.1) holds, and the statement (c.1.2) holds if and only if the statement (c.3.2) holds. Hence the statement (c.1) holds if and only if the statement (c.3) holds.

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s Let us show that (Γ, υ(z)) ∈ Rˆ ρ,K if and only if (Γ, T ) ∈ Rρs . Let (Γ, υ(z)) ∈ / s ˆ Rρ,K . Then at least one of the statements (c.1) and (c.2) holds. Hence at least one / Rρs . of the statements (c.2) and (c.3) holds. Therefore (Γ, T ) ∈ / Rρs . Let (Γ, T ) ∈ Then at least one of the statements (c.2) and (c.3) holds. Hence at least one of the s . statements (c.1) and (c.2) holds. Therefore (Γ, υ(z)) ∈ / Rˆ ρ,K s s ˆ One can show that (Γ, υ(z)) ∈ Rρ,K if and only if (Γ, z) ∈ Rˆ ρ,K . Thus, the statement (c) of the theorem holds. 

10.2 Arbitrary Classes of Information Systems and One-Element Classes of Information Systems Let ρ = (F, k) be a signature and K be a nonempty class of information systems of the signature ρ. We define an information system V (K ) of the signature ρ in the following way: A V (K ) = U ∈K {(a, U ) : a ∈ AU }, and f V (K ) ((a, U )) = f U (a) for any f ∈ F and (a, U ) ∈ A V (K ) . Denote K˜ = {V (K )}. Theorem 10.2 Let ρ = (F, k) be a signature and K be a nonempty class of informad d a a s s tion systems of the signature ρ. Then Rˆ ρ,K = Rˆ ρ, , Rˆ ρ,K = Rˆ ρ, , and Rˆ ρ,K = Rˆ ρ, . K˜ K˜ K˜ Proof Let b ∈ {d, a, s}. We now define the set ρb . If b = s, then ρb = ρ0−1 (K ). If b = s, then ρb = ρ . Let z ∈ ρb and Γ ∈ Cρ . From Theorem 10.1 it follows b b if and only if (Γ, Tρ (z, K )) ∈ Rρb , and (Γ, z) ∈ Rˆ ρ, if and only that (Γ, z) ∈ Rˆ ρ,K K˜ if (Γ, Tρ (z, K˜ )) ∈ Rρb . One can show that Tρ (z, K ) = Tρ (z, K˜ ). Therefore (Γ, z) ∈ b if and only if (Γ, z) ∈ Rˆ b .  Rˆ ρ,K ρ, K˜

Corollary 10.1 Let ρ be a signature, K be a nonempty class of information systems of the signature ρ, ψ be a complexity function of the signature ρ, τ = (ρ, K , ψ), ˆτ =  ˆ τ˜ and  ˆτ = ˆ τ˜ . and τ˜ = (ρ, K˜ , ψ). Then 

10.3 Upper and Lower Bounds Remind that, for an arbitrary function f , Arg f is the domain of f . Let S be a nonempty set, a : S → ω and b : S → ω. We now define partial functions ψ ab : ω → ω and ϕ ba : ω → ω. Let n ∈ ω. Then ψ ab (n) = max{a(s) : s ∈ S, b(s) ≤ n} , ϕ ba (n) = min{b(s) : s ∈ S, a(s) ≥ n} .

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The function ψ ab will be called total from above if, for any n ∈ ω, the set {a(s) : s ∈ S, b(s) ≤ n} is finite. The notation ψ ab (n) = ∞ will mean that the set {a(s) : s ∈ S, b(s) ≤ n} is infinite. The notation ψ ab (n) = ∅ will mean that the set {a(s) : s ∈ S, b(s) ≤ n} is empty. If the function ψ ab is not total from above, then we denote by l(ψ ab ) the minimum number n such that ψ ab (n) = ∞. Let n, m ∈ ω and n ≤ m. Set ω(n) = {c : c ∈ ω, c ≥ n} and ω(n, m) = {n, n + 1, . . . , m}. Remark 10.1 One can show that (a) For any m, n ∈ Arg ψ ab , if m ≤ n, then ψ ab (m) ≤ ψ ab (n). (b) Typ ψ ab = ω if and only if the function ψ ab is not total from above. (c) If Typ ψ ab = ω, then Arg ψ ab = ∅ or there exist m, n ∈ ω such that m ≤ n and Arg ψ ab = ω(m, n). (d) If Typ ψ ab = ω, then there exists n ∈ ω such that Arg ψ ab = ω(n). Remark 10.2 One can show that (a) For any m, n ∈ Arg ϕ ba , if m ≤ n, then ϕ ba (m) ≤ ϕ ba (n). (b) If typ ϕ ba = ω, then there exists n ∈ ω such that Arg ϕ ba = ω(0, n). (c) If typ ϕ ba = ω, then Arg ϕ ba = ω. We denote by F Z the set of functions f having the following properties: (a) There exists m ∈ ω such that Arg f = ω(m). (b) f (n) ∈ ω for any n ∈ Arg f . (c) If c, d ∈ Arg f and c ≤ d, then f (c) ≤ f (d). (d) There is no c ∈ ω such that f (n) ≤ c for any n ∈ Arg f . Remark 10.3 One can show that (a) If Typ ψ ab = ω and Typ ψ ab = ε, then ψ ab ∈ F Z . (b) If typ ϕ ba = ω and typ ϕ ba = ε, then ϕ ba ∈ F Z . Proposition 10.1 For any n ∈ ω, (a) If ψ ab (t) < n for any t ∈ Arg ψ ab and the function ψ ab is total from above, then the value ϕ ba (n) is indefinite. (b) If ψ ab (t) < n for any t ∈ Arg ψ ab and the function ψ ab is not total from above, then ϕ ba (n) = l(ψ ab ). (c) If there exists t ∈ Arg ψ ab such that ψ ab (t) ≥ n, then ϕ ba (n) = min{t : t ∈ Arg ψ ab , ψ ab (t) ≥ n}. Proof Let n ∈ ω. Let ψ ab (t) < n for any t ∈ Arg ψ ab and the function ψ ab be total from above. Then a(s) < n for any s ∈ S. Hence the value ϕ ba (n) is indefinite. Let ψ ab (t) < n for any t ∈ Arg ψ ab and the function ψ ab be not total from above. It is clear that, for any s ∈ S, if a(s) ≥ n, then b(s) ≥ l(ψ ab ). Therefore if the value ϕ ba (n) is definite, then ϕ ba (n) ≥ l(ψ ab ). Evidently, {a(s) : s ∈ S, b(s) = l(ψ ab )} is an infinite set. Hence the value ϕ ba (n) is definite and ϕ ba (n) ≤ l(ψ ab ). Thus, ϕ ba (n) = l(ψ ab ). Let the set {t : t ∈ Arg ψ ab , ψ ab (t) ≥ n} be not empty and m be the minimum number from this set. Then m is the minimum number having the following proper ties: there exists s ∈ S such that b(s) = m and a(s) ≥ n. Hence ϕ ba (n) = m.

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Proposition 10.2 For any n ∈ ω, ψ ab (n) = max{t : t ∈ Arg ϕ ba , ϕ ba (t) ≤ n} . Proof Let n ∈ ω. Let {t : t ∈ Arg ϕ ba , ϕ ba (t) ≤ n} = ∅. Then b(s) > n for any s ∈ S. Hence the value ψ ab (n) is indefinite. Let {t : t ∈ Arg ϕ ba , ϕ ba (t) ≤ n} be an infinite set. Then the set {a(s) : s ∈ S, b(s) ≤ n} is infinite. Hence the value ψ ab (n) is indefinite. Let {t : t ∈ Arg ϕ ba , ϕ ba (t) ≤ n} be a finite nonempty set. Let m be the maximum number from this set. Then m is the maximum number such that there exists s ∈ S  for which a(s) = m and b(s) ≤ n. Evidently, ψ ab (n) = m. Let f be a partial function from ω to ω, and G f = {(n, f (n)) : n ∈ Arg f }. The set G f will be called the graph of the function f . / Arg ψ ab and (n − 1) ∈ / Arg ψ ab , then For n ∈ ω, we define the set G ψ ab (n). If n ∈ / Arg ψ ab and (n − 1) ∈ Arg ψ ab , then G ψ ab (n) = {(n, ψ ab (n − G ψ ab (n) = ∅. If n ∈ 1) + i) : i ∈ ω \ {0}}. If n ∈ Arg ψ ab and (n − 1) ∈ / Arg ψ ab , then G ψ ab (n) = ab ab {(n, 0), (n, 1), . . . , (n, ψ (n))}. If n ∈ Arg ψ and (n − 1) ∈ Arg ψ ab , then G ψ ab (n) = ∅ if ψ ab (n) = ψ ab (n − 1), and G ψ ab (n) = {(n, ψ ab (n − 1) + 1), (n, ψ ab (n − 1) + 2), . . . , (n, ψ ab (n))} if ψ ab (n) > ψ ab (n  − 1). Denote G ψ ab = n∈ω G ψ ab (n). The set G ψ ab will be called the lower neighborhood of the function ψ ab graph. Let us define a mapping sm : ω2 → ω2 . For any pair (x, y) ∈ ω2 , the equality sm (x, y) = (y, x) holds. Evidently, sm is the reflection over the line given by the 2 equation x = y. Let us extend the mapping sm to the set 2ω . Let A ⊆ ω2 . Then sm A = {sm a¯ : a¯ ∈ A}. Proposition 10.3 If Arg ψ ab = ∅, then G ϕ ba = sm G ψ ab . Proof We now show that sm G ψ ab ⊆ G ϕ ba . Let n ∈ / Arg ψ ab , (n − 1) ∈ Arg ψ ab , and (n, i) ∈ G ψ ab (n). Let us show that ϕ ba (i) = n. Evidently, n = l(ψ ab ), the function ψ ab is not total from above, and ψ ab (t) < i for any t ∈ Arg ψ ab . By Proposition 10.1, ϕ ba (i) = l(ψ ab ) = n. / Arg ψ ab , and (n, i) ∈ G ψ ab (n). We show that Let n ∈ Arg ψ ab , (n − 1) ∈ ba ϕ (i) = n. Evidently, the set {t : t ∈ Arg ψ ab , ψ ab (t) ≥ i} is a nonempty set, and n is the minimum number from this set. By Proposition 10.1, ϕ ba (i) = n. Let n ∈ Arg ψ ab , (n − 1) ∈ Arg ψ ab , ψ ab (n) > ψ ab (n − 1), and (n, i) ∈ G ψ ab (n). We show that ϕ ba (i) = n. Evidently, the set {t : t ∈ Arg ψ ab , ψ ab (t) ≥ i}

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is a nonempty set, and n is the minimum number from this set. By Proposition 10.1, ϕ ba (i) = n. Thus, sm G ψ ab ⊆ G ϕ ba . Let us show that G ϕ ba ⊆ sm G ψ ab . Let i ∈ Arg ϕ ba . Using Proposition 10.1 we conclude that either ψ ab (t) < i for any t ∈ Arg ψ ab and the function ψ ab is not total from above, or {t : t ∈ Arg ψ ab , ψ ab (t) ≥ i} is a nonempty set. Let ψ ab (t) < i for any t ∈ Arg ψ ab and the function ψ ab be not total from above. Using Proposition 10.1 we obtain ϕ ba (i) = l(ψ ab ). We show that (i, l(ψ ab )) ∈ / Arg ψ ab . Since Arg ψ ab = ∅, we have l(ψ ab ) − 1 ∈ sm G ψ ab . Evidently, l(ψ ab ) ∈ ab ab ab Arg ψ and i > ψ (l(ψ ) − 1). Hence (l(ψ ab ), i) ∈ G ψ ab (l(ψ ab )) ⊆ G ψ ab and (i, l(ψ ab )) ∈ sm G ψ ab . Let {t : t ∈ Arg ψ ab , ψ ab (t) ≥ i} be a nonempty set and n be the minimum number from this set. Using Proposition 10.1 we obtain ϕ ba (i) = n. We show that / Arg ψ ab . Then, evidently, (i, n) ∈ sm G ψ ab . Evidently, n ∈ Arg ψ ab . Let (n − 1) ∈ ab (n, i) ∈ G ψ ab . Let (n − 1) ∈ Arg ψ . Since n is the minimum number from the set {t : t ∈ Arg ψ ab , ψ ab (t) ≥ i}, we have ψ ab (n − 1) < i. Hence (n, i) ∈ G ψ ab (n).  Therefore (i, n) ∈ sm G ψ ab . Thus, G ϕ ba ⊆ sm G ψ ab . For n ∈ ω, we define a set G ϕ ba (n). If n ∈ / Arg ϕ ba , then G ϕ ba (n) = ∅. If / Arg ϕ ba , then G ϕ ba (n) = {(n, ϕ ba (n) + i) : i ∈ ω}. Let n ∈ Arg ϕ ba and (n + 1) ∈ ba n ∈ Arg ϕ and (n + 1) ∈ Arg ϕ ba . If ϕ ba (n) = ϕ ba (n + 1), then G ϕ ba (n) = ∅. If then G ϕ ba (n) = {(n, ϕ ba (n)), (n, ϕ ba (n) + 1), . . . , (n, ϕ ba (n) < ϕ ba (n + 1), ba ϕ (n + 1) − 1)}.  Denote G ϕ ba = n∈ω G ϕ ba (n). The set G ϕ ba will be called the upper neighborhood of the function ϕ ba graph. Proposition 10.4 G ψ ab = sm G ϕ ba . Proof First, we show that sm G ϕ ba ⊆ G ψ ab . Let n ∈ Arg ϕ ba , (n + 1) ∈ / Arg ϕ ba , ab ba and (n, i) ∈ G ϕ ba (n). Show that ψ (i) = n. Evidently, ϕ (t) ≤ i for any t ∈ Arg ϕ ba . Hence {t : t ∈ Arg ϕ ba , ϕ ba (t) ≤ i} = Arg ϕ ba . It is clear that Arg ϕ ba is a finite nonempty set and max Arg ϕ ba = n. By Proposition 10.2, ψ ab (i) = n. Let n ∈ Arg ϕ ba , (n + 1) ∈ Arg ϕ ba , ϕ ba (n) < ϕ ba (n + 1), and (n, i) ∈ G ϕ ba (n). Let us show that ψ ab (i) = n. Evidently, {t : t ∈ Arg ϕ ba , ϕ ba (t) ≤ i} is a finite nonempty set and n is the maximum number from this set. By Proposition 10.2, ψ ab (i) = n. Thus, sm G ϕ ba ⊆ G ψ ab . We now show that G ψ ab ⊆ sm G ϕ ba . Let i ∈ Arg ψ ab . Using Proposition 10.2 we conclude that {t : t ∈ Arg ϕ ba , ϕ ba (t) ≤ i} is a finite nonempty set and ψ ab (i) = n, where n is the maximum number from the considered set. Let us show that (i, n) ∈ sm G ϕ ba . It is clear that n ∈ Arg ϕ ba . Let (n + 1) ∈ / Arg ϕ ba . Then, taking into account that i ≥ ϕ ba (n), we obtain (n, i) ∈ G ϕ ba . Let (n + 1) ∈ Arg ϕ ba . Taking into account that n is the maximum number from the set {t : t ∈ Arg ϕ ba , ϕ ba (t) ≤ i} we obtain ϕ ba (n + 1) > i. Evidently, ϕ ba (n) ≤  i. Hence (n, i) ∈ G ϕ ba . Therefore (i, n) ∈ sm G ϕ ba . Thus, G ψ ab ⊆ sm G ϕ ba .

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Proposition 10.5 The following statements hold: (a) Typ ψ ab = ε if and only if typ ϕ ba = ω. (b) Typ ψ ab = ω if and only if typ ϕ ba = ε. (c) Typ ψ ab = λ if and only if typ ϕ ba = μ. (d) Typ ψ ab = χ if and only if typ ϕ ba = γ . Proof (a) Let Typ ψ ab = ε. It means that Arg ψ ab is an infinite set and there exists c ∈ ω such that ψ ab (n) ≤ c for any n ∈ Arg ψ ab . Using Proposition 10.1 we conclude that, for any n ∈ ω, if n > c, then the value ϕ ba (n) is indefinite. Hence typ ϕ ba = ω. Let typ ϕ ba = ω. Evidently, Arg ϕ ba = ∅. Set m = max{ϕ ba (t) : t ∈ Arg ϕ ba } and c = max{t : t ∈ Arg ϕ ba }. Let n ∈ ω and n ≥ m. By Proposition 10.2, the value ψ ab (n) is definite. Hence Arg ψ ab is an infinite set. Using Proposition 10.2 we obtain ψ ab (n) ≤ c for any n ∈ Arg ψ ab . Therefore Typ ψ ab = ε. (b) Let Typ ψ ab = ω. One can show that in this case the function ψ ab is not total from above. By Proposition 10.1, Arg ϕ ba = ω and ϕ ba (n) ≤ l(ψ ab ) for any n ∈ ω. Hence typ ϕ ba = ε. Let typ ϕ ba = ε. It means that Arg ϕ ba is an infinite set and there exists c ∈ ω such that ϕ ba (n) ≤ c for any n ∈ Arg ϕ ba . Using Proposition 10.2 we conclude that, for any n ∈ ω, if n ≥ c, then the value ψ ab (n) is indefinite. Hence Typ ψ ab = ω. (c) Let Typ ψ ab = λ. By Proposition 10.1, Arg ϕ ba = ω and ϕ ba (n) = min{t : t ∈ Arg ψ ab , ψ ab (t) ≥ n} for any n ∈ Arg ϕ ba . Since Typ ψ ab = λ, there exists m ∈ ω such that, for any n ∈ Arg ψ ab , if n ≥ m, then ψ ab (n) < n. Let n ≥ m. It is clear that, for any t ∈ Arg ψ ab , if t ≤ n, then ψ ab (t) < n. Therefore ϕ ba (n) > n. Hence typ ϕ ba = μ. Let typ ϕ ba = μ. Using the statements (a) and (b) of the theorem we obtain Typ ψ ab = ε and Typ ψ ab = ω. Therefore Arg ψ ab is an infinite set, and there is no c ∈ ω such that ψ ab (n) ≤ c for any n ∈ Arg ψ ab . Taking into account that typ ϕ ba = μ we conclude that there exists m ∈ ω such that, for any n ∈ Arg ϕ ba , if n ≥ m, then ϕ ba (n) > n. Let n ∈ Arg ψ ab and n ≥ m. Then, for any t ∈ Arg ϕ ba , if t ≥ n, then ϕ ba (t) > n. By Proposition 10.2, ψ ab (n) = max{t : t ∈ Arg ϕ ba , ϕ ba (t) ≤ n}. Therefore ψ ab (n) < n. Hence Typ ψ ab = λ. (d) Let Typ ψ ab = χ . Using the statements (a), (b), and (c) of the theorem we obtain typ ϕ ba = ω, typ ϕ ba = ε, and typ ϕ ba = μ. Therefore typ ϕ ba = γ . Let typ ϕ ba = γ . Using the statements (a), (b), and (c) of the theorem we obtain  Typ ψ ab = ε, Typ ψ ab = ω, and Typ ψ ab = λ. Therefore Typ ψ ab = χ . Let R be the set of real numbers. For an arbitrary c ∈ R, denote R(c) = {d : d ∈ R, d ≥ c}. We denote by F R the set of functions f having the following properties: (a) There exists c ∈ R(0) such that R(c) = Arg f . (b) f (d) ∈ R(0) for any d ∈ R(c). (c) { f (d) : d ∈ R(c)} = R( f (c)). (d) For any d, t ∈ R(c), if d < t, then f (d) < f (t). Let f ∈ F R. We denote by f −1 the inverse function for the function f . One can show that f −1 ∈ F R.

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We denote by G R the set of functions satisfying the following conditions: (a) Arg f ⊆ R. (b) f (c) ∈ R for any c ∈ Arg f . Let f, g ∈ G R. The notation f  g will mean that there exists m ∈ ω such that ω(m) ⊆ Arg f , ω(m) ⊆ Arg g, and f (n) ≤ g(n) for any n ∈ ω(m). The notation g  f is equivalent to the notation f  g. Proposition 10.6 Let Typ ψ ab = ε, Typ ψ ab = ω, and f ∈ F R. Then the following statements hold: (a) If ψ ab  f , then ϕ ba  f −1 . (b) If ψ ab  f , then ϕ ba  f −1 + 1. Proof (a) Let ψ ab  f . Then there exists d ∈ ω such that ω(d) ⊆ Arg ψ ab , ω(d) ⊆ Arg f , and ψ ab (t) ≤ f (t) for any t ∈ ω(d). Set c = f (d) + 1. Evidently, ω(c) ⊆ Arg f −1 . Taking into account that Typ ψ ab = ε and using Proposition 10.5 we obtain typ ϕ ba = ω. By Remark 10.2, ω(c) ⊆ Arg ϕ ba . Let n ∈ ω(c). Show that ϕ ba (n) ≥ f −1 (n). Taking into account that Typ ψ ab = ε and Typ ψ ab = ω, and using Proposition 10.1 we obtain ϕ ba (n) = min{t : t ∈ Arg ψ ab , ψ ab (t) ≥ n}. Let ϕ ba (n) = t. Then ψ ab (t) ≥ n. Evidently, ψ ab (d) ≤ f (d) < c ≤ n. Using Remark 10.1 we obtain t > d. Therefore ψ ab (t) ≤ f (t) and f (t) ≥ n. Let f −1 (n) = t1 . Since f (t1 ) = n and f (t) ≥ n, we have f (t) ≥ f (t1 ). Therefore t ≥ t1 . Taking into account that ϕ ba (n) = t and f −1 (n) = t1 we obtain ϕ ba (n) ≥ f −1 (n). Hence ϕ ba  f −1 . (b) Let ψ ab  f . Then there exists d ∈ ω such that ω(d) ⊆ Arg ψ ab , ω(d) ⊆ Arg f , and ψ ab (t) ≥ f (t) for any t ∈ ω(d). Set c = f (d) + 1. Evidently, ω(c) ⊆ Arg f −1 . Taking into account that Typ ψ ab = ε and using Proposition 10.5 we obtain typ ϕ ba = ω. By Remark 10.2, Arg ϕ ba = ω. Therefore ω(c) ⊆ Arg ϕ ba . Let n ∈ ω(c). Show that ϕ ba (n) ≤ f −1 (n) + 1. Taking into account that Typ ψ ab = ε and Typ ψ ab = ω, and using Proposition 10.1 we obtain ϕ ba (n) = min{t : t ∈ Arg ψ ab , ψ ab (t) ≥ n}. Let ϕ ba (n) = t. If t = 0, then, evidently, ϕ ba (n) = 0 ≤ f −1 (n) + 1. Let t > 0. Then ψ ab (t − 1) < n. Let f −1 (n) = t1 . Then f (t1 ) = n. Taking into account that f (d) < n we obtain d < t1 . Hence ψ ab ( t1 ) ≥ f ( t1 ) ≥ n. From the inequalities ψ ab (t − 1) < n and ψ ab ( t1 ) ≥ n and Remark 10.1 it follows that t − 1 < t1 . Taking into account that t − 1 ∈ ω we obtain t − 1 < t1 and t < t1 + 1. Since ϕ ba (n) = t and f −1 (n) = t1 , we have ϕ ba (n) < f −1 (n) + 1.  Hence ϕ ba  f −1 + 1. Proposition 10.7 Let typ ϕ ba = ε, typ ϕ ba = ω, and f ∈ F R. Then the following statements hold: (a) If ϕ ba  f , then ψ ab  f −1 . (b) If ϕ ba  f , then ψ ab  f −1 − 1. Proof (a) Let ϕ ba  f . Then there exists d ∈ ω such that ω(d) ⊆ Arg ϕ ba , ω(d) ⊆ Arg f , and ϕ ba (t) ≥ f (t) for any t ∈ ω(d). Taking into account that typ ϕ ba = ε and using Proposition 10.5 we obtain Typ ψ ab = ω. From here and from Remark 10.1 it follows that there exists m ∈ ω such that Arg ψ ab = ω(m). Set c = max{ f (d) + 1, m}. Evidently, ω(c) ⊆ Arg ψ ab and ω(c) ⊆ Arg f −1 . Let n ∈ ω(c). Show that

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ψ ab (n) ≤ f −1 (n). Using Proposition 10.2 we obtain ψ ab (n) = max{t : t ∈ Arg ϕ ba , ϕ ba (t) ≤ n}. Let ψ ab (n) = t. Then ϕ ba (t) ≤ n. Let f −1 (n) = t1 . Then f (t1 ) = n. Taking into account that f (d) < n we obtain t1 > d. Hence, for any t2 ∈ ω, if t2 > t1 , then ϕ ba (t2 ) ≥ f (t2 ) > f (t1 ) = n. Therefore t ≤ t1 . Taking into account that ψ ab (n) = t and f −1 (n) = t1 we obtain ψ ab (n) ≤ f −1 (n). Therefore ψ ab  f −1 . (b) Let ϕ ba  f . Then there exists d ∈ ω such that ω(d) ⊆ Arg ϕ ba , ω(d) ⊆ Arg f , and ϕ ba (t) ≤ f (t) for any t ∈ ω(d). Taking into account that typ ϕ ba = ε and using Proposition 10.5 we obtain Typ ψ ab = ω. From here and from Remark 10.1 it follows that there exists m ∈ ω such that Arg ψ ab = ω(m). Set c = max{ f (d) + 1, m}. Evidently, ω(c) ⊆ Arg ψ ab and ω(c) ⊆ Arg f −1 . Let n ∈ ω(c). Show that ψ ab (n) ≥ f −1 (n) − 1. Using Proposition 10.2 we obtain ψ ab (n) = max{t : t ∈ Arg ϕ ba , ϕ ba (t) ≤ n}. Let ψ ab (n) = t. Taking into account that typ ϕ ba = ω and using Remark 10.2 we obtain Arg ϕ ba = ω. Therefore ϕ ba (t) ≤ n and ϕ ba (t + 1) > n. Let f −1 (n) = t1 . Then f (t1 ) = n. Taking into account that n > f (d) we obtain t1 > d. Since d ∈ ω, we have t1 ≥ d. Therefore ϕ ba ( t1 ) ≤ f ( t1 ) ≤ f (t1 ) = n. Taking into account that ϕ ba (t + 1) > n and using Remark 10.2 we obtain t1 < t + 1. Since t + 1 ∈ ω, we have t1 < t + 1. Since f −1 (n) = t1 and ψ ab (n) = t, we  have ψ ab (n) > f −1 (n) − 1. Hence ψ ab  f −1 − 1. Lemma 10.1 The following two statements are equivalent: (a) There exists m ∈ ω \ {0} such that ψ ab (n) = n for any n ∈ ω(m). (b) There exists m ∈ ω \ {0} such that ϕ ba (n) = n for any n ∈ ω(m). Proof Let the statement (a) hold and n ∈ ω(m). Using Proposition 10.1 we obtain ϕ ba (n) = n. Hence the statement (b) holds. Let the statement (b) hold and n ∈ ω(m). Using Proposition 10.2 we obtain  ψ ab (n) = n. Hence the statement (a) holds. The following two statements characterize some relationships between the set F Z and the set F R. Lemma 10.2 Let f ∈ F R and Arg f = R(c). Then there exist functions g1 , g2 ∈ F Z such that Arg g1 = Arg g2 = ω( c) and, for any n ∈ ω( c), the following inequalities hold: g1 (n) ≤ f (n) ≤ g2 (n) and g2 (n) − g1 (n) ≤ 1. Proof We now define functions g1 and g2 . Let n ∈ ω( c). Then g1 (n) = f (n) and g2 (n) = f (n). One can show that g1 , g2 ∈ F Z , Arg g1 = Arg g2 = ω( c) and, for any n ∈ ω( c), the inequalities g1 (n) ≤ f (n) ≤ g2 (n) and g2 (n) − g1 (n) ≤ 1 hold.  Lemma 10.3 Let f ∈ F Z , d = min{t : t ∈ Arg f, f (t) > 0}, ε ∈ R, and 0 < ε ≤ 1 . Then there exist functions g1 , g2 ∈ F R such that Arg g1 = Arg g2 = R(d) and, for 2 any n ∈ ω(d), the following inequalities hold: g1 (n) ≤ f (n) ≤ g2 (n) and g2 (n) − g1 (n) ≤ ε.

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Proof Let m 0 , m 1 , m 2 , . . . be a sequence of numbers from ω having the following properties: (a) m 0 = d. (b) m i < m i+1 for any i ∈ ω. (c) {m 1 , m 2 , . . .} = {m : m ∈ ω(d + 1), f (m) > f (m − 1)}. For any i ∈ ω, we define a function g2 on the set Ri = {x : x ∈ R, m i ≤ x ≤ m i+1 }. If m i+1 = m i + 1, then g2 (x) = ( f (m i+1 ) − f (m i ))(x − m i ) + f (m i ) for any x ∈ Ri . Let m i+1 > m i + 1. Set Ri− = {x : x ∈ R, m i ≤ x ≤ m i+1 − 1} and i) Ri+ = {x : x ∈ R, m i+1 − 1 ≤ x ≤ m i+1 }. For any x ∈ Ri− , set g2 (x) = mε(x−m + i+1 −m i −1 + f (m i ) and, for any x ∈ Ri , set g2 (x) = ( f (m i+1 ) − f (m i ) − ε)(x − m i+1 + 1) + f (m i ) + ε. Let us define a function g1 . For any x ∈ R(d), set g1 (x) = g2 (x) − ε. One can show that g1 , g2 ∈ F R, Arg g1 = Arg g2 = R(d) and, for any n ∈ ω(d),  the inequalities g1 (n) ≤ f (n) ≤ g2 (n) and g2 (n) − g1 (n) ≤ ε hold.

References 1. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity, Global approach. Fundam. Inform. 25(2), 201–214 (1996) 2. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity, Local approach. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets IV, Lecture Notes in Computer Science, vol. 3700, pp. 125–143. Springer, Berlin (2005)

Chapter 11

Functions on Main Diagonal and Below

In this chapter, for restricted sccf-triples τ , we study functions Ψˆ τbc , b, c ∈ {i, d, a, s}, located in the matrix of upper local bounds for the triple τ on the main diagonal and below. For each of these functions, we list all possible upper types and consider criterion for each such type. In several cases, we give upper and lower bounds for the considered functions. Some similar results were obtained in [2] for types of functions different from the upper types considered in this book. For various kinds of restricted sccf-triples τ , functions Ψˆ τdi and Ψˆ τai were studied in [1, 3, 4].

11.1 Function Ψˆ τi i Lemma 11.1 Let τ = (ρ, K , ψ) be a sccf-triple and z ∈ Σρ . Then i d a Ψˆ ρ,K (z) ≥ Ψˆ ρ,K (z) ≥ Ψˆ ρ,K (z) i d a s and if z ∈ Σρ0−1 (K ), then ψˆ ρ,K (z) ≥ Ψˆ ρ,K (z) ≥ Ψˆ ρ,K (z) ≥ Ψˆ ρ,K (z).

Proof Let z = (ν, f 1 , . . . , f n ). Denote T = Tρ (z, K ). Evidently, T ∈ Mρ . Using a d (z) and ψρd (T ) = ψˆ ρ,K (z). From these Theorem 10.1 we obtain ψρa (T ) = ψˆ ρ,K d a (z). By Lemma equalities and from Proposition 6.1 it follows that ψˆ ρ,K (z) ≥ ψˆ ρ,K d 3.1, ψρ (T ) ≤ Θρ,ψ (T ). Evidently, the word α = f 1 · · · f n is an unconditional test for the table T . Therefore Θρ,ψ (T ) ≤ ψ(α). Taking into account that ψρd (T ) = d i i d ψˆ ρ,K (z) and ψ(α) = ψˆ ρ,K (z) we obtain ψˆ ρ,K (z) ≥ ψˆ ρ,K (z). Let z ∈ Σρ0−1 (K ). 0−1 a Then, evidently, T ∈ Mρ . By Proposition 6.1, ψρ (T ) ≥ ψρs (T ). Moreover, from a s Theorem 10.1 it follows that ψˆ ρ,K (z) = ψρa (T ) and ψˆ ρ,K (z) = ψρs (T ). Therefore a s ψˆ ρ,K (z) ≥ ψˆ ρ,K (z).  © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_11

137

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Let b1 , c1 , b2 , c2 ∈ {i, d, a, s} and τ = (ρ, K , ψ) be a sccf-triple. The notation Ψˆ τb1 c1 ≺ Ψˆ τb2 c2 will mean that, for any n ∈ ω, the following conditions hold: (a) If the value Ψˆ τb1 c1 (n) is definite, then either Ψˆ τb2 c2 (n) = ∞ or the value Ψˆ τb2 c2 (n) is definite and Ψˆ τb1 c1 (n) ≤ Ψˆ τb2 c2 (n). (b) If Ψˆ τb1 c1 (n) = ∞, then Ψˆ τb2 c2 (n) = ∞. Lemma 11.2 Let τ = (ρ, K , ψ) be a sccf-triple. Then the following statements hold: (a) Ψˆ τbi ≺ Ψˆ τbd ≺ Ψˆ τba for any b ∈ {i, d, a}, and Ψˆ τsi ≺ Ψˆ τsd ≺ Ψˆ τsa ≺ Ψˆ τss . (b) Ψˆ τsb ≺ Ψˆ τab ≺ Ψˆ τdb ≺ Ψˆ τib for any b ∈ {i, d, a, s}. Proof The statement of the lemma follows immediately from the definition of the  functions Ψˆ τbc , b, c ∈ {i, d, a, s}, and from Lemma 11.1. Lemma 11.3 Let τ = (ρ, K , ψ) be a sccf-triple and b ∈ {i, d, a, s}. Then Typ Ψˆ τbb = ε or Typ Ψˆ τbb = χ , and the following statements hold: (a) If b = s, then Typ Ψˆ τbb = ε if and only if there exists c ∈ ω such that b ˆ ψρ,K (z) ≤ c for any z ∈ Σρ . s (z) ≤ c for any (b) Typ Ψˆ τss = ε if and only if there exists c ∈ ω such that ψˆ ρ,K 0−1 z ∈ Σρ (K ). Proof Let b ∈ {i, d, a, s}. We define the set Σρb in the following way: if b = s, then Σρb = Σρ , and if b = s, then Σρb = Σρ0−1 (K ). b Let there exist c ∈ ω such that Ψˆ ρ,K (z) ≤ c for any z ∈ Σρb . Let n ∈ ω and n ≥ c. Then, evidently, the value Ψˆ τbb (n) is definite, and the inequality Ψˆ τbb (n) ≤ c holds. Hence Typ Ψˆ τbb = ε. Let Typ Ψˆ τbb = ε. Then, evidently, there exists c ∈ ω such that b Ψˆ ρ,K (z) ≤ c for any z ∈ Σρb . b (z) ≤ c for any z ∈ Σρb . Then the set Let there be no c ∈ ω such that ψˆ ρ,K b b B = {ψˆ ρ,K (z) : z ∈ Σρ } is infinite. Evidently, Ψˆ τbb (n) = n for any n ∈ B. Hence  Typ Ψˆ τbb = χ . Proposition 11.1 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψˆ τii = χ and there exists c ∈ ω \ {0} such that n − c < Ψˆ τii (n) ≤ n for any n ∈ Arg Ψˆ τii . Proof Let ρ = (F, k), f ∈ F, and ψ( f ) = min{ψ( f  ) : f  ∈ F}. Let ψ( f ) = c. Using the property Λ3 of the function ψ we obtain c ∈ ω \ {0}. For m ∈ ω \ {0}, we denote by z m the tuple (νm , f, . . . , f ) in which the element f repeats m times and νm : E km → {{1}}. Evidently, z m ∈ Σρ . Using the property Λ3 of the function ψ we i conclude that there is no d ∈ ω such that ψˆ ρ,K (z m ) ≤ d for any m ∈ ω \ {0}. From here and from Lemma 11.3 it follows that Typ Ψˆ τii = χ . Let n ∈ Arg Ψˆ τii . Using the i property Λ2 of the function ψ we obtain n ≥ c. Evidently, ψˆ ρ,K (z 1 ) = c. Let m be the i i (z m+1 ) > n. maximum number from ω \ {0} such that ψˆ ρ,K (z m ) ≤ n. Evidently, ψˆ ρ,K i i (z m ) + Using the property Λ1 of the function ψ we obtain ψˆ ρ,K (z m+1 ) ≤ ψˆ ρ,K i ii ii ˆ ˆ ˆ c. Hence ψρ,K (z m ) > n − c and Ψτ (n) ≥ n − c. The inequality Ψτ (n) ≤ n is obvious. 

11.2 Function Ψˆ τdi

139

11.2 Function Ψˆ τd i Let τ = (ρ, K , ψ) be a sccf-triple. Denote Mρ,K = {Tρ (z, K ) : z ∈ Σρ } and 0−1 = {Tρ (z, K ) : z ∈ Σρ0−1 (K )}. Mρ,K We now define a function Sψ : Mρ → ω. Let T ∈ Mρ . If Δ(T ) = ∅, then ¯ = min{ψ(α) : α ∈ Sψ (T ) = 0. Let Δ(T ) = ∅ and δ¯ ∈ Δ(T ). Then Sψ (T, δ) ¯ ¯ ¯ Ωρ (T ), Δ(T α) = {δ}} and Sψ (T ) = max{Sψ (T, δ) : δ ∈ Δ(T )}. Define a function m ψ : Mρ → ω. For T ∈ Mρ , let m ψ (T ) = max{ψ( f ) : f ∈ P(T )}. We now define partial functions Sτ and Nτ from ω to ω. Let n ∈ ω. Then Sτ (n) = max{Sψ (T ) : T ∈ Mρ,K , m ψ (T ) ≤ n} and Nτ (n) = max{Nρ (T ) : T ∈ Mρ,K , m ψ (T ) ≤ n}. Define a function Sˆτ : Mρ,K → ω. For T ∈ Mρ,K , let Sˆτ (T ) = max{Sψ (T  ) :  T ∈ Mρ,K , P(T  ) ⊆ P(T )}. Lemma 11.4 Let τ = (ρ, K , ψ) be a restricted sccf-triple and T ∈ Mρ,K . Then Mρ,ψ (T ) ≤ 2 Sˆτ (T ). Proof Let ρ = (F, k), dim T = n, μT (1) = f 1 , . . . , μT (n) = f n , and δ¯ = (δ1 , . . . , δn ) ∈ E kn . For each σ¯ = (σ1 , . . . , σn ) ∈ Δ(T ), we denote by β(σ¯ ) a word from Ωρ (T ) such that Δ(Tβ(σ¯ )) = {σ¯ } and ψ(β(σ¯ )) = Sψ (T, σ¯ ). Using the property Λ2 of the function ψ we obtain ψ( f i ) ≤ Sψ (T, σ¯ ) ≤ Sψ (T ) for any pair ( f i , σi ) ∈ χ (β(σ¯ )). Let there exist σ¯ ∈ Δ(T ) such that the words β(σ¯ ) and ( f 1 , δ1 ) · · · ( f n , δn ) are compatible. Then the word β(σ¯ ) satisfies the following conditions: Tβ(σ¯ ) ∈ Mρ C , ¯ ≤ χ (β(σ¯ )) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )} and ψ(β(σ¯ )) ≤ Sψ (T ). Hence Mρ,ψ (T, δ) ˆ Sψ (T ) ≤ 2 Sτ (T ). Let, for each σ¯ ∈ Δ(T ), the words β(σ¯ ) and ( f 1 , δ1 ) · · · ( f n , δn ) are incompatible. Then there exists a subset { f i1 , . . . , f im } of the set { f 1 , . . . , f n } having the following properties: ψ( f i j ) ≤ Sψ (T ) for j = 1, . . . , m, and Δ(T ( f i1 , δi1 ) · · · ( f im , δim )) = ∅. ¯ ≤ Sψ (T ) ≤ 2 Sˆτ (T ). Let If Δ(T ( f i1 , δi1 )) = ∅, then, evidently, Mρ,ψ (T, δ) Δ(T ( f i1 , δi1 )) = ∅ . Then there exists t ∈ {1, . . . , m − 1} such that Δ(T ( f i1 , δi1 ) · · · ( f it , δit )) = ∅ and Δ(T ( f i1 , δi1 ) · · · ( f it+1 , δit+1 )) = ∅ . Let us consider the schema z = (ν, f i1 , . . . , f it ), where ν : E kt → {{1}}. Set T  = Tρ (z, K ) and δ¯ = (δi1 , . . . , δit ). Evidently, δ¯ ∈ Δ(T  ). Let α be a word from Ωρ (T  ) such that Δ(T  α) = {δ¯ } and ψ(α) = Sψ (T  , δ¯ ). Taking into account that P(T  ) ⊆ P(T ) we obtain ψ(α) ≤ Sψ (T  ) ≤ Sˆτ (T ). One can show that Δ(T α) = Δ(T ( f i1 , δi1 ) · · · ( f it , δit )). Set γ = α( f it+1 , δit+1 ). Then χ (γ ) ⊆ {( f 1 , δ1 ), . . . ,

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( f n , δn )} and Δ(T γ ) = ∅. Taking into account that the function ψ has the property Λ1, ψ(α) ≤ Sˆτ (T ), and ψ(( f it+1 , δit+1 )) ≤ Sψ (T ) ≤ Sˆτ (T ) we obtain ψ(γ ) ≤ ¯ ≤ 2 Sˆτ (T ). By Lemma 3.5, Mρ,ψ (T ) ≤ 2 Sˆτ (T ).  2 Sˆτ (T ). Hence Mρ,ψ (T, δ) Lemma 11.5 Let τ = (ρ, K , ψ) be a restricted sccf-triple, ρ = (F, k), and T ∈ Mρ,K . Then Nρ (T ) ≤ (k dim T ) Sψ (T ) . Proof If Nρ (T ) = 1, then using the property Λ4 of the function ψ we obtain Sψ (T ) = 0, and hence the statement of the lemma holds. Let Nρ (T ) > 1. Then using the property Λ3 we obtain Sψ (T ) > 0. Denote m = Sψ (T ). Taking into account that ψ has the property Λ3 we conclude that, for any δ¯ ∈ Δ(T ), there exist elements f 1 , . . . , f m ∈ P(T ) and numbers σ1 , . . . , σm ∈ E k such that Δ(T ( f 1 , σ1 ) · · · ( f m , ¯ Hence there exists a one-to-one correspondence between the set Δ(T ) σm )) = {δ}. and a set B of pairs of type (( f 1 , . . . , f m ), (σ1 , . . . , σm )), where f 1 , . . . , f m ∈ P(T ) and σ1 , . . . , σm ∈ E k . It is clear that |B| ≤ (dim T )m k m . Therefore Nρ (T ) ≤  (k dim T ) Sψ (T ) . Lemma 11.6 Let τ = (ρ, K , ψ) be a restricted sccf-triple, ρ = (F, k) be a signature, and there exists t ∈ ω such that Nτ (t) = ∞. Then Ψˆ τdi (n) ≥ logk n − logk t for any n ∈ Arg Ψˆ τdi . Proof If n ≤ t, then, evidently, the statement of the lemma holds. Let n > t. Set m = nt . Since Nτ (t) = ∞, there exists a table T ∈ Mρ,K such that m ψ (T ) ≤ t and Nρ (T ) ≥ k m . Let { f 1 , . . . , f p } be a minimum (relative to the cardinality)  subset of the set P(T ) having the following property: Δ(T ( f 1 , δ1 ) · · · ( f p , δ p )) ≤ 1 for any δ1 , . . . , δ p ∈ E k . Since Nρ (T ) ≥ k m , we have p ≥ m. Set z = (ν, f 1 , . . . , f m ), where ν : E km → P(ω) and, for any δ¯1 , δ¯2 ∈ E km , if δ¯1 = δ¯2 , then ν(δ¯1 ) ∩ ν(δ¯2 ) = ∅. Denote T  = Tρ (z, K ). One can show that, for any i ∈ {1, . . . , m}, there exist tuples δ¯1 , δ¯2 ∈ Δ(T  ) which are different only in ith component. Therefore Θρ,ψ (T  ) = m. By Theorem 3.3, ψρd (T  ) ≥ logk (m + 1) ≥ logk ( nt ). From Theorem 10.1 it follows d that ψˆ ρ,K (z) = ψρd (T  ). Taking into account that the function ψ has the property Λ1 i (z) ≤ n. Hence Ψˆ τdi (n) ≥ logk n − logk t.  we obtain ψˆ ρ,K Lemma 11.7 Let τ = (ρ, K , ψ) be a restricted sccf-triple, ρ = (F, k), T ∈ Mρ,K , Sψ (T ) = n > 0, and m ψ (T ) ≤ t. Then the following statements hold: i d (a) There exist schemes z 1 , z 2 ∈ Σρ0−1 (K ) such that ψˆ ρ,K (z 1 ) = ψˆ ρ,K (z 1 ) = a s i d a s ˆ ˆ ˆ ˆ ˆ ˆ ψρ,K (z 1 ) = n, ψρ,K (z 1 ) ≤ t, and ψρ,K (z 2 ) = ψρ,K (z 2 ) = ψρ,K (z 2 ) = ψρ,K (z 2 ) = n. (b) If ψ = h and n > 1, then there exists a table T  ∈ Mρ,K such that Sh (T  ) = n − 1. Proof Let dim T = r , μT (1) = f 1 , . . . , μT (r ) = fr and δ¯ = (δ1 , . . . , δr ) ∈ E kr . Let ¯ = Sψ (T ), α ∈ Ωρ (T ), ψ(α) = Sψ (T, δ) ¯ and Δ(T α) = {δ}. ¯ δ¯ ∈ Δ(T ), Sψ (T, δ) Taking into account that ψ(α) = n > 0 and the function ψ has the property

11.2 Function Ψˆ τdi

141

Λ4 we obtain α = λ. Let α = ( f i1 , δi1 ) · · · ( f im , δim ). Set z 1 = (ν1 , f i1 , . . . , f im ), where ν1 : E km → {{0}, {1}} and, for any σ¯ = (σ1 , . . . , σm ) ∈ E km , if (σ1 , . . . , σm ) = (δi1 , . . . , δim ), then ν1 (σ¯ ) = {0} and if (σ1 , . . . , σm ) = (δi1 , . . . , δim ), then ν1 (σ¯ ) = {1}. Set z 2 = (ν2 , f i1 , . . . , f im ), where ν2 : E km → {{0}, {1}} and, for any σ¯ = (σ1 , . . . , σm ) ∈ E km , if (σ1 , . . . , σm ) = (δi1 , . . . , δim ), then ν2 (σ¯ ) = {1} and if (σ1 , . . . , σm ) = (δi1 , . . . , δim ), then ν2 (σ¯ ) = {0}. Denote T1 = Tρ (z 1 , K ) and T2 = Tρ (z 2 , K ). Evidently, dim T1 = dim T2 and μT1 ≡ μT2 . Let dim T1 = p and μT1 (1) = f j1 , . . . , μT1 ( p) = f j p . Set δ¯ = (δ j1 , . . . , δ j p ). It is clear that Δ(T1 ) = Δ(T2 ) and δ¯ ∈ Δ(T1 ). One can show that Sψ (T1 , δ¯ ) = Sψ (T2 , δ¯ ) = n. From here and from the property Λ4 of the function ψ it follows that |Δ(T1 )| ≥ 2. Therefore z 1 , z 2 ∈ Σρ0−1 (K ). One can show that Mρ,ψ (T1 , δ¯ ) = Mρ,ψ (T2 , δ¯ ) = n. Using Theorems a s 6.1, 6.2, and 10.1 we obtain ψˆ ρ,K (z 1 ) ≥ n and ψˆ ρ,K (z 2 ) ≥ n. From these inequali i ˆ (z 2 ) = n it follows that ities, Lemma 11.1 and obvious equalities ψρ,K (z 1 ) = ψˆ ρ,K i d a i d a (z 2 ) = ψˆ ρ,K (z 2 ) = ψˆ ρ,K (z 1 ) = ψˆ ρ,K (z 1 ) = ψˆ ρ,K (z 1 ) = n and ψˆ ρ,K (z 2 ) = ψˆ ρ,K s ˆ ψρ,K (z 2 ) = n. Let σ¯ = (σ1 , . . . , σt ) ∈ Δ(T1 ) and νT1 (σ¯ ) = {1}. Then, evidently, σ¯ = δ¯ . Let σi = δ ji . It is clear that T1 ( f ji , σi ) ∈ Mρ C . Therefore Mρ,ψ (T1 , σ¯ ) ≤ t. Using Theorems 6.2 and 10.1 we obtain s ψˆ ρ,K (z 1 ) ≤ t .

Let ψ = h and n > 1. Then m = n. Set z  = (ν  , f i1 , . . . , f im−1 ), where ν  : → {{0}}. Denote T  = Tρ (z  , K ) and δ¯ = (δi1 , . . . , δim−1 ). One can show that dim T  = m − 1 = n − 1 and Sh (T  , δ¯ ) = m − 1 = n − 1. Therefore Sh (T  ) = n − 1.  E km−1

Corollary 11.1 Let τ = (ρ, K , h) be a restricted sccf-triple, Typ Sτ = ε, and b ∈ {i, d, a, s}. Then, for any n ∈ ω \ {0}, the value Ψˆ τbi (n) is definite and is equal to n. Proof Let n ∈ ω \ {0}. Since Typ Sτ = ε, there exists a table T ∈ Mρ,K such that Sh (T ) ≥ n. Using Lemma 11.7 we conclude that there exists a schema z ∈ Σρ0−1 (K ) for which hˆ iρ,K (z) = hˆ bρ,K (z) = n. Hence if the value Ψˆ τbi (n) is definite, then Ψˆ τbi (n) ≥ n. Using the existence of the schema z and Lemma 11.1 we conclude that the value Ψˆ τbi (n) is definite and is at most n. Hence Ψˆ τbi (n) = n.  Lemma 11.8 Let τ = (ρ, K , ψ) be a restricted sccf-triple and ρ = (F, k). Then the following statements are equivalent: d (a) There exists c ∈ ω such that ψˆ ρ,K (z) ≤ c for any z ∈ Σρ . (b) Typ Nτ = ε and Typ Sτ = ε. Proof Let the statement (b) hold. Then there exist c1 , c2 ∈ ω \ {0} such that, for each table T ∈ Mρ,K , the inequalities Nρ (T ) ≤ c1 and Sψ (T ) ≤ c2 hold and, hence, the inequality Sˆτ (T ) ≤ c2 holds. Let z ∈ Σρ . Set T = Tρ (z, K ). Then Nρ (T ) ≤ c1

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and Sˆτ (T ) ≤ c2 . By Lemma 11.4, Mρ,ψ (T ) ≤ 2c2 . Using these inequalities, Corollary 5.2, and the fact that the function ψ has the properties Λ1 and Λ4 we obtain ψρd (T ) ≤ 2c2 log2 c1 . From this inequality and from Theorem 10.1 it follows d that ψˆ ρ,K (z) ≤ 2c2 log2 c1 . Since z is an arbitrary schema from Σρ , the statement (a) holds. Let the statement (b) do not hold. Let us show that the statement (a) does not d (z) ≤ c for any schema hold. Assume the contrary: there exists c ∈ ω such that ψˆ ρ,K z ∈ Σρ . Let Typ Nτ = ε. Then there exists a table T ∈ Mρ,K such that Nρ (T ) > k c . Let z = (ν, f 1 , . . . , f n ) ∈ Σρ and T = Tρ (z, K ). Set z  = (ν  , f 1 , . . . , f n ), where ν  : E kn → P(ω) and, for any δ¯1 , δ¯2 ∈ E kn , if δ¯1 = δ¯2 , then ν  (δ¯1 ) ∩ ν  (δ¯2 ) = ∅. Denote T  = Tρ (z  , K ). It is clear that T  is a diagnostic table such that Nρ (T  ) > k c . Taking into account that the function ψ has the property Λ3 and using Proposition 3.2 we obtain ψρd (T  ) ≥ logk Nρ (T  ) > c. From this inequality and from Theorem 10.1 it d follows that ψˆ ρ,K (z  ) > c which contradicts the assumption. Let Typ Sτ = ε. Then there exists a table T ∈ Mρ,K such that Sψ (T ) > c. By d Lemma 11.7, there exists a schema z ∈ Σρ such that Ψˆ ρ,K (z) > c which contradicts the assumption. Thus, the statement (a) does not hold.  Lemma 11.9 Let τ = (ρ, K , ψ) be a restricted sccf-triple, ρ = (F, k), Typ Nτ = ε, and Typ Nτ = ω. Then Typ Sτ = ε. Proof Assume that Typ Nτ = ε, Typ Nτ = ω, and Typ Sτ = ε. Then there exists c ∈ ω \ {0} such that Sψ (T ) ≤ c for any table T ∈ Mρ,K . By Lemma 11.7, there i exists z ∈ Z ρ such that ψˆ ρ,K (z) ≤ c. Set T = Tρ (z, K ). Taking into account that the function ψ has the property Λ2 we obtain m ψ (T ) ≤ c. Hence the set {T : T ∈ Mρ,K , m ψ (T ) ≤ c} is nonempty. Since Typ Nτ = ω, the value Nτ (c) is definite. Let Nτ (c) = t. Then, for any table T ∈ Mρ,K , if m ψ (T ) ≤ c, then Nρ (T ) ≤ t. Since Typ Nτ = ε, there exists a table T ∈ Mρ,K such that Nρ (T ) > t. Let { f i : f i ∈ P(T ), ψ( f i ) ≤ c} = { f 1 , . . . , f n }. Set z  = (ν, f 1 , . . . , f n ), where ν : E kn → {{1}}, and T  = Tρ (z  , K ). Taking into account that the function ψ has the property Λ2 and the inequality Sψ (T ) ≤ c holds one can show that Nρ (T  ) = Nρ (T ). Therefore m ψ (T  ) ≤ c and Nρ (T  ) > t. The obtained contradiction shows that  Typ Sτ = ε. Proposition 11.2 Let τ = (ρ, K , ψ) be a restricted sccf-triple and ρ = (F, k). Then the following statements hold: (a) Either Typ Ψˆ τdi = ε or Typ Ψˆ τdi = λ, or Typ Ψˆ τdi = χ , and Ψˆ τdi (n) ≤ n for any n ∈ Arg Ψˆ τdi . (b) Typ Ψˆ τdi = ε if and only if Typ Nτ = ε and Typ Sτ = ε. (c) Typ Ψˆ τdi = λ if and only if Typ Nτ = ε and Typ Sτ = ε. (d) Typ Ψˆ τdi = χ if and only if Typ Sτ = ε. (e) If Typ Ψˆ τdi = λ, then there exist c1 , c2 , c3 ∈ ω such that, for any n ∈ Arg Ψˆ τdi ,

11.2 Function Ψˆ τdi

143

1 log2 n − c1 ≤ Ψˆ τdi (n) ≤ c2 log2 n + c3 . log2 k d i Proof From Lemma 11.1 it follows that ψˆ ρ,K (z) ≤ ψˆ ρ,K (z) for any schema z ∈ Σρ . di di ˆ ˆ Therefore Typ Ψτ = ω and Ψτ (n) ≤ n for any n ∈ Arg Ψˆ τdi . Thus, the statement (a) holds. The next statement follows immediately from Lemma 11.8: (b*) If Typ Nτ = ε and Typ Sτ = ε, then Typ Ψˆ τdi = ε. Let us prove the following statement: (c*) If Typ Nτ = ε and Typ Sτ = ε, then Typ Ψˆ τdi = λ and there exist c1 , c2 , c3 ∈ ω such that, for any n ∈ Arg Ψˆ τdi ,

1 log2 n − c1 ≤ Ψˆ τdi (n) ≤ c2 log2 n + c3 . log2 k

(11.1)

Let Typ Nτ = ε and Typ Sτ = ε. By Lemma 11.9, Typ Nτ = ω. Therefore there exists t ∈ ω such that Nτ (t) = ∞. By Lemma 11.6, there exists c1 ∈ ω such that Ψˆ τdi (n) ≥ log1 k log2 n − c1 for any n ∈ Arg Ψˆ τdi . Since Typ Sτ = ε, there exists 2 m ∈ ω \ {0} such that, for any table T ∈ Mρ,K , the inequality Sψ (T ) ≤ m holds and, hence, the inequality Sˆτ (T ) ≤ m holds. Let n ∈ Arg Ψˆ τdi and z be an arbitrary i schema from Σρ for which ψˆ ρ,K (z) ≤ n. Denote T = Tρ (z, K ). Since the function i (z) ≤ n it follows that dim T ≤ n. ψ has the property Λ3, from the inequality ψˆ ρ,K ˆ Moreover, Sψ (T ) ≤ m and Sτ (T ) ≤ m. By Lemma 11.4, Mρ,ψ (T ) ≤ 2 Sˆτ (T ) ≤ 2m. Using Lemma 11.5 we obtain Nρ (T ) ≤ (k dim T ) Sψ (T ) ≤ (nk)m . From these inequalities, from the fact that the function ψ has properties Λ1 and Λ4, and from Corollary 5.2 it follows that the inequality ψρd (T ) ≤ 2m 2 log2 n + 2m 2 log2 k holds. By d Theorem 10.1, ψˆ ρ,K (z) ≤ 2m 2 log2 n + 2m 2 log2 k. Taking into account that z is i (z) ≤ n and n is an arbitrary number an arbitrary schema from Σρ such that ψˆ ρ,K di di ˆ ˆ from Arg Ψτ we conclude that Ψτ (n) ≤ c2 log2 n + c3 for any n ∈ Arg Ψˆ τdi , where   c2 = 2m 2 and c3 = 2m 2 log2 k . By (11.1), Typ Ψˆ τdi = ε and Typ Ψˆ τdi = χ . From the statement (a) it follows that Typ Ψˆ τdi = ω. Hence Typ Ψˆ τdi = λ. Thus, the statement (b*) holds. Let us prove the following statement: (d*) If Typ Sτ = ε, then Typ Ψˆ τdi = χ . Let Typ Sτ = ε. Then the set B = {Sψ (T ) : T ∈ Mρ,K } is infinite. Let n ∈ B and T be a table from Mρ,K such that Sψ (T ) = n. By Lemma 11.7, there exists a schema i d z ∈ Σρ such that ψˆ ρ,K (z) = ψˆ ρ,K (z) = n. Hence Ψˆ τdi (n) = n. Taking into account that B is an infinite set we conclude that Typ Ψˆ τdi = χ . Thus, the statement (d*) holds. Let us prove the statement (b). If Typ Nτ = ε and Typ Sτ = ε, then Typ Ψˆ τdi = ε by the statement (b*). Let Typ Ψˆ τdi = ε. From this equality and from the statement (d*) it follows that Typ Sτ = ε. From the last two equalities and from the statement (c*) it follows that Typ Nτ = ε. Thus, the statement (b) holds.

