Discrete-Time and Discrete-Space Dynamical Systems [1st ed. 2020] 978-3-030-25971-6, 978-3-030-25972-3

Table of contents :
Front Matter ....Pages i-xiv
Front Matter ....Pages 1-1
Preliminaries (Kuize Zhang, Lijun Zhang, Lihua Xie)....Pages 3-33
Different Types of Discrete-Time and Discrete-Space Dynamical Systems (Kuize Zhang, Lijun Zhang, Lihua Xie)....Pages 35-56
Front Matter ....Pages 57-57
Invertibility and Nonsingularity of Boolean Control Networks (Kuize Zhang, Lijun Zhang, Lihua Xie)....Pages 59-86
Observability of Boolean Control Networks (Kuize Zhang, Lijun Zhang, Lihua Xie)....Pages 87-104
Detectability of Boolean Control Networks (Kuize Zhang, Lijun Zhang, Lihua Xie)....Pages 105-115
Observability and Detectability of Large-Scale Boolean Control Networks (Kuize Zhang, Lijun Zhang, Lihua Xie)....Pages 117-142
Front Matter ....Pages 143-143
Observability of Nondeterministic Finite-Transition Systems (Kuize Zhang, Lijun Zhang, Lihua Xie)....Pages 145-163
Detectability of Nondeterministic Finite-Transition Systems (Kuize Zhang, Lijun Zhang, Lihua Xie)....Pages 165-175
Front Matter ....Pages 177-177
Detectability of Finite-State Automata (Kuize Zhang, Lijun Zhang, Lihua Xie)....Pages 179-192
Front Matter ....Pages 193-193
Detectability of Labeled Petri Nets (Kuize Zhang, Lijun Zhang, Lihua Xie)....Pages 195-209
Front Matter ....Pages 211-211
Generalized Reversibility of Cellular Automata (Kuize Zhang, Lijun Zhang, Lihua Xie)....Pages 213-220
Back Matter ....Pages 221-222

Citation preview

Communications and Control Engineering

Kuize Zhang Lijun Zhang Lihua Xie

Discrete-Time and Discrete-Space Dynamical Systems

Communications and Control Engineering Series Editors Alberto Isidori, Roma, Italy Jan H. van Schuppen, Amsterdam, The Netherlands Eduardo D. Sontag, Boston, USA Miroslav Krstic, La Jolla, USA

Communications and Control Engineering is a high-level academic monograph series publishing research in control and systems theory, control engineering and communications. It has worldwide distribution to engineers, researchers, educators (several of the titles in this series find use as advanced textbooks although that is not their primary purpose), and libraries. The series reflects the major technological and mathematical advances that have a great impact in the fields of communication and control. The range of areas to which control and systems theory is applied is broadening rapidly with particular growth being noticeable in the fields of finance and biologically-inspired control. Books in this series generally pull together many related research threads in more mature areas of the subject than the highly-specialised volumes of Lecture Notes in Control and Information Sciences. This series’s mathematical and control-theoretic emphasis is complemented by Advances in Industrial Control which provides a much more applied, engineering-oriented outlook. Indexed by SCOPUS and Engineering Index. Publishing Ethics: Researchers should conduct their research from research proposal to publication in line with best practices and codes of conduct of relevant professional bodies and/or national and international regulatory bodies. For more details on individual ethics matters please see: https://www.springer.com/gp/authors-editors/journal-author/journal-authorhelpdesk/publishing-ethics/14214

More information about this series at http://www.springer.com/series/61

Kuize Zhang Lijun Zhang Lihua Xie •

•

Discrete-Time and Discrete-Space Dynamical Systems

123

Kuize Zhang School of Electrical Engineering and Computer Science KTH Royal Institute of Technology Stockholm, Sweden

Lijun Zhang School of Marine Science and Technology Northwestern Polytechnical University Xi’an, China

Lihua Xie School of Electrical and Electronic Engineering Nanyang Technological University Singapore, Singapore

ISSN 0178-5354 ISSN 2197-7119 (electronic) Communications and Control Engineering ISBN 978-3-030-25971-6 ISBN 978-3-030-25972-3 (eBook) https://doi.org/10.1007/978-3-030-25972-3 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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

Preface

Discrete-time and discrete-space dynamical systems are widely used in various areas, e.g., decision-making or computation. For instance, Boolean control networks were initially proposed to model genetic regulatory networks; finite automata and labeled Petri nets as models of discrete-event systems have been applied to many engineering fields such as manufacturing processes, production scheduling; nondeterministic finite-transition systems have been applied to model checking and automated synthesis of cyber-physical systems; cellular automata have been used in quantum physics, biological dynamics as well as computational mathematics, just to name a few. Let us introduce what discrete-time and discrete-space dynamical systems mean in this book (transitioned from the well-known dynamical systems over Euclidean spaces). It seems widely accepted nowadays that the world consists of time elapsing “continuously” and space arranged “seamlessly”. Evolution of a process within this setting can be described by nonlinear differential equations in a (locally) Euclidean space. Such description is intuitive but explicit solutions are not easy to obtain, which makes it difficult to analyze their long-term behavior. To bypass the obstacle of finding an explicit solution, one can discretize time to generate sequences of points (called trajectories) by iterations of maps. Another difficulty of analyzing long-term behavior of a dynamical system over a continuous state space lies in its continuity, since it is almost impossible to separate the crucial locations from the redundant ones. To overcome this dilemma, space is also discretized to guarantee every sequence has a convergent subsequence. For example, this property holds in any finite metric space; in addition, every sequence has an increasing subsequence (and thus convergent (possibly to infinity)) in the countable metric space Nn (as a subspace of Rn ); this property even holds in some uncountable spaces, like the Cantor space, where every point can be regarded as a mapping from Zn to a common alphabet. It is worth mentioning that none of the three kinds of spaces above is locally Euclidean; indeed, they have topological dimension 0 (in the sense of Čech–Lebesgue covering dimension) while Euclidean spaces or general manifolds that are not the singleton have positive topological dimensions. For dynamical systems over these zero-dimensional spaces, explicit solutions can be found without v

vi

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any restriction on the system maps and therefore the focus is on the spaces. In this book, such dynamical systems are called discrete-time and discrete-space dynamical systems. What makes it challenging to study these systems mainly lies in the unrestricted system maps. Among diverse fundamental properties (controllability, observability, detectability, and stabilizability) in control theory, observability and detectability stand out: both of them deduce internal information out of external information. Compared to controllability and stabilizability which focus directly on trajectories, observability and detectability are somewhat indirect; nevertheless, the latter provide basis for quantitative analysis of long-term behavior as well as prerequisites for (automated) controller synthesis. Exploring long-term behavior is a long-lasting topic in dynamical systems. In this book, we will study decidability and complexity of observability and detectability as well as their variants for different kinds of discrete-time and discrete-space dynamical systems, such as the aforementioned Boolean control networks, finite automata, and labeled Petri nets. Various methods will be introduced to study different kinds of systems. Fundamental characterization for observability and detectability will (1) provoke new related studies such as state estimation and automated synthesis; (2) help reveal relations and essential differences among different types of systems. For instance, for finite automata, strong detectability is verifiable in polynomial time while weak detectability is PSPACE-complete; in contrast for labeled Petri nets, strong detectability is EXPSPACE-hard while weak detectability is even undecidable. We will investigate another “indirect” property that is called “invertibility” or “reversibility”. It means for control systems that an output sequence allows uniquely determining the corresponding input sequence, but it means for dynamical systems without control that every trajectory has a unique backward-in-time extension. For Boolean control networks, a tool (equivalent to cellular automata) beyond them is demanding to characterize invertibility, which in turn reveals the importance of invertibility in unveiling some link between Boolean control networks and cellular automata. Finally, let us point out several potential applications of fundamental properties of discrete-time and discrete-space dynamical systems (particularly of such systems with finitely many states) in formal verification and synthesis of hybrid (control) systems. Since the verification problem for basic properties such as controllability and observability of (infinite-state) hybrid systems is mostly formidable and it is likely that many properties are undecidable. By constructing a finite-state system as an approximation which (bi)simulates a given hybrid system while preserving some useful property, one can benefit in verifying this property over the finite approximating system instead. Automated synthesis can be dealt with in an analogous way. From this perspective, results presented in this book can also be related to the active field of formal verification and synthesis emerging in the last two decades.

Preface

vii

This book will first introduce (in Chap. 1) basic mathematical preliminaries such as graph theory, the semitensor product of matrices, finite automata, and topology to support the study throughout the book, and also the differences between different types of zero-dimensional spaces and Euclidean spaces based on these preliminaries. Second, it will discuss (in Chap. 2) different types of discrete-time and discrete-space dynamical systems (Boolean control networks, finite automata, nondeterministic finite-transition systems, Petri nets, and cellular automata), highlighting their similarities and differences. These essential differences show that there exists no unified method available to deal with these types of dynamical systems. Then in the main parts (the remaining chapters), it will collect a series of recent fundamental results in control-theoretic and topological dynamical problems of discrete-time and discrete-space dynamical systems, e.g., invertibility, observability, detectability, reversibility, etc., by developing new techniques. In addition, the book will also contain some practical applications of these problems in systems biology, etc. In order to study different types of systems, various methods, e.g., a

Chapter 9 Detectability of FSAs

Chapter 2.3 FSAs

Chapter 10 Detectability of LPNs

Chapter 2.4 LPNs

Chapter 1.1 Graph theory

Chapter 1.3 Finite automata

Chapter 1.2 Semitensor product

Chapter 1.4 Topology

Chapter 2.2 NFTSs

Chapter 2.1 BCNs

Chapter 3.2 Invertibility of BCNs

Chapter 2.5 CAs

Chapter 7 Observability of NFTSs Chapter 8 Detectability of NFTSs

Chapter 4 Observability of BCNs Chapter 5 Detectability of BCNs

Chapter 3.3 Nonsingularity of BCNs Chapter 6 Observability and detectability of large-scale BCNs

Chapter 11 Generalized reversibility of CAs

Fig. 0.1 Reading flow of the book

viii

Preface

semitensor product method, a graph-theoretic method, a finite-automaton method, a topological method, etc., will be adopted. The book is aiming at bringing the reader new understanding of discrete-time and discrete-space dynamical systems. While reading the book, the reader could refer to the reading flow shown in Fig. 0.1 through arrow lines of the same type. We did not introduce the mathematical tools used to handle labeled Petri nets in Chap 1, but introduced them in Chap. 10 when labeled Petri nets were studied, because such tools quite depend on the labeled Petri nets themselves. We are in debt to Dr. Shaoshuai Mou at Purdue University, USA, Dr. Rong Su at Nanyang Technological University, Singapore, Dr. Karl Henrik Johansson at KTH Royal Institute of Technology, Sweden, Dr. Ting Liu and Dr. Daizhan Cheng both at Academy of Mathematics and Systems Science, Chinese Academy of Sciences, PR China, Dr. Majid Zamani at University of Colorado Boulder, USA, and Dr. Alessandro Giua at University of Cagliari, Italy, who have all coauthored with us a few papers, which have been included in this book. We would like to thank Dr. Chuang Xu at University of Copenhagen, Denmark, and Mr. Ping Sun and Mr. Zhenkun Wang at Harbin Engineering University, PR China, who helped a lot in proofreading the manuscripts. We are also indebted to Mr. Oliver Jackson for his patient support. Stockholm, Sweden Xi’an, China Singapore

Kuize Zhang Lijun Zhang Lihua Xie

Contents

Part I

Introduction

1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Graph Theory . . . . . . . . . . . . . . . . . . . . . . 1.2 The Semitensor Product of Matrices . . . . . . 1.3 Finite Automata and Regular Languages . . 1.4 Topological Spaces and Discrete Spaces . . . 1.4.1 Topological Spaces . . . . . . . . . . . . 1.4.2 Separability . . . . . . . . . . . . . . . . . . 1.4.3 Compactness . . . . . . . . . . . . . . . . . 1.4.4 Connectedness . . . . . . . . . . . . . . . . 1.4.5 Metric Spaces . . . . . . . . . . . . . . . . 1.4.6 Topological Dimensions . . . . . . . . . 1.4.7 Several Discrete Topological Spaces Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2

Different Types of Discrete-Time and Discrete-Space Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Boolean Control Networks . . . . . . . . . . . . . . . . 2.2 Nondeterministic Finite-Transition Systems . . . . 2.3 Finite-State Automata . . . . . . . . . . . . . . . . . . . . 2.4 Labeled Petri Nets . . . . . . . . . . . . . . . . . . . . . . 2.5 Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Invertibility and Nonsingularity of Boolean Control Networks . . . . 3.1 Notions of Invertibility and Nonsingularity . . . . . . . . . . . . . . . .

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Part II 3

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Boolean Control Networks

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3.2

Invertibility Characterization . . . . . . . . . . . . . . . . . . . . 3.2.1 The Map from the Infinite Input Sequences Space to the Infinite Output Sequences Space . . 3.2.2 Invertibility Verification . . . . . . . . . . . . . . . . . . 3.2.3 Invariance of Invertibility . . . . . . . . . . . . . . . . . 3.2.4 Inverse Boolean Control Networks of Invertible Boolean Control Networks . . . . . . . . . . . . . . . . Nonsingularity Characterization . . . . . . . . . . . . . . . . . . 3.3.1 Nonsingularity Graphs . . . . . . . . . . . . . . . . . . . 3.3.2 Nonsingularity Verification . . . . . . . . . . . . . . . . Relationship Between Invertibility and Nonsingularity . Application to the Mammalian Cell Cycle . . . . . . . . . . 3.5.1 Invertibility of the Mammalian Cell Cycle . . . . . 3.5.2 Further Discussion . . . . . . . . . . . . . . . . . . . . . .

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Observability of Boolean Control Networks . . . . . . . . . . . . . . . 4.1 Notions of Observability . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Observability Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Verifying Different Notions of Observability . . . . . . . . . . . 4.3.1 Verifying Multiple-Experiment Observability . . . . . . 4.3.2 Verifying Strong Multiple-Experiment Observability 4.3.3 Verifying Single-Experiment Observability . . . . . . . 4.3.4 Verifying Arbitrary-Experiment Observability . . . . . 4.4 Complexity Analysis of Verifying Observability . . . . . . . . . 4.5 Pairwise Nonequivalence of Different Notions of Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5

Detectability of Boolean Control Networks . . . . . . . . . . . 5.1 Notions of Detectability . . . . . . . . . . . . . . . . . . . . . 5.2 Detectability Graphs . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Verifying Different Notions of Detectability . . . . . . . 5.3.1 Verifying Single-Experiment Detectability . . . 5.3.2 Verifying Arbitrary-Experiment Detectability . 5.4 Complexity of Verifying Detectability . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6

Observability and Detectability of Large-Scale Boolean Control Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Dependency Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Node Aggregations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Observability Verification Based on a Node Aggregation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

6.4

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6.3.1 Advantage of the Aggregation Method . . . . . . 6.3.2 Limitation of the Aggregation Method . . . . . . 6.3.3 An Aggregation Algorithm and Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detectability Verification Based a Node Aggregation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Advantage of the Aggregation Method . . . . . . 6.4.2 Limitation of the Aggregation Method . . . . . . Application to the T-Cell Receptor Kinetics . . . . . . . . 6.5.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Observability Analysis Based on Acyclic Node Aggregations . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Detectability Analysis Based on Acyclic Node Aggregations . . . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . . 140 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Part III

Nondeterministic Finite-Transition Systems

7

Observability of Nondeterministic Finite-Transition Systems 7.1 Notions of Observability . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Verifying Multiple-Experiment Observability . . . . . . . . . 7.3 Verifying Strong Multiple-Experiment Observability . . . . 7.4 Verifying Single-Experiment Observability . . . . . . . . . . . 7.5 Verifying Arbitrary-Experiment Observability . . . . . . . . . 7.6 Complexity Analysis of Verifying Observability . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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145 146 148 152 155 157 158 163

8

Detectability of Nondeterministic Finite-Transition Systems 8.1 Notions of Detectability . . . . . . . . . . . . . . . . . . . . . . . 8.2 Verifying Arbitrary-Experiment Detectability . . . . . . . . 8.3 Verifying Single-Experiment Detectability . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Detectability of Finite-State Automata . . . . . . . . . . . . . . . . . 9.1 Notions of Detectability . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Verifying Strong Detectability . . . . . . . . . . . . . . . . . . . . 9.2.1 A Polynomial-Time Verification Algorithm . . . . . 9.2.2 Another Polynomial-Time Verification Algorithm Under Assumption 2 . . . . . . . . . . . . . . . . . . . . .

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Part IV 9

Finite-State Automata

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9.3 Verifying Weak Detectability . . . . . . . . . . . 9.4 Complexity of Deciding Weak Detectability . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Detectability of Labeled Petri Nets . . . . . . . . . . . . . . . . . 10.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Notions of Detectability . . . . . . . . . . . . . . . . . . . . . 10.3 Decidability and Complexity of Strong Detectability . 10.4 Decidability of Weak Detectability . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part V

Part VI

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Labeled Petri Nets

Cellular Automata

11 Generalized Reversibility of Cellular Automata 11.1 Cellular Automata . . . . . . . . . . . . . . . . . . . 11.2 Drazin Inverses . . . . . . . . . . . . . . . . . . . . . 11.3 Generalized Reversibility . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Abbreviations

; N Z Zþ R R0 Rn Rmn Lmn jEj 2E Id ordx ðaÞ ordðaÞ ba DðaÞ dimðXÞ A\B A[B AnB AB Br ðxÞ sup E inf E
E or IntðEÞ E or CloðEÞ diamðEÞ distðx; EÞ ½a; b

Empty set Set of natural numbers Set of integers Set of positive integers Set of real numbers Set of nonnegative real numbers Set of n-dimensional real column vectors Set of m  n real matrices Set of m  n logical matrices Cardinality of set E Power set of set E Identity map Order of a at x Order of a b is finer than a Minimum of ordðbÞ for all b  a Covering dimension of topological space X Intersection of sets A and B Union of sets A and B Disjoint union of sets A and B Difference of sets A and B Set A is a subset of set B Ball centered at x with radius r Supremum of set E Infimum of set E Interior of set E Closure of set E Diameter of set E Distance of point x to set E Closed interval with endpoints a and b (a  b)

xiii

xiv

½a; b ða; bÞ ½a; bÞ ða; b S Sþ Sx ½u d SZ D lcmðp; qÞ gcdðp; qÞ In din 1n Dn (D2 ¼: D) AT Coli ðAÞ ColðAÞ Rowi ðAÞ RowðAÞ n A1    An

Abbreviations

Set of integers no less than a and no greater than b (a  b) Open interval with endpoints a and b (a  b) ½a; bnfbg ½a; bnfag Set of words over alphabet S Set of words over alphabet S excluding the empty word  Set of configurations over alphabet S Cylinder determined by pattern u Symbolic space over alphabet S f0; 1g Least common multiple of positive integers p and q Greatest common divisor of positive integers p and q Identity matrix of order n i-th column of the identity matrix In Pn i i¼1 dn Set of the columns of the identity matrix In Transpose of matrix A i-th column of matrix A Set of columns of matrix A i-th row of matrix A Set of rows of matrix A Semitensor product Kronecker product 2 3 A1 6 7 .. 4 5 . An

Part I

Introduction

Chapter 1

Preliminaries

In this chapter, we introduce basic theoretical tools used throughout the book, that is, graph theory, the semitensor product of matrices, finite automata, and topology. As in the reading flow shown in Fig. 0.1, topology is used to study invertibility (Sect. 3.2) of Boolean control networks and generalized reversibility (Chap. 11) of cellular automata; graph theory is used to characterize nonsingularity (Sect. 3.3) of Boolean control networks, observability (Sect. 4.2) and detectability (Sect. 5.2) of Boolean control networks and large-scale Boolean control networks (Chap. 6); the semitensor product of matrices is used to give an intuitive matrix representation for Boolean control networks (Chap. 3); the theory of finite automata is used to investigate observability (Sect. 4.2) and detectability (Sect. 5.2) of Boolean control networks, observability (Chap. 7) and detectability (Chap. 8) of nondeterministic finite-transition systems, and detectability of finite-state automata (Chap. 9); some existing theoretical results in labeled Petri nets are used to study detectability of labeled Petri nets (Chap. 10). In Sects. 1.1, 1.2, and 1.3, we briefly introduce basic concepts and results in graph theory, the semitensor product of matrices, and finite automata. Then in Sect. 1.4, we introduce discrete spaces, and compare them with Euclidean spaces mainly from the perspective of topological dimensions and several topological properties. We call a topological space discrete if it has topological dimension 0, and continuous if it has topological dimension greater than 0 (a positive integer or ∞). The discrete spaces studied in this book are finite metric spaces, the countable spaces Zn (Nn ), and the Cantor space, which cover the state spaces of all the dynamical systems studied in this book. Boolean control networks, finite automata, and nondeterministic finitetransition systems are over finite metric spaces, Petri nets are over the countable space Nn , cellular automata are over the Cantor space. In order to distinguish the dynamical systems studied in this book from dynamical systems over Euclidean spaces, we adopt the concept of topological dimension. We will introduce this essential difference between Euclidean spaces and discrete spaces step by step.

© Springer Nature Switzerland AG 2020 K. Zhang et al., Discrete-Time and Discrete-Space Dynamical Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-25972-3_1

3

4

1 Preliminaries

1.1 Graph Theory A directed graph is a 2-tuple (V, E), where a finite set V denotes its vertex set, and E ⊂ V × V denotes its edge set. Given two vertices v1 , v2 ∈ V, if (v1 , v2 ) ∈ E, then we say “there is an edge from v1 to v2 ”, and also denote (v1 , v2 ) by v1 → v2 . Vertices v1 and v2 are called tail and head of edge v1 → v2 , respectively. Vertex v1 is called a parent of v2 , and similarly vertex v2 is called a child of v1 . Given v0 , v1 , . . . , v p ∈ V, if for all i ∈ [0; p − 1], (vi , vi+1 ) ∈ E, then v0 → · · · → v p is called a path, and p is called the length of the path. Particularly, if v0 = v p , path v0 → · · · → v p is called a cycle. Cycle v0 → · · · → v p is called simple if v0 , . . . , v p−1 are pairwise different. An edge from a vertex to itself is called a self-loop. Given vertices v0 , . . . , v p ∈ V, denote {v0 , . . . , v p } by V p . The subgraph of graph (V, E) generated by V p is defined as graph (V p , E p ), where E p = (V p × V p ) ∩ E. Two vertices v and v are called strongly connected if there exist a path from v to v and a path from v to v, i.e., v and v belong to some cycle. A strongly connected component is a vertex that does not belong to any cycle or a subgraph generated by a subset of vertices, where every two of these vertices are strongly connected, and no vertex of them and no vertex outside them are strongly connected. Let G be a directed graph, let G be / E for every obtained from G by adding edge v1 → v2 if (v2 , v1 ) ∈ E but (v1 , v2 ) ∈ two vertices v1 , v2 . Two vertices v and v are called weakly connected in G if they are strongly connected in G . A weakly connected component of G is defined by a strongly connected component of G . A strongly (weakly) connected component of G is called generated by a vertex v if v belongs to the component. In a directed graph (V, E), we define an equivalence relation ∼ as follows: for all vertices u, v ∈ V, u and v have the relation (denoted by u ∼ v) if and only if u = v, or, they are strongly connected. Then a strongly connected component generated by a vertex v ∈ V is the subgraph ([v], ([v] × [v]) ∩ E), which is also generated by its equivalence class [v] = {u ∈ V|u ∼ v}. See Figs. 1.1, 1.2, and 1.3 for illustration. A weighted directed graph (V, E, W, 2 ) is a directed graph (V, E) such that each edge e ∈ E is labeled by a weight w ⊂ , represented by a function W : E → 2 , where  is a finite set of weights. Given vertex v ∈ V, | ∪u∈V,(v,u)∈E W((v, u))| is called the outdegree of vertex v, denoted by outdeg(v); similarly, | ∪u∈V,(u,v)∈E W((u, v))| is called the indegree of vertex v, denoted by indeg(v). A subgraph is called complete if in the subgraph each vertex has outdegree ||. See Fig. 1.4 for an illustrative example.

Fig. 1.1 A directed graph

1.2 The Semitensor Product of Matrices

5

Fig. 1.2 Strongly connected components of the directed graph in Fig. 1.1

Fig. 1.3 Weakly connected components of the directed graph in Fig. 1.1

Fig. 1.4 A weighted directed graph, where V = {A, B, C, D, E},  = {1, 2, 3, 4}, indeg(A) = 2, outdeg(A) = 3, indeg(E) = 1, outdeg(E) = 4

1.2 The Semitensor Product of Matrices Matrices (all matrices considered in this book are finite dimensional) can be seen as linear operators/transformations on sets (e.g., topological/linear spaces). Then products of matrices are compositions of operators on sets. The conventional matrix product applies to a pair (A, B) of matrices satisfying that the number of columns of A and the number of rows of B are equal, and preserves many properties, such as the associative law, the distributive law, and several inverse-order laws (e.g., (AB)T = B T A T ). Is it possible to extend the concept of matrix product and meanwhile preserve the basic properties of the conventional product? Taking the Kronecker product, for example, the Kronecker product applies to any pair of matrices, and preserves the associative law, but does not preserve inverse-order laws (e.g., one has (A ⊗ B)T = A T ⊗ B T but does not necessarily have (A ⊗ B)T = B T ⊗ A T ). And furthermore, the Kronecker product is not a generalization of the conventional matrix product, because

6

1 Preliminaries

when the number of columns of matrix A equals the number of rows of matrix B, one usually does not have AB = A ⊗ B. In 2001, (Cheng 2001) proposed a generalization of the conventional matrix product called the semitensor product (STP) of matrices, which applies to any pair of matrices, and preserves the associate law, the distributive law, several inverse-order laws, etc. During the following years, STP has been applied into many fields, e.g., analysis and control of Boolean control networks (Cheng et al. 2011), Morgen’s problem (Cheng 2001), symmetry of dynamical systems (Cheng et al. 2007), differential geometry and Lie algebras (Cheng and Zhang 2003), dynamic and static games (Qi 2016; Cheng et al. 2016), just to name a few. In particular, using STP, logic variables are represented as vectors, and then logic operations are represented as the STP of vectors and the so-called logical matrices (a special class of Boolean matrices). Based on this intuitive representation, an algebraic framework for Boolean control networks has been constructed (Cheng and Qi 2009). Under this framework, many control-theoretic problems, e.g., controllability (Cheng and Qi 2009; Zhao et al. 2010), observability (Cheng and Qi 2009; Fornasini and Valcher 2013; Xu and Hong 2013a; Li et al. 2014), identifiability (Cheng and Zhao 2011; Zhang et al. 2017), reconstructibility (Fornasini and Valcher 2013; Zhang et al. 2016), invertibility (Zhang et al. 2015), pinning control (Lu 2016), output tracking (Li et al. 2015), robust control (Li et al. 2016), synchronization (Xu and Hong 2013b), decomposition (Zou and Zhu 2015, 2017), optimal control (Zhao et al. 2011; Fornasini and Valcher 2014; Wu and Shen 2017), etc., have been investigated. Definition 1.1 Let A ∈ Rm×n , B ∈ R p×q , and l = lcm(n, p) be the least common multiple of n and p. The STP of A and B is defined as    B ⊗ I pl , A  B = A ⊗ I nl where ⊗ denotes the Kronecker product. Proposition 1.1 (Associative law) Let A ∈ Rm×n , B ∈ R p×q , and C ∈ Rr ×s . Then we have A  (B  C) = (A  B)  C. Proof Denote lcm(n, p) = nn 1 = pp1 , lcm(q, r ) = qq1 = rr1 , lcm(r, qp1 ) = qp1 p2 = rr2 , and lcm(n, pq1 ) = nn 2 = pq1 q2 . Then one has (A  B)  C =((A ⊗ In 1 )(B ⊗ I p1 ))  C =(((A ⊗ In 1 )(B ⊗ I p1 )) ⊗ I p2 )(C ⊗ Ir2 ) =(A ⊗ In 1 ⊗ I p2 )(B ⊗ I p1 ⊗ I p2 )(C ⊗ Ir2 ) =(A ⊗ In 1 p2 )(B ⊗ I p1 p2 )(C ⊗ Ir2 ) and

(1.1)

1.2 The Semitensor Product of Matrices

7

A  (B  C) =A  ((B ⊗ Iq1 )(C ⊗ Ir1 )) =(A ⊗ In 2 )(((B ⊗ Iq1 )(C ⊗ Ir1 )) ⊗ Iq2 ) =(A ⊗ In 2 )(B ⊗ Iq1 ⊗ Iq2 )(C ⊗ Ir1 ⊗ Iq2 )

(1.2)

=(A ⊗ In 2 )(B ⊗ Iq1 q2 )(C ⊗ Ir1 q2 ). By the associativity law of the least common multiple, one has that lcm(qn, lcm( pq, pr )) = lcm(lcm(qn, pq), pr ), lcm(qn, p lcm(q, r )) = lcm(q lcm(n, p), pr ), lcm(qn, pqq1 ) = lcm(qpp1 , pr ), q lcm(n, pq1 ) = p lcm(qp1 , r ), lcm(r, qp1 ) lcm(n, pq1 ) = , p q q 1 q 2 = p1 p2 . Furthermore,

lcm(n, p) lcm(r, qp1 ) n qp1 lcm(n, p) lcm(n, pq1 ) = n pp1 lcm(n, pq1 ) = n = n2,

(1.3)

lcm(q, r ) lcm(n, pq1 ) r pq1 lcm(q, r ) lcm(r, qp1 ) = r qq1 lcm(r, qp1 ) = r = r2 .

(1.4)

n 1 p2 =

r 1 q2 =

Consequently, (A  B)  C = A  (B  C).  We have proved that STP preserves the associative law, hence later on we can omit the symbol  and write A  B as AB for short. Proposition 1.2 (Inverse-order law) Let A ∈ Rm×n and B ∈ R p×q . Then

8

1 Preliminaries

(A  B)T = B T  A T . Proof Let α = lcm(n, p). Then  T A ⊗ Iα/n B ⊗ Iα/ p T  T  A ⊗ Iα/n = B ⊗ Iα/ p    = B T ⊗ Iα/ p A T ⊗ Iα/n

(A  B)T =



= B T  AT .  Proposition 1.3 (Pseudocommutative law) Let A ∈ Rm×n and z ∈ Rt . Then A  z T = z T  (It ⊗ A), z  A = (It ⊗ A)  z. Proof We only verify z  A = (It ⊗ A)  z. The other naturally holds by the inverse-order law. z  A = (z ⊗ Im ) A = (It ⊗ A) (z ⊗ In ) = (It ⊗ A)  z.  Definition 1.2 The swap matrix, W[m,n] , is an mn × mn matrix defined by   W[m,n] := δn1 δm1 , δn2 δm1 , . . . , δnn δm1 . . . , δn1 δmm , δn2 δmm , . . . , δnn δmm . Proposition 1.4 Let W[m,n] be a swap matrix, P ∈ Rm , and Q ∈ Rn . Then −1 T = W[m,n] = W[n,m] , W[m,n]

W[m,n] P Q = Q P,

(1.6)

P Q W[m,n] = Q P . T

T

T

(1.5)

T

Proof By definition, we have   W[m,n] W[n,m] = δn1 δm1 , δn1 δm2 , . . . , δn1 δmm , . . . , δnn δm1 , δnn δm2 , . . . , δnn δmm = Imn . −1 Then we have W[n,m] W[m,n] = Imn , and W[m,n] = W[n,m] .

(1.7)

1.2 The Semitensor Product of Matrices

9

T As W[m,n] is a logical invertible matrix, it satisfies W[m,n] W[m,n] = Imn , which −1 T implies W[m,n] = W[m,n] . Hence (1.5) holds. Let P = [ p1 , . . . , pm ]T and Q = [q1 , . . . , qn ]T . Then we have

PQ =



pi q j δmi δnj ,

i∈[1;m], j∈[1;n]

W[m,n] P Q =



pi q j δnj δmi = Q P.

i∈[1;m], j∈[1;n]

Hence (1.6) holds. Similarly, by the inverse-order law, (1.7) also holds.



Definition 1.3 The matrix Mkr = δk1 ⊕ · · · ⊕ δkk is called the power-reducing matrix. Particularly, we denote M2r := Mr . By definition, the following proposition holds. Proposition 1.5 For power-reducing matrix Mkr , we have P 2 = M kr P for each P ∈ k . A matrix A ∈ Rm×n is called Boolean if each entry of the matrix is either 0 or 1. More particularly, a Boolean matrix A ∈ Rm×n is called logical if each of its columns is some column of the identity matrix Im . Hence, swap matrices and power-reducing matrices are logical matrices. The set of m × n logical matrices is denoted by Lm×n . Hence, the STP of any two logical matrices is still a logical matrix. Identifying 1 ∼ δ21 , 0 ∼ δ22 , where δ2i is the i-th column of the identity matrix I2 , using STP, a Boolean function f : Dn → D can be uniquely represented by a logical matrix. Proposition 1.6 For a Boolean function f : Dn → D, there exists a unique logical matrix F ∈ L2×2n such that f (x1 , . . . , xn ) = F x˜1  · · ·  x˜n , where xi ∈ D, x˜i ∈ , and xi ∼ x˜i , i ∈ [1; n]. Definition 1.4 Let A ∈ Rm×n , B ∈ R p×n . The Khatri–Rao product of A and B is defined by A ∗ B = [Col1 (A)  Col1 (B), . . . , Coln (A)  Coln (B)]. Proposition 1.7 For two Boolean functions F A1  · · ·  An and G A1  · · ·  An , where Ai ∈ , i ∈ [1; n], F, G ∈ L2×2n , one has

10

1 Preliminaries

(F A1 · · · An )  (G A1 · · · An ) = H A1 · · · An , where H = F ∗ G.

1.3 Finite Automata and Regular Languages A nondeterministic finite automata (NFA) is a quintuple A = (Q, , δ, q0 , F), where Q is a finite set of states,  is a finite set (called alphabet), elements of  are called letters (Kari 2016a) (also called events Ramadge and Wonham 1987; Shu et al. 2007), q0 ∈ Q is the initial state, F ⊂ Q is the set of accepting states (also called final states Kari 2016a) or marker states (Ramadge and Wonham 1987), δ ⊂ Q ×  × Q is the transition relation. Finite sequences of letters are called words (Kari 2016a) (also called strings Sipser 1996). Given a word u in  ∗ , a subword of u is defined as a subsequence of u. Elements of relation δ are called transitions. More generally, we extend σ to σ ⊂ Q ×  + × Q in the usual way (recall that  + =  ∗ \ {}): for all states q1 , qn+1 ∈ Q and words σ1 . . . σn ∈  + , (q1 , σ1 . . . σn , qn+1 ) ∈ δ if and only if there exist states q2 , . . . , qn ∈ Q such that (qi , σi , qi+1 ) ∈ δ for all 1 ≤ i ≤ n. A word σ1 . . . σn ∈  + is called accepted by NFA (Q, , δ, q0 , F), if there exist states q1 , . . . , qn ∈ Q such that qn ∈ F and (qi , σi+1 , qi+1 ) ∈ δ for all 0 ≤ i ≤ n − 1. Particularly  is accepted by the NFA if can be equivalently represented by a function and only if q0 ∈ F. Recall that δ δ : Q ×  + → 2 Q as δ(q, wa) = r ∈δ(q,w) δ(r, a) for all q ∈ Q, all w ∈  + , and all a ∈ . Initially, the NFA is in state q0 ; after read a word σ1 . . . σn ∈  + , the NFA changes its state according to its transition relation δ nondeterministically. For an alphabet , a subset L of  ∗ is called a formal language (or language for short). A formal language L ⊂  ∗ is called recognized by an NFA A = (Q, , δ, q0 , F) if each word of L is accepted by the NFA and no word of  ∗ \ L is accepted by the NFA. A formal language recognized by an NFA is called regular. The regular language recognized by NFA A is denoted by L(A). An NFA (Q, , δ, q0 , F) is called a deterministic finite automaton (DFA) if for all q1 , q2 , q2 ∈ Q and σ ∈ , (q1 , σ, q2 ) ∈ δ and (q1 , σ, q2 ) ∈ δ imply q2 = q2 . In this case, we also regard δ as a partial function from Q ×  ∗ to Q. Later on, we will use δ as a relation or a partial function alternatively for convenience. Proposition 1.8 Each regular language is recognized by a DFA. Proof Let A = (Q, , δ, q0 , F) be an NFA. Now we use the classical subset construction (a.k.a. powerset construction) technique to construct a DFA A = (Q  , , δ  , q0 , F  ) satisfying L(A) = L(A ). Set Q  = 2 Q \ {∅}, q0 = {q0 }, F  = {E ⊂ Q|E ∩ F = ∅}, for all q1 , q2 ∈ Q  and σ ∈ , (q1 , σ, q2 ) ∈ δ  if and only if q2 = {q ∈ Q|(∃q  ∈ q1 )[(q  , σ, q) ∈ δ]}.

1.3 Finite Automata and Regular Languages

11

Fig. 1.5 An NFA (left) and a DFA (right), where they recognize the same regular language, double circles denote final states

Remove each q  ∈ 2 Q that is not accessible from q0 , i.e., there exists no w ∈  + such that (q0 , w, q  ) ∈ δ  , and all transitions starting from q  . It is clear that A is a  DFA and L(A) = L(A ). By Proposition 1.8, one sees that given an NFA, one can construct a DFA which is usually of exponential size of the NFA recognizing the same formal language. Example 1.1 Consider an NFA shown in the left of Fig. 1.5. To its right is a DFA that recognizes the same regular language as the NFA recognizes. Word 01 is accepted by the NFA, but neither  nor 00 is accepted by the NFA. One sees that every NFA can be represented as a weighted directed graph in a natural way (e.g., Fig. 1.5), where the initial state is marked by a single arrow (e.g., s1 and {s1 } in Fig. 1.5), weights and double circles/rectangles are used to denote letters and final states, respectively. In this sense, we call a DFA complete if the corresponding weighted directed graph is complete. For ease of the subsequent writing, we design an algorithm for computing the powerset construction (a DFA) of a weighted directed graph as a slight generalization of an NFA, where the obtained DFA contains only states reachable from the initial state. Intuitively, Algorithm 1.1 starts from a subset V of vertices of G, tracks the sets of states that are consistent with the letter reading of G (as a generalized NFA), and 1 records newly finally results in a DFA AV . Symbol S records the states of AV , Stemp 2 occurred states of AV , and Stemp records the newly occurred states that have not been recorded by S. In the FOR structure, for each newly found state s of AV (in 1 ), for each letter j, the algorithm finds the new transition (s, j, s j ) (if it exists), Stemp 2 puts the transition into transition relation δ, and puts s j into S and Stemp if s j ∈ / S. 1 After the for structure, it clears Stemp and puts the new found states (i.e., states in 2 1 2 Stemp ) into Stemp , and finally clears Stemp . V V Because S ⊂ 2 and 2 is finite, after executing these steps for finitely many 1 becomes empty, and the algorithm terminates. For the process of Algotimes, Stemp rithm 1.1 returning a DFA, see the next example for an illustration. Example 1.2 Consider a weighted directed graph G shown in Fig. 1.6. The process of Algorithm 1.1 receiving G and {a} and returning the corresponding DFA is shown in Fig. 1.7.

12

1 Preliminaries

Algorithm 1.1 Require: A weighted directed graph G = (V, E, W, 2 ) and a subset V ⊂ V of vertices Ensure: A DFA AV as the powerset construction of G 1: Let S, S, , δ, and V be the state set, the final state set, the alphabet, the transition relation, and the initial state of AV , respectively 1 2 2: S := {V }, Stemp := {V }, Stemp := ∅ 1 3: while Stemp = ∅ do 1 4: for all s ∈ Stemp and j ∈  do 5: s j := {vs ∈ V|there is v ∈ s such that (v, vs ) ∈ E and j ∈ W((v, vs ))} 6: if s j = ∅ and s j ∈ / S then 2 2 7: S := S ∪ {s j }, Stemp := Stemp ∪ {s j }, δ := δ ∪ {(s, j, s j )} 8: else 9: if s j = ∅ then 10: δ := δ ∪ {(s, j, s j )} 11: end if 12: end if 13: end for 1 2 , S2 14: Stemp := Stemp temp := ∅ 15: end while Fig. 1.6 A directed weighted graph

In several of the following sections, in addition to repetitively using Algorithm 1.1, we will also repetitively use a variant of the powerset construction. The special construction deals with a special finite automaton containing a special state  such that all transitions from  are self-loops with weight being the whole alphabet. When constructing a new automaton, all generated states containing  are regarded as the same. In detail, the new algorithm is illustrated in Algorithm 1.2. Compared to Algorithm 1.1, the new algorithm contains three more lines 5, 6, and 7, in which all generated states containing  are changed to the same state consisting of . Example 1.3 Consider a weighted directed graph G shown in Fig. 1.8. The process of Algorithm 1.2 receiving G, {a}, and c (the special vertex) and returning the corresponding DFA is shown in Fig. 1.9. Note that not all formal languages are regular. Proposition 1.9 Language {a i bi |i ∈ N} that consists of all words that begin with any number of a’s, followed by equally many b’s, is not regular. Proof Suppose on the contrary that there exists a DFA G = (Q, {a, b}, δ, q0 , F) recognizing {a i bi |i ∈ N}. Then G accepts word a |Q| b|Q| . By the Pigeonhole Principle,

1.3 Finite Automata and Regular Languages

13

Fig. 1.7 Process of Algorithm 1.1 returning the powerset construction of the weighted directed graph in Fig. 1.6

Fig. 1.8 A directed weighted graph

Fig. 1.9 Process of Algorithm 1.2 returning the powerset construction of the weighted directed graph in Fig. 1.8

14

1 Preliminaries

Algorithm 1.2 Require: A weighted directed graph G = (V, E, W, 2 ), a subset V ⊂ V of vertices, and a special vertex  Ensure: A DFA AV as the powerset construction of G 1: Let S, S, , δ, and V be the state set, the final state set, the alphabet, the transition relation, and the initial state of AV , respectively 1 2 2: S := {V }, Stemp := {V }, Stemp := ∅ 1 3: while Stemp = ∅ do 1 and j ∈  do 4: for all s ∈ Stemp 5: s j := {vs ∈ V|there is v ∈ s such that (v, vs ) ∈ E and j ∈ W((v, vs ))} 6: if  ∈ s j then 7: s j := {} 8: end if 9: if s j = ∅ and s j ∈ / S then 2 2 10: S := S ∪ {s j }, Stemp := Stemp ∪ {s j }, δ := δ ∪ {(s, j, s j )} 11: else 12: if s j = ∅ then 13: δ := δ ∪ {(s, j, s j )} 14: end if 15: end if 16: end for 1 2 , S2 17: Stemp := Stemp temp := ∅ 18: end while

there exist 1 ≤ i < j ≤ |Q| such that δ(q0 , a i ) = δ(q0 , a j ). Thus G accepts each  word a |Q|+( j−i)n b|Q| , where n ∈ N, leading to a contradiction. An NFA can be extended to an -NFA by adding spontaneous transitions, i.e., transitions of states without reading a letter. Formally, an -NFA is a quintuple A = (Q, , δ, q0 , F), where Q is still a finite set of states, q0 ∈ Q is still the initial state, and F ⊂ Q is still a subset of final states, δ is a function from Q × ( ∪ {}) to 2 Q . For a state q ∈ Q, denote -CLOSURE(q) := {q} ∪ δ(q, ), which denotes the set of all states that the -NFA can enter from q without reading any letter. More generally, for a subset Q  of Q, denote -CLOSURE(Q  ) := q∈Q  -CLOSURE(q). δ is extended to δˆ : Q ×  ∗ → 2 Q recursively as follows: 1. For each state q ∈ Q,

ˆ ) = -CLOSURE(q). δ(q,

2. For each state q ∈ Q, each word w ∈  ∗ , and each letter a ∈ , ⎛ ˆ wa) = -CLOSURE ⎝ δ(q,



ˆ r ∈δ(q,w)

⎞ δ(r, a)⎠ .

1.3 Finite Automata and Regular Languages

15

Similar to an NFA, for an -NFA A = (Q, , δ, q0 , F), the language recognized ˆ 0 , w) ∩ F = ∅}. We say a state q ∈ Q is by A is defined by L(A) := {w ∈  ∗ |δ(q ˆ p, w). reachable from a state p ∈ Q through a w ∈  ∗ path if q ∈ δ( Proposition 1.10 Each -NFA recognizes a regular language. Proof To prove this, we only need to prove that for each -NFA A, there exists an NFA that recognizes L(A). Given an -NFA A = (Q, , δ, q0 , F), we next construct an NFA A = (Q, , δ  , q0 , F  ) that recognizes L(A). As above, δ : Q × ( ∪ {}) → ˆ a) for all q ∈ Q and 2 Q is extended to δˆ : Q ×  ∗ → 2 Q . We set δ  (q, a) = δ(q,   + Q a ∈ , and extend δ to δ : Q ×  → 2 as the case for an NFA. Next we prove, using mathematical induction on |w|, that ˆ w) δ  (q, w) = δ(q, for each state q ∈ Q and each nonempty word w ∈  + . 1. Let |w| = 1. Then w is a letter and ˆ w) δ  (q, w) = δ(q, by definition. 2. Assume that the claim has been proved for u ∈  + and consider w = ua, where a ∈ . Then for each q ∈ Q, we have δ  (q, ua) =



δ  (r, a) =

r ∈δ  (q,u)

=

ˆ r ∈δ(q,u)



ˆ a) δ(r,

ˆ r ∈δ(q,u)

⎛

-CLOSURE ⎝ ⎛

ˆ ua) = -CLOSURE ⎝ δ(q,



ˆ r  ∈δ(r,)



⎞    δ r ,a ⎠,

(1.8)

⎞

δ(r, a)⎠ .

(1.9)

ˆ r ∈δ(q,u)

It is clear that the right-hand sides of (1.8) and (1.9) are equal. It holds that (1.9) is ˆ u) through an a subset of (1.8), because each state x that is reachable from r ∈ δ(q, ˆ ). On the other a path is trivially reachable from r through an a path as r ∈ δ(r, direction, we have (1.8) is a subset of (1.9), because every state x that is reachable ˆ ) ∩ δ(q, ˆ u) through ˆ u) through an a path is reachable from r  ∈ δ(r, from r ∈ δ(q, an a path. ˆ w) for all q ∈ Q and all w ∈  + . Finally, we We have shown that δ  (q, w) = δ(q,    / L(A) and F  = F ∪ {q0 } otherwise. set the final state set F of A as F = F if  ∈ / F, thus q0 ∈ / F  , and  ∈ / L(A ). We also have for Assume  ∈ / L(A), then q0 ∈  ˆ each nonempty word w, δ (q0 , w) = δ(q0 , w). Then by F = F  , we have A accepts w if and only if A accepts w. Consequently L(A) = L(A ).

16

1 Preliminaries

Fig. 1.10 An -NFA (left), an NFA (middle), and a DFA (right) recognizing the same language

Assume  ∈ L(A), then  ∈ L(A ). For a nonempty word w ∈  + , if w ∈ ˆ 0 , w) ∩ F = ∅, δ  (q0 , w) ∩ F  = ∅, w ∈ L(A ). If w ∈ L(A ), then L(A), then δ(q  δ (q0 , w) ∩ F  = ∅. If in A a final state q  in F is reachable from q0 through a w ˆ 0 , w) ∩ F = δ  (q0 , w) ∩ F = ∅, w ∈ L(A). Next assume that the final path, then δ(q state q0 is reachable from q0 in A through a w path. If q0 ∈ F, then w ∈ L(A); else / F, by  ∈ L(A), a state q  ∈ F is reachable from q0 in A through an  path, if q0 ∈  hence w ∈ L(A). Consequently, we also have L(A) = L(A ). Example 1.4 Consider the -NFA, NFA, and DFA shown in Fig. 1.10. The languages recognized by them are the same by Propositions 1.8 and 1.10. An NFA is a classical computation model. We say a problem can be solvable by an NFA if when the problem is encoded to a formal language, the language can be recognized by the NFA. That is, all instances of the problem are encoded to words over the same alphabet. Each positive instance is encoded to a word accepted by the NFA, but each negative instance is encoded to a word that is not accepted by the NFA. For example, if we consider the problem whether a given NFA is a DFA, then a DFA is a positive instance, and NFAs that are not deterministic are negative instances. Let us briefly recall basic concepts on decidability and complexity (Linz 2006; Sipser 1996; Stearns et al. 1965). Given two sets A and B such that B ⊂ A, a decision problem refers to as whether there exists an algorithm (equivalently defined by a halting Turing machine) for determining whether a given a ∈ A belongs to B. A decision problem is called decidable if an algorithm for solving this problem exists, and called undecidable otherwise. For example, the well-known Turing machine halting problem is undecidable. Decidable problems can be classified into several classes according to the complexity of the algorithms solving them. For example, P (resp., NP, PSPACE, NPSPACE, EXPTIME, EXPSPACE) denotes the class of problems solvable by polynomial-time (resp. nondeterministic polynomial-time, polynomialspace, nondeterministic polynomial-space, exponential-time, exponential-space) algorithms. It is known that PSPACE = NPSPACE, PEXPTIME, PSPACE  EXPSPACE, and P ⊂ NP ⊂ PSPACE ⊂ EXPTIME. It is widely conjectured that all these containments are proper. A decision problem is called NP (resp. PSPACE, EXPTIME,

1.3 Finite Automata and Regular Languages

17

EXPSPACE)-hard if every NP (resp. PSPACE, EXPTIME, EXPSPACE) problem can be reduced to it by a polynomial-time algorithm. Hence, there exists no polynomialtime algorithm for determining an NP (resp. PSPACE)-hard problem unless P = NP (resp. PSPACE). There exists no polynomial-space (hence polynomial-time) algorithm for solving an EXPSPACE-hard problem. There exists no polynomial-time algorithm for solving an EXPTIME-hard problem. A problem is called NP (resp. PSPACE, EXPTIME, EXPSPACE)-complete if it belongs to NP (resp. PSPACE, EXPTIME, EXPSPACE) and is NP (resp. PSPACE, EXPTIME, EXPSPACE)-hard.