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Let us prove the statement (c). If Typ Nτ = ε and Typ Sτ = ε, then Typ Ψˆ τdi = λ by the statement (c*). Let Typ Ψˆ τdi = λ. From this equality and from the statement (d*) it follows that Typ Sτ = ε. From the last two equalities and from the statement (b*) it follows that Typ Nτ = ε. Thus, the statement (c) holds. Let us prove the statement (d). If Typ Sτ = ε, then Typ Ψˆ τdi = χ by the statement (d*). Let Typ Ψˆ τdi = χ . Let us show that Typ Sτ = ε. Assume the contrary. If Typ Nτ = ε, then Typ Ψˆ τdi = ε by the statement (b*). If Typ Nτ = ε, then Typ Ψˆ τdi = λ by the statement (c*). The obtained contradiction shows that Typ Sτ = ε. Thus, the statement (d) holds. Let Typ Ψˆ τdi = λ Using the statements (c) and (c*) we conclude that there exist c1 , c2 , c3 ∈ ω such that (11.1) holds for any n ∈ Arg Ψˆ τdi . Thus, the statement (e) holds. 

11.3 Function Ψˆ τd d Lemma 11.10 Let τ = (ρ, K , ψ) be a restricted sccf-triple and there exists t ∈ ω such that Nτ (t) = ∞. Then Ψˆ τdd (n) ≥ n − t for any n ∈ Arg Ψˆ τdd . Proof Let T1 , T2 ∈ Mρ,K and P(T1 ) ⊆ P(T2 ). One can see that Nρ (T1 ) ≤ Nρ (T2 ). Using this inequality and the fact that Nτ (t) = ∞ one can show that there exists a sequence T1 , T2 , . . . of tables from Mρ,K having the following properties: (a) For any i ∈ ω \ {0}, dim Ti = i, Ti is a diagnostic table, m ψ (Ti ) ≤ t, and P(Ti ) ⊆ P(Ti+1 ). (b) The set {Nρ (Ti ) : i ∈ ω \ {0}} is infinite. Using the fact that the function ψ has the property Λ3 and using Proposition 3.2 we conclude that the set {ψρd (Ti ) : i ∈ ω \ {0}} is infinite. Taking into account that the function ψ has the property Λ1 one can show that ψρd (Ti+1 ) ≤ ψρd (Ti ) + t for any i ∈ ω \ {0}. Let n ∈ Arg Ψˆ τdd . If n ≤ t, then, evidently, the inequality Ψˆ τdd (n) ≥ n − t holds. Let n > t and m be the minimum number from ω \ {0} for which ψρd (Tm ) ≥ n. One can show that ψρd (T1 ) ≤ t < n. Therefore m > 1. Evidently, ψρd (Tm−1 ) < n. Since ψρd (Tm ) ≤ ψρd (Tm−1 ) + t, we have ψρd (Tm−1 ) ≥ n − t. We denote by z a schema d from Σρ such that Tρ (z, K ) = Tm−1 . Using Theorem 10.1 we obtain ψˆ ρ,K (z) < n d dd ˆ ˆ  and ψρ,K (z) ≥ n − t. Therefore Ψτ (n) ≥ n − t. Proposition 11.3 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then the following statements hold: (a) Either Typ Ψˆ τdd = ε or Typ Ψˆ τdd = χ , and Ψˆ τdd (n) ≤ n for any n ∈ Arg Ψˆ τdd . (b) Typ Ψˆ τdd = ε if and only if Typ Nτ = ε and Typ Sτ = ε. Proof Evidently, Ψˆ τdd (n) ≤ n for any n ∈ Arg Ψˆ τdd . From here and from Lemma 11.3 the statement (a) follows. Statement (b) follows from Lemmas 11.3 and 11.8. 

11.4 Functions Ψˆ τai , Ψˆ τad , Ψˆ τaa , Ψˆ τsi , Ψˆ τsd , Ψˆ τsa , and Ψˆ τss

145

11.4 Functions Ψˆ τai , Ψˆ τad , Ψˆ τaa , Ψˆ τsi , Ψˆ τsd , Ψˆ τsa , and Ψˆ τss Lemma 11.11 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then the following statements hold: (a) ψρa (T ) ≤ Sψ (T ) for any table T ∈ Mρ,K . 0−1 (b) ψρs (T ) ≤ Sψ (T ) for any table T ∈ Mρ,K . ¯ ≤ Sψ (T, δ) ¯ ≤ Proof Let T ∈ Mρ,K and δ¯ ∈ Δ(T ). One can show that Mρ,ψ (T, δ) Sψ (T ). From here and from Theorems 6.1 and 6.2 the statements (a) and (b) follow.  Lemma 11.12 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then the following three statements are equivalent: a (a) There exists a constant c ∈ ω such that ψˆ ρ,K (z) ≤ c for any z ∈ Σρ . s ˆ (b) There exists a constant c ∈ ω such that ψρ,K (z) ≤ c for any z ∈ Σρ0−1 (K ). (c) Typ Sτ = ε. Proof Let the statement (c) hold. Using Lemma 11.11 and Theorem 10.1 we conclude that the statements (a) and (b) hold. Let the statement (c) do not hold. Using Lemma 11.7 we conclude that the statements (a) and (b) do not hold.  Lemma 11.13 Let τ = (ρ, K , ψ) be a restricted sccf-triple, and there exist t ∈ ω such that Sτ (t) = ∞. Then Ψˆ τss (n) ≥ n − t for any n ∈ Arg Ψˆ τss . Proof Let n ∈ Arg Ψˆ τss . If n ≤ t, then, evidently, the inequality Ψˆ τss (n) ≥ n − t holds. Let n > t. Since Sτ (t) = ∞, there exists a table T ∈ Mρ,K such that m ψ (T ) ≤ t and Sψ (T ) > n. Let P(T ) = { f 1 , . . . , f m }. Let T1 , T2 , . . . , Tm be a sequence of tables from Mρ,K having the following property: P(Ti ) = { f 1 , . . . , f i } for i = 1, . . . , m. Using the inequality m ψ (T ) ≤ t and the fact that the function ψ has the property Λ1 one can show that Sψ (Ti+1 ) ≤ Sψ (Ti ) + t for i = 1, . . . , m − 1. Let i be the minimum number from the set {1, . . . , m} such that Sψ (Ti ) ≥ n. One can show that Sψ (T1 ) ≤ t. Therefore i > 1. Then Sψ (Ti−1 ) < n. Taking into account that Sψ (Ti ) ≤ Sψ (Ti−1 ) + t we obtain n − t ≤ Sψ (Ti−1 ) < n. Using Lemma 11.7 we s conclude that there exists a schema z ∈ Σρ0−1 (K ) such that n − t ≤ ψˆ ρ,K (z) < n.  Hence Ψˆ τss (n) ≥ n − t. Lemma 11.14 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then there exists m ∈ ω such that, for any n ∈ ω, if n ≥ m, then the values Ψˆ τai (n), Ψˆ τad (n), Ψˆ τaa (n), Ψˆ τsi (n), Ψˆ τsd (n), Ψˆ τsa (n), and Ψˆ τss (n) are definite and Ψˆ τai (n) = Ψˆ τad (n) = Ψˆ τaa (n) = Ψˆ τsi (n) = Ψˆ τsd (n) = Ψˆ τsa (n) = Ψˆ τss (n). Proof Let ρ = (F, k). Denote B = {Ψˆ τai , Ψˆ τad , Ψˆ τaa , Ψˆ τsi , Ψˆ τsd , Ψˆ τsa , Ψˆ τss }. Let ϕ ∈ B. Using Lemma 11.1 we obtain Typ ϕ = ω. Hence there exists a number m ϕ ∈ ω

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11 Functions on Main Diagonal and Below

such that Arg ϕ = ω(m ϕ ). Set m = max{m ϕ : ϕ ∈ B}. Let n ∈ ω and n ≥ m. We now show that Ψˆ τsi (n) = Ψˆ τaa (n) = Ψˆ τss (n). Let b ∈ {a, s}. By Lemma 11.2, Ψˆ τsi (n) ≤ Ψˆ τbb (n). Let us show that Ψˆ τbb (n) ≤ Ψˆ τsi (n). Let bb Ψˆ τ (n) = t. If t = 0, then, evidently, the considered inequality holds. Let t > 0. By Theorem 10.1, there exists a table T ∈ Mρ,K such that ψρb (T ) = t and if b = s, 0−1 . Using Theorems 6.1 and 6.2 we conclude that there exists δ¯ ∈ then T ∈ Mρ,K ¯ Let dim T = p, μT (1) = f 1 , . . . , μT ( p) = Δ(T ) such that ψρb (T ) = Mρ,ψ (T, δ). f p , δ¯ = (δ1 , . . . , δ p ), and α be a word from Ωρ (T ) having the following properties: T α ∈ Mρ C , χ (α) ⊆ {( f 1 , δ1 ), . . . , ( f p , δ p )}, and ψ(α) = t. Taking into account that t > 0 and the function ψ has the property Λ4 we obtain α = λ. Let α = ( f i1 , δi1 ) · · · ( f ir , δir ). Set z  = (ν, f i1 , . . . , f ir ), where ν : E kr → {{0}, {1}} and, for any σ¯ ∈ E kr , if σ¯ = (δi1 , . . . , δir ), then ν(σ¯ ) = {1} and if σ¯ = (δi1 , . . . , δir ), then ν(σ¯ ) = {0}. Denote T  = Tρ (z  , K ). Let dim T  = l and μT  (1) = f j1 , . . . , μT  (l) = f jl . Set δ¯ = (δ j1 , . . . , δ jl ). Evidently, δ¯ ∈ Δ(T  ) and νT  (δ¯ ) = {1}. Let β ∈ Ωρ (T  ), χ (β) ⊆ {( f j1 , δ j1 ), . . . , ( f jl , δ jl )}, and T  β ∈ Mρ C . Then, evidently, Δ(T  β) = {δ¯ }. Using this equality one can show that Tβ = T α. Hence ψ(β) ≥ t. Therefore Mρ,ψ (T  , δ¯ ) ≥ t. By Theorem 6.2, ψρs (T  ) ≥ t. Using Theorem 10.1 we s i s obtain ψˆ ρ,K (z  ) ≥ t. Evidently, ψˆ ρ,K (z  ) = t. By Lemma 11.1, ψˆ ρ,K (z  ) = t. Hence Ψˆ τsi (n) ≥ t. Therefore Ψˆ τsi (n) = Ψˆ τbb (n). Let ϕ ∈ B, n ∈ ω, and n ≥ m. Using Lemma 11.2 we obtain Ψˆ τsi ≺ ϕ ≺ Ψˆ τaa or si Ψˆ τ ≺ ϕ ≺ Ψˆ τss . Taking into account the equalities Ψˆ τsi (n) = Ψˆ τss (n) = Ψˆ τaa (n) we conclude that ϕ(n) = Ψˆ τsi (n).  Proposition 11.4 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then the following statements hold: (a) There exists m ∈ ω such that, for any n ∈ ω, if n ≥ m, then the values Ψˆ τai (n), Ψˆ τad (n), Ψˆ τaa (n), Ψˆ τsi (n), Ψˆ τsd (n), Ψˆ τsa (n), and Ψˆ τss (n) are definite and Ψˆ τai (n) = Ψˆ τad (n) = Ψˆ τaa (n) = Ψˆ τsi (n) = Ψˆ τsd (n) = Ψˆ τsa (n) = Ψˆ τss (n). (b) For any function ϕ ∈ {Ψˆ τai , Ψˆ τad , Ψˆ τaa , Ψˆ τsi , Ψˆ τsd , Ψˆ τsa , Ψˆ τss }, the following three statements hold: (b.1) ϕ(n) ≤ n for any n ∈ Arg ϕ. (b.2) Typ ϕ = ε or Typ ϕ = χ . (b.3) Typ ϕ = ε if and only if Typ Sτ = ε. Proof Statement (a) follows from Lemma 11.14. Statement (b) follows from Lemmas 11.1–11.14. 

References 1. Alsolami, F., Azad, M., Chikalov, I., Moshkov, M.: Decision and Inhibitory Trees and Rules for Decision Tables with Many-valued Decisions, Intelligent Systems Reference Library, vol. 156. Springer, Cham (2020)

References

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2. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity, Local approach. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets IV, Lecture Notes in Computer Science, vol. 3700, pp. 125–143. Springer, Berlin (2005) 3. Moshkov, M.: Time complexity of decision trees. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III, Lecture Notes in Computer Science, vol. 3400, pp. 244–459. Springer, Berlin (2005) 4. Moshkov, M., Zielosko, B.: Combinatorial Machine Learning - A Rough Set Approach. Studies in Computational Intelligence, vol. 360. Springer, Berlin (2011)

Chapter 12

Functions Over Main Diagonal

In this chapter, for restricted sccf-triples τ , we study functions Ψˆ τbc , b, c ∈ {i, d, a, s}, located in the matrix of upper local bounds for the triple τ over the main diagonal. For each of these functions, we list all possible upper types and consider criterion for each such type. In several cases, we give upper and lower bounds for the considered functions. Some similar results were obtained in [2] for types of functions different from the upper types considered in this book.

12.1 Functions Ψˆ τi d , Ψˆ τi a , and Ψˆ τi s Proposition 12.1 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψˆ τid = Typ Ψˆ τia = Typ Ψˆ τis = ω and Arg Ψˆ τid = Arg Ψˆ τia = Arg Ψˆ τis = ∅. Proof Let ρ = (F, k) and f ∈ F. For j ∈ ω \ {0}, we denote by z j the tuple j (ν j , f, . . . , f ) in which the element f is repeated exactly j times, ν j : E k → P(ω), j ¯ = {1} for any δ¯ ∈ E k . Taking into account that the function ψ has the propand ν j (δ) d a s erty Λ4 one can show that ψˆ ρ,K (z j ) = ψˆ ρ,K (z j ) = ψˆ ρ,K (z j ) = 0. Since the function i id ˆ ˆ ψ has the property Λ3, we have ψρ,K (z) ≥ j. Hence Ψτ (0) = ∞, Ψˆ τia (0) = ∞, and Ψˆ τis (0) = ∞. From these equalities it follows that the considered statement holds. 

12.2 Function Ψˆ τas Lemma 12.1 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Sτ = ε or Typ Sτ = χ , or Typ Sτ = ω.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_12

149

150

12 Functions Over Main Diagonal

Proof Let Typ Sτ = ε and Typ Sτ = ω. We show that, for any m ∈ ω, there exists n ∈ ω such that n > m and Sτ (n) ≥ n. Since Typ Sτ = ε, there exists a table T ∈ Mρ,K such that Sψ (T ) > m. Let Sψ (T ) = n. By Lemma 11.7, there exists a schema i a z ∈ Σρ such that ψˆ ρ,K (z) = n and ψˆ ρ,K (z) = n. Set T  = Tρ (z, K ). Using Theorem 10.1 we obtain ψρa (T  ) = n. From this equality and from Lemma 11.11 it follows i that Sψ (T  ) ≥ n. From the equality ψˆ ρ,K (z) = n and from the property Λ2 of the  function ψ it follows that m ψ (T ) ≤ n. Hence Sτ (n) ≥ n. Therefore Typ Sτ = χ .  Lemma 12.2 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then the following statements hold: (a) For any n ∈ ω, if n ∈ Arg Sτ , then n ∈ Arg Ψˆ τas and Sτ (n) = Ψˆ τas (n). (b) For any n ∈ ω, if Sτ (n) = ∞, then Ψˆ τas (n) = ∞. Proof (a) Assume that ρ = (F, k). Let n ∈ Arg St and Sτ (n) = t. First, we consider the case, when t = 0. Since n ∈ Arg Sτ , there exists z ∈ Σρ0−1 (K ) such that s ψˆ ρ,K (z) ≤ n. Therefore if Ψˆ τas (n) = ∞, then Ψˆ τas (n) ∈ ω and, evidently, Ψˆ τas (n) ≥ t = 0. Let now t > 0. Then there exists a table T ∈ Mρ,K for which m ψ (T ) ≤ n and Sψ (T ) = t. By Lemma 11.7, there exists a schema z ∈ Σρ0−1 (K ) such that s a ψˆ ρ,K (z) ≤ n and ψˆ ρ,K (z) = t. Hence if Ψˆ τas (n) = ∞, then the value Ψˆ τas (n) is definite and Ψˆ τas (n) ≥ t = Sτ (n). s (z) ≤ n. Set We now show that Ψˆ τas (n) ≤ Sτ (n). Let z ∈ Σρ0−1 (K ) and ψˆ ρ,K 0−1 T = Tρ (z, K ). Evidently, T ∈ Mρ,K . By Theorem 10.1, ψρs (T ) ≤ n. Let dim T = p, μT (1) = f 1 , . . . , μT ( p) = f p , and δ¯ = (δ1 , . . . , δ p ) ∈ Δ(T ). Let us show that ¯ ≤ Sτ (n). If T ∈ Mρ C , then the considered inequality holds. Let T ∈ Mρ,ψ (T, δ) / Mρ C . Since ψρs (T ) ≤ n, there exists a strongly nondeterministic decision tree Γ for the table T such that ψ(Γ ) ≤ n. Using Lemma 3.2 we conclude that P(Γ ) is a test for the table T . Since T ∈ / Mρ C , P(Γ ) = ∅. Let P(Γ ) = { f i1 , . . . , f ir }. Taking into account that the function ψ has the property Λ2 we obtain ψ( f ) ≤ n for any f ∈ P(Γ ). Since P(Γ ) is a test for the table T , we have T ( f i1 , δi1 ) · · · ( f ir , δir ) ∈ Mρ C . Set z  = (ν, f i1 , . . . , f ir ), where ν : E kr → {{1}}, T  = Tρ (z  , K ), and δ¯ = (δi1 , . . . , δir ). It is clear that m ψ (T  ) ≤ n. Therefore Sψ (T  , δ¯ ) ≤ Sψ (T  ) ≤ Sτ (n). Let α ∈ Ωρ (T  ), Δ(T  α) = {δ¯ }, and ψ(α) ≤ Sτ (n). One can show that T α = ¯ ≤ Sτ (n). By Theorem 6.1, ψρa (T ) ≤ T ( f i1 , δi1 ) · · · ( f ir , δir ). Therefore Mρ,ψ (T, δ) Sτ (n). Therefore Ψˆ τas (n) ≤ Sτ (n). Thus, the statement (a) holds. (b) Let Sτ (n) = ∞. Then the set {Sψ (T ) : T ∈ Mρ,K , m ψ (T ) ≤ n} is infinite. By a s (z) : z ∈ Σρ0−1 (K ), ψˆ ρ,K (z) ≤ n} is infinite. Therefore the Lemma 11.7, the set {ψˆ ρ,K statement (b) holds.  Proposition 12.2 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψˆ τas = ε or Typ Ψˆ τas = χ , or Typ Ψˆ τas = ω, and the following statements hold: (a) Typ Ψˆ τas = ε if and only if Typ Sτ = ε. (b) Typ Ψˆ τas = χ if and only if Typ Sτ = χ . (c) Typ Ψˆ τas = ω if and only if Typ Sτ = ω.

12.2 Function Ψˆ τas

151

(d) If Typ Ψˆ τas = χ , then there exists m ∈ ω such that, for any n ∈ ω, if n ≥ m, then the values Ψˆ τas (n) and Sτ (n) are definite and Ψˆ τas (n) = Sτ (n). Proof Using Lemma 12.2 one can show that Typ Ψˆ τas = Typ Sτ . From this equality and from Lemma 12.1 it follows that Typ Ψˆ τas ∈ {ε, χ , ω} and the statements (a), (b), and (c) hold. Let Typ Ψˆ τas = χ . Then Typ Sτ = χ . Therefore there exists m ∈ ω for which Arg Sτ = ω(m). By Lemma 12.2, ω(m) ⊆ Arg Ψˆ τas and Ψˆ τas (n) = Sτ (n) for any n ∈ ω(m).  Corollary 12.1 Let τ = (ρ, K , ψ) be a restricted sccf-triple and Typ Ψˆ τas = χ . Then a polynomial P1 , such that Ψˆ τas (n) ≤ P1 (n) for any n ∈ Arg Ψˆ τas , exists if and only if there exists a polynomial P2 such that Sτ (n) ≤ P2 (n) for any n ∈ Arg Sτ .

12.3 Function Ψˆ τds Let G be a finite undirected graph without loops in which nodes are colored with two colors. An edge of the graph will be called many-colored if this edge is incident to nodes with different colors. By induction on the number of nodes, it is not difficult to prove that the nodes of G can be colored with two colors such that at least one-half of the graph G edges will be many-colored. Lemma 12.3 Let τ = (ρ, K , ψ) be a restricted sccf-triple, ρ = (F, k), and T1 ∈ 0−1 such that P(T2 ) ⊆ P(T1 ) and Mρ,K \ Mρ C . Then there exists a table T2 ∈ Mρ,K ψρd (T2 ) ≥

logk Nρ (T1 ) Sˆτ (T1 )

− 2.

Proof We denote by T a table with the minimum dimension from Mρ,K having the following properties: P(T ) ⊆ P(T1 ) and Nρ (T ) = Nρ (T1 ). Let dim T = n and / Mρ C , we have Nρ (T1 ) > 1. Using this μT (1) = f 1 , . . . , μT (n) = f n . Since T1 ∈ inequality one can show that, for any element f i ∈ P(T ), there exist two tuples δ¯0 ( f i ) and δ¯1 ( f i ) from Δ(T ) which are different only in the component with number i. We denote by G an undirected graph such that the set of nodes of G coincides with the set {δ¯0 ( f i ), δ¯1 ( f i ) : f i ∈ P(T )}, and the set of edges coincides with the set {r ( f i ) : f i ∈ P(T )}. For each f i ∈ P(T ), the edge r ( f i ) is incident to nodes δ¯0 ( f i ) and δ¯1 ( f i ). As it was mentioned above, there exists a coloring of nodes of G with two colors (for example, black and white) such that at least one-half of the graph G edges are many-colored. We now define a mapping ν : E kn → {{0}, {1}}. Let δ¯ ∈ E kn . ¯ = {1}. If the tuple δ¯ is a node of the graph G colored with the white color, then ν(δ) ¯ Otherwise, ν(δ) = {0}. Set z = (ν, f 1 , . . . , f n ) and T2 = Tρ (z, K ). Evidently, T2 ∈ 0−1 , Nρ (T2 ) = Nρ (T1 ), and P(T2 ) ⊆ P(T1 ). One can show that Θρ,h (T2 ) ≥ n2 . Mρ,K Taking into account that the function ψ has the property Λ3 and using Theorem 3.3 we obtain ψρd (T ) ≥ logk ( n2 ). Since P(T2 ) ⊆ P(T1 ), we have Sψ (T2 ) ≤ Sˆτ (T1 ). ˆ

By Lemma 11.5, Nρ (T1 ) = Nρ (T2 ) ≤ (kn) Sψ (T2 ) ≤ (kn) Sτ (T1 ) . Taking into account

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that Nρ (T1 ) > 1 we obtain Sˆτ (T1 ) > 0 and logk Nρ (T1 ) ≤ Sˆτ (T1 )(1 + logk n). Hence log N (T ) log N (T )  logk n ≥ Sˆk (Tρ ) 1 − 1. Therefore ψρd (T2 ) ≥ Sˆk (Tρ ) 1 − 2. τ

τ

1

1

Lemma 12.4 Let τ = (ρ, K , ψ) be a restricted sccf-triple and T ∈ Mρ,K . Then Sψ (T ) ≤ (Nρ (T ) − 1)m ψ (T ). Proof One can show that Sh (T ) ≤ Nρ (T ) − 1. The considered statement follows from this inequality and from properties Λ1 and Λ4 of the function ψ.  Proposition 12.3 Let τ = (ρ, K , ψ) be a restricted sccf-triple and ρ = (F, k). Then Typ Ψˆ τds ∈ {ε, χ , ω} and the following statements hold: (a) Typ Ψˆ τds = ε if and only if Typ Nτ = ε and Typ Sτ = ε. (b) Typ Ψˆ τds = χ if and only if Typ Nτ = ω and Typ Sτ = ε. (c) Typ Ψˆ τds = ω if and only if Typ Nτ = ω. (d) If Typ Ψˆ τds = χ , then there exists m ∈ ω such that, for any n ∈ ω(n), the values Sτ (n), Nτ (n), and Ψˆ τds (n) are definite, and the inequalities Ψˆ τds (n) ≥ Sτ (n), log N (n) Ψˆ τds (Sτ (n)) ≥ Skτ (n)τ − 2, and Ψˆ τds (n) ≤ 2Sτ (n) logk Nτ (n) hold. Proof First, we prove the following statement: (a*) If Typ Nτ = Typ Sτ = ε, then Typ Ψˆ τds = ε. d Let Typ Nτ = Typ Sτ = ε. By Lemma 11.8, there exists c ∈ ω such that ψˆ ρ,K (z) 0−1 ds ˆ ≤ c for any z ∈ Σρ (K ). Hence Typ Ψτ = ε. Therefore the statement (a*) holds. We now prove the following statement: (b*) If Typ Nτ = ω and at least one of the conditions Typ Nτ = ε, Typ Sτ = ε holds, then Typ Ψˆ τds = χ and there exists m ∈ ω such that, for any n ∈ ω(n), the values Sτ (n), Nτ (n), and Ψˆ τds (n) are definite and the following inequalities hold: Ψˆ τds (n) ≥ Sτ (n) , Ψˆ τds (Sτ (n)) ≥

logk Nτ (n) −2, Sτ (n)

Ψˆ τds (n) ≤ 2Sτ (n) logk Nτ (n) .

(12.1) (12.2) (12.3)

Let Typ Nτ = ω and at least one of the conditions Typ Nτ = ε, Typ Sτ = ε hold. Since the function ψ has the property Λ4 and at least one of the conditions Typ Nτ = ε, Typ Sτ = ε holds, there exists a table T ∈ Mρ,K such that Nρ (T ) ≥ 2. Set m = m ψ (T ). Then, evidently, ω(m) ⊆ Arg Nτ and Nτ (n) ≥ 2 for any n ∈ ω(m). Let n ∈ ω(m). Then the set {Sψ (T ) : T ∈ Mρ,K , m ψ (T ) ≤ n} is nonempty. Using Lemma 12.4 and the relation Typ Nτ = ω we obtain ω(m) ⊆ Arg Sτ . Using the inequality Nτ (n) ≥ 2 and the fact that the function ψ has the property Λ4 we have Sτ (n) > 0. d s (z) : z ∈ Σρ0−1 (K ), ψˆ ρ,K (z) ≤ n}. By Theorem Let n ∈ ω(m). Set B = {ψˆ ρ,K 0−1 d s 10.1, B = {ψρ (T ) : T ∈ Mρ,K , ψρ (T ) ≤ n}. Taking into account that the function 0−1 and ψρs (T ) ≤ n. ψ has the property Λ4 one can show that B = ∅. Let T ∈ Mρ,K

12.3 Function Ψˆ τds

153

We now show that ψρd (T ) ≤ 2Sτ (n) log2 Nτ (n). If T ∈ Mρ C , then, evidently, ψρd (T ) = 0. It is clear that Sτ (n) ≥ 0 and Nτ (n) ≥ 1. Therefore the considered inequality holds. Let T ∈ / Mρ C . Since ψρs (T ) ≤ n, there exists a strongly nondeterministic decision tree Γ for the table T such that ψ(Γ ) ≤ n. By Lemma 3.2, the set P(Γ ) is a test for the table T . Since T ∈ / Mρ C , we have P(Γ ) = ∅. Let dim T = m, μT (1) = f 1 , . . . , μT (m) = f m , and P(Γ ) = { f i1 , . . . , f ir }. Taking into account that the function ψ has the property Λ2 we obtain ψ( f i j ) ≤ n for j = 1, . . . , r . We now define a mapping ν : E kr → P(ω). Let δ¯ = (δ1 , . . . , δr ) ∈ E kr . ¯ = {0}. Let Δ(T γ ) = ∅. Since Set γ = ( f i1 , δ1 ) · · · ( f ir , δr ). If Δ(T γ ) = ∅, then ν(δ) ¯ is the minP(Γ ) is a test for the table T , we have T γ ∈ Mρ C . In this case, ν(δ) imum number from the set Π (T γ ). Set z = (ν, f i1 , . . . , f ir ) and T  = Tρ (z, K ). Evidently, m ψ (T  ) ≤ n. Therefore Nρ (T  ) ≤ Nτ (n) and Sˆτ (T  ) ≤ Sτ (n). From the last inequality and from Lemma 11.4 it follows that Mρ,ψ (T  ) ≤ 2Sτ (n). Using these inequalities, Corollary 5.2, and the fact that the function ψ has properties Λ1 and Λ4 we obtain ψρd (T  ) ≤ 2Sτ (n) log2 Nτ (n). One can show that any deterministic decision tree for the table T  is a deterministic decision tree for the table T . Therefore ψρd (T ) ≤ ψρd (T  ) ≤ 2Sτ (n) log2 Nτ (n). Taking into account that T is an arbitrary 0−1 table from Mρ,K such that ψρs (T ) ≤ n we conclude that the value Ψˆ τds (n) is definite and (12.3) holds. Using Lemma 11.9 we obtain Typ Sτ = ε. Using the relation Typ Sτ = ε and Proposition 11.4 we obtain Typ Ψˆ τss = χ . Using this equality, the relation ω(m) ⊆ Arg Ψˆ τds , and Lemma 11.2 we conclude that Typ Ψˆ τds = χ . Let n ∈ ω(m). Then Sτ (n) > 0. By Lemma 11.7, there exists a schema z ∈ Σρ0−1 (K ) d s such that ψˆ ρ,K (z) = Sτ (n) and ψˆ ρ,K (z) ≤ n. Hence the inequality (12.1) holds. The value Nτ (n) is definite, Nτ (n) ≥ 2, and there exists a table T1 ∈ Mρ,K \ Mρ C such that m ψ (T1 ) ≤ n and Nρ (T1 ) = Nτ (n). By Lemma 12.3, there exists a table log N (T ) 0−1 such that P(T2 ) ⊆ P(T1 ) and ψρd (T2 ) ≥ Sˆk (Tρ ) 1 − 2. It is clear that T2 ∈ Mρ,K τ 1 log N (n) Sˆτ (T1 ) ≤ Sτ (n). Therefore ψρd (T2 ) ≥ Skτ (n)τ − 2. Since P(T2 ) ⊆ P(T1 ), we have m ψ (T2 ) ≤ n. Hence Sψ (T2 ) ≤ Sτ (n). By Lemma 11.11, ψρs (T2 ) ≤ Sτ (n). Let z be a schema from Σρ0−1 (K ) for which T2 = Tρ (z, K ). Using Theorem 10.1 we obtain log N (n) d s ψˆ ρ,K (z) ≥ Skτ (n)τ − 2 and ψˆ ρ,K (z) ≤ Sτ (n). Hence (12.2) holds. Thus, the statement (b*) holds. Let us prove the following statement: (c*) If Typ Nτ = ω, then Typ Ψˆ τds = ω. Let Typ Nτ = ω. Then there exists t ∈ ω such that Nτ (t) = ∞. Evidently, either the value Sτ (t) is definite or Sτ (t) = ∞. Let the value Sτ (t) be definite and Sτ (t) = 0−1 , m ψ (T ) ≤ t}. Let T ∈ B. Then, evidently, Sψ (T ) ≤ m. Set B = {T : T ∈ Mρ,K Sτ (t) ≤ m. By Lemma 11.11, ψρs (T ) ≤ m. Using the equalities Sτ (t) = m, Nτ (t) = ∞ and Lemma 12.3 one can show that there is no constant c ∈ ω such that ψρd (T ) ≤ c for any T ∈ B. By Theorem 10.1, Ψˆ τds (m) = ∞. Let Sτ (t) = ∞. Then, using Lemma 11.7, one can show that Ψˆ τds (t) = ∞. Thus, the statement (c*) holds.

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From the statements (a*), (b*), and (c*) it follows that Typ Ψˆ τds ∈ {ε, χ , ω}. Let Typ Ψˆ τds = ε. Using the statement (c*) we obtain Typ Nτ = ω. Using this relation and the statement (b*) we conclude that Typ Sτ = Typ Nτ = ε. From here and from the statement (a*) the statement (a) follows. Let Typ Ψˆ τds = χ . Using the statements (a*) and (c*) we obtain Typ Nτ = ω, and at least one of the conditions Typ Sτ = ε, Typ Nτ = ε holds. Assume that Typ Sτ = 0−1 . By Lemma ε. Then there exists c1 ∈ ω such that Sψ (T ) ≤ c1 for any table T ∈ Mρ,K 0−1 s 11.11, ψρ (T ) ≤ c1 for any table T ∈ Mρ,K . From here and from Theorem 10.1 it s (z) ≤ c1 for any schema z ∈ Σρ0−1 (K ). Since Typ Ψˆ τds = χ , there follows that ψˆ ρ,K d (z) ≤ c2 for any schema z ∈ Σρ0−1 (K ). Hence is no constant c2 ∈ ω such that ψˆ ρ,K Ψˆ τds (c1 ) = ∞ which is impossible. Therefore Typ Sτ = ε. Thus, if Typ Ψˆ τds = χ , then Typ Nτ = ω and Typ Sτ = ε. Let Typ Nτ = ω and Typ Sτ = ε. Then using the statement (b*) we obtain Typ Ψˆ τds = χ . Thus, the statement (b) holds. Let Typ Ψˆ τds = ω. Using the statement (a*) we conclude that at least one of the conditions Typ Sτ = ε, Typ Nτ = ε holds. Using the statement (b*) we obtain Typ Nτ = ω. From here and from the statement (c*) the statement (c) follows. Let Typ Ψˆ τds = χ . Using the statements (b) and (b*) we conclude that the statement (d) holds.  Corollary 12.2 Let τ = (ρ, K , ψ) be a restricted sccf-triple and Typ Ψˆ τds = χ . Then the following statements are equivalent: (a) There exists a polynomial P0 such that Ψˆ τds  P0 . (b) There exist polynomials P1 and P2 such that Sτ  P1 and Nτ  2 P2 . Proof Let the statement (b) hold. Then, using Proposition 12.3, we conclude that there exists m ∈ ω such that Ψˆ τds (n) ≤ 2P1 (n)P2 (n) for any n ∈ ω(m). Hence the statement (a) holds. Let the statement (b) do not hold. We show that the statement (a) does not hold. Assume the contrary: there exists a polynomial P0 such that Ψˆ τds  P0 . Let there is no polynomial P1 such that Sτ  P1 . By Proposition 12.3, there exists m ∈ ω such that Sτ (n) ≤ Ψˆ τds (n) ≤ P0 (n) for any n ∈ ω(m), which is impossible. Let there exist a polynomial P1 such that Sτ  P1 , and there be no polynomial P2 such that Nτ  2 P2 . By Proposition 12.3, there exists m ∈ ω such that logk Nτ (n) ≤ ˆ ds Ψˆ τds (Sτ (n))Sτ (n) + 2 for any n ∈ ω(m). Therefore Nτ (n) ≤ 2(log2 k)(Ψτ (Sτ (n))Sτ (n)+2) for any n ∈ ω(m). Taking into account that there exist polynomials P0 and P1 such that Ψˆ τds  P0 and Sτ  P1 one can show that there exist nondecreasing polynomials P0 and P1 such that Ψˆ τds  P0 and Sτ  P1 . Hence there exists m  ∈ ω such that    Nτ (n) ≤ 2(log2 k)(P0 (P1 (n))P1 (n)+2) for any n ∈ ω(m  ), which is impossible. Hence the statement (a) does not hold. 

12.4 Function Ψˆ τda

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12.4 Function Ψˆ τda Let ρ = (F, k) be a signature, T ∈ Mρ , dim T = n and μT (1) = f 1 , . . . , μT (n) = ˆ ) the set of nonempty words f i1 · · · f im from P(T )∗ such f n . We denote by P(T ˆ ). Set Δ(T, α) = {(δi1 , . . . , δim ) : that i 1 < i 2 < · · · < i m . Let α = f i1 · · · f im ∈ P(T (δ1 , . . . , δn ) ∈ Δ(T )}. We now define the value Z ρ (T ). If Nρ (T ) ≤ 1, then Z ρ (T ) = 0. Let Nρ (T ) ≥ 2. Then Z ρ (T ) is the maximum m ∈ {1, . . . , n} such that there exist ˆ ) and sets B1 , . . . , Bm ⊆ E k which have the following properties: a word α ∈ P(T |α| = m, |B1 | = |B2 | = · · · = |Bm | = 2 and |Δ(T, α) ∩ (B1 × B2 × · · · × Bm )| = 2m . Note that this parameter is similar to the independence dimension [1]. Let τ = (ρ, K , ψ) be a sccf-triple. We now define a function Z τ with Arg Z τ ⊆ ω. Let n ∈ ω. Then Z τ (n) = max{Z ρ (T ) : T ∈ Mρ,K , m ψ (T ) ≤ n}. We now define a r r : Σρ → ω. Let z ∈ Σρ . Then lρ,K (z) = l r (Tρ (z, K )). function lρ,K The following statement was published in [3]. It is similar to results from [4]. Lemma 12.5 Let ρ = (F, k) be a signature, T ∈ Mρ , and Δ(T ) = ∅. Then 2 Z ρ (T ) ≤ Nρ (T ) ≤ (k 2 dim T ) Z ρ (T ) . Proof Let n ∈ ω \ {0} and t ∈ ω. Denote Nρ (n, t) = max{Nρ (T ) : T ∈ Mρ , dim T ≤ n, Z ρ (T ) ≤ t}. One can see that the value Nρ (n, t) is definite and the following inequalities hold: Nρ (n, 0) ≤ 1 ,

(12.4)

Nρ (1, t) ≤ k .

(12.5)

Let us show that, for any n ∈ ω \ {0} and t ∈ ω, the following inequality holds: Nρ (n + 1, t + 1) ≤ Nρ (n, t + 1) + k 2 Nρ (n, t) .

(12.6)

One can show that there exists a table T ∈ Mρ such that dim T = n + 1, Z ρ (T ) ≤ t + 1, and Nρ (T ) = Nρ (n + 1, t + 1). Let μT (1) = f 1 , . . . , μT (n + 1) = f n+1 . Set α = f 1 f 2 · · · f n . Let δ¯ = (δ1 , . . . , δn ) ∈ Δ(T, α). Denote ¯ = {δ : δ ∈ E k , (δ1 , . . . , δn , δ) ∈ Δ(T )} . η(δ) We define tables T0 and Tδ1 δ2 , where δ1 , δ2 ∈ E k and δ1 = δ2 , in the following way: dim T0 = dim Tδ1 ,δ2 = n, μT0 (1) = μTδ1 ,δ2 (1) = f 1 , . . . , μT0 (n) = μTδ1 ,δ2 (n) =   ¯  = 1} and Δ(Tδ1 ,δ2 ) = f n , νT0 ≡ {0}, νTδ1 ,δ2 ≡ {0}, Δ(T0 ) = {δ¯ : δ¯ ∈ Δ(T, α), η(δ) ¯ δ2 ∈ η(δ)}. ¯ { δ¯ : δ¯ ∈ Δ(T, α), δ1 ∈ η(δ), One can show that Nρ (T ) ≤ Nρ (T0 ) + δ1 ,δ2 ∈E k ,δ1 =δ2 Nρ (Tδ1 ,δ2 ) and Z ρ (T0 ) ≤ t + 1. Let δ1 , δ2 ∈ E k and δ1  = δ2 . We now show that Z ρ (Tδ1 ,δ2 ) ≤ t. Assume the contrary. Then, as it is not difficult to see, Z ρ (T ) ≥ t + 2 which is impossible. Therefore Z ρ (Tδ1 ,δ2 ) ≤ t. Taking into

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account that Nρ (T ) = Nρ (n + 1, t + 1) we obtain Nρ (n + 1, t + 1) ≤ Nρ (n, t + 1) + k 2 Nρ (n, t). Hence (12.6) holds. Let us show that, for any n ∈ ω \ {0} and t ∈ ω, Nρ (n, t) ≤ k 2t n t .

(12.7)

From (12.4) and (12.5) it follows that (12.7) holds if n = 1 or t = 0. We prove the inequality (12.7) by induction on the value n + t. If n + t ≤ 2, then, as it follows from (12.4) and (12.5), the inequality (12.7) holds. Assume that r ∈ ω, r ≥ 2, and if n + t ≤ r , then (12.7) holds. Let n  ∈ ω \ {0}, t  ∈ ω, and n  + t  ≤ r + 1. If n  = 1 or t  = 0, then (12.7) holds. Let n  = n + 1 and t  = t + 1, where n ∈ ω \ {0} and t ∈ ω. Using (12.6) and the inductive hypothesis we obtain Nρ (n + 1, t + 1) ≤ k 2(t+1) n t+1 + k 2(t+1) n t ≤ k 2(t+1) (n + 1)t+1 . Hence (12.7) holds. Let T ∈ Mρ and Δ(T ) = ∅. The inequality Nρ (T ) ≥ 2 Z ρ (T ) is obvious. The  inequality Nρ (T ) ≤ (k 2 dim T ) Z ρ (T ) follows from (12.7). Lemma 12.6 Let τ = (ρ, K , ψ) be a restricted sccf-triple, ρ = (F, k), T1 ∈ Mρ,K , ≤ Z ρ (T1 ). Then Z ρ (T1 ) > 0, and n be the maximum number from ω such that n(n+1) 2 a d ˆ there exists a problem schema z ∈ Σρ such that ψρ,K (z) ≤ 3m ψ (T1 ), ψˆ ρ,K (z) ≥ n(n+1) r n − 1, and lρ,K (z) ≥ 2 . . Since Z ρ (T1 ) ≥ r (n), there exists Proof For an arbitrary m ∈ ω, set r (m) = m(m+1) 2 ˆ · · · f ∈ P(T ) and sets B , . . . , B a word α = f r(n) 1 r (n) ⊆ E k such that |B1 | = |B2 | =   1 · · ·  Br (n)  = 2 and Δ(T, α) ∩ (B1 × B2 × · · · × Br (n) ) = 2r (n) . Let, for the definiteness, B1 = · · · = Br (n) = E 2 . For an arbitrary m ∈ ω \ {0}, we denote by G m a directed graph depicted in Fig. 12.1. The set of nodes of G m coincides with the set {1, . . . , r (m)} and is divided into m layers. Let i be a node of G m which does not belong to mth layer. We denote by l(i) the left successor of the node i and by p(i) we denote the right successor of the node i. We now define a mapping ν : E kr (n) → P(ω). Let δ¯ = (δ1 , . . . , δr (n) ) ∈ E kr (n) . ¯ ⊆ {0, 1, . . . , r (n)} and 0 ∈ ν(δ) ¯ if and only if δ1 = 0. Let i ∈ {1, . . . , Then ν(δ)

Fig. 12.1 Graph G m

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¯ if and only if δi = 0 and r (n)}. If i is not a node of nth layer in G n , then i ∈ ν(δ) ¯ if and only if δi = 0. δl(i) = δ p(i) = 0. If i is a node of nth layer, then i ∈ ν(δ) Let us show that, for any δ¯ = (δ1 , . . . , δr (n) ) ∈ E kr (n) , at least one of the following three statements holds: (a.1) δ1 = 0. (a.2) There exists i ∈ {1, . . . , r (n)} such that i is not a node of nth layer, δi = 0, and δl(i) = δ p(i) = 0. (a.3) There exists i ∈ {1, . . . , r (n)} such that i is a node of nth layer and δi = 0. If δ1 = 0, then the statement (a.1) holds. Let δ1 = 0. Let us consider the node 1 of the graph G n . If 1 is a node of nth layer, then the statement (a.3) holds. Let 1 be not a node of nth layer. If δl(1) = 0 and δ p(1) = 0, then the statement (a.2) holds. Let at least one of the considered equalities do not hold. Let, for example, δl(1) = 0. Then consider the node l(1) and so on. Since G n is a finite acyclic graph, in some step, the statement (a.2) or the statement (a.3) will hold. Set z = (ν, f 1 , . . . , fr (n) ) and T2 = Tρ (z, K ). It is clear that T2 ∈ Mρ,K , dim T2 = r (n), and m ψ (T2 ) ≤ m ψ (T1 ). Evidently, E 2r (n) ⊆ Δ(T2 ). Let δ¯ = (δ1 , . . . , δr (n) ). Taking into account that at least one of the statements (a.1), (a.2), and (a.3) holds, and ¯ ≤ 3m ψ (T2 ) ≤ the function ψ has the property Λ1 we conclude that Mρ,ψ (T, δ) ¯ 3m ψ (T1 ). Taking into account that δ is an arbitrary tuple from Δ(T2 ) and using a (z) ≤ 3m ψ (T1 ). Theorem 6.1 we obtain ψρa (T2 ) ≤ 3m ψ (T1 ). By Theorem 10.1, ψˆ ρ,K Let m ∈ ω \ {0}. A complete path in the graph G m is a path from the node 1 to a node of m-th layer. We correspond to a complete path ξ in the graph G m a tuple ¯ ) = (δ1 , . . . , δr (m) ) ∈ E 2r (m) in the following way. Let i ∈ {1, . . . , r (m)}. Then δ(ξ δi = 1 if and only if the path ξ passes through the node i. We denote by Ξ (G m ) the set of complete paths in the graph G m . It is clear that |Ξ (G m )| = 2m−1 . We now define a table T m ∈ Mσ , where σ = (Φ, 2) and Φ = {ϕi : i ∈ ω}, as fol¯ ): lows: dim T m = r (m), μT m (1) = ϕ1 , . . . , μT m (r (m)) = ϕr (m) , and Δ(T m ) = {δ(ξ ξ ∈ Ξ (G m )}. Let ξ ∈ Ξ (G m ) and i be the terminal node of the path ξ . Then ¯ )) = {i}. νT m (δ(ξ   ¯ : δ¯ ∈ Δ(T m α}. Evidently, T m α ∈ Let α ∈ Ωσ (T m ). Set η(T m α) = {νT m (δ) Mσ C if and only if η(T m α) ≤ 1. We now prove the following statement: (b) Let m ∈ ω \ {0}, α ∈ Ωσ (T m ), χ (α) ⊆ {(ϕ1 , 0), . . . , (ϕr (m) , 0)}, and Δ(T m α) = ∅ . Then η(T m α) ≥ max{1, m − |α|}, where |α| is the length of the word α in the alphabet {(ϕ1 , 0), . . . , (ϕr (m) , 0)}. We prove this statement by induction on m. If m = 1, then, evidently, the statement holds. Let, for some m 0 ≥ 1, the statement hold. Let us show that the statement holds for m = m 0 + 1. If α = λ, then, evidently, the considered inequality holds. Let α = λ. If |α| ≥ m − 1, then, evidently, the considered inequality holds. Let |α| < m − 1. Then there exist words β1 and β2 having the following properties: χ (β1 β2 ) = χ (α), |β1 β2 | = |α|, χ (β1 ) ⊆ {(ϕ1 , 0), . . . , (ϕr (m−1) , 0)} and χ (β2 ) ⊆ {(ϕr (m−1)+1 , 0), . . . , (ϕr (m) , 0)}. Evidently, T m α = T m β1 β2 . Since Δ(T m α) = ∅, we

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have Δ(T m β1 ) = ∅. Using this relation it is not difficult to show that Δ(T m−1 β1 ) = ∅. From here and from the inductive hypothesis it follows that η(T m−1 β1 ) ≥  ¯ max{1, m − 1 − |β1 |}. Let { j1 , . . . , js } = δ∈Δ(T m−1 β ) ν T m−1 β1 (δ) and j1 < · · · < ¯ 1 js . Then, as it is not difficult to see, l( j ) < p( j ) < p( j ) < · · · < p( js ) and 1 1 2  m ¯ m ν ( δ). Hence η(T β1 ) ≥ s + 1 ≥ {l( j1 ), p( j1 ), . . . , p( js )} ⊆ δ∈Δ(T m ¯ β1 ) T β1 m m m − |β1 |. One can show that η(T β1 β2 ) ≥ η(T β1 ) − |β2 | ≥ m − |β1 | − |β2 | = m − |α|. Taking into account that m − |α| > 1 we obtain η(T m α) ≥ max{1, m − |α|}. Thus, the statement (b) holds. Let us show that h dσ (T n ) ≥ n − 1. If n = 1, then, evidently, the considered inequality holds. Let n ≥ 2. We now describe a (P(T n ), σ )-tree Dn . Each complete path in the tree Dn contains exactly n nodes. Let ξ = w1 , d1 , . . . , dn−1 , wn be an arbitrary complete path in the tree Dn and, for j = 1, . . . , n − 1, the edge d j be labeled with a pair (ϕi j , δ j ). Then δ j = 0 if and only if Δ(T n (ϕi1 , δ1 ) · · · (ϕi j−1 , δ j−1 )(ϕi j , 0)) = ∅. Let w be a terminal node of the tree Dn . Then, evidently, Δ(T n ζ (w)) = ∅ and h(ζ (w)) = n − 1. Let w be a node of the tree Dn which is not terminal. If ζ (w) = λ, / Mσ C . Let ζ (w) = λ and ζ (w) = (ϕi1 , δ1 ) · · · (ϕit , δt ). Set then, evidently, T n ζ (w) ∈ B = { j : j ∈ {1, . . . , t}, δ j = 0}. We now define a word α ∈ Ωσ (T n ). If B = ∅, then α = λ. If B = ∅ and B = { j (1), . . . , j (s)}, then α = (ϕi j (1) , 0) · · · (ϕi j (s) , 0). Evidently, |α| ≤ n − 2 and T n ζ (w) = T n α. Using the statement (b) we obtain / Mσ C . Thus, the tree Dn is a proof-tree for the η(T n ζ (w)) ≥ 2. Hence T n ζ (w) ∈ bound h dσ (T n ) ≥ n − 1. By Theorem 3.4, h dσ (T n ) ≥ n − 1. Let us show that ψρd (T2 ) ≥ n − 1. We define a table T3 ∈ Mρ as follows: dim T3 = dim T n , Δ(T3 ) = Δ(T n ), νT3 ≡ νT n and μT3 (1) = f 1 , . . . , μT3 (r (n)) = fr (n) . Evidently, h dρ (T3 ) = h dρ (T n ). Therefore h dρ (T3 ) ≥ n − 1. One can show that dim T3 = dim T2 , μT3 ≡ μT2 , Δ(T3 ) ⊆ Δ(T2 ), and νT3 is the restriction of the mapping νT2 to the set Δ(T3 ). Therefore any deterministic decision tree for the table T2 is a deterministic decision tree for the table T3 . Hence h dρ (T2 ) ≥ h dρ (T3 ) ≥ n − 1. Taking into account that the function ψ has the property Λ3 we obtain ψρd (T2 ) ≥ n − 1. d By Theorem 10.1, ψˆ ρ,K (z) ≥ n − 1. One can show that, for any i ∈ {1, . . . , r (n)}, there is a tuple δ¯ ∈ Δ(T2 ) such that r ¯ = {i}. Therefore l r (T2 ) ≥ n(n+1) and lρ,K (z) ≥ n(n+1) .  ν(δ) 2 2 Let ρ = (F, k) be a signature. Let T ∈ Mρ , dim T = n, and μT (1) = f 1 , . . . , μT (n) = f n . We now define the value Q ρ (T ). If Δ(T ) = E kn , then Q ρ (T ) = 0. Let Δ(T ) = E kn ¯ = min{|α| : α ∈ Ωρ (T ), χ (α) and δ¯ = (δ1 , . . . , δn ) ∈ E kn \ Δ(T ). Then Q ρ (T, δ) ¯ : δ¯ ∈ E kn \ ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}, Δ(T α) = ∅} and Q ρ (T ) = max{Q ρ (T, δ) Δ(T )}. Let τ = (ρ, K , ψ) be a sccf-triple. We now define the function Q τ with Arg Q τ ⊆ ω. Let n ∈ ω. Then Q τ (n) = max{Q ρ (T ) : T ∈ Mρ,K , m ψ (T ) ≤ n}. Lemma 12.7 Let τ = (ρ, K , ψ) be a restricted sccf-triple, ρ = (F, k), and T ∈ a d Mρ,K . Then there exists a schema z ∈ Σρ such that ψˆ ρ,K (z) ≤ m ψ (T ), ψˆ ρ,K (z) ≥ r Q ρ (T ) − 1, and lρ,K (z) ≥ Q ρ (T ).

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Proof Let dim T = n and μT (1) = f 1 , . . . , μT (n) = f n . Let Q ρ (T ) ≤ 1. Set z = (ν, f 1 , . . . , f n ), where ν : E kn → {{0}}. Taking onto account that the function ψ has a d r the property Λ4 one can show that ψˆ ρ,K (z) = ψˆ ρ,K (z) = 0. It is clear that lρ,K (z) = 1. Hence if Q ρ (T ) ≤ 1, then the statement of the lemma holds. Let Q ρ (T ) ≥ 2. Then there exists a tuple (δ1 , . . . , δn ) ∈ E kn \ Δ(T ) such that ¯ = Q ρ (T ). Let α ∈ Ωρ (T ), χ (α) ⊆ {( f 1 , δ1 ), . . . , ( f n , δn )}, Δ(T α) = ∅, Q ρ (T, δ) ¯ Let α = ( f i1 , δi1 ) · · · ( f im , δim ). Set δ¯ = (δi1 , . . . , δim ). We now and |α| = Q ρ (T, δ). define a mapping ν : E km → P(ω). Let σ¯ = (σ1 , . . . , σm ) ∈ E km . If σ¯ = δ¯ , then ν(σ¯ ) = {0}. Let σ¯ = δ¯ . Then ν(σ¯ ) = { j : j ∈ {1, . . . , m}, δi j = σ j }. Set z = (ν, f i1 , . . . , f im ) and T  = Tρ (z, K ). Taking into account the choice of the word α we conclude that f i1 , . . . , f im are pairwise different elements from F, and Δ(T ( f i1 , δi1 ) · · · ( f i j−1 , δi j−1 )( f i j+1 , δi j+1 ) · · · ( f im , δim )) = ∅ for any j ∈ {1, . . . , m}. Therefore dim T  = m, δ¯ ∈ / Δ(T  ) and, for any j ∈ {1, . . . , m}, there exists a tuple ¯δ j ∈ Δ(T  ) which is different from the tuple δ¯ only in jth component. Evidently, r (z) ≥ Q ρ (T ). Let us show that Mρ,h (T  , δ¯ ) ≥ m − 1. νT  (δ¯ j ) = { j}. Therefore lρ,K Let β ∈ Ωρ (T ), χ (β) ⊆ {( f i1 , δi1 ), . . . , ( f im , δim )}, and T  β ∈ Mρ C . Assume that |β| ≤ m − 2. Then there exist j1 , j2 ∈ {1, . . . , m} such that j1 = j2 , δ¯ j1 ∈ Δ(T  β), / Mρ C which is impossible. Therefore |β| ≥ m − 1 and δ¯ j2 ∈ Δ(T  β). Hence T  β ∈ and Mρ,h (T  , δ¯ ) ≥ m − 1. By Lemma 3.5, Mρ,h (T  ) ≥ m − 1. From this inequality and from Theorem 3.1 it follows that h dρ (T  ) ≥ m − 1. Since the function ψ has the d property Λ3, we have ψρd (T  ) ≥ m − 1. By Theorem 10.1, ψˆ ρ,K (z) ≥ m − 1. Since d ˆ m = Q ρ (T ), we have ψρ,K (z) ≥ Q ρ (T ) − 1. Let σ¯ = (σ1 , . . . , σm ) ∈ Δ(T  ). Then σ¯ = δ¯ and there exists j ∈ {1, . . . , m} such that σ j = δi j . Evidently, T  ( f j , σ j ) ∈ Mρ C . Therefore Mρ,ψ (T  , σ¯ ) ≤ m ψ (T  ) ≤ m ψ (T ). Taking into account that σ¯ is an arbitrary tuple from Δ(T  ) and using Theorem 6.1 we obtain ψρa (T  ) ≤ m ψ (T ). From this inequality and from a Theorem 10.1 it follows that ψˆ ρ,K (z) ≤ m ψ (T ).  Let ρ = (F, k) be a signature and T ∈ Mρ . A cover for the table T is an arbitrary finite subset B of the set Ωρ (T ) having the following properties: (a) If B = ∅, then Δ(T  ) = ∅. (b) If B = ∅, then α∈B Δ(T α) = Δ(T ). A cover B for the table T will be called uncancelable if each subset of the set B, which is not equal to B, is not a cover for T . Let ψ be a complexity function of the signature ρ and B be a finite subset of the set Ωρ (T ). We now define the value ψ(B). If B = ∅, then ψ(B) = 0. If B = ∅, then ψ(B) = max{ψ(α) : α ∈ B}. Let n ∈ ω. We denote by Iψ (n, T ) the maximum cardinality of an uncancelable cover B for the table T such that ψ(B) ≤ n. Lemma 12.8 Let τ = (ρ, K , ψ) be a restricted sccf-triple, T ∈ Mρ,K , and n ∈ ω. Then the value Iψ (n, T ) is definite and 1 ≤ Iψ (n, T ) ≤ Nρ (T ). Proof Evidently, Δ(T ) = ∅. One can show that {λ} is an uncancelable cover for the table T and ψ({λ}) = 0 ≤ n. Let B be an arbitrary uncancelable cover for the table T . ¯ ∈B Since Δ(T ) = ∅, we have |B| ≥ 1. For any δ¯ ∈ Δ(T ), there exists a word β(δ)

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¯ Denote D = {β(δ) ¯ : δ¯ ∈ Δ(T )}. Evidently, |D| ≤ Nρ (T ), such that δ¯ ∈ Δ(Tβ(δ)). D ⊆ B, and D is a cover for the table T . Taking into account that B is an uncancelable cover for the table T we obtain B = D. Hence 1 ≤ |B| ≤ Nρ (T ). Thus, the value Iψ (n, T ) is definite and 1 ≤ Iψ (n, T ) ≤ Nρ (T ).  Let τ = (ρ, K , ψ) be a sccf-triple. We now define a function Iτ with Arg Iτ ⊆ ω. Let n ∈ ω. Then Iτ (n) = max{Iψ (n, T ) : T ∈ Mρ,K , m ψ (T ) ≤ n}. One can show that, for any n, m ∈ Arg Iτ , if n ≤ m, then Iτ (n) ≤ Iτ (m), and Typ Iτ = ω if and only if there exists n ∈ ω such that Iτ (n) = ∞. If Typ Iτ = ω, then Arg Iτ = ∅ or there exist n, m ∈ ω, n ≤ m, such that Arg Iτ = ω(n, m). If Typ Iτ = ω, then there exists n ∈ ω such that Arg Iτ = ω(n). Lemma 12.9 Let τ = (ρ, K , ψ) be a restricted sccf-triple, ρ = (F, k), m ∈ ω \ {0}, a (z) ≤ T ∈ Mρ,K , and m ψ (T ) ≤ m. Then there exists a schema z ∈ Σρ such that ψˆ ρ,K d r ˆ m, ψρ,K (z) ≥ logk Iψ (m, T ), and lρ,K (z) ≥ Iψ (m, T ). Proof Let dim T = n, μT (1) = f 1 , . . . , μT (n) = f n , and B be an uncancelable cover for the table T such that ψ(B) ≤ n and |B| = Iψ (m, T ). Taking into account that Δ(T ) = ∅ we obtain B = ∅. Let B = {α1 , . . . , αt }, where t = Iψ (m, T ). We ¯ = {0}. / Δ(T ), then ν(δ) now define a mapping ν : E kn → P(ω). Let δ¯ ∈ E kn . If δ¯ ∈ ¯ = {i : i ∈ {1, . . . , t}, δ¯ ∈ Δ(T αi )}. Set z = (ν, f 1 , . . . , f n ) If δ¯ ∈ Δ(T ), then ν(δ) and T  = Tρ (z, K ). Evidently, dim T  = dim T and Δ(T  ) = Δ(T ) Since B is an uncancelable cover for the table T , for any i ∈ {1, . . . , t}, there exists δ¯i ∈ Δ(T  ) r (z) ≥ Iψ (m, T ). Taking into such that ν(δ¯i ) = {i}. Therefore l r (T  ) ≥ t and lρ,K account that the function ψ has the property Λ3 and using Theorem 3.2 we obtain ψρd (T  ) ≥ logk t. From this inequality, from the equality t = Iψ (m, T ), and from d Theorem 10.1 it follows that ψˆ ρ,K (z) ≥ logk Iψ (m, T ). Let δ¯ ∈ Δ(T  ). Then there exists a word αi ∈ B such that δ¯ ∈ Δ(T αi ). Evidently, T  αi ∈ Mρ C and ψ(αi ) ≤ m. ¯ ≤ m. Taking into account that δ¯ is an arbitrary tuple from Therefore Mρ,ψ (T  , δ) Δ(T  ) and using Theorem 6.1 we obtain ψρa (T  ) ≤ m. From this inequality and from a Theorem 10.1 it follows that ψˆ ρ,K (z) ≤ m.  Lemma 12.10 Let τ = (ρ, K , ψ) be a restricted sccf-triple, Typ Nτ = ε, and Typ Sτ = ε . Then Typ Iτ = ω. Proof Using Lemma 11.9 we obtain Typ Nτ = ω. Therefore there exists m ∈ ω such that Nτ (m) = ∞. Since Typ Sτ = ε, there exists c ∈ ω such that Sψ (T ) ≤ c for any table T ∈ Mρ,K . Set n = max{m, c} and D = {Iψ (n, T ) : T ∈ Mρ,K , m ψ (T ) ≤ n}. Since Nτ (m) = ∞ and n ≥ m, we have D = ∅. Assume that D is a finite set and t is the maximum number from D. Since Nτ (m) = ∞, there exists a table T ∈ Mρ,K such that m ψ (T ) ≤ m ≤ n and Nρ (T ) > t. Since Sψ (T ) ≤ c ≤ n, for any δ¯ ∈ ¯ = {δ} ¯ and ψ(α(δ)) ¯ ≤ ¯ ∈ Ωρ (T ) such that Δ(T α(δ)) Δ(T ), there exists a word α(δ) ¯ ¯ n. Set B = {α(δ) : δ ∈ Δ(T )}. Evidently, |B| = Nρ (T ) > t, ψ(B) ≤ n, and B is

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an uncancelable cover for the table T . Therefore Iψ (n, T ) > t which contradicts the assumption on the finiteness of the set D. Hence D is an infinite set. Therefore  Iτ (n) = ∞. Thus, Typ Iτ = ω. Proposition 12.4 Let τ = (ρ, K , ψ) be a restricted sccf-triple and ρ = (F, k). Then Typ Ψˆ τda ∈ {ε, χ , ω} and the following statements hold: (a) Typ Ψˆ τda = ε if and only if Typ Nτ = ε and Typ Sτ = ε. (b) Typ Ψˆ τda = χ if and only if Typ Iτ = ω and Typ Sτ = ε. (c) Typ Ψˆ τda = ω if and only if Typ Iτ = ω. (d) If Typ Ψˆ τda = χ , then there exists m ∈ ω such that, for any n ∈ ω(m), the values Ψˆ τda (n), Ψˆ τda (3n), Z τ (n), Q τ (n), and Iτ (n) are definite and the follow1 ing inequalities hold: Ψˆ τda (n) ≥ Q τ (n) − 1, Ψˆ τda (3n) ≥ (2Z τ (n)) 2 − 3, Iτ (n) ≥ 1, Ψˆ τda (n) ≥ logk Iτ (n), and Ψˆ τda (n) ≤ max{n, n Q τ (n)}Z τ (n) log2 (k 2 n Iτ (n)). Proof First, we prove the following statement: (a*) If Typ Nτ = Typ Sτ = ε, then Typ Ψˆ τda = ε. d (z) ≤ c for any schema z ∈ Σρ . By Lemma 11.8, there exists c ∈ ω such that ψˆ ρ,K da ˆ Hence Typ Ψτ = ε. Thus, the statement (a*) holds. We now prove the following statement: (b*) If Typ Iτ = ω and at least one of the relations Typ Nτ = ε, Typ Sτ = ε holds, then Typ Ψˆ τda = χ , Typ Sτ = ε, and Arg Ψˆ τda = ω. Let Typ Iτ = ω and at least one of the relations Typ Nτ = ε, Typ Sτ = ε hold. Taking into account that Typ Iτ = ω and using Lemma 12.10 we conclude that at least one of the relations Typ Nτ = ε, Typ Sτ = ε holds. Hence Typ Sτ = ε. Let us d a (z) : z ∈ Σρ , ψˆ ρ,K (z) ≤ n}. By show that Arg Ψˆ τda = ω. Let n ∈ ω. Set Dn = {ψˆ ρ,K d a Theorem 10.1, Dn = {ψρ (T ) : T ∈ Mρ,K , ψρ (T ) ≤ n}. Evidently, the set Mρ,K ∩ Mρ C is not empty. Let T be a table from this set. Taking into account that the function ψ has the property Λ4 one can show that ψρa (T ) = 0. Hence the set Dn is nonempty. Let us show that the set Dn is finite. Let T ∈ Mρ,K and ψρa (T ) ≤ n. If T ∈ Mρ C , / Mρ C . Then there exists a then, as it is not difficult to show, ψρd (T ) = 0. Let T ∈ nondeterministic decision tree Γ for the table T such that ψ(Γ ) ≤ n. Evidently, the set {π(ξ ) : ξ ∈ Ξ (Γ )} is a cover for the table T . It is clear that there exists a subset A of the set Ξ (Γ ) such that the set B = {π(ξ ) : ξ ∈ A} is an uncancelable cover for the table T . We denote by Γ  a schema obtained from Γ by removal of all nodes and edges which do not belong to the paths from A. One can show that Γ  is a nondeterministic decision tree for the table T . Since T ∈ / Mρ C , we have P(Γ  ) = ∅. Let P(Γ  ) = { f 1 , . . . , f m }. Set z = (ν, f 1 , . . . , f m ), where ν : E kn → {{0}}, and T  = Tρ (z, K ). Since ψ(Γ ) ≤ n, we have ψ(Γ  ) ≤ n. Taking into account that the function ψ has the property Λ2 we obtain m ψ (T  ) ≤ n. Since the set B is an uncancelable cover for the table T , one can show that B is an uncancelable cover for the table T  . Using the inequality ψ(B) ≤ n we obtain |B| ≤ Iψ (n, T  ). Since Typ Iτ = ω and m ψ (T  ) ≤ n, the value Iτ (n) is definite and Iψ (n, T  ) ≤ Iτ (n). Therefore |B| ≤ that ψ has the property Λ3 and ψ(α) ≤ n Iτ (n). Let α ∈ B. Taking into account  we obtain |α| ≤ n. Hence  P(Γ  ) = m ≤ n Iτ (n). Set β = f 1 f 2 · · · f m . Taking into

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account that ψ( f i ) ≤ n for any i ∈ {1, . . . , m}, and the function ψ has the property Λ1 we conclude that ψ(β) ≤ n 2 Iτ (n). By Lemma 3.2, P(Γ  ) is a test for the table T . Hence β is an unconditional test for the table T . Therefore Θρ,ψ (T ) ≤ n 2 Iτ (n). Hence all numbers from Dn are at most n 2 Iτ (n). Therefore the value Ψˆ τda (n) is definite. Thus, Arg Ψˆ τda = ω. Using the relation Typ Sτ = ε and Proposition 11.4 we obtain Typ Ψˆ τaa = χ . By Lemma 11.2, Typ Ψˆ τda = χ . From Lemma 12.9 it follows that the following statement holds: (c*) If Typ Iτ = ω, then Typ Ψˆ τda = ω. Let Typ Ψˆ τda = ε. From the statement (c*) it follows that Typ Iτ = ω. Using this inequality and the statement (b*) we obtain Typ Nτ = ε and Typ Sτ = ε. From here and from the statement (a*) the statement (a) follows. Let Typ Ψˆ τda = χ . Using the statements (a*) and (c*) we conclude that Typ Iτ = ω, and at least one of the conditions Typ Sτ = ε, Typ Nτ = ε holds. Using the statement (b*) we obtain Typ Sτ = ε. Let Typ Iτ = ω and Typ Sτ = ε. Using the statement (b*) we obtain Typ Ψˆ τda = χ . Hence the statement (b) holds. Let Typ Ψˆ τda = ω. Using the statement (a*) we conclude that at least one of the following two conditions holds: Typ Sτ = ε and Typ Nτ = ε. Using the statement (b*) we obtain Typ Iτ = ω. From here and from the statement (c*) the statement (c) follows. Let Typ Ψˆ τda = χ . Using the statements (b) and (b*) we obtain Arg Ψˆ τda = ω and Typ Iτ = ω. Assume that Typ Q τ = ω or Typ Z τ = ω. Then, using Lemmas 12.6 and 12.7, we obtain Typ Ψˆ τda = ω which is impossible. Hence Typ Q τ = ω and Typ Z τ = ω. Therefore there exists m ∈ ω such that, for any n ∈ ω(m), the values Ψˆ τda (n), Ψˆ τda (3n), Z τ (n), Q τ (n), and Iτ (n) are definite. Let n ∈ ω(m). By Lemma 12.8, Iτ (n) ≥ 1. From Lemma 12.9 it follows that Ψˆ τda (n) ≥ logk Iτ (n). By Lemma 12.7, Ψˆ τda (n) ≥ Q τ (n) − 1. Let Z τ (n) = 0. Then, evidently, the inequality 1 Ψˆ τda (3n) ≥ (2Z τ (n)) 2 − 3 holds. Let Z τ (n) > 0 and m be the maximum number m(m+1) from ω such that 2 ≤ Z τ (n). By Lemma 12.6, there exists a schema z ∈ Σρ such a d that ψˆ ρ,K (z) ≤ 3n and ψˆ ρ,K (z) ≥ m − 1. Evidently, (m+1)(m+2) > Z τ (n). Therefore 2 1 1 da ˆ 2 2 m > (2Z τ (n)) − 2. Hence Ψτ (3n) ≥ (2Z τ (n)) − 3. a (z) ≤ n. Set T = Tρ (z, K ). By Theorem 10.1, ψρa (T ) ≤ n. Let z ∈ Σρ and ψˆ ρ,K Let us show that   ψρd (T ) ≤ max{n, n Q τ (n)}Z τ (n) log2 k 2 n Iτ (n) . If T ∈ Mρ C , then, as it is not difficult to show, ψρd (T ) = 0. Therefore the considered inequality holds. Let T ∈ / Mρ C . One can show that there exists a nondeterministic decision tree Γ for the table T such that ψ(Γ ) ≤ n, and the set B = {π(ξ ) : ξ ∈ Ξ (Γ )} is an uncancelable cover for the table T . Since T ∈ / Mρ C , we have P(Γ ) = ∅. Let P(Γ ) = { f 1 , . . . , f m }, Ξ (Γ ) = {ξ1 , . . . , ξt }, and i j be the number attached to the terminal node of the path ξ j for j = 1, . . . , t. Let us define a mapping ν : E km → P(ω). Let δ¯ = (δ1 , . . . , δm ) ∈ E km . Set α = ( f 1 , δ1 ) · · · ( f m , δm ). If, for ¯ = {0}. Othany j ∈ {1, . . . , t}, the words α and π(ξ j ) are incompatible, then ν(δ)

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¯ is equal to the set of such j ∈ {1, . . . , t} that the words α and π(ξ j ) erwise, ν(δ) are compatible. Set z  = (ν, f 1 , . . . , f m ) and T  = Tρ (z  , K ). One can show that any deterministic decision tree for the table T  is a deterministic decision tree for the table T . Therefore ψρd (T ) ≤ ψρd (T  ). One can show that m ψ (T  ) ≤ n and B is an uncancelable cover for the table T  . Therefore t = |B| ≤ Iτ (n). Taking into account that ψ is restricted we conclude that |α| ≤ n for any α ∈ B. Hence m = |P(Γ )| ≤ n Iτ (n).   Z (T  )  2  Z (n) By Lemma 12.5, Nρ (T  ) ≤ k 2 n Iτ (n) ρ ≤ k n Iτ (n) τ . Let δ¯ ∈ E km . If δ¯ ∈ Δ(T  ), then there exists a word α ∈ B such that δ¯ ∈ Δ(T  α). Evidently, ψ(α) ≤ ¯ ≤ n. If δ¯ ∈ / Δ(T  ), then, evidently, n and T  α ∈ Mρ C . Therefore Mρ,ψ (T  , δ)  ¯  ¯  Mρ,h (T , δ) ≤ Q ρ (T , δ) ≤ Q ρ (T ) ≤ Q τ (n). Taking into account that m ψ (T  ) ≤ n ¯ ≤ n Q τ (n). By and ψ is a restricted complexity function we obtain Mρ,ψ (T  , δ)  Lemma 3.5, Mρ,ψ (T ) ≤ max{n, n Q τ (n)}. Using Corollary 5.2 and the fact that ψ is a restricted complexity function we obtain ψρd (T  ) ≤ Mρ,ψ (T  ) log2 Nρ (T  ) ≤   max{n, n Q τ (n)}Z τ (n) log2 k 2 n Iτ (n) . As it was mentioned above, ψρd (T ) ≤ d ψρd (T  ). By Theorem 10.1, ψˆ ρ,K (z) = ψρd (T ). Thus, the statement (d) holds.  Corollary 12.3 Let τ = (ρ, K , ψ) be a restricted sccf-triple, ρ = (F, k), and Typ Ψˆ τda = χ . Then the following two statements are equivalent: (a) There exists a polynomial P0 such that Ψˆ τda  P0 . (b) There exist polynomials P1 , P2 , and P3 such that Z τ  P1 , Q τ  P2 , and Iτ  2 P3 . Proof Let the statement (b) hold. Then, using Proposition 12.4, we conclude that there exists m ∈ ω such that, for any n ∈ ω(m), Ψˆ τda (n) ≤ (n(P2 (n) + 1))P1 (n)(P3 (n) + k 2 n) . Hence the statement (a) holds. Let the statement (b) do not hold. Let us show that the statement (a) does not hold. Assume the contrary: there exists a polynomial P0 such that Ψˆ τda  P0 . By Propo2 sition 12.4, Q τ  P0 + 1, Iτ  2 P0 log2 k , and Z τ  P1 , where P1 (x) = (P0 (3x)+3) . 2 Hence the statement (b) holds which is impossible. The obtained contradiction shows that the statement (a) does not hold.  Lemma 12.11 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψˆ τds = ω if and only if Typ Ψˆ τda = ω and Typ Ψˆ τas = ω. Proof Let Typ Ψˆ τds = ω. By Proposition 12.3, Typ Nτ = ω. From here and from Lemma 12.8 it follows that Typ Iτ = ω. By Proposition 12.4, Typ Ψˆ τda = ω. Assume that Typ Ψˆ τas = ω. Using Lemma 11.2 we conclude that Typ Ψˆ τds = ω which is impossible. Hence Typ Ψˆ τas = ω.

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Assume now that Typ Ψˆ τda = ω and Typ Ψˆ τas = ω. Let n ∈ ω. One can show that s (z) = 0 ≤ n. Let z be an arbitrary there exists a schema z ∈ Σρ0−1 (K ) such that ψˆ ρ,K s 0−1 schema from Σρ (K ) such that ψˆ ρ,K (z) ≤ n. Then, evidently, the value Ψˆ τas (n) a (z) ≤ Ψˆ τas (n). Therefore the value Ψˆ τda (Ψˆ τas (n)) is definite and is definite and ψˆ ρ,K d ψˆ ρ,K (z) ≤ Ψˆ τda (Ψˆ τas (n)). Taking into account that z is an arbitrary schema from s 0−1 (z) ≤ n we conclude that the value Ψˆ τds (n) is definite.  Σρ (K ) such that ψˆ ρ,K

References 1. Laskowski, M.: Vapnik-Chervonenkis classes of definable sets. J. Lond. Math. Soc. 45, 377–384 (1992) 2. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity. Local approach. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets IV, Lecture Notes in Computer Science, vol. 3700, pp. 125–143. Springer, Berlin (2005) 3. Moshkov, M.: Time complexity of decision trees. In: Peters, J.F., Skowron, A., (eds.) Transactions on Rough Sets III, Lecture Notes in Computer Science, vol. 3400, pp. 244–459. Springer, Berlin (2005) 4. Sauer, N.: On the density of families of sets. J. Comb. Theory, Ser. A 13(1), 145–147 (1972)

Chapter 13

Local Upper Types of Restricted Sccf-Triples

In this chapter, we describe all possible six local upper types of restricted sccf-triples. For each of these six types, we consider the criterion of its implementation and give an example of a restricted sccf-triple with this type. Some similar results were obtained in [1] for types of functions different from the upper types considered in this book.

13.1 Possible Local Upper Types Let us consider the following six matrices:

i Tp1 = d a s

i χ ε ε ε

d ω ε ε ε

a ω ε ε ε

s ω ε ε ε

i Tp2 = d a s

i χ λ ε ε

d ω χ ε ε

a ω ω ε ε

s ω ω ε ε

i Tp3 = d a s

i χ χ χ χ

d ω χ χ χ

a ω ω χ χ

s ω ω χ χ

i Tp4 = d a s

i χ χ χ χ

d ω χ χ χ

a ω χ χ χ

s ω χ χ χ

i Tp5 = d a s

i χ χ χ χ

d ω χ χ χ

a ω ω χ χ

s ω ω ω χ

i Tp6 = d a s

i χ χ χ χ

d ω χ χ χ

a ω χ χ χ

s ω ω ω χ

Proposition 13.1 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψˆ τ ∈ {Tp1, Tp2, Tp3, Tp4, Tp5, Tp6} .

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_13

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13 Local Upper Types of Restricted Sccf-Triples

Proof Using Proposition 11.1 we obtain Typ Ψˆ τii = χ . By Proposition 12.1, Typ Ψˆ τid = Typ Ψˆ τia = Typ Ψˆ τis = ω . Let b ∈ {d, a, s}. Using Lemma 11.3 we obtain Typ Ψˆ τbb ∈ {ε, χ }. Let us consider the following three cases: (a) Typ Ψˆ τdd = ε, (b) Typ Ψˆ τdd = χ and Typ Ψˆ τaa = ε, and (c) Typ Ψˆ τdd = Typ Ψˆ τaa = χ . Evidently, for any restricted sccf-triple τ , one of cases (a), (b), and (c) holds. (a). Let Typ Ψˆ τdd = ε. By Proposition 11.3, Typ Nτ = Typ Sτ = ε. Using Proposition 12.3 we obtain Typ Ψˆ τds = ε. By Proposition 12.4, Typ Ψˆ τda = ε. Let bc ∈ {di, ai, ad, aa, as, si, sd, sa, ss}. Using Lemma 11.2 we obtain Ψˆ τbc ≺ Ψˆ τds or Ψˆ τbc ≺ Ψˆ τda . From here it follows that Typ Ψˆ τbc = ε. Thus, Typ Ψˆ τ = Tp1. (b). Let Typ Ψˆ τdd = χ and Typ Ψˆ τaa = ε. By Proposition 11.4, Typ Sτ = ε. Using Proposition 11.3 we obtain Typ Nτ = ε. Let bc ∈ {ai, ad, aa, si, sd, sa, ss}. By Proposition 11.4, Typ Ψˆ τbc = ε. Using Proposition 11.2 we obtain Typ Ψˆ τdi = λ. By Proposition 12.2, Typ Ψˆ τas = ε. By Lemma 11.9, Typ Nτ = ω. From this equality and from Proposition 12.3 it follows that Typ Ψˆ τds = ω. By Lemma 12.10, Typ Iτ = ω. From this equality and from Proposition 12.4 it follows that Typ Ψˆ τda = ω. Thus, Typ Ψˆ τ = Tp2. (c). Let Typ Ψˆ τdd = Typ Ψˆ τaa = χ . Using Proposition 11.4 we obtain Typ Sτ = ε. Let bc ∈ {ai, ad, aa, si, sd, sa, ss}. By Proposition 11.4, Typ Ψˆ τbc = χ . By Proposition 11.2, Typ Ψˆ τdi = χ . Let bc ∈ {ds, da, as}. Using Lemma 11.2 we obtain Typ Ψˆ τbc ∈ {χ , ω}. Using Lemma 12.11 we conclude that either Typ Ψˆ τds = Typ Ψˆ τda = Typ Ψˆ τas = χ or Typ Ψˆ τds = Typ Ψˆ τda = Typ Ψˆ τas = ω, or Typ Ψˆ τds = ω, Typ Ψˆ τda = χ , Typ Ψˆ τas = ω, or Typ Ψˆ τds = Typ Ψˆ τda = ω, Typ Ψˆ τas = χ . Thus, Typ Ψˆ τ ∈ {Tp3, Tp4, Tp5, Tp6}.  Proposition 13.2 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψˆ τ = Tp1 if and only if Typ Nτ = Typ Sτ = ε. Proof Let Typ Ψˆ τ = Tp1. Then Typ Ψˆ τdd = ε. By Proposition 11.3, Typ Nτ = Typ Sτ = ε. Let Typ Nτ = Typ Sτ = ε. By Proposition 11.3, Typ Ψˆ τdd = ε. From this equality  and from Proposition 13.1 it follows that Typ Ψˆ τ = Tp1. Proposition 13.3 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψˆ τ = Tp2 if and only if Typ Nτ = ε and Typ Sτ = ε. Proof Let Typ Ψˆ τ = Tp2. Then Typ Ψˆ τdi = λ. By Proposition 11.2, Typ Nτ = ε and Typ Sτ = ε. Let Typ Nτ = ε and Typ Sτ = ε. By Proposition 11.2, Typ Ψˆ τdi = λ. From this  equality and from Proposition 13.1 it follows that Typ Ψˆ τ = Tp2.

13.1 Possible Local Upper Types

167

Proposition 13.4 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψˆ τ = Tp3 if and only if Typ Nτ = ω and Typ Sτ = χ . Proof Let Typ Ψˆ τ = Tp3. Then Typ Ψˆ τds = ω and Typ Ψˆ τas = χ . By Proposition 12.3, Typ Nτ = ω. Using Proposition 12.2 we obtain Typ Sτ = χ . Let Typ Nτ = ω and Typ Sτ = χ . By Propositions 12.2 and 12.3, Typ Ψˆ τas = χ and Typ Ψˆ τds = ω. From these equalities and from Proposition 13.1 it follows that  Typ Ψˆ τ = Tp3. Proposition 13.5 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψˆ τ = Tp4 if and only if Typ Nτ = ω and Typ Sτ = ε. Proof Let Typ Ψˆ τ = Tp4. Then Typ Ψˆ τds = χ . By Proposition 12.3, Typ Nτ = ω and Typ Sτ = ε. Let Typ Nτ = ω and Typ Sτ = ε. By Proposition 12.3, Typ Ψˆ τds = χ . From this  equality and from Proposition 13.1 it follows that Typ Ψˆ τ = Tp4. Proposition 13.6 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψˆ τ = Tp5 if and only if Typ Iτ = Typ Sτ = ω. Proof Let Typ Ψˆ τ = Tp5. Then Typ Ψˆ τda = Typ Ψˆ τas = ω. By Propositions 12.2 and 12.4, Typ Iτ = Typ Sτ = ω. Let Typ Iτ = Typ Sτ = ω. By Propositions 12.2 and 12.4, Typ Ψˆ τda = Typ Ψˆ τas = ω. From these equalities and from Proposition 13.1 it follows that Typ Ψˆ τ = Tp5.  Proposition 13.7 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψˆ τ = Tp6 if and only if Typ Iτ = ω and Typ Nτ = ω. Proof Let Typ Ψˆ τ = Tp6. Then Typ Ψˆ τda = χ and Typ Ψˆ τds = ω. By Propositions 12.3 and 12.4, Typ Iτ = ω and Typ Nτ = ω. Let Typ Iτ = ω and Typ Nτ = ω. By Proposition 12.3, Typ Ψˆ τds = ω. Using Lemma 12.10 we obtain Typ Sτ = ε. By Proposition 12.4, Typ Ψˆ τda = χ . From the equalities Typ Ψˆ τds = ω and Typ Ψˆ τda = χ , and from Proposition 13.1 it follows that  Typ Ψˆ τ = Tp6.

13.2 Examples of Restricted Sccf-Triples We denote by Z the set of integers and by Q the set of rational numbers. Let i ∈ Z. We now define functions pi : Q → E 2 and li : Q → E 2 . Let q ∈ Q. Then  pi (q) =

1, if q = i , 0, if q = i ,

 li (q) =

1, if q ≥ i , 0, if q < i .

168

13 Local Upper Types of Restricted Sccf-Triples

Denote F0 = { f i : i ∈ ω \ {0}} and G 0 = {gi : i ∈ ω \ {0}}. Let us define three signatures as follows: ρ0 = ({ f 0 }, 2), ρ1 = (F0 , 2), and ρ2 = (F0 ∪ G 0 , 2). For n ∈ ω \ {0}, we denote by en the mapping from E 2n to {{1}}. For m ∈ ω \ {0}, we ¯ = { j}, define a mapping dm : E 2m → P(ω). Let δ¯ = (δ1 , . . . , δm ) ∈ E 2m . Then dm (δ) where j is the number from ω such that δ¯ is the binary representation of j. In examples, in the capacity of complexity functions we will use only functions each of which is a weighted depth. Evidently, for the definition of a weighted depth of the signature ρ = (F, k) it is sufficient to define it on the set F. Let m ∈ ω \ {0}. We define an information system U(1,m) of the signature ρ0 as follows: U(1,m) = (Q, γ(1,m) ) and γ(1,m) ( f 0 ) = l0 . We define a weighted depth (1, m)ψ of the signature ρ0 as follows: (1, m)ψ( f 0 ) = m. Denote τ (1, m) = (ρ0 , {U(1,m) }, (1, m)ψ) . Lemma 13.1 For any m, n ∈ ω \ {0}, the sccf-triple τ (1, m) is restricted, Typ Ψˆ τ (1,m) = Tp1 , Ψˆ τii(1,1) (n) = n, and if n > 1, then Ψˆ τii(1,m) (nm − 1) ≤ nm − 1 − (m − 1). Proof One can show that Nρ0 (T ) ≤ 2 and S(1,m)ψ (T ) ≤ m for any table T ∈ Mρ0 ,{U(1,m) } . Therefore Typ Nτ (1,m) = Typ Sτ (1,m) = ε. By Proposition 13.2, Typ Ψˆ τ (1,m) = Tp1 . We denote by z n the tuple (en , f 0 , . . . , f 0 ), where the element f 0 repeats exactly n times. Evidently, (1, m)Ψˆ ρi0 ,{U(1,m) } (z n ) = mn. Therefore Ψˆ τii(1,1) (n) = n. Let z ∈ Σρ0 . Then, evidently, (1, m)Ψˆ ρi ,{U } (z) = m j, where j ∈ ω \ {0}. There0

(1,m)

fore if n > 1, then the value Ψˆ τii(1,1) (nm − 1) is definite and Ψˆ τii(1,m) (nm − 1) ≤ m(n − 1) ≤ nm − 1 − (m − 1).  Let m ∈ ω \ {0}. We define an information system U(2,1,m) of the signature ρ1 as follows: U(2,1,m) = (Q, γ(2,1,m) ) and γ(2,1,m) ( f i ) = li for any i ∈ ω \ {0}. We define a weighted depth (2, 1, m)ψ of the signature ρ1 as follows: (2, 1, m)ψ( f i ) = m for any i ∈ ω \ {0}. Denote τ (2, 1, m) = (ρ1 , {U(2,1,m) }, (2, 1, m)ψ). Lemma 13.2 For any m, n ∈ ω \ {0}, the sccf-triple τ (2, 1, m) is restricted, Typ Ψˆ τ (2,1,m) = Tp2 ,   ˆ di Ψˆ τii(2,1,1) (n) = n, Ψˆ τdd (2,1,1) (n) = n, Ψτ (2,1,1) (n) ≤ log2 n + 1, and if n > 1, then Ψˆ τii(2,1,m) (nm − 1) ≤ (n − 1)m and Ψˆ τdd (2,1,m) (nm − 1) ≤ (n − 1)m. Proof One can show that S(2,1,m)ψ (T ) ≤ 2m for any table T ∈ Mρ1 ,{U(2,1,m) } . Hence Typ Sτ (2,1,m) = ε. Set z n = (en , f 1 , . . . , f n ) and Tn = Tρ1 (z n , {U(2,1,m) }). One can

13.2 Examples of Restricted Sccf-Triples

169

show that Nρ1 (Tn ) = n + 1. Hence Typ Nτ (2,1,m) = ε. By Proposition 13.3, Typ Ψˆ τ (2,1,m) = Tp2. Evidently, (2, 1, 1)ψˆ ρi 1 ,{U(2,1,1) } (z n ) = n. Therefore Ψˆ τii(2,1,1) (n) = n. Set σn = (dn ,   f 1 , . . . , f n ). One can show that (2, 1, 1)ψˆ ρd ,{U } (σn ) = log2 n + 1. Hence (2,1,1)

1

ˆi Ψˆ τdd (2,1,1) (n) = n. Let z ∈ Σρ1 and (2, 1, 1)ψρ1 ,{U(2,1,1) } (z) ≤ n. Then z = (ν, f i 1 , . . . ,   f it ), where t ≤ n. One can show that (2, 1, 1)ψˆ ρd ,{U } (z) ≤ log2 t + 1. Hence 1

(2,1,1)

  Ψˆ τdi(2,1,1) (n) ≤ log2 n + 1 . Set Bm = {m j : j ∈ ω}. One can show that {(2, 1, m)Ψˆ ρi1 ,{U(2,1,m) } (z) : z ∈ Σρ1 } ⊆ Bm and {(2, 1, m)Ψˆ ρd1 ,{U(2,1,m) } (z) : z ∈ Σρ1 } ⊆ Bm . Therefore if n > 1, then Ψˆ τii(2,1,m) (nm − 1) ≤ (n − 1)m and Ψˆ τdd (2,1,m) (nm − 1) ≤ (n − 1)m.



Let m ∈ ω \ {0}. We define an information system U(3,1,m) of the signature ρ2 as follows: U(3,1,m) = (Q, γ(3,1,m) ), and γ(3,1,m) (gi ) = l0 and γ(3,1,m) ( f i ) = li for any i ∈ ω \ {0}. We define a weighted depth (3, 1, m)ψ of the signature ρ2 as follows: (3, 1, m)ψ(gi ) = im and (3, 1, m)ψ( f i ) = m for any i ∈ ω \ {0}. Denote τ (3, 1, m) = (ρ2 , {U(3,1,m) }, (3, 1, m)ψ). Lemma 13.3 For any m, n ∈ ω \ {0}, the sccf-triple τ (3, 1, m) is restricted, Typ Ψˆ τ (3,1,m) = Tp3 , ˆ ss Ψˆ τii(3,1,1) (n) = Ψˆ τdi(3,1,1) (n) = Ψˆ τdd (3,1,1) (n) = Ψτ (3,1,1) (n) = n, and if n > 1, then Ψˆ τii(3,1,m) (nm − 1) ≤ (n − 1)m and Ψˆ τdd (3,1,m) (nm − 1) ≤ (n − 1)m. Proof Set z n = (en , f 1 , . . . , f n ) and Tn = Tρ2 (z n , {U(3,1,m) }). Evidently, m (3,1,m)ψ (Tn ) ≤ m and Nρ2 (Tn ) = n + 1. Hence Typ Nτ (3,1,m) = ω. Set σn = (d1 , gn ) and Tn = Tρ2 (σn , {U(3,1,m) }) . One can show that S(3,1,m)ψ (Tn ) = nm. Therefore Typ Sτ (3,1,m) = ε. Let T ∈ Mρ2 ,{U(3,1,m) }

170

13 Local Upper Types of Restricted Sccf-Triples

and m (3,1,m)ψ (Tn ) ≤ n. Then, as it is not difficult to show, S(3,1,m)ψ (T ) ≤ 2n. Hence Typ Sτ (3,1,m) = ω. Using Lemma 12.1 we obtain Typ Sτ (3,1,m) = χ . By Proposition 13.4, Typ Ψˆ τ (3,1,m) = Tp3. ({U(3,1,1) }). Let b ∈ {i, d, s}. One can show that Evidently, σn ∈ Σρ0−1 2 (3, 1, 1)Ψˆ ρb2 ,{U(3,1,1) } (σn ) = n . ˆ ss Hence Ψˆ τii(3,1,1) (n) = Ψˆ τdi(3,1,1) (n) = Ψˆ τdd (3,1,1) (n) = Ψτ (3,1,1) (n) = n. Set Bm = {m j : j ∈ ω}. One can show that {(3, 1, m)Ψˆ ρi2 ,{U(3,1,m) } (z) : z ∈ Σρ2 } ⊆ Bm and {(3, 1, m)Ψˆ ρd ,{U } (z) : z ∈ Σρ2 } ⊆ Bm . Therefore if n > 1, then 2

(3,1,m)

Ψˆ τii(3,1,m) (nm − 1) ≤ (n − 1)m and Ψˆ τdd (3,1,m) (nm − 1) ≤ (n − 1)m.



Let m ∈ ω \ {0}. We define an information system U(4,m) of the signature ρ1 as follows: U(4,m) = (Q, γ(4,m) ) and γ(4,m) ( f i ) = l0 for any i ∈ ω \ {0}. We define a weighted depth (4, m)ψ of the signature ρ1 as follows: (4, m)ψ( f i ) = im for any i ∈ ω \ {0}. Denote τ (4, m) = (ρ1 , {U(4,m) }, (4, m)ψ). Lemma 13.4 For any n, m ∈ ω \ {0}, the sccf-triple τ (4, m) is restricted, Typ Ψˆ τ (4,m) = Tp4 , ˆ ss Ψˆ τii(4,1) (n) = Ψˆ τdi(4,1) (n) = Ψˆ τdd (4,1) (n) = Ψτ (4,1) (n) = n, Ψˆ τii(4,m) (nm − 1) ≤ (n − 1)m.

and

if

n > 1,

then

Proof Evidently, Typ Nτ = ε. Set σn = (d1 , f n ) and Tn = Tρ1 (σn , {U(4,m) }). It is clear that S(4,m)ψ (Tn ) = nm. Therefore Typ Sτ (4,m) = ε. By Proposition 13.5, Typ Ψˆ τ (4,m) = Tp4. ({U(4,1) }). Let b ∈ {i, d, s}. One can show that Evidently, σn ∈ Σρ0−1 1 (4, 1)Ψˆ ρb2 ,{U(4,1) } (σn ) = n . ˆ ss Hence Ψˆ τii(4,1) (n) = Ψˆ τdi(4,1) (n) = Ψˆ τdd (4,1) (n) = Ψτ (4,1) (n) = n. Set Bm = {m j : j ∈ ω}. One can show that {(4, m)Ψˆ ρi1 ,{U(4,m) } (z) : z ∈ Σρ1 } ⊆ Bm . Therefore if n > 1, then Ψˆ τii(4,m) (nm − 1) ≤ (n − 1)m.



Let m ∈ ω \ {0}. We define an information system U(5,m) of the signature ρ2 as follows: U(5,m) = (Q, γ(5,m) ), γ(5,m) (gi ) = p−i and γ(5,m) ( f i ) = li for any i ∈ ω \ {0}.

13.2 Examples of Restricted Sccf-Triples

171

We define a weighted depth (5, m)ψ of the signature ρ2 as follows: (5, m)ψ(gi ) = (5, m)ψ( f i ) = m for any i ∈ ω \ {0}. Denote τ (5, m) = (ρ2 , {U(5,m) }, (5, m)ψ). Lemma 13.5 For any n, m ∈ ω \ {0}, the sccf-triple τ (5, m) is restricted, Typ Ψˆ τ (5,m) = Tp5 , ˆ ss Ψˆ τii(5,1) (n) = Ψˆ τdi(5,1) (n) = Ψˆ τdd and if n > 1, then (5,1) (n) = Ψτ (5,1) (n) = n, ii di ˆ ˆ Ψτ (5,m) (nm − 1) ≤ (n − 1)m, Ψτ (5,m) (nm − 1) ≤ (n − 1)m, Ψˆ τdd (5,m) (nm − 1) ≤ (n − 1)m, and Ψˆ τss(5,m) (nm − 1) ≤ (n − 1)m. Proof Set z n = (en , g1 , . . . , gn ) and Tn = Tρ2 (z n , {U(5,m) }). It is not difficult to see that m (5,m)ψ (Tn ) = m and (Tn ) = {(0, . . . , 0), (1, 0, . . . , 0), . . . , (0, . . . , 0, 1)}. One can show that S(5,m)ψ (Tn ) = nm. Hence Typ Sτ (5,m) = ω. Set σn = (en , f 1 , . . . , f n ) and Tn = Tρ2 (σn , {U(5,m) }). It is not difficult to see that m (5,m)ψ (Tn ) = m. One can show that I(5,m)ψ (2m, Tn ) = n + 1. Hence Typ Iτ (5,m) = ω. By Proposition 13.6, Typ Ψˆ τ (5,m) = Tp5. As it was mentioned above, S(5,1)ψ (Tn ) = n. By Lemma 11.7, there exists a schema z ∈ Σρ0−1 ({U(5,1) }) such that (5, 1)ψˆ ρi 2 ,{U(5,1) } (z) = (5, 1)ψˆ ρd2 ,{U(5,1) } (z) = 2 s ˆ ss (5, 1)ψˆ ρ2 ,{U(5,1) } (z) = n. Hence Ψˆ τii(5,1) (n) = Ψˆ τdi(5,1) (n) = Ψˆ τdd (5,1) (n) = Ψτ (5,1) (n) = n. Set Bm = {m j : j ∈ ω}. One can show that {(5, m)ψˆ ρi 2 ,{U(5,m) } (z) : z ∈ Σρ2 } ⊆ Bm , {(5, m)ψˆ ρd ,{U } (z) : z ∈ Σρ2 } ⊆ Bm , and 2

(5,m)

({U(5,1) })} ⊆ Bm . {(5, m)ψˆ ρs2 ,{U(5,m) } (z) : z ∈ Σρ0−1 2 Therefore if n > 1, then Ψˆ τii(5,m) (nm − 1) ≤ (n − 1)m, Ψˆ τdi(5,m) (nm − 1) ≤ (n − 1)m, ˆ ss Ψˆ τdd  (5,m) (nm − 1) ≤ (n − 1)m, and Ψτ (5,m) (nm − 1) ≤ (n − 1)m. Let m ∈ ω \ {0}. We define an information system U(6,1,m) of the signature ρ1 as follows: U(6,1,m) = (Q, γ(6,1,m) ) and γ(6,1,m) ( f i ) = pi for any i ∈ ω \ {0}. We define a weighted depth (6, 1, m)ψ of the signature ρ1 in the following way: (6, 1, m)ψ( fi ) = m for any i ∈ ω \ {0}. Denote τ (6, 1, m) = (ρ1 , {U(6,1,m) }, (6, 1, m)ψ). Lemma 13.6 For any n, m ∈ ω \ {0}, the sccf-triple τ (6, 1, m) is restricted, Typ Ψˆ τ (6,1,m) = Tp6 , ˆ ii ˆ di ˆ dd ˆ ss Ψˆ τda (6,1,m) (n) ≤ n, Ψτ (6,1,1) (n) = Ψτ (6,1,1) (n) = Ψτ (6,1,1) (n) = Ψτ (6,1,1) (n) = n, and if n > 1, then Ψˆ τii(6,1,m) (nm − 1) ≤ (n − 1)m, Ψˆ τdi(6,1,m) (nm − 1) ≤ (n − 1)m, Ψˆ τdd (6,1,m) (nm − 1) ≤ (n − 1)m , ˆ ss Ψˆ τda (6,1,m) (nm − 1) ≤ (n − 1)m, and Ψτ (6,1,m) (nm − 1) ≤ (n − 1)m.

172

13 Local Upper Types of Restricted Sccf-Triples

Proof Set σn = (en , f 1 , . . . , f n ) and Tn = Tρ1 (σn , {U(6,1,m) }). It is not difficult to see that m (6,1,m)ψ (Tn ) = m and Nρ1 (Tn ) = n + 1. Therefore Typ Nτ (6,1,m) = ω. Let T ∈ Mρ1 ,{U(6,1,m) } and m (6,1,m)ψ (T ) ≤ n. We now show that I(6,1,m)ψ (n, T ) ≤ n + 1. Let dim T = t and μT (1) = f i1 , . . . , μT (t) = f it . Let B be an uncancelable cover for the table T , (6, 1, m)ψ(B) ≤ n, and |B| = I(6,1,m)ψ (n, T ). One can show that (T ) = {δ¯0 , δ¯1 , . . . , δ¯t }, where δ¯0 is the tuple from E 2t filled with 0, and δ¯i is the tuple obtained from δ¯0 by substitution of 1 for ith component, i = 1, . . . , t. Since B is a cover for the table T , there exists a word α ∈ B such that δ¯0 ∈ (T α). Evidently, χ (α) ⊆ {( f i1 , 0), . . . , ( f it , 0)}. Therefore |(T α)| ≥ t − |α| + 1. Taking into account that B is an uncancelable cover we obtain |B| ≤ |(T )| − |(T α)| + 1 ≤ (t + 1) − (t − |α| + 1) + 1 = |α| + 1. Since the function (6, 1, m)ψ has the property 3 and (6, 1, m)ψ(α) ≤ n, we have |α| ≤ n. Hence I(6,1,m)ψ (n, T ) = |B| ≤ n + 1. Thus, Typ Iτ (6,1,m) = ω. By Proposition 13.7, Typ Ψˆ τ (6,1,m) = Tp6. One can show that S(6,1,1)ψ (Tn ) = n. Using Lemma 11.7 we conclude that there ({U(6,1,1) }) such that exists a schema z ∈ Σρ0−1 1 (6, 1, 1)Ψˆ ρi1 ,{U(6,1,1) } (z) = (6, 1, 1)Ψˆ ρd1 ,{U(6,1,1) } (z) = (6, 1, 1)Ψˆ ρs1 ,{U(6,1,1) } (z) = n . ˆ ss Hence Ψˆ τii(6,1,1) (n) = Ψˆ τdi(6,1,1) (n) = Ψˆ τdd (6,1,1) (n) = Ψτ (6,1,1) (n) = n. da Evidently, n ∈ Arg Ψˆ τ (6,1,m) . Let z ∈ Σρ1 and ψˆ ρa1 ,{U(6,1,m) } (z) ≤ n. We now show that ψˆ ρd ,{U } (z) ≤ n. Set T = Tρ1 (z, {U(6,1,m) }). Using Theorem 10.1 we obtain 1

(6,1,m)

ψˆ ρa1 ,{U(6,1,m) } (z) = (6, 1, m)ψρa1 (T ) and ψˆ ρd1 ,{U(6,1,1) } (z) = (6, 1, m)ψρd1 (T ). dim T = t and μT (1) = f i1 , . . . , μT (t) = f it . One can show that

Let

(T ) = {δ¯0 , δ¯1 , . . . , δ¯t } , where δ¯0 is the tuple from E 2t filled with 0, and δ¯i is the tuple obtained from δ¯0 by substitution of 1 for ith component, i = 1, . . . , t. Let (, T ) ∈ Rρa1 and (6, 1, m)ψ() ≤ n. Then there exists ξ ∈ () such that δ¯0 ∈ (T π(ξ )). If π(ξ ) = λ, then, as it is not difficult to see, T ∈ Mρ1 C and hence (6, 1, m)ψρd1 (T ) = 0 < n. Let π(ξ ) = λ and π(ξ ) = ( f j1 , 0) · · · ( f jm , 0). Set α = f j1 · · · f jm . Evidently, (6, 1, m)ψ(α) ≤ n. One can show that α is an unconditional test for the table T . Therefore ρ1 ,(6,1,m)ψ (T ) ≤ n. By Lemma 3.1, (6, 1, m)ψρd1 (T ) ≤ ρ1 ,(6,1,m)ψ (T ) ≤ n. Hence (6, 1, m)ψˆ ρd1 ,{U(6,1,1) } (z) ≤ n and Ψˆ τda (6,1,m) (n) ≤ n. Set Bm = {m j : j ∈ ω}. Let b ∈ {i, d, a}. One can show that {(6, 1, m)ψˆ ρb1 ,{U(6,1,m) } (z) : z ∈ Σρ1 } ⊆ Bm and {(6, 1, m)ψˆ ρs1 ,{U(6,1,m) } (z) : z ∈ Σρ0−1 ({U(6,1,m) })} ⊆ Bm . Therefore if n > 1 and 1 bc ∈ {ii, di, dd, da, ss}, then Ψˆ τbc(6,1,m) (nm − 1) ≤ (n − 1)m. 