1.4 Topological Spaces and Discrete Spaces 1.4.1 Topological Spaces Let X be a set, a collection τ of subsets of X is called a topology over X if it satisfies the following three open-set axioms: • ∅, X ∈ τ . • The union of any collection of elements of τ belongs to τ . • The intersection of any finite collection of elements of τ belongs to τ . A topological space is an ordered pair (X, τ ), where X is a set and τ is a topology over X . When τ is understood, we simply call X a topological space or a space. Elements of a topology τ over X are called open subsets of X . A subset U of topological space X is called closed if its complement X \ U is open, and called clopen if it is both closed and open. Given a subset U of X , the closure of U , denoted by U or Clo(U ), is the intersection of all closed subsets of X containing U ; the interior of U , denoted by U˚ or Int(U ), is the union of all open subsets of X contained in U . For a topological space (X, τ ), as usual we call a subset Y of X a topological subspace equipped with the subspace topology {U |U = Y ∩ V, V ∈ τ }. A topology can be generated by a collection of open subsets. Let X be a set, a collection B of subsets of X is called a topological base over X if it satisfies the following conditions: • X = ∪ B∈B B. • For all B1 , B2 ∈ B and x ∈ B1 ∩ B2 , there exists B3 ∈ B such that x ∈ B3 ⊂ B1 ∩ B2 . Let X be a set and B a topological base over X . One sees that the nonempty intersection of any finitely many elements of B is a union of elements of B. Then the collection of ∅ and all unions of elements of B form a topology over X , which is called the topology generated by B. A topological space is called second-countable if its topology can be generated by a countable topological base.

18

1 Preliminaries

Fig. 1.11 T1 space, Hausdorff space, and normal space

Let X and Y be two topological spaces. A map f : X → Y is said to be continuous if for each open subset V of Y , its preimage f −1 (V ) := {x ∈ X | f (x) ∈ V } is open in X . In particular, map f : X → Y is called a homeomorphism if f is bijective and both f and f −1 are continuous. In this sense, X is called homeomorphic to Y .

1.4.2 Separability A topological space (X, τ ) is called T1 if every singleton of X is closed, equivalently, for every pair of distinct points x and y of X , there exists a neighborhood Ux of x that does not contain y. For a point x of X , a neighborhood of x is a subset of X that contains an open subset of X containing x. More specifically, a topological space X is called Hausdorff if for every pair of distinct points x and y in X , there exist disjoint open subsets Ux and Vy of X separating them, i.e., Ux ∩ Vy = ∅, x ∈ Ux , and y ∈ Vy . A topological space X is called normal if for every two disjoint nonempty closed subsets A and B of X , there exist disjoint open subsets U A and VB of X such that A ⊂ U A and B ⊂ VB (see Fig. 1.11 for an illustration). Let X be a topological space. A sequence x1 , x2 , . . . of points of X is called convergent to a point x of X (x is called a limit of the sequence) if for every open subset U of X containing x, there exists a positive integer N such that xi ∈ U for all i > N . In a Hausdorff space, each convergent sequence x1 , x2 , . . . has a unique limit x, and denote limi→∞ xi = x. Given topological spaces X and Y and a continuous map f : X → Y , for each convergent sequence (xi )i∈N in X , we have limi→∞ f (xi ) = f (limi→∞ xi ).

1.4.3 Compactness A cover of a topological space X is a collection of subsets of X whose union equals X . Given a cover α of X , a subcover of α is a subset β of α such that β is also a cover of X . A cover α of X is called an open cover if all elements of α are open, and finite if α consists of finitely many subsets of X . Let α and β be two covers of X , α is said to be finer than β, denoted by α  β, if for each element A of α, there exists B ∈ β such that A ⊂ B. In this sense, α is called a refinement of β. For covers α, β, γ of

1.4 Topological Spaces and Discrete Spaces

19

space X , if α  β and β  γ, then α  γ. A topological space X is called compact if every open cover of X has a finite subcover.

1.4.4 Connectedness A topological space X is said to be connected if the only clopen subsets of X are ∅ and X , equivalently, X is not connected if and only if there exist disjoint nonempty open subsets U and V of X satisfying X = U ∪ V . A topological space X is called path-connected if for every two distinct points x and y of X , there exists a continuous map f : [0, 1] → X such that f (0) = x and f (1) = y, where [0, 1] is equipped with the topology generated by the topological base ((a, b) ∩ [0, 1])a,b∈R with a 0 there exists δ > 0 such that for all x, y ∈ X , d1 (x, y) < δ implies d2 ( f (x), f (y)) < . Uniformly continuous functions are continuous. Proposition 1.12 Metric spaces are Hausdorff and normal. Proof Let (X, d) be a metric space. For every two distinct points x and y in X , the disjoint open balls B d(x,y) (x) and B d(x,y) (y) separate them. That is, X is Hausdorff. 2 2 Let U be a nonempty subset of X . The distance of a point x ∈ X to U is defined as dist(x, U ) = inf d(x, u) ∈ R. u∈U

We claim that the map x → dist(x, U ) is uniformly continuous on X . For all x, y ∈ X , and all u ∈ U , d(x, u) ≤ d(x, y) + d(y, u), d(y, u) ≤ d(y, x) + d(x, u), then dist(x, U ) ≤ d(x, y) + dist(y, U ), dist(y, U ) ≤ d(y, x) + dist(x, U ), | dist(x, U ) − dist(y, U )| ≤ d(x, y).

1.4 Topological Spaces and Discrete Spaces

21

Hence, the map x → dist(x, U ) is uniformly continuous. Suppose that A1 and A2 are disjoint nonempty closed subsets of X . The map f : X → R defined by f (x) := dist(x, A1 ) − dist(x, A2 ) is continuous. Note that for all x ∈ X , dist(x, Ai ) = 0 if and only if x ∈ Ai , i = 1, 2. Then the open subsets U := {x ∈ X | f (x) < 0} and V := {x ∈ X | f (x) > 0} are disjoint and contain A1 and A2 , respectively. Consequently, the space X is normal.  Example 1.6 Consider metric space (Rn , d), where d is the Euclidean metric. The space is Hausdorff, normal (by Proposition 1.12), path-connected and connected (by Proposition 1.11), but not compact. The space is second-countable. Actually, the set of all open balls with rational radii and rational centers (i.e., components of centers are rational) is a countable topological base for the topology induced by metric d.

1.4.6 Topological Dimensions Next we introduce the topological dimension dim(X ) of a topological space X . Throughout this book, we call a topological space discrete if it has topological dimension 0, and call a space continuous if it has a positive or infinite topological dimension. We set ∞ + ∞ = ∞, ∞ + a = a + ∞ = ∞ and a < ∞ for all a ∈ R as usual. Let α = (Ai )i∈I be a collection of subsets of a set X indexed by a set I . For each x ∈ X , the order of α at point x is defined as ord x (α) := −1 + |{i ∈ I |x ∈ Ai }| ∈ {−1} ∪ N ∪ {∞}. The order of α is then defined as ord(α) := sup ord x (α). x∈X

(If X = ∅, we write ord(α) = −1.) Intuitively, ord(α) is the greatest integer n (or ∞ if such an integer does not exist) such that there exist n + 1 distinct elements i 0 , . . . , i n ∈ I satisfying Ai0 ∩ · · · ∩ Ain = ∅. Then α is a cover of X if and only if ord x (α) ≥ 0 for all x ∈ X . A cover α = (Ai )i∈I of X is called a partition of X if Ai ∩ A j = ∅ for any distinct i, j ∈ I . A cover α of X is a partition if and only if ord x (α) = 0 for all x ∈ X . We next use F OC(X ) to denote the set of all finite open covers of X .

22

1 Preliminaries

Definition 1.5 Let α = (Ui )i∈I be a finite open cover of a topological space X . The number D(α) is defined by D(α) :=

min

βα β∈F OC(X )

ord(β).

By definition of D(α), we have D(α) ≤ ord(α) ≤ −1 + |I |, as α  α; D(α) ∈ {−1} ∪ N; and D(α) ≤ n if and only if there exists β ∈ F OC(X ) such that β  α and ord(β) ≤ n. Proposition 1.13 Let X be a topological space and α, β be in F OC(X ) such that α  β. Then D(α) ≥ D(β). Proof This conclusion follows from the transitivity of refinement.



Definition 1.6 Let X be a topological space. Its topological dimension dim(X ) ∈ {−1} ∪ N ∪ {∞} is defined by dim(X ) :=

sup

D(α).

α∈F OC(X )

ˇ The topological dimension dim(X ) is also called the Cech–Lebesgue covering dimension of X . For two topological spaces X and Y such that X is homeomorphic to Y , by definition one has dim(X ) = dim(Y ). Then the topological dimension is a topological invariant. Example 1.7 One has dim(X ) = −1 if and only if X = ∅. Example 1.8 Consider a set X with its discrete topology 2 X and α = (Ai )i∈[1;n] ∈ c (i) n Ai j ) j∈[0;2n −1] , where for each F OC(X ). Consider another collection β = (∩i=1 j ∈ [0; 2n − 1], c j (i) ∈ {0, 1}, Ai0 and Ai1 denote Ai and X \ Ai , respectively, the concatenation c j (1) . . . c j (n) is the binary representation of j. Then β  α and β is a finite open partition of X . Moreover, ord(β) = 0, D(α) = 0. Consequently, dim(X ) = 0. Proposition 1.14 Let X be a topological space and α = (Ui )i∈I ∈ F OC(X ). Then one has ord(β). D(α) = min β=(Vi )i∈I ∈F OC(X ) Vi ⊂Ui ,i∈I

Proof We only need to prove that for all α = (Ui )i∈I , β = (V j ) j∈J ∈ F OC(X ) with β  α, there is γ = (Wi )i∈I ∈ F OC(X ) such that Wi ⊂ Ui for all i ∈ I and ord x (γ) ≤ ord x (β) for all x ∈ X . Since β  α, there is a map φ : J → I such that V j ⊂ Uφ( j) for all j ∈ J . = Choose W i j∈φ −1 (i) V j , where i ∈ I . Then each Wi is open in X , γ covers X since i∈I Wi ⊃ j∈J Vi = X , and ord x (γ) ≤ ord x (β) for all x ∈ X since for all Vi1 , Wi2 , Vi3 , Wi4 with Vi1 ⊂ Wi2 , Vi3 ⊂ Wi4 , and i 2 = i 4 , we have i 1 = i 3 . 

1.4 Topological Spaces and Discrete Spaces

23

Proposition 1.15 Let X be a connected T1 topological space containing at least two points. Then dim(X ) ≥ 1. Proof Choose two distinct points x and y of X . Since X is T1 , the subsets X \ {x} and X \ {y} are open in X . Consider α = {X \ {x}, X \ {y}} ∈ F OC(X ). The connectedness of X implies that for every β = (U, V ) ∈ F OC(X ) with U ⊂ X \ {x} and V ⊂ X \ {y}, ord(β) ≥ 1. Hence D(α) ≥ 1 by Proposition 1.14. Moreover, dim(X ) ≥ D(α) ≥ 1.  Remark 1.1 As each Euclidean space Rn with n > 1 is connected and Hausdorff (hence T1 ), Proposition 1.15 implies that each Euclidean space has topological dimension at least 1. Next we prove that each positive-length closed interval of the real line R has topological dimension 1. To this end, we need a number of results on the topological dimensions of compact metric spaces. Let (X, d) be a metric space. The diameter diam(Y ) of a subset Y ⊂ X is diam(Y ) := sup d(y1 , y2 ) ∈ [0, ∞]. y1 ,y2 ∈Y

The mesh of a cover α = (Ai )i∈I of X is defined by mesh(α) := sup diam(Ai ) ∈ [0, ∞]. i∈I

Remark 1.2 If α and β are covers of a metric space such that β  α, then mesh(β) ≤ mesh(α). Proposition 1.16 Consider a compact metric space X and its open cover α = (Ui )i∈I . There is a positive real number λ (called a Lebesgue number of α) such that for each Y ⊂ X satisfying diam(Y ) ≤ λ, Y ⊂ Ui for some i ∈ I . Proof For every x ∈ X , we choose i(x) ∈ I such that x ∈ Ui(x) . Since Ui(x) is open in X , there is r x > 0 such that B2rx (x) ⊂ Ui(x) . Then open balls (Brx (x))x∈X cover X . Since X is compact, there is a finite subset X  of X such that the balls (Brx (x))x∈X  cover X . Denote λ := min x∈X  r x > 0. Suppose that Y ⊂ X satisfies diam(Y ) ≤ λ and choose an arbitrary point y ∈ Y . Then there exists a point x  ∈ X  such that d(y, x  ) < r x  . Then we have Y ⊂ Brx  +λ (x  ) ⊂ B2rx  (x  ) ⊂ Ui(x  ) . Consequently, λ satisfies the condition in this proposition.



Proposition 1.17 Consider a compact metric space X and choose n ∈ N. Then the following items are equivalent:

24

1 Preliminaries

1. dim(X ) ≤ n; 2. for every  > 0, there is α ∈ F OC(X ) such that mesh(α) ≤  and ord(α) ≤ n; 3. for each k ∈ N, there exists αk ∈ F OC(X ) such that ord(αk ) ≤ n, and limk→∞ mesh(αk ) = 0. Proof Clearly, conditions 2 and 3 are equivalent. Let  > 0. Since X is compact, there exists α ∈ F OC(X ) consisting of open balls of radius 2 . Then mesh(α) ≤ . If dim(X ) ≤ n, then D(α) ≤ n and there exists β ∈ F OC(X ) satisfying ord(β) = D(α) and β  α. Hence, ord(β) ≤ n and mesh(β) ≤ mesh(α) ≤ . This shows that condition 1 implies condition 2. Let α be in F OC(X ) and λ > 0 be a Lebesgue number of α. If condition 2 holds, then there exists β ∈ F OC(X ) satisfying mesh(β) ≤ λ and ord(β) ≤ n. By Proposition 1.16, we have β  α. Then D(α) ≤ ord(β) ≤ n and hence dim(X ) ≤ n. This shows that condition 2 implies condition 1.  Next we prove that dim([0, 1]) = 1. Since for all real numbers a < b, [a, b] is homeomorphic to [0, 1], we then have dim([a, b]) = 1. Proposition 1.18 dim([0, 1]) = 1. Proof As [0, 1] is connected, T1 , and has cardinality greater than 2, by Proposition 1.15, we have dim([0, 1]) ≥ 1. Choose 2 ≤ k ∈ N. Consider points xi = 2ki for all i ∈ [0; 2k]. Let αk be in F OC([0, 1]) consisting of the intervals [x0 , x2 ), (x2k−2 , x2k ], and all the intervals of the form (xi , xi+2 ) for i ∈ [1; 2k − 3]. We then have ord(αk ) = 1. On the other hand, limk→∞ mesh(αk ) = limk→∞ k1 = 0. This implies dim([0, 1]) ≤ 1 by Proposition 1.17. 

1.4.7 Several Discrete Topological Spaces Consider a finite metric space (X, d). It is evident to see that the metric d induces the discrete topology. Hence, each finite metric space has topological dimension 0. In the main contexts of the book, Boolean control networks, nondeterministic finitetransition systems, and finite automata are dynamical systems over finite metric spaces. Consider a countable set Zn , where n ∈ Z+ . Over Zn , the Euclidean metric d also induces the discrete topology. In this sense, Zn (and hence any nonempty subset of Zn least 2) has topological dimension 0. Petri nets are dynamical systems over such discrete spaces. Now we introduce the Cantor set, which also has topological dimension 0 but its intrinsic topology is not the discrete topology. Later on, we prove that the Cantor set is compact, totally disconnected, and perfect, showing that the Cantor set is remarkably different from a Euclidean space and a topological space with the discrete topology.

1.4 Topological Spaces and Discrete Spaces

25

Cellular automata are dynamical systems over the Cantor space that is homeomorphic to the Cantor set. For a closed interval [a, b] ⊂ R, where a < b, we define     a + 2b 2a + b ∪ ,b . T ([a, b]) := a, 3 3 That is, T ([a, b]) is obtained from [a, b] by removing its middle third part. More generally, for every subset A ⊂ R which is the union of finitely many ([ai , bi ])1≤i≤k of pairwise disjoint closed intervals, we set T (A) :=

k

T ([ai , bi ]).

i=1

Let us define inductively a sequence (Cn )n∈N of closed subsets of [0, 1] by setting C0 := [0, 1], Cn+1 := T (Cn ) for all n ∈ N. We then have    2 1 ∪ C1 = 0, ,1 , 3 3         2 1 2 7 8 1 ∪ , ∪ , ∪ ,1 , C2 = 0, 9 9 3 3 9 9 .... 

Note that the set Cn is the union of 2n pairwise disjoint closed intervals of length These intervals are the connected components of Cn (see Fig. 1.12). The set  Cn C := n∈N

Fig. 1.12 Construction of the Cantor set

1 . 3n

26

1 Preliminaries

is called the Cantor ternary set or simply the Cantor set. Later on, we assume that the Cantor set C is equipped with the topology generated by the topological base (C ∩ (a, b))a,b∈R with a k j . Proposition 1.21 The symbolic space {0, 1}N is compact. Proof First, we show that each sequence (ci )i∈N in {0, 1}N has a convergent subsequence. We next choose integers 0 ≤ i 0 < i 1 < · · · such that (ci j ) j∈N is a convergent subsequence of (ci )i∈N (see Table 1.1). We choose i 0 satisfying c j (0) = ci0 (0) for infinitely many j ∈ N. Suppose i 0 , . . . , i k−1 have been chosen, then we choose i k satisfying cik (l) = cik−1 (l) for all 0 ≤ l ≤ k − 1, and c j (k) = cik (k) for infinitely many

1.4 Topological Spaces and Discrete Spaces

27

Table 1.1 Construction of a convergent subsequence (ci j ) j∈N of a sequence (ci )i∈N in {0, 1}N , where si ∈ {0, 1}, i ∈ N ci0 ci1 ... cik−1 cik ... 0 1 .. . k−1 k .. .

s0

s0 s1

... ...

s0 s1

s0 s1

... ...

sk−1

sk−1 sk

... ...

j ∈ N. The existence of such a subsequence (ci j ) j∈N naturally follows from the fact each infinite sequence whose elements are chosen from a common finite set must have an infinite constant subsequence. Choose c ∈ {0, 1}N such that c( j) = ci j ( j) for all j ∈ N, then one has lim j→∞ ci j = c. Second, we show that {0, 1}N is compact. By the countable base consisting of cylinders, we only need to prove that each countable cover consisting of cylinders of {0, 1}N has a finite subcover of {0, 1}N , since each open subset of {0, 1}N is a union of cylinders, and there are totally countably many cylinders. Let (Ui )i∈I be an j j open cover of {0, 1}N , where I is a set, Ui = ∪ j∈N Vi , and Vi is a cylinder. If the j N countable cover (Vi )i∈I, j∈N of {0, 1} has a finite subcover, then for each cylinder of the finite subcover, choose Ui containing the cylinder, the finitely many Ui ’s form a finite subcover of {0, 1}N . Suppose on the contrary there exists a cover (Vi )i∈N of {0, 1}N , where each Vi is a exist infinitely cylinder, such that (Vi )i∈N has no finite subcover of {0, 1}N . Then there k −1 V j =: Wk = ∅. many integers 0 < i 0 < i 1 < · · · such that for all k ∈ N, Vik \ ij=0 For each k ∈ N, choose ck ∈ Wk ⊂ Vik . We have proved that each sequence in {0, 1}N has a converging subsequence. Choose a convergent subsequence (ck j ) j∈N of (ck )k∈N , and denote lim j→∞ ck j =: c. Then c ∈ Vl for some l ∈ N, and ck j ∈ Vl for infinitely  many j ∈ N, which contradicts the construction of (ck )k∈N . Now we can prove that the Cantor set is homeomorphic to the symbolic space {0, 1}N . Proposition 1.22 The map ϕ : {0, 1}N → C defined by ϕ(u) :=

∞  2u(k) k=0

3k+1

for all u ∈ {0, 1}N is a homeomorphism from the symbolic space {0, 1}N onto the Cantor set C.

28

1 Preliminaries

Proof Define f n+1 (u(0), . . . , u(n)) :=

n  2u(k) k=0

3k+1

,

 where n ∈ N. Recall that C = n∈N Cn , where Cn is the union of 2n pairwise disjoint closed of length 31n . Then f n+1 (u(0), . . . , u(n)) is the minimal value n intervals n−i ofthe ( i=0 2 u(i))-th closed interval of Cn+1 . Moreover, ϕ(u) belongs to the n 2n−i u(i))-th closed interval of Cn+1 . Note that ϕ(u) belongs to Cn+1 for each ( i=0 n ∈ N, ϕ(u) ∈ C. On the other hand, for each n ∈ N, we have ∞  1 1 1 = < n, n+k n 3 2 · 3 3 k=1

hence ϕ is injective. Each real number x ∈ [0, 1] admits a ternary expansion, that is, a sequence (vk )k∈N ∈ {0, 1, 2}N such that ∞  vk x= . k+1 3 k=0 We also write this equality under the form x = 0.v0 v1 . . . vk . . .. For all n ∈ N, all w0 , . . . , wn ∈ {0, 2}, the minimal value of the ( closed interval of Cn+1 has the unique ternary expansion

n i=0

2n−1−i wi )-th

0.w0 w1 . . . wn 0000 . . ., the maximal value has exactly two ternary expansions 0.w0 w1 . . . wn 2222 . . . and

 w−1 .w0 w1 . . . wn 0000 . . .,

 where w−1 .w0 w1 . . . wn = 0.w0 w1 . . . wn + 0. 0 . . . 0 1, and wk = 1 for some −1 ≤

k ≤ n. For example, taking n = 0, we have 0 = 0.0000 . . .,

n

1 = 0.1000 . . . = 0.0222 . . ., 3

2 = 0.2000 . . ., 1 = 1.0000 . . . = 0.2222 . . .. 3

1.4 Topological Spaces and Discrete Spaces

29

The above arguments imply that Cn+1 consists of all numbers x ∈ [0, 1] that admit a ternary expansion (wk )k∈N such that wk ∈ {0, 2} for all k ≤ n. Moreover, the Cantor set C is the set consisting of the numbers x ∈ [0, 1] that admit a ternary expansion whose bits all belong to {0, 2}. That is, the map ϕ is surjective and hence bijective. For each u ∈ {0, 1}+ , one has |ϕ(u) − ϕ(v)| ≤

∞  k=|u|

|ϕ(u) − ϕ(v)| ≥

2 3k+1

=

1 for each v ∈ [u], 3|u|

2 for each v ∈ {0, 1}N \ [u]. 3|u|

As lim|u|→∞ 31|u| = 0, and [u] is open in {0, 1}N , the map ϕ is continuous. Since {0, 1}N is compact, for each closed subset U of {0, 1}N , U is compact in {0, 1}N . By the continuity of ϕ, ϕ(U ) is compact in C, and hence closed in C since C is Hausdorff by Proposition 1.19. That is, ϕ−1 is also continuous. Consequently, ϕ is a homeomorphism.  Corollary 1.1 The Cantor set C is uncountable. From Propositions 1.19, 1.20, and 1.22, the following Proposition 1.23 follows. Proposition 1.23 The symbolic space {0, 1}N is totally disconnected and has topological dimension 0. Proposition 1.24 The symbolic space {0, 1}N (and hence the Cantor set) is perfect. Proof Since each nonempty open subset of {0, 1}N is a union of cylinders and cylinders are uncountable subsets of {0, 1}N , the symbolic space (and hence the Cantor set) is perfect.  Now we have obtained several fundamental properties of the Cantor set and the symbolic space {0, 1}N . Actually, in the literature, a topological space is usually called a Cantor space if it is homeomorphic to the Cantor set. Formally, one of Brouwer’s results induces the following definition. Definition 1.7 A topological space is called a Cantor space if it is nonempty, metrizable, compact, totally disconnected, and perfect. By Proposition 1.22, the symbolic space {0, 1}N is metrizable, hence is a Cantor space. Consider the symbolic space S N , where S is an arbitrary finite set having at least two elements. The concept of cylinders for {0, 1}N is naturally extended to those for SN. d Consider the symbolic space S Z , where S is an arbitrary finite set having at least two elements, and d ∈ Z+ . Similar to {0, 1}N , a map c : Zd → S is also called a configuration. For each finite subset (called window) D ⊂ Zd , a map p : D → S is

30

1 Preliminaries

called a pattern, D is called the domain of p, which is denoted by D =: dom( p). For d a pattern p : D → S, a cylinder is a subset [ p] := {c ∈ S Z |c(i) = p(i)∀i ∈ D}. A Zd cylinder [ p] can also be written as [c] D if c ∈ S and c(i) = p(i) for all i ∈ D. Here d cylinders are also clopen. The topology of the symbolic space S Z is generated by a topological base consisting of all cylinders. Hence, the symbolic space is also secondd countable. A special cylinder [c] D is called a ball if c ∈ S Z , and D = {(z 1 , . . . , z d ) ∈ d Z | max(|z 1 |, . . . , |z d |) ≤ r ∈ N}. It is evident to see that each cylinder is a finite union of balls, hence the family of all balls is also a topological base generating d d the topology of the symbolic space S Z . Let c, c ∈ S Z be two configurations, and d  D ⊂ Z a window. We say that c and c agree on D if c(d) = c (d) for each d ∈ D. d Similar to the symbolic space {0, 1}N , a sequence (ci )i∈N in S Z converges to a Zd d configuration c ∈ S if and only if for each j ∈ Z , there exists k j ∈ N such that cl ( j) = c( j) for all l > k j . d Cellular automata are dynamical systems over the symbolic space S Z . Next we Zd show that the symbolic space S is homeomorphic to the Cantor set, hence is also a Cantor space. Proposition 1.25 The symbolic space {0, 1}N is homeomorphic to the symbolic space S N , where S is a finite set having at least two elements. Proof Let S = {0, 1, . . . , k}, where k ≥ 2. Let ψ : S → {0, 1}∗ be such that ψ(k) = 1k , ψ(i) = 1i 0, where 0 ≤ i < k. For example, S = {0, 1, 2, 3}, ψ(3) = 111, ψ(2) = 110, ψ(1) = 10, ψ(0) = 0. Then for each word u ∈ {0, 1}∗ , there exists a unique word v ∈ S ∗ such that ψ(v(0)) . . . ψ(v(|v| − 1)) = u. This induces a bijection ϕ from S N to {0, 1}N : for each c ∈ S N , ϕ(c) = ψ(c(0))ψ(c(1)) . . . ; for each e ∈ {0, 1}N , ϕ−1 (e) = c, where c ∈ S N is the unique configuration in S N such that ψ(c(0))ψ(c(1)) · · · = e. One sees that ϕ maps each cylinder in S N to a cylinder in {0, 1}N , and ϕ−1 maps each cylinder in {0, 1}N to a finite union of cylinders in S N . Hence, ϕ is a homeomorphism.  Proposition 1.26 The symbolic space S N is homeomorphic to the symbolic space S Z , where S is a finite set having at least two elements. Proof Consider the map ϕ : S N → S Z defined by ϕ(u) = v, ⎧ if i = 0, ⎨ u0 vi = u 2i−1 if i > 0, ⎩ u −2i if i < 0, where u = u 0 u 1 u 2 . . . ∈ S N , v = . . . v−2 v−1 v0 v1 v2 . . . ∈ S Z . It is clear that ϕ is bijective. One also sees that ϕ maps each cylinder in S N to a cylinder in S Z , and ϕ−1 maps each cylinder of S Z to a finite union of cylinders in S N . Hence, ϕ is a homeomorphism. 