13.3 Main Statements

173

13.3 Main Statements Lemma 13.7 Let σ = (G, 2) be a signature, V be an information system of the signature σ , ϕ be a weighted depth of the signature σ , and ρ = (F, k) be a signature such that |F| ≥ |G|. Then there exist an information system U of the signature ρ and a weighted depth ψ of the signature ρ such that Ψˆ (σ,{V },ϕ) = Ψˆ (ρ,{U },ψ) and Φˆ (σ,{V },ϕ) = Φˆ (ρ,{U },ψ) . Proof a subset F of the set F such that   Let V = (A, γ1 ) and g0 ∈ G. Choose  F  = |G|. Let η be a bijection of the set F to the set G. We define an information system U of the signature ρ as follows: U = (A, γ2 ), γ2 ( f ) = γ1 (η( f )) for any f ∈ F , and γ2 ( f ) = γ1 (g0 ) for any f ∈ F \ F . We define a weighted depth ψ of the signature ρ in the following way: ψ( f ) = ϕ(η( f )) for any f ∈ F , and ψ( f ) = ϕ(g0 ) for any f ∈ F \ F . Denote τ1 = (σ, {V }, ϕ) and τ2 = (ρ, {U }, ψ). Set Dτ1 = i ˆd ˆa {(ψˆ σ,{V } (z), ψσ,{V } (z), ψσ,{V } (z)) : z ∈ Σσ }, i 0−1 ˆd ˆa ˆs = {(ψˆ σ,{V Dτ0−1 } (z), ψσ,{V } (z), ψσ,{V } (z), ψσ,{V } (z)) : z ∈ Σσ ({V })} , 1 i ˆd ˆa Dτ2 = {(ψˆ ρ,{U } (z), ψρ,{U } (z), ψρ,{U } (z)) : z ∈ Σρ } and i 0−1 ˆd ˆa ˆs = {(ψˆ ρ,{U Dτ0−1 } (z), ψρ,{U } (z), ψρ,{U } (z), ψρ,{U } (z)) : z ∈ Σρ ({U })} . 2

One can show that Dτ1 = Dτ2 and Dτ0−1 = Dτ0−1 . Therefore Ψˆ τ1 = Ψˆ τ2 and Φˆ τ1 = 1 2 ˆ Φτ 2 .  Lemma 13.8 Let τ = (ρ, K , ψ) be a restricted sccf-triple, ρ = (F, k), and F be a finite set. Then Typ Ψˆ τ = Tp1. Proof Let n = |F| and m = max{ψ( f ) : f ∈ F}. Taking into account that the function ψ has the property 1 one can show that Nρ (T ) ≤ k n and Sψ (T ) ≤ nm for any table T ∈ Mρ,K . Hence Typ Nτ = Typ Sτ = ε. By Proposition 13.2,  Typ Ψˆ τ = Tp1. Theorem 13.1 Let ρ = (F, k) be a signature. Then the following statements hold: (a) If F is a finite set, then Typ Ψˆ τ = Tp1 for any restricted sccf-triple τ = (ρ, K , ψ). (b) If F is an infinite set, then Typ Ψˆ τ ∈ {Tp1, Tp2, Tp3, Tp4, Tp5, Tp6} for any restricted sccf-triple τ = (ρ, K , ψ), and, for any i ∈ {1, 2, 3, 4, 5, 6}, there exists a restricted sccf-triple τ = (ρ, K , ψ) such that Typ Ψˆ τ = Tpi. Proof Statement (a) follows from Lemma 13.8. Statement (b) follows from Proposition 13.1 and Lemmas 13.1–13.7. 

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13 Local Upper Types of Restricted Sccf-Triples

Reference 1. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity, Local approach. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets IV, Lecture Notes in Computer Science, vol. 3700, pp. 125–143. Springer, Berlin (2005)

Chapter 14

Bounds Inside Types

In this chapter, for a given signature ρ and each possible local upper type of restricted sccf-triples Tpi, i ∈ {1, . . . , 6}, we consider the set Wˆ ρ (i) of restricted sccf-triples τ with Typ Ψˆ τ = Tpi. For each pair (b, c) ∈ {i, d, a, s}2 such that in the matrix Tpi at the intersection of the row with index b and the column with index c either λ or χ stays, we study upper and lower bounds on the function Ψˆ τbc true for any sccf-triple τ ∈ Wˆ ρ (i). Earlier [1], we did not investigate common lower and upper bounds for sccf-triples of a given type. Let ρ = (F, k) be a signature and i ∈ {1, . . . , 6}. We denote by Wˆ ρ (i) the set of restricted sccf-triples τ = (ρ, K , ψ) such that Typ Ψˆ τ = Tpi. Let i ∈ {1, . . . , 6} and b, c ∈ {i, d, a, s}. We denote by Ψˆ X ρbc (i) the set of all functions f ∈ G R such that Ψˆ τbc  f for any triple τ ∈ Wˆ ρ (i). We denote by Ψˆ Yρbc (i) the set of all functions f ∈ G R such that Ψˆ τbc  f for any triple τ ∈ Wˆ ρ (i). We now define the sets Ψ¨ X ρbc (i) ⊆ Ψˆ X ρbc (i) and Ψ¨ Yρbc (i) ⊆ Ψˆ Yρbc (i). Let f ∈ Ψˆ X ρbc (i). Then f ∈ Ψ¨ X ρbc (i) if and only if there exists a triple τ ∈ Wˆ ρ (i) such that Ψˆ τbc  f . Let f ∈ Ψˆ Yρbc (i). Then f ∈ Ψ¨ Yρbc (i) if and only if there exists a triple τ ∈ Wˆ ρ (i) such that Ψˆ τbc  f . For any i ∈ {1, . . . , 6} and any pair (b, c) ∈ {i, d, a, s}2 such that in the matrix Tpi at the intersection of the row with index b and the column with index c the element λ or the element χ stays, we study the sets Ψˆ X ρbc (i) and Ψˆ Yρbc (i).

ˆ ρ (1) 14.1 Set W Proposition 14.1 Let ρ = (F, k) be a signature. Then x ∈ Ψ¨ X ρii (1), x − f (x) ∈ Ψˆ Yρii (1) for any function f ∈ F R, and x − c ∈ / Ψˆ Yρii (1) for any constant c ∈ ω.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_14

175

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14 Bounds Inside Types

Proof Let f ∈ F R. By Proposition 11.1, x − f (x) ∈ Ψˆ Yρii (1) and x ∈ Ψˆ X ρii (1). Let c ∈ ω. Using Lemmas 13.1 and 13.7 we obtain x ∈ Ψ¨ X ρii (1) and x − c ∈ / Ψˆ Yρii (1). 

ˆ ρ (2) 14.2 Set W j

For j ∈ ω \ {0}, set F j = { f i : i ∈ ω \ {0}}. Let m ∈ ω \ {0}. Denote ρ m = (F 1 ∪ j · · · ∪ F m , 2). Let j ∈ {1, . . . , m} and i ∈ m \ {0}. We now define a function pi : m m Q → E 2 . Let σ¯ = (σ1 , . . . , σm ) ∈ Q . Then  j

pi (σ¯ ) =

0, if σ j < i , 1, if σ j ≥ j .

We define an information system U(2,2,m) of the signature ρ m as follows: U(2,2,m) = (Q m , γ(2,2,m) ) j

j

and γ(2,2,m) ( f i ) = pi for any j ∈ {1, . . . , m} and i ∈ ω \ {0}. We define a weighted j depth (2, 2, m)ψ of the signature ρ m in the following way: (2, 2, m)ψ( f i ) = 1 for m any j ∈ {1, . . . , m} and i ∈ ω \ {0}. Denote τ (2, 2, m) = (ρ , {U(2,2,m) }, (2, 2, m)ψ). Lemma 14.1 For any n, m ∈ ω \ {0}, the sccf-triple τ (2, 2, m) is restricted, Typ Ψˆ τ (2,2,m) = Tp2 , and Ψˆ τdi(2,2,m) (nm) ≥ m log2 (nm) − m log2 m. Proof One can see that S(2,2,m)ψ (T ) ≤ 2m for any table T ∈ Mρ m ,{U(2,2,m) } . Therefore Typ Sτ (2,2,m) = ε. Set z nm = (dnm , f 11 , . . . , f n1 , f 12 , . . . , f n2 , . . . , f 1m , . . . , f nm ) and Tnm = Tρ m (z nm , {U(2,2,m) }). One can show that Nρ m (Tnm ) = (n + 1)m . Hence Typ Nτ (2,2,m) = ε. By Proposition 13.3, Typ Ψˆ τ (2,2,m) = Tp2. It is clear that Tnm is a diagnostic table, and the function (2, 2, m)ψ has the property Λ3. Using Proposition 3.2 we obtain (2, 2, m)ψρdm (Tnm ) ≥ log2 Nρ m (Tnm ) = m log2 (n + 1) ≥ m log2 (nm) − m log2 m. By Theorem 10.1, (2, 2, m)ψˆ ρdm ,{U(2,2,m) } (z nm ) ≥ m log2 (nm) − m log2 m. Evidently, (2, 2, m)ψˆ ρi m ,{U(2,2,m) } (z nm ) = nm. Hence Ψˆ τdi(2,2,m) (nm)  ≥ m log2 (nm) − m log2 m. Proposition 14.2 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: / (a) x ∈ Ψ¨ X ρii (2), x − f (x) ∈ Ψˆ Yρii (2) for any function f ∈ F R, and x − c ∈ ii ˆ Ψ Yρ (2) for any constant c ∈ ω.

14.2 Set Wˆ ρ (2)

177

(b) f (x) log2 (x) ∈ Ψˆ X ρdi (2) for any function f ∈ F R, c log2 (x) ∈ / Ψˆ X ρdi (2) for any constant c ∈ ω, log1 k log2 (x) − f (x) ∈ Ψˆ Yρdi (2) for any function f ∈ F R, and 2 2 + log2 (x) ∈ / Ψˆ Yρdi (2). (c) x ∈ Ψ¨ X ρdd (2), x − f (x) ∈ Ψˆ Yρdd (2) for any function f ∈ F R, and x − c ∈ / dd ˆ Ψ Yρ (2) for any constant c ∈ ω. Proof (a) Let f ∈ F R. By Proposition 11.1, x ∈ Ψˆ X ρii (2) and x − f (x) ∈ Ψˆ Yρii (2). Let c ∈ ω. By Lemmas 13.2 and 13.7, x ∈ Ψ¨ X ρii (2) and x − c ∈ / Ψˆ Yρii (2). Thus, the statement (a) holds. (b) Let f ∈ F R. By Proposition 11.2, f (x) log2 (x) ∈ Ψˆ X ρdi (2) and log1 k log2 (x)   2 − f (x) ∈ Ψˆ Yρdi (2). By Lemmas 13.2 and 13.7, 2 + log2 (x) ∈ / Ψˆ Yρdi (2). Therefore 2 + log2 (x) ∈ / Ψˆ Yρdi (2). Let c ∈ ω. By Lemmas 14.1 and 13.7, c log2 (x) ∈ / di Ψˆ X ρ (2). Thus, the statement (b) holds. (c) Using Proposition 11.3 we obtain x ∈ Ψˆ X ρdd (2). By Lemmas 13.2 and 13.7, x ∈ Ψ¨ X ρdd (2). Let τ ∈ Wˆ ρ (2). By Proposition 13.3, Typ Nτ = ε and Typ Sτ = ε. From here and from Lemma 11.9 it follows that Typ Nτ = ω. Using Lemma 11.10 we obtain x − f (x) ∈ Ψˆ Yρdd (2) for any function f ∈ F R. Let c ∈ ω. By Lemmas 13.2 and 13.7, x − c ∈ / Ψˆ Yρdd (2). Thus, the statement (c) holds. 

ˆ ρ (3) 14.3 Set W We define an information system U(3,2) of the signature ρ2 as follows: U(3,2) = (Q, γ(3,2) ), γ(3,2) (gi ) = l0 and γ(3,2) ( f i ) = li for any i ∈ ω \ {0}. We define a weighted depth (3, 2)ψ of the signature ρ2 in the following way: (3, 2)ψ( f i ) = 1 for any i ∈ ω \ {0}, (3, 2)ψ(g1 ) = 1, and (3, 2)ψ(gi+1 ) = 2(3,2)ψ(gi ) for any i ∈ ω \ {0}. Denote τ (3, 2) = (ρ2 , {U(3,2) }, (3, 2)ψ). Lemma 14.2 The sccf-triple τ (3, 2) is restricted, Typ Ψˆ τ (3,2) = Tp3, and the relation Ψˆ τdi(3,2)  log2 (x + 1) + 3 does not hold. Proof Set z n1 = (en , f 1 , . . . , f n ) and Tn1 = Tρ2 (z n1 , {U(3,2) }). One can see that Nρ2 (Tn1 ) = n + 1 and m (3,2)ψ (Tn1 ) = 1. Therefore Typ Nτ (3,2) = ω. Set z n2 = (e1 , gn ) and Tn2 = Tρ2 (z n2 , {U(3,2) }) . One can show that S(3,2)ψ (Tn2 ) = (3, 2)ψ(gn ). Therefore Typ Sτ (3,2) = ε. Let T ∈ Mρ2 ,{U(3,2) } .

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14 Bounds Inside Types

One can see that S(3,2)ψ (T ) ≤ m (3,2)ψ (T ) + 1. Hence Typ Sτ (3,2) = ω. By Lemma 12.1, Typ Sτ (3,2) = χ . Using Proposition 13.4 we obtain Typ Ψˆ τ (3,2) = Tp3. Let i ∈ ω \ {0, 1}. Denote n(i) = (3, 2)ψ(gi ) − 1. Let z ∈ Σρ2 and (3, 2)ψˆ ρi 2 ,{U(3,2) } (z) ≤ n(i) . Then, evidently, |P(z)| ≤ n(i) and (3, 2)ψ(q) ≤ log2 (n(i) + 1) for any q ∈ P(z). Using these inequalities one can show that (3, 2)ψˆ ρd2 ,{U(3,2) } (z) ≤ 1 + max{log2 (n(i)   + 1), log2 n(i) + 1} < log2 (n(i) + 1) + 3. Hence the relation Ψˆ τdi(3,2)  log2 (x + 1) + 3 does not hold.  Let f ∈ F R. We define an information system U(3,3, f ) of the signature ρ2 as follows: U(3,3, f ) = (Q, γ(3,3, f ) ), γ(3,3, f ) (gi ) = l0 and γ(3,3, f ) ( f i ) = li for any i ∈ ω \ {0}. We define a weighted depth (3, 3, f )ψ of the signature ρ2 in the following way: (3, 3, f )ψ( f i ) = 1 for any i ∈ ω \ {0}, (3, 3, f )ψ(g1 ) = 1, and, for any i ∈ ω \ {0}, (3, 3, f )ψ(gi+1 ) = t + 1, where t is the minimum number from the set ω ∩ Arg f such that min{t, f (t) − 2} > (3, 3, f )ψ(gi ). Denote τ (3, 3, f ) = (ρ2 , {U(3,3,2) }, (3, 3, f )ψ). Lemma 14.3 For any function f ∈ F R, the sccf-triple τ (3, 3, f ) is restricted, Typ Ψˆ τ (3,3, f ) = Tp3 , and the relations Ψˆ τas(3,3, f )  f and Ψˆ τss(3,3, f )  f do not hold. Proof Let f ∈ F R. Similarly to the proof of Lemma 14.2, one can show that Typ Ψˆ τ (3,3, f ) = Tp3. Let i ∈ ω \ {0, 1}. Set n(i) = (3, 3, f )ψ(gi ) − 1. Let T ∈ Mρ2 ,{U(3,3, f ) } and m (3,3, f )ψ (T ) ≤ n(i). Then, evidently, m (3,3, f )ψ (T ) ≤ (3, 3, f )ψ(gi−1 ) < f (n(i)) − 2. It is clear that S(3,3, f )ψ (T ) ≤ max{2, m (3,3, f )ψ (T ) + 1} < f (n(i)). Therefore Sτ (3,3, f ) (n(i)) < f (n(i)). Hence the relation Sτ (3,3, f )  f does not hold. Since Typ Ψˆ τ (3,3, f ) = Tp3 , we have Typ Ψˆ τas(3,3, f ) = χ . By Proposition 12.2, the relation Ψˆ τas(3,3, f )  f does not hold. Since Typ Ψˆ τ (3,3, f ) = Tp3, we have Typ Ψˆ τss(3,3, f ) = χ . By Lemma 11.2, Ψˆ τss(3,3, f )  Ψˆ τas(3,3, f ) . Therefore the relation Ψˆ τss(3,3, f )  f does not hold.  Let f ∈ F R, c ∈ R(0), and Arg f = R(c). Set m = c + 1. We define an information system U(3,4, f ) of the signature ρ2 as follows: U(3,4, f ) = (Q, γ(3,4, f ) ), γ(3,4, f ) (gi ) = p−i and γ(3,4, f ) ( f i ) = li for any i ∈ ω \ {0}. One can show that there exists a sequence a1 , a2 , . . . having the following properties: (a) ai ∈ ω and  ai ≤ ai+1 for any i ∈ ω \ {0}. (b) f (n) < i∈ω\{0},ai ≤n ai < ∞ for any n ∈ ω(m).

14.3 Set Wˆ ρ (3)

179

We define a weighted depth (3, 4, f )ψ of the signature ρ2 as follows: (3, 4, f )ψ( f i ) = 1 and (3, 4, f )ψ(gi ) = ai for any i ∈ ω \ {0}. Denote τ (3, 4, f ) = (ρ2 , {U(3,4, f ) }, (3, 4, f )ψ) .

Lemma 14.4 For any function f ∈ F R, the sccf-triple τ (3, 4, f ) is restricted, Typ Ψˆ τ (3,4, f ) = Tp3 and the relation Ψˆ τas(3,3, f )  f does not hold. Proof One can show that Typ Nτ (3,4, f ) = ω. One can see that the function (3, 4, f )ψ is not bounded from above on the set G 0 and, for any n ∈ ω, the set {gi : gi ∈ G 0 , (3, 4, f )ψ(gi ) ≤ n} is finite. Using these facts one can show that Typ Sτ (3,4, f ) = ε and Typ Sτ (3,4, f ) = ω. From these relations and from Lemma 12.1 it follows that Typ Sτ (3,4, f ) = χ . By Proposition 13.4, Typ Ψˆ τ (3,4, f ) = Tp3. Let n ∈ ω(m) and {g1 , . . . , gt } = {gi : gi ∈ G 0 , (3, 4, f )ψ(gi ) ≤ n}. Set z n = (et , g1 , . . . , gt ) and Tn = T ρ2 (z n , {U(3,4, f ) }). Evidently, m (3,4, f )ψ (Tn ) ≤ n. One can t (3, 4, f )ψ(gi ) > f (n). Taking into account that show that S(3,4, f )ψ (Tn ) = i=1 Typ Sτ (3,4, f ) = χ we conclude that the value Sτ (3,4, f ) (n) is definite and Sτ (3,4, f ) (n) > f (n). Since n is an arbitrary number from ω(m), we have that the relation Sτ (3,4, f )  f does not hold. Since Typ Ψˆ τ (3,4, f ) = Tp3, we have Typ Ψˆ τas(3,4, f ) = χ . By Proposition 12.2,  the relation Ψˆ τas(3,4, f )  f does not hold. Proposition 14.3 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) x ∈ Ψ¨ X ρii (3), x − f (x) ∈ Ψˆ Yρii (3) for any function f ∈ F R, and x − c ∈ / ii ˆ Ψ Yρ (3) for any constant c ∈ ω. (b) x ∈ Ψ¨ X ρdi (3), log1 k log2 x − f (x) ∈ Ψˆ Yρdi (3) for any function f ∈ F R, and 2 log2 (x + 1) + 3 ∈ / Ψˆ Yρdi (3). (c) x ∈ Ψ¨ X ρdd (3), x − f (x) ∈ Ψˆ Yρdd (3) for any function f ∈ F R, and x − c ∈ / Ψˆ Yρdd (3) for any constant c ∈ ω. (d) F R ∩ Ψˆ X ρas (3) = ∅, F R ∩ Ψˆ Yρas (3) = ∅, and m ∈ Ψˆ Yρas (3) for any constant m ∈ ω. (e) If bc ∈ {ai, ad, aa, si, sd, sa, ss}, then x ∈ Ψ¨ X ρbc (3), F R ∩ Ψˆ Yρbc (3) = ∅, and m ∈ Ψˆ Yρbc (3) for any constant m ∈ ω.

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14 Bounds Inside Types

Proof (a) Let f ∈ F R. By Proposition 11.1, x ∈ Ψˆ X ρii (3) and x − f (x) ∈ Ψˆ Yρii (3). Let c ∈ ω. Using Lemmas 13.3 and 13.7 we obtain x ∈ Ψ¨ X ρii (3) and x − c ∈ / Ψˆ Yρii (3). (b) Using Proposition 11.2 and Lemmas 13.3, 13.7 we obtain x ∈ Ψ¨ X ρdi (3). By Proposition 13.4, Typ Nτ = ω for any triple τ ∈ Wˆ ρ (3). Using this equality and Lemma 11.6 we obtain log1 k log2 x − f (x) ∈ Ψˆ Yρdi (3) for any function f ∈ F R. 2 By Lemma 14.2, log2 (x + 1) + 3 ∈ / Ψˆ Yρdi (3). (c) Using Proposition 11.3 and Lemmas 13.3, 13.7 we obtain x ∈ Ψ¨ X ρdd (3), and x −c ∈ / Ψˆ Yρdd (3) for any constant c ∈ ω. By Proposition 13.4, Typ Nτ = ω for any triple τ ∈ Wˆ ρ (3). Using this equality and Lemma 11.10 we obtain x − f (x) ∈ Ψˆ Yρdd (3) for any function f ∈ F R. (d) Using Lemmas 14.4 and 13.7 we obtain F R ∩ Ψˆ X ρas (3) = ∅. By Lemmas 14.3 and 13.7, F R ∩ Ψˆ Yρas (3) = ∅. Let τ ∈ Wˆ ρ (3). Then Typ Ψˆ τas = χ , and Ψˆ τas is a nondecreasing function. Therefore m ∈ Ψˆ Yρas (3) for any constant m ∈ ω. (e) Let bc ∈ {ai, ad, aa, si, sd, sa, ss}. By Proposition 11.4 and Lemmas 13.3 and 13.7, x ∈ Ψ¨ X ρbc (3). Let τ ∈ Wˆ ρ (3). Then Typ Ψˆ τbc = χ , and Ψˆ τbc is a nondecreasing function. Therefore m ∈ Ψˆ Yρbc (3) for any constant m ∈ ω. Using Proposition 11.4 and Lemmas 14.3 and 13.7 we obtain F R ∩ Ψˆ Yρbc (3) = ∅. 

ˆ ρ (4) 14.4 Set W Lemma 14.5 Let ρ = (F, k) be a signature with an infinite set F, U = (A, γ ) be an information system of the signature ρ such that γ ( f ) ≡ const on A for any f ∈ F, and ψ be a weighted depth of the signature ρ such that the set { f : f ∈ F, ψ( f ) ≤ n} is finite for any n ∈ ω. Then τ = (ρ, {U }, ψ) is a restricted sccf-triple such that Typ Ψˆ τ = Tp4. Proof Evidently, τ is a restricted sccf-triple. One can show that Typ Nτ = ω. One  can show also that Typ Sτ = ε. By Proposition 13.5, Typ Ψˆ τ = Tp4. Let f ∈ F R. We define an information system U(4,2, f ) of the signature ρ1 as follows: U(4,2, f ) = (Q, γ(4,2, f ) ) and γ(4,2, f ) ( f i ) = l0 for any i ∈ ω \ {0}. One can show that there exists a sequence a1 , a2 , . . . having the following properties: for any i ∈ ω \ {0}, ai ∈ ω \ {0}, ai < ai+1 − 1, and ai < f (ai+1 − 1). We define a weighted depth (4, 2, f )ψ of the signature ρ1 in the following way: (4, 2, f )ψ( f i ) = ai for any i ∈ ω \ {0}. Denote τ (4, 2, f ) = (ρ1 , {U(4,2, f ) }, (4, 2, f )ψ). Lemma 14.6 For any function f ∈ F R, the sccf-triple τ (4, 2, f ) is restricted, Typ Ψˆ τ (4,2, f ) = Tp4

14.4 Set Wˆ ρ (4)

181

and, for any b ∈ {d, a, s} and c ∈ {i, d, a, s}, the relation Ψˆ τbc(4,2, f )  f does not hold. Proof By Lemma 14.5, Typ Ψˆ τ (4,2, f ) = Tp4. Let i ∈ ω \ {0, 1}. Let z ∈ Σρ1 and (4, 2, f )ψˆ ρa1 ,{U(4,2, f ) } (z) ≤ ai+1 − 1. One can show that (4, 2, f )ψˆ ρd1 ,{U(4,2, f ) } (z) ≤ ai < ˆ da f (ai+1 − 1). Evidently, the value Ψˆ τda (4,2, f ) (ai+1 − 1) is definite and Ψτ (4,2, f ) (ai+1 − da 1) < f (ai+1 − 1). Hence the relation Ψˆ τ (4,2, f )  f does not hold. In the same way, we can prove that the relation Ψˆ τds(4,2, f )  f does not hold. Let bc ∈ {di, dd, ai, ad, aa, as, si, sd, sa, ss}. By Lemma 11.2, Ψˆ τbc(4,2, f ) ≺ da Ψˆ τ (4,2, f ) or Ψˆ τbc(4,2, f ) ≺ Ψˆ τds(4,2, f ) . Using these relations one can show that the relation Ψˆ τbc(4,2, f )  f does not hold.  Let f ∈ F R. Then there exists c ∈ R(0) such that Arg f = R(c). Set m = c + 1. We define an information system U(4,3, f ) of the signature ρ1 as follows: U(4,3, f ) = (Q, γ(4,3, f ) ) and γ(4,3, f ) ( f i ) = pi for any i ∈ ω \ {0}. One can show that there exists a sequence a1 , a2 , . . . having the following properties: {0} and ai ≤ ai+1 for any i ∈ ω \ {0}. (a) ai ∈ ω \ (b) f (n) < i∈ω\{0},ai ≤n ai < ∞ for any n ∈ ω(m). We define a weighted depth (4, 3, f )ψ of the signature ρ1 as follows: (4, 3, f )ψ( f i ) = ai for any i ∈ ω \ {0}. Denote τ (4, 3, f ) = (ρ1 , {U(4,3, f ) }, (3, 4, f )ψ). Lemma 14.7 For any function f ∈ F R, the sccf-triple τ (4, 3, f ) is restricted, Typ Ψˆ τ (4,3, f ) = Tp4 , and the relations Ψˆ τas(4,3, f )  f and Ψˆ τds(4,3, f )  f do not hold. Proof By Lemma 14.5, Typ Ψˆ τ (4,3, f ) = Tp4. Let n ∈ ω(m) and { f 1 , . . . , f t } = { f i : f i ∈ F0 , ai ≤ n}. Set z = (et , f 1 , . . . , f t ) and Tn = T ρ1 (z n , {U(4,3,   f ) }). Evidently, t ai = i∈ω\{0},ai ≤n ai > m (4,3, f )ψ (Tn ) ≤ n. One can show that S(4,3, f )ψ (Tn ) = i=1 f (n). Hence Sτ (4,3, f ) (n) > f (n). Therefore the relation Sτ (4,3, f ) (n)  f does not hold. By Proposition 12.2, the relation Ψˆ τas(4,3, f )  f does not hold. Using Lemma 11.2 we obtain Ψˆ τas(4,3, f ) ≺ Ψˆ τds(4,3, f ) . Therefore the relation Ψˆ τds(4,3, f )  f does not hold.  Let f ∈ F R, c ∈ R(0), and Arg f = R(c). Set m = c + 1. We define an information system U(4,4, f ) of the signature ρ1 as follows: U(4,4, f ) = (Q, γ(4,4, f ) ) and γ(4,4, f ) ( f i ) = li for any i ∈ ω \ {0}. One can show that there exists a sequence a1 , a2 , . . . having the following properties: (a) ai ∈ ω \ {0} and ai ≤ ai+1 for any i ∈ ω \ {0}. (b) 2 f (2n) < |{i : i ∈ ω \ {0}, ai ≤ n}| < ∞ for any n ∈ ω(m).

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14 Bounds Inside Types

We define a weighted depth (4, 4, f )ψ of the signature ρ1 as follows: (4, 4, f )ψ( f i ) = ai for any i ∈ ω \ {0}. Denote τ (4, 4, f ) = (ρ1 , {U(4,4, f ) }, (4, 4, f )ψ). Lemma 14.8 For any function f ∈ F R, the sccf-triple τ (4, 4, f ) is restricted, Typ Ψˆ τ (4,4, f ) = Tp4 , and the relation Ψˆ τda (4,4, f )  f does not hold. Proof By Lemma 14.5, Typ Ψˆ τ (4,4, f ) = Tp4. Let n ∈ ω(m) and { f 1 , . . . , f t } = { f i : f i ∈ F0 , ai ≤ n}. Set z = (dt , f 1 , . . . , f t ) and Tn = Tρ1 (z n , {U(4,4, f ) }). One can show that (4, 4, f )ψρa1 (Tn ) ≤ 2n, Nρ1 (Tn ) = t + 1 > 2 f (2n) , and Tn is a diagnostic table. Using Proposition 3.2 we obtain (4, 4, f )ψρd1 (Tn ) ≥ log2 Nρ1 (Tn ) > f (2n). ˆ da By Theorem 10.1, Ψˆ τda (4,4, f ) (2n) > f (2n). Hence the relation Ψτ (4,4, f )  f does not hold.  Proposition 14.4 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) x ∈ Ψ¨ X ρii (4), x − f (x) ∈ Ψˆ Yρii (4) for any function f ∈ F R, and x − m ∈ / Ψˆ Yρii (4) for any constant m ∈ ω. (b) x ∈ Ψ¨ X ρdi (4), F R ∩ Ψˆ Yρdi (4) = ∅, and m ∈ Ψˆ Yρdi (4) for any constant m ∈ ω. (c) x ∈ Ψ¨ X ρdd (4), F R ∩ Ψˆ Yρdd (4) = ∅, and m ∈ Ψˆ Yρdd (4) for any constant m ∈ ω. (d) F R ∩ Ψˆ X ρda (4) = ∅, F R ∩ Ψˆ Yρda (4) = ∅, and m ∈ Ψˆ Yρda (4) for any constant m ∈ ω. (e) F R ∩ Ψˆ X ρds (4) = ∅, F R ∩ Ψˆ Yρds (4) = ∅, and m ∈ Ψˆ Yρds (4) for any constant m ∈ ω. (f) F R ∩ Ψˆ X ρas (4) = ∅, F R ∩ Ψˆ Yρas (4) = ∅, and m ∈ Ψˆ Yρas (4) for any constant m ∈ ω. (g) If bc ∈ {ai, ad, aa, si, sd, sa, ss}, then x ∈ Ψ¨ X ρbc (4), F R ∩ Ψˆ Yρbc (4) = ∅, and m ∈ Ψˆ Yρbc (4) for any constant m ∈ ω. Proof (a) Let f ∈ F R. By Proposition 11.1, x ∈ Ψˆ X ρii (4) and x − f (x) ∈ Ψˆ Yρii (4). Let m ∈ ω. By Lemmas 13.4 and 13.7, x ∈ Ψ¨ X ρii (4) and x − m ∈ / Ψˆ Yρii (4). (b) Using Proposition 11.2 and Lemmas 13.4 and 13.7 we obtain x ∈ Ψ¨ X ρdi (4). By Lemmas 14.6 and 13.7, F R ∩ Ψˆ Yρdi (4) = ∅. Let τ ∈ Wˆ ρ (4). Then Typ Ψˆ τdi = χ and Ψˆ τdi is a nondecreasing function. Therefore m ∈ Ψˆ Yρdi (4) for any constant m ∈ ω. (c) Using Proposition 11.3 and Lemmas 13.4 and 13.7 we obtain x ∈ Ψ¨ X ρdd (4). By Lemmas 14.6 and 13.7, F R ∩ Ψˆ Yρdd (4) = ∅. Evidently, m ∈ Ψˆ Yρdd (4) for any constant m ∈ ω.

14.4 Set Wˆ ρ (4)

183

(d) By Lemmas 14.8 and 13.7, F R ∩ Ψˆ X ρda (4) = ∅. By Lemmas 14.6 and 13.7, F R ∩ Ψˆ Yρda (4) = ∅. Evidently, m ∈ Ψˆ Yρda (4) for any constant m ∈ ω. (e) Using Lemmas 14.7 and 13.7 we obtain F R ∩ Ψˆ X ρds (4) = ∅. By Lemmas 14.6 and 13.7, F R ∩ Ψˆ Yρds (4) = ∅. Evidently, m ∈ Ψˆ Yρds (4) for any constant m ∈ ω. (f) By Lemmas 14.7 and 13.7, F R ∩ Ψˆ X ρas (4) = ∅. From Lemmas 14.6 and 13.7 it follows that F R ∩ Ψˆ Yρas (4) = ∅. Evidently, m ∈ Ψˆ Yρas (4) for any constant m ∈ ω. (g) Let bc ∈ {ai, ad, aa, si, sd, sa, ss}. Using Proposition 11.4 and Lemmas 13.4 and 13.7 one can show that x ∈ Ψ¨ X ρbc (4). By Lemmas 14.6 and 13.7, F R ∩ Ψˆ Yρbc (4) = ∅. Evidently, m ∈ Ψˆ Yρbc (4) for any constant m ∈ ω. 

ˆ ρ (5) 14.5 Set W Proposition 14.5 Let ρ = (F, k) be a signature with an infinite set F, and bc ∈ {ii, di, dd, ai, ad, aa, si, sd, sa, ss}. Then x ∈ Ψ¨ X ρbc (5), x − f (x) ∈ Ψˆ Yρbc (5) for any function f ∈ F R, and x − m ∈ / Ψˆ Yρbc (5) for any constant m ∈ ω. Proof Let f ∈ F R and τ ∈ Wˆ ρ (5). By Proposition 13.6, Typ Sτ = ω. Using Lemma 11.13 we conclude that there exists t ∈ ω such that Ψˆ τss (n) ≥ n − t for any n ∈ Arg Ψˆ τss . Therefore Ψˆ τss  x − f (x). By Lemma 11.14, Ψˆ τsi  x − f (x). Using Lemma 11.2 we obtain Ψˆ τsi ≺ Ψˆ τbc . Therefore Ψˆ τbc  x − f (x). Hence x − f (x) ∈ Ψˆ Yρbc (5). By Propositions 11.1–11.4, x ∈ Ψˆ X ρbc (5). By Lemmas 13.5 and 13.7, and Proposition 11.4, x ∈ Ψ¨ X ρbc (5) and x − m ∈ / Ψˆ Yρbc (5) for any constant m ∈ ω. 

ˆ ρ (6) 14.6 Set W Let f ∈ F R. Then there exists c ∈ R(0) such that Arg f = R(c). Set m = c + 1. We define an information system U(6,2, f ) of the signature ρ2 as follows: U(6,2, f ) = (Q, γ(6,2, f ) ), γ(6,2, f ) (gi ) = p−i and γ(6,2, f ) ( f i ) = li for any i ∈ ω \ {0}. One can show that there exists a sequence a1 , a2 , . . . having the following properties: (a) ai ∈ ω \ {0} and ai ≤ ai+1 for any i ∈ ω \ {0}. (b) a1 = 1. (c) 2 f (2n) < |{i : i ∈ ω \ {0}, ai ≤ n}| < ∞ for any n ∈ ω(m). We define a weighted depth (6, 2, f )ψ of the signature ρ2 as follows: (6, 2, f )ψ(gi ) = 1

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14 Bounds Inside Types

and (6, 2, f )ψ( f i ) = ai for any i ∈ ω \ {0}. Denote τ (6, 2, f ) = (ρ2 , {U(6,2, f ) }, (6, 2, f )ψ) .

Lemma 14.9 For any function f ∈ F R, the sccf-triple τ (6, 2, f ) is restricted, Typ Ψˆ τ (6,2, f ) = Tp6 and the relation Ψˆ τda (6,2, f )  f does not hold. Proof Set z n1 = (en , g1 , . . . , gn ) and Tn1 = Tρ2 (z n1 , {U(6,2, f ) }). One can show that Nρ2 (Tn1 ) = n + 1 and m (6,2, f )ψ (Tn1 ) = 1. Therefore Typ Nτ (6,2, f ) = ω. Let n ∈ ω \ {0}. Set r (n) = |{i : i ∈ ω \ {0}, ai ≤ n}|. Let T ∈ Mρ2 ,{U(6,2, f ) } , m (6,2, f )ψ (T ) ≤ n, and P(T ) = {gi1 , . . . , gic , f j1 , . . . , f jt }. Evidently, t ≤ r (n). We denote by δ¯0 the tuple (0, . . . , 0) ∈ E 2c+t . One can see that Nρ2 (T ) = c + t + 1 and δ¯0 ∈ Δ(T ). Let B be an uncancelable cover for the table T , (6, 2, f )ψ(B) ≤ n, and |B| = I(6,2, f )ψ (n, T ). Since B is a cover for the table T , there exists a word α ∈ B such that δ¯0 ∈ Δ(T α). Evidently, χ (α) ⊆ {(gi1 , 0), . . . , (gic ), ( f j1 , 0), . . . , ( f jt )}. Using this relation one can show that |Δ(T α)| ≥ c − |α| + 1. Taking into account that B is an uncancelable cover we obtain |B| ≤ |Δ(T )| − |Δ(T α)| + 1 ≤ c + t + 1 − (c − |α| + 1) + 1 = t + |α| + 1 ≤ r (n) + |α| + 1. Since the function (6, 2, f )ψ has the property Λ3 and (6, 2, f )ψ(α) ≤ n, we have |α| ≤ n. Hence I(6,2, f )ψ (n, T ) = |B| ≤ r (n) + n + 1. Thus, Typ Iτ (6,2, f )ψ = ω. By Proposition 13.7, Typ Ψˆ τ (6,2, f ) = Tp6. Let n ∈ ω(m) and {i : i ∈ ω \ {0}, ai ≤ n} = {1, . . . , t}. Set z n2 = (dt , f 1 , . . . , f t ) and Tn2 = Tρ2 (z n2 , {U(6,2, f ) }). One can show that (6, 2, f )ψρa2 (Tn2 ) ≤ 2n, Nρ2 (Tn2 ) = t + 1 > 2 f (2n) + 1, and Tn2 is a diagnostic table. By Proposition 3.2, (6, 2, f )ψρd2 (Tn2 ) ≥ log2 Nρ2 (Tn2 ) > f (2n). By Theorem 10.1, Ψˆ τda (6,2, f ) (2n) > f (2n). Hence the rela f does not hold.  tion Ψˆ τda (6,2, f ) Proposition 14.6 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) If bc ∈ {ii, di, dd, ai, ad, aa, si, sd, sa, ss}, then x ∈ Ψ¨ X ρbc (6), x − f (x) ∈ Ψˆ Yρbc (6) for any function f ∈ F R, and x − m ∈ / Ψˆ Yρbc (6) for any constant m ∈ ω. (b) F R ∩ Ψˆ X ρda (6) = ∅, x − f (x) ∈ Ψˆ Yρda (6) for any function f ∈ F R, and x −m ∈ / Ψˆ Yρda (6) for any constant m ∈ ω. Proof Denote B = {ii, di, dd, ai, ad, aa, si, sd, sa, ss}. Let bc ∈ B ∪ {da}, f ∈ F R, and τ ∈ Wˆ ρ (6). Then Typ Ψˆ τas = ω. By Proposition 12.2, Typ Sτ = ω. By Lemma 11.13, there exists t ∈ ω such that Ψˆ τss (n) ≥ n − t for any n ∈ Arg Ψˆ τss .

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185

Therefore Ψˆ τss  x − f (x). By Lemma 11.14, Ψˆ τsi  x − f (x). Using Lemma 11.2 we conclude that Ψˆ τsi ≺ Ψˆ τbc . Therefore Ψˆ τbc  x − f (x). Hence x − f (x) ∈ / Ψˆ Yρbc (6) for Ψˆ Yρbc (6). By Lemmas 13.6 and 13.7, and Proposition 11.4, x − m ∈ any constant m ∈ ω. Let bc ∈ B. By Propositions 11.1–11.4, x ∈ Ψˆ X ρbc (6). Using Lemmas 13.6 and 13.7, and Proposition 11.4 we conclude that x ∈ Ψ¨ X ρbc (6). By Lemmas 14.9 and 13.7, F R ∩ Ψˆ X ρda (6) = ∅. 

Reference 1. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity. Local approach. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets IV, Lecture Notes in Computer Science, vol. 3700, pp. 125–143. Springer, Berlin (2005)

Chapter 15

Matrices of Lower Local Bounds

In this chapter, we describe all possible six local lower types tp1, . . . , tp6 of restricted sccf-triples which correspond to the local upper types Tp1, . . . , Tp6, respectively. Some similar results were obtained in [1] for types of functions different from the upper and lower types considered in this book. For a given signature ρ, each local lower type tpi, i ∈ {1, . . . , 6}, and each pair (b, c) ∈ {i, d, a, s}2 such that in the matrix tpi at the intersection of the row with index b and the column with index c either μ or γ stays, we study upper and lower ˆ ˆ bc bounds on the function  τ true for any sccf-triple τ ∈ Wρ (i).

15.1 Possible Local Lower Types Proposition 15.1 Let τ = (ρ, K , ψ) be a restricted sccf-triple and b, c ∈ {i, d, a, s}. Then the following statements hold: ˆ cb ˆ bc (a) typ  τ = ε if and only if Typ τ = ω. bc ˆ τ = γ if and only if Typ  ˆ τcb = χ . (b) typ  bc ˆ τ = μ if and only if Typ  ˆ τcb = λ. (c) typ  ˆ cb ˆ bc (d) typ  τ = ω if and only if Typ τ = ε. Proof The considered statements follow immediately from Proposition 10.5.



Let us consider the following six matrices

i tp1 = d a s

i γ ε ε ε

d ω ω ω ω

a ω ω ω ω

s ω ω ω ω

i tp2 = d a s

i γ ε ε ε

d μ γ ε ε

a ω ω ω ω

s ω ω ω ω

i tp3 = d a s

i γ ε ε ε

d γ γ ε ε

a γ γ γ γ

s γ γ γ γ

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_15

187

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15 Matrices of Lower Local Bounds

i tp4 = d a s

i γ ε ε ε

d γ γ γ γ

a γ γ γ γ

s γ γ γ γ

i tp5 = d a s

i γ ε ε ε

d γ γ ε ε

a γ γ γ ε

s γ γ γ γ

i tp6 = d a s

i γ ε ε ε

d γ γ γ ε

a γ γ γ ε

s γ γ γ γ

Proposition 15.2 Let τ = (ρ, K , ψ) be a restricted sccf-triple and i ∈ {1, 2, 3, 4, 5, ˆ τ = tpi. ˆ τ = Tpi if and only if typ  6}. Then Typ  Proof The considered statement follows immediately from Proposition 15.1.



Theorem 15.1 Let ρ = (F, k) be a signature. Then the following statements hold: ˆ τ = tp1 for any restricted sccf-triple τ = (a) If F is a finite set, then typ  (ρ, K , ψ). ˆ τ ∈ {tp1, tp2, tp3, tp4, tp5, tp6} for any (b) If F is an infinite set, then typ  restricted sccf-triple τ = (ρ, K , ψ), and, for any i ∈ {1, 2, 3, 4, 5, 6}, there exists ˆ τ = tpi. a restricted sccf-triple τ = (ρ, K , ψ) such that typ  Proof The considered statements follow immediately from Theorem 13.1 and Proposition 15.2. 

15.2 Auxiliary Statements Let ρ = (F, k) be a signature, i ∈ {1, . . . , 6} and b, c ∈ {i, d, a, s}. We denote by ˆ ˆ ρbc (i) the set of functions f ∈ G R such that  ˆ bc X τ  f for any triple τ ∈ Wρ (i). ˆ ρbc (i) the set of functions f ∈ G R such that  ˆ bc We denote by Y τ  f for any triple bc bc ¨ ρ (i) ⊆ X ˆ ρ (i) and Y ¨ ρbc (i) ⊆ Y ˆ ρbc (i). τ ∈ Wˆ ρ (i). We now define the sets X ¨ ρbc (i) if and only if there exists a triple τ ∈ Wˆ ρ (i) ˆ ρbc (i). Then f ∈ X Let f ∈ X ˆ bc ˆ bc ¨ bc such that  τ  f . Let f ∈ Yρ (i). Then f ∈ Yρ (i) if and only if there exists a bc ˆτ  f. triple τ ∈ Wˆ ρ (i) such that  Let i ∈ {1, 2, 3, 4, 5, 6} and (b, c) ∈ {i, d, a, s}2 . We denote by Tp(i, b, c) the element which stays in the matrix Tpi at the intersection of the row with index b and the column with index c. We denote by tp(i, b, c) the element which stays in the matrix tpi at the intersection of the row with index b and the column with index c. Lemma 15.1 Let ρ = (F, k) be a signature, (b, c) ∈ {i, d, a, s}2 , i ∈ {1, 2, 3, 4, ¨ X ρbc (i). Then x ∈ 5, 6}, Wˆ ρ (i) = ∅, Tp(i, b, c) = ω, Tp(i, b, c) = ε, and x ∈  ¨ ρcb (i). Y Proof By Proposition 15.2, tp(i, c, b) = ω and tp(i, c, b) = ε. Let τ ∈ Wˆ ρ (i). Then ˆ τbc  x. Evidently, x ∈ F R and x −1 = x. By Proposition 10.6,  ˆ cb  τ  x. Hence cb bc ˆ ¨ ˆ x ∈ Yρ (i). Since x ∈  X ρ (i), there exists a triple τ ∈ Wρ (i) and number m ∈ ω

15.2 Auxiliary Statements

189

ˆ τbc (n) = n for any n ∈ ω(m). Using Lemma 10.1 we obtain  ˆ bc such that  τ  x. cb ¨ ρ (i).  Hence x ∈ Y Lemma 15.2 Let ρ = (F, k) be a signature, (b, c) ∈ {i, d, a, s}2 , i ∈ {1, 2, 3, 4, ˆ ρbc (i) for any constant m ∈ ω. 5, 6}, Tp(i, b, c) = ω, Tp(i, b, c) = ε and x − m ∈ / Y cb ˆ ρ (i) for any constant m ∈ ω. Then x + m ∈ / X ˆ ρcb (i). Let Proof Assume the contrary: let there exist m ∈ ω such that x + m ∈ X −1 ˆ cb = x − m. τ ∈ Wˆ ρ (i). Then  τ  x + m. Evidently, x + m ∈ F R and (x + m) Using Proposition 15.2 we obtain tp(i, c, b) = ω and tp(i, c, b) = ε. By Proposition ˆ ρbc (i) which is impossible. ˆ τbc  x − m − 1. Hence x − m − 1 ∈ Y  10.7,  Lemma 15.3 Let ρ = (F, k) be a signature, (b, c) ∈ {i, d, a, s}2 , i ∈ {1, 2, 3, 4, ˆ ρbc (i) for any 5, 6}, Tp(i, b, c) = ω, Tp(i, b, c) = ε, Wˆ ρ (i) = ∅, and x − f (x) ∈ Y ˆ ρcb (i) for any function f ∈ F R. function f ∈ F R. Then x + f (x) ∈ X Proof By Proposition 15.2, tp(i, c, b) = ω and tp(i, c, b) = ε. Let τ ∈ Wˆ ρ (i). We ˆ τbc  x − m. Assume the now show that there exists a constant m ∈ ω such that  bc ˆ τ  x − m do not hold. Then contrary: let, for any constant m ∈ ω, the relation  there exists an infinite sequence n 1 , n 2 , . . . having the following properties: ˆ τbc , n i < n i+1 , and n i −  ˆ τbc (n i ) < n i+1 −  ˆ τbc (n i+1 ) for any i ∈ (a) n i ∈ Arg  ω \ {0}. ˆ τbc (n 1 ) ≥ 2. (b) n 1 −  Let us define a function ϕ : ω → ω. Let n ∈ ω. If n < n 1 , then ϕ(n) = 1. If ˆ τbc (n i ) − 1. Evidently, ϕ ∈ n i ≤ n < n i+1 for some i ∈ ω \ {0}, then ϕ(n) = n i −  F Z and Arg ϕ = ω. Using Lemma 10.3 we conclude that there exists a function g ∈ F R such that Arg g = R(0) and g(n) ≤ ϕ(n) for any n ∈ ω. Evidently, g(n i ) < ˆ τbc (n i ) for any i ∈ ω \ {0} and, hence,  ˆ τbc (n i ) < n i − g(n i ). Therefore the ni −  ˆ τbc  x − g does not hold which is impossible. Hence there exists a constant relation  ˆ τbc  x − m. By Proposition 10.6,  ˆ cb ˆ cb m ∈ ω such that  τ  x + m + 1. Hence τ  cb ˆ ρ (i) for any function x + f (x) for any function f ∈ F R. Thus, x + f (x) ∈ X f ∈ F R.  Lemma 15.4 Let ρ = (F, k) be a signature, (b, c) ∈ {i, d, a, s}2 , i ∈ {1, 2, 3, 4, ˆ X ρbc (i) = ∅. Then 5, 6}, Tp(i, b, c) = ω, Tp(i, b, c) = ε, Wˆ ρ (i) = ∅, and F R ∩  ˆ ρcb (i) = ∅. F R ∩ Y ˆ ρcb (i). Proof Assume that the considered statement does not hold. Let f ∈ F R ∩ Y By Proposition 15.2, tp(i, c, b) = ω and tp(i, c, b) = ε. Using Proposition 10.7 we ˆ X ρbc (i) which is impossible.  obtain f −1 ∈ F R ∩  Lemma 15.5 Let ρ = (F, k) be a signature, (b, c) ∈ {i, d, a, s}2 , i ∈ {1, 2, 3, 4, ˆ ρbc (i) = ∅. Then 5, 6}, Tp(i, b, c) = ω, Tp(i, b, c) = ε, Wˆ ρ (i) = ∅, and F R ∩ Y ˆ ρcb (i) = ∅. F R ∩ X

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15 Matrices of Lower Local Bounds

ˆ ρcb (i). Proof Assume that the considered statement does not hold. Let f ∈ F R ∩ X By Proposition 15.2, tp(i, c, b) = ω and tp(i, c, b) = ε. Using Proposition 10.7 we ˆ ρbc (i) which is impossible.  obtain f −1 − 1 ∈ F R ∩ Y The next statement is obvious. Lemma 15.6 Let ρ = (F, k) be a signature, (b, c) ∈ {i, d, a, s}2 , i ∈ {1, 2, 3, 4, ˆ ρbc (i) for any 5, 6}, tp(i, b, c) = ω, tp(i, b, c) = ε, and Wˆ ρ (i) = ∅. Then m ∈ Y constant m ∈ ω. Lemma 15.7 Let ρ = (F, k) be a signature with an infinite set F. Then ˆ ρid (2) 2(x+ f (x)) log2 k + 1 ∈ X ˆ ρid (2), 2 f (x) ∈ Y ˆ ρid (2) for any function f ∈ for any function f ∈ F R, 2x−3 ∈ / X x id ˆ ρ (2) for any constant m ∈ ω \ {0}. F R, and 2 m ∈ / Y x

Proof Let τ ∈ Wˆ ρ (2). By Proposition 13.3, Typ Nτ = ε and Typ Sτ = ε. Using Proposition 11.2 we conclude that there exist c1 , c2 ∈ ω such that log1 k log2 x − 2 ˆ τdi  c2 log2 x. One can show that 1 log2 x − c1 , c2 log2 x ∈ F R, c1   log2 k



1 log2 x − c1 log2 k

−1

= 2(x+c1 ) log2 k ,

(x+c1 ) log2 k ˆ id + 1. Hence and (c2 log2 x)−1 = 2 c2 . By Proposition 10.6, 2 c2   τ 2 x

x

ˆ ρid (2) 2(x+ f (x)) log2 k + 1 ∈ X ˆ ρid (2) for any function f ∈ F R. and 2 f (x) ∈ Y x−3 ˆ ρid (2). Let τ ∈ Wˆ ρ (2). Then  ˆ di . Evidently, 2x−3 ∈ Assume that 2x−3 ∈ X τ 2  x−3 −1 ˆ τdi  2 + log2 x. Therefore = 3 + log2 x. By Proposition 10.7,  F R and 2 ˆ ρdi (2) which contradicts Proposition 14.2. Hence 2x−3 ∈ ˆ ρid (2). 2 + log2 x ∈ Y / X x id ˆ ρ (2). Let τ ∈ Wˆ ρ (2). Assume that there exists m ∈ ω \ {0} such that 2 m ∈ Y  x −1 x x di ˆ τ  2 m . Evidently, 2 m ∈ F R and 2 m = m log2 x. By Proposition 10.7, Then  di di ˆ ˆ τ  m log2 x. Therefore m log2 x ∈  X ρ (2) which contradicts Proposition 14.2. x ˆ ρid (2) for any constant m ∈ ω \ {0}. Thus, 2 m ∈ / Y  x

Lemma 15.8 Let ρ = (F, k) be a signature with an infinite set F. Then ˆ ρid (3) 2(x+ f (x)) log2 k + 1 ∈ X ˆ ρid (3). for any function f ∈ F R, and 2x−4 − 1 ∈ / X

15.2 Auxiliary Statements

191

Proof Let τ ∈ Wˆ ρ (3). By Proposition 13.4, Typ Nτ = ω. Using Lemma 11.6 we ˆ τdi  1 log2 x − c. Eviconclude that there exists a constant c ∈ ω such that  log2 k  −1 1 1 = 2(x+c) log2 k . By Proposidently, log k log2 x − c ∈ F R and log k log2 x − c 2 2 (x+c) log2 k ˆ id ˆ ρid (3) for any function 10.6,  + 1. Hence 2(x+ f (x)) log2 k + 1 ∈ X τ 2 tion f ∈ F R. ˆ ρid (3). Let τ ∈ Wˆ ρ (3). Then 2x−4 − 1   ˆ id Assume that 2x−4 − 1 ∈ X τ . Evi  −1 = 4 + log2 (x + 1). Using Proposition dently, 2x−4 − 1 ∈ F R and 2x−4 − 1 ˆ τid  3 + log2 (x + 1). Therefore 3 + log2 (x + 1) ∈ Y ˆ ρdi (3) which 10.7 we obtain  ˆ ρid (3). contradicts Proposition 14.3. Hence 2x−4 − 1 ∈ / X 

15.3 Bounds Inside Types Let ρ = (F, k) be a signature. For each i ∈ {1, 2, 3, 4, 5, 6} and each pair (b, c) ∈ ˆ ρbc (i) {i, d, a, s}2 such that tp(i, b, c) = ω and tp(i, b, c) = ε, we study the sets X bc ˆ ρ (i). and Y ˆ ρii (1) for any Proposition 15.3 Let ρ = (F, k) be a signature. Then x + f (x) ∈ X ii ˆ ρ (1) for any constant m ∈ ω, and x ∈ Y ¨ ρii (1). function f ∈ F R, x + m ∈ / X Proof The considered statement follows from Proposition 14.1 and Lemmas 15.1– 15.3.  Proposition 15.4 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: ˆ ρii (2) for any ˆ ρii (2) for any function f ∈ F R, x + m ∈ / X (a) x + f (x) ∈ X ¨ ρii (2). constant m ∈ ω, and x ∈ Y (x+ f (x)) log2 k ˆ ρid (2) for any function f ∈ F R, 2x−3 ∈ ˆ ρid (2), (b) 2 + 1 ∈ X / X x x ˆ ρid (2) for any function f ∈ F R, and 2 m ∈ ˆ ρid (2) for any constant 2 f (x) ∈ Y / Y m ∈ ω \ {0}. ˆ ρdd (2) for any ˆ ρdd (2) for any function f ∈ F R, x + m ∈ / X (c) x + f (x) ∈ X dd ¨ ρ (2). constant m ∈ ω, and x ∈ Y Proof The considered statements follow from Proposition 14.2 and Lemmas 15.1– 15.7.  Proposition 15.5 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: ˆ ρii (3) for any ˆ ρii (3) for any function f ∈ F R, x + m ∈ / X (a) x + f (x) ∈ X ¨ ρii (3). constant m ∈ ω, and x ∈ Y (x+ f (x)) log2 k ˆ ρid (3) for any function f ∈ F R, 2x−4 − 1 ∈ ˆ ρid (3), (b) 2 + 1 ∈ X / X ¨ ρid (3). and x ∈ Y

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15 Matrices of Lower Local Bounds

ˆ ρdd (3) for any function f ∈ F R, x + m ∈ ˆ ρdd (3) for any (c) x + f (x) ∈ X / X ¨ ρdd (3). constant m ∈ ω, and x ∈ Y sa ˆ ρ (3) = ∅, F R ∩ Y ˆ ρsa (3) = ∅, and m ∈ Y ˆ ρsa (3) for any constant (d) F R ∩ X m ∈ ω. ¨ ρbc (3). ˆ ρbc (3) = ∅ and x ∈ Y (e) If bc ∈ {ia, da, aa, is, ds, as, ss}, then F R ∩ X Proof The considered statements follow from Proposition 14.3 and Lemmas 15.1– 15.6 and 15.8.  Proposition 15.6 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: ˆ ρii (4) for any function f ∈ F R, x + m ∈ ˆ ρii (4) for any (a) x + f (x) ∈ X / X ii ¨ ρ (4). constant m ∈ ω, and x ∈ Y id ˆ ¨ ρid (4). (b) F R ∩ X ρ (4) = ∅ and x ∈ Y dd ˆ ρ (4) = ∅ and x ∈ Y ¨ ρdd (4). (c) F R ∩ X ˆ ρad (4) = ∅, F R ∩ Y ˆ ρad (4) = ∅, and m ∈ Y ˆ ρad (4) for any constant (d) F R ∩ X m ∈ ω. ˆ ρsd (4) = ∅, and m ∈ Y ˆ ρsd (4) for any constant ˆ ρsd (4) = ∅, F R ∩ Y (e) F R ∩ X m ∈ ω. ˆ ρsa (4) = ∅, and m ∈ Y ˆ ρsa (4) for any constant ˆ ρsa (4) = ∅, F R ∩ Y (f) F R ∩ X m ∈ ω. ¨ ρbc (4). ˆ ρbc (4) = ∅ and x ∈ Y (g) If bc ∈ {ia, da, aa, is, ds, as, ss}, then F R ∩ X Proof The considered statements follow from Proposition 14.4 and Lemmas 15.1– 15.6.  Proposition 15.7 Let ρ = (F, k) be a signature with an infinite set F, and bc ∈ ˆ ρbc (5) for any function {ii, id, dd, ia, da, aa, is, ds, as, ss}. Then x + f (x) ∈ X ˆ ρbc (5) for any constant m ∈ ω, and x ∈ Y ¨ ρbc (5). f ∈ F R, x + m ∈ / X Proof The considered statement follows from Proposition 14.5 and Lemmas 15.1– 15.3.  Proposition 15.8 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: ˆ ρbc (6) for (a) If bc ∈ {ii, id, dd, ia, da, aa, is, ds, as, ss}, then x + f (x) ∈ X ˆ ρbc (6) for any constant m ∈ ω, and x ∈ Y ¨ ρbc (6). any function f ∈ F R, x + m ∈ / X ˆ ρad (6) for any function f ∈ F R, x + m ∈ ˆ ρad (6) for any (b) x + f (x) ∈ X / X ad ad ˆ ρ (6) = ∅, and m ∈ Y ˆ ρ (6) for any constant m ∈ ω. constant m ∈ ω, F R ∩ Y Proof The considered statements follow from Proposition 14.6 and Lemmas 15.1– 15.4 and 15.6. 

Reference

193

Reference 1. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity, Local approach. In: Peters, J.F., Skowron, A., (eds.) Transactions on Rough Sets IV, Lecture Notes in Computer Science, vol. 3700, pp. 125–143. Springer, Berlin (2005)

Chapter 16

Algorithmic Problems. Local Approach

In this chapter, we study algorithmic problems related to the local approach to the investigation of decision trees: problems of computation of the minimum complexity of deterministic, nondeterministic, and strongly nondeterministic decision trees, problems of construction of decision trees with the minimum complexity, and the problem of solvability of systems of equations over information systems. We study relationships among these problems and describe all variants of algorithmic problem behavior. We also discuss an algorithm for decision table construction and two algorithms that construct deterministic decision trees. Some results for deterministic decision trees considered in this chapter were published in [1, 2].

16.1 Relationships Among Algorithmic Problems A sccf-triple τ = (ρ, K , ψ) will be called a restricted enumerated sccf-triple if ρ is an enumerated signature and ψ is a computable restricted complexity function of the signature ρ. Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple, ρ = (F, k), and F = { f i : i ∈ ω}. Let b ∈ {d, a, s}. We define a set of schemes Στb in the following way: Στs = Σρ0−1 (K ) and Στd = Στa = Σρ . We now define two algorithmic ˆ b (τ ). ˆ b (τ ) and Des problems: Com b ˆ (τ ): for a given schema z ∈ Στb , it is required to compute the Problem Com b (z). value ψˆ ρ,K ˆ b (τ ): for a given schema z ∈ Στb , it is required to construct a Problem Des b b and ψ(Γ ) = ψˆ ρ,K (z). schema Γ ∈ Cρ such that (Γ, z) ∈ Rˆ ρ,K A word α ∈ Ωρ will be called τ -realizable if there exists an information system U = (AU , γU ) ∈ K such that the set AU (α) is nonempty. We now define an algorithmic problem E x(τ ).

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_16

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Problem E x(τ ): for a given word α ∈ Ωρ , it is required to verify if α is a τ realizable word. This problem is related to the solvability of systems of equations over information systems. An element f i ∈ F will be called τ -constant if there exists δ ∈ E k such that, for any system U ∈ K and for any element a ∈ AU , the equality γU ( f i )(a) = δ holds. A sccf-triple τ will be called degenerate if each element from F is τ -constant, and nondegenerate otherwise. Theorem 16.1 Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple. Then the following statements hold: ˆ d (τ ), Des ˆ d (τ ), (a) If the problem E x(τ ) is decidable, then the problems Com a a s s ˆ ˆ ˆ (τ ) are decidable. ˆ (τ ), Des (τ ), Com (τ ), and Des Com (b) If τ is a nondegenerate sccf-triple, and the problem E x(τ ) is undecidable, ˆ d (τ ), Com ˆ a (τ ), Des ˆ a (τ ), Com ˆ s (τ ), and Des ˆ s (τ ) ˆ d (τ ), Des then the problems Com are undecidable. (c) If τ is a degenerate sccf-triple and the problem E x(τ ) is undecidable, then ˆ a (τ ) are undecidable, and the problems Des ˆ s (τ ), ˆ d (τ ) and Des the problems Des d a s ˆ ˆ ˆ (τ ), Com (τ ), and Com (τ ) are decidable. Com Proof (a) Let the problem E x(τ ) be decidable. One can show that there exists an algorithm A which, for a given schema z ∈ Σρ , constructs the table Tρ (z, K ). Taking into account that the function ψ has the property Λ2 and using Proposition 8.6 we conclude that the problems Com d (ρ, ψ), Des d (ρ, ψ), Com a (ρ, ψ), Des a (ρ, ψ), Com s (ρ, ψ), and Des s (ρ, ψ) are decidable. Using Theorem 10.1 and the existence ˆ d (τ ), Com ˆ a (τ ), ˆ d (τ ), Des of the algorithm A we conclude that the problems Com ˆ s (τ ), and Des ˆ s (τ ) are decidable. ˆ a (τ ), Com Des (b) Let τ be a nondegenerate sccf-triple and the problem E x(τ ) be undecidˆ b (τ ) is undecidable. able. Let b ∈ {d, a, s}. We now show that the problem Com Assume the contrary. Let us show that the problem E x(τ ) is decidable. Since τ is a nondegenerate sccf-triple, there exists an element f i0 ∈ F which is not τ constant. Let α ∈ Ωρ . If α = λ, then, evidently, α is a τ -realizable word. Let α = λ. If α is inconsistent, then, evidently, α is not τ -realizable. Let α be a nonempty consistent word and α = ( f i1 , δi1 ) · · · ( f in , δin ). For each δ ∈ E k , we define a schema z δ = (ν, f i0 , f i1 , . . . , f in ) as follows: νδ : E kn+1 → {{0}, {1}} and, for any σ¯ ∈ E kn+1 , if σ¯ = (δ, δi1 , . . . , δin ), then νδ (σ¯ ) = {0}, and if σ¯ = (δ, δi1 , . . . , δin ), then νδ (σ¯ ) = {1}. Taking into account that the element f i0 is not τ -constant one can show ˆ b (τ ), for each that z δ ∈ Σρ0−1 (K ) ⊆ Στb . Using the decidability of the problem Com b δ ∈ E k , we find the value ψˆ ρ,K (z δ ). Taking into account that the function ψ has properties Λ3 and Λ4, and f i0 is an element which is not τ -constant one can show b (z δ ) = 0 for any δ ∈ E k . Thus, that the word α is not τ -realizable if and only if ψˆ ρ,K ˆ b (τ ) is the problem E x(τ ) is decidable which is impossible. Hence the problem Com undecidable. Taking into account that ψ is a computable function we conclude that ˆ b (τ ) is undecidable. the problem Des

16.1 Relationships Among Algorithmic Problems

197

(c) Let τ be a degenerate sccf-triple and the problem E x(τ ) is undecidable. Since τ is a degenerate sccf-triple, Nρ (T ) = 1 for any table T ∈ Mρ,K . Taking into account that the function ψ has the property Λ4 and using Theorem 10.1 we conclude that, for b (z) = 0 holds. Hence any b ∈ {d, a, s} and for any schema z ∈ Σρb , the equality ψˆ ρ,K d a s ˆ ˆ ˆ the problems Com (τ ), Com (τ ), and Com (τ ) are decidable. We denote by Γ0 the schema which contains the root, a terminal node labeled with the number 1, and the edge leaving the root and entering the terminal node. Evidently, for any problem ˆ s (τ ). Therefore schema z ∈ Σρ0−1 (K ), the schema Γ0 is a solution of the problem Des ˆ s (τ ) is decidable. Let b ∈ {d, a}. Let us show that the problem the problem Des ˆ b (τ ) is undecidable. Assume the contrary. Let us show that the problem E x(τ ) is Des decidable. Let α ∈ Ωρ . If α = λ, then, evidently, the word α is τ -realizable. Let α = λ and α = ( f i1 , δ1 ) · · · ( f in , δn ). We now define a mapping ν : E kn → {{0}, {1}}. Let σ¯ ∈ E kn . If σ¯ = (δ1 , . . . , δn ), then ν(σ¯ ) = {1}. If σ¯ = (δ1 , . . . , δn ), then ν(σ¯ ) = {0}. Set z = (ν, f i1 , . . . , f in ). Evidently, z δ ∈ Στb . Let Γ ∈ Cρ and the schema Γ is a ˆ b (τ ) for the problem schema z. Taking into account that solution of the problem Des the function ψ has properties Λ3 and Λ4, and τ is degenerate one can show that Γ consists of the root, a terminal node labeled with a number r and the edge leaving the root and entering the terminal node. One can show that r = 1 if and only if the word α is τ -realizable. Thus, the problem E x(τ ) is decidable which is impossible. ˆ b (τ ) is undecidable.  Hence the problem Des Theorem 16.2 Let ρ = (F, k) be an enumerated signature. Then the following statements hold: (a) There exists a restricted enumerated sccf-triple τ1 = (ρ, K 1 , ψ1 ) for which the problem E x(τ1 ) is decidable. (b) There exists a degenerate restricted enumerated sccf-triple τ2 = (ρ, K 2 , ψ2 ) for which the problem E x(τ2 ) is undecidable. (c) There exists a nondegenerate restricted enumerated sccf-triple τ3 = (ρ, K 3 , ψ3 ) for which the problem E x(τ3 ) is undecidable. Proof Let F = { f i : i ∈ ω}, and B be a subset of the set ω which is not recursive. We define sccf-triples τ1 = (ρ, K 1 , ψ1 ), τ2 = (ρ, K 2 , ψ2 ), and τ3 = (ρ, K 3 , ψ3 ) in the following way: ψ1 = ψ2 = ψ3 = h, K 1 = {U1 }, K 2 = {U2 }, and K 3 = {U3 }. We define the information system U1 of the signature ρ as follows: AU1 = {0} and f iU1 (0) = 0 for any i ∈ ω. We define the information system U2 of the signature ρ as follows: AU2 = {0} and, for any i ∈ ω, if i ∈ B, then f iU2 (0) = 1, and if i ∈ / B, U2 then f i (0) = 0. We define the information system U3 of the signature ρ as follows: AU3 = {0, 1}, f 0U3 (0) = 0, f 0U3 (1) = 1, and, for any i ∈ ω \ {0}, if i − 1 ∈ B, then f iU3 (0) = f iU3 (1) = 1, and if i − 1 ∈ / B, then f iU3 (0) = f iU3 (1) = 0. One can show that the problem E x(τ1 ) is decidable, τ2 is degenerate, and τ3 is nondegenerate. Assume that at least one of the problems E x(τ2 ) and E x(τ3 ) is decidable. Then, as it is not difficult to show, the set B is recursive which is impossible. 

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16.2 Algorithm for Table Tρ (z, K ) Construction Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple, ρ = (F, k), and F = { f i : i ∈ ω}. Let the problem E x(τ ) be decidable and Bτ be an algorithm solving this problem. We now describe an algorithm VB τ which, for a given schema z ∈ Σρ , constructs the table VB τ (z) = Tρ (z, K ). Algorithm VB τ . Step 1. For a given problem schema z, we construct the normal form υ(z) of the problem schema z. Let υ(z) = (ν, f i1 , . . . , f in ). We construct a tree containing exactly one node, mark this node by the word λ, and pass to the second step. Assume that t steps were made. We denote by D a finite directed tree with the root constructed in the step t. Step (t + 1). Let each terminal node in the tree D be labeled with a tuple from E kn . We define a table T ∈ Mρ as follows: dim T = n, μT (1) = f i1 , . . . , μT (n) = f in , the set Δ(T ) coincides with the set of tuples attached to terminal nodes of D, and νT is the restriction of the mapping ν to the set Δ(T ). Set VB τ (z) = T . The work of the algorithm VB τ is completed. Let, in the tree D, there be terminal nodes which are not labeled with tuples from E kn . We choose a terminal node w of the tree D which is labeled with a word α ∈ Ωρ . Let |α| = n and α = ( f i1 , δ1 ) · · · ( f in , δn ). Instead of the word α, we mark the node w by the tuple (δ1 , . . . , δn ) and pass to the step (t + 2). Let |α| < n and α = ( f i1 , δ1 ) · · · ( f ir , δr ) (if α = λ, then r = 0). Using the algorithm Bτ we find the set E(α) of numbers δ ∈ E k such that the word α( f ir +1 , δ) is τ -realizable. We remove the word α attached to w. For each δ ∈ E(α), we add to the tree D the node w(δ) and the edge leaving w and entering w(δ), and mark the node w(δ) by the word α( f ir +1 , δ). We pass to the step (t + 2). Remark 16.1 Evidently, the word λ is τ -realizable. One can show that if a word α ∈ Ωρ is τ -realizable and f i ∈ F, then there exists δ ∈ E k such that the word α( f ir +1 , δ) is τ -realizable. Therefore the set E(α), defined under the description of the step (t + 1), is nonempty. Proposition 16.1 Let ρ = (F, k) be an enumerated signature, τ = (ρ, K , ψ) be a restricted enumerated sccf-triple for which the problem E x(τ ) is decidable, and Bτ be an algorithm solving the problem E x(τ ). Then, for any problem schema z ∈ Σρ , the table VB τ (z) constructed by the algorithm VB τ coincides with the table Tρ (z, K ), the number of algorithm VB τ steps is at most 2 + dim Tρ (z, K )Nρ (Tρ (z, K )), and the number of algorithm VB τ calls to the algorithm Bτ is at most k dim Tρ (z, K )Nρ (Tρ (z, K )) . Proof Let z ∈ Σρ . Simple analysis of the algorithm VB τ shows that VB τ (z) = Tρ (z, K ).

16.2 Algorithm for Table Tρ (z, K ) Construction

199

Let t be the number of steps which the algorithm VB τ makes, when it constructs the table VB τ (z). We denote by D the tree constructed in the step (t − 1), and L(D) the number of nodes in the tree D. One can show that t = L(D) + 2. Evidently, the number of terminal nodes in the tree D coincides with Nρ (Tρ (z, K )), and in each complete path in the tree D there are exactly |P(z)| nodes. Taking into account that |P(z)| = dim Tρ (z, K ) we obtain L(D) ≤ dim Tρ (z, K )Nρ (Tρ (z, K )). Therefore the number of algorithm VB τ steps is at most 2 + dim Tρ (z, K )Nρ (Tρ (z, K )). Evidently, during each step (with the exception of the first and the last ones) the algorithm VB τ makes k calls to the algorithm Bτ . During the first and the last steps, the algorithm VB τ does not use the algorithm Bτ . Therefore the number of algorithm  VB τ calls to the algorithm Bτ is at most k dim Tρ (z, K )Nρ (Tρ (z, K )). Corollary 16.1 Let ρ = (F, k) be an enumerated signature, τ = (ρ, K , ψ) be a restricted enumerated sccf-triple for which the problem E x(τ ) is decidable, and there exist a constant m ∈ ω such that Z ρ (T ) ≤ m for any table T ∈ Mρ,K . Let Bτ be an algorithm solving the problem E x(τ ). Then, for any problem schema z ∈ Σρ , the number of algorithm VB τ steps during the process of the table Tρ (z, K ) construction is at most k 2m |P(z)|m+1 + 2. Proof Evidently, dim Tρ (z, K ) = |P(z)|. By Lemma 12.5,  m Nρ (Tρ (z, K )) ≤ k 2 dim Tρ (z, K ) . Using Proposition 16.1 we conclude that the number of steps is at most 2 + k 2m |P(z)|m+1 .



16.3 Algorithms for Construction of Decision Tree Schemes

Let ρ = (F, k) be an enumerated signature and A be an algorithm that, for a given table T ∈ Mρ , constructs a deterministic decision tree A(T ) for the table T . Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple for which the problem E x(τ ) is decidable and Bτ be an algorithm solving this problem. We now describe an algorithm Aτ which, for a given problem schema z ∈ Σρ , constructs a schema Aτ (z) ∈ Cρ that corresponds deterministically to the problem schema z relative to the class K . Algorithm Aτ . For a given problem schema z ∈ Σρ , we construct the decision table Tρ (z, K ) using the algorithm VB τ and the decision tree schema A(Tρ (z, K )) using the algorithm A. Then Aτ (z) = A(Tρ (z, K )).

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Using Theorem 10.1 we conclude that the schema Aτ (z) corresponds deterministically to the problem schema z relative to the class K . di di ˆ τ,A ˆ τ,A with Arg  ⊆ ω. Let n ∈ ω. Then We now define a function  di i ˆ τ,A  (n) = max{ψ(Aτ (z)) : z ∈ Σρ , ψˆ ρ,K (z) ≤ n} .

We denote by ψY the algorithm Yρ,Nρ ,R Hρ2 ,ψ (see Sect. 5.6). Proposition 16.2 Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple and ρ (F, k). Then the following statements hold: di ˆ τdi = ε, then there exists a constant c1 ∈ ω such that  ˆ τ,ψY ˆ τdi (a) If Typ   c1 . di ˆ τ,ψY ˆ τdi = λ, then there exist constants c2 , c3 ∈ ω such that  (b) If Typ   2 ˆ τdi + c3 . c2  di ˆ τ,ψY ˆ τdi = χ , then there exists a constant c4 ∈ ω such that  (c) If Typ   4 ˆ τdi . c4 

= +  

ˆ τdi = ε. By Proposition 11.2, there exist m 1 , m 2 ∈ ω \ {0} such Proof (a) Let Typ  ˆ τdi , z ∈ that Sψ (T ) ≤ m 1 and Nρ (T ) ≤ m 2 for any table T ∈ Mρ,K . Let n ∈ Arg  i Σρ , and ψˆ ρ,K (z) ≤ n. Set T = Tρ (z, K ). By Lemma 11.4, Mρ,ψ (T ) ≤ 2m 1 . Taking into account that the function ψ has properties Λ1–Λ4 and using Theorem 5.4 we   obtain ψ(Yρ,Nρ ,R Hρ2 ,ψ (T )) ≤ 4m 21 (1 + log2 m 2 )2 . Denote c1 = 4m 21 (1 + log2 m 2 )2 . di ˆ τ,ψY (n) is definite and is at most Using Theorem 10.1 we conclude that the value  di ˆ τ,ψY ˆ τdi (n) + c1 . Thus, the statement (a) holds. c1 . Therefore  (n) ≤  ˆ τdi = λ. Using Proposition 11.2 we conclude that there exist m ∈ (b) Let Typ  ˆ τdi \ {0, 1}, ω \ {0} such that Sψ (T ) ≤ m for any table T ∈ Mρ,K . Let n ∈ Arg  i ˆ z ∈ Σρ , and ψρ,K (z) ≤ n. Set T = Tρ (z, K ). By Lemma 11.4, Mρ,ψ (T ) ≤ 2m. Since ψ is a restricted complexity function, we have dim T ≤ n. By Lemma 11.5, Nρ (T ) ≤ (kn)m . Using Theorem 5.4 and the fact that ψ is restricted we obtain ψ(Yρ,Nρ ,R Hρ2 ,ψ (T )) ≤ (2m)2 (1 + m log2 (kn))2 . By Proposition 11.2, there exists a ˆ τdi (n) log2 k + r log2 k for any n ∈ Arg  ˆ τdi . constant r ∈ ω such that log2 n ≤  Using these inequalities one can show that there exist constants c2 and c3 from ω ˆ τdi (n))2 + c3 . Hence depending on m, k, and r such that ψ(Yρ,Nρ ,R Hρ2 ,ψ (T )) ≤ c2 ( di di 2 ˆ τ (n)) + c3 . Thus, the statement ˆ τ,ψY (n) is definite and is at most c2 ( the value  (b) holds. i ˆ τdi ,  ˆ τdi ≥ 1, z ∈ Σρ , and ψˆ ρ,K ˆ τdi = χ . Let n ∈ Arg  (z) ≤ n. Set (c) Let Typ  d T = Tρ (z, K ). By Theorem 3.1, Mρ,ψ (T ) ≤ ψρ (T ). By Theorem 10.1, Mρ,ψ (T ) ≤ ˆ τdi (n). Let z = (ν, f i1 , . . . , f im ). Let ν  : E km → P(ω) and, for any δ¯1 , δ¯2 ∈ E km ,  if δ¯1 = δ¯2 , then ν  (δ¯1 ) ∩ ν  (δ¯2 ) = ∅. Set z  = (ν  , f i1 , . . . , f im ) and T  = Tρ (z  , K ). One can see that T  is a diagnostic table. Taking into account that ψ is a restricted complexity function and using Proposition 3.2 we obtain ψρd (T  ) ≥ logk Nρ (T  ). Evii ˆ τdi (n). Evidently, Nρ (T  ) = dently, ψˆ ρ,K (z  ) ≤ n. By Theorem 10.1, logk Nρ (T  ) ≤ 

16.3 Algorithms for Construction of Decision Tree Schemes

201

ˆ τdi (n) log2 k. Using Theorem 5.4 and the fact that Nρ (T ). Therefore log2 Nρ (T ) ≤  ˆ τdi (n))2 (1 +  ˆ τdi (n) log2 k)2 ≤ ψ is restricted we obtain ψ(Yρ,Nρ ,R Hρ2 ,ψ (T )) ≤ ( 2 ˆ di 4 (2 log2 k) (τ (n)) . di ˆ τ,ψY ˆ τdi (n))4 , where c4 = Hence the value  (n) is definite and is at most c4 (   2 (2 log2 k) . Thus, the statement (c) holds.  Let ρ be an enumerated signature and ψ be a computable complexity function of the signature ρ. We now describe an algorithm ψU which, for a given table T ∈ Mρ , constructs a deterministic decision tree ψU (T ) for the table T . Algorithm ψU . Let T ∈ Mρ . We define a table Tˆ ∈ Mρ as follows: dim Tˆ = dim T , Δ(Tˆ ) = ¯ = {m}, where m is the minimum Δ(T ), and μTˆ ≡ μT . Let δ¯ ∈ Δ(Tˆ ). Then νTˆ (δ) ¯ number from the set νT (δ). Let us apply the algorithm Uρ,R Hρ2 ,ψ (see Sect. 4.6) to the table Tˆ . Then ψU (T ) = Uρ,R H 2 ,ψ (Tˆ ). ρ

It is clear that any deterministic decision tree for the table Tˆ is a deterministic decision tree for the table T . Therefore the schema ψU (T ) is a deterministic decision tree for the table T . Proposition 16.3 Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple and ρ (F, k). Then the following statements hold: di ˆ τ,ψU ˆ τdi ˆ τdi = ε, then there exists a constant c1 ∈ ω such that   (a) If Typ  c1 . di ˆ τ,ψU ˆ τdi = λ, then there exist constants c2 , c3 ∈ ω such that  (b) If Typ  ˆ τdi + c3 . c2  di ˆ τ,ψU ˆ τdi = χ , then there exists a constant c4 ∈ ω such that  (c) If Typ   3 ˆ τdi . c4  di ˆ τdi = χ and ψ = h, then  ˆ τ,ψU ˆ τdi . (d) If Typ  

= +  

Proof Let T ∈ Mρ and Δ(T ) = ∅. Evidently, Tˆ ∈ Mρ (1). Using Proposition 4.1, Corollary 4.2 and the fact that ψ is a restricted complexity function we obtain ψ(ψU (T )) ≤ 3(Mρ,ψ (Tˆ ))2 log2 Nρ (T ) .

(16.1)

ˆ τdi = ε. By Proposition 11.2, there exist m 1 , m 2 ∈ ω \ {0} such that (a) Let Typ  ˆ τdi . Let z ∈ Sψ (T ) ≤ m 1 and Nρ (T ) ≤ m 2 for any table T ∈ Mρ,K . Let n ∈ Arg  i ˆ ˆ Σρ and ψρ,K (z) ≤ n. Set T = Tρ (z, K ). Evidently, T ∈ Mρ,K . By Lemma 11.4, Mρ,ψ (T ) ≤ 2m 1 . By (16.1), ψ(ψU (T )) ≤ 12m 21 log2 m 2 . Denote   c1 = 12m 21 log2 m 2 . di di ˆ τ,ψU ˆ τ,ψU ˆ τdi + Evidently, the value  (n) is definite and is at most c1 . Therefore   c1 . Thus, the statement (a) holds.

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16 Algorithmic Problems. Local Approach

ˆ τdi = λ. By Proposition 11.2, there exists m ∈ ω such that Sψ (T ) ≤ (b) Let Typ  i ˆ τdi \ {0}, z ∈ Σρ , and ψˆ ρ,K m for any table T ∈ Mρ,K . Let n ∈ Arg  (z) ≤ n. Set T = Tρ (z, K ). Evidently, Tˆ ∈ Mρ,K . By Lemma 11.4, Mρ,ψ (Tˆ ) ≤ 2m. Since ψ is a restricted complexity function, we have dim T ≤ n. Using Lemma 11.5 we obtain Nρ (T ) ≤ (kn)m . By (16.1), ψ(ψU (T )) ≤ 12m 3 log2 n + 12m 3 log2 k. From Proposition 11.2 it follows that there exists a constant r ∈  ω such that log2 n ≤ ˆ τdi (n) log2 k + r log2 k for any n ∈ Arg  ˆ τdi . Denote c2 = 12m 3 log2 k and c3 =    ˆ τdi (n) + c3 . Hence the value 12m 3 log2 k + 12m 3r log2 k . Then ψ(ψU (T )) ≤ c2  di di ˆ τ (n) + c3 . Thus, the statement (b) holds. ˆ τ,ψU (n) is definite and is at most c2   i di ˆ τdi , z ∈ Σρ , and ψˆ ρ,K ˆ τ = χ . Let n ∈ Arg  (z) ≤ n. Set T = (c) Let Typ  Tρ (z, K ). Let z = (ν, f i1 , . . . , f im ). We now define a mapping ν1 : E km → P(ω). ¯ = {m}, where m is the minimum number from the set ν(δ). ¯ Let δ¯ ∈ E km . Then ν1 (δ) i (z 1 ) ≤ n. ThereSet z 1 = (ν1 , f i1 , . . . , f im ). Evidently, Tˆ = Tρ (z 1 , K ) and ψˆ ρ,K d ˆ τdi (n). By Theorem 3.1, Mρ,ψ (Tˆ ) ≤ ψρd (Tˆ ). From Theorem fore ψˆ ρ,K (z 1 ) ≤  d ˆ τdi (n). Let ν2 : 10.1 it follows that ψρd (Tˆ ) = ψˆ ρ,K (z 1 ). Therefore Mρ,ψ (Tˆ ) ≤  m m E k → P(ω) and, for any δ¯1 , δ¯2 ∈ E k , if δ¯1 = δ¯2 , then ν2 (δ¯1 ) ∩ ν2 (δ¯2 ) = ∅. Set z 2 = (ν2 , f i1 , . . . , f im ) and T2 = Tρ (z 2 , K ). One can show that T2 is a diagnostic table. Taking into account that ψ is a restricted complexity function and i (z 2 ) ≤ n. using Proposition 3.2 we obtain ψρd (T2 ) ≥ logk Nρ (T2 ). Evidently, ψˆ ρ,K di ˆ By Theorem 10.1, logk Nρ (T2 ) ≤ τ (n). Evidently, Nρ (T2 ) = Nρ (T ). Therefore  3 ˆ τdi (n) log2 k. By (16.1), ψ(ψU (T )) ≤ (3 log2 k)  ˆ τdi (n) . Denote log2 Nρ (T ) ≤   3   di ˆ τdi (n) . ˆ τ,ψU c4 = 3 log2 k . Evidently, the value  (n) is definite and is at most c4  Thus, the statement (c) holds. ˆ τdi = χ and ψ = h. By Proposition 11.2, Typ Sτ = ε. Let n ∈ (d) Let Typ  di ˆ τ , z ∈ Σρ , and hˆ iρ,K (z) ≤ n. Set T = Tρ (z, K ). Evidently, dim T ≤ n. Using Arg  the description of the algorithm Uρ,R Hρ2 ,h we conclude that, in each complete path in the schema hU (T ), nodes, which are not neither the root nor a terminal node, are labeled with pairwise different elements from the set P(T ). Therefore di ˆ τ,hU (n) is definite and is at most n. Using Corolh(hU (T )) ≤ n. Hence the value  di di ˆ τ,hU ˆ τdi (n). Thus, the statement ˆ (n) ≤  lary 11.1 we obtain τ (n) = n. Therefore  (d) holds. 

References 1. Moshkov, M.: Decision Trees. Theory and Applications (in Russian). Nizhny Novgorod University Publishers, Nizhny Novgorod (1994) 2. Moshkov, M.: Time complexity of decision trees. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III, Lecture Notes in Computer Science, vol. 3400, pp. 244–459. Springer, Berlin (2005)

Part III

Decision Trees for Problems. Global Approach

In this part, we develop global approach to the study of decision trees for problems, where decision trees can use arbitrary attributes from the considered information system. This part consists of nine chapters. In Chap. 17, we discuss main notions and notation for the global approach to the study of decision trees for problems. We consider different parameters of problems related to complexity of decision trees, relationships among these parameters, and upper and lower types of these relationships. In Chap. 18, we consider some reductions which will be used later in the investigations of decision trees in the framework of the global approach. We consider relationships among decision trees for problems and decision trees for decision tables. We prove that, instead of arbitrary classes of information systems, we can consider classes containing only one information system. We discuss some operations on sccf-triples, relationships between matrices of upper local and global bounds for sccf-triples, and possibilities to transfer results from one signature to another. In Chap. 19, for arbitrary and for the restricted sccf-triples τ , we study functions τbc , b, c ∈ {i, d, a, s}, located in the matrix of upper global bounds for the triple τ on the main diagonal and below. For each of these functions, we list all possible upper types and consider criterion for each such type (with the exception of the function τdi ). In several cases, we give bounds for the considered functions. In Chap. 20, for arbitrary and for the restricted sccf-triples τ , we study functions τbc , b, c ∈ {i, d, a, s}, located in the matrix of upper global bounds for the triple τ over the main diagonal. For each of these functions, we list all possible upper types and consider criterion for each such type. In several cases, we give upper and lower bounds for the considered functions. In Chap. 21, we describe all possible 11 global upper types Tp1, . . . , Tp11 of sccf-triples and all possible 10 global upper types Tp1, . . . , Tp10 of restricted sccftriples. For each of the last 10 global upper types, we consider the criterion of its implementation for restricted sccf-triples. In Chap. 22, for each i ∈ {1, . . . , 11}, we prove that there exists a sccf-triple τ such that Typ τ = Tpi. We also prove that, for each i ∈ {1, . . . , 10}, there exists a restricted sccf-triple τ such that Typ τ = Tpi. In Chap. 23, for a given signature ρ and each possible global upper type of restricted sccf-triples Tpi, i ∈ {1, . . . , 10}, we consider the set Wρ (i) of restricted

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sccf-triples τ with Typ τ = Tpi. For each pair (b, c) ∈ {i, d, a, s}2 such that in the matrix Tpi at the intersection of the row with index b and the column with index c either λ or χ stays, we study upper and lower bounds on the function τbc true for any sccf-triple τ ∈ Wρ (i). In Chap. 24, we describe all possible 11 global lower types tp1, . . . , tp11 of sccftriples which correspond to the global upper types Tp1, . . . , Tp11, respectively. We also describe all possible 10 global lower types tp1, . . . , tp10 of restricted sccf-triples which correspond to the global upper types Tp1, . . . , Tp10 . For a given signature ρ, each global lower type tpi, i ∈ {1, . . . , 10}, and each pair (b, c) ∈ {i, d, a, s}2 such that in the matrix tpi at the intersection of the row with index b and the column with index c either μ or γ stays, we study upper and lower bounds on the function bc τ true for any sccf-triple τ ∈ Wρ (i). In Chap. 25, we study algorithmic problems related to the global approach to the investigation of decision trees: problems of computation of the minimum complexity of deterministic, nondeterministic, and strongly nondeterministic decision trees, problems of construction of decision trees with the minimum complexity, and the problem of solvability of systems of equations over information systems. We study relationships among these problems. We also discuss the notion of a proper weighted depth for which the problems of computation of the minimum complexity of decision trees and problems of construction of decision trees with the minimum complexity are decidable if the problem of solvability of systems of equations over information systems is decidable.

Chapter 17

Basic Definitions and Notation

In this chapter, we discuss main notions and notation for the global approach to the study of decision trees for problems. We consider different parameters of problems related to complexity of decision trees, relationships among these parameters, and upper and lower types of these relationships. Note that the upper types of functions considered in this book were introduced in [2]. Types of functions slightly different from upper and lower types were investigated in [1, 3].

17.1 Complexity Functions Let ρ = (F, k) be a signature, K be a nonempty class of information systems of the signature ρ, and ψ be a complexity function of the signature ρ. d a s ⊆ Cρ × ρ , Rρ,K ⊆ Cρ × ρ , and Rρ,K ⊆ We now define the relations Rρ,K d 0−1 Then (Γ, z) ∈ Rρ,K if and only if the Cρ × ρ (K ). Let (Γ, z) ∈ Cρ × ρ . schema Γ corresponds deterministically to the problem schema z relative to the class a if and only if the schema Γ corresponds K . The pair (Γ, z) belongs to the set Rρ,K nondeterministically to the problem schema z relative to the class K . Let (Γ, z) ∈ s if and only if the schema Γ corresponds strongly Cρ × ρ0−1 (K ). Then (Γ, z) ∈ Rρ,K nondeterministically to the problem schema z relative to the class K . i d a : ρ → ω, ψρ,K : ρ → ω, ψρ,K : ρ → ω, We now define the functions ψρ,K s i 0−1 (z) = and ψρ,K : ρ (K ) → ω. Let z ∈ ρ and z = (ν, f 1 , . . . , f n ). Then ψρ,K d d a ψ( f 1 · · · f n ), ψρ,K (z) = min{ψ(Γ ) : (Γ, z) ∈ Rρ,K } and ψρ,K (z) = min{ψ(Γ ) : a s s (Γ, z) ∈ Rρ,K }. Let z ∈ ρ0−1 (K ). Then ψρ,K (z) = min{ψ(Γ ) : (Γ, z) ∈ Rρ,K }.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_17

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17.2 Matrices of Upper and Lower Global Bounds for Sccf-Triple Let ρ be a signature, K be a nonempty class of information systems of the signature ρ, and ψ be a complexity function of the signature ρ. Denote τ = (ρ, K , ψ). Let b, c ∈ {i, d, a, s}. We define partial functions Ψτbc : ω → ω and Φτbc : ω → ω as follows. Let n ∈ ω. If b = s and c = s, then b c (z) : z ∈ ρ , ψρ,K (z) ≤ n} , Ψτbc (n) = max{ψρ,K b c (z) : z ∈ ρ , ψρ,K (z) ≥ n} . Φτbc (n) = min{ψρ,K

If b = s or c = s, then b c (z) : z ∈ ρ0−1 (K ), ψρ,K (z) ≤ n} , Ψτbc (n) = max{ψρ,K b c (z) : z ∈ ρ0−1 (K ), ψρ,K (z) ≥ n} . Φτbc (n) = min{ψρ,K

We denote by Ψτ a matrix with four rows and four columns in which rows from the top to the bottom and columns from the left to the right are labeled with indices i, d, a, s, and at the intersection of the row with index b ∈ {i, d, a, s} and the column with index c ∈ {i, d, a, s} the function Ψτbc is placed. The matrix Ψτ will be called the matrix of upper global bounds for the triple τ . We denote by Φτ a matrix with four rows and four columns in which rows from the top to the bottom and columns from the left to the right are labeled with indices i, d, a, s, and at the intersection of the row with index b ∈ {i, d, a, s} and the column with index c ∈ {i, d, a, s} the function Φτbc is placed. The matrix Φτ will be called the matrix of lower global bounds for the triple τ .

17.3 Types of Sccf-Triples Let ρ be a signature, K be a nonempty class of information systems of the signature ρ, and ψ be a complexity function of the signature ρ. Denote τ = (ρ, K , ψ). We denote by Typ Ψτ the matrix with four rows and four columns in which rows from the top to the bottom and columns from the left to the right are labeled with indices i, d, a, s, and at the intersection of the row with index b ∈ {i, d, a, s} and the column with index c ∈ {i, d, a, s} the value Typ Ψτbc is placed. The matrix Typ Ψτ will be called the global upper type of the triple τ . We denote by typ Φτ the matrix with four rows and four columns in which rows from the top to the bottom and columns from the left to the right are labeled with indices i, d, a, s, and at the intersection of the row with index b ∈ {i, d, a, s} and the column with index c ∈ {i, d, a, s} the value typ Φτbc is placed. The matrix typ Φτ will be called the global lower type of the triple τ .

References

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References 1. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity, Global approach. Fundam. Inform. 25(2), 201–214 (1996) 2. Moshkov, M.: Unimprovable upper bounds on time complexity of decision trees. Fundam. Inform. 31(2), 157–184 (1997) 3. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity, Local approach. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets IV, Lecture Notes in Computer Science, vol. 3700, pp. 125–143. Springer, Berlin (2005)

Chapter 18

Some Reductions

In this chapter, we consider some reductions which will be used later in the investigations of decision trees in the framework of the global approach. We consider relationships among decision trees for problems and decision trees for decision tables. We prove that, instead of arbitrary classes of information systems, we can consider classes containing only one information system. We discuss some operations on sccf-triples, relationships between matrices of upper local and global bounds for sccf-triples, and possibilities to transfer results from one signature to another. These techniques were not considered previously [1, 2].

18.1 Problem Schemes and Decision Tables Let ρ = (F, k) be a signature, K be a nonempty class of information systems of the signature ρ, z = (ν, f 1 , . . . , f n ) ∈ Σρ , and B be a finite subset of the set F. We now define a problem schema α(z, B) ∈ Σρ . If B = ∅, then α(z, B) = z. Let B = ∅ and B = { f n+1 , . . . , f m }. Then α(z, B) = (γ , f 1 , . . . , f m ), where γ : E km → P(ω) and γ (δ1 , . . . , δm ) = ν(δ1 , . . . , δn ) for any (δ1 , . . . , δm ) ∈ E km . One can show that ϕ(z,U ) ≡ ϕ(α(z,B),U ) for any information system U ∈ K . Evidently, z ∈ Σρ0−1 (K ) if and only if α(z, B) ∈ Σρ0−1 (K ). Theorem 18.1 Let ρ = (F, k) be a signature, K be a nonempty class of information systems of the signature ρ, z ∈ Σρ , Γ ∈ Cρ , and B be a finite subset of the set F such that P(Γ ) ⊆ B ∪ P(z). Then the following statements hold: a if and only if (Γ, Tρ (α(z, B), K )) ∈ Rρa . (a) (Γ, z) ∈ Rρ,K d (b) (Γ, z) ∈ Rρ,K if and only if (Γ, Tρ (α(z, B), K )) ∈ Rρd . s 0−1 (c) If z ∈ Σρ (K ), then (Γ, z) ∈ Rρ,K if and only if (Γ, Tρ (α(z, B), K )) ∈ Rρs .

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_18

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a Proof Using Theorem 10.1 we conclude that (Γ, α(z, B)) ∈ Rρ,K if and only if a (Γ, Tρ (α(z, B), K )) ∈ Rρ . Using the relation ϕ(z,U ) ≡ ϕ(α(z,B),U ) , which holds for a if and only if (Γ, z) ∈ any system U ∈ K , we conclude that (Γ, α(z, B)) ∈ Rρ,K a a Rρ,K . Therefore (Γ, z) ∈ Rρ,K if and only if (Γ, Tρ (α(z, B), K )) ∈ Rρa . Thus, the statement (a) holds. The statements (b) and (c) can be proved in the same way. Note that for the statement (c) proof we must use the following fact: z ∈ Σρ0−1 (K ) if and  only if α(z, B) ∈ Σρ0−1 (K ).

18.2 Arbitrary Classes of Information Systems and One-Element Classes of Information Systems Let ρ = (F, k) be a signature and K be a nonempty class of information systems of the signature ρ. The one-element class K˜ = {V (K )} is defined in the same way as in Sect. 10.2. Theorem 18.2 Let ρ = (F, k) be a signature and K be a nonempty class of informad d a a s s = Rρ, , Rρ,K = Rρ, , and Rρ,K = Rρ, . tion systems of the signature ρ. Then Rρ,K K˜ K˜ K˜ Proof Let b ∈ {d, a, s}, Γ ∈ Cρ , z ∈ Σρ , and if b = s, then z ∈ Σρ0−1 (K ). From b Theorem 18.1 it follows that (Γ, z) ∈ Rρ,K if and only if (Γ, Tρ (α(z, P(Γ )), K )) ∈ b b Rρ . By Theorem 18.1, (Γ, z) ∈ Rρ, K˜ if and only if (Γ, Tρ (α(z, P(Γ )), K˜ )) ∈ Rρb . One can show that Tρ (α(z, P(Γ )), K ) = Tρ (α(z, P(Γ )), K˜ ). Therefore (Γ, z) ∈ b b if and only if (Γ, z) ∈ Rρ, .  Rρ,K K˜ Corollary 18.1 Let ρ = (F, k) be a signature, K be a nonempty class of information systems of the signature ρ, ψ be a complexity function of the signature ρ, τ = (ρ, K , ψ), and τ˜ = (ρ, K˜ , ψ). Then Ψτ = Ψτ˜ and Φτ = Φτ˜ .

˜ 18.3 Set R ˜ = R ∪ {∅, ∞}, where R is the set of real numbers, and ∅ and ∞ are Denote R ˜ as follows. Let b, c ∈ R. ˜ special symbols. We define a linear order < ˜ on the set R ˜ ∪ {(∅, a), (∅, ∞), (a, ∞) : a ∈ Then bc ˜ means that the relation cm. that the considered inequality holds.  Let z ∈ Σρ and z = (ν, f 1 , . . . , f n ). Let t ∈ {1, 2}. We now define a tuple z (t) . Set α = f 1 · · · f n . If α (t) = λ, then z (t) = (ν). Let α (t) = λ and α = f i1 · · · f im . Then z (t) = (ν (t) , f i1 , . . . , f im ), where ν (t) : E 2m → P(ω) and, for any δ¯ = (δ1 , . . . , δm ) ∈ ¯ = ν(σ¯ ) holds, where the tuple σ¯ = (σ1 , . . . , σn ) is obtained E 2m , the equality ν (t) (δ) from the tuple δ¯ in the following way. Let l ∈ {1, . . . , n}. If l = i j for some j ∈ / {i 1 , . . . , i m }. If fl = p, then σl = 0, and if fl = p, {1, . . . , m}, then σl = δ j . Let l ∈ then  0, if t = 1 , σl = 1, if t = 2 . / Σρt , and if α (t) = λ, then z (t) ∈ Σρt . Evidently, if α (t) = λ, then z (t) ∈ Let z ∈ Σρ , t ∈ {1, 2}, and b ∈ {d, a, s}. We now define a value Btb (z) ∈ {∅} ∪ ω. Let b ∈ {d, a}. Then

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 Btb (z) =

∅,

ψtb (z (t) ),

/ Σρt , if z (t) ∈ if z (t) ∈ Σρt .

Let b = s. Then  Bts (z)

=

∅,

ψts (z (t) ),

/ Σρ0−1 ({Ut }) , if z (t) ∈ t (t) if z ∈ Σρ0−1 ({Ut }) . t

  ˜ Lemma 18.3 (a) Let z ∈ Σρ and b ∈ {d, a}. Then ψ b (z) t, we have, evidently, |α| ≥ 2. Let α = ( f 1 , δn ) · · · and ψρ,K s (z β ) < n. ( f m , δm ). Set β = ( f 1 , δn ) · · · ( f m−1 , δm−1 ). Evidently, β ∈ Ωt and ψρ,K Taking into account that ψ is a restricted complexity function one can show that s s s (z α ) ≤ ψρ,K (z β ) + t. Therefore ψρ,K (z β ) ≥ n − t. Hence Ψτss (n) ≥ n − t.  ψρ,K Lemma 19.12 Let ρ = (F, k) be a signature, τ = (ρ, K , ψ) be a restricted sccftriple, Typ Nτ = ε, and Typ Nτ = ω. Then Typ Pτ = χ . s Proof Assume that Typ Pτ = ε. Then there exists c ∈ ω such that ψρ,K (z α ) ≤ c for any word α ∈ Ωτ . Let t ∈ ω. Since Typ Nτ = ε, there exists a table T ∈ Mρ,K such that Nρ (T ) ≥ t. Let dim T = n and μT (1) = f 1 , . . . , μT (n) = f n . For each δ¯ = ¯ = ( f 1 , δ1 ) · · · ( f n , δn ). Let δ¯ ∈ Δ(T ), (Γδ¯ , z α(δ) (δ1 , . . . , δn ) ∈ Δ(T ), set α(δ) ¯ )∈  s l P(Γ ) = {ϕ , . . . , ϕ }. Set z = (ν , ϕ , . . . , ϕl ) Rρ,K , and ψ(Γδ¯ ) ≤ c. Let δ∈Δ(T ¯ ¯ 1 l 1 δ ) k and T = Tρ (z, K ). One can see that Nρ (T ) ≥ Nρ (T ) ≥ t. Taking into account that ψ is a restricted complexity function we obtain m ψ (T ) ≤ c. Since t is an arbitrary number from ω, we have Nτ (c) = ∞ which is impossible. Hence Typ Pτ = ε. Let us show that Typ Pτ = ω. Let n ∈ ω, α ∈ Ωτ , and m ψ (α) ≤ n. Set T = Tρ (z α , K ). Taking into account that ψ is a restricted complexity function and using Theorem s (z α ) ≤ 6.2 one can show that ψρs (T ) ≤ n Nρ (T ) ≤ n Nτ (n). By Theorem 18.1, ψρ,K n Nτ (n). Therefore Pτ (n) ≤ n Nτ (n) and Typ Pτ = ω. Using Lemma 19.10 we obtain  Typ Pτ = χ .

Proposition 19.4 Let ρ = (F, k) be a signature and τ = (ρ, K , ψ) be a sccf-triple. Then the following statements hold: (a) There exists m ∈ ω such that, for any n ∈ ω, if n ≥ m, then the values Ψτai (n), ad Ψτ (n), Ψτaa (n), Ψτsi (n), Ψτsd (n), Ψτsa (n), and Ψτss (n) are definite, and Ψτai (n) = Ψτad (n) = Ψτaa (n) = Ψτsi (n) = Ψτsd (n) = Ψτsa (n) = Ψτss (n). (b) For any function ϕ ∈ {Ψτai , Ψτad , Ψτaa , Ψτsi , Ψτsd , Ψτsa , Ψτss }, the following four statements hold: (b1) ϕ(n) ≤ n for any n ∈ Arg ϕ. (b2) Typ ϕ ∈ {ε, χ }. s (z) ≤ c (b3) Typ ϕ = ε if and only if there exists a constant c ∈ ω such that ψρ,K for any problem schema z ∈ Σρ0−1 (K ). (b4) If τ is a restricted sccf-tuple, then Typ ϕ = ε if and only if Typ Pτ = ε. Proof Statement (a) follows from Lemma 19.9. Statement (b1)—from Lemma 19.1. Statements (b2) and (b3) follow from Lemmas 19.9 and 19.4. Let τ be a restricted sccf-triple. Let Typ ϕ = ε. By Lemma 19.9, Typ Ψτss = ε. From here and from Lemma 19.4 it follows that Typ Pτ = ε. Let Typ Pτ = ε. Then s (z α ) ≤ t for any α ∈ Ωτ . Let z ∈ Σρ , there exists a constant t ∈ ω such that ψρ,K

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T = Tρ (z, K ), dim T = n, and μT (1) = f 1 , . . . , μT (n) = f n . One can show that a s a (z) ≤ max{ψρ,K (z ( f1 ,δ1 )···( fn ,δn ) ) : (δ1 , . . . , δn ) ∈ Δ(T )}. Therefore ψρ,K (z) ≤ ψρ,K aa  t. Hence Typ Ψτ = ε. Using Lemma 19.9 we obtain Typ ϕ = ε.

References 1. Alsolami, F., Azad, M., Chikalov, I., Moshkov, M.: Decision and Inhibitory Trees and Rules for Decision Tables with Many-valued Decisions. Intelligent Systems Reference Library, vol. 156. Springer, Cham (2020) 2. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity, Global approach. Fundam. Inform. 25(2), 201–214 (1996) 3. Moshkov, M.: Time complexity of decision trees. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III, Lecture Notes in Computer Science, vol. 3400, pp. 244–459. Springer, Berlin (2005) 4. Moshkov, M., Zielosko, B.: Combinatorial Machine Learning - A Rough Set Approach. Studies in Computational Intelligence, vol. 360. Springer, Berlin (2011)

Chapter 20

Functions over Main Diagonal

In this chapter, for arbitrary and for the restricted sccf-triples τ , we study functions Ψτbc , b, c ∈ {i, d, a, s}, located in the matrix of upper global bounds for the triple τ over the main diagonal. For each of these functions, we list all possible upper types and consider criterion for each such type. In several cases, we give upper and lower bounds for the considered functions. Some similar results were obtained in [1] for types of functions different from the upper types considered in this book.