1.4 Topological Spaces and Discrete Spaces

31

Proposition 1.27 The symbolic space S Z is homeomorphic to the symbolic space d S Z , where S is a finite set having at least two elements, and d ∈ Z+ . As a result, Zd S is a Cantor space. Proof Similar to Proposition 1.26, we can construct a bijection from S Z to S Z that d  maps each ball in S Z to a finite union of balls in S Z and vice versa. d

Notes In Sect. 1.4 of this chapter, we compare discrete spaces with Euclidean spaces mainly from the perspective of topological dimensions, where the topological dimension ˇ was introduced by Cech (1933). Discrete spaces, e.g., finite metric spaces, Zd , and Zd the Cantor space S have topological dimension 0. However, Euclidean spaces have positive topological dimensions, and particularly each positive-length closed interval in the real line has topological dimension 1. This shows that discrete spaces have essentially different topological properties compared to Euclidean spaces. We have chosen sufficient materials showing an essential difference between discrete spaces and Euclidean spaces, e.g., the discrete spaces we will consider in this book have topological dimension 0, while Euclidean spaces have positive topological dimensions. For further reading, e.g., an n-dimensional Euclidean space (i.e., the number of linearly independent vectors generating the space is n) has topological dimension n for each n ∈ Z+ , we refer the reader to Coornaert (2015), etc. Further results on symbolic dynamics can be found in K˚urka (2003), Lind and Marcus (1995), Kari (2016). On the other hand, Euclidean spaces are connected. However, discrete spaces considered in this book are totally disconnected. This also shows a remarkable difference between Euclidean spaces and discrete spaces: a discrete space cannot be locally Euclidean, hence cannot be a manifold. Normally, derivatives and calculus are basic tools that are used to define and study dynamical systems over Euclidean spaces. However, these tools seem pale in dealing with dynamical systems over discrete spaces. Despite this, limit and topology apply to dynamical systems over discrete spaces. Hence, many good properties of such systems have been derived (Kari 2016; K˚urka 2003). Moreover, plenty of particular techniques have been developed in dynamical systems over discrete spaces (Kari 2016; Cassandras and Lafortune 2010; Reutenauer 1990; Cheng et al. 2011; Wonham and Cai 2019).

References Cassandras CG, Lafortune S (2010) Introduction to discrete event systems. 2nd edn. Springer Publishing Company, Incorporated, Berlin Cheng D (2001) Semi-tensor product of matrices and its application to Morgen’s problem. Sci China Ser: Inf Sci 44(3):195–212

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Cheng D, Qi H (2009) Controllability and observability of Boolean control networks. Automatica 45(7):1659–1667 Cheng D, Qi H, Li Z (2011) Analysis and control of Boolean networks: a semi-tensor product approach. Springer, London Cheng D, Yang G, Xi Z (2007) Nonlinear systems possessing linear symmetry. Int J Robust Nonlinear Control 17(1):51–81 Cheng D, Zhang L (2003) On semi-tensor product of matrices and its applications. Acta Math Appl Sin 19(2):219–228 Cheng D, Zhao Y (2011) Identification of Boolean control networks. Automatica 47(4):702–710 Cheng D et al (2016) On decomposed subspaces of finite games. IEEE Trans Autom Control 61(11):3651–3656 Coornaert M (2015) Topological dimension and dynamical systems. Springer International Publishing, Berlin ˇ ˇ Cech E (1933) Pˇríspˇevek k teorii dimense. Casopis pro pˇestování matematiky a fysiky 62(8):277– 291 Fornasini E, Valcher ME (2013) Observability, reconstructibility and state observers of Boolean control networks. IEEE Trans Autom Control 58(6):1390–1401 Fornasini E, Valcher ME (2014) Optimal control of Boolean control networks. IEEE Trans Autom Control 59(5):1258–1270 Kari J (2016a) A lecture note on automata and formal languages. http://users.utu.fi/jkari/automata/ Kari J (2016b) Cellular automata. http://users.utu.fi/jkari/ca2016/ K˚urka P (2003) Topological and symbolic dynamics. Société mathématique de France Li H, Wang Y, Xie L (2015) Output tracking control of Boolean control networks via state feedback: constant reference signal case. Automatica 59:54–59 Li H, Xie L, Wang Y (2016) On robust control invariance of Boolean control networks. Automatica 68:392–396 Li R, Yang M, Chu T (2014) Observability conditions of Boolean control networks. Int J Robust Nonlinear Control 24(17):2711–2723 Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambridge University Press, New York Linz P (2006) An introduction to formal language and automata. Jones and Bartlett Publishers Inc, USA Lu J et al (2016) On pinning controllability of Boolean control networks. IEEE Trans Autom Control 61(6):1658–1663 Qi H et al (2016) Vector space structure of finite evolutionary games and its application to strategy profile convergence. J Syst Sci Complex 29(3):602–628 Ramadge PJ, Wonham WM (1987) Supervisory control of a class of discrete event processes. SIAM J Control Optim 25(1):206–230 Reutenauer C (1990) The mathematics of petri nets. Prentice-Hall Inc, Upper Saddle River Shu S, Lin F, Ying H (2007) Detectability of discrete event systems. IEEE Trans Autom Control 52(12):2356–2359 Sipser M (1996) Introduction to the theory of computation, 1st edn. International Thomson Publishing, Stamford Stearns RE, Hartmanis J, Lewis PM (1965) Hierarchies of memory limited computations. In: 6th annual symposium on switching circuit theory and logical design (SWCT 1965), pp 179-190 Wonham WM, Cai K (2019) Supervisory control of discrete-event systems. Springer International Publishing, Berlin Wu Y, Shen T (2017) Policy iteration algorithm for optimal control of stochastic logical dynamical systems. IEEE Trans Neural Netw Learn Syst 99, 1–6 (2017) Xu X, Hong Y (2013a) Observability analysis and observer design for finite automata via matrix approach. IET Control Theory Appl 7(12):1609–1615 Xu X, Hong Y (2013b) Solvability and control design for synchronization of Boolean networks. J Syst Sci Complex 26(6):871–885

References

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Zhang K, Zhang L, Su R (2016) A weighted pair graph representation for reconstructibility of Boolean control networks. SIAM J Control Optim 54(6):3040–3060 Zhang K, Zhang L, Xie L (2015) Invertibility and nonsingularity of Boolean control networks. Automatica 60:155–164 Zhang Z, Leifeld T, Zhang P (2017) Identification of Boolean control networks incorporating prior knowledge. In: 2017 IEEE 56th annual conference on decision and control (CDC), pp 5839–5844 Zhao Y, Li Z, Cheng D (2011) Optimal control of logical control networks. IEEE Trans Autom Control 56(8):1766–1776 Zhao Y, Qi H, Cheng D (2010) Input-state incidence matrix of Boolean control networks and its applications. Syst Control Lett 59(12):767–774 Zou Y, Zhu J (2017) Graph theory methods for decomposition w.r.t. outputs of Boolean control networks. J Syst Sci Complex 30(3):519–534 Zou Y, Zhu J (2015) Kalman decomposition for Boolean control networks. Automatica 54:65–71

Chapter 2

Different Types of Discrete-Time and Discrete-Space Dynamical Systems

In this chapter, we introduce basic concepts and properties of discrete-time and discrete-space dynamical systems which will be discussed in this book, including Boolean control networks, nondeterministic finite-transition systems, finite automata, labeled Petri nets, and cellular automata.

2.1 Boolean Control Networks During the past decades, a research direction called systems biology aiming at studying not individual genes, proteins, or cells but complex interactions between them has gradually emerged. It requires using a holistic approach to biological research, hence results in the study from a systematic point of view. To this end, computational and mathematical modeling of complex biological systems is fundamental. The development of the Human Genome Project is such an example. In the early 1960s, Jacob and Monod showed that any cell contains a number of “regulatory” genes that act as switches, where these switches can turn one another on and off (Waldrop 1993). This indicates that a genetic network is acting in a Boolean manner. Boolean networks (BNs), initiated by Kauffman (1969) to model genetic regulatory networks in 1969, are a proper tool in describing the Boolean manner. In a BN, nodes can be in one of two discrete states “1” and “0”, which represent a gene state “on” (high concentration of a protein) and “off” (low concentration), respectively. Every node updates its state according to a Boolean function of the network node states. When external regulation or perturbation are considered, BNs are naturally extended to Boolean control networks (BCNs) (Ideker et al. 2001). According to the intrinsic function principle of BNs/BCNs, they are discrete-time and discrete-space dynamical systems. Although a BN or a BCN is a simplified model of a genetic regulatory network, they can be used to characterize many important complex phenomena of biological systems, e.g., cell cycles (Fauré et al. 2006), cell apoptosis (Sridharan et al. 2012). For related interesting topics in control problems and dynamics of © Springer Nature Switzerland AG 2020 K. Zhang et al., Discrete-Time and Discrete-Space Dynamical Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-25972-3_2

35

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(probabilistic) Boolean networks, we recommend monographs such as Akutsu (2018), Cheng et al. (2011), Kunz and Stoffel (1997), Rosin (2015), Shmulevich and Dougherty (2010). A BN is formulated as the following logical form: x1 (t + 1) = f 1 (x1 (t), . . . , xn (t)), x2 (t + 1) = f 2 (x1 (t), . . . , xn (t)), .. .

(2.1)

xn (t + 1) = f n (x1 (t), · · · , xn (t)),

where t = 0, 1, 2, . . . denote discrete time steps; xi (t) ∈ D := {0, 1} denotes the value of state node xi at time step t, i ∈ [1; n]; f i : Dn → D is a Boolean function, i ∈ [1; n]. A BN (2.1) can be briefly denoted as x(t + 1) = f (x(t)),

(2.2)

where t = 0, 1, 2, . . . ; x(t) ∈ Dn stands for the state at time step t; f : Dn → Dn is a Boolean mapping. A BCN is formulated as the following logical form: x1 (t + 1) = f 1 (x1 (t), . . . , xn (t), u 1 (t), . . . , u m (t)), x2 (t + 1) = f 2 (x1 (t), . . . , xn (t), u 1 (t), . . . , u m (t)), .. . xn (t + 1) = f n (x1 (t), . . . , xn (t), u 1 (t), . . . , u m (t)), y1 (t) = h 1 (x1 (t), . . . , xn (t)),

(2.3)

y2 (t) = h 2 (x1 (t), . . . , xn (t)), .. . yq (t) = h n (x1 (t), . . . , xn (t)), where t = 0, 1, 2, . . . denote discrete time steps; xi (t), u j (t) and yk (t) ∈ D denote values of state node xi , input node u j , and output node yk at time step t, respectively, i ∈ [1; n], j ∈ [1; m], k ∈ [1; q]; f i : Dm+n → D and h j : Dn → D are Boolean functions, i ∈ [1; n], j ∈ [1; q]. A BCN (2.3) is represented in the compact form x(t + 1) = f (x(t), u(t)), y(t) = h(x(t)),

(2.4)

2.1 Boolean Control Networks

37

Fig. 2.1 Graph associated with BCN (2.5)

Fig. 2.2 State-transition graph of BCN (2.5)

where t = 0, 1, 2, . . . ; x(t) ∈ Dn , u(t) ∈ Dm , and y(t) ∈ Dq stand for the state, input, and output of the BCN at time step t; f : Dn+m → Dn and h : Dn → Dq are Boolean mappings. Example 2.1 (A simple BCN) A(t + 1) = B(t) ∧ u(t), B(t + 1) = ¬A(t), y(t) = A(t),

(2.5)

where t ∈ N, A(t), B(t), u(t), y(t) ∈ D (Figs. 2.1 and 2.2). The study on control-theoretic properties of BCNs dates back to 2007, when the problem of verifying controllability of a BCN was proved to be NP-hard in the number of nodes by Akutsu et al. (2007). They also pointed out that “One of the major goals of systems biology is to develop a control theory for complex biological systems.” Hence, it is of both theoretical and practical importance to study the control problems of BCNs. In 2009, Cheng and Qi used the semitensor product (STP) of matrices (proposed by Cheng 2001) to build a control-theoretic framework for BCNs (Cheng and Qi 2009). The STP of matrices is a natural generalization of the conventional matrix product. The STP applies to every pair of finite-dimensional matrices and preserves the associative law, the distributive law, etc. This framework makes it very convenient to use many basic results in matrix theory to deal with BCNs. For a detailed introduction to the STP and how to deal with control problems of BCNs under the framework, we refer the reader to the monograph (Cheng et al. 2011). Here, we only show related results that will be used in this book. The STP can provide a holistic and intuitive equivalent representation for a BN/BCN; hence although the STP can be used to transform a BN/BCN into a linear/bilinear algebraic form (2.6)/(2.7), it will not change the essence of nonlinearity

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of a BN/BCN. The essence of nonlinearity of BCNs will be revealed in Chap. 4 by studying observability. Next, we introduce the necessary basic properties of the STP, more properties can be found in Cheng et al. (2011). Identifying 1 ∼ δ21 , 0 ∼ δ22 , using the STP of matrices, by Propositions 1.6 and 1.7, a BN (2.1) can be represented equivalently as the following algebraic form: x(t + 1) = L x(t),

(2.6)

where t = 0, 1, . . . ; x(t) ∈ 2n ; L ∈ L2n ×2n . Similarly, a BCN (2.3) can be equivalently represented in the following algebraic form: x(t + 1) = Lu(t)x(t), (2.7) y(t) = H x(t), where t = 0, 1, . . . ; x(t) ∈ 2n , u(t) ∈ 2m , and y(t) ∈ 2q ; L ∈ L2n ×2n+m ; H ∈ L2q ×2n . For the procedure of transforming the algebraic form of a BN/BCN back to its logical form, we refer the reader to Cheng and Qi (2009), Cheng et al. (2011). Controllability is a basic control property, which implies that external control inputs can be chosen to drive each state to each state. Definition 2.1 A BCN (2.7) is called controllable if for all states x0 , xd ∈ 2n , if x(0) = x0 , then x( p) = xd for some p ∈ Z+ and some inputs u(0), u(1), . . . , u( p − 1) ∈ 2m . Proposition 2.1 A BCN (2.7) is controllable if and only if all entries of n 2 −1

(L12m )k

k=0

are positive. Proof Define a directed graph G = (2n , E), where 2n is the vertex set and E ⊂ 2n × 2n is the edge set. Set E is specified as follows: for all x, x  ∈ 2n , (x, x  ) ∈ E if and only if there exists u ∈ 2m such that x  = Lux. Then matrix L12m is the adjacent matrix of the graph. Moreover, for each p ∈ Z+ , we have the (i, j)-th entry j of matrix (L12m ) p is positive if and only if there exists a p-length path from δ2n to i n δ2n , where i, j ∈ [1; 2 ]. Given two states x0 , xd ∈ 2n and assume x(0) = x0 , if x( p) = xd for some p ∈ Z+ and some inputs u(0), u(1), . . . , u( p − 1) ∈ 2m , then x( p  ) = xd for some positive integer p  ≤ 2n − 1 and some inputs u  (0), u  (1), . . . , u  ( p  − 1) ∈ 2m , since there exist exactly 2n states. In other words, if in G there is a p-length path from x0 to xd , then there exists a path from x0 to xd with length no greater than 2n − 1. Based on the above discussion, a BCN (2.7) is controllable if and only if graph G is strongly connected. Hence, we conclude the proposition. 

2.2 Nondeterministic Finite-Transition Systems

39

2.2 Nondeterministic Finite-Transition Systems Nondeterministic finite-transition systems (NFTSs) are nondeterministic and nontotal systems with finitely many states, inputs, and outputs. They play a fundamental role in the verification and control of hybrid systems (Tabuada 2009; Lin 2014; Belta et al. 2017), model checking (Baier and Katoen 2008), mobile robot motion planning and control (Lin and Antsaklis 2014), control of linear systems (Kloetzer and Belta 2008), as well as automated verification or synthesis for hybrid systems (Tabuada 2009; Belta et al. 2017). An NFTS  is a sextuple (X, X 0 , U, →, Y, h) consisting of • • • • • •

a finite set X of states, a set X 0 ⊂ X of initial states, a finite set U of inputs, a transition relation →⊂ X × U × X , a finite set Y of outputs/labels, and an output map h : X × U → Y .

Elements of → are called transitions. For each (x, u, x  ) ∈→, x  is called a successor of x under u, and x is called a predecessor of x  under u. Transition relation →⊂ X × U × X can be represented as a function from X × U to 2 X : for all x, x  ∈ X and u ∈ U , (x, u, x  ) ∈→ if and only if x  ∈→ (x, u). An NFTS is called deterministic if for all x ∈ X and u ∈ U , | → (x, u)| ≤ 1; nondeterministic if it is not necessarily deterministic; total if for all x ∈ X and u ∈ U , | → (x, u)| ≥ 1; and non-total if it is not necessarily total. Several important models are special types of NFTSs. We give a brief explanation as follows. Nondeterministic machines investigated in Alur et al. (1995) are total NFTSs satisfying X 0 = X . Mealy machines (Lee and Yannakakis 1994) are deterministic and total NFTSs satisfying X 0 = X . Moore machines (Moore 1956) are Mealy machines such that h does not depend on U , i.e., for all x ∈ X and u 1 , u 2 ∈ U , h(x, u 1 ) = h(x, u 2 ). BCNs are Moore machines whose numbers of states, inputs, and outputs are powers of 2. Recall that X ∗ and X ω denote the set of words over X and the set of configurations over X , respectively. For each α ∈ X ∗ ∪ X ω , |α| denotes the length of α, and |α| = ∞ if α ∈ X ω . U ∗ , U ω , Y ∗ , and Y ω are described analogously. For each α ∈ X ∗ (α ∈ X ω ), for all integers 0 ≤ i ≤ j ≤ |α| − 1 (0 ≤ i ≤ j), we use α[i, j] to denote α(i)α(i + 1) . . . α( j) for short, where α(i) denotes the i-th element of α. h : X × U → Y is naturally extended to h : ((X ∗ ∪ X ω ) \ {}) × ((U ∗ ∪ U ω ) \ {}) → Y ∗ ∪ Y ω that is defined for all α ∈ (X ∗ ∪ X ω ) \ {} and β ∈ (U ∗ ∪ U ω ) \ {} as h(α, β) = h(α(0), β(0))h(α(1), β(1)) . . . h(α(l), β(l)) if α ∈ X ∗ or β ∈ U ∗ , where l = min{|α| − 1, |β| − 1}; h(α, β) = h(α(0), β(0))h(α(1), β(1)) . . . ∈ Y ω otherwise.

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2 Different Types of Discrete-Time and Discrete-Space Dynamical Systems

Fig. 2.3 State-transition graph of the NFTS in Example 2.2

A state sequence α ∈ X ∗ is called a run of the system over input sequence β ∈ U ∪ U ω if |α| − 1 ≤ |β|, α(0) ∈ X 0 , and for all i ∈ [0; |α| − 2], (α(i), β(i), α(i + 1)) ∈→. A state sequence α ∈ X ω is called a run of the system over input sequence β ∈ U ω if for all i ∈ N, (α(i), β(i), α(i + 1)) ∈→. A run α ∈ X ∗ over input sequence β ∈ U ∗ ∪ U ω is called maximal if |α| − 1 = |β| or (α(|α| − 1), β(|α| − / for any α ∈ X . Every run α ∈ X ω over input sequence β ∈ U ω is also 1), α ) ∈→ called maximal. Transition relation → is naturally extended to →∗ ⊂ X × U ∗ × X as follows: for all x, x  ∈ X , (x, , x  ) ∈→∗ if and only if x = x  ; for all x, x  ∈ X , α ∈ U ∗ , and α ∈ U , (x, αα , x  ) ∈→∗ if and only if there exists x  ∈ X such that (x, α, x  ) ∈→∗ and (x  , α , x  ) ∈→. A state-transition graph of an NFTS is defined by a directed graph whose vertices correspond to the states of the NFTS and whose edges correspond to state transitions. Each edge is labeled with the input and output associated with the transition, a state that is the head of an edge (arrow), where the edge has no tail, means an initial state, and each virtual “transition” (i.e., “transition” with no next-step state) is represented as a dotted arrow. We give an example to depict these concepts. ∗

Example 2.2 Consider NFTS (X, X 0 , U, →, Y, h), where X = {a, b, c}, X 0 = X , U = Y = {0, 1}, →= {(a, 1, a), (a, 0, b), (a, 0, c), (b, 0, b), (b, 1, c)}, h(a, 0) = h(b, 0) = h(c, 0) = 0, h(a, 1) = h(b, 1) = h(c, 1) = 1, see Fig. 2.3. For input sequence 000 ∈ U ∗ , state sequences ac and abbb are both maximal runs over 000, abb is a run over 000, but not a maximal run over 000. h(ac, 000) = h(a, 0)h(c, 0) = 00, h(abbb, 000) = h(a, 0)h(b, 0)h(b, 0) = 000.