20.1 Functions Ψτi d , Ψτi a , and Ψτi s Proposition 20.1 Let ρ = (F, k) be a signature, τ = (ρ, K , ψ) be a sccf-triple, and ϕ ∈ {Ψτid , Ψτia , Ψτis }. Then Typ ϕ ∈ {ε, ω} and the following statements hold: (a) Typ ϕ = ε if and only if there exists a constant c ∈ ω such that ψ(α) ≤ c for any α ∈ F ∗ . (b) If τ is a restricted sccf-triple, then Typ ϕ = ω and Arg ϕ = ∅. Proof Let ϕ = Ψτib , where b ∈ {d, a, s}. Denote B = {ψ(α) : α ∈ F ∗ }. Let the set B be finite. Then, evidently, there exists i (z) ≤ r for any problem schema z ∈ Στb . By Lemma a constant r ∈ ω such that ψρ,K 19.3, Typ ϕ = ε. Let B be infinite. Let α ∈ F ∗ and α = λ. We now correspond a problem schema σα ∈ Στb to the word α. Let α = f 1 · · · f n . Then σα = (νkn , f 1 , . . . , f n ), where i b (σα ) = ψ(α), and ψρ,K (σα ) ≤ ψ(λ). νkn : E kn → {{1}}. Evidently, σα ∈ Στb , ψρ,K Therefore ϕ(ψ(λ)) = ∞. Hence Typ ϕ ∈ {ε, ω} and the statement (a) holds. Let τ be a restricted sccf-triple and f ∈ F. For n ∈ ω \ {0}, we denote by z n the tuple (νkn , f, . . . , f ), where f repeats exactly n times. Evidently, z n ∈ Στb . Taking i (z n ) ≥ n and into account that ψ is a restricted complexity function we obtain ψρ,K b ψρ,K (z n ) ≤ 0. Hence ϕ(0) = ∞. Therefore Typ ϕ = ω and Arg ϕ = ∅.  © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_20

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20.2 Function Ψτas Lemma 20.1 Let ρ = (F, k) be a signature and τ = (ρ, K , ψ) be a restricted sccftriple. Then the following statements hold: (a) For any n ∈ ω, if n ∈ Arg Pτ , then n ∈ Arg Ψτas and Pτ (n) = Ψτas (n). (b) For any n ∈ ω, if Pτ (n) = ∞, then Ψτas (n) = ∞. Proof (a) Let n ∈ Arg Pτ and Pτ (n) = t. Let t = 0. Evidently, if Ψτas (n) = ∞, then the value Ψτas (n) is definite and Ψτas (n) ≥ t = 0. Let t > 0. Then there exists s (z α ) = t. Let α = ( f 1 , δ1 ) · · · ( f m , δm ). Set α ∈ τ such that m ψ (α) ≤ n and ψρ,K

z = (ν, f 1 , . . . , f m ), where, for any σ¯ ∈ E km , if σ¯ = (δ1 , . . . , δm ), then ν(σ¯ ) = {0} and, if σ¯ = (δ1 , . . . , δm ), then ν(σ¯ ) = {1}. Taking into account that t > 0 and ψ is a restricted complexity function one can show that z ∈ Σρ0−1 (K ). It is not difficult to s a s (z ) ≤ n and ψρ,K (z ) ≥ ψρ,K (z ) = t. Hence if Ψτas (n) = ∞, then prove that ψρ,K as as n ∈ Arg Ψτ and Ψτ (n) ≥ t. s (z) We now show that Ψτas (n) = ∞ and Ψτas (n) ≤ Pτ (n). Let z ∈ Σρ0−1 (K ), ψρ,K s ≤ n, (Γ, z) ∈ Rρ,K , and ψ(Γ ) ≤ n. Let P(Γ ) = { f 1 , . . . , f m }. Then, evidently, there exists a mapping ν : E km → {{0}, {1}} such that, for the problem schema z = (ν, f 1 , . . . , f m ) for any information system U ∈ K , the relation ϕ(z,U ) ≡ ϕ(z ,U ) a s holds. Set T = Tρ (z , K ). One can show that ψρ,K (z ) ≤ max{ψρ,K (z ( f1 ,δ1 )···( fm ,δm ) ) : (δ1 , . . . , δm ) ∈ Δ(T )}. Since ψ(Γ ) ≤ n and ψ is a restricted complexity funca (z ) ≤ Pτ (n). Evition, we have ψ( f i ) ≤ n for any f i ∈ P(Γ ). Therefore ψρ,K a a

as as dently, ψρ,K (z) = ψρ,K (z ). Therefore Ψτ (n) = ∞ and Ψτ (n) ≤ Pτ (n). Hence Ψτas (n) = Pτ (n). Thus, the statement (a) holds. s (z α ) > 0. In (b) Let n ∈ ω and Pτ (n) = ∞. Let α ∈ τ , m ψ (α) ≤ n, and ψρ,K the same way as it was made under the proof of the statement (a) one can show that s a (z ) ≤ n and ψρ,K (z ) ≥ there exists a problem schema z ∈ Σρ0−1 (K ) such that ψρ,K s as  ψρ,K (z α ). Therefore Ψτ (n) = ∞. Proposition 20.2 Let ρ = (F, k) be a signature and τ = (ρ, K , ψ) be a sccf-triple. Then Typ Ψτas ∈ {ε, χ , ω}, and if τ is a restricted sccf-triple, then the following statements hold: (a) Typ Ψτas = ε if and only if Typ Pτ = ε. (b) Typ Ψτas = χ if and only if Typ Pτ = χ . (c) Typ Ψτas = ω if and only if Typ Pτ = ω. (d) If Typ Ψτas = χ , then there exists m ∈ ω such that, for any n ∈ ω, if n ≥ m, then the values Ψτas (n) and Pτ (n) are definite and Ψτas (n) = Pτ (n). Proof Let Typ Ψτas = ε and Typ Ψτas = ω. Then, evidently, Typ Ψτaa = ε. By Lemma 19.9, Typ Ψτss = ε. From here and from Proposition 19.4 it follows that Typ Ψτss = χ . By Lemma 19.2, Typ Ψτas = χ . Hence Typ Ψτas ∈ {ε, χ , ω}. Let τ be a restricted sccf-triple. Using Lemmas 19.10 and 20.1 we obtain Typ Ψτas = Typ Pτ . Thus, the statements (a), (b), and (c) hold. Let Typ Ψτas = χ . Then Typ Pτ = χ and there exists m ∈ ω such that Arg Pτ = ω(m). Using Lemma 20.1 we conclude that ω(m) ⊆ Arg Ψτas and Ψτas (n) = Pτ (n) for any n ∈ ω(m). 

20.3 Function Ψτds

235

Corollary 20.1 Let τ = (ρ, K , ψ) be a restricted sccf-triple and Typ Ψτas = χ . A polynomial p1 exists such that Ψτas (n) ≤ p1 (n) for any n ∈ Arg Ψτas if and only if a polynomial p2 exists such that Pτ (n) ≤ p2 (n) for any n ∈ Arg Pτ .

20.3 Function Ψτds Let ρ = (F, k) be a signature and τ = (ρ, K , ψ) be a restricted sccf-triple. We define a function Rτ with Arg Rτ ⊆ ω in the following way: for any n ∈ ω, d (z) : z ∈ Σρ0−1 (K ), m ψ (z) ≤ n} . Rτ (n) = max{ψρ,K

Lemma 20.2 Let ρ = (F, k) be a signature, τ = (ρ, K , ψ) be a restricted sccftriple, Typ Rτ = ε and Typ Rτ = ω. Then Typ Pτ = ε. Proof Assume that Typ Pτ = ε. Using Proposition 19.4 we conclude that there exists s (z) ≤ c for any problem schema z ∈ Σρ0−1 (K ). Let a constant c ∈ ω such that ψρ,K s 0−1 z ∈ Σρ (K ), (Γ, z) ∈ Rρ,K , and ψ(Γ ) ≤ c. Let P(Γ ) = { f 1 , . . . , f m }. One can show that there exists a mapping ν : E km → {{0}, {1}} such that, for the problem schema z = (ν, f 1 , . . . , f m ) for any information system U ∈ K , the relation ϕ(z,U ) ≡ ϕ(z ,U ) holds. Evidently, z ∈ Σρ0−1 (K ). Since ψ is a restricted complexity function d d and ψ(Γ ) ≤ c, we have m ψ (z ) ≤ c. Evidently, ψρ,K (z ) = ψρ,K (z). Taking into account that Typ Rτ = ε and z is an arbitrary problem schema from Σρ0−1 (K ) we  conclude that Rτ (c) = ∞ which is impossible. Therefore Typ Pτ = ε. Lemma 20.3 Let ρ = (F, k) be a signature, τ = (ρ, K , ψ) be a restricted sccftriple, Typ Nτ = ω, and Typ Rτ = ω. Then Typ Z τ = ω. Proof Let t ∈ ω and Nτ (t) = ∞. Taking into account that Typ Rτ = ω one can show that t ∈ Arg Rτ . Let Rτ (t) = c. Assume that Typ Z τ = ω. Then, as it is not difficult to show, c ∈ Arg Z τ . Let Z τ (c) = m, T ∈ Mρ,K , and m ψ (T ) ≤ t. Let dim T = n, Nρ (T ) ≥ 2, and μT (1) = f 1 , . . . , μT (n) = f n . Evidently, there exist mappings ν1 , . . . , ν log2 Nρ (T ) having the following properties: ν j : E kn → {{0}, {1}},    z j = (ν j , f 1 , . . . , f n ) ∈ Σρ0−1 (K ) for any j ∈ 1, . . . , log2 Nρ (T ) , and, for any    δ¯1 , δ¯2 ∈ Δ(T ), if δ¯1 = δ¯2 , then there exist j1 , j2 ∈ 1, . . . , log2 Nρ (T ) such    that ν j1 (δ¯1 ) = ν j2 (δ¯2 ). Evidently, for any j ∈ 1, . . . , log2 Nρ (T ) , the inequality d (z j ) and m ψ (z j ) ≤ t holds and there exists a schema Γ j such that (Γ j , z j ) ∈ Rρ,K ψ(Γ j ) ≤ c. Taking into account that ψ is a restricted complexity function we obtain    log2 Nρ (T ) h(Γ j ) ≤ c. Therefore  P(Γ j ) ≤ ck c−1 . Let P(Γ j ) = { f i1 , . . . , f il }. j=1   c−1 l Evidently, l ≤ log2 Nρ (T ) ck . Set zˆ = (νk , f i1 , . . . , f il ) and Tˆ = Tρ (ˆz , K ). Evidently, m ψ (Tˆ ) ≤ c. One can show that Nρ (Tˆ ) ≥ Nρ (T ). By Lemma 12.5,  Z (Tˆ ) Nρ (Tˆ ) ≤ k 2 l ρ . Taking into account that m ψ (Tˆ ) ≤ c we obtain Z ρ (Tˆ ) ≤ m. m m     Nρ (T ) ≤ ck c+1 Therefore Nρ (Tˆ ) ≤ k 2 log2 Nρ (T ) ck c−1 . Hence

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 m 1 + log2 Nρ (T ) . Evidently, there exists a constant d ∈ ω such that, for any r ∈ ω,  m  m if r > d, then r > ck c+1 1 + log2 r . Therefore Nρ (T ) ≤ d. Hence Nτ (t) ≤ d  which is impossible. Thus, Typ Z τ = ω. Proposition 20.3 Let ρ = (F, k) be a signature and τ = (ρ, K , ψ) be a sccf-triple. Then Typ Ψτds ∈ {ε, χ , ω} and, if τ is a restricted sccf-triple, then the following statements hold: (a) Typ Ψτds = ε if and only if Typ Rτ = ε. (b) Typ Ψτds = χ if and only if Typ Rτ = ε and Typ Rτ = ω. (c) Typ Ψτds = ω if and only if Typ Rτ = ω. (d) If Typ Ψτds = χ , then there exists m ∈ ω such that, for any n ∈ ω(m), values Rτ (n), Ψτds (n), and Pτ (n) belong to ω, and the following inequalities hold: Ψτds (n) ≤ Rτ (n) ≤ Ψτds (Pτ (n)) . Proof Let Typ Ψτds = ε and Typ Ψτds = ω. Then, evidently, Typ Ψτss = ε. By Proposition 19.4, Typ Ψτss = χ . Using Lemma 19.2 we obtain Typ Ψτds = λ. Hence Typ Ψτds = χ . Thus, Typ Ψτds ∈ {ε, χ , ω}. Let τ be a restricted sccf-triple. We now prove the following statement: (a*) If Typ Rτ = ε, then Typ Ψτds = ε. d (z) ≤ c for any Let Typ Rτ = ε. Then there exists a constant c such that ψρ,K 0−1 ds problem schema z ∈ Σρ (K ). By Lemma 19.3 , Typ Ψτ = ε. Thus, the statement (a*) holds. We now prove the following statement: (b*) If Typ Rτ = ε and Typ Rτ = ω, then Typ Ψτds = χ and there exists m ∈ ω such that, for any n ∈ ω(m), values Rτ (n), Ψτds (n), and Pτ (n) belong to the set ω and the following inequalities hold: Ψτds (n) ≤ Rτ (n) ,

(20.1)

Rτ (n) ≤ Ψτds (Pτ (n)) .

(20.2)

Let Typ Rτ = ε and Typ Rτ = ω. Since Typ Rτ = ω, there exists m ∈ ω such s s (z) ≤ n, (Γ, z) ∈ Rρ,K , and that Arg Rτ = ω(m). Let n ∈ ω(m), z ∈ Στs , ψρ,K d ψ(Γ ) ≤ n. We now show that ψρ,K (z) ≤ Rτ (n). Let P(Γ ) = ∅. Then, as it is not d . Taking into account that ψ is a restricted complexity difficult to show, (Γ, z) ∈ Rρ,K d function we obtain ψρ,K (z) = 0. Evidently, Rτ (n) ≥ 0. Let P(Γ ) = ∅ and P(Γ ) = { f 1 , . . . , fr }. Then, evidently, there exists a mapping ν : E kr → {{0}, {1}} such that, for the problem schema z = (ν, f 1 , . . . , fr ), ϕ(z,U ) ≡ ϕ(z ,U ) for any system U ∈ K . Evidently, z ∈ Z τs . Taking into account that ψ is a restricted complexity function d (z ) ≤ we obtain m ψ (z ) ≤ n. Since Typ Rτ = ω, we have n ∈ Arg Rτ . Hence ψρ,K d d d

Rτ (n). Evidently, ψρ,K (z ) = ψρ,K (z). Therefore ψρ,K (z) ≤ Rτ (n). One can show that Ψτds (n) = ∅. Hence n ∈ Arg Ψτds (n) and Ψτds (n) ≤ Rτ (n). Thus, for any n ∈ ω(n), the values Ψτds (n) and Rτ (n) belong to the set ω, and the inequality (20.1) holds.

20.3 Function Ψτds

237

Therefore, Typ Ψτds = ω. By Lemma 20.2, Typ Pτ = ε. From here and from Proposition 19.4 it follows that Typ Ψτss = χ . Using Lemma 19.2 we obtain Typ Ψτds = ε. Hence Typ Ψτds = χ . Let n ∈ ω(m). Then, evidently, Pτ (n) = ∅. Assume that Pτ (n) = ∞. Then, using Proposition 20.2, we obtain Typ Ψτas = ω. From this equality and from Lemma 19.2 it follows that Typ Ψτds = ω which is impossible. Therefore Pτ (n) ∈ ω. Let z ∈ Σρ0−1 (K ) and m ψ (z) ≤ n. Set T = Tρ (z, K ). Let dim T = r and μT (1) = f 1 , . . . , μT (r ) = fr . One can show that s s (z) ≤ max{ψρ,K (z( f 1 , δ1 ) · · · ( fr , δr )) : ψρ,K

(δ1 , . . . , δr ) ∈ Δ(T ), νT ((δ1 , . . . , δr )) = {1}} ≤ Pτ (n) . d (z) ≤ Ψτds (Pτ (n)). Hence Rτ (n) ≤ Ψτds (Pτ (n)). Thus, the statement Therefore ψρ,K (b*) holds. We now prove the following statement: (c*) If Typ Rτ = ω, then Typ Ψτds = ω. Let Typ Rτ = ω. Then there exists t ∈ ω such that Rτ (t) = ∞. Evidently, Pτ (t) = ∅. Let Pτ (t) = ∞. By Proposition 20.2, Typ Ψτas = ω. Using Lemma 19.2 we obtain Typ Ψτds = ω. Let Pτ (t) ∈ ω and Pτ (t) = c. Let z ∈ Σρ0−1 (K ) and m ψ (z) ≤ t. As s (z) ≤ Pτ (t) ≤ c. Hence in the proof of the statement (b*), one can show that ψρ,K ds ds Ψτ (c) = ∞ and Typ Ψτ = ω. Thus, the statement (c*) holds. Let Typ Rτ = ε. Using the statement (a*) we obtain Typ Ψτds = ε. Let Typ Ψτds = ε. By the statement (b*), Typ Rτ = ε or Typ Rτ = ω. Using the statement (c*) we obtain Typ Rτ = ω. Therefore Typ Rτ = ε. Thus, the statement (a) holds. Let Typ Rτ = ε and Typ Rτ = ω. Using the statement (b*) we obtain Typ Ψτds = χ . Let Typ Ψτds = χ . By the statements (a*) and (c*), Typ Rτ = ε and Typ Rτ = ω. Thus, the statement (b) holds. Let Typ Rτ = ω. Using the statement (c*) we obtain Typ Ψτds = ω. Let Typ Ψτds = ω. By the statement (b*), Typ Rτ = ε or Typ Rτ = ω. Using the statement (a*) we obtain Typ Rτ = ε. Therefore Typ Rτ = ω. Thus, the statement (c) holds. Let Typ Ψτds = χ . Using the statements (b) and (b*) we conclude that the statement (d) holds. 

Corollary 20.2 Let τ = (ρ, K , ψ) be a restricted sccf-triple and Typ Ψτds = χ . Then the following two statements are equivalent: (a) There exists a polynomial p1 such that Ψτds  p1 . (b) There exists a polynomial p2 such that Rτ  p2 . Proof Assume that the statement (b) holds. Using Proposition 20.3 we obtain Ψτds  Rτ  p2 . Thus, the statement (a) holds. Assume now that the statement (b) does not hold. By Proposition 20.3, Typ Rτ = ε and Typ Rτ = ω. From here and from Lemma 20.2 it follows that Typ Pτ = ε. Assume that Typ Pτ = ω. Then, using Proposition 20.2, we obtain Typ Ψτas = ω. From this equality and from Lemma 19.2 it follows that Typ Ψτds = ω which is impossible. Therefore Typ Pτ = ω. By Lemma 19.10, Typ Pτ = χ . Assume that

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there is no polynomial p3 such that Pτ  p3 . Assume also that the statement (a) holds. By Proposition 20.2, Typ Ψτas = χ and Pτ  Ψτas . Taking into account that Typ Ψτas = Typ Ψτds = χ and using Lemma 19.2 we obtain Ψτas  Ψτds . Therefore Pτ  p1 which is impossible. Thus, the statement (a) does not hold. Let there exist a polynomial p3 such that Pτ  p3 . We now show that the statement (a) does not hold. Assume the contrary. Let there exist a polynomial p1 such that Ψτds  p1 . Taking into account that there are polynomials p1 and p3 such that Pτ  p3 and Ψτds  p1 one can show that there exist nondecreasing polynomials p1 and p3 such that Pτ  p3

and Ψτds  p1 . Using Proposition 20.3 we obtain Rτ  Ψτds (Pτ )  p1 ( p3 ) which is impossible. Thus, the statement (a) does not hold. 

20.4 Function Ψτda Lemma 20.4 Let ρ = (F, k) be a signature, τ = (ρ, K , ψ) be a restricted sccfd r (z) ≥ logk lρ,K (z). triple, and z ∈ Σρ . Then ψρ,K d d and ψ(Γ ) = ψρ,K (z). Set T = Tρ (z, K ) and Proof Let (Γ, z) ∈ Rρ,K

T1 = Tρ (α(z, P(Γ ), K ) . d (z) = ψρd (T1 ). By Theorem 3.2, ψρd (T1 ) ≥ Using Theorem 18.1 we obtain ψρ,K d r logk l r (T1 ). One can show that l r (T1 ) = l r (T ) = lρ,K (z). Thus, ψρ,K (z) ≥ r (z).  logk lρ,K

Lemma 20.5 Let ρ = (F, k) be a signature, τ = (ρ, K , ψ) be a restricted sccftriple, T ∈ Mρ,K , Z ρ (T ) > 0, and n be the maximum number from ω such that n(n+1) a ≤ Z ρ (T ). Then there exists a problem schema z ∈ Σρ such that ψρ,K (z) ≤ 2

 n(n+1) d 3m ψ (T ) and ψρ,K . (z) ≥ logk 2 Proof The statement of the lemma follows immediately from Lemmas 12.6, 18.9, and 20.4.  Lemma 20.6 Let ρ = (F, k) be a signature, τ = (ρ, K , ψ) be a restricted sccftriple, T ∈ Mρ,K , and Q ρ (T ) > 0. Then there exists a problem schema z ∈ Σρ a d such that ψρ,K (z) ≤ m ψ (T ) and ψρ,K (z) ≥ logk Q ρ (T ). Proof The statement of the lemma follows immediately from Lemmas 12.7, 18.9, and 20.4.  Lemma 20.7 Let ρ = (F, k) be a signature, τ = (ρ, K , ψ) be a restricted sccftriple, m ∈ ω \ {0}, T ∈ Mρ,K , and m ψ (T ) ≤ m. Then there exists a problem schema a d (z) ≤ m and ψρ,K (z) ≥ logk Iψ (m, T ). z ∈ Σρ such that ψρ,K Proof The statement of the lemma follows immediately from Lemmas 12.9, 18.9, and 20.4. 

20.4 Function Ψτda

239

Lemma 20.8 Let τ = (ρ, K , ψ) be a restricted sccf-triple such that Typ Nτ = ω. Then Typ Iτ = ω. Proof Let n ∈ ω, T ∈ Mρ,K , and m ψ (T ) ≤ n. By Lemma 12.8, Iψ (n, T ) ≤ Nρ (T ). Taking into account that Typ Nτ = ω and m ψ (T ) ≤ n we obtain n ∈ Arg Nτ and  Nρ (T ) ≤ Nτ (n). Hence Iψ (n, T ) ≤ Nτ (n). Thus, Typ Iτ = ω. Lemma 20.9 Let τ = (ρ, K , ψ) be a restricted sccf-triple, ρ = (F, k), Typ Nτ = ω, and Typ Pτ = ω. Then Typ Iτ = ω. Proof Since Typ Nτ = ω, there exists m ∈ ω such that Typ Nτ (m) = ∞. By this equality, Pτ (m) = ∅. Taking into account that Typ Pτ = ω we obtain Pτ (m) ∈ ω. Let Pτ (m) = c. Set n = max{m, c}. Let T ∈ Mρ,K , m ψ (T ) ≤ m, dim T = r , and ¯ = ( f 1 , δ1 ) · · · μT (1) = f 1 , . . . , μT (r ) = fr . Let δ¯ = (δ1 , . . . , δr ) ∈ Δ(T ). Set α(δ) s ( fr , δr ). Then there exists a schema Γδ¯ ∈ Cρ such that (Γδ¯ , z α(δ) ¯ ) ∈ Rρ,K and , there exists a problem schema z ∈ Σρ such ψ(Γδ¯ ) ≤ c ≤ n. Since T ∈ Mρ,K



that T = Tρ (z, K ). Set z = α z, δ∈Δ(T ¯ ) P(Γδ¯ ) and T = Tρ (z , K ). Taking into

account that  ψ is a restricted complexity function one can show that m ψ (T ) ≤ n. Set B = δ∈Δ(T ¯ ) {π(ξ ) : ξ ∈ (Γδ¯ )}. Evidently, B is a cover for the table T , and ψ(B) ≤ n. It is clear that there exists an uncancelable cover B1 for the table T such that B1 ⊆ B. One can show that |B1 | ≥ |Δ(T )| = Nρ (T ). Therefore Iψ (n, T ) ≥  Nρ (T ). Taking into account that Nτ (m) = ∞ we obtain Iτ (n) = ∞. Proposition 20.4 Let ρ = (F, k) be a signature and τ = (ρ, K , ψ) be a sccf-triple. Then Typ Ψτda ∈ {ε, χ , ω} and, if τ is a restricted sccf-triple, then the following statements hold: (a) Typ Ψτda = ε if and only if Typ Nτ = ε. (b) Typ Ψτda = χ if and only if Typ Nτ = ε and Typ Iτ = ω. (c) Typ Ψτda = ω if and only if Typ Iτ = ω. (d) If Typ Ψτda = χ , then there exists m ∈ ω such that, for each n ∈ ω(n), the values Ψτda (n), Ψτda (3n), Z τ (n), Q τ (n), and Iτ (n) are definite, and the inequal1 ities Ψτda (n) ≥ logk max{1, Q τ (n)}, Ψτda (3n) ≥ logk max{1, Z τ (n) − (2Z τ (n)) 2 − da da 1}, Iτ (n) ≥ 1, Ψτ (n) ≥ logk Iτ (n), and Ψτ (n) ≤ max{n, n Q τ (n)}Z τ (n) logk (k 2 n Iτ (n)) hold. Proof Let Typ Ψτda = ε and Typ Ψτda = ω. Then, evidently, Typ Ψτaa = ε. By Proposition 19.4, Typ Ψτaa = χ . Using Lemma 19.2 we obtain Typ Ψτda = λ. Hence Typ Ψτda = χ . Thus, Typ Ψτda ∈ {ε, χ , ω}. Let τ be a restricted sccf-triple. We now prove the following statement: (a*) If Typ Nτ = ε, then Typ Ψτda = ε. Let Typ Nτ = ε. By Proposition 19.2, there exists a constant c ∈ ω such that d (z) ≤ c for any problem schema z ∈ Σρ . From here and from Lemma 19.3 it ψρ,K follows that Typ Ψτda = ε. Thus, the statement (a*) holds. We now prove the following statement: (b*) If Typ Nτ = ε and Typ Iτ = ω, then Typ Ψτda = χ and Typ Ψˆ τda = χ .

240

20 Functions over Main Diagonal

Let Typ Nτ = ε and Typ Iτ = ω. By Lemma 12.10, Typ Sτ = ε. From Proposition 12.4 it follows that Typ Ψˆ τda = χ . By Proposition 18.4, Typ Ψτda = ω. Taking into account that Typ Nτ = ε and using Proposition 19.2 and Lemma 19.3 we obtain d that there is no constant c ∈ ω such that ψρ,K (z) ≤ c for any problem schema z ∈ Σρ . da aa Since Typ Ψτ = ω, we have Typ Ψτ = ε. From here and from Proposition 19.4 it follows that Typ Ψτaa = χ . By Lemma 19.2, Typ Ψτda = ε and Typ Ψτda = λ. Therefore Typ Ψτda = χ . Thus, the statement (b*) holds. We now prove the following statement: (c*) If Typ Iτ = ω, then Typ Ψτda = ω. Let Typ Iτ = ω. Then there exists m ∈ ω such that the set {Iψ (n, T ) : T ∈ Mρ,K , m ψ (T ) ≤ m} is infinite. By Lemma 20.7, Ψτda (m) = ∞. Thus, the statement (c*) holds. Let Typ Nτ = ε. Using the statement (a*) we obtain Typ Ψτda = ε. Let Typ Ψτda = ε. Using the statement (b*) we conclude that Typ Nτ = ε or Typ Iτ = ω. By the statement (c*), Typ Iτ = ω. Therefore Typ Nτ = ε. Thus, the statement (a) holds. Let Typ Nτ = ε and Typ Iτ = ω. By the statement (b*), Typ Ψτda = χ . Let Typ Ψτda = χ . Using the statements (a*) and (c*) we obtain Typ Nτ = ε and Typ Iτ = ω. Thus, the statement (b) holds. Let Typ Iτ = ω. By the statement (c*), Typ Ψτda = ω. Let Typ Ψτda = ω. Using the statement (b*) we conclude that Typ Nτ = ε or Typ Iτ = ω. By the statement (a*), Typ Nτ = ε. Therefore Typ Iτ = ω. Thus, the statement (c) holds. Let Typ Ψτda = χ . Using the statement (b*) we obtain Typ Ψˆ τda = χ . From this equality and from Proposition 12.4 it follows that there exists m ∈ ω such that, for each n ∈ ω(m), the values Ψˆ τda (n), Ψˆ τda (3n), Z τ (n), Q τ (n), and Iτ (n) are definite, and the inequalities Iτ (n) ≥ 1 and Ψˆ τda (n) ≤ max{n, n Q τ (n)}Z τ (n) logk (k 2 n Iτ (n)) hold. Taking into account that Typ Ψτda = χ and using Lemma 18.9 we conclude that the values Ψτda (n) and Ψτda (3n) are definite. From Proposition 18.4 it follows that Ψτda (n) ≤ Ψˆ τda (n). By Lemma 20.7, Ψτda (n) ≥ logk Iτ (n). From Lemma 20.6 it follows that Ψτda (n) ≥ logk max{1, Q τ (n)}. Let Z τ (n) = 0. Then, evidently, 1 Ψτda (3n) ≥ logk max{1, Z τ (n) − (2Z τ (n)) 2 − 1}. Let Z τ (n) > 0, and m be the maxm(m+1) ≤ Z τ (n). By Lemma 20.5, Ψτda (3n) ≥ imum number from ω such that 2

 1 . Evidently, m < (2Z τ (n)) 2 and (m+1)(m+2) logk m(m+1) = m(m+1) + (m + 1) > 2 2 2 Z τ (n). Therefore

m(m+1) 2

1

> Z τ (n) − (2Z τ (n)) 2 − 1. Hence 1

Ψτda (3n) ≥ logk max{1, Z τ (n) − (2Z τ (n)) 2 − 1} .  Corollary 20.3 Let ρ = (F, k) be a signature, τ = (ρ, K , ψ) be a restricted sccftriple, and Typ Ψτda = χ . Then the following statements hold: (a) If there exist polynomials p1 , p2 , and p3 such that Z τ  p1 , Q τ  p2 , and Iτ  2 p3 , then there exists a polynomial p4 such that Ψτda (n)  p4 .

20.4 Function Ψτda

241

(b) If there is no polynomial p1 such that Z τ  2 p1 , or there is no polynomial p2 such that Q τ  2 p2 , or there is no polynomial p3 such that Iτ  2 p3 , then there is no polynomial p4 such that Ψτda (n)  p4 . Lemma 20.10 Let τ = (ρ, K , ψ) be a sccf-triple, Typ Ψτda = ω, and Typ Ψτas = ω. Then Typ Ψτds = ω and Ψτds (n) ≤ Ψτda (Ψτas (n)) for any n ∈ Arg Ψτds . s Proof Let n ∈ ω and Ψτds (n) = ∅. Let z ∈ Στs and ψρ,K (z) ≤ n. Then, evidently, d da as da as Ψτ (Ψτ (n)) ∈ ω and ψρ,K (z) ≤ Ψτ (Ψτ (n)). Hence Ψτds (n) ∈ ω and Ψτds (n) ≤ Ψτda (Ψτas (n)). 

Reference 1. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity, Global approach. Fundam. Inform. 25(2), 201–214 (1996)

Chapter 21

Possible Global Upper Types of Sccf-Triples

In this chapter, we describe all possible 11 global upper types Tp1, . . . , Tp11 of sccf-triples and all possible 10 global upper types Tp1, . . . , Tp10 of restricted sccftriples. For each of the last 10 global upper types, we consider the criterion of its implementation for restricted sccf-triples. Some similar results were obtained in [1] for types of functions different from the upper types considered in this book.

21.1 Possible Global Upper Types Let us consider the following five matrices:

i Tp7 = d a s

i χ λ ε ε

d ω χ ε ε

a ω ω ε ε

s ω ε ε ε

i Tp10 = d a s

i Tp8 = d a s i χ χ χ χ

d ω χ χ χ

a ω ω χ χ

s ω χ χ χ

i χ χ ε ε

d ω χ ε ε

a ω ω ε ε

s ω ε ε ε

i Tp11 = d a s

i Tp9 = d a s i ε ε ε ε

d ε ε ε ε

a ε ε ε ε

i χ χ ε ε

d ω χ ε ε

a ω ω ε ε

s ω ω ε ε

s ε ε ε ε

Proposition 21.1 Let τ = (ρ, K , ψ) be a sccf-triple. Then Typ Ψτ ∈ {Tp1, Tp2, Tp3, Tp4, Tp5, Tp6, Tp7, Tp8, Tp9, Tp10, Tp11} .

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_21

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21 Possible Global Upper Types of Sccf-Triples

Proof We will consider four cases: (a) Typ Ψτii = Typ Ψτdd = Typ Ψτaa = ε. (b) Typ Ψτii = χ and Typ Ψτdd = Typ Ψτaa = ε. (c) Typ Ψτii = Typ Ψτdd = χ and Typ Ψτaa = ε. (d) Typ Ψτii = Typ Ψτdd = Typ Ψτaa = χ . Let b ∈ {i, d, a}. By Lemma 19.4, Typ Ψτbb ∈ {ε, χ }. Using Lemmas 19.1 and 19.4 we conclude that, if Typ Ψτaa = χ , then Typ Ψτdd = χ , and, if Typ Ψτdd = χ , then Typ Ψτii = χ . Therefore, for any sccf-triple τ , one of the cases (a), (b), (c), (d) holds. (a) Let Typ Ψτii = Typ Ψτdd = Typ Ψτaa = ε. By Proposition 19.4, Typ Ψτss = ε. Using Lemmas 19.4 and 19.3 we conclude that Typ Ψτbc = ε for any b, c ∈ {i, d, a, s}. Therefore Typ Ψτ = Tp11. (b) Let Typ Ψτii = χ and Typ Ψτdd = Typ Ψτaa = ε. Using Proposition 19.1 we conclude that there is no constant c ∈ ω such that ψ(α) ≤ c for any α ∈ F ∗ . By Proposition 20.1, Typ Ψτid = Typ Ψτia = Typ Ψτis = ω. From Proposition 19.4 it follows that Typ Ψτss = ε. By Lemmas 19.4 and 19.3, for any b, c ∈ {i, d, a, s}, if b = i, then Typ Ψτbc = ε. Therefore Typ Ψτ = Tp1. (c) Let Typ Ψτii = Typ Ψτdd = χ and Typ Ψτaa = ε. By Proposition 19.1, there is no constant c ∈ ω such that ψ(α) ≤ c for any α ∈ F ∗ . Using Proposition 20.1 we obtain Typ Ψτid = Typ Ψτia = Typ Ψτis = ω. From Proposition 19.4 it follows that Typ Ψτss = ε. By Lemmas 19.4 and 19.3, Typ Ψτbc = ε for any b ∈ {a, s} and c ∈ {i, d, a, s}. Taking into account that Typ Ψτdd = χ and Typ Ψτaa = ε, and using Lemma 19.4 one can show that Typ Ψτda = ω. Using the equality TypΨτdd = χ and Lemmas 19.4 and 19.3 we conclude that Typ Ψτdi = ε. From here and from Proposition 19.2 it follows that Typ Ψτdi ∈ {λ, χ }. Using the equality Typ Ψτss = ε and s (z) ≤ c for Lemma 19.4 we conclude that there exists a constant c ∈ ω such that ψρ,K d s any problem schema z ∈ Στ . If there exists a constant m ∈ ω such that ψρ,K (z) ≤ m s ds for any problem schema z ∈ Στ , then using Lemma 19.3 we obtain Typ Ψτ = ε. If such constant does not exist, then, evidently, Typ Ψτds = ω. Thus, Typ Ψτds ∈ {ε, ω}. Therefore Typ Ψτ ∈ {Tp2, Tp7, Tp8, Tp9}. (d) Let Typ Ψτii = Typ Ψτdd = Typ Ψτaa = χ . By Proposition 19.1, there is no constant c ∈ ω such that ψ(α) ≤ c for any α ∈ F ∗ . Using Proposition 20.1 we obtain Typ Ψτid = Typ Ψτia = Typ Ψτis = ω. Taking into account that Typ Ψτaa = χ and using Proposition 19.4 we conclude that Typ Ψτbc = χ for any pair (b, c) ∈ {(a, i), (a, d), (a, a), (s, i), (s, d), (s, a), (s, s)}. Using Lemma 19.2, Proposition 19.2, and the equality Typ Ψτai = χ , we obtain Typ Ψτdi = χ . Let (b, c) ∈ {(d, a), (d, s), (a, s)} . Using the equalities Typ Ψτaa = Typ Ψτss = χ and Lemma 19.2 we obtain Typ Ψτbc = ε. By Propositions 20.2–20.4, Typ Ψτbc ∈ {χ , ω}. From Lemma 19.2 it follows that, if Typ Ψτas = ω, then Typ Ψτds = ω. By Lemma 20.10, if Typ Ψτda = ω and Typ Ψτas =  ω, then Typ Ψτds = ω. Therefore Typ Ψτ ∈ {Tp3, Tp4, Tp5, Tp6, Tp10}.

21.1 Possible Global Upper Types

245

Proposition 21.2 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψτ ∈ {Tp1, Tp2, Tp3, Tp4, Tp5, Tp6, Tp7, Tp8, Tp9, Tp10} . Proof Using Proposition 20.1 we obtain Typ Ψτ = Tp11. By Proposition 21.1,  Typ Ψτ ∈ {Tp j : j = 1, . . . , 10}.

21.2 Criteria for Equalities Typ Ψτ = Tp j Proposition 21.3 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψτ = Tp1 if and only if Typ Nτ = ε. Proof Let Typ Ψτ = Tp1. Then Typ Ψτdd = ε. By Proposition 19.3, Typ Nτ = ε. Let Typ Nτ = ε. By Proposition 19.3, Typ Ψτdd = ε. From this equality and Proposition  21.2 it follows that Typ Ψτ = Tp1. Proposition 21.4 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψτ = Tp2 if and only if Typ Ψτdi = λ and Typ Rτ = ω. Proof Let Typ Ψτ = Tp2. Then Typ Ψτdi = λ and Typ Ψτds = ω. By Proposition 20.3, Typ Rτ = ω. Let Typ Ψτdi = λ and Typ Rτ = ω. By Proposition 20.3,  Typ Ψτds = ω. Using Proposition 21.2 we obtain Typ Ψτ = Tp2. Proposition 21.5 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψτ = Tp3 if and only if Typ Pτ = χ and Typ Rτ = ω. Proof Let Typ Ψτ = Tp3. Then Typ Ψτds = ω and Typ Ψτas = χ . By Propositions 20.2 and 20.3, Typ Pτ = χ and Typ Rτ = ω. Let Typ Pτ = χ and Typ Rτ = ω. By Propositions 20.2 and 20.3 , Typ Ψτds = ω and Typ Ψτas = χ . From these equalities  and from Proposition 21.2 it follows that Typ Ψτ = Tp3. Proposition 21.6 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψτ = Tp4 if and only if Typ Nτ = ε and Typ Nτ = ω. Proof Let Typ Ψτ = Tp4. Then Typ Ψτda = Typ Ψτas = χ . By Propositions 20.2 and 20.4, Typ Pτ = χ , Typ Nτ = ε, and Typ Iτ = ω. By Lemma 20.9, Typ Nτ = ω. Let Typ Nτ = ε and Typ Nτ = ω. By Lemma 20.8, Typ Iτ = ω. Using Lemma 19.12 we obtain Typ Pτ = χ . By Propositions 20.2 and 20.4, Typ Ψτda = Typ Ψτas = χ . From these equalities and from Proposition 21.2 it follows that Typ Ψτ = Tp4.  Proposition 21.7 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψτ = Tp5 if and only if Typ Iτ = Typ Pτ = ω. Proof Let Typ Ψτ = Tp5. Then Typ Ψτda = Typ Ψτas = ω. By Propositions 20.2 and 20.4, Typ Iτ = Typ Pτ = ω. Let Typ Iτ = Typ Pτ = ω. By Propositions 20.2 and 20.4, Typ Ψτda = Typ Ψτas = ω. From these equalities and from Proposition 21.2 it follows that Typ Ψτ = Tp5. 

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21 Possible Global Upper Types of Sccf-Triples

Proposition 21.8 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψτ = Tp6 if and only if Typ Iτ = ω and Typ Pτ = ω. Proof Let Typ Ψτ = Tp6. Then Typ Ψτda = χ and Typ Ψτas = ω. By Propositions 20.2 and 20.4, Typ Iτ = ω and Typ Pτ = ω. Let Typ Iτ = ω and Typ Pτ = ω. Assume that Typ Nτ = ε. Then, using Proposition 21.3, we obtain Typ Ψτas = ε. From this equality and from Proposition 20.2 it follows that Typ Pτ = ε which is impossible. Therefore Typ Nτ = ε. By Propositions 20.2 and 20.4 , Typ Ψτas = ω and Typ Ψτda = χ . From these equalities and from Proposition 21.2 it follows that  Typ Ψτ = Tp6. Proposition 21.9 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψτ = Tp7 if and only if Typ Ψτdi = λ and Typ Rτ = ε. Proof Let Typ Ψτ = Tp7. Then Typ Ψτdi = λ and Typ Ψτds = ε. By Proposition 20.3, Typ Rτ = ε. Let Typ Ψτdi = λ and Typ Rτ = ε. By Proposition 20.3,  Typ Ψτds = ε. Using Proposition 21.2 we obtain Typ Ψτ = Tp7. Proposition 21.10 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψτ = Tp8 if and only if Typ Ψτdi = χ and Typ Rτ = ε. Proof Let Typ Ψτ = Tp8. Then Typ Ψτdi = χ and Typ Ψτds = ε. By Proposition 20.3, Typ Rτ = ε. Let Typ Ψτdi = χ and Typ Rτ = ε. Using Proposition 20.3 we  obtain Typ Ψτds = ε. By Proposition 21.2, Typ Ψτ = Tp8. Proposition 21.11 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψτ = Tp9 if and only if Typ Ψτdi = χ , Typ Rτ = ω, and Typ Pτ = ε. Proof Let Typ Ψτ = Tp9. Then Typ Ψτdi = χ , Typ Ψτds = ω, and Typ Ψτas = ε. From Propositions 20.2 and 20.3 it follows that Typ Rτ = ω and Typ Pτ = ε. Let Typ Ψτdi = χ , Typ Rτ = ω, and Typ Pτ = ε. By Propositions 20.2 and 20.3, Using Proposition 21.2 we obtain Typ Ψτas = ε and Typ Ψτds = ω.  Typ Ψτ = Tp9. Proposition 21.12 Let τ = (ρ, K , ψ) be a restricted sccf-triple. Then Typ Ψτ = Tp10 if and only if Typ Iτ = ω, Typ Rτ = ε, and Typ Rτ = ω. Proof Let Typ Ψτ = Tp10. Then Typ Ψτda = ω and Typ Ψτds = χ . From Propositions 20.3 and 20.4 it follows that Typ Iτ = ω, Typ Rτ = ε, and Typ Rτ = ω. Let Typ Iτ = ω, Typ Rτ = ε, and Typ Rτ = ω. By Propositions 20.3 and 20.4, Typ Ψτda = ω and Typ Ψτds = χ . From these equalities and from Proposition 21.2  it follows that Typ Ψτ = Tp10. It was already mentioned that we have no criteria for the equalities Typ Ψτdi = λ and Typ Ψτdi = χ in general case. However, such criteria are known for the case, when ψ = h (Theorem 5.1 from [2]) and for the case, when ψ is a weighted depth (Theorem 5.4 from [2]).

21.3 Global Upper Type of Composition of Simple Sccf-Triples Table 21.1 Global upper type of composition of simple sccf-triples 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 2 2 2 3 3 5 5 2 3 3 3 3 3 5 5 3 4 4 3 3 4 5 6 10 5 5 5 5 5 5 5 5 6 6 5 5 6 5 6 5 7 7 2 3 10 5 5 7 8 8 9 3 10 5 5 8 9 9 9 3 3 5 5 9 10 10 3 3 10 5 5 10

247

8 8 9 3 10 5 5 8 8 9 10

9 9 9 3 3 5 5 9 9 9 3

10 10 3 3 10 5 5 10 10 3 10

21.3 Global Upper Type of Composition of Simple Sccf-Triples Proposition 21.13 Let τ1 and τ2 be simple sccf-triples, τ be an arbitrary composition of sccf-triples τ1 and τ2 , Typ Ψτ1 = Tp j1 , Typ Ψτ2 = Tp j2 , and l be the number which is placed in Table 21.1, at the intersection of the row with number j1 and the column with number j2 . Then τ is a simple sccf-triple and Typ Ψτ = Tpl. Proof The considered statement follows immediately from Proposition 18.2.



References 1. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity, Global approach. Fundam. Inform. 25(2), 201–214 (1996) 2. Moshkov, M.: Time complexity of decision trees. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III, Lecture Notes in Computer Science, vol. 3400, pp. 244–459. Springer, Berlin (2005)

Chapter 22

Realizable Global Upper Types of Sccf-Triples

In this chapter, for each i ∈ {1, . . . , 11}, we prove that there exists a sccf-triple τ such that Typ Ψτ = Tpi. We also prove that, for each i ∈ {1, . . . , 10}, there exists a restricted sccf-triple τ such that Typ Ψτ = Tpi. Some similar results were obtained in [1] for types of functions different from the upper types considered in this book. We repeat here some definitions from Sect. 13.2. Set F0 = { f i : i ∈ ω \ {0}} and G 0 = {gi : i ∈ ω \ {0}}. We define three signatures as follows: ρ0 = ({ f 0 }, 2), ρ1 = (F0 , 2), and ρ2 = (F0 ∪ G 0 , 2). For n ∈ ω \ {0}, we denote by en the mapping from the set E 2n to the set {{1}}. For n ∈ ω \ {0}, we define a mapping dn from E 2n to P(ω) ¯ = { j}, where j is the number from as follows. Let δ¯ = (δ1 , . . . , δn ) ∈ E 2n . Then dn (δ) ¯ ω such that δ is the binary representation of j. Let i ∈ Z. We now define functions pi : Q → E 2 and li : Q → E 2 . Let q ∈ Q. Then   1, if q = i , 1, if q ≥ i , pi (q) = li (q) = 0, if q = i , 0, if q < i . Let f ∈ F R and m ∈ ω. We denote by f − m the restriction of the function f (x) − m to the set {r : r ∈ Arg f, f (r ) − m ≥ 0}. It is clear that f − m ∈ F R. For i ∈ {1, . . . , 10}, we denote by W s (i) the set of all simple sccf-triples τ such that Typ Ψτ = Tpi.

22.1 Sccf-Triples from W s (1) Let m ∈ ω \ {0}. We define the information system U(1,m) of the signature ρ0 as in Sect. 13.2: U(1,m) = (Q, γ(1,m) ) and γ(1,m) ( f 0 ) = l0 . We define a weighted depth (1, m)ψ of the signature ρ0 in the following way: (1, m)ψ( f 0 ) = m. Denote τ (1, m) = (ρ0 , {U(1,m) }, (1, m)ψ).

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_22

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22 Realizable Global Upper Types of Sccf-Triples

Lemma 22.1 For any n, m ∈ ω \ {0}, τ (1, m) ∈ W s (1), Ψτii(1,1) (n) = n and, if n > 1, then Ψτii(1,m) (nm − 1) ≤ nm − 1 − (m − 1). Proof One can show that Typ Nτ (1,m) = ε. By Proposition 21.3, Typ Ψτ (1,m) = Tp1. Evidently, τ (1, m) is a simple sccf-triple. Therefore τ (1, m) ∈ W s (1). Set z n = (en , f 0 , . . . , f 0 ), where the element f 0 is repeated exactly n times. It is clear that (1, m)ψρi 0 ,{U(1,m) } (z n ) = mn. Therefore Ψτii(1,1) (n) = n. Let z ∈ Σρ0 . Then, evidently, (1, m)ψρi 0 ,{U(1,m) } (z n ) = m j where j ∈ ω \ {0}. Therefore, if n > 1, then the value Ψτii(1,m) (nm − 1) is definite and does not exceed m(n − 1). 

22.2 Sccf-Triples from W s (7) We define an information system U(7) of the signature ρ1 as follows: U(7) = (ω, γ(7) ). Let f i ∈ F0 . Set B( f i ) = {n : n ∈ ω, γ(7) ( f i )(n) = 0}. Then, for any f i ∈ F0 , B( f i ) is a finite nonempty set and, for any finite nonempty subset B of the set ω, there exists f i ∈ F0 such that B( f i ) = B. ( p) · · · f i p the Let α ∈ F0∗ , α = λ, and α = f i1 · · · f i p . We denote by α˜ = f i(1) 1 word obtained from α by the insertion of additional indices of letters which ( p) , . . . , fi p } allow us to distinguish the occurrences of equal letters. Let D ⊆ { f i(1) 1 and |D| = n > 0. The set D will be called a proper n-tuple if there exists (j ) (j ) its ordering D = { f i j 1 , . . . , f i jn n } and a partition ω = A1 ∪ · · · ∪ An+1 such that 1 B( f i jt ) = A1 ∪ · · · ∪ At for t = 1, . . . , n (it is possible that Ai = ∅ for some i ∈ {1, . . . ,n + 1}. We now define a function l : ω \ {0} → ω. Let n ∈ ω \ {0}. Then l(n) = log2 n + 1. (j ) (j ) Let D = { f i j 1 , . . . , f i jn n } be a proper n-tuple. Set z = (dn , f i j1 , . . . , f i jn ). One 1 can show that (22.1) h dρ1 ,{U(7) } (z) ≤ l(n) . ( p)

, . . . , f i p } into nonempty subsets with Let q ∈ ω \ {0}. A partition of the set { f i(1) 1 at most q elements in each will be called a q-partition of the word α. Let r ∈ ω \ {0, 1}. We now define a complexity function (7, 1, r )ψ of the signature ρ1 . Let α ∈ F0∗ and t (α) be the maximum number of proper triples in a 3-partition of the word α. Then (7, 1, r )ψ(α) = r |α| − t (α). Denote τ (7, 1, r ) = (ρ1 , {U(7) }, (7, 1, r )ψ). Let m ∈ ω \ {0, 1}. We now define a complexity function (7, 2, m)ψ of the signature ρ1 . By rm we denote the minimum number from ω such that rm ≥ 4m2 l(rm ). For log2 n + n ∈ ω \ {0} set tm (n) = n − 2ml(n).  Let n ≥ 2. Then tm(n) = n − 2m(   + 1). If log n = log (n − 1) , 1) and tm (n − 1) = n − 1 − 2m( log2 (n− 1) 2 2   then tm (n − 1) = tm (n) − 1. If log2 n = log2 (n − 1) + 1, then tm (n − 1) = tm (n) + 2m − 1 > tm (n). Evidently, tm (rm ) > 1 and tm (1) = 1 − 2m < 0. Let n 0

22.2 Sccf-Triples from W s (7)

251

be the maximum number from ω such that n 0 < rm and tm (n 0 ) ≤ 0. Then, evidently, tm (n 0 ) < tm (n 0 + 1). Taking into account that tm (n 0 + 1) > 0 we conclude that tm (n 0 + 1) = 1. Set sm = n 0 + 1 and Nm = {sm , sm+1 , . . . , rm }. Let α ∈ F0∗ , R be an arbitrary rm -partition of the word α, R = D1 ∪ · · · ∪ Dq , and i ∈ {1, . . . , q}. We now define a value tm (Di ). Let |Di | = n. If n ∈ Nm and Di is a proper n-tuple, then tm (Di ) = tm (n). If n ∈ / Nm or Di is not a proper n-tuple, q then tm (Di ) = 0. Denote tm (R) = i=1 tm (Di ) and tm (α) = max tm (R), where the maximum is taken among all rm -partitions R of the word α. Then (7, 2, m)ψ(α) = rm (|α| − tm (α)). Denote τ (7, 2, m) = (ρ1 , {U(7) }, (7, 2, m)ψ). Lemma 22.2 Let r ∈ ω \ {0, 1} and n ∈ ω \ {0}. Then τ (7, 1, r ) ∈ W s (7), Ψτdi(7,1,r ) (r n)

  1 n, ≥ r− 3

dd ds if n ∈ Arg Ψτdd (7,1,2) , then Ψτ (7,1,2) (n) ≥ n − 1 and, for any n ∈ ω, n ∈ Arg Ψτ (7,1,2) ds and Ψτ (7,1,2) (n) ≤ 2.

Proof Denote by ψ the function (7, 1, r )ψ. Let us show that ψ is a restricted complexity function. Let α1 , α2 ∈ F0∗ and f i ∈ F0 . Then, evidently, ψ(α1 ) + ψ(α2 ) − ψ(α1 α2 ) = t (α1 α2 ) − t (α1 )− t (α2 ) ≥ 0. Therefore ψ has the property Λ1. Evidently, ψ(α1 f i α2 ) − ψ(α1 α2 ) = r − (t (α1 f i α2 ) − t (α1 α2 )). It is clear that t (α1 f i α2 ) ≤ t (α1 α2 ) + 1 . Hence ψ(α1 f i α2 ) ≥ ψ(α1 α2 ). Using this inequality one can show that the function ψ has the property Λ2. One can notice that t (α1 ) ≤ |α1 | /3. Taking into account that r ≥ 2 we conclude that ψ(α1 ) ≥ 53 |α1 |. Therefore ψ has the property Λ3. Evidently, ψ(λ) = 0. Hence the function ψ has the property Λ4. Thus, ψ is a restricted complexity function, and τ (7, 1, r ) is a simple sccf-triple. ({U(7) }). Set B1 = {n : n ∈ ω, ϕ(z,U(7) ) (n) = {1}} and B0 = ω \ B1 . Let z ∈ Σρ0−1 1 If B1 = ω, then ψρd1 ,{U(7) } (z) = 0. Let B1 = ω. Then either B1 or B0 is a finite nonempty set. Let, for the definiteness, B1 is a finite nonempty set. Then there exists f i ∈ F1 such that B( f i ) = B1 . Using this equality one can show that ψρd1 ,{U(7) } (z) ≤ r . Hence Typ Rτ (7,1,r ) = ε. Evidently, for any n ∈ ω, n ∈ Arg Ψτds(7,1,2) and Ψτds(7,1,2) (n) ≤ 2. Let z ∈ Σρ1 and z = (ν, f i1 , . . . , f im ). Set α = f i1 · · · f im . Let t (α) > 0 and t (α) = t. Set z  = (dm , f i1 , . . . , f im ). It is clear that l(3) = 2. Using (22.1) one can show that h dρ1 ,{U(7) } (z  ) ≤ m − t. Therefore ψρd1 ,{U(7) } (z  ) ≤ r (m − t). Evidently, ψρd1 ,{U(7) } (z) ≤ ψρd1 ,{U(7) } (z  ) and ψρi 1 ,{U(7) } (z) = r m − t > r (m − t). Hence ψρd1 ,{U(7) } (z) < ψρi 1 ,{U(7) } (z) . Let t (α) = 0. Then ψρi 1 ,{U(7) } (z) = r m. Let m ≥ 3. Set T = Tρ1 (z, {U(7) }) and p = Nρ1 (T ). Evidently, there exists a partition ω = A1 ∪ · · · ∪ A p having the

252

22 Realizable Global Upper Types of Sccf-Triples

 following properties: for any i ∈ {1, . . . , p}, Ai = ∅ and a∈Ai ϕ(z,U(7) ) (a) = ∅, and for any i ∈ {1, . . . , p − 1}, Ai is a finite set. Then there exist f j1 , . . . , f j p−1 ∈ F0 such that B( f ji ) = A1 ∪ · · · ∪ Ai for i = 1, . . . , p − 1. One can show that there exists a schema Γ ∈ Cρ1 such that (Γ, z) ∈ Rρd1 ,{U(7) } , P(Γ ) ⊆ { f j1 , . . . , f j p−1 }, and h(Γ ) ≤

log2 p . Evidently, p ≤ 2m . Therefore h(Γ ) ≤ m. Let ξ ∈ Ξ(Γ ). If |π(ξ )| < m,

then ψ(π(ξ )) < r m. Let |π(ξ )| = m. It is clear that t (π(ξ )) = ψρd1 ,{U(7) } (z)

|π(ξ )| 3

≥ 1. There-

ψρi 1 ,{U(7) } (z).

< fore ψ(π(ξ )) = r m − t (π(ξ )) < r m. Hence Let z ∈ Σρ1 , ψρi 1 ,{U(7) } (z) > 3r , and z = (ν, f i1 , . . . , f im ). Then, evidently, m ≥ 3 and, by proved above, ψρd1 ,{U(7) } (z) < ψρi 1 ,{U(7) } (z). Hence Typ Ψτdi(7,1,r ) = χ . Evidently, Typ Nτ (7,1,r ) = ε. By Proposition 19.2, Typ Ψτdi(7,1,r ) = λ. Using Proposition 21.9 we obtain Typ Ψτ (7,1,r ) = Tp7. Hence τ (7, 1, r ) ∈ W s (7). Taking into account that Typ Nτ (7,1,r ) = ε and using Proposition 21.6 we obtain Typ Nτ (7,1,r ) = ω. By Lemma 20.3, Typ Z τ (7,1,r ) = ω. Using this equality and Lemma 12.5 one can show that there exist f i1 , . . . , f in ∈ F0 such that Nρ1 ,{U(7) } (z 1 ) = 2n for the problem schema z 1 = (en , f i1 , . . . , f in ). By Lemma 19.5, there exists a mapping ν : E 2n → P(ω) such that h dρ1 ,{U(7) } (z 2 ) ≥ n for the problem schema z 2 = (ν, f i1 , . . . , f in ). Using this inequality one can prove that ψρd1 ,{U(7) } (z 2 ) ≥ r n − n/3 = (r − 1/3)n. Evidently, ψρi 1 ,{U(7) } (z 2 ) ≤ r n. Hence Ψτdi(7,1,r ) (r n) ≥ (r − 1/3)n. Since Typ Nτ (7,1,r ) = ω, we have Nτ (7,1,r ) (r ) = ∞. Therefore Nτ (7,1,2) (2) = ∞. By dd  Lemma 19.8, if n ∈ Arg Ψτdd (7,1,2) , then Ψτ (7,1,2) (n) ≥ n − 1. Lemma 22.3 Let m ∈ ω \ {0, 1}. Then τ (7, 2, m) ∈ W s (7), and there exists a constant cm ∈ ω such that, for any n ∈ ω(cm ), the value Ψτdi(7,2,m) (n) is definite and the inequality Ψτdi(7,2,m) (n) < n/m holds. Proof We denote by ψ the function (7, 2, m)ψ. Let us show that ψ is a restricted complexity function. Let α1 , α2 ∈ F0∗ and f i ∈ F0 . Evidently, ψ(α1 ) + ψ(α2 ) − ψ(α1 α2 ) = rm (tm (α1 α2 ) − tm (α1 )− tm (α2 )). One can show that tm (α1 α2 ) ≥ tm (α1 ) + tm (α2 ) . Therefore the function ψ has the property Λ1. Set β = α1 f i α2 . We now show that tm (α1 α2 ) ≥ tm (β) − 1. Let the letter f i , staying between α1 and α2 in the word β, is the letter number j of the word β. Let R be a rm -partition of the word β such that tm (R) = tm (β), R = D1 ∪ · · · ∪ Dq and let, for the definite( j) ness, f i ∈ D1 . If tm (D1 ) = 0, then one can show that t (β) = tm (α1 α2 ). Let ( j) tm (D1 ) > 0. Set D0 = D1 \ { f i }. Let |D1 | = n. Then, evidently, D0 is a proper (n − 1)-tuple. If n − 1 ∈ Nm , then taking into account that tm (R) = tm (β) we obtain / Nm . Then n = sm . tm (D0 ) = tm (D1 ) − 1. Hence tm (α1 α2 ) ≥ tm (β) − 1. Let n − 1 ∈ Hence tm (D0 ) = 0 and tm (D1 ) = 1. Therefore tm (α1 α2 ) ≥ tm (β) − 1. Evidently, ψ(α1 f i α2 ) − ψ(α1 α2 ) = rm (1 − tm (β) + tm (α1 α2 )) ≥ 0. Using this inequality one can show that the function ψ has the property Λ2. Let R be a rm -partition of the word α1 such that tm (R) = tm (α1 ). Let R = D1 ∪ · · · ∪ Dq . Then ψ(α1 ) =

22.2 Sccf-Triples from W s (7)

253

q rm (|α1 | − tm (α1 )) = i=1 rm (|Di | − tm (Di )). We now show that |Di | − tm (Di ) ≥ 1 for any i ∈ {1, . . . , q}. Let tm (Di ) = 0. Then taking into account that Di = ∅ we conclude that the considered inequality holds. Let tm (Di ) > 0. Set n = |Di |. Then |Di | − tm (Di ) = n − tm (n) = n − n + 2ml(n) > 1. Hence ψ(α1 ) ≥ qrm . Since |Di | ≤ rm for i = 1, . . . , q, we have ψ(α1 ) ≥ |α1 |. Therefore ψ has the property Λ3. Evidently, ψ(λ) = 0. Hence the function ψ has the property Λ4. Thus, ψ is a restricted complexity function, and τ (7, 2, m) is a simple sccf-triple. In the same way as in the proof of Lemma 22.2 we can show that Typ Rτ (7,2,m) = ε. Let z = (ν, f i1 , . . . , f in ) ∈ Σρ1 . Set α = f i1 · · · f in . Then ψ(α) = rm (n − tm (α)). Let R = D1 ∪ · · · ∪ Dq be a rm -partition of the word α such that tm (R) = tm (α). We assume, for the definiteness, that tm (Di ) > 0 for i = 1, . . . , p and tm (Di ) = 0 for i = p + 1, . . . , q. Let |Di | = n i for i = 1, . . . , p, and D p+1 ∪ · · · ∪ Dq = { f j1 , . . . , f jk }. Set z 0 = (dk , f j1 , . . . , f jk ). We now define a problem schema z i for i = 1, . . . , p. Let Di = { f a1 , . . . , f ani }. Then z i = (dni , f a1 , . . . , f ani ). Taking into account that ψ is a restricted complexity function, one can show that ψρd1 ,{U(7) } (z) ≤ ψρd1 ,{U(7) } (z 0 ) +

p

ψρd1 ,{U(7) } (z i ) .

i=1

Using (22.1) we conclude that ψρd1 ,{U(7) } (z i ) ≤ rm l(n i ) for i = 1, . . . , p. In the same way as in the proof of Lemma 22.2 one can show that there exists a schema Γ ∈ Cρ1 such that (Γ, z 0 ) ∈ Rρd1 ,{U(7) } , |π(ξ )| = k for any ξ ∈ Ξ (Γ ), and any subset of the set P(Γ ) with cardinality rm is a proper rm -tuple. Set u = k − k/rm  rm . rm (u + k/rm  2ml(rm )) ≤ rm Then, evidently,  ψ(Γ ) ≤ rm (kd− k/rm  tm (rm)) = p rm rm u + k/rm  2m . Hence ψρ1 ,{U(7) } (z) ≤ rm u + i=1 l(n i ) + k/rm  2m ≤ rm   p k rm + i=1 l(n i ) + 2m . Evidently,    p p

(n i − tm (n i )) = rm k + 2ml(n i ) . k+

 ψρi 1 ,{U(7) } (z)

= ψ(α) = rm

i=1 ψρi

,{U

i=1

} (z)

n Hence ψρd1 ,{U(7) } (z) ≤ 1 2m(7) + rm2 . Therefore Ψτdi(7,2,m) (n) ≤ 2m + rm2 for any n ∈ di 2 Arg Ψτ (7,2,m) . Set cm = 2mrm + 1. Let n ∈ ω(cm ). It is clear that the value Ψτdi(7,2,m) (n) n is definite and Ψτdi(7,2,m) (n) ≤ 2m + rm2 < mn . Hence Typ Ψτdi(7,2,m) = χ . Evidently, Typ Nτ (7,2,m) = ε. Using Proposition 19.2 we obtain Typ Ψτdi(7,2,m) = λ. Using Propo sition 21.9 we conclude that Typ Ψτ (7,2,m) = Tp7. Thus, τ (7, 2, m) ∈ W s (7).

22.3 Sccf-Triples from W s (2) and W s (9) We define the sccf-triple τ (2.1.1) = (ρ1 , {U(2.1.1) }, (2, 1, 1)ψ) in the same way as in Sect. 13.2. Evidently, (2, 1, 1)ψ = h. Let n ∈ ω \ {0}. We now define a problem schema σn = (νn , f 1 , . . . , f n ) ∈ Σρ1 . Let δ¯ = (δ1 , . . . , δn ) ∈ E 2n . Let there exists k ∈

254

22 Realizable Global Upper Types of Sccf-Triples

¯ = {0} {0, . . . , n} such that δ1 = · · · = δk = 0 and δk+1 = · · · = δn = 1. Then νn (δ) ¯ = {1} if k is odd. If such k does not exist, then νn (δ) ¯ = {0}. if k is even, and νn (δ) Evidently, σn ∈ Στs(2,1,1) . Lemma 22.4 The sccf-triple τ (2, 1, 1) belongs to the set W s (2). For any n ∈ ω \ {0}, n ∈ Arg Ψτdi(2,1,1) , Ψτdi(2,1,1) (n) = l(n), h dρ1 ,{U(2,1,1) } (σn ) = l(n). For any n ∈ ω \ {0}, n ∈ Arg Ψτas(2,1,1) and Ψτas(2,1,1) (n) ≤ 2. For any n ∈ ω \ {0}, n ∈ Arg Ψτaa (2,1,1) and Ψτaa (2,1,1) (n) ≤ 2. Proof Let n ∈ ω \ {0}. Evidently, n ∈ Arg Ψτdi(2,1,1) . Let z ∈ Σρ1 and h iρ1 ,{U(2,1,1) } (z) ≤ n. Then h dρ1 ,{U(2,1,1) } (z) ≤ l(n). Therefore Ψτdi(2,1,1) (n) ≤ l(n). Evidently, h iρ1 ,{U(2,1,1) } (σn ) = n . Therefore h dρ1 ,{U(2,1,1) } (σn ) ≤ l(n). Let Γ ∈ Cρ1 , (Γ, σn ) ∈ Rρd1 ,{U(2,1,1) } and h(Γ ) = h dρ1 ,{U(2,1,1) } (σn ). Set T = Tρ1 (α(σn , P(Γ )), {U(2,1,1) }). Using Theorem 18.1 we obtain h dρ1 (T ) = h dρ1 ,{U(2,1,1) } (σn ). One can show that Θρ,h (T ) = n. By Theorem 3.3, h dρ1 (T )

≥ log2 (n + 1). Therefore h dρ1 (T ) ≥ log2 (n + 1) = l(n). Hence h dρ1 ,{U(2,1,1) } (σn ) = l(n) and Ψτdi(2,1,1) (n) = l(n). Evidently, Typ Ψτdi(2,1,1) = λ. It is clear that m h (σn ) = 1 for any n ∈ ω \ {0}. Therefore Rτ (2,1,1) (1) = ∞ and Typ Rτ (2,1,1) = ω. By Proposition 21.4, Typ Ψτ (2,1,1) = Tp2. Let n ∈ ω \ {0}. Evidently, n ∈ Arg Ψτas(2,1,1) . Let z ∈ Στs(2,1,1) . One can show that a ψρ1 ,{U(2,1,1) } (z) ≤ 2. Therefore Ψτas(2,1,1) (n) ≤ 2. Let n ∈ ω \ {0}. Evidently, n ∈ Arg Ψτaa (2,1,1) . Let z ∈ Σρ1 . One can show that a  ψρ1 ,{U(2,1,1) } (z) ≤ 2. Therefore Ψτaa (2,1,1) (n) ≤ 2. Let w : ω → ω be a nondecreasing function such that w(0) = w(1) = w(2) = w(3) = 0 and l(n) + 1 ≤ w(n) ≤ n for any n ∈ ω(4). Set D(w) = {w(n) : n ∈ ω(4)}. Let d ∈ D(w). Evidently, d > 0. One can show that the function l has the value d on exactly 2d−1 numbers 2d−1 , . . . , 2d − 1. Let us show that w  the function  has the value d on at most 2d−1 numbers. Let w(m) = d. Then d ≥ log2 m + 2 and d > log2 m + 1. Therefore m < 2d−1 . Let i 1 , . . . , i m be all numbers in the increasing order on which the function w has the value d. Let n 1 , . . . , n k be all numbers in the increasing order on which the function l has the value d. As it was mentioned above, m ≤ k. Set G(d) = {gn 1 , . . . , gn m }. For each gn j ∈ G(d), we define the value r (gn j ) as follows: r (gn 1 ) = i 1 , . . . , r (gn m ) = i m . Let j ∈ {1, . . . , m}. Since w(i j ) = d and w(n) ≤ n for any n ∈ ω \ {0}, we have d ≤ i j . Therefore r (gn j ) ≥ l(n j ) .

(22.2)

22.3 Sccf-Triples from W s (2) and W s (9)

255

 Set G w = d∈D(w) G(d) and ρw = (F0 ∪ G w , 2). We define an information system U(2,3,w) of the signature ρw as follows: U(2,3,w) = (Q, (2, 3, w)γ ), (2, 3, w)γ ( f i ) = li for any i ∈ ω \ {0}, and (2, 3, w)γ (gn ) = ϕ(σn ,U(2,1,1) ) for any gn ∈ G w . We define a weighted depth (2, 3, w)ψ of the signature ρw as follows: (2, 3, w)ψ( f i ) = 1 for any f i ∈ F0 , and (2, 3, w)ψ(gn ) = r (gn ) for any gn ∈ G w . Denote τ (2, 3, w) = (ρw , U(2,3,w) , (2, 3, w)ψ). Lemma 22.5 Let w : ω → ω be a nondecreasing function such that w(0) = w(1) = w(2) = w(3) = 0 and l(n) + 1 ≤ w(n) ≤ n for any n ∈ ω(4). Then the following statements hold: di (a) For any n ∈ ω \ {0}, n ∈ Arg Ψτdi(2,3,w) , n ∈ Arg Ψτdd (2,3,w) , w(n) ≤ Ψτ (2,3,w) (n) ≤ dd w(n) + 2, and Ψτ (2,3,w) (n) = n. (b) If Typ w + 2 = λ, then τ (2, 3, w) ∈ W s (2). (c) If Typ w = χ , then τ (2, 3, w) ∈ W s (9).

Proof Set ρ = ρw , U = U(2,3,w) , ψ = (2, 3, w)ψ, and τ = τ (2, 3, w). Let gn ∈ G w . Set κn = (e, gn ), where e((δ)) = {δ} for any (δ) ∈ E 21 . Using (22.1) and Lemma 22.4 d we conclude that there exists Γn ∈ Cρ such that P(Γn ) ⊆ F0 , (Γn , κn ) ∈ Rρ,{U }, d d and ψ(Γn ) ≤ ψ(gn ). Let z ∈ Σρ , Γ ∈ Cρ , (Γ, z) ∈ Rρ,{U } , and ψ(Γ ) = ψρ,{U } (z). Taking into account that ψ(Γn ) ≤ ψ(gn ) for any gn ∈ G w and ψ is a weighted depth it is not difficult to transform Γ by changing of all gn to Γn such that for the d obtained schema Γ  the following conditions hold: P(Γ  ) ⊆ F0 , (z, Γ  ) ∈ Rρ,{U }, d  and ψ(Γ ) = ψ(Γ ). From here and from Lemma 22.4 it follows that ψρ,{U } (κn ) = d l(n) for any gn ∈ G w , and ψρ,{U } (σn ) = l(n) for any n ∈ ω \ {0}. Let n ∈ ω \ {0}. If n ≤ 3, then, evidently, n ∈ Arg Ψτdi and Ψτdi (n) ≥ 0 = w(n). Let n ≥ 4 and w(n) = d. Then there exists gi ∈ G w such that ψ(gi ) = n and l(i) = d. i d di di Evidently, ψρ,{U } (κi ) = n and ψρ,{U } (κi ) = d. Hence n ∈ Arg Ψτ and Ψτ (n) ≥ w(n). i We now show that Ψτdi (n) ≤ w(n) + 2. Let z ∈ Σρ and ψρ,{U } (z) ≤ n. Let, for the definiteness, z = (ν, f i1 , . . . , f im , g j1 , . . . , g js ) and k = max{ j1 , . . . , js }. One can show that there exists a mapping ν  : E 2m+k → P(ω) such that, for the problem d schema z  = (ν  , f i1 , . . . , f im , f 1 , . . . , f k ), ϕ(z,U ) ≡ ϕ(z  ,U ) . Therefore ψρ,{U } (z) = d d   ψρ,{U } (z ). Using Lemma 22.4 we obtain ψρ,{U } (z ) ≤ l(m + k). We now evaluate the number k. Evidently, m + ψ(gk ) ≤ n. Let ψ(gk ) = a. Then w(a) = l(k). Evidently, a ≤ n − m. Since w is a nondecreasing function, l(k) ≤ w(n − m). Therefore log2 k ≤ w(n − m) − 1. Hence log2 k < w(n − m) and k < 2w(n−m) . d  taking into account that l is a Using the inequality ψρ,{U } (z ) ≤ l(m + k) and    d  nondecreasing function we obtain ψρ,{U } (z ) ≤ log2 m + 2w(n−m) + 1. Set c =     d d   max m, 2w(n−m) . Then ψρ,{U } (z ) ≤ log2 (2c) + 1. Let c = m. Then ψρ,{U } (z ) ≤     d  log2 m + 2 ≤ log2 n + 2 = l(n) + 1 ≤ w(n). Let c = 2w(n−m) . Then ψρ,{U } (z )   d w(n−m)+1 + 1 = w(n − m) + 2 ≤ w(n) + 2. Therefore ψρ,{U } (z) = ≤ log2 2 d  (z ) ≤ w(n) + 2. Hence Ψτdi (n) ≤ w(n) + 2. ψρ,{U }

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Let n ∈ ω \ {0}. Evidently, n ∈ Arg Ψτdd . One can show that Nτ (1) = ∞. Using Lemma 19.8 we obtain Ψτdd (n) ≥ n. It is clear that Ψτdd (n) ≤ n. Therefore Ψτdd (n) = n. d Let n ∈ ω \ {0}. As it was mentioned above, σn ∈ Στs and ψρ,{U } (σn ) = l(n). It is clear that m ψ (σn ) = 1. Hence Rτ (1) = ∞ and Typ Rτ = ω. One can show that s s ψρ,{U } (z) ≤ 2 for any problem schema z ∈ Στ . Therefore Typ Pτ = ε. Evidently, τ is a simple sccf-triple. Let Typ w + 2 = λ. By proved above, Typ Ψτdi = λ and Typ Rτ = ω. Using Proposition 21.4 we obtain Typ Ψτ = Tp2. Therefore τ ∈ W s (2). Let Typ w = χ . By proved above, Typ Ψτdi = χ , Typ Rτ = ω, and Typ Pτ = ε.  Using Proposition 21.11 we obtain Typ Ψτ = Tp9. Therefore τ ∈ W s (9). Corollary 1 The sets W s (2) and W s (9) are nonempty sets.

22.4 Sccf-Triples from W s (8) Let U(7) be the information system of the signature ρ1 defined in Sect. 22.2. Denote τ (8) = (ρ1 , {U(7) }, h). Lemma 22.6 The sccf-triple τ (8) belongs to the set W s (8) and, for any n ∈ ω \ {0}, n ∈ Arg Ψτdi(8) and Ψτdi(8) (n) = n. Proof In the same way as in the proof of Lemma 22.2 one can show that Typ Rτ (8) = ε. Let n ∈ ω \ {0}. In the proof of Lemma 22.2 we shown that there exists a problem schema z = (ν, f i1 , . . . , f in ) ∈ Σρ1 such that h dρ1 ,{U(7) } (z) ≥ n. Evidently, h iρ1 ,{U(7) } (z) = n. Therefore Ψτdi(8) (n) = n. Hence Typ Ψτdi(8) = χ . By Proposition 21.10, Typ Ψτ (8) = Tp8. It is clear that τ (8) is a simple sccf-triple. Therefore  τ (8) ∈ W s (8).

22.5 Sccf-Triples from W s (4) We define an information system U(4,5,1) of the signature ρ1 as follows: U(4,5,1) = (Q, γ(4,5,1) ) and γ(4,5,1) ( f i ) = pi for any i ∈ ω \ {0}. We now define a weighted depth (4, 5, 1)ψ of the signature ρ1 : for any i ∈ ω \ {0}, (4, 5, 1)ψ( f i ) = i. Denote τ (4, 5, 1) = (ρ1 , {U(4,5,1) }, (4, 5, 1)ψ). Lemma 22.7 The sccf-triple τ (4, 5, 1) belongs to the set W s (4) and, for any n ∈ ω \ {0}, n ∈ Arg Ψτsi(4,5,1) and Ψτsi(4,5,1) (n) = n. Proof Evidently, Typ Nτ (4,5,1) = ε and Typ Nτ (4,5,1) = ω. By Proposition 21.6, Typ Ψτ (4,5,1) = Tp4 .

22.5 Sccf-Triples from W s (4)

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It is clear that τ (4, 5, 1) is a simple sccf-triple. Hence τ (4, 5, 1) ∈ W s (4). Let n ∈ ω \ {0}. Set z n = (e, f n ), where e : E 21 → P(ω), and e((δ)) = {δ} for any δ ∈ E 2 . Evidently, z n ∈ Στs(4,5,1) and ψρi 1 ,{U(4,5,1) } (z n ) = n. Therefore n ∈ Arg Ψτsi(4,5,1) . One can show that ψρs1 ,{U(4,5,1) } (z n ) = n. Thus, Ψτsi(4,5,1) (n) = n.  Let f ∈ F R. We define an information system U(4,6, f ) of the signature ρ1 as follows: U(4,6, f ) = (Q, γ(4,6, f ) ) and γ(4,6, f ) ( f i ) = pi for any i ∈ ω \ {0}. Taking into account that the function f is not bounded from above one can show that there exists , a2 , . . . such that, for any i ∈ ω \ {0}, ai ∈ ω \ {0}, ai ∈ Arg f , ai < a sequence a1 ai+1 − 1, and ij=1 a j < f (ai+1 − 1). We define a weighted depth (4, 6, f )ψ of the signature ρ1 as follows: (4, 6, f )ψ( f i ) = ai for any i ∈ ω \ {0}. Denote τ (4, 6, f ) = (ρ1 , {U(4,6, f ) }, (4, 6, f )ψ). Lemma 22.8 For any function f ∈ F R, τ (4, 6, f ) ∈ W s (4) and, for any b ∈ {d, a, s} and c ∈ {i, d, a, s}, the relation Ψτbc(4,6, f )  f does not hold. Proof Set U = U(4,6, f ) , ψ = (4, 6, f )ψ, and τ = τ (4, 6, f ). It is not difficult to show that Typ Nτ = ε and Typ Nτ = ω. By Proposition 21.6, Typ Ψτ = Tp4. It is clear that τ is a simple sccf-triple. Therefore τ ∈ W s (4). Let b ∈ {a, s}, i ∈ ω \ {0}, z ∈ Στb , and ψρb1 ,{U } (z) ≤ ai+1 − 1. Let (Γ, z) ∈ Rρb1, {U } and ψ(Γ ) ≤ ai+1 − 1. Set T = Tρ1 (α(z, P(Γ )), {U }). Using Theorem 18.1 we obtain (Γ, T ) ∈ Rρb1 . From here and from Lemma 3.2 it follows that the set P(Γ ) is a test for the table T . Let P(Γ ) = { f i1 , . . . , f im }. Set α = f i1 · · · f im . Taking into account that ψ is a restricted complexity function we conclude that ψ( fi j ) ≤ ai+1 − 1  for j = 1, . . . , m. Therefore ψ(α) ≤ ij=1 a j < f (ai+1 − 1). Evidently, α is an unconditional test for the table T . Therefore Θρ,ψ (T ) < f (ai+1 − 1). By Lemma 3.1, ψρd1 (T ) < f (ai+1 − 1). Using Theorem 18.1 we obtain ψρd1 ,{U } (z) < f (ai+1 − 1). It is clear that ai+1 − 1 ∈ Arg Ψτdb . Hence Ψτdb (ai+1 − 1) < f (ai+1 − 1). Therefore the relation Ψτdb  f does not hold. Let b ∈ {d, a, s} and c ∈ {i, d, a, s}. Using Lemma 19.2 we conclude that ˜ τda (n) Ψτbc (n) 0 and {i : i ∈ ω \ {0}, ai ≤ n} = {1, . . . , m}. Evidently, m = 2q(n) − 1. Set z = (dm , f 1 , . . . , f m ). One can show that ψρa2 ,{U } (z) ≤ n and Nρ2 ,{U } (z) = m + 1 = 2q(n) . In the same way as in the proof of Lemma 19.5 we obtain ψρd2 ,{U } (z) ≥ log2 Nρ2 ,{U } (z) = q(n). Therefore Ψτda (n) ≥ q(n). Let z ∈ Σρ2 and ψρa2 ,{U } (z) ≤ n. We will show that ψρd2 ,{U } (z) ≤ nq(n). If q(n) = 0, then ψρa2 ,{U } (z) = 0 and ψρd2 ,{U } (z) = 0 = nq(n). Let q(n) > 0. Let (Γ, z) ∈ Rρa2 ,{U } and ψ(Γ ) = ψρa2 ,{U } (z). Taking into account that ψ is a restricted complexity function we conclude that, if ρ ∈ P(Γ ), then ψ(ρ) ≤ n. Hence P(Γ ) ⊆ { f 1 , . . . , f m , g1 , . . . , gm−1 }, where {1, . . . , m} = {i : i ∈ ω \ {0}, ai ≤ n}. It is not difficult to show that there exists a mapping ν : E 2m → P(ω) such that, for the problem schema z  = (ν, f 1 , . . . , f m ), for any b ∈ Q, ϕ(z  ,U ) (b) ⊆ ϕ(z,U ) (b). Therefore ψρd2 ,{U } (z) ≤ ψρd2 ,{U } (z  ). One can show that there exists Γ  ∈ Cρ2 such that (Γ  , z  ) ∈   Rρd2 ,{U } , P(Γ  ) ⊆ { f 1 , . . . , f m }, and h(Γ  ) ≤ l(m) = log2 m + 1 = q(n). Evidently, ψ(Γ  ) ≤ nq(n). Therefore ψρd2 ,{U } (z) ≤ nq(n) and Ψτda (n) ≤ nq(n). 

22.6 Sccf-Triples from W s (6) Let τ (6.1.1) be the sccf-triple defined in Sect. 13.2. Lemma 22.11 The sccf-triple τ (6, 1, 1) belongs to the set W s (6) and, for any n ∈ da ω \ {0}, n ∈ Arg Ψτda (6,1,1) and Ψτ (6,1,1) (n) ≤ n. Proof By Lemma 13.6, Typ Ψˆ τ (6.1.1) = Tp6. Using Proposition 13.7 we obtain Typ Iτ (6,1,1) = ω and Typ Nτ (6,1,1) = ω. By Lemma 20.9, Typ Pτ (6,1,1) = ω. Using Proposition 21.8 we obtain Typ Ψτ (6.1.1) = Tp6. Evidently, τ (6, 1, 1) is a simple sccf-triple. Therefore τ (6, 1, 1) ∈ W s (6).

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Let n ∈ ω \ {0}. It is not difficult to show that n ∈ Arg Ψτda (6,1,1) . Using Proposition da da da ˆ ˆ ˜ Ψτ (6,1,1) (n). By Lemma 13.6, Ψτ (6,1,1) (n) ≤ n. Therefore 18.4 we obtain Ψτ (6,1,1) (n)< (n) ≤ n.  Ψτda (6,1,1) We denote by τ (6, 3) a composition of sccf-triples τ (6, 1, 1) and τ (4, 5). Let q ∈ F Z , Arg q = ω, and q(0) = 0. We denote by τ (6, 4, q) a composition of sccftriples τ (6, 1, 1) and τ (4, 8, q). Lemma 22.12 The following statements hold: (a) The sccf-triple τ (6, 3) belongs to the set W s (6) and, for any n ∈ ω \ {0}, n ∈ Arg Ψτsi(6,3) and Ψτsi(6,3) (n) = n. (b) Let q ∈ F Z , Arg q = ω, and q(0) = 0. Then τ (6, 4, q) ∈ W s (6) and, for any n ∈ ω \ {0}, n ∈ Arg Ψτda (6,4,q) and q(n) ≤ Ψτda (6,4,q) (n) ≤ n(q(n) + 2) . Proof (a) Using Lemmas 22.7, 22.11 and Proposition 21.13 we obtain τ (6, 3) ∈ W s (6). By Lemma 22.7 and Proposition 18.1, for any n ∈ ω \ {0}, n ∈ Arg Ψτsi(6,3) and Ψτsi(6,3) (n) ≥ n. Using Lemma 19.1 we conclude that Ψτsi(6,3) (n) ≤ n. Therefore Ψτsi(6,3) (n) = n. (b) Using Lemmas 22.10, 22.11 and Proposition 21.13 we obtain τ (6, 4, q) ∈ W s (6). By Lemmas 22.10, 22.11 and Proposition 18.1, for any n ∈ ω \ {0}, n ∈ da Arg Ψτda (6,4,q) and q(n) ≤ Ψτ (6,4,q) (n) ≤ max{nq(n), n} + 1. Evidently, max{nq(n), n} + 1 ≤ n(q(n) + 2) . 

22.7 Sccf-Triples from W s (3) We denote by τ (3, 5) a composition of sccf-triples τ (2, 1, 1) and τ (4, 5). Evidently log2 x ∈ F R. Denote by τ (3, 6) a composition of sccf-triples τ (2, 1, 1) and τ (4, 6, log2 x). Let f ∈ F R. Evidently, f − 4 ∈ F R. We denote by τ (3, 7, f ) a composition of sccf-triples τ (2, 1, 1) and τ (4, 6, f − 4). Let g ∈ F Z , Arg g = ω, and g(0) = 0. We denote by τ (3, 8, g) a composition of sccf-triples τ (2, 1, 1) and τ (4, 7, g). Lemma 22.13 The following statements hold: (a) The sccf-triple τ (3, 5) belongs to the set W s (3) and, for any n ∈ ω \ {0}, n ∈ Arg Ψτsi(3,5) and Ψτsi(3,5) (n) = n.

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(b) The sccf-triple τ (3, 6) belongs to the set W s (3) and the relation Ψτdi(3,6) (n)  3 + log2 n does not hold. (c) For any function f ∈ F R, τ (3, 7, f ) ∈ W s (3) and, for any b ∈ {a, s}, c ∈ {i, d, a, s}, the relation Ψτbc(3,7, f )  f does not hold. (d) Let g ∈ F Z , Arg g = ω, and g(0) = 0. Then τ (3, 8, g) ∈ W s (3) and, for any n ∈ ω \ {0}, n ∈ Arg Ψτas(3,8,g) and g(n) ≤ Ψτas(3,8,g) (n) ≤ g(n) + n + 3. Proof (a) Using Lemmas 22.4, 22.7 and Proposition 21.13 we obtain τ (3, 5) ∈ W s (3). By Proposition 18.1 and Lemmas 19.1, 22.7, for any n ∈ ω \ {0}, n ∈ Arg Ψτsi(3,5) and Ψτsi(3,5) (n) = n. (b) Using Lemmas 22.4, 22.8 and Proposition 21.13 we obtain τ (3, 6) ∈ W s (3). By Lemma 22.8, there exists an infinite set D ⊆ ω \ {0} such that, for any n ∈ D, n ∈ Arg Ψτdi(4,6,log2 x) and Ψτdi(4,6,log2 x) (n) < log2 n. Using Lemma 22.4 and Proposition 18.1 we conclude that, for any n ∈ D, n ∈ Arg Ψτdi(3,6) and Ψτdi(3,6) (n) ≤ log2 n + 2 < log2 n + 3. Thus, the relation Ψτdi(3,6) (n)  3 + log2 n does not hold. (c) Let f ∈ F R, b ∈ {a, s}, and c ∈ {i, d, a, s}. Using Lemmas 22.4, 22.8 and Proposition 21.13 we obtain τ (3, 7, f ) ∈ W s (3). By Lemma 22.8, there exists an infinite set D ⊆ ω such that, for any n ∈ D, n ∈ Arg f , f (n) > 4, n ∈ Arg Ψτbc(4,6, f −4) , and Ψτbc(4,6, f −4) (n) < f (n) − 4. Using Lemmas 19.2 and 22.4 we conclude that there exists an infinite set D1 ⊆ D such that, for any n ∈ D1 , n ∈ Arg Ψτbc(2,1,1) and Ψτbc(2,1,1) (n) ≤ 2. By Proposition 18.1, for any n ∈ D1 , n ∈ Arg Ψτbc(3,7, f ) and Ψτbc(3,7, f ) (n) ≤ max{ f (n) − 4, 2} + 1 ≤ f (n) − 1 < f (n). Thus, the relation Ψτbc(3,7, f )  f does not hold. (d) Using Lemmas 22.4, 22.9 and Proposition 21.13 we obtain τ (3, 8, g) ∈ W s (3). From Lemmas 22.4, 22.9 and Proposition 18.1 it follows that, for any n ∈ ω \ {0},  n ∈ Arg Ψτas(3,8,g) and g(n) ≤ Ψτas(3,8,g) (n) ≤ g(n) + n + 3.

22.8 Sccf-Triples from W s (5) We denote by τ (5, 1, 1) a composition of sccf-triples τ (3, 5) and τ (6, 3). Lemma 22.14 The sccf-triple τ (5, 1, 1) belongs to the set W s (5) and, for any n ∈ ω \ {0}, n ∈ Arg Ψτsi(5,1,1) and Ψτsi(5,1,1) (n) = n. Proof Using Lemmas 22.12, 22.13 and Proposition 21.13 we obtain τ (5, 1, 1) ∈ W s (5). Let n ∈ ω \ {0}. From Proposition 18.1 and Lemmas 22.12, 22.13 it follows  that n ∈ Arg Ψτsi(5,1,1) and Ψτsi(5,1,1) (n) = n.

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22.9 Sccf-Triples from W s (10) We denote by τ (10, 1) a composition of sccf-triples τ (7, 2, 2) and τ (4, 5). x ∈ F R. We denote by τ (10, 2, m) a composition Let m ∈ ω \ {0}. Evidently, 2m x ). of sccf-triples τ (7, 2, 2m) and τ (4, 6, 2m Let f ∈ F R. Evidently, f − 4 ∈ F R. We denote by τ (10, 3, f ) a composition of sccf-triples τ (7, 1, 2) and τ (4, 6, f − 4). Let g ∈ F Z , Arg g = ω, and g(0) = 0. We denote by τ (10, 4, g) a composition of sccf-triples τ (7, 1, 2) and τ (4, 7, g). Lemma 22.15 The following statements hold: (a) The sccf-triple τ (10, 1) belongs to the set W s (10) and, for any n ∈ ω \ {0}, n ∈ Arg Ψτsi(10,1) and Ψτsi(10,1) (n) = n. (b) Let m ∈ ω \ {0}. Then τ (10, 2, m) ∈ W s (10) and there exists an infinite subset D of the set ω such that, for any n ∈ D, n ∈ Arg Ψτdi(10,2,m) and Ψτdi(10,2,m) (n) < mn . (c) Let f ∈ F R. Then τ (10, 3, f ) ∈ W s (10) and if bc ∈ {ds, ai, ad, aa, as, si, sd, sa, ss} , then the relation Ψτbc(10,3, f )  f does not hold. (d) Let g ∈ F Z , Arg g = ω, and g(0) = 0. Then τ (10, 4, g) ∈ W s (10) and, for any n ∈ ω, n ∈ Arg Ψτds(10,4,g) , n ∈ Arg Ψτas(10,4,g) , and g(n) ≤ Ψτas(10,4,g) (n) ≤ Ψτds(10,4,g) (n) ≤ g(n) + n + 2 . Proof (a) Using Lemmas 22.3, 22.7 and Proposition 21.13 we obtain τ (10, 1) ∈ W s (10). By Lemma 22.7 and Proposition 18.1, for any n ∈ ω \ {0}, n ∈ Arg Ψτsi(10,1) and Ψτsi(10,1) (n) = n. (b) Using Lemmas 22.3, 22.8 and Proposition 21.13 we obtain τ (10, 2, m) ∈ W s (10). By Lemma 22.3, there exists c ∈ ω such that, for any n ∈ ω(c), n ∈ n . Using Lemma 22.8 we conclude that there Arg Ψτdi(7,2,2m) and Ψτdi(7,2,2m) (n) < 2m exists an infinite set D  ⊆ ω such that, for any n ∈ D  , n ∈ Arg Ψτdi(4,6, x ) and Ψτdi(4,6,

x 2m )

(n)
4, and Ψτbc(4,6, f −4) (n) < f (n) − 4. Set D = D  ∩ ω(m). Evidently, D is an infinite set. By Proposition 18.1, for any

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n ∈ D, n ∈ Arg Ψτbc(10,3, f ) and Ψτbc(10,3, f ) (n) ≤ f (n) − 1 < f (n). Thus, the relation Ψτbc(10,3, f )  f does not hold. (d) Using Lemmas 22.2, 22.9 and Proposition 21.13 we obtain τ (10, 4, g) ∈ W s (10). From Lemmas 22.2, 22.9, 19.2 and Proposition 18.1 it follows that, for any n ∈ ω, n ∈ Arg Ψτds(10,4,g) , n ∈ Arg Ψτas(10,4,g) , and g(n) ≤ Ψτas(10,4,g) (n) ≤  Ψτds(10,4,g) (n) ≤ g(n) + n + 2.

22.10 Sccf-Triples τ with Typ Ψτ = Tp11 Lemma 22.16 Let ρ = (F, k) be a signature and τ = (ρ, K , ψ) be a sccf-triple. Then Typ Ψτ = Tp11 if and only if there exists a constant m ∈ ω such that ψ(α) ≤ m for any word α ∈ F ∗ . Proof Let there exist a constant m ∈ ω such that ψ(α) ≤ m for any word α ∈ F ∗ . By Lemma 19.3, Typ Ψτbc = ε for any b, c ∈ {i, d, a, s}. Hence Typ Ψτ = Tp11. Let there be no constant m ∈ ω such that ψ(α) ≤ m for any word α ∈ F ∗ . By  Proposition 19.1, Typ Ψτii = χ . Therefore Typ Ψτ = Tp11.

22.11 Main Statements Lemma 22.17 Let ρ = (F, k) be a signature with a finite set F, and τ = (ρ, K , ψ) be a sccf-triple. Then the following statements hold: (a) If there exists a constant c ∈ ω such that ψ(α) ≤ c for any α ∈ F ∗ , then Typ Ψτ = Tp11. (b) If there is no constant c ∈ ω such that ψ(α) ≤ c for any α ∈ F ∗ , then Typ Ψτ = Tp1. (c) If τ is a restricted sccf-triple, then Typ Ψτ = Tp1. Proof (a) Statement (a) follows from Lemma 22.16. (b) Let there be no constant c ∈ ω such that ψ(α) ≤ c for any α ∈ F ∗ . By Proposition 19.1, Typ Ψτii = χ . Let bc ∈ {id, ia, is}. Using Proposition 20.1 we obtain Typ Ψτbc = ω. Let b ∈ {d, a, s}, z ∈ Στb and F = { f 1 , . . . , f n }. One can b (z) ≤ ψ( f 1 . . . f n ). Using Lemma 19.3 we conclude that, for any show that ψρ,K b, c ∈ {i, d, a, s}, Typ Ψτbc = ε. Therefore Typ Ψτ = Tp1. (c) Let τ be a restricted sccf-triple. Then there is no constant c ∈ ω such that  ψ(α) ≤ c for any α ∈ F ∗ . Using the statement (b) we obtain Typ Ψτ = Tp1. Theorem 22.1 Let ρ = (F, k) be a signature. Then the following statements hold: (a) If F is a finite set, then for any sccf-triple τ = (ρ, K , ψ), Typ Ψτ ∈ {Tp1, Tp11} and, for any j ∈ {1, 11}, there exists a sccf-triple τ = (ρ, K , ψ) for which Typ Ψτ = Tp j.

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(b) If F is an infinite set, then, for any sccf-triple τ = (ρ, K , ψ), Typ Ψτ ∈ {Tp1, Tp2, Tp3, Tp4, Tp5, Tp6, Tp7, Tp8, Tp9, Tp10, Tp11} and, for any j ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, there exists a sccf-triple τ = (ρ, K , ψ) for which Typ Ψτ = Tp j . Proof (a) Statement (a) follows from Lemma 22.17. (b) Let F be an infinite set and τ = (ρ, K , ψ) be a sccf-triple. Using Proposition 21.1 we obtain Typ Ψτ ∈ {Tp j : j ∈ {1, 2, . . . , 11}}. Let j ∈ {1, 2, . . . , 11}. Using Lemmas 22.1, 22.2, 22.5, 22.6, 22.7, 22.11, 22.13, 22.14, 22.15, 22.16, and 18.10 we conclude that there exists a sccf-triple τ = (ρ, K , ψ) for which Typ Ψτ = Tp j.  Theorem 22.2 Let ρ = (F, k) be a signature. Then the following statements hold: (a) If F is a finite set, then for any restricted sccf-triple τ = (ρ, K , ψ), Typ Ψτ = Tp1. (b) If F is an infinite set, then for any restricted sccf-triple τ = (ρ, K , ψ), Typ Ψτ ∈ {Tp1, Tp2, Tp3, Tp4, Tp5, Tp6, Tp7, Tp8, Tp9, Tp10} and, for any j ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, there exists a restricted sccf-triple τ = (ρ, K , ψ) for which Typ Ψτ = Tp j. Proof (a) Statement (a) follows from Lemma 22.17. (b) Let F be an infinite set and τ = (ρ, K , ψ) be a restricted sccf-triple. Using Proposition 21.2 we obtain Typ Ψτ ∈ {Tp j : j ∈ {1, 2, . . . , 10}}. Let j ∈ {1, 2, . . . , 10}. Using Lemmas 22.1, 22.2, 22.5, 22.6, 22.7, 22.11, 22.13, 22.14, 22.15, and 18.10 we conclude that there exists a restricted sccf-triple τ = (ρ, K , ψ) for which  Typ Ψτ = Tp j.

Reference 1. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity. Global approach. Fundam. Inform. 25(2), 201–214 (1996)

Chapter 23

Bounds Inside Types

In this chapter, for a given signature ρ and each possible global upper type of restricted sccf-triples Tpi, i ∈ {1, . . . , 10}, we consider the set Wρ (i) of restricted sccf-triples τ with Typ Ψτ = Tpi. For each pair (b, c) ∈ {i, d , a, s}2 such that in the matrix Tpi at the intersection of the row with index b and the column with index c either λ or χ stays, we study upper and lower bounds on the function Ψτbc true for any sccf-triple τ ∈ Wρ (i). Earlier [1], we did not investigate common lower and upper bounds for sccf-triples of a given type. Let ρ = (F, k) be a signature and i ∈ {1, . . . , 10}. Denote Wρ (i) the set of all restricted sccf-triples τ = (ρ, K, ψ) for which Typ Ψτ = Tpi. Let i ∈ {1, . . . , 10}, b ∈ {i, d , a, s}, and c ∈ {i, d , a, s}. Denote Ψ Xρbc (i) the set of all functions f ∈ GR satisfying the following condition: for any sccf-triple τ ∈ Wρ (i), Ψτbc  f . Denote Ψ Yρbc (i) the set of all functions f ∈ GR satisfying the following condition: for any sccf-triple τ ∈ Wρ (i), Ψτbc  f . We now define sets Ψ˜ Xρbc (i) ⊆ Ψ Xρbc (i) and Ψ˜ Yρbc (i) ⊆ Ψ Yρbc (i). Let f ∈ Ψ Xρbc (i). Then f ∈ Ψ˜ Xρbc (i) if and only if there exists a sccf-triple τ ∈ Wρ (i) for which Ψτbc  f . Let f ∈ Ψ Yρbc (i). Then f ∈ Ψ˜ Yρbc (i) if and only if there exists a sccf-triple τ ∈ Wρ (i) for which Ψτbc  f . For each i ∈ {1, . . . , 10} and each pair (b, c) ∈ {i, d , a, s}2 such that at the intersection of the row with index b and the column with index c in the matrix Tpi there is the element λ or the element χ , we study the sets Ψ Xρbc (i) and Ψ Yρbc (i).

23.1 Auxiliary Statements Lemma 23.1 Let ρ = (F, k) be a signature with an infinite set F, i ∈ {1, . . . , 10}, bc ∈ {ii, di, dd , ai, ad , aa, si, sd , sa, ss}, and m ∈ ω. Then x − m ∈ / Ψ Yρbc (i). Proof Using Theorem 22.2 we obtain Wρ (i) = ∅. Let τ ∈ Wρ (i). If Typ Ψτbc = ε / Ψ Yρbc (i). then, evidently, the relation x − m Ψτbc does not hold. Therefore x − m ∈ © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_23

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Let Typ Ψτbc = ε. Using Lemma 19.1 we obtain that Typ Ψτbc ∈ {λ, χ } and, for any n ∈ Arg Ψτbc , Ψτbc (n) ≤ n. Let us consider the sccf-triple τ  = (m + 2) ⊗ τ . Using Lemma 18.7 and Proposition 18.3 we obtain τ  ∈ Wρ (i), Ψτbc ∈ {λ, χ }, there exists r ∈ ω for which Arg Ψτbc = ω(r), and, for any n ∈ ω(r), Ψτbc (n) ≤ n and Ψτbc (n) ∈ {i(m + 2) : i ∈ ω}. Let n ∈ ω(r) and n ≥ 1. Then, evidently, n(m + 2) − 1 ∈ ω(r) and Ψτbc (n(m + 2) − 1) ≤ (n − 1)(m + 2) = n(m + 2) − 1 − (m + / Ψ Yρbc (i).  1). Therefore the relation x − m  Ψτbc does not hold and x − m ∈ Lemma 23.2 Let ρ = (F, k) be a signature with an infinite set F and j ∈ {1, . . . , 10}. Then x ∈ Ψ˜ Xρii (j), for any function f ∈ FR, x − f (x) ∈ Ψ Yρii (j) and, for any constant c ∈ ω, x − c ∈ / Ψ Yρii (j). Proof Let f ∈ FR. Using Proposition 19.1 we obtain x ∈ Ψ Xρii (j) and x − f (x) ∈ Ψ Yρii (j). Let c ∈ ω. Using Lemma 23.1 we obtain x − c ∈ / Ψ Yρii (j). From the results obtained in Chap. 22 it follows that there exists a sccf-triple τ = ((G, 2), {U }, ψ) ∈ W s (j) such that |G| ≤ |F|. Let a ∈ AU , p be a some symbol, and κ = (a, a, p). Set τ  = τ ⊕κ τ . Using Proposition 21.13 we obtain Typ Ψτ  = Tpj. Let n ∈ ω \ {0}. Denote zn the tuple (dn , (p, 0), . . . , (p, 0)) in which (p, 0) appears exactly n times. Evidently, zn ∈ Στd . Let τ  = (ρ  , {V }, ψ). It is clear that ψρi  ,{V } (zn ) = n. Hence, for   any n ∈ ω \ {0}, Ψτii (n) = n. Let ρ  = (G  , 2). Evidently, G   ≤ |F|. Using Lemma 18.10 we obtain there exists a sccf-triple τ  ∈ Wρ (j) such that, for any n ∈ ω \ {0},  Ψτii (n) = n. Therefore x ∈ Ψ˜ Xρii (j). Lemma 23.3 Let ρ = (F, k) be a signature with a finite set F. Then x ∈ Ψ˜ Xρii (1), / for any function f ∈ FR, x − f (x) ∈ Ψ Yρii (1) and, for any constant c ∈ ω, x − c ∈ Ψ Yρii (1). Proof Let f ∈ FR. Using Proposition 19.1 we obtain that x ∈ Ψ Xρii (1) and x − f (x) ∈ Ψ Yρii (1). Evidently, |F| ≥ 1. Using Lemmas 22.1 and 18.10 we obtain that x ∈ Ψ˜ Xρii (1) and, for any constant c ∈ ω, x − c ∈ / Ψ Yρii (1).  Lemma 23.4 Let ρ = (F, k) be a signature with an infinite set F and j ∈ {2, 3, 5, 6, 7, 8, 9, 10}. Then, for any function f ∈ FR, x − f (x) ∈ Ψ Yρdd (j). Proof Let τ ∈ Wρ (j). Using Propositions 21.3 and 21.6 we obtain Typ Nτ = ω. Let f ∈ FR. Using Lemma 19.8 we obtain Ψτdd  x − f (x). Therefore x − f (x) ∈  Ψ Yρdd (j).

23.2 Set Wρ (1) Proposition 23.1 Let ρ = (F, k) be a signature. Then x ∈ Ψ˜ Xρii (1), for any function / Ψ Yρii (1). f ∈ FR, x − f (x) ∈ Ψ Yρii (1) and, for any constant c ∈ ω, x − c ∈

23.2 Set Wρ (1)

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Proof The considered statement follows immediately from Lemmas 23.2 and 23.3. 

23.3 Set Wρ (2) Proposition 23.2 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) x ∈ Ψ˜ Xρii (2), for any function f ∈ FR, x − f (x) ∈ Ψ Yρii (2) and, for any constant c ∈ ω, x − c ∈ / Ψ Yρii (2). di (b) x − 1 ∈ Ψ Xρ (2) and x − 4 ∈ / Ψ Xρdi (2), for any function f ∈ FR, logk x − f (x) ∈ Ψ Yρdi (2) and 4 + log2 x ∈ / Ψ Yρdi (2). dd (c) x ∈ Ψ˜ Xρ (2), for any function f ∈ FR, x − f (x) ∈ Ψ Yρdd (2) and, for any constant c ∈ ω, x − c ∈ / Ψ Yρdd (2). Proof (a) Statement (a) follows from Lemma 23.2. (b) Let τ ∈ Wρ (2). Then Typ Ψτdi = λ. Therefore x − 1 ∈ Ψ Xρdi (2). Using Lemmas 22.5 and 18.10 we obtain x − 4 ∈ / Ψ Xρdi (2). Let f ∈ FR. Let τ ∈ Wρ (2). From Propositions 21.3 and 21.6 it follows that Typ Nτ = ω. By Lemma 19.6, / logk x − f (x) ∈ Ψ Yρdi (2). Using Lemmas 22.4 and 18.10 we obtain 4 + log2 x ∈ Ψ Yρdi (2). (c) Evidently, x ∈ Ψ Xρdd (2). Using Lemmas 22.5 and 18.10 we obtain x ∈ Ψ˜ Xρdd (2). Let f ∈ FR and c ∈ ω. From Lemmas 23.4 and 23.1 it follows that x − f (x) ∈ Ψ Yρdd (2) and x − c ∈ / Ψ Yρdd (2). 

23.4 Set Wρ (3) Proposition 23.3 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) x ∈ Ψ˜ Xρii (3), for any function f ∈ FR, x − f (x) ∈ Ψ Yρii (3) and, for any constant c ∈ ω, x − c ∈ / Ψ Yρii (3). (b) x ∈ Ψ˜ Xρdi (3), for any function f ∈ FR, logk x − f (x) ∈ Ψ Yρdi (3), and 4 + log2 x ∈ / Ψ Yρdi (3). (c) x ∈ Ψ˜ Xρdd (3), for any function f ∈ FR, x − f (x) ∈ Ψ Yρdd (3) and, for any constant c ∈ ω, x − c ∈ / Ψ Yρdd (3). as (d) FR ∩ Ψ Xρ (3) = ∅, FR ∩ Ψ Yρas (3) = ∅ and, for any constant m ∈ ω, m ∈ Ψ Yρas (3). (e) If bc ∈ {ai, ad , aa, si, sd , sa, ss}, then x ∈ Ψ˜ Xρbc (3), FR ∩ Ψ Yρbc (3) = ∅ and, for any constant m ∈ ω, m ∈ Ψ Yρbc (3).

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Proof Denote B = {ai, ad , aa, si, sd , sa, ss}. Let bc ∈ {di, dd } ∪ B. Using Lemma 19.1 we obtain x ∈ Ψ Xρbc (3). By Lemmas 22.13, 18.10, and 19.2, x ∈ Ψ˜ Xρbc (3). (a) Statement (a) follows from Lemma 23.2. (b) From Propositions 21.3 and 21.6 it follows that Typ Nτ = ω. By Lemma 19.6, for any f ∈ FR, logk x − f (x) ∈ Ψ Yρdi (3). Using Lemmas 22.13 and 18.10 we obtain 4 + log2 x ∈ / Ψ Yρdi (3). (c) Let f ∈ FR and c ∈ ω. From Lemma 23.4 it follows that x − f (x) ∈ Ψ Yρdd (3). From Lemma 23.1 it follows that x − c ∈ / Ψ Yρdd (3). (d) Using the statement (d) of Lemmas 22.13, 18.10 and 10.2 we obtain FR ∩ Ψ Xρas (3) = ∅. By Lemmas 22.13 and 18.10, FR ∩ Ψ Yρas (3) = ∅. Let m ∈ ω. It is clear that m ∈ Ψ Yρas (3). (e) Let bc ∈ B. Using Lemmas 22.13 (the statement (c)) and 18.10 we obtain  FR ∩ Ψ Yρbc (3) = ∅. Let m ∈ ω. Evidently, m ∈ Ψ Yρbc (3).