2.3 Finite-State Automata Discrete-event systems (DESs) under the framework of finite-state automata were initiated by Ramadge and Wonham (1987). The study on DESs has gradually become an active research branch in control theory during the past thirty years (Cassandras and Lafortune 2010; Wonham and Cai 2019). A DES can be regarded as an abstraction

2.3 Finite-State Automata

41

of the logic layer of a hybrid system, i.e., the transitions of physical models of the hybrid system, and each state of the DES can be regarded as a physical model of the hybrid system. A hybrid system consists of a logic layer and several physical models, where the logic layer decides which physical model is active. When physical models of a hybrid system perform sufficiently well (e.g., controllable and observable), an abstracted DES can take the place of the hybrid system in some sense. On the other hand, due to the complexity of hybrid systems, many problems could be undecidable, while most problems for finite automata are decidable, so using finite automata to give approximate solutions to some complex problems of hybrid systems is effective from the perspective of practical applications, for example, automated synthesis for hybrid systems based on finite-automaton abstractions (Belta et al. 2017; Tabuada 2009). A (nondeterministic) DES can be modeled as a quadruple G = (Q, , δ, Q 0 ),

(2.8)

where Q is a finite set of states, Q 0 ⊂ Q is the set of initial states,  is a finite set of events, and δ ⊂ Q ×  × Q is the transition relation. Note that the above G (we call it a finite-state automaton to distinguish it with a finite automaton) can be obtained from an NFA by removing the final state set F and replacing the initial state q0 by an initial state set Q 0 ⊂ Q, meaning that one knows that the initial state can only be in Q 0 but may not know what exactly it is. A transition (q1 , σ, q2 ) ∈ δ for some states q1 , q2 ∈ Q and event σ ∈  is interpreted as when the system is in state q1 and event σ occurs, the system goes to q2 . The set  of events is partitioned into two disjoint parts o and uo , i.e.,  = o ∪· uo , where o denotes the set of observable events, and uo the set of unobservable events. Under this framework, a deterministic DES is formulated as the above (2.8) satisfying for all q, q  , q  ∈ Q and all σ ∈ , if (q, σ, q  ) ∈ δ and (q, σ, q  ) ∈ δ then q  = q  . A deterministic DES can be obtained from a DFA also by removing its final state set and replacing its initial state by an initial state set. Note that a deterministic DES is not necessarily a DFA. The formal language generated by a DES G = (Q, , δ, Q 0 ) is defined as / L(G) := {w ∈  + |δ(q0 , w) = ∅ for some q0 ∈ Q 0 } ∪ {}. Now add a new state  ∈  → q0 for all q0 ∈ Q 0 into G, then we obtain an -NFA Q and transitions  − A = (Q ∪ {}, , δ  , , Q). One directly sees that L(G) = L(A ). Then the languages generated by DESs modeled by finite-state automata are always regular by Proposition 1.10. Furthermore, by this proposition, there exists an NFA whose states are all final recognizing L(G). Example 2.3 Consider the DES, -NFA, and NFA shown in Fig. 2.4. The languages generated by the DES, recognized by the -NFA and the NFA are the same by Proposition 1.10. Observation to a DES is described as observation to events, which is formulated as a projection P :  ∗ → o∗ defined for each word σ1 . . . σn ∈  ∗ as

42

2 Different Types of Discrete-Time and Discrete-Space Dynamical Systems

Fig. 2.4 A DES (left), an -NFA (middle), and an NFA (right), where the languages generated by the DES, recognized by the -NFA and the NFA are the same

P(σ1 . . . σn ) = P(σ1 ) . . . P(σn ), 

where P(σ) =

σ if σ ∈ o ,  if σ ∈ uo ∪ {}.

In a similar way, the projection P can be naturally extended to the set of infinite sequences of events.

2.4 Labeled Petri Nets A labeled Petri net is one of several mathematical modeling languages for the description of distributed systems (Petri 1962), and also a popular model for implementing the supervisory control framework or different frameworks beyond the supervisory control (Giua and Silva 2017). Petri nets seem to be invented in 1939 by Carl Petri for the purpose of describing chemical processes. Finite-state automata consist of finitely many states and events, while labeled Petri nets contain finitely many events (also called transitions) but at most infinitely countably many states (also called markings). As a result, a Petri net has a stronger expressive power, and meanwhile, the analysis of the net becomes more difficult. For example, if checking a property for finite-state automata is decidable, checking the same property may be undecidable for Petri nets, e.g., the language inclusion problem (Stockmeyer and Meyer 1973; Hack 1976). Despite being more complex, because of plenty of fundamental results developed by computer scientists and mathematicians (cf. Esparza 1998; Reutenauer 1990, etc.), various control properties have been characterized until now by also developing new techniques (Seatzu et al. 2013; Giua and Silva 2017). In addition, Petri nets have been applied to many practical problems, e.g., automated guided vehicle coordination, piston rod robotic assembly cells, hybrid control

2.4 Labeled Petri Nets

43

systems (Moody and Antsaklis 1998), automated manufacturing systems (Li and Zhou 2009), resource allocation systems (Reveliotis 2017), just to name a few. Recall that for a finite set S, S ∗ and S ω denote the set of words over S and configurations over S, respectively. For a word (configuration) s ∈ S ∗ (S ω ), a word s  ∈ S ∗ is called a prefix of s, denoted as s   s, if there exists another word (configuration) s  ∈ S ∗ (S ω ) such that s = s  s  . For a word s ∈ S ∗ , where S = {s1 , . . . , sn }, (s)(si ) denotes the number of occurrences of si in s, i ∈ [1; n]. A net is a quadruple N = (P, T, Pr e, Post), where • P is a finite set of places graphically represented by circles; • T is a finite set of transitions graphically represented by squares; P ∪ T = ∅, P ∩ T = ∅; • Pr e : P × T → N and Post : P × T → N are the pre- and post-incidence functions that specify the arcs directed from places to transitions, and vice versa. Graphically Pr e( p, t) is the weight of the arc p → t and Post ( p, t) is the weight of the arc t → p for all ( p, t) ∈ P × T . The incidence function is defined by C = Post − Pr e. A marking is a map M : P → N that assigns to each place of a net a natural number of tokens, graphically represented by black dots. For a marking M ∈ N P , a transition t ∈ T is called enabled at M if M( p) ≥ Pr e( p, t) for any p ∈ P, and is denoted by M[t, where as usual N P denotes the set of maps from P to N. An enabled transition t at M may fire and yield a new making M  ( p) = M( p) + C( p, t) for all p ∈ P, written as M[tM  . As usual, we assume that at each marking and each time step, at most one transition fires. At a marking M, if a sequence t1 . . . tn of transitions fire one by one and yield a sequence of markings M1 , . . . , Mn , we write M[t1 . . . tn , and furthermore, denote the firing sequence by M[t1 M1 · · · [tn Mn or M[t1 · · · tn Mn for short. In this case, we say t1 . . . tn is enabled at M. The set T (N , M0 ) := {s ∈ T ∗ |M0 [s} is used to denote the set of transition sequences enabled at M0 . Particularly we have M0 [M0 . A pair (N , M0 ) is called a Petri net or a place/transition net (P/T net), where N = (P, T, Pr e, Post) is a net, M0 : P → N is called the initial marking. The Petri net evolves initially at M0 as transition sequences fire. Denote the set of reachable markings of the Petri net by R(N , M0 ) := {M ∈ N P |∃s ∈ T ∗ , M0 [sM  }. For a Petri net (N , M0 ), R(N , M0 ) is at most countably infinite.

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2 Different Types of Discrete-Time and Discrete-Space Dynamical Systems

A labeled Petri net is a quadruple (N , M0 , , ), where N is a net, M0 is an initial marking,  is an alphabet (a finite set of labels), and  : T →  ∪ {} is a labeling function that assigns to each transition t ∈ T a symbol of  or the empty word , which means when a transition t fires, the firing can be observed if (t) ∈ ; and cannot be observed if (t) = . Particularly, a labeling function  : T →  is called -free, and a P/T net with an -free labeling function is called an -free labeled P/T net. A Petri net is actually an -free labeled P/T net with an injective labeling function. A labeling function  : T →  ∪ {} can be recursively extended to  : T ∗ →  ∗ as (st) = (s)(t) for all s ∈ T ∗ and t ∈ T . Particularly we set () = . For a labeled P/T net G = (N , M0 , , ), the language generated by G is denoted by L(G) := {σ ∈  ∗ |∃s ∈ T ∗ , M0 [s, (s) = σ}, i.e., the set of labels of finite-transition sequences enabled at the initial marking M0 . We also say for each σ ∈ L(G), G generates σ. The labeling function  : T ∗ →  ∗ can be extended to  : T ∗ ∪ T ω →  ∗ ∪  ω as (t1 t2 . . . ) = (t1 )(t2 ) . . . for every t1 t2 . . . ∈ T ω . For σ ∈  ω , we say G generates σ if there exists an infinite transition sequence s ∈ T ω such that (s) = σ and every prefix of s is enabled at M0 . The ω-language generated by G, i.e., the set of infinite labeling sequences generated by G, is denoted by Lω (G) := {σ ∈  ω |∃s ∈ T ω , (s) = σ, M0 [s   for every prefix s  of s}. Note that for a labeled P/T net G = (N , M0 , , ), when we observe a label sequence σ ∈  ∗ , there may exist infinitely many firing transition sequences labeled by σ. However, for an -free labeled P/T net, when we observe a label sequence σ, there exist at most finitely many firing transition sequences labeled by σ. Denote by    M(G, σ) := M ∈ N P  ∃s ∈ T ∗ , M0 [sM, (s) = σ , the set of markings in which G can be when σ is observed. Then for each σ ∈  ∗ , M(G, σ) is finite for an -free labeled P/T net G. Compared to finite-state automata, labeled Petri nets are more general models. For an NFA A = (Q, , δ, q0 , Q), we can construct a labeled Petri net G = (N = (P, T, Pr e, Post), M0 , , ) generating L(A). Specifically, we set P = Q, M0 (q0 ) = 1, M0 (q) = 0 for any q ∈ Q \ {q0 }; for any transition (q, σ, q  ) ∈ δ, add a transition tq,σ,q  with label σ and arcs q → tq,σ,q  → q  both with weight 1 into G. Then we have L(A) = L(G). We have shown that for each DES G  modeled by an NFA, there exists an NFA whose states are all final recognizing L(G  ). Hence, there exists a labeled Petri net G  generating L(G  ).

2.4 Labeled Petri Nets

45

Fig. 2.5 A labeled Petri net Fig. 2.6 A Petri net

Example 2.4 Consider a labeled Petri net G = (N , M0, , ) shown in Fig. 2.5,  where N = (P, T, Pr e, Post), P = {s0 , s1 , s2 , s3 }, T = t01 , t02 , t03 , t11 , t12 , t13 , t21 , t31 , each arc is with weight 1, M0 (s0 ) = 1, M0 (s1 ) = M0 (s2 ) = M0 (s3 ) = 0,  = {0, 1}, the label of each transition is the number in “()” following the transition. One firing sequence is       (1, 0, 0, 0) t01 (0, 1, 0, 0) t12 (0, 0, 0, 1) t31 (0, 0, 0, 1), where the components of markings are in the order s0 , s1 , s2 , s3 . It is clear that the language generated by the labeled Petri net equals the language recognized by the NFA in Fig. 2.4. Example 2.5 An evolution of the Petri net in Fig. 2.6 is t1

t2

t4

t4

→ (2, 0, 0) − → (0, 1, 0) − → (0, 1, 0) − → (0, 1, 0). (1, 0, 0) −

46

2 Different Types of Discrete-Time and Discrete-Space Dynamical Systems

2.5 Cellular Automata A cellular automaton (CA) consists of a regular arrangement of countably infinitely many cells whose states update according to the same updating rule. Formally a CA is defined by a triple

d S Z , f, D , where elements of Zd are called cells, elements of S are called states, f : S n → S is a mapping which is called the local updating rule, n ∈ Z+ , and D = {z 1 , . . . , z n } is an ordered finite subset of Zd with cardinality n that is called the neighborhood. d Recall that elements of S Z are called configurations. The map f induces a global d d d updating rule G : S Z → S Z : for each configuration c ∈ S Z and each cell z ∈ Zd , G(c)(z) = f (c(z + z 1 ), . . . , c(z + z n )). Example 2.6 Consider the elementary CA with Wolfram rule 110,1 i.e., the CA specified by  {0, 1}Z , f, {−1, 0, 1} , where f (1, 1, 1) = 0, f (0, 1, 1) = 1,

f (1, 1, 0) = 1, f (0, 1, 0) = 1,

f (1, 0, 1) = 1, f (0, 0, 1) = 1,

f (1, 0, 0) = 0, f (0, 0, 0) = 0,

(01101110) is the binary representation of 110. The space-time diagram of elementary CA 110 with the initial configuration (· · · 0101101110 · · · ) having only finitely many cells in state 1 (such configurations are called finite) is shown in Fig. 2.7, where each line denotes a configuration, white (black) cells denote state 1 (0), and the top line denotes the initial configuration. A CA has another definition, i.e., a discrete-time dynamical system consists of a pair of a symbolic space and a continuous function on the space that commutes with any transition of the space. d d For each vector v ∈ Zd , a translation τv : S Z → S Z is defined by τv (c)(z) = c(z + v) for all c ∈ S Z and z ∈ Zd . Particularly, when d = 1, the translation τ1 is called the left shift, since it shifts each cell of each configuration c ∈ S Z one position to the left. Formally, a CA is a pair d

A = SZ , F , d

1 Elementary

CAs are exactly with rules 0~255.

2.5 Cellular Automata

47

Fig. 2.7 Space-time diagram of elementary CA 110 with a finite initial configuration

where S Z is a symbolic space, and F : S Z → S Z is a continuous function satisfying F ◦ τv = τv ◦ F for any v ∈ Zd , ◦ means composition of functions, S is again a finite set of states, d ∈ Z+ is called the dimension of the CA. Long-term dynamical d behavior of a CA is generated by repetitive actions of F on S Z . The above two definitions of CAs are equivalent. That is, the following proposition holds. d

d

d

Proposition 2.2 Let S Z be a symbolic space and F : S Z → S Z a function. The d pair (S Z , F) is a CA if and only if there exists a finite ordered subset D = {z 1 , . . . , z n } d of Zd with cardinality n ∈ Z+ and a function f : S n → S such that for each c ∈ S Z and each z ∈ Zd , F(c)(z) = f (c(z + z 1 ), . . . , c(z + z n )). d

d

d

Proof (if:) We need to prove that F defined as above is continuous and commutes with any translation. d For each vector v ∈ Zd , each configuration c ∈ S Z , and each cell z ∈ Zd , we have (F ◦ τv )(c)(z) = F(τv (c))(z) = f (τv (c)(z + z 1 ), . . . , τv (c)(z + z n )) = f (c(z + z 1 + v), . . . , c(z + z n + v)) = f (c(z + v + z 1 ), . . . , c(z + v + z n )) = F(c)(z + v) = τv (F(c))(z) = (τv ◦ F)(c)(z). That is, F ◦ τv = τv ◦ F. Let W = {w1 , . . . , wm } be a finite subset of Zd with cardinality m ∈ Z+ , and p : W → S a pattern. Consider cylinder [ p]. To prove the continuity of F, we only d need to prove that F −1 ([ p]) is open in S Z . Denote pi := p|{wi } , i ∈ [1; m]. Then

48

2 Different Types of Discrete-Time and Discrete-Space Dynamical Systems

[ p] =

i∈[1;m] [ pi ].

We have F −1 ([ p]) =

 i∈[1;m]

=



F −1 ([ pi ]) 

i∈[1;m]

  [q] , q

where q runs over all patterns in S {wi +z1 ,...,wi +zn } satisfying f (q(wi + z 1 ), . . . , q(wi + z n )) = pi (wi ). Hence F −1 ([ p]) is open in S Z . d where 0 is the zero (only if:) Let c ∈ S Z and pattern p : {0} → S, 0 → F(c)(0), d vector of Zd . Since F −1 ([ p]) is open in S Z , it can be written as i∈I Ui , where I is −1 an index set and each Ui is a cylinder. Also because  [ p] is closed, F ([ p]) is also −1 closed, and hence compact. Hence F ([ p]) = j∈[1; p] Ui j for some p ∈ Z+ . Since each Ui j is a cylinder, we have Ui j = [ pi j ], where pi j is the corresponding pattern, d  ∈ S Z , if c belongs to [ pi j ] for some j ∈ [1; p], then then for every configuration c  F(c )(0) = F(c)(0). Denote j∈[1; p] dom( pi j ) =: Ds (a finite subset of Zd ), where  s = F(c)(0). Denote s∈S Ds =: {z 1 , . . . , z n } (a finite subset of Z D ). Then there d exists a function g : S n → S such that for each e ∈ S Z , d

F(e)(0) = g(e(z 1 ) . . . , e(z n )). By F ◦ τv = τv ◦ F for each v ∈ Zd , we have for each e ∈ S Z and z ∈ Zd , d

F(e)(z) = τz (F(e))(0) = (τz ◦ F)(e)(0) = (F ◦ τz )(e)(0) = F(τz (e))(0) = g(τz (e)(z 1 ), . . . , τz (e)(z n )) = g(e(z 1 + z), . . . , e(z n + z)).  Corollary 2.1 Let S Z be a symbolic space and F : S Z → S Z a function. Then F is continuous and commutes with the left shift if and only if there exists a finite ordered subset D = {z 1 , . . . , z n } of Z with cardinality n ∈ Z+ and a function f : S n → S such that for each c ∈ S Z and each z ∈ Z, F(c)(z) = f (c(z + z 1 ), . . . , c(z + z n )).

2.5 Cellular Automata

49

Proof By Proposition 2.2, we only need to prove that if F commutes with the left shift τ1 , then it commutes with any shift. By F ◦ τ1 = τ1 ◦ F, we have F ◦ τ2 = F ◦ τ1 ◦ τ1 = τ1 ◦ F ◦ τ1 = τ1 ◦ τ1 ◦ F = τ2 ◦ F. Hence by induction, we have F ◦ τv = τv ◦ F for any v ∈ Z+ . For each v ∈ Z+ , we have τv ◦ τ−v = τ−v ◦ τv = Id, F = F ◦ (τ−v ◦ τv ) = F ◦ τ−v ◦ τv = (τ−v ◦  τv ) ◦ F = τ−v ◦ F ◦ τv . Consequently, we have F ◦ τ−v = τ−v ◦ F. Before going to the next step, we briefly introduce the concept of Turing machines and also show that CAs are more general than deterministic Turing machines. In particular, we will show that deterministic Turing machines are subsystems of onedimensional CAs. It is widely accepted that Turing machines are a formal definition for semi-algorithms, and halting Turing machines are a formal definition for algorithms (Sipser 1996). It is also known that each Turing machine can be simulated by a deterministic Turing machine; hence, CAs are more general than Turing machines in the sense of computation. A (deterministic) Turing machine is specified by (Q, q0 , qa , qr , , , b, δ), where Q is a finite set of states; q0 , qa , qr ∈ Q are the initial, accepting, and rejecting states, respectively, and satisfy qa = qr (both qa and qr are also called halting);  is a finite tape alphabet;  ⊂  is an input alphabet; b ∈  \  is a blank tape symbol; and δ : Q ×  → Q ×  × {−, +} is a transition function satisfying that for all γ ∈  we must have δ(qa , γ) = (qa , γ, +) and δ(qr , γ) = (qr , γ, +). A Turing machine (Fig. 2.8) consists of a tape and a control unit. The tape is a bi-infinite sequence of cells, each capable of storing a letter from the tape alphabet . The tape positions are indexed by Z. The control unit stores a state in Q. After a Turing machine started to run, the control unit read, rewrote the symbol in the tape position to which the control unit was pointing, changed its states, and meanwhile moved one position to the left or the right. Specifically, let a configuration be (q, i, t) ∈ Q × Z ×  Z , where the configuration is interpreted as that the control unit stores state q, pointed to the tape position i, and the tape stores letter t ( j) in position j ∈ Z. Denote δ(q, t (i)) = (q  , r  , ∗), then the control unit changed the symbol t (i) to r  , changed its own state from q to q  , and moved one position to the left if ∗ = − and one position to the right if ∗ = +. If we specify i = and specify t  ∈  Z as



i + 1 if ∗ = +, i − 1 if ∗ = −,

50

2 Different Types of Discrete-Time and Discrete-Space Dynamical Systems

Fig. 2.8 Sketch of a Turing machine

t  ( j) =



t ( j) if j = i, if j = i, r

then we can write the above move by (q, i, t)  (q  , i  , t  ).

(2.9)

Initially in a Turing machine, an input word w ∈  ∗ was stored in the tape, all the other positions of the tape stored the blank tape symbol b, the control unit stored the initial state q0 , pointed to the position of the leftmost letter of w, then the machine started to run step by step according to the above move (2.9) until either of the two halting states qa or qr appears. That is, once started to run, a Turing machine would halt when qa or qr appears, or run forever otherwise. After starting to run from word w ∈  ∗ , if a Turing machine halts with qa , then w is called accepted by the machine, and called rejected otherwise. The formal language recognized by a Turing machine is defined by the set of words accepted by the machine. Such languages are called recursively enumerable. A Turing machine that halts on every word w ∈  ∗ is called halting. The formal languages recognized by halting Turing machines are called recursive. Now we show how a deterministic Turing machine can be simulated by a onedimensional CA. Intuitively, the configuration of the target CA consists of two tracks the first one of which is the tape with letters and the second one of which is obtained by expanding the state to a bi-infinite track by adding a new state not in Q. Specifically, given a deterministic Turing machine (Q, q0 , qa , qr , , , b, δ), a one-dimensional

2.5 Cellular Automata

51

CA simulating the machine can be specified by  ( × (Q ∪ {B}))Z , f, {−1, 0, 1} , where B ∈ / Q, the local updating rule f satisfies that for all (γ1 , q1 ), (γ2 , q2 ), (γ3 , q3 ) ∈  × (Q ∪ {B}) satisfying at least two of q1 , q2 , q3 equal B, we have f ((γ1 , q1 ), (γ2 , q2 ), (γ3 , q3 )) ⎧ (γ2 , q2 ) if q1 = q2 = q3 = B, ⎪ ⎪    ⎪  ⎪ (γ ⎪ 2 , B) if q1 = q3 = B, q2 ∈ Q, δ(q2 , γ2 ) = q2 , γ2 , + or − , ⎪ ⎨ (γ2 , q1 ) if q1 ∈ Q, q2 = q3 = B, δ(q1 , γ1 ) = q1 , γ1 , + , = (γ2 , q2 ) if q1 ∈ Q, q2 = q3 = B, δ(q1 , γ1 ) = q1 , γ1 , − , ⎪ ⎪ ⎪ ⎪ (γ , q ) if q3 ∈ Q, q1 = q2 = B, δ(q3 , γ3 ) = q3 , γ3 , + , ⎪ ⎪ ⎩ 2 2 (γ2 , q3 ) if q3 ∈ Q, q1 = q2 = B, δ(q3 , γ3 ) = q3 , γ3 , − . Example 2.7 Consider the following Turing machine (a two-state busy beaver): • Q = {A, B, qa , qr }, where A = q0 is the initial state, and qa and qr are the accepting and rejecting states. •  = {0, 1}, where 0 is the blank symbol, and  = {1}. • The transition function δ satisfies (A, 0) → (B, 1, +), (A, 1) → (B, 1, −), (B, 0) → (A, 1, −), (B, 1) → (qa , 1, +). With the empty input word (on the initially blank tape), the machine halts after six moves: A ··· 0 0 0 0 0 0 0 0 ··· 

B ··· 0 0 0 0 1 0 0 0 ···



A ··· 0 0 0 0 1 1 0 0 ···



B ··· 0 0 0 0 1 1 0 0 ···



A ··· 0 0 0 1 1 1 0 0 ···



B ··· 0 0 1 1 1 1 0 0 ···

52

2 Different Types of Discrete-Time and Discrete-Space Dynamical Systems



qa . ··· 0 0 1 1 1 1 0 0 ···

Next, we show some fundamental topological properties of CAs. Definition 2.2 A configuration c ∈ S Z is called v-periodic, where v ∈ Zd , if c = τv (c). Particularly a configuration c is called totally periodic if there exist k1 , . . . , kd ∈ Z+ such that c is ki ei -periodic for all i ∈ [1; d], where ei = Rowi (Id ). d

It is easy to see that for a CA (S Z , F), if a configuration c ∈ S Z is ki ei -periodic for all i ∈ [1; d] with k1 , . . . , kd ∈ Z+ , then F(c) is also ki ei -periodic for all i ∈ [1; d], d since F commutes with any translation of S Z . d

d

Definition 2.3 A CA (S Z , F) is called injective (resp. surjective, bijective) if F is d injective (resp. surjective, bijective). A CA (S Z , F) is called reversible if F is a d bijective and (S Z , F −1 ) is also a CA. d

Proposition 2.3 If a CA (S Z , F) is injective, then it is surjective, bijective, and d (S Z , F −1 ) is also a CA. d