23.5 Set Wρ (4) Proposition 23.4 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) x ∈ Ψ˜ Xρii (4), for any function f ∈ FR, x − f (x) ∈ Ψ Yρii (4) and, for any constant m ∈ ω, x − m ∈ / Ψ Yρii (4). (b) If bc ∈ {di, dd , ai, ad , aa, si, sd , sa, ss}, then x ∈ Ψ˜ Xρbc (4), FR ∩ Ψ Yρbc (4) = ∅ and, for any m ∈ ω, m ∈ Ψ Yρbc (4). (c) If bc ∈ {da, ds, as}, then FR ∩ Ψ Xρbc (4) = ∅, FR ∩ Ψ Yρbc (4) = ∅ and, for any m ∈ ω, m ∈ Ψ Yρbc (4). Proof (a) Statement (a) follows from Lemma 23.2. Denote B = {di, dd , ai, ad , aa, si, sd , sa, ss}. Let bc ∈ B ∪ {da, ds, as}. Using Lemmas 22.8 and 18.10 we obtain FR ∩ Ψ Yρbc (4) = ∅. (b) Let bc ∈ B. Using Lemma 19.1 we obtain x ∈ Ψ Xρbc (4). From Lemmas 22.7, 18.10, and 19.2 it follows that x ∈ Ψ˜ Xρbc (4). Let m ∈ ω. Evidently, m ∈ Ψ Yρbc (4). (c) Let bc ∈ {da, ds, as}. Using Lemmas 22.9, 22.10, 18.10, and 10.2 we obtain  FR ∩ Ψ Xρbc (4) = ∅. Let m ∈ ω. It is clear that m ∈ Ψ Yρbc (4).

23.6 Set Wρ (5) Proposition 23.5 Let ρ = (F, k) be a signature with an infinite set F, and bc ∈ {ii, di, dd , ai, ad , aa, si, sd , sa, ss}. Then x ∈ Ψ˜ Xρbc (5), for any function f ∈ FR, x − f (x) ∈ Ψ Yρbc (5) and, for any constant m ∈ ω, x − m ∈ / Ψ Yρbc (5).

23.6 Set Wρ (5)

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Proof Using Lemma 19.1 we obtain x ∈ Ψ Xρbc (5). From Lemmas 22.14, 18.10, and 19.2 it follows that x ∈ Ψ˜ Xρbc (5). Let m ∈ ω. Using Lemma 23.1 we obtain x − m∈ / Ψ Yρbc (5). Let f ∈ FR. Let τ ∈ W s (5). Then Typ Ψτas = ω. Using Proposition 20.2 we obtain Typ Pτ = ω. From Lemma 19.11 it follows that there exists t ∈ ω for which Ψτss  x − t. Therefore Ψτss  x − f (x). Using Lemma 19.9 we obtain Ψτsi  x − f (x). By Lemma 19.2, Ψτbc  x − f (x). Therefore x − f (x) ∈ Ψ Yρbc (5). 

23.7 Set Wρ (6) Proposition 23.6 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) If bc ∈ {ii, di, dd , ai, ad , aa, si, sd , sa, ss}, then x ∈ Ψ˜ Xρbc (6), for any function f ∈ FR, x − f (x) ∈ Ψ Yρbc (6) and, for any constant m ∈ ω, x − m ∈ / Ψ Yρbc (6). (b) FR ∩ Ψ Xρda (6) = ∅, for any function f ∈ FR, x − f (x) ∈ Ψ Yρda (6) and, for any constant m ∈ ω, x − m ∈ / Ψ Yρda (6). Proof Denote B = {ii, di, dd , ai, ad , aa, si, sd , sa, ss}. Let bc ∈ B ∪ {da} and f ∈ FR. Let τ ∈ Wρ (6). Then Typ Ψτas = ω. Using Proposition 20.2 we obtain Typ Pτ = ω. From Lemma 19.11 it follows that there exists t ∈ ω such that, for any n ∈ Arg Ψτss , Ψτss (n) ≥ n − t. Therefore Ψτss  x − f (x). By Lemma 19.9, Ψτsi  x − f (x). Using ˜ τbc (n). Therefore Ψτbc  x − f (x). Lemma 19.2 we obtain, for any n ∈ ω, Ψτsi (n) 1, then x/c ∈ / Ψ Xρdi (7), for any function f ∈ FR, x/f (x) ∈ Ψ Yρdi (7) and, for any constant m ∈ ω \ {0}, x/m ∈ / Ψ Yρdi (7).

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(c) x ∈ Ψ Xρdd (7), x − 1 ∈ / Ψ Xρdd (7), for any function f ∈ FR, x − f (x) ∈ Ψ Yρdd (7) and, for any constant m ∈ ω, x − m ∈ / Ψ Yρdd (7). Proof The statement (a) follows from Lemma 23.2. (b) Let τ ∈ Wρ (7). Then Typ Ψτdi = λ. Therefore Ψτdi  x − 1. Hence x − 1 ∈ / Ψ Xρdi (7). Let c ∈ R and c > 1. Using Lemmas 22.2 and 18.10 we obtain x/c ∈ Ψ Xρdi (7). Let f ∈ FR. Let τ ∈ Wρ (7). Then Typ Ψτds = ε. Using Proposition 20.3 we obtain Typ Rτ = ε. From Propositions 21.3 and 21.6 it follows that Typ Nτ = ω. By Lemma 20.3, Typ Zτ = ω. From this equality and Lemma 19.7 it follows that there exists t ∈ ω \ {0} such that, for any n ∈ Arg Ψτdi , Ψτdi (n) ≥ n/t logk 2. Therefore the relation Ψτdi  x/f (x) holds. Hence x/f (x) ∈ Ψ Yρdi (7). Let m ∈ ω \ {0, 1}. Using Lemmas 22.3 and 18.10 we obtain x/m ∈ / Ψ Yρdi (7). Therefore x ∈ / Ψ Yρdi (7). dd dd (c) Evidently, x ∈ Ψ Xρ (7). Let τ ∈ Wρ (7). Then Typ Ψτ = χ . Therefore the relation Ψτdd  x − 1 does not hold. Hence x − 1 ∈ / Ψ Xρdd (7). Let f ∈ FR. From Lemma 23.4 it follows that x − f (x) ∈ Ψ Yρdd (7). Let m ∈ ω. By Lemma 23.1, x − m∈ / Ψ Yρdd (7). 

23.9 Set Wρ (8) Proposition 23.8 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) x ∈ Ψ˜ Xρii (8), for any function f ∈ FR, x − f (x) ∈ Ψ Yρii (8) and, for any constant m ∈ ω, x − m ∈ / Ψ Yρii (8). (b) x ∈ Ψ˜ Xρdi (8), for any function f ∈ FR, x/f (x) ∈ Ψ Yρdi (8) and, for any constant m ∈ ω, x − m ∈ / Ψ Yρdi (8). (c) x ∈ Ψ˜ Xρdd (8), for any function f ∈ FR, x − f (x) ∈ Ψ Yρdd (8) and, for any constant m ∈ ω, x − m ∈ / Ψ Yρdd (8). Proof The statement (a) follows from Lemma 23.2. Let bc ∈ {di, dd }. Using Lemma 19.1 we obtain x ∈ Ψ Xρbc (8). From Lemmas 22.6, 18.10, and 19.2 it follows that x ∈ Ψ˜ Xρbc (8). (b) Let τ ∈ Wρ (8). Then Typ Ψτds = ε. Using Proposition 20.3 we obtain Typ Rτ = ε. From Propositions 21.3 and 21.6 it follows that Typ Nτ = ω. By Lemma 20.3, Typ Zτ = ω. From this equality and Lemma 19.7 it follows that there exists t ∈ ω \ {0} such that, for any n ∈ Arg Ψτdi , Ψτdi (n) ≥ n/t logk 2. Let f ∈ FR. Then, evidently, Ψτdi  x/f (x). Hence x/f (x) ∈ Ψ Yρdi (8). Let m ∈ ω. Using Lemma 23.1 we obtain x − m ∈ / Ψ Yρdi (8). (c) Let m ∈ ω. Using Lemma 23.1 we obtain x − m ∈ / Ψ Yρdd (8). Let f ∈ FR. dd Using Lemma 23.4 we obtain x − f (x) ∈ Ψ Yρ (8). 

23.10 Set Wρ (9)

271

23.10 Set Wρ (9) Proposition 23.9 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) x ∈ Ψ˜ Xρii (9), for any function f ∈ FR, x − f (x) ∈ Ψ Yρii (9) and, for any constant m ∈ ω, x − m ∈ / Ψ Yρii (9). (b) x ∈ Ψ˜ Xρdi (9), for any function f ∈ FR, logk x − f (x) ∈ Ψ Yρdi (9), and 4 + log2 x ∈ / Ψ Yρdi (9). (c) x ∈ Ψ˜ Xρdd (9), for any function f ∈ FR, x − f (x) ∈ Ψ Yρdd (9) and, for any constant m ∈ ω, x − m ∈ / Ψ Yρdd (9). Proof The statement (a) follows from Lemma 23.2. Let bc ∈ {di, dd }. Evidently, x ∈ Ψ Xρbc (9). Let w : ω → ω, w(0) = w(1) = w(2) = w(3) = 0 and, for any n ∈ ω(4), w(n) = n. It is clear that, for any n ∈ ω(4), l(n) + 2 ≤ w(n) ≤ n. Using Lemma 22.5 we obtain τ (2, 3, w) ∈ W s (9) and, for any n ∈ ω(4), Ψτbc(2,3,w) (n) = n. By Lemma 18.10, x ∈ Ψ˜ Xρbc (9). (b) Let f ∈ FR. Let τ ∈ Wρ (9). From Propositions 21.3 and 21.6 it follows that Typ Nτ = ω. Using Lemma 19.6 we obtain logk x − f (x) ∈ Ψ Yρdi (9). It is not difficult to show that there exists a nondecreasing function w : ω → ω possessing the following properties: w(0) = w(1) = w(2) = w(3) = 0, for any n ∈ ω(4), l(n) + 1 ≤ w(n) ≤ n, Typ w = χ , and there exists an infinite subset D of the set ω such that, for any n ∈ D, w(n) = l(n) + 1. Using Lemma 22.5 we obtain τ (2, 3, w) ∈ Wρ (9) and, / Ψ Yρdi (9). for any n ∈ D, Ψτdi(2,3,w) (n) ≤ l(n) + 3. By Lemma 18.10, 4 + log2 x ∈ (c) Let f ∈ FR and m ∈ ω. Using Lemmas 23.4 and 23.1 we obtain x − f (x) ∈ / Ψ Yρdd (9).  Ψ Yρdd (9) and x − m ∈

23.11 Set Wρ (10) Proposition 23.10 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) x ∈ Ψ˜ Xρii (10), for any function f ∈ FR, x − f (x) ∈ Ψ Yρii (10) and, for any constant m ∈ ω, x − m ∈ / Ψ Yρii (10). (b) x ∈ Ψ˜ Xρdi (10), for any function f ∈ FR, x/f (x) ∈ Ψ Yρdi (10) and, for any constant m ∈ ω \ {0}, x/m ∈ / Ψ Yρdi (10). (c) x ∈ Ψ˜ Xρdd (10), for any function f ∈ FR, x − f (x) ∈ Ψ Yρdd (10) and, for any constant m ∈ ω, x − m ∈ / Ψ Yρdd (10). (d) If bc ∈ {ds, as}, then FR ∩ Ψ Xρbc (10) = ∅, FR ∩ Ψ Yρbc (10) = ∅ and, for any m ∈ ω, m ∈ Ψ Yρbc (10). (e) If bc ∈ {ai, ad , aa, si, sd , sa, ss}, then x ∈ Ψ˜ Xρbc (10), FR ∩ Ψ Yρbc (10) = ∅ and, for any m ∈ ω, m ∈ Ψ Yρbc (10).

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23 Bounds Inside Types

Proof The statement (a) follows from Lemma 23.2. Denote B = {ai, ad , aa, si, sd , sa, ss}. Let bc ∈ {di, dd } ∪ B. By Lemma 19.1, x ∈ Ψ Xρbc (10). Using the statement (a) of Lemma 22.15 and Lemmas 19.2 and 18.10 we obtain x ∈ Ψ˜ Xρbc (10). (b) Let f ∈ FR. Let τ ∈ Wρ (10). Then Typ Ψτds = χ . Using Proposition 20.3 we obtain Typ Rτ = ε and Typ Rτ = ω. From Propositions 21.3 and 21.6 it follows that Typ Nτ = ω. By Lemma 20.3, Typ Zτ = ω. From this equality and Lemma 19.7 it follows that there exists t ∈ ω \ {0} such that, for any n ∈ Arg Ψτdi , Ψτdi (n) ≥

n/t logk 2. Therefore the relation Ψτdi  x/f (x) holds. Hence x/f (x) ∈ Ψ Yρdi (10). Let m ∈ ω \ {0}. Using the statement (b) of Lemmas 22.15 and 18.10 we obtain x/m ∈ / Ψ Yρdi (10). (c) Let f ∈ FR and m ∈ ω. Using Lemma 23.4 we obtain x − f (x) ∈ Ψ Yρdd (10). From Lemma 23.1 it follows that x − m ∈ / Ψ Yρdd (10). (d) Let bc ∈ {ds, as}. Using the statement (d) of Lemma 22.15 and Lemmas 18.10 and 10.2 we obtain FR ∩ Ψ Xρbc (10) = ∅. Using the statement (c) of Lemma 22.15 and Lemma 18.10 we obtain FR ∩ Ψ Yρbc (10) = ∅. Let m ∈ ω. It is clear that m ∈ Ψ Yρbc (10). (e) Let bc ∈ B. Using the statement (c) of Lemma 22.15 and Lemma 18.10 we  obtain FR ∩ Ψ Yρbc (10) = ∅. Let m ∈ ω. It is clear that m ∈ Ψ Yρbc (10).

Reference 1. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity. Global Approach. Fundam. Inform. 25(2), 201–214 (1996)

Chapter 24

Matrices of Lower Global Bounds

In this chapter, we describe all possible 11 global lower types tp1, . . . , tp11 of sccftriples which correspond to the global upper types Tp1, . . . , Tp11, respectively. We also describe all possible 10 global lower types tp1, . . . , tp10 of restricted sccftriples which correspond to the global upper types Tp1, . . . , Tp10. Some similar results were obtained in [1] for types of functions different from the upper and lower types considered in this book. For a given signature ρ, each global lower type tpi, i ∈ {1, . . . , 10}, and each pair (b, c) ∈ {i, d, a, s}2 such that in the matrix tpi at the intersection of the row with index b and the column with index c either μ or γ stays, we study upper and lower bounds on the function Φτbc true for any sccf-triple τ ∈ Wρ (i).

24.1 Possible Global Lower Types Proposition 24.1 Let τ = (ρ, K , ψ) be a sccf-triple, b ∈ {i, d, a, s}, and c ∈ {i, d, a, s}. Then the following statements hold: (a) (b) (c) (d)

typ Φτbc typ Φτbc typ Φτbc typ Φτbc

= ε if and only if Typ Ψτcb = ω. = γ if and only if Typ Ψτcb = χ . = μ if and only if Typ Ψτcb = λ. = ω if and only if Typ Ψτcb = ε.

Proof The considered statements follow immediately from Proposition 10.5.



Let us consider the following five matrices

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_24

273

274

24 Matrices of Lower Global Bounds

i tp7 = d a s

i γ ε ε ε

d a μω γ ω ε ω ωω

s ω ω ω ω

i tp10 = d a s

i tp8 = d a s i γ ε ε ε

d γ γ ε γ

a γ γ γ γ

s γ γ γ γ

i γ ε ε ε

d γ γ ε ω

a ω ω ω ω

s ω ω ω ω

i tp11 = d a s

i tp9 = d a s i ω ω ω ω

d ω ω ω ω

a ω ω ω ω

i γ ε ε ε

d γ γ ε ε

a ω ω ω ω

s ω ω ω ω

s ω ω ω ω

Proposition 24.2 Let τ = (ρ, K , ψ) be a sccf-triple and j ∈ {1, 2, . . . , 11}. Then Typ Ψτ = Tp j if and only if typ Φτ = tp j. Proof The considered statement follows immediately from Proposition 24.1.



Theorem 24.1 Let ρ = (F, k) be a signature. Then the following statements hold: (a) If F is a finite set, then for any sccf-triple τ = (ρ, K , ψ), typ Φτ ∈ {tp1, tp11} and, for any j ∈ {1, 11}, there exists a sccf-triple τ = (ρ, K , ψ) for which typ Φτ = tp j. (b) If F is an infinite set, then for any sccf-triple τ = (ρ, K , ψ), typ Φτ ∈ {tp1, tp2, . . . , tp11} and, for any j ∈ {1, 2, . . . , 11}, there exists a sccf-triple τ = (ρ, K , ψ) for which typ Φτ = tp j. Proof The considered statements follow immediately from Theorem 22.1 and Proposition 24.2.  Theorem 24.2 Let ρ = (F, k) be a signature. Then the following statements hold: (a) If F is a finite set, then for any restricted sccf-triple τ = (ρ, K , ψ), typ Φτ = tp1. (b) If F is an infinite set, then for any restricted sccf-triple τ = (ρ, K , ψ), typ Φτ ∈ {tp1, tp2, . . . , tp10} and, for any j ∈ {1, 2, . . . , 10}, there exists a restricted sccf-triple τ = (ρ, K , ψ) for which typ Φτ = tp j. Proof The considered statements follow immediately from Theorem 22.2 and Proposition 24.2. 

24.2 Auxiliary Statements Let ρ = (F, k) be a signature, j ∈ {1, 2, . . . , 10}, b ∈ {i, d, a, s}, and c ∈ {i, d, a, s}. Denote Φ X ρbc ( j) the set of all functions f ∈ G R satisfying the following condition: for any sccf-triple τ ∈ Wρ ( j), Φτbc  f . Denote ΦYρbc ( j) the set of all functions

24.2 Auxiliary Statements

275

f ∈ G R satisfying the following condition: for any sccf-triple τ ∈ Wρ ( j), Φτbc  ˜ ρbc ( j) ⊆ ΦYρbc ( j). Let f ∈ f . We now define sets Φ˜ X ρbc ( j) ⊆ Φ X ρbc ( j) and ΦY bc bc ˜ Φ X ρ ( j). Then f ∈ Φ X ρ ( j) if and only if there exists a sccf-triple τ ∈ Wρ ( j) for ˜ ρbc ( j) if and only if there exists a which Φτbc  f . Let f ∈ ΦYρbc (i). Then f ∈ ΦY bc sccf-triple τ ∈ Wρ (i) for which Φτ  f . Let i ∈ {1, . . . , 11} and (b, c) ∈ {i, d, a, s}2 . Denote Tp( j, b, c) the element of the matrix Tp j at the intersection of the row with index b and the column with index c. Denote tp( j, b, c) the element of the matrix tp j at the intersection of the row with index b and the column with index c. Lemma 24.1 Let ρ = (F, k) be a signature, (b, c) ∈ {i, d, a, s}2 , j ∈ {1, 2, . . . , 10}, Wρ ( j) = ∅, Tp( j, b, c) = ω, Tp( j, b, c) = ε, and m ∈ ω. Then the following statements hold: (a) Let x − m ∈ Ψ X ρbc ( j). Then x + m ∈ ΦYρcb ( j). / ΦYρcb ( j). (b) Let x − m ∈ / Ψ X ρbc ( j). Then x + m ∈ Proof Using Proposition 24.2 we obtain tp( j, c, b) = ε and tp( j, c, b) = ω. (a) Let τ ∈ Wρ ( j). Then Ψτbc  x − m. Evidently, x − m ∈ F R and (x − m)−1 = x + m (we consider here the restriction of x − m to R(m) and the restriction of x + m to R(0)). From Proposition 10.6 it follows that Φτcb  x + m. Therefore x + m ∈ ΦYρcb ( j). (b) Let us assume that the considered statement does not hold. Let x + m ∈ ΦYρcb ( j) and τ ∈ Wρ ( j). Then Φτcb  x + m. Using Proposition 10.7 we obtain / Ψτbc  x − m. Hence x − m ∈ Ψ X ρbc ( j) but this is impossible. Therefore x + m ∈  ΦYρcb ( j). Lemma 24.2 Let ρ = (F, k) be a signature, (b, c) ∈ {i, d, a, s}2 , j ∈ {1, 2, . . . , 10}, ˜ ρcb ( j). Wρ ( j) = ∅, Tp( j, b, c) = ω, Tp( j, b, c) = ε, and x ∈ Ψ˜ X ρbc ( j). Then x ∈ ΦY Proof By Proposition 24.2, tp( j, c, b) = ω and tp( j, c, b) = ε. Using Lemma 24.1 we obtain x ∈ ΦYρcb ( j). Since x ∈ Ψ˜ X ρbc ( j), there exists a sccf-triple τ ∈ Wρ ( j) and a number m ∈ ω such that, for any n ∈ ω(m), Ψτbc (n) = n. From Lemma 10.1 ˜ ρcb ( j).  it follows that Φτcb  x. Hence x ∈ ΦY Lemma 24.3 Let ρ = (F, k) be a signature, (b, c) ∈ {i, d, a, s}2 , j ∈ {1, 2, . . . , 10}, / Wρ ( j) = ∅, Tp( j, b, c) = ε, Tp( j, b, c) = ω, and, for any constant m ∈ ω, x − m ∈ / Φ X ρcb ( j). Ψ Yρbc ( j). Then, for any constant m ∈ ω, x + m ∈ Proof Let us assume the contrary. Let there exist m ∈ ω for which x + m ∈ Φ X ρcb ( j). Let τ ∈ Wρ ( j). Then Φτcb  x + m. Evidently, the restriction of x + m to R(0) belongs to F R and (x + m)−1 = x − m. Using Proposition 24.2 we obtain tp( j, c, b) = ω and tp( j, c, b) = ε. From Proposition 10.7 it follows that Ψτbc  x −  m − 1. Hence x − m − 1 ∈ Ψ Yρbc ( j) but this is impossible.

276

24 Matrices of Lower Global Bounds

Lemma 24.4 Let ρ = (F, k) be a signature, (b, c) ∈ {i, d, a, s}2 , j ∈ {1, 2, . . . , 10}, Wρ ( j) = ∅, Tp( j, b, c) = ω,Tp( j, b, c) = ε, and, for any function f ∈ F R, x − f (x) ∈ Ψ Yρbc ( j). Then, for any function f ∈ F R, x + f (x) ∈ Φ X ρcb ( j). Proof Using Proposition 24.2 we obtain tp( j, c, b) = ω and tp( j, c, b) = ε. Let τ ∈ Wρ ( j). Let us show that there is a constant m ∈ ω for which Ψτbc  x − m. Let us assume the contrary: for each constant m ∈ ω, the relation Ψτbc  x − m does not hold. Then there is an infinite sequence n 1 , n 2 , . . . possessing the following properties: (a) For each i ∈ ω \ {0}, n i ∈ Arg Ψτbc , n i < n i+1 , and n i − Ψτbc (n i ) < n i+1 − Ψτbc (n i+1 ). (b) n 1 − Ψτbc (n 1 ) ≥ 2. We now define a function ϕ : ω → ω. Let n ∈ ω. If n ≤ n 1 , then ϕ(n) = 1. If, for some i ∈ ω \ {0}, n i ≤ n < n i+1 , then ϕ(n) = n i − Ψτbc (n i ) − 1. Evidently, ϕ ∈ F Z and Arg ϕ = ω. Using Lemma 10.3 we obtain there exists a function g ∈ F R such that Arg g = R(0) and, for any n ∈ ω, g(n) ≤ ϕ(n). It is clear that, for any i ∈ ω \ {0}, g(n i ) ≤ ϕ(n i ) < n i − Ψτbc (n i ) and, hence, Ψτbc (n i ) < n i − g(n i ). Therefore the relation Ψτbc  x − g does not hold, but this is impossible. Hence there exists a constant m ∈ ω for which Ψτbc  x − m. Using Proposition 10.6 we obtain Φτcb  x + m + 1. Hence, for any function f ∈ F R, Φτcb  x + f (x). Thus, for any  function f ∈ F R, x + f (x) ∈ Φ X ρcb ( j). Lemma 24.5 Let ρ = (F, k) be a signature, (b, c) ∈ {i, d, a, s}2 , j ∈ {1, 2, . . . , 10}, Wρ ( j) = ∅, Tp( j, b, c) = ω,Tp( j, b, c) = ε, and F R ∩ Ψ X ρbc ( j) = ∅. Then F R ∩ ΦYρcb ( j) = ∅. Proof Let us assume that the considered statement does not hold. Let f ∈ F R ∩ ΦYρcb ( j). Using Proposition 24.2 we obtain tp( j, c, b) = ε and tp( j, c, b) = ω. By  Proposition 10.7, f −1 ∈ F R ∩ Ψ X ρbc ( j) but this is impossible. Lemma 24.6 Let ρ = (F, k) be a signature, (b, c) ∈ {i, d, a, s}2 , j ∈ {1, 2, . . . , 10}, Wρ ( j) = ∅, Tp( j, b, c) = ω,Tp( j, b, c) = ε, and F R ∩ Ψ Yρbc ( j) = ∅. Then F R ∩ Φ X ρcb ( j) = ∅. Proof Let us assume that the considered statement does not hold. Using Proposition 24.2 we obtain tp( j, c, b) = ε and tp( j, c, b) = ω. Let f ∈ F R ∩ Φ X ρcb ( j). By  Proposition 10.7, f −1 − 1 ∈ F R ∩ Ψ Yρbc ( j) but this is impossible. Lemma 24.7 Let ρ = (F, k) be a signature, (b, c) ∈ {i, d, a, s}2 , j ∈ {1, 2, . . . , 10}, Wρ ( j) = ∅, tp( j, b, c) = ε, and tp( j, b, c) = ω. Then, for any constant m ∈ ω, m ∈ ΦYρbc ( j). Proof The proof is obvious.



Lemma 24.8 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold:

24.2 Auxiliary Statements

277

(a) Let c ∈ R and c > 1. Then cx ∈ / ΦYρid (7). (b) Let j ∈ {7, 8, 10} and f ∈ F R. Then x f (x) ∈ Φ X ρid ( j). (c) Let j ∈ {7, 10} and m ∈ ω \ {0}. Then mx ∈ / Φ X ρid ( j). Proof (a) Let us assume that there exists c ∈ R such that c > 1 and cx ∈ ΦYρid (7). Let τ ∈ Wρ (7). Then Φτid  cx. Evidently, cx ∈ F R and (cx)−1 = x/c. Using Proposition 10.7 we obtain Ψτdi  x/c. Hence x/c ∈ Ψ X ρdi (7) but this contradicts to Proposition 23.7. Therefore cx ∈ / ΦYρid (7). (b) Let τ ∈ Wρ ( j). Then Typ Ψτds = ω. Using Proposition 20.3 we obtain Typ Rτ = ω. By Propositions 21.3 and 21.6, Typ Nτ = ω. Using Lemma 20.3 we obtain Typ Z τ = ω. From this equality and Lemma 19.7 it follows that there exists t ∈ ω \ {0} such that, for any n ∈ Arg Ψτdi , Ψτdi (n) ≥ n/t logk 2. Therefore there exists m ∈ ω \ {0} for which Ψτdi  x/m. From Proposition 10.6 it follows that Φτid  mx + 1. Therefore Φτid  x f (x). Hence x f (x) ∈ Φ X ρid ( j). (c) Let us assume that mx ∈ Φ X ρid ( j). Let τ ∈ Wρ ( j). Then Φτid  mx. Using Proposition 10.7 we obtain Ψτdi  x/m − 1. Hence Ψτdi  x/(2m). Therefore / x/(2m) ∈ Ψ Yρdi ( j) which contradicts to Propositions 23.7 and 23.10. Hence mx ∈  Φ X ρid ( j). Lemma 24.9 Let ρ = (F, k) be a signature with an infinite set F and j ∈ {2, 3, 9}. / Φ X ρid ( j) and, for any function f ∈ F R, f (x)2x log2 k ∈ Φ X ρid ( j). Then 2x−5 ∈ Proof Let us assume that 2x−5 ∈ Φ X ρid ( j). Let τ ∈ Wρ ( j). Then Φτid  2x−5 . Evi−1  = 5 + log2 x. Using Proposition 10.7 we obtain dently, 2x−5 ∈ F R and 2x−5 Ψτdi  4 + log2 x. Therefore 4 + log2 x ∈ Ψ Yρdi ( j), but this contradicts to Propositions 23.2, 23.3, and 23.9. Hence 2x−5 ∈ / Φ X ρid ( j). Let τ ∈ Wρ ( j). By Propositions 21.3 and 21.6, Typ Nτ = ω. Using Lemma 19.6 we obtain there exists t ∈ ω \ {0} such that, for any n ∈ Arg Ψτdi , Ψτdi (n) ≥ logk (n/t). Therefore Ψτdi  logk (x/t). Evidently, logk (x/t) ∈ F R and (logk (x/t))−1 = tk x = t2x log2 k . From Proposition 10.6 it follows that Φτid  t2x log2 k + 1. Let f ∈ F R. Then, evidently, Φτid  f (x)2x log2 k . Hence f (x)2x log2 k ∈ Φ X ρid ( j). 

24.3 Bounds Inside Types Let ρ = (F, k) be a signature, j ∈ {1, . . . , 10} and Wρ ( j) = ∅. For each pair (b, c) ∈ {i, d, a, s}2 such that tp(i, b, c) = ω and tp(i, b, c) = ε, we study the sets Φ X ρbc ( j) and ΦYρbc ( j). Proposition 24.3 Let ρ = (F, k) be a signature. Then, for any function f ∈ F R, ˜ ρii (1). / Φ X ρii (1), and x ∈ ΦY x + f (x) ∈ Φ X ρii (1), for any constant m ∈ ω, x + m ∈ Proof The considered statement follows immediately from Proposition 23.1 and Lemmas 24.2–24.4. 

278

24 Matrices of Lower Global Bounds

Proposition 24.4 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) For any function f ∈ F R, x + f (x) ∈ Φ X ρii (2), for any constant m ∈ ω, x + ˜ ρii (2). m∈ / Φ X ρii (2), and x ∈ ΦY (b) For any function f ∈ F R, f (x)2x log2 k ∈ Φ X ρid (2), 2x−5 ∈ / Φ X ρid (2), x + 1 ∈ id id / ΦYρ (2). ΦYρ (2), and x + 4 ∈ (c) For any function f ∈ F R, x + f (x) ∈ Φ X ρdd (2), for any constant m ∈ ω, x + ˜ ρdd (2). m∈ / Φ X ρdd (2), and x ∈ ΦY Proof The considered statement follows immediately from Proposition 23.2 and Lemmas 24.1–24.4 and 24.9.  Proposition 24.5 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) For any function f ∈ F R, x + f (x) ∈ Φ X ρii (3), for any constant m ∈ ω, x + ˜ ρii (3). m∈ / Φ X ρii (3), and x ∈ ΦY (b) For any function f ∈ F R, f (x)2x log2 k ∈ Φ X ρid (3), 2x−5 ∈ / Φ X ρid (3), and x ∈ ˜ ρid (3). ΦY (c) For any function f ∈ F R, x + f (x) ∈ Φ X ρdd (3), for any constant m ∈ ω, x + ˜ ρdd (3). m∈ / Φ X ρdd (3), and x ∈ ΦY sa (d) F R ∩ Φ X ρ (3) = ∅, F R ∩ ΦYρsa (3) = ∅ and, for any m ∈ ω, m ∈ ΦYρsa (3). ˜ ρbc (3). (e) If bc ∈ {ia, da, aa, is, ds, as, ss}, then F R ∩ Φ X ρbc (3) = ∅ and x ∈ ΦY Proof The considered statement follows immediately from Proposition 23.3 and Lemmas 24.2–24.7 and 24.9.  Proposition 24.6 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) For any function f ∈ F R, x + f (x) ∈ Φ X ρii (4), for any constant m ∈ ω, x + ˜ ρii (4). m∈ / Φ X ρii (4), and x ∈ ΦY (b) If bc ∈ {id, dd, ia, da, aa, is, ds, as, ss}, then F R ∩ Φ X ρbc (4) = ∅ and x ∈ ˜ ρbc (4). ΦY (c) If bc ∈ {ad, sd, sa}, then F R ∩ Φ X ρbc (4) = ∅, F R ∩ ΦYρbc (4) = ∅ and, for any m ∈ ω, m ∈ ΦYρbc (4). Proof The considered statement follows immediately from Proposition 23.4 and Lemmas 24.2–24.7.  Proposition 24.7 Let ρ = (F, k) be a signature with an infinite set F and bc ∈ {ii, id, dd, ia, da, aa, is, ds, as, ss}. Then, for any function f ∈ F R, x + f (x) ∈ Φ X ρbc (5), ˜ ρbc (5). for any constant m ∈ ω, x + m ∈ / Φ X ρbc (5), and x ∈ ΦY

24.3 Bounds Inside Types

279

Proof The considered statement follows immediately from Proposition 23.5 and Lemmas 24.2–24.4.  Proposition 24.8 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) If bc ∈ {ii, id, dd, ia, da, aa, is, ds, as, ss}, then for any function f ∈ F R, x + ˜ ρbc (6). / Φ X ρbc (6), and x ∈ ΦY f (x) ∈ Φ X ρbc (6), for any constant m ∈ ω, x + m ∈ ad (b) For any function f ∈ F R, x + f (x) ∈ Φ X ρ (6), for any constant m ∈ ω, x + m∈ / Φ X ρad (6), F R ∩ ΦYρad (6) = ∅ and, for any m ∈ ω, m ∈ ΦYρad (6). Proof The considered statement follows immediately from Proposition 23.6 and Lemmas 24.2–24.5 and 24.7.  Proposition 24.9 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) For any function f ∈ F R, x + f (x) ∈ Φ X ρii (7), for any constant m ∈ ω, x + ˜ ρii (7). m∈ / Φ X ρii (7), and x ∈ ΦY (b) For any function f ∈ F R, x f (x) ∈ Φ X ρid (7), for any constant m ∈ ω \ {0}, / ΦYρid (7). mx ∈ / Φ X ρid (7), x + 1 ∈ ΦYρid (7) and, if c ∈ R and c > 1, then cx ∈ (c) For any function f ∈ F R, x + f (x) ∈ Φ X ρdd (7), for any constant m ∈ ω, x + / ΦYρdd (7). m∈ / Φ X ρdd (7), x ∈ ΦYρdd (7), and x + 1 ∈ Proof The considered statement follows immediately from Proposition 23.7 and Lemmas 24.1–24.4 and 24.8.  Proposition 24.10 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) For any function f ∈ F R, x + f (x) ∈ Φ X ρii (8), for any constant m ∈ ω, x + ˜ ρii (8). m∈ / Φ X ρii (8), and x ∈ ΦY (b) For any function f ∈ F R, x f (x) ∈ Φ X ρid (8), for any constant m ∈ ω, x + m ∈ ˜ ρid (8). Φ X ρid (8), and x ∈ ΦY (c) For any function f ∈ F R, x + f (x) ∈ Φ X ρdd (8), for any constant m ∈ ω, x + ˜ ρdd (8). m∈ / Φ X ρdd (8), and x ∈ ΦY Proof The considered statement follows immediately from Proposition 23.8 and Lemmas 24.2–24.4 and 24.8.  Proposition 24.11 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) For any function f ∈ F R, x + f (x) ∈ Φ X ρii (9), for any constant m ∈ ω, x + ˜ ρii (9). m∈ / Φ X ρii (9), and x ∈ ΦY (b) For any function f ∈ F R, f (x)2x log2 k ∈ Φ X ρid (9), 2x−5 ∈ / Φ X ρid (9), and x ∈ ˜ ρid (9). ΦY

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(c) For any function f ∈ F R, x + f (x) ∈ Φ X ρdd (9), for any constant m ∈ ω, x + ˜ ρdd (9). m∈ / Φ X ρdd (9), and x ∈ ΦY Proof The considered statement follows immediately from Proposition 23.9 and Lemmas 24.2–24.4 and 24.9.  Proposition 24.12 Let ρ = (F, k) be a signature with an infinite set F. Then the following statements hold: (a) For any function f ∈ F R, x + f (x) ∈ Φ X ρii (10), for any constant m ∈ ω, x + ˜ ρii (10). m∈ / Φ X ρii (10), and x ∈ ΦY (b) For any function f ∈ F R, x f (x) ∈ Φ X ρid (10), for any constant m ∈ ω \ {0}, ˜ ρid (10). mx ∈ / Φ X ρid (10), and x ∈ ΦY (c) For any function f ∈ F R, x + f (x) ∈ Φ X ρdd (10), for any constant m ∈ ω, x + ˜ ρdd (10). m∈ / Φ X ρdd (10), and x ∈ ΦY (d) If bc ∈ {sd, sa}, then F R ∩ Φ X ρbc (10) = ∅, F R ∩ ΦYρbc (10) = ∅ and, for any m ∈ ω, m ∈ ΦYρbc (10). (e) If bc ∈ {ia, da, aa, is, ds, as, ss}, then F R ∩ Φ X ρbc (10) = ∅ and x ∈ ˜ ρbc (10). ΦY Proof The considered statement follows immediately from Proposition 23.10 and Lemmas 24.2–24.8. 

Reference 1. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity, Global approach. Fundam. Inform. 25(2), 201–214 (1996)

Chapter 25

Algorithmic Problems. Global Approach

In this chapter, we study algorithmic problems related to the global approach to the investigation of decision trees: problems of computation of the minimum complexity of deterministic, nondeterministic, and strongly nondeterministic decision trees, problems of construction of decision trees with the minimum complexity, and the problem of solvability of systems of equations over information systems. We study relationships among these problems. We also discuss the notion of a proper weighted depth for which the problems of computation of the minimum complexity of decision trees and problems of construction of decision trees with the minimum complexity are decidable if the problem of solvability of systems of equations over information systems is decidable. Some results for deterministic decision trees considered in this chapter were published in [1, 2].

25.1 Some Relationships Among Algorithmic Problems Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple. Let b ∈ {d, a, s}. We now define the algorithmic problems Com b (τ ) and Des b (τ ). Problem Com b (τ ): for a given schema z ∈ Στb , it is required to compute the value b ψρ,K (z). Problem Des b (τ ): for a given schema z ∈ Στb , it is required to construct a schema b b and ψ(Γ ) = ψρ,K (z). Γ ∈ Cρ such that (Γ, z) ∈ Rρ,K Theorem 25.1 Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple, where ρ = (F, k). Then the following statements hold: (a) If the problem E x(τ ) is undecidable and τ is a nondegenerate sccf-triple, then the problems Com d (τ ), Des d (τ ), Com a (τ ), Des a (τ ), Com s (τ ), and Des s (τ ) are undecidable. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_25

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(b) If the problem E x(τ ) is undecidable and τ is a degenerate sccf-triple, then the problems Des d (τ ) and Des a (τ ) are undecidable, and the problems Com d (τ ), Com a (τ ), Com s (τ ), and Des s (τ ) are decidable. Proof (a) Let τ be a nondegenerate sccf-triple and the problem E x(τ ) be undecidable. Let b ∈ {d, a, s}. We now show that the problem Com b (τ ) is undecidable. Assume the contrary. Let us show that the problem E x(τ ) is decidable. Since τ is a nondegenerate sccf-triple, there exists an element f i0 ∈ F which is not τ -constant. Let α ∈ Ωρ . If α = λ, then, evidently, α is a τ -realizable word. Let α = λ. If α is inconsistent, then, evidently, α is not τ -realizable. Let α be a nonempty consistent word and α = ( f i1 , δi1 ) · · · ( f in , δin ). For each δ ∈ E k , we define a schema z δ = (ν, f i0 , f i1 , . . . , f in ) as follows: νδ : E kn+1 → {{0}, {1}} and, for any σ¯ ∈ E kn+1 , if σ¯ = (δ, δi1 , . . . , δin ), then νδ (σ¯ ) = {0}, and if σ¯ = (δ, δi1 , . . . , δin ), then νδ (σ¯ ) = {1}. Taking into account that the element f i0 is not τ -constant one can show that z δ ∈ Σρ0−1 (K ) ⊆ Στb . Using the decidability of the problem Com b (τ ), for each b δ ∈ E k , we find the value ψρ,K (z δ ). Taking into account that the function ψ has properties Λ3 and Λ4, and f i0 is an element which is not τ -constant one can show b (z δ ) = 0 for any δ ∈ E k . Thus, that the word α is not τ -realizable if and only if ψρ,K the problem E x(τ ) is decidable which is impossible. Hence the problem Com b (τ ) is undecidable. Taking into account that ψ is a computable function we conclude that the problem Des b (τ ) is undecidable. (b) Let τ be a degenerate sccf-triple and the problem E x(τ ) be undecidable. Since τ is a degenerate sccf-triple, Nρ (T ) = 1 for any table T ∈ Mρ,K . Taking into account that the function ψ has the property Λ4 and using Theorem 10.1 we conclude that, for b (z) = 0 holds. Hence any b ∈ {d, a, s} and for any schema z ∈ Σρb , the equality ψρ,K d a s the problems Com (τ ), Com (τ ), and Com (τ ) are decidable. We denote by Γ0 the schema which contains the root, a terminal node labeled with the number 1, and the edge leaving the root and entering the terminal node. Evidently, for any problem schema z ∈ Σρ0−1 (K ), the schema Γ0 is a solution of the problem Des s (τ ). Therefore the problem Des s (τ ) is decidable. Let b ∈ {d, a}. Let us show that the problem Des b (τ ) is undecidable. Assume the contrary. We now show that the problem E x(τ ) is decidable. Let α ∈ Ωρ . If α = λ, then, evidently, the word α is τ -realizable. Let α = λ and α = ( f i1 , δ1 ) · · · ( f in , δn ). We now define a mapping ν : E kn → {{0}, {1}}. Let σ¯ ∈ E kn . If σ¯ = (δ1 , . . . , δn ), then ν(σ¯ ) = {1}. If σ¯ = (δ1 , . . . , δn ), then ν(σ¯ ) = {0}. Set z = (ν, f i1 , . . . , f in ). Evidently, z ∈ Στb . Let Γ ∈ Cρ and the schema Γ is a solution of the problem Des b (τ ) for the problem schema z. Taking into account that the function ψ has properties Λ3 and Λ4, and τ is degenerate one can show that Γ consists of the root, a terminal node labeled with a number r and the edge leaving the root and entering the terminal node. One can show that r = 1 if and only if the word α is τ -realizable. Thus, the problem E x(τ ) is decidable which is impossible.  Hence the problem Des b (τ ) is undecidable. Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple and b ∈ {d, a, s}. We now define the algorithmic problem R b (τ ).

25.1 Some Relationships Among Algorithmic Problems

283

Problem R b (τ ): for given schema z ∈ Στb and schema Γ ∈ Cρ it is required to b . recognize if (Γ, z) ∈ Rρ,K Lemma 25.1 Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple, the problem E x(τ ) be decidable, and b ∈ {s, a, d}. Then the problem R b (τ ) is decidable. Proof Let z ∈ Στb and Γ ∈ Cρ . Since the problem E x(τ ) is decidable, there is an algorithm which, for given z ∈ Στb and Γ ∈ Cρ , constructs the decision table b if and only if Tρ (α(z, P(Γ )), K ). By Theorem 18.1, (Γ, z) ∈ Rρ,K (Γ, Tρ (α(z, P(Γ )), K )) ∈ Rρb . It is easy to show that the relation Rρb is decidable.



Theorem 25.2 Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple, the problem E x(τ ) be decidable, and b ∈ {s, a, d}. Then the problem Com b (τ ) is decidable if and only if the problem Des b (τ ) is decidable. Proof Let the problem Des b (τ ) be decidable and z ∈ Στb . Using the decidability of b and the problem Des b (τ ) we construct a schema Γ ∈ Cρ such that (Γ, z) ∈ Rρ,K b ψ(Γ ) = ψρ,K (z). Using the computability of the function ψ we compute the value b ψ(Γ ). It coincides with the value ψρ,K (z). Hence the problem Com b (τ ) is decidable. b Let the problem Com (τ ) be decidable. Using Lemma 25.1 and the fact that the b is decidable. We problem E x(τ ) is decidable we conclude that the relation Rρ,K know that the function ψ is computable on the set Cρ . One can show that there exists an algorithm which enumerates all elements of the set Cρ . Let z ∈ Στb . Using b (z). Among the decidability of the problem Com b (τ ) we compute the value ψρ,K b listed schemes from the set Cρ , we choose a schema Γ such that (Γ, z) ∈ Rρ,K and b b  ψ(Γ ) = ψρ,K (z). Hence the problem Des (τ ) is decidable.

25.2 Proper Weighted Depth Let ρ = (F, k) be an enumerated signature, F = { f i : i ∈ ω}, and ψ be a computable weighted depth of the signature ρ. We will say that ψ is a proper weighted depth if, for any nonempty class K of information systems of the signature ρ such that the problem E x(τ ) is decidable for the sccf-triple τ = (ρ, K , ψ), the problems Com b (τ ) and Des b (τ ) are decidable for any b ∈ {d, a, s}. For i ∈ ω, we denote ωψ (i) = { j : j ∈ ω, ψ( f j ) = i}. Define a partial  function  Hψ : ω → ω as follows. Let i ∈ ω. If ωψ (i) is a finite set then Hψ (i) = ωψ (i). If ωψ (i) is an infinite set, then the value of Hψ (i) is indefinite. Denote by Arg Hψ the domain of Hψ . Lemma 25.2 Let ρ = (F, k) be an enumerated signature, F = { f i : i ∈ ω}, ψ be a computable weighted depth of the signature ρ, Arg Hψ = ω, and the function Hψ be recursive. Then ψ is a proper weighed depth.

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Proof Let b ∈ {d, a, s}. Let K be a nonempty class of information systems of the signature ρ such that the problem E x(τ ) is decidable for the sccf-triple τ = (ρ, K , ψ). By Lemma 25.1, the problem R b (τ ) is decidable. Since Arg Hψ = ω and the function Hψ is recursive, there exists an algorithm which, for a given number r ∈ ω constructs the set { f i : f i ∈ F, ψ( f i ) ≤ r }. Using this fact it is not difficult to show that there exists an algorithm which, for a given number r ∈ ω and a finite nonempty subset D of the set ω, constructs the set Cρ (r, D) of all schemes of decision trees Γ of the signature ρ satisfying the following conditions: h(Γ ) ≤ r , ψ( f i ) ≤ r for any element f i ∈ P(Γ ), each terminal node of the tree Γ is labeled with a number from the set D, and, for any two different complete paths ξ1 and ξ2 of the scheme Γ , π(ξ1 ) = π(ξ2 ).  ¯ Let us prove Let z ∈ Στb and z = (ν, f i1 , . . . , f in ). Denote D(z) = δ∈E ¯ kn ν(δ). i that the set Cρ (ψρ,K (z), D(z)) contains a decision tree schema Γ such that (Γ, z) ∈ b b Rρ,K and ψ(Γ ) = ψρ,K (z). One can show that there exists a decision tree schema Γ b b of the signature ρ for which (Γ, z) ∈ Rρ,K , ψ(Γ ) = ψρ,K (z), all terminal nodes are labeled with numbers from the set D(z), and, for any two different complete paths b i (z) ≤ ψρ,K (z). ξ1 and ξ2 of the scheme Γ , π(ξ1 ) = π(ξ2 ). By Lemma 19.1, ψρ,K i i Therefore ψ(Γ ) ≤ ψρ,K (z). Using this inequality we obtain h(Γ ) ≤ ψρ,K (z) and i i ψ( f i ) ≤ ψρ,K (z) for any attribute f i ∈ P(Γ ). Therefore Γ ∈ Cρ (ψρ,K (z), D(z)). b We now describe an algorithm which solves the problem Des (τ ). Let z ∈ i (z) and construct the set D(z). Construct the set Στb . Compute the value ψρ,K i Cρ (ψρ,K (z), D(z)). With the help of algorithm which solves the problem R b (τ ) i b we find a decision tree schema Γ ∈ Cρ (ψρ,K (z), D(z)) such that (Γ, z) ∈ Rρ,K i b and ψ(Γ ) = min{ψ(G) : G ∈ Cρ (ψρ,K (z), D(z)), (G, z) ∈ Rρ,K }. It is clear that b b ψ(Γ ) = ψρ,K (z). So the problem Des (τ ) is decidable. Using Theorem 25.2 we conclude that the problem Com b (τ ) is also decidable. Taking into account that b is an arbitrary index from the set {d, a, s} and K is an arbitrary nonempty class of information systems of the signature ρ such that the problem E x(τ ) is decidable for the sccf-triple τ = (ρ, K , ψ), we obtain that ψ is proper weight function.  Lemma 25.3 Let ρ be an enumerated signature, ψ be a computable weighted depth of the signature ρ, Arg Hψ = ω, and the function Hψ be not recursive. Then ψ is not a proper weighed depth. Proof In the proof of Lemma 5.41 from [2] it was shown that there exists an information system U of the signature ρ such that, for the sccf-triple τ = (ρ, {U }, ψ), the problem E x(τ ) is decidable but the problem Des d (τ ) is undecidable. Therefore ψ is not a proper weighed depth.  Lemma 25.4 Let ρ be an enumerated signature, ψ be a computable weighted depth of the signature ρ, and Arg Hψ = ω. Then ψ is a not proper weighed depth. Proof In the proof of Lemma 5.42 from [2] it was shown that there exists an information system U of the signature ρ such that, for the sccf-triple τ = (ρ, {U }, ψ),

25.2 Proper Weighted Depth

285

the problem E x(τ ) is decidable but the problem Com d (τ ) is undecidable. Therefore ψ is not a proper weighed depth.  Theorem 25.3 Let ρ be an enumerated signature and ψ be a computable weighted depth of the signature ρ. Then ψ is a proper weighed depth if and only if Arg Hψ = ω and Hψ is a recursive function. Proof The considered statement follows immediately from Lemmas 25.2–25.4.  Let ρ = (F, k) be an enumerated signature and F = { f i : i ∈ ω}. We now consider examples of proper weighted depths of the signature ρ. Let ϕ : ω → ω \ {0} be a total recursive nondecreasing function which is unbounded from above. Then the ϕ ϕ weighted depthψ √ such that ψ ( f i ) = ϕ(i) for any i ∈ ω is proper. In particular, 2 x x + 1 are positive total recursive nondecreasing functions which x + 1, 2 , and are unbounded from above.

References 1. Moshkov, M.: Decision Trees. Theory and Applications (in Russian). Nizhny Novgorod University Publishers, Nizhny Novgorod (1994) 2. Moshkov, M.: Time complexity of decision trees. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III, Lecture Notes in Computer Science, vol. 3400, pp. 244–459. Springer, Berlin (2005)

Final Remarks

The aim of this book is to compare four parameters of problems over information systems: complexity of problem representation and complexity of deterministic, nondeterministic, and strongly nondeterministic decision trees for problem solving. First, we created tools for the study of decision trees: lower and upper bounds on complexity and algorithms for construction of decision trees for decision tables with many-valued decisions. Next, we studied the local approach to the investigation of decision trees that allows us to use in the decision trees only attributes from the problem representation. Finally, we studied the global approach to the investigation of decision trees that allows us to use arbitrary attributes from the considered information system. For both approaches, we described all types of relationships among the four parameters of problems. We also discussed a number of algorithmic problems related to the local and the global approaches. Future study will be devoted to the comparative analysis of deterministic and nondeterministic acyclic programs over various bases.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4

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Notation

A an , 107 AU , 119 AU (α), 119 Aτ , 199 Arg f , 123 α(A), 38 α(z, B), 209 α (t) , 211 α, ˜ 250

B B(f , g, A), 58 Bτbc (n), 211 Btb (z), 215

C Coma (ρ, ψ), 105 Comd (ρ, ψ), 105 Coms (ρ, ψ), 105 Coma (τ ), 281 Comd (τ ), 281 Coms (τ ), 281 ˆ a (τ ), 195 Com ˆ d (τ ), 195 Com ˆ s (τ ), 195 Com Cρ , 19 Cρ0 (T ), 66 Cρd , 71 χ(α), 25

D D1 , 94 D2 , 94 D3 , 94 Dρ , 70 Dρ (η), 80 Des(ρ, ψ), 112 Desa (ρ, ψ), 105 Desd (ρ, ψ), 105 Dess (ρ, ψ), 105 Desa (τ ), 281 Desd (τ ), 281 Dess (τ ), 281 ˆ a (τ ), 195 Des ˆ d (τ ), 195 Des ˆ Dess (τ ), 195 d [α], 41 dn , 95 dρ (T ), 31 dρ (T , m), 31 dm , 168 Δ(T ), 18 Δ(T , 1), 86 Δ(T , α), 155 Δ(T , m), 31

E EV (f ), 95 Ek , 18 Ex(τ ), 196 en , 168 η(D, α), 41 ε2 , 93

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M. Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4

289

290 F FR, 132 FZ, 129 f −1 , 132 F0 , 168 F1r , 94 F2r , 94 F3r , 94 F4r , 94 f U , 119

G G(α, T ), 85 GHρ (T ), 38 GR, 133 G f , 130 G 0 , 168 G ρ (α), 27 G w , 255 G ϕ ba , 131 G ϕ ba (n), 131 G ψ ab , 130 G ψ ab (n), 130 γ (f ), 119 γD (α), 41 γ(1,m) , 168 γ(2,1,m) , 168 γ(2,2,m) , 176 γ(2,3,w) , 255 γ(3,1,m) , 169 γ(3,2) , 177 γ(3,3,f ) , 178 γ(3,4,f ) , 179 γ(4,2,f ) , 180 γ(4,3,f ) , 181 γ(4,4,f ) , 182 γ(4,5,1) , 256 γ(4,6,f ) , 257 γ(4,7,g) , 258 γ(4,8,q) , 259 γ(4,m) , 170 γ(5,m) , 171 γ(6,1,m) , 171 γ(6,2,f ) , 184 γ(7) , 250 γU , 119

H H V , 114 Hρm (T ), 38 Hψ , 283

Notation a , 94 hU d , 94 hU s , 94 hU h, 21 ha , 94 hd , 94 hs , 94

I [i]2 , 22 ∞, 210 Iψ (n, T ), 159 Iτ , 160

J Jρ (T ), 67

K Kρ (T ), 67 Kρ (Γ ), 67 Kρ (w), 67 ˜ 128 K, kn , 95 κρ (T , α), 72

L