Proof Let CA (S Z , F) be injective. Since for each configuration c ∈ S Z that is ki ei -periodic for all i ∈ [1; d] with k1 , . . . , kd ∈ Z+ , then F(c) is also ki ei -periodic for all i ∈ [1; d]. Hence F is injective and then bijective on the set P{k1 ,...,kd } of all ki ei -periodic configurations for all i ∈ [1; d], since P{k1 ,...,kd } is a finite set. We then have F is bijective on the set P of all totally periodic configurations. d We now show that F is surjective. We are given a configuration c ∈ S Z . Let d ci ∈ S Z be (2i − 1)e j -periodic for all j ∈ [1; d] such that ci and c agree on {(z 1 , . . . , z d ) ∈ Nd |(∀k ∈ [1; d])[1 − i ≤ z k ≤ i − 1]}, where i ∈ Z+ . Then limi→∞ ci = c. As shown above, for each ci , there exists a totally periodic cond d figuration ei ∈ S Z such that F(ei ) = ci . By compactness of S Z , there is a convergent subsequence (ei j ) j∈Z+ of (ei )i∈Z+ . Denote lim j→∞ ei j =: e, then we have F(e) = F(lim j→∞ ei j ) = lim j→∞ F(ei j ) = lim j→∞ ci j = c. That is, F is surjective and hence bijective. d d Let X be a closed subset of S Z . Since S Z is compact, X is also compact. Then d d F(X ) is compact by continuity of F, and hence closed in S Z since S Z is Hausdorff. d That is, F −1 is continuous. Let τ be a translation of S Z . Then F ◦ τ = τ ◦ F and d  τ ◦ F −1 = F −1 ◦ τ . That is, (S Z , F −1 ) is a CA. d

d

Long-term behavior of CAs is expressed as properties of limit sets. Definition 2.4 Consider a CA (S Z , F). Its limit set is defined by d

 :=

∞ 

d

F n SZ .

n=0

The following proposition shows a few simple but fundamental properties of d d limit sets. A subshift of S Z is any X ⊂ S Z that is closed, nonempty, and translation

2.5 Cellular Automata

53

invariant (i.e., τv (X ) = X for any v ∈ Zd ). The whole configuration space S Z is also called a fullshift. d

Proposition 2.4 Let (S Z , F) be a CA and  its limit set. Then d

(1)  is a subshift. (2) F() = . (3) For every configuration c, we have c ∈  if and only if there is a sequence . . . , c−2 , c−1 , c0 of configurations such that c0 = c and F(ci ) = ci+1 for all i < 0. (In other words, elements of  are exactly the configurations that belong to two-way infinite orbits.) (4) The limit set  is finite if and only if  contains only one configuration.

n Zd Zd Proof (1): As  = ∞ n=0 F (S ), an intersection of closed subsets of S ,  is closed. d Consider a uniform configuration c ∈ S Z (i.e., c(z 1 ) = c(z 2 ) for all z 1 , z 2 ∈ Zd ), we naturally have there exist positive integers m 1 < m 2 such that F m 1 (c) = F m 2 (c), since F n (c) is uniform for each n ∈ Z+ . Then F m 1 (c) ∈ , i.e.,  is nonempty. d Let τ be a translation of S Z and c ∈ . Then for all n ∈ Z+ , c = F n (e) for some d e ∈ S Z , so τ (c) = τ (F n (e)) = F n (τ (e)), which implies τ (c) ∈ F n (S Z ). This is true for all n, so τ (c) ∈ . We conclude that τ () ⊂ . Then we also have τ −1 () ⊂ , hence  = τ (τ −1 ()) ⊂ τ (). d d (2): Let c ∈ . Then we have c ∈ F n (S Z ) for all n ∈ N; F(c) ∈ F n+1 (S Z ) for

∞ d all n ∈ N. Hence F(c) ∈ n=0 F n+1 (S Z ) = . That is, F() ⊂ . Let c ∈ . Then d

F −1 (c) ∩ F n S Z =: An d

is nonempty and closed in S Z for each n ∈ N. We also have d

A1 ⊃ A2 ⊃ · · · . Then we have F

−1

(c) ∩  =

∞ 

(2.10)

An = ∅.

n=1

Zd On the contrary, we assume that ∞ \ An )n∈Z+ is an open n=1 An = ∅. Then (S d d d Z Z Zd cover of S . By compactness of S and (2.10), we have S \ Am = S Z for some m ∈ Z+ . Hence we obtain Am = ∅, a contradiction. That is, each configuration c ∈  has a preimage in . We conclude  ⊂ F(). (3): If c belongs to some two-way infinite orbit, then F −n (c) is nonempty for any n ∈ N, so c ∈ . Conversely, suppose c0 ∈ . It follows from (2) that there exists c−1 ∈  such that F(c−1 ) = c0 . By iterating the reasoning we obtain a sequence . . . , c−2 , c−1 , c0 of configurations in  satisfying F(c−(n+1) ) = c−n for any n ∈ N.

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2 Different Types of Discrete-Time and Discrete-Space Dynamical Systems

(4): We only need to prove the “only if” part. Suppose that  is a finite set with cardinality greater than 1. Let a uniform configuration cq be in , where cq (z) = q ∈ S for all z ∈ Zd . Such a uniform configuration must exist. Since || > 1, there exists a configuration e ∈  such that e(z) = a ∈ S with a = q for some z ∈ Zd . d By definition, for each n ∈ Z+ , there exist cn , en ∈ S Z such that F n (cn ) = c and n F (en ) = e. Combining cn and en we get a new configuration f n such that F n ( f n ) has all cells (z 1 , . . . , z d ) ∈ Zd in state q when z 1 > 0 and also has one cell in state a. This furthermore implies that for each n ∈ Z+ , there exists a vector vn ∈ Zd such that F n (τvn ( f n )) has all cells (z 1 , . . . , z d ) ∈ Zd in state q when z 1 > 0 and also has cell 0 in state other than q. Denote    d B = c ∈ S Z  c(z 1 , . . . , z d ) = q with z 1 > 0 and c(0) = q , d

B ∩ F n S Z = ∅

we have

for all n ∈ Z+ . Define patterns as follows: pvs : {v} → {s}, v → s, where v ∈ Zd , s ∈ S. Then we have ⎞ ⎛    s   q ⎝ p0 B= pv ⎠ s∈S\{q} v∈{ (z 1 ,...,z d )∈Zd |z 1 >0} is closed in S Z . By similar argument as that for (2) we have B ∩  = ∅. Arbitrarily choosing a configuration c in B ∩ , we have d

c, τe1 (c), τ2e1 (c), . . . , τne1 (c), . . . are pairwise different and belong to , where e1 = Row1 (Id ). That is,  is infinite. 

Notes For controllability of Boolean control networks, Proposition 2.1 is a simpler equivalent version of Zhao et al. (2010, Theorem 3.3) with a simpler proof. The definition of cellular automata using local functions was first constructed in the 1940s to describe the growth of crystals by Ulam (2012) and to build selfreplicating systems by von Neumann and Burks (1966), while the definition based on topology was given by Hedlund in 1969 (Hedlund 1969). The one-dimensional version of Proposition 2.2 (called the Curtis–Hedlund–Lyndon theorem) was proved in Hedlund (1969), and also discovered by Curtis and Lyndon, which was pointed out by Hedlund. The Curtis–Hedlund–Lyndon theorem has been called “one of the

2.5 Cellular Automata

55

fundamental results in symbolic dynamics”. Proposition 2.2 as a generalization of the Curtis–Hedlund–Lyndon theorem to lattices was proved by Richardson soon later in 1972 (Richardson 1972). It can be even further generalized from lattices to discrete groups (Ceccherini-Silberstein and Coornaert 2010). We also refer the reader to Hadeler and Müller (2017), Kari (2016) for extensive reading.

References Akutsu T (2018) Algorithms for analysis, inference, and control of boolean networks. World Scientific, Singapore Akutsu T et al (2007) Control of Boolean networks: hardness results and algorithms for tree structured networks. J Theor Biol 244(4):670–679 Alur R, Courcoubetis C, Yannakakis M (1995) Distinguishing tests for nondeterministic and probabilistic machines. In: Proceedings of the twenty-seventh annual ACM symposium on theory of computing. STOC ’95. ACM, Las Vegas, Nevada, USA, pp 363–372 Baier C, Katoen JP (2008) Principles of model checking. The MIT Press, Cambridge Belta C, Yordanov B, Gol EA (2017) Formal methods for discrete-time dynamical systems. Springer International Publishing AG, Berlin Cassandras CG, Lafortune S (2010) Introduction to discrete event systems, 2nd edn. Springer Publishing Company, Incorporated, Berlin Ceccherini-Silberstein T, Coornaert M (2010) Cellular automata and groups. Springer monographs in mathematics. Springer, Berlin Cheng D (2001) Semi-tensor product of matrices and its application to Morgen’s problem. Sci China Ser: Inf Sci 44(3):195–212 Cheng D, Qi H (2009) Controllability and observability of Boolean control networks. Automatica 45(7):1659–1667 Cheng D, Qi H, Li Z (2011) Analysis and control of Boolean networks: a semi-tensor product approach. Springer, London Esparza J (1998) Decidability and complexity of Petri net problems - an introduction. In: Reisig W, Rozenberg G (eds) Lectures on petri nets i: basic models: advances in petri nets. Springer, Berlin, pp 374–428 Fauré A et al (2006) Dynamical analysis of a generic Boolean model for the control of the mammalian cell cycle. In: Bioinformatics 22(14):e124 Giua A, Silva M (2017) Modeling, analysis and control of discrete event systems: a petri net perspective. IFAC-PapersOnLine 50(1):1772–1783 Hack M (1976) Petri net languages. Technical report. Cambridge, MA, USA Hadeler KP, Müller J (2017) Cellular automata: analysis and applications. Springer monographs in mathematics. Springer, Cham Hedlund GA (1969) Endomorphisms and automorphisms of the shift dynamical system. Math Syst Theory 3(4):320–375 Ideker T, Galitski T, Hood L (2001) A new approach to decoding life: systems biology. Annu Rev Genomics Hum Genet 2(1):343–372 Kari J (2016) Cellular automata. http://users.utu.fi/jkari/ca2016/ Kauffman SA (1969) Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol 22(3):437–467 Kloetzer M, Belta C (2008) A fully automated framework for control of linear systems from temporal logic specifications. IEEE Trans Autom Control 53(1):287–297 Kunz W, Stoffel D (1997) Reasoning in Boolean networks. Springer, Boston Lee D, Yannakakis M (1994) Testing finite-state machines: state identification and verification. IEEE Trans Comput 43(3):306–320

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Li Z, Zhou M (2009) Deadlock resolution in automated manufacturing systems: a novel petri net approach. 1st edn. Springer Publishing Company, Incorporated, Berlin Lin H (2014) Mission accomplished: an introduction to formal methods in mobile robot motion planning and control. Unmanned Syst 02(02):201–216 Lin H, Antsaklis PJ (2014) Hybrid dynamical systems: an introduction to control and verification. Found Trends Syst Control 1(1):1–172 Moody JO, Antsaklis PJ (1998) Supervisory control of discrete event systems using petri nets. Kluwer Academic Publishers, Norwell Moore EF (1956) Gedanken-experiments on sequential machines. Autom Stud, Ann Math Stud 34:129–153 von Neumann J, Burks AW (1966) Theory of self-reproducing automata. University of Illinois Press, Champaign Petri CA (1962) Kommunikation mit Automaten. PhD thesis. University of Bonn Pickover CA (2012) The math book: from pythagoras to the 57th dimension, 250 milestones in the history of mathematics. Sterling Milestones, Sterling Ramadge PJ, Wonham WM (1987) Supervisory control of a class of discrete event processes. SIAM J Control Optim 25(1):206–230 Reutenauer C (1990) The mathematics of petri nets. Prentice-Hall Inc, Upper Saddle River Reveliotis SA (2017) Logical control of complex resource allocation systems. Now Publishers Inc, Boston Richardson D (1972) Tessellations with local transformations. J Comput Syst Sci 6(5):373–388 Rosin DP (2015) Dynamics of complex autonomous Boolean networks. Springer International Publishing, Berlin Seatzu C, Silva M, van Schuppen J (ed) (2013) Control of discrete-event systems: automata and petrinet perspectives. Lecture notes in control and information sciences, vol 433. Springer, London, p 478 Shmulevich I, Dougherty ER (2010) Probabilistic Boolean networks: the modeling and control of gene regulatory networks. SIAM Sipser M (1996) Introduction to the Theory of Computation, 1st edn. International Thomson Publishing, Stamford Sridharan S et al (2012) Boolean modeling and fault diagnosis in oxidative stress response. In: BMC genomics 13(Suppl 6), S4:1–16 Stockmeyer LJ, Meyer AR (1973) Word problems requiring exponential time (preliminary report). In: Proceedings of the fifth annual ACM symposium on theory of computing. STOC’73. ACM, New York, NY, USA, pp 1–9 Tabuada P (2009) Verification and control of hybrid systems: a symbolic approach. 1st edn. Springer Publishing Company, Incorporated, Berlin Waldrop MM (1993) Complexity: the emerging science at the edge of order and chaos. Simon & Schuster Wonham WM, Cai K (2019) Supervisory control of discrete-event systems. Springer International Publishing, Berlin Zhao Y, Qi H, Cheng D (2010) Input-state incidence matrix of Boolean control networks and its applications. Syst Control Lett 59(12):767–774

Part II

Boolean Control Networks

Chapter 3

Invertibility and Nonsingularity of Boolean Control Networks

As stated before, initially a Boolean control network (BCN) (see 1 in Fig. 3.1) was in a state, then as inputs were fed into the BCN one by one, state transitions occurred successively, yielding a sequence of outputs. What may interest us is: Could the above process be reversed? That is, whether there exists another BCN (see 2 in Fig. 3.1) that reverses the inputs and outputs of 1 . In this chapter,1 we prove a series of fundamental results on this problem, and apply these results to the mammalian cell cycle.

3.1 Notions of Invertibility and Nonsingularity Invertibility is a classical control property. If a control system satisfies the property, then for each initial state, one can recover the input by using an output; then furthermore, one can recover the input sequence by using the output sequence. This property reflects one intrinsic property of a control system. For an invertible control system S1 , one can find another control system S2 that receives the output sequence of S1 , and then returns the corresponding input sequence of S1 . In order to describe the invertibility phenomenon of BCNs, we define invertibility of a BCN by the bijectivity of a map from the set of infinite input sequences to the set of infinite output sequences generated by the BCN. In this case, we can always make sure that if a BCN is invertible, then the input sequence can be recovered by an output sequence. Moreover, we can also construct a new inverse system that receives an output sequence, and returns the corresponding input sequence. More 1 Parts of Sects. 3.2, 3.3, and 3.5.1 were reproduced from Zhang et al. (2015) with permission @ 2015

Elsevier Ltd. Some of the material in Sect. 3.5.2 were reproduced from Zhang et al. (2017) with permission @ 2017 IEEE. © Springer Nature Switzerland AG 2020 K. Zhang et al., Discrete-Time and Discrete-Space Dynamical Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-25972-3_3

59

60

3 Invertibility and Nonsingularity of Boolean Control Networks

Fig. 3.1 Two Boolean control networks

generally, we define nonsingularity of a BCN by injectivity of the map. If a BCN is nonsingular, then we can always determine the input sequence by using an output sequence. However, an inverse system may not always exist. Since we will consider the set of infinite input sequences, and such a set can form a symbolic space (Proposition 1.26). We may try to use the theory of symbolic spaces to deal with invertibility. After trying the above idea, we find that the theory of symbolic spaces is a suitable tool. In the sequel, we use the theory to prove fundamental propositions that will be used to establish a verification method for invertibility.

3.2 Invertibility Characterization 3.2.1 The Map from the Infinite Input Sequences Space to the Infinite Output Sequences Space The concept of invertibility is closely related to the map from the space of input sequences to the space of output sequences. In this part, we prove a series of propositions on the continuity, injectivity, and surjectivity of the map. These propositions are necessary preliminaries for defining and characterizing invertibility. Recall the logical form of BCN (2.4) as follows: x(t + 1) = f (x(t), u(t)), y(t) = h(x(t)),

(3.1)

where t = 0, 1, 2, . . . ; x(t) ∈ Dn , u(t) ∈ Dm , and y(t) ∈ Dq ; f : Dn+m → Dn and h : Dn → Dq . Recall 2m = M, 2n = N , 2q = Q, and the algebraic form of a BCN as follows: x(t + 1) = Lu(t)x(t), y(t) = H x(t),

(3.2)

where t = 0, 1, . . . ; x(t) ∈  N , u(t) ∈  M , and y(t) ∈  Q ; L ∈ L N ×(N M) ; H ∈ L Q×N .

3.2 Invertibility Characterization

61

Fig. 3.2 Input-state-output-time transfer graph of BCN (3.2), where subscripts denote time steps, x0 , x1 , . . . denote states, u 0 , u 1 , . . . denote inputs, y0 , y1 , . . . denote outputs, and arrows infer dependence, reproduced from Zhang et al. (2015) with permission @ 2015 Elsevier Ltd.

The input-state-output-time transfer graph of BCN (3.2) is shown in Fig. 3.2. For (3.2), we define the following map from the set of input sequences to the sets of state sequences and output sequences: 1. For each x0 ∈  N and each p ∈ Z+ , L xp0 : ( M ) p → ( N ) p , u 0 . . . u p−1 → x1 . . . x p ; Hxp0 : ( N ) p → ( Q ) p , x1 . . . x p → y1 . . . y p ; (H L)xp0

=

Hxp0

◦

L xp0 , u 0

(3.3)

. . . u p−1 → y1 . . . y p .

2. For each x0 ∈  N , L Nx0 : ( M )N → ( N )N , u 0 u 1 . . . → x1 x2 . . . ; HxN0 : ( N )N → ( Q )N , x1 x2 . . . → y1 y2 . . . ; (H L)Nx0 3.

=

HxN0

◦

L Nx0 , u 0 u 1

(3.4)

. . . → y1 y2 . . . .

L N :  N ( M )N → ( N )N , x0 u 0 u 1 . . . → x0 x1 x2 . . . ; H N :  N ( N )N →  Q ( Q )N , x0 x1 x2 . . . → y0 y1 y2 . . . ; N

N

(3.5)

N

(H L) = H ◦ L , x0 u 0 u 1 . . . → y0 y1 y2 . . . . The spaces of infinite input sequences, infinite state trajectories and infinite output trajectories are ( M )N , L N ( N ( M )N ) and (H L)N ( N ( M )N ), respectively. The following Proposition 3.1 is a key result in establishing an equivalent test criterion for invertibility. Proposition 3.1 For each state x ∈  N , the maps L Nx , HxN , and (H L)Nx are continuous. Proof We first prove that for each state x ∈  N , the map L Nx is continuous.

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3 Invertibility and Nonsingularity of Boolean Control Networks

Recall that the sets ( N )N , ( M )N , and ( Q )N form symbolic spaces (cf. Sect. 1.4.7 for introduction). For ( M )N , the set of cylinders ([U [0, p]])U ∈( M )N , p∈N forms a countable topological base for the symbolic space ( M )N . Similarly, the set of cylinders ([X [0, p]]) X ∈( N )N , p∈N (resp. ([Y [0, p]])Y ∈( Q )N , p∈N ) form a countable topological base for the symbolic space ( N )N (resp. ( Q )N ). Let x be in  N . We now show that L Nx is continuous. For each p ∈ Z+ and each X ∈ ( N ) p , we obtain a cylinder [X ] as a subset of ( N )N . We have 

L Nx

−1

([X ]) =



[U ],

U

where U runs over all words of ( M ) p satisfying L x (U ) = X . That is, the map L Nx is continuous. Similarly the map HxN is continuous, and hence (H L)Nx = HxN ◦ L Nx is also continuous.  p

On the injectivity properties of the map (H L)Nx , the following proposition holds. Proposition 3.2 The following four items satisfy the implications (i)⇒(ii) ⇒(iii) ⇒(iv). (i) For each x ∈  N , the map (H L)1x is injective. p (ii) For each x ∈  N and each p ∈ Z+ , the map (H L)x is injective. p (iii) For each x ∈  N , there is an integer p ∈ Z+ such that the map (H L)x is injective. (iv) For each x ∈  N , the map (H L)Nx is injective. Proof (ii) ⇒(iii): Obvious. (i)⇒(ii): We prove this implication by mathematical induction. Assume that (i) holds, and assume for some p ∈ Z+ , for each x ∈  N and each 1 ≤ l ≤ p, p+1 (H L)lx is injective. Then for each x ∈  N , (H L)x is injective. The reason is stated as follows. Arbitrarily given x  ∈  N and distinct U1 , U2 ∈ ( M )N satisfying U1 [0, p] = U2 [0, p], if U1 [0, p − 1] = U2 [0, p − 1], then by the assumption, it holds that p p (H L)x  (U1 [0, p − 1]) = (H L)x  (U2 [0, p − 1]); else if U1 [ p] = U2 [ p], then by the assumption, we have p

p

(H L)x  (U1 [0, p − 1]) = (H L)x  (U2 [0, p − 1]), p+1

p+1

(H L)x  (U1 [0, p]) = (H L)x  (U2 [0, p]). Thus (ii) holds. (iii) ⇒(iv): Arbitrarily given x ∈  N and distinct a, b ∈ ( M )N . Denote min{ j ∈ N|a( j) = b( j)} := k.

3.2 Invertibility Characterization

Then

63

L Nx (a)[0, k − 1] = L Nx (b)[0, k − 1].

Denote L Nx (a)(k − 1) =: x  . If k = 0, set L Nx (a)(−1) = x. There is p  ∈ Z+ such p that (H L)x  is injective. Then we have p

p

(H L)x  (a[k, k + p  − 1]) = (H L)x  (b[k, k + p  − 1]). Hence (H L)Nx (a) = (H L)Nx (b). That is, (H L)Nx is injective.



On the surjectivity properties of the map (H L)Nx , the following proposition holds. Proposition 3.3 The following three items are equivalent: (i) For each x ∈  N , the map (H L)Nx is surjective. p (ii) For each x ∈  N and each p ∈ Z+ , the map (H L)x is surjective. 1 (iii) For each x ∈  N , the map (H L)x is surjective. p

Proof (i)⇒(ii): Assume that for some x ∈  N and p ∈ Z+ , (H L)x is not surjective. p Then there exists e ∈ ( Q ) p that has no preimage under (H L)x . And then for any c ∈ ( Q )N , ec ∈ ( Q )N has no preimage under (H L)Nx . (ii)⇒(iii): Obvious. (iii)⇒(ii): Similar to the proof of (i)⇒(ii) in Proposition 3.2, it can be proved by mathematical induction. We omit the similar proof. (ii)⇒(i): This proof is obtained by compactness (Proposition 1.2.1) of the space ( M )N of infinite input sequences. Arbitrarily choose c ∈ ( Q )N . Assume that (ii) holds. Then for each i ∈ N, c[0, i] i+1 by ei , has a preimage under (H L)i+1 x . Denote a preimage of c[0, i] under (H L)x i+1  that is, (H L)x (ei ) = c[0, i]. Construct configuration sequence e0 , e1 , . . . such that for each i ∈ N, ei = ei (δ 1M )ω . Then (H L)Nx (ei )[0, i] = c[0, i]. The compactness of the configuration space ( M )N shows that there exists a converging subsequence {el(i) } of {ei }, where l : N → N is a strictly monotonically increasing function. Denote limi→∞ el(i) by e. We then have c = lim (H L)Nx (ei ) = lim (H L)Nx (el(i) ) = (H L)Nx ( lim el(i) ) = (H L)Nx (e) i→∞

i→∞

i→∞

by continuity of (H L)Nx (see Proposition 3.1). That is, c has a preimage e. Hence  (H L)Nx is surjective. Proposition 3.4 For each x ∈  N , if (H L)Nx is injective, then L Nx is injective. Proposition 3.5 For each x ∈  N , if (H L)Nx is surjective, then HxN is surjective. If for each x ∈  N , (H L)Nx is surjective, then min{m, n} ≥ q. Next, we give the key proposition that will be used to give an equivalent test criterion for invertibility. This proposition is obtained by using compactness of the space of infinite input sequences.

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3 Invertibility and Nonsingularity of Boolean Control Networks

Proposition 3.6 Consider a BCN (3.2). The following three items are equivalent: (i) For each x ∈  N , (H L)Nx is bijective. p (ii) m = q, n ≥ q, and for each x ∈  N and each p ∈ Z+ , (H L)x is bijective. 1 (iii) m = q, n ≥ q, and for each x ∈  N , (H L)x is bijective. Proof (i)⇒(ii): Assume that (i) holds. Then by Propositions 3.3, 3.4 and 3.5, we have for each x ∈  N , L Nx is injective, HxN is surjective; for each x ∈  N and each p p ∈ Z+ , the map (H L)x is surjective; and min{m, n} ≥ q. We claim that m = q. Suppose the contrary: m > q. Then for each x and each p p ∈ Z+ , (H L)x is not injective. Due to the fact that for each x ∈  N and each p p ∈ Z+ , the map (H L)x is surjective, hereinafter fix x, we construct two input sequences (ci )i∈N ⊂ ( M )N and (ei )i∈N ⊂ ( M )N as follows: • c0 = u 0 (δ 1M )ω , e0 = u 0 (δ 1M )ω such that u 0 = u 0 and (H L)1x (u 0 ) = (H L)1x (u 0 ) =: y1 . Denote L 1x (u 0 ) =: x1 and L 1x (u 0 ) =: x1 . Then x1 = x1 by injectivity of L Nx . • Arbitrarily choose y2 ∈  Q . Then there exist u 1 , u 1 ∈  M such that (H L)1x1 (u 1 ) = (H L)1x  (u 1 ) = y2 . Denote L 1x1 (u 1 ) =: x2 and L 1x  (u 1 ) =: x2 . Construct 1 1 c1 = u 0 u 1 (δ 1M )ω , e1 = u 0 u 1 (δ 1M )ω . • ... • Arbitrarily choose yi+1 ∈  Q . Then there exist u i , u i ∈  M such that  . (H L)1xi (u i ) = (H L)1x  (u i ) = yi+1 . Denote L 1xi (u i ) =: xi+1 and L 1x  (u i ) =: xi+1 i i 1 ω   1 ω Construct ci = u 0 . . . u i (δ M ) , ei = u 0 . . . u i (δ M ) . • ... Now consider the input sequences (ci )i∈N and (ei )i∈N . By compactness of ( M )N , there exist converging subsequences (cl(i) )i∈N and (el(i) )i∈N , where l : N → N is a strictly monotonically increasing function. Denote limi→∞ cl(i) and limi→∞ el(i) by c and e, respectively. By construction of (ci )i∈N and (ei )i∈N , we have c(0) = u 0 = e(0) = u 0 , i.e. c = e; and we also have for all i ∈ N, (H L)Nx (cl(i) )[0, l(i)] = (H L)Nx (el(i) )[0, l(i)], then by continuity of (H L)Nx (see Proposition 3.1), lim

(H L)Nx

  cl(i) = (H L)Nx

= lim

(H L)Nx

  el(i) = (H L)Nx

i→∞

i→∞



 lim cl(i)

i→∞



lim el(i)

i→∞

= (H L)Nx (c)



(3.6) =

(H L)Nx (e).

Hence (H L)Nx is not injective, which is a contradiction. That is, m = q. p Furthermore, by surjectivity of the map (H L)x for each x ∈  N and each p ∈ Z+ , p we have for each x ∈  N and each p ∈ Z+ , the map (H L)x is also injective, hence bijective. (ii)⇒(i): Directly follows from Propositions 3.2 and 3.3. (ii)⇒(iii): Obvious. (iii)⇒(ii): Also directly follows from Propositions 3.2 and 3.3.  Now we are ready to present the main results on invertibility of BCN (3.2).

3.2 Invertibility Characterization

65

3.2.2 Invertibility Verification We call a BCN (3.2) nonsingular if its inputs can be uniquely determined by its initial state and outputs; and invertible if its inputs can be uniquely recovered by its initial state and outputs. Definition 3.1 A BCN (3.2) is said to be nonsingular if for each initial state x0 ∈  N , the map (H L)Nx0 is injective. Definition 3.2 A BCN (3.2) is said to be invertible if for each initial state x0 ∈  N , the map(H L)Nx0 is bijective. We give a description for the physical meaning of nonsingularity and invertibility of BCNs. Recall in frequency domain, a classical linear control system with its transfer function A(s) from its space U of inputs to its space Y of outputs, is called left/right invertible, if there is a system whose transfer function is B(s) from Y to U such that B(s)A(s) = IdU /A(s)B(s) = IdY , where IdU and IdY denote the identity operators on U and Y, respectively. Particularly if U = Y, then A(s) is called invertible if A(s)B(s) = B(s)A(s) = IdU . From this perspective, invertibility and nonsingularity of BCNs can be seen as the above invertibility and left invertibility, respectively. Besides, surjectivity of (H L)Nx for each x ∈  N (characterized in Proposition 3.3) can be seen as the above right invertibility. Here we only investigate invertibility. Nonsingularity is left to investigate later on. Now we give an equivalent test criterion for invertibility of BCN (3.2). Theorem 3.1 A BCN (3.2) is invertible if and only if m = q, n ≥ q, and for each i ∈ [1; N ], rank(H L W[N ,M] δ iN ) = M. That is, a BCN (3.1) is invertible if and only if m = q, n ≥ q, and for each x ∈ Dn , the map h( f (x, u)) : Dm → Dq is bijective. Proof By Proposition 3.6, BCN (3.2) is invertible if and only if m = q, n ≥ q, and for all x0 ∈  N , (H L)1x0 is bijective. By BCN (3.2) we have y(t + 1) = H L W[N ,M] x(t)u(t). Then (H L)1x0 is bijective if and only if rank(H L W[N ,M] x0 ) = M. That is, Theorem 3.1 holds.  In Theorem 3.1, the complexity of computing H L W[N ,M] δ iN = H (L(W[N ,M] δ iN )) is M(M · 1) = M 2 = 22m , since H, L , W[N ,M] and δ iN are all logical matrices. It is easy to obtain that judging whether rank(H L W[N ,M] δ iN ) = M holds is equivalent to judging whether H L W[N ,M] δ iN has M distinct columns. Since there are totally N states, the overall computational cost of Theorem 3.1 is N M 2 = 2n+2m . Hence the computational complexity of using Theorem 3.1 to verify invertibility of BCN (3.2) is O(2n+2m ).

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3.2.3 Invariance of Invertibility Invariance is an important property in characterizing topological properties for dynamical systems, and can be used to classify dynamical systems. For example, if a linear time-invariant control system is controllable, then it is still controllable under coordinate transformations. Here we prove that invertibility of BCN (3.2) remains invariant under coordinate transformations. The invariance of controllability of BCN (3.2) under coordinate transformations can be proved similarly. We adopt the concept of logical coordinate transformation given in Cheng et al. (2010): Given logical variables x1 , . . . , x p ∈ , and Boolean functions z i : p p (x1 , . . . , x p ) → z i (x1 , . . . , x p ) ∈ , i = 1, . . . , p. Denote z = i=1 z i , x = i=1 xi and z = T x, where T ∈ L2 p ×2 p . z = T x is called a logical coordinate transformation, if T is nonsingular. If z = T x is a logical coordinate transformation, then its inverse transformation is x = T −1 z = T T z. Theorem 3.2 Invertibility of BCN (3.2) remains invariant under logical coordinate transformations. ¯ Then (3.2) is equivalent to Proof Consider BCN (3.2). Denote L W[N ,M] =: L. x(t + 1) = L¯ x(t)u(t), y(t) = H x(t).

(3.7)

Let z = T x be a logical coordinate transformation, where z ∈  N , T ∈ L N ×N is nonsingular. Substituting x = T −1 z into BCN (3.7), we have z(t + 1) = T L¯ T −1 z(t)u(t), y(t) = H T −1 z(t).

(3.8)

By Theorem 3.1, a BCN (3.8) is invertible if and only if m = q, n ≥ q, and for ¯ −1 δ iN ) = M. all i ∈ [1; N ], rank(H LT ¯ ¯ Denote L = [ L 1 , . . . , L¯ N ], where L¯ i ∈ L N ×M , i = 1, . . . , N . Because permutation matrix T −1 has N distinct columns, we have   ¯ −1 = L¯ l(1) , . . . , L¯ l(N ) , LT where l : [1; N ] → [1; N ] is a permutation (bijection). ¯ iN ) = M if and only if for all i ∈ [1; N ], Then for all i ∈ [1; N ], rank(H Lδ −1 i ¯ rank(H LT δ N ) = M. Based on the above analysis, invertibility of BCN (3.2) remains invariant under logical coordinate transformations. 

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67

3.2.4 Inverse Boolean Control Networks of Invertible Boolean Control Networks Until now, we have obtained the test criterion for invertibility of BCN (3.2). Then what is the dynamics of its inverse network like? In this subsection, we design an algorithm to construct the inverse BCN for an invertible BCN (3.2). In this subsection, we assume that BCN (3.2) is invertible, and denote L W[N ,M] = ¯ Then by Theorem 3.1, we have m = q, n ≥ q, and for all i ∈ [1; N ], H Lδ ¯ iN ∈ L. L Q×M is nonsingular. Define the following operator (·)−1 for H L ∈ L Q×(N M) as follows:   (H L)−1 := (δ 1N )T W[M,N ] (H L)T , . . . , (δ NN )T W[M,N ] (H L)T W[Q,N ] .

(3.9)

Then (H L)−1 ∈ L M×(Q N ) . The following theorem gives a simple algorithm to calculate (H L)−1 . Theorem 3.3 Consider BCN (3.2). Assume that m = q, n ≥ q, and for all i ∈ [1; N ], H L W[N ,M] δ iN ∈ L Q×M is nonsingular. Denote   H H L = ⎡L 1H , . . . , L M ⎤ H ) Row1 (L 1H ) Row1 (L 2H ) · · · Row1 (L M ⎢ ⎥ .. .. .. .. =⎣ ⎦ . . . . H H H (L ) Row (L ) · · · Row (L ) Row Q 1  Q Q 2 M  = Rowi L Hj

, 1 ≤ i ≤ Q, 1 ≤ j ≤ M,

where L iH ∈ L Q×N , i = 1, . . . , M. Then (H L)−1 = (ai j ), 1 ≤ i ≤ M, 1 ≤ j ≤ Q, where ai j = Row j (L iH ). Proof For all i ∈ [1; N ], we have ⎡

⎤ (L 1H )T ⎢ ⎥ (δ iN )T W[M,N ] (H L)T = (δ iN )T W[M,N ] ⎣ ... ⎦ ⎡

⎤

H T ) (L M

Row1 ((L 1H )T ) ⎢ ⎥ .. ⎢ ⎥ . ⎢ ⎥ ⎡ ⎤ H T ⎥ ⎢ Row1 ((L M Rowi ((L 1H )T ) ) ) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ .. .. =(δ iN )T ⎢ ⎥=⎣ ⎦. . . ⎢ ⎥ H T ⎢Row N ((L H )T )⎥ Rowi ((L M ) ) 1 ⎢ ⎥ ⎢ ⎥ .. ⎣ ⎦ . H T ) ) Row N ((L M

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Fig. 3.3 Input-state-output-time transfer graph of the inverse BCN of an invertible BCN (3.2), where subscripts denote time steps, x0 , x1 , . . . denote states, y1 , y2 , . . . denote inputs, u 0 , u 1 , . . . denote outputs, and arrows infer dependence, reproduced from Zhang et al. (2015) with permission @ 2015 Elsevier Ltd.

Fig. 3.4 Input-state-output-time transfer graph of the inverse BCN of an invertible BCN (3.2) when n = q, where subscripts denote time steps, x0 , x1 , . . . denote states, y1 , y2 , . . . denote inputs, u 0 , u 1 , . . . denote outputs, and arrows infer dependence, reproduced from Zhang et al. (2015) with permission @ 2015 Elsevier Ltd.

Then Eq. (3.9) shows that ⎤ Row1 ((L 1H )T ) · · · Row N ((L 1H )T ) ⎥ ⎢ .. .. .. =⎣ ⎦ W[Q,N ] . . . H T H T Row1 ((L M ) ) · · · Row N ((L M ) ) ⎤ ⎡ Col1 (L 1H )T · · · Col N (L 1H )T ⎥ ⎢ .. .. .. =⎣ ⎦ W[Q,N ] . . . ⎡

(H L)−1

H T H T ) · · · Col N (L M ) Col1 (L M ⎡ ⎤⎞ T ⎛ H ) Col1 (L 1H ) · · · Col1 (L M ⎢ ⎥⎟ ⎜ .. .. .. = ⎝W[N ,Q] ⎣ ⎦⎠ . . . H ) Col N (L 1H ) · · · Col N (L M

3.2 Invertibility Characterization

69

⎤⎞T H T ) Row1 (L 1H )T · · · Row1 (L M ⎥⎟ ⎜⎢ .. .. .. = ⎝⎣ ⎦⎠ . . . H T H T Row Q (L 1 ) · · · Row Q (L M ) ⎤ ⎡ Row1 (L 1H ) · · · Row Q (L 1H ) ⎥ ⎢ .. .. .. =⎣ ⎦. . . . ⎛⎡

H H ) · · · Row Q (L M ) Row1 (L M

 Now we are ready to design an algorithm to construct the inverse BCN for an invertible BCN (3.2). By Theorem 3.1, Eq. (3.2) and Fig. 3.2, we have u i = (H L)−1 yi+1 xi , i ∈ N,

(3.10)

which is shown in Fig. 3.3. ¯ the new inputs, and u i := y¯ (i + 1), the new Hereinafter, we denote yi+1 := u(i), outputs, respectively, i ∈ N. The states remain the same. Then Eq. (3.10) is equivalent to Eq. (3.11). ¯ t ∈ N. y¯ (t + 1) = (H L)−1 u(t)x(t), (3.11) To finish the construction of the inverse BCN, what is left is to recover x(t). According to Eqs. (3.2) and (3.11), we have x(t + 1) = L y¯ (t + 1)x(t) = L(H L)−1 u(t)x(t)x(t) ¯ = L(H L)−1 u(t)M ¯ Nr x(t) = L(H L)−1 (I Q ⊗ M Nr )u(t)x(t), ¯

(3.12)

where L(H L)−1 (I Q ⊗ M Nr ) ∈ L N ×(Q N ) . The following theorem is itself an algorithm to construct the inverse BCN for an invertible BCN (3.2). Theorem 3.4 If a BCN (3.2) is invertible, then its inverse BCN is as follows (shown in Fig. 3.3): x(t + 1) = L(H L)−1 (I Q ⊗ M Nr )u(t)x(t), ¯ (3.13) y¯ (t + 1) = (H L)−1 u(t)x(t), ¯ where x ∈  N , u¯ ∈  Q , y¯ ∈  M , t = 0, 1, . . . . It is easy to get that for an invertible BCN (3.2), for every x0 ∈  N , the restriction of the mapping HxN0 on L Nx0 (( M )N ) is injective. Due to the fact that n > q may hold, HxN0 is not necessarily injective, hence x(t) cannot be recovered only by y(t) usually.

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However, if n = q, based on Theorem 3.1, we can show that x(t) can be recovered by y(t) by proving the following Theorem 3.5. We will give a novel and direct proof by using the properties of the STP and swap matrices repetitively. Recall that a swap matrix W[m,n] is the unique (mn) × (mn) matrix such that j j W[m,n] δmi  δn = δn  δmi for all 1 ≤ i ≤ m and 1 ≤ j ≤ n (Proposition 1.4). By −1 T definition, W[m,n] ∈ L(mn)×(mn) , W[m,n] is invertible and W[m,n] = W[m,n] = W[n,m] . We will use the following proposition repetitively. We omit the similar proof to the proposition as the one to Proposition 1.4. Proposition 3.7 For a matrix A ∈ R(mnq)× p , label its row blocks by (11, 12, . . . , 1n, 21, 22, . . . , 2n, . . . , m1, m2, . . . , mn), where each i j denotes consecutive q rows, then W[m,n] A equals the matrix that is obtained by reordering the row blocks of A as (11, 21, . . . , m1, 12, 22, . . . , m2, . . . , 1n, 2n, . . . , mn). The relationship between a matrix A ∈ R p×(mnq) and W[m,n] can be got analogously. Theorem 3.5 If a BCN (3.2) is invertible and n = q, then for all i ∈ [1; N ], L(H L)−1 (I Q ⊗ M Nr )W[N ,Q] δ iN = H T . Proof First, by Proposition 3.7, we observe that W[N ,N ] (I N ⊗ M Nr )W[N ,N ] =W[N ,N ] (I N ⊗ (δ 1N ⊕ δ 2N ⊕ · · · ⊕ δ NN ))W[N ,N ] ⎡ 1 1 T ⎤ δ N (δ N ) .. ⎢ ⎥ ⎢ ⎥ ⎢ N .N T ⎥ ⎢δ N (δ N ) ⎥ ⎢ ⎥ 1 1 T ⎢ ⎥ δ N (δ N ) ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎢ ⎥ W[N ,N ] =W[N ,N ] ⎢ N N T ⎥ δ (δ ) N N ⎢ ⎥ ⎢ ⎥ . .. ⎢ ⎥ ⎢ ⎥ 1 1 T⎥ ⎢ δ N (δ N ) ⎥ ⎢ ⎢ ⎥ .. ⎣ ⎦ . δ NN (δ NN )T

3.2 Invertibility Characterization

⎡

71

⎤

δ 1N (δ 1N )T

⎥ ⎥ ⎥ ⎥ ⎥ δ 1N (δ 1N )T ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ W[N ,N ] ⎥ 2 2 T⎥ δ N (δ N ) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ δ 1N (δ 1N )T ⎢ ⎢ .. ⎢ . ⎢ ⎢ ⎢ 2 2 T ⎢ δ N (δ N ) ⎢ ⎢ δ 2N (δ 2N )T ⎢ ⎢ .. . =⎢ ⎢ ⎢ ⎢ ⎢ .. ⎢ . ⎢ ⎢δ N (δ N )T ⎢ N N ⎢ δ NN (δ NN )T ⎢ ⎢ .. ⎣ . ⎡

δ NN (δ NN )T

⎤

diag N (δ 1N (δ 1N )T ) ⎢ diag N (δ 2N (δ 2N )T ) ⎥ ⎢ ⎥ =⎢ ⎥ W[N ,N ] .. ⎣ ⎦ . diag N (δ NN (δ NN )T )

= diag N (δ 1N ) ⊕ diag N (δ 2N ) ⊕ · · · ⊕ diag N (δ NN ), ⎡

where

∗

(3.14)

⎤

⎢ ⎥ diag N (∗) := ⎣ . . . ⎦ ∗ is a diagonal block matrix consisting of N copies of ∗. For matrix I N ⊗ (δ 1N ⊕ δ 2N ⊕ · · · ⊕ δ NN ), label its row blocks by (11, 12, . . . , 1N , 21, 22, . . . , 2N , . . . , N 1, N 2, . . . , N N ), where each i j denotes consecutive N rows, then W[N ,N ] (I N ⊗ (δ 1N ⊕ δ 2N ⊕ · · · ⊕ δ NN )) equals the matrix that is obtained by reordering the row blocks of I N ⊗ (δ 1N ⊕ δ 2N ⊕ · · · ⊕ δ NN ) as (11, 21, . . . , N 1, 12, 22, . . . , N 2, . . . , 1N , 2N , . . . , N N ). ⎡

Similarly, W[N ,N ]

⎤ diag N (δ 1N (δ 1N )T ) 2 2 T ⎢ diag N (δ N (δ N ) ) ⎥ ⎢ ⎥ reorders the columns of ⎢ ⎥ from .. ⎣ ⎦ . diag N (δ NN (δ NN )T )

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3 Invertibility and Nonsingularity of Boolean Control Networks

(11, 12, . . . , 1N , 21, 22, . . . , 2N , . . . , N 1, N 2, . . . , N N ) to (11, 21, . . . , N 1, 12, 22, . . . , N 2, . . . , 1N , 2N , . . . , N N ). By Theorem 3.1, we have M = N = Q and for each i ∈ [1; N ], H L W[N ,N ] δ iN is invertible. Then H is invertible, and for each i ∈ [1; N ], L W[N ,N ] δ iN =: L i is also invertible. Since each L i is a logical matrix, L i L iT = I N . Second we verify that for all i, j ∈ [1; N ], j

L(H L)−1 (I N ⊗ M Nr )W[N ,N ] δ iN δ N

(3.15)

= Col j (H T ) = Row j (H ). By (3.9) and (3.14), we have j

(L(H L)−1 (I N ⊗ M Nr )W[N ,N ] δ iN δ N )T

⎡

⎤ H L W[N ,N ] δ 1N ⎢ ⎥ T j .. =(δ N )T (δ iN )T (W[N ,N ] (I N ⊗ M Nr )W[N ,N ] )T ⎣ ⎦L . H L W[N ,N ] δ NN

⎤ H L W[N ,N ] δ 1N ⎢ ⎥ T j .. =(δ N )T (δ iN )T (diag N (δ 1N ) ⊕ diag N (δ 2N ) ⊕ · · · ⊕ diag N (δ NN ))T ⎣ ⎦L . ⎡

⎤ H L W[N ,N ] δ 1N ⎥ T (i−1)N + j T ⎢ .. =(δ iN )T (δ N 2 ) ⎣ ⎦L .

⎡

H L W[N ,N ] δ NN

H L W[N ,N ] δ NN

j

=(δ iN )T (δ N )T H L W[N ,N ] δ iN L T =(δ iN )T Row j (H )L W[N ,N ] δ iN W[N ,N ] (L W[N ,N ] )T =(δ iN )T (Row j (H )L i )W[N ,N ] (L W[N ,N ] )T =((δ iN )T ((Row j (H )L i ) ⊗ I N )W[N ,N ] )(L W[N ,N ] )T =(Row j (H )L i )(δ iN )T (L W[N ,N ] )T =(Row j (H )L i )(L W[N ,N ] δ iN )T = Row j (H )L i L iT = Row j (H ).  By Theorem 3.5, when n = q, Theorem 3.4 can be simplified to the following Theorem 3.6.

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73

Theorem 3.6 If BCN (3.2) is invertible and n = q, then its inverse BCN is as follows (shown in Fig. 3.4): x(t + 1) = H T u(t), ¯ y¯ (t + 1) = (H L)−1 u(t)x(t), ¯ t ∈ N.

(3.16)

We give an example to illustrate the foregoing theorems. Example 3.1 Consider the following BCN x(t + 1) = Lu(t)x(t), y(t) = H x(t),

(3.17)

where t ∈ N, x(t), u(t), y(t) ∈ 4 , m = q = n = 2, L = δ4 [2, 3, 3, 4, 3, 2, 1, 3, 1, 4, 2, 2, 4, 1, 4, 1], H = δ4 [1, 2, 3, 4], H L = δ4 [2, 3, 3, 4, 3, 2, 1, 3, 1, 4, 2, 2, 4, 1, 4, 1], H L W[4,4] = δ4 [2, 3, 1, 4, 3, 2, 4, 1, 3, 1, 2, 4, 4, 3, 2, 1]. ¯ We have H Lδ ¯ 41 = δ4 [2, 3, 1, 4], H Lδ ¯ 42 = δ4 [3, 2, 4, 1], Denote L W[4,4] =: L. 3 4 ¯ ¯ H Lδ4 = δ4 [3, 1, 2, 4], and H Lδ4 = δ4 [4, 3, 2, 1] are all nonsingular. By Theorem 3.1, BCN (3.17) is invertible. Here n = q, we can use either Eq. (3.13) or Eq. (3.16) to construct its inverse BCN. By Theorem 3.3, for all i ∈ [1; 4], (H L)−1 =δ4 [3, 4, 2, 4, 1, 2, 3, 3, 2, 1, 1, 2, 4, 3, 4, 1], L(H L)−1 (I4 ⊗ M4r ) =δ4 [1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4] =δ4 [1, 2, 3, 4](I4 ⊗ 14T ), L(H L)−1 (I4 ⊗ M4r )W[4,4] δ4i =δ4 [1, 2, 3, 4]((I4 ⊗ 14T )W[4,4] )δ4i =δ4 [1, 2, 3, 4](14T ⊗ I4 )δ4i =δ4 [1, 2, 3, 4]I4 =H T . Hence no matter we choose either Eq. (3.13) or Eq. (3.16), the inverse BCN of BCN (3.17) is

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x(t + 1) = H T u(t), ¯ y¯ (t + 1) = (H L)−1 u(t)x(t), ¯

(3.18)

where t ∈ N, x(t), u(t), ¯ y¯ (t) ∈ 4 . Consider the inverse BCN (3.18). Choose initial state x(0) = δ41 , input ¯ = δ41 , u(2) ¯ = δ42 , u(3) ¯ = δ42 , u(4) ¯ = δ43 , u(5) ¯ = δ44 , u(6) ¯ = sequence u(0) ¯ = δ41 , u(1) ¯ = δ44 . Substituting these state and inputs into Eq. (3.18), we get the correδ43 , u(7) sponding output sequence y¯ (1) = δ43 , y¯ (2) = δ43 , y¯ (3) = δ41 , y¯ (4) = δ42 , y¯ (5) = δ41 , ¯ = δ41 , y¯ (8) = δ41 . y¯ (6) = δ44 , u(7) Regarding the above output sequence as the input sequence of BCN (3.17), that is, u(i) = y¯ (i + 1), i = 0, 1, . . . , 7, and substituting x(0) and u(0), u(1), . . . , u(7) into Eq. (3.17), we get the corresponding output sequence y(i) = u(i ¯ − 1), i = 1, 2, . . . , 8.

3.3 Nonsingularity Characterization 3.3.1 Nonsingularity Graphs Although nonsingularity is a direct generalization of invertibility, the method of dealing with invertibility is not suitable for nonsingularity. Unlike the pairwise equivalence of items stated in Proposition 3.6, the items stated in Proposition 3.2 are not equivalent (see Example 3.2). Hence to verify nonsingularity, we will choose a totally different way. In order to verify nonsingularity, we define a directed graph that is called a nonsingularity graph. By using the graph, an equivalent test criterion for nonsingularity will be given. In Chap. 4, in order to characterize observability of BCNs, and in Chap. 5, in order to study detectability of BCNs, we will define two new types of weighted directed graphs that are different from the nonsingularity graph. In addition, in order to study observability and detectability, we will combine newly defined graphs with finite automata. Differently, to characterize nonsingularity, the nonsingularity graph is enough. Definition 3.3 Consider a BCN (3.2). A weighted directed graph G N = (V, E, W), where V denotes the vertex set, E ⊂ V × V denotes the edge set, and W : E → {+, −} denotes the weight function, is called the nonsingularity graph of the BCN if V = {(x, x  )|x, x  ∈  N , H x = H x  }; for all (x1 , x1 ), (x2 , x2 ) ∈ V, ((x1 , x1 ), (x2 , x2 )) ∈ E if and only if there exist u 1 , u 1 ∈  M such that Lu 1 x1 = x2 and Lu 1 x1 = x2 ; for all edges e = ((x1 , x1 ), (x2 , x2 )) ∈ E,

3.3 Nonsingularity Characterization

75

⎧ ⎨ +, if there exist u 1 , u 1 ∈  M such that u 1 = u 1 , Lu 1 x1 = x2 , Lu 1 x1 = x2 , W(e) = ⎩ −, otherwise. Hereinafter, we call each vertex (x, x) ∈  N ×  N a diagonal vertex. We also call a BCN (3.2) singular if it is not nonsingular, i.e. there is x0 ∈  N such that the mapping (H L)Nx0 is not injective.

3.3.2 Nonsingularity Verification Theorem 3.7 A BCN (3.2) is singular if and only if in its nonsingularity graph, there is a diagonal vertex v and a cycle C such that there is a path P from v to a vertex in C, and there is an edge with weight + in P ∪ C. Proof (if:) Denote the path P and the cycle C by v → p1 → p2 → · · · → pk and pk → c1 → c2 → · · · → cl → pk , respectively. Construct an infinite vertex sequence as v p1 p2 . . . pk−1 ( pk c1 c2 . . . cl )ω ∈ ( N ×  N )N , where (·)ω means the concatenation of infinite copies of ·. Since there is an edge with weight + in P ∪ C, there exist distinct U1 , U2 ∈ ( M )N such that ∗ ( pk∗ c1∗ c2∗ . . . cl∗ )ω , L Nx0 (U1 ) = x0 p1∗ p2∗ . . . pk−1 # L Nx0 (U2 ) = x0 p1# p2# . . . pk−1 ( pk# c1# c2# . . . cl# )ω ,

and (H L)Nx0 (U1 ) = (H L)Nx0 (U2 ), where x0 , pi∗ , pi# , c∗j , c#j ∈  N , (x0 , x0 ) = v, ( pi∗ , pi# ) = pi , (c∗j , c#j ) = c j , i = 1, 2, . . . , k, j = 1, 2, . . . , l. That is to say, BCN (3.2) is singular. (only if:) Assume that BCN (3.2) is singular. There exist x0 ∈  N and distinct U1 , U2 ∈ ( M )N such that (H L)Nx0 (U1 ) = (H L)Nx0 (U2 ). Let min{ j ∈ N|U1 ( j) = U2 ( j)} be k, (x0 , x0 ) be v0 , and (L Nx0 (U1 )(i), L Nx0 (U2 )(i)) be vi , i = 0, 1, . . . , k, respectively. We use v0 → v1 → · · · → vk → · · · =: P  to denote the path of the nonsingularity graph of BCN (3.2) generated by U1 , U2 and v0 . Since the graph has a finite number of distinct vertices, from the pigeonhole principle, P  has a cycle after vertex vk . Denote such a cycle by vk+c → vk+c+1 → · · · → vk+c+l → vk+c , where c, l ∈ Z+ . Then the vertex sequence v0 v1 . . . vk vk+1 . . . vk+c−1 (vk+c vk+c+1 . . . vk+c+l )ω

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3 Invertibility and Nonsingularity of Boolean Control Networks

forms a new path of the nonsingularity graph. Denote v0 by v, v1 . . . vk vk+1 . . . vk+c−1 vk+c by P and vk+c vk+c+1 . . . vk+c+l vk+c by C, respectively. P ∪ C has an edge with weight +.  Since in the nonsingularity graph of BCN (3.2), the vertices (u, v) and (v, u) are mirror images of each other, they can be merged. After merging all such vertices, the new graph is called reduced nonsingularity graph. By Theorem 3.7, the following theorem directly follows. Theorem 3.8 A BCN (3.2) is singular if and only if in its reduced nonsingularity graph, there is a diagonal vertex v and a cycle C such that there is a path P from v to a vertex in C, and there is an edge with weight + in P ∪ C. Example 3.2 Consider the following BCN: x(t + 1) = δ4 [1, 1, 1, 3, 2, 2, 2, 4]u(t)x(t), y(t) = δ4 [1, 2, 3, 3]x(t),

(3.19)

where t ∈ N, x(t), y(t) ∈ 4 , u(t) ∈ , L W[4,2] = δ4 [1, 2, 1, 2, 1, 2, 3, 4], H = δ4 [1, 2, 3, 3], H L W[4,2] = δ4 [1, 2, 1, 2, 1, 2, 3, 3]. By H L W[4,2] δ41 = H L W[4,2] δ42 = H L W[4,2] δ43 = δ4 [1, 2], one has (H L)1δ1 ,

(H L)1δ2 , and (H L)1δ3 are injective. Observe that for all p ∈ Z+ , 4

4

4

p

p

4

4

(H L)δ4 (δ22 · · · δ22 δ21 ) = (H L)δ4 (δ22 · · · δ22 δ22 ). p

Then we have (H L)δ4 is not injective for any p ∈ Z+ . That is, for BCN (3.19), neither 4 (i), nor (ii), nor (iii) of Proposition 3.2 holds. Next, we use Theorem 3.8 to prove that BCN (3.19) is nonsingular. That is, (iv) of Proposition 3.2 holds. Hence the four items stated in Proposition 3.2 are not equivalent. The reduced nonsingularity graph of BCN (3.19) is shown in Fig. 3.5. In Fig. 3.5, there is only one edge with weight +, (δ44 , δ44 ) → (δ43 , δ44 ), and the only diagonal vertex that goes to this edge is (δ44 , δ44 ). But there is no cycle after (δ43 , δ44 ), and there is no cycle passing through (δ44 , δ44 ) and this edge. Hence by Theorem 3.8, BCN (3.19) is nonsingular. To test the conditions in Theorem 3.8, we can first find a cycle, and then search backward each path pointing to the cycle, and finally check whether there exists an edge with weight +. Such a procedure can be implemented by using the well-known depth-first algorithm (Jungnickel 2013) with time computational complexity linear in the size of the reduced nonsingularity graph. Hence the computational complexity of using Theorem 3.8 to verify nonsingularity of BCN (3.2) is O(22n+m ).

3.3 Nonsingularity Characterization

77

Similar to invertibility, according to Theorem 3.8, one sees that nonsingularity also remains invariant under coordinate transformations. By Theorem 3.8, the following corollary holds. Corollary 3.1 A BCN (3.2) is singular if in its reduced nonsingularity graph, there is a cycle having a diagonal vertex and an edge with weight +. Note that Corollary 3.1 provides a sufficient but not necessary condition for singularity of BCN (3.2). See the following example. Example 3.3 Consider the following BCN: x(t + 1) = δ4 [1, 1, 1, 3, 2, 2, 2, 4]u(t)x(t), y(t) = δ4 [3, 1, 3, 3]x(t),

(3.20)

where t ∈ N, x(t), y(t) ∈ 4 , u(t) ∈ . The reduced nonsingularity graph of BCN (3.20) is shown in Fig. 3.6. By Theorem 3.8, Fig. 3.6 shows that BCN (3.20) is singular, since (δ44 , δ44 ) → (δ43 , δ44 ) → (δ41 , δ44 ) → (δ41 , δ44 ) is a path in which a diagonal vertex (δ44 , δ44 ) goes into the cycle (δ41 , δ44 ) → (δ41 , δ44 ), and the edge of the cycle has weight +. Besides, the cycle (δ41 , δ44 ) → (δ41 , δ44 ) is the unique cycle that has an edge with weight +, which does not satisfy the condition in Corollary 3.1.

Fig. 3.5 Reduced nonsingularity graph of BCN (3.19), where the number i j in each circle denotes the j vertex (δ4i , δ4 ), reproduced from Zhang et al. (2015) with permission @ 2015 Elsevier Ltd.

Fig. 3.6 Reduced nonsingularity graph of BCN (3.20), where the number i j in each circle denotes the j vertex (δ4i , δ4 ), reproduced from Zhang et al. (2015) with permission @ 2015 Elsevier Ltd.

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3 Invertibility and Nonsingularity of Boolean Control Networks

Fig. 3.7 Reduced nonsingularity graph of BCN (3.21), where numbers i j in circles denote vertices j {δ4i , δ4 }

3.4 Relationship Between Invertibility and Nonsingularity By Definitions 3.1 and 3.2, if a BCN is invertible then it is also nonsingular. When does invertibility coincide with nonsingularity? Next, we solve this problem. Theorem 3.1 shows that if a BCN (3.2) is invertible, then m = q ≤ n. Hence invertibility can coincide with nonsingularity only if m = q ≤ n. We first show that even if m = q ≤ n, these two concepts do not coincide by a counterexample (see Example 3.4), and second prove that if m = q = n, they coincide (Theorem 3.9). Example 3.4 Consider the following BCN: x(t + 1) = δ4 [1, 3, 4, 2, 2, 4, 3, 1]u(t)x(t), y(t) = δ2 [1, 1, 2, 2]x(t),

(3.21)

where t ∈ N, q = m = 1, n = 2, y(t), u(t) ∈ , x(t) ∈ 4 , δ2 [1, 1, 2, 2] =: H , δ4 [1, 3, 4, 2, 2, 4, 3, 1] =: L. Since δ2 [1, 1, 2, 2]δ4 [1, 2, 3, 4, 4, 3, 2, 1]δ41 = δ2 [1, 1] is not nonsingular, the BCN (3.21) is not invertible by Theorem 3.1. Next we prove BCN (3.21) is nonsingular. Its reduced nonsingularity graph is is shown in Fig. 3.7. By Fig. 3.7 and Theorem 3.8, for each state δ4i ∈ 4 , (H L)∞ δ4i injective, i.e., the BCN is nonsingular. Theorem 3.9 Consider a BCN (3.2). If m = q = n, then the BCN is invertible if and only if it is nonsingular. Proof We only need to prove the “if” part, as the “only if” part naturally holds. We are given a nonsingular BCN (3.2) satisfying m = q = n. Denote the reduced nonsingularity graph of the BCN by G N = (V, E, W). Assume the BCN is not invertible, then by Theorem 3.1, either (1) there exists δ iN ∈  N such that L W[N ,N ] δ iN j is not invertible, or (2) for each j ∈ [1; N ], L W[N ,N ] δ N is invertible, and H is not invertible. If (1) holds, then there exist distinct i 1 , i 2 ∈ [1; N ] such that L W[N ,N ] j j j δ iN δ iN1 = L W[N ,N ] δ iN δ iN2 =: δ N1 , i.e., ((δ iN , δ iN ), (δ N1 , δ N1 )) ∈ E and W(((δ iN , δ iN ), j1 j1 (δ N , δ N ))) = +. Since for every diagonal vertex v in V, there exists at least one diagonal vertex v ∈ V such that (v, v ) ∈ E, and there are totally finitely many

3.4 Relationship Between Invertibility and Nonsingularity

79

vertices, there exists a cycle C consisting of diagonal vertices in G N and a path j j P in G N from (δ N1 , δ N1 ) to C, then the BCN is not nonsingular by Theorem 3.8. If (2) holds, then there exist distinct i 1 , i 2 ∈ [1; N ] such that H δ iN1 = H δ iN2 , i.e., (δ iN1 , δ iN2 ) ∈ V, ((δ 1N , δ 1N ), (δ iN1 , δ iN2 )) ∈ E, and W(((δ 1N , δ 1N ), (δ iN1 , δ iN2 ))) = +. We also have ((δ iN1 , δ iN2 ), (δ 1N , δ 1N )) ∈ E. Then the cycle (δ 1N , δ 1N ) → (δ iN1 , δ iN2 ) → (δ 1N , δ 1N ) contains an edge with weight +. By Theorem 3.8, the BCN is not nonsingular. 

3.5 Application to the Mammalian Cell Cycle The cell cycle contains a series of molecular events leading to the reproduction of the genome of a cell (Synthesis phase) and its division into two daughter cells (Mitosis phase). The Synthesis and Mitosis phases are preceded by two gap phases, called G1 and G2, respectively, in which RNA and proteins are synthesized. Mammalian cell division is tightly controlled in order to make it be coordinated with the overall growth of the organism, as well as answer specific needs, such as wound healing. This coordination is achieved through extracellular positive and negative signals whose balance decides whether a cell will divide or remain in a resting state (a fifth phase, G0). The positive signals or growth factors ultimately elicit the activation of protein CycD in the cell. Thus CycD can be regarded as the control input. A more detailed introduction to the mammalian cell cycle can be found in Fauré et al. (2006), etc. This part focuses on a holistic input-output behavior of the mammalian cell cycle.

3.5.1 Invertibility of the Mammalian Cell Cycle In Fauré et al. (2006), the dynamics of the core network regulating the mammalian cell cycle is formulated as a BCN model shown as follows: ¯ ∧ x¯3 (t) ∧ x¯4 (t) ∧ x¯9 (t)) ∨ (x5 (t) ∧ u(t) ¯ ∧ x¯9 (t)), x1 (t + 1) =(u(t) x2 (t + 1) =(x¯1 (t) ∧ x¯4 (t) ∧ x¯9 (t)) ∨ (x5 (t) ∧ x¯1 (t) ∧ x¯9 (t)), x3 (t + 1) =x2 (t) ∧ x¯1 (t), x4 (t + 1) =(x2 (t) ∧ x¯1 (t) ∧ x¯6 (t) ∧ (x7 (t) ∧ x8 (t))) ∨ (x4 (t) ∧ x¯1 (t) ∧ x¯6 (t) ∧ (x7 (t) ∧ x8 (t))), x5 (t + 1) =(u(t) ¯ ∧ x¯3 (t) ∧ x¯4 (t) ∧ x¯9 (t)) ¯ ∧ x¯9 (t)), ∨ (x5 (t) ∧ (x3 (t) ∧ x4 (t)) ∧ u(t) x6 (t + 1) =x9 (t),

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3 Invertibility and Nonsingularity of Boolean Control Networks

x7 (t + 1) =(x¯4 (t) ∧ x¯9 (t)) ∨ x6 (t) ∨ (x5 (t) ∧ x¯9 (t)), x8 (t + 1) =x¯7 (t) ∨ (x7 (t) ∧ x8 (t) ∧ (x6 (t) ∨ x4 (t) ∨ x9 (t))), x9 (t + 1) =x¯6 (t) ∧ x¯7 (t),

(3.22)

where ¯·, ∧, ∨ denote logical operators: negation, conjunction, and disjunction, respectively; t ∈ N; u(t), xi (t) ∈ D, i ∈ [1; 9]. The BCN model (3.22) consists of one input node, CycD, and nine state nodes (proteins), Rb, E2F, CycE, CycA, p27, Cdc20, Cdh1, UbcH10, CycB, which are represented as ten Boolean variables u, x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 , respectively, showing their activation and inactivation. Protein Cdc20 is responsible for the metaphase-to-anaphase transition: it activates separase through the destruction of its inhibitor securin; this activation elicits the cleavage of the cohesin complexes that maintain the cohesion between the sister chromatids, thus leading to their separation. Hence Cdc20 plays a central role in the division of cells. Based on the above introduction, if at any time step, both activation and inactivation of Cdc20 can be achieved, one may fully control the division of cells. As CycD is the control input of the BCN model, a natural idea is designing a CycD sequence to obtain any Cdc20 sequence. If this idea is reconsidered in a backward way, that is, determining the CycD sequence by using a Cdc20 sequence, and Cdc20 is regarded as the output node, then it is just the concept of invertibility. Based on this idea, a natural problem arises: Problem 3.1 Can one obtain any Cdc20 sequence by designing a CycD sequence? If the BCN (3.22) is invertible when Cdc20 is regarded as the unique output node, then one can obtain any Cdc20 sequence by designing CycD sequences so as to control the division of the mammalian cell cycle. Note that in order to make the BCN (3.22) invertible, one of the necessary conditions is that there is exactly one output node by Theorem 3.1. The computation results are shown in Table 3.1. By Table 3.1 and Theorem 3.1, one sees that no matter which protein is chosen as the unique output node, BCN (3.22) is not invertible. This result tells us that, unfortunately, one cannot fully control the division of the mammalian cell cycle. That is, the answer to Problem 3.1 is “No”. On the other hand, in the algebraic form of BCN (3.22), each entry of the first 9 rows of the matrix L12 ∈ R512×512 equals 0. Then by Proposition 2.1, the BCN (3.22) is not controllable. Note that we formulate the control of the mammalian cell cycle as invertibility. Why not controllability? First, BCN (3.22) is not controllable. Second, controllability involves driving a state (nine proteins) to another state at some future time step, while it is powerless to obtain a target output sequence. While obtaining a target output sequence is just what we want. Based on the above analysis we conclude that the Cdc20 sequence cannot be obtained arbitrarily by designing CycD sequences. This implies that the cell cycle cannot be fully controlled based on the BCN model (3.22).

3.5 Application to the Mammalian Cell Cycle Table 3.1 Consider BCN (3.22), when the unique output is chosen as xi , then the corresponding H L W[N ,M] shown in Theorem 3.1 is on its right-hand side

81

Output

The corresponding H L W[N ,M] in Theorem 3.1

x1 x2 x3 x4 x5 x6 x7 x8 x9

δ2 [2, 2, . . . δ2 [2, 2, . . . δ2 [2, 2, . . . δ2 [2, 2, . . . δ2 [2, 2, . . . δ2 [1, 1, . . . δ2 [1, 1, . . . δ2 [1, 1, . . . δ2 [2, 2, . . .

3.5.2 Further Discussion Now that we cannot obtain any Cdc20 sequence, we pay attention to looking for periodic Cdc20 sequences to control BCN (3.22) to some extent. If we could find such periodic Cdc20 sequences, although one cannot fully control the mammalian cell cycle (proved in Sect. 3.5.1), one may do it to some extent. In the sequel, we show a class of periodic Cdc20 sequences with alternative active and inactive Cdc20, and the process of finding them. We first characterize the space of state trajectories of BCN (3.2). Denote L¯ := j L1 M ∈ R N ×N and Fs := {δ iN δ N |i, j ∈ [1; N ], L¯ ji > 0}. Intuitively, L¯ ji > 0 means j that there is a control input u 0 ∈  M such that δ N = Lu 0  δ iN . Then it follows that the space of state trajectories is characterized as      L N  N ( M )N = X ∈ ( N )N  ∀i ∈ N, X (i)X (i + 1) ∈ Fs .

(3.23)

As X is of infinite length, one cannot determine whether X is a state trajectory or not algorithmically. However, one can determine whether any prefix of X is a prefix of some state trajectory by (3.23) by checking whether the concatenation of every two consecutive states of the prefix belongs to Fs , where a prefix of X is any finite subsequence of X that starts at X (0). Second we consider the space of output trajectories. The space of output trajectories is the image of the space of state trajectories under the mapping H N . Given a state x0 ∈  N and an output y0 ∈  Q such that y0 = H x0 , and a finite output sequence Y ∈ ( Q ) p for some p ∈ Z+ , one can check whether Y has a preimage X ∈ ( N ) p p under the mapping Hx0 such that x0 X is a prefix of a state trajectory. If x0 X is a prefix of a state trajectory, there is an input sequence that makes y0 Y appear. The procedure is formulated in Algorithm 3.1. Given a BCN (3.2), a state x0 ∈  N , and an output y0 ∈  Q such that y0 = H x0 , and a finite output sequence Y ∈ ( Q ) p for some p ∈ Z+ , Algorithm 3.1 generates a

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3 Invertibility and Nonsingularity of Boolean Control Networks

Algorithm 3.1 An algorithm to determine whether a given finite output sequence is a prefix of an output trajectory for a BCN (3.2) Require: Given a BCN (3.2), a state x0 ∈  N , an output y0 ∈  Q such that y0 = H x0 , and a finite output sequence Y ∈ ( Q ) p for some p ∈ Z+ Ensure: “Yes”, if y0 Y is a prefix of an output trajectory; and “No”, otherwise 1: Set {x0 } = X0 , i = 1 and Xi = {x ∈  N |H x = Y (i)} 2: while Xi = ∅ & i ≤ p do 3: Denote Xi = {x1i , . . . , xki i } and Xi−1 = {x1i−1 , . . . , xki−1 } i−1 4: Set j = 1 5: while j ≤ ki do 6: Set k = 1 and num = 0 7: while k ≤ ki−1 do 8: if xki−1 x ij ∈ Fs then 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24:

num = num + 1, add an arc from xki−1 to x ij end if k =k+1 end while if num = 0 then Remove x ij from Xi end if j = j +1 end while i = i + 1, set Xi = {x ∈  N |H x = Y (i)} end while if Xi = ∅ then return “No”, stop else return “Yes”, stop end if

directed graph with node set X0 ∪ · · · ∪ X p for some p  ≤ p and the corresponding arcs from some node in Xi to some node in Xi+1 , i = 0, . . . , p  − 1. If Algorithm 3.1 returns “Yes”, then p  = p and X p = ∅, and each path of the tree that starts at x0 p and ends at a node in X p has a preimage in ( M ) p under the mapping L x0 . Each of such preimages is a finite input sequence that makes the finite output sequence y0 Y appear. Now consider the BCN (3.22). Choose protein Cdc20 as the unique output node. Given any finite Cdc20 sequence and any initial states of the nine proteins, one can use Algorithm 3.1 to determine whether there is a finite CycD sequence that makes the Cdc20 sequence appear. Now we try to find an input sequence that makes a periodic Cdc20 sequence appear. Specifically, during a consecutive period of time, Cdc20 is inactive, necessary RNA and proteins are synthesized; and then during another consecutive period of time, Cdc20 is active, sister chromatids separate. We arrange that these two types of consecutive period of time appear alternately. Since the BCN (3.22) is not invertible, it is quite challenging to find such an input sequence. One candidate we can find is shown in Table 3.2. Here is the process of looking for it. First, we choose an

3.5 Application to the Mammalian Cell Cycle

83

Table 3.2 A finite input sequence U , an initial state x0 and the state sequence X and output sequence Y determined by U and x0 , where subscripts denote time steps; u 0 = δ22 , u 1 = δ22 , . . . , u 12 = δ21 ; 378 , . . . , x = δ 378 ; y = δ 2 , . . . , y = δ 2 1/2 means u i can be either δ21 or δ22 ; x0 = δ512 13 13 2 2 512 0 U X Y

x0 378 y0 2

u0 2

u1 2

u2 2

u3 2

u4 2

u5 1

u6 1

u7 1

u8 1/2

u9 1/2

u 10 1/2

u 11 1/2

u 12 1

x1 44 y1 2

x2 236 y2 2

x3 236 y3 2

x4 236 y4 2

x5 236 y5 2

x6 508 y6 2

x7 380 y7 2

x8 284 y8 2

x9 416 y9 2

x10 477 y10 2

x11 469 y11 1

x12 498 y12 1

x13 378 y13 2

arbitary output sequence. Then we check whether there exists a corresponding input sequence which could generate the output sequence. If yes and its first and last state are the same, the output sequence and input sequence are what we desire. Repeat the above process until we reach the sequence in Table 3.2. From Table 3.2, one sees that x0 = x13 , that is, if we choose an input trajectory as the concatenation of countably infinitely many copies of the finite sequence u 0 . . . u 12 , we will get a state trajectory as the concatenation of countably infinitely many copies of the sequence x0 . . . x12 and an output trajectory as the concatenation of countably infinite copies of the sequence y0 . . . y12 . That is, at time steps 13k + i, Cdc20 is inactive, where k ∈ N, i ∈ [0; 10]; and at time steps 13k + i, Cdc20 is active, where k ∈ N, i ∈ [11; 12]. On the other 236 , i.e., 011101100 (corresponding to hand, one sees that x2 = x3 = x4 = x5 = δ512 236 is a proteins Rb, E2F, CycE, CycA, p27, Cdc20, Cdh1, UbcH10, CycB) and δ512 fixed point (essentially the unique attractor) of the BN obtained from BCN (3.22) by setting u(t) ≡ δ22 (i.e., absence of protein CycD). Hence in the above periodic state 236 is replaced by any positive integer, the trajectory, if the number of consecutive δ512 new state sequence is still a state trajectory. Formally, set X k := x0 x1 x2 . . . x2 x6 . . . x12 ,   k

then for all k1 , k2 , . . . ∈ Z+ , X k1 X k2 . . . is a state trajectory. Consequently, set Yk := δ22 . . . δ22 ,   k

then for all k1 , k2 , . . . ∈ N \ [0; 7], Yk1 δ21 δ21 Yk2 δ21 δ21 . . . is an output trajectory. One also 284 416 477 469 498 378 316 284 → δ512 → δ512 → δ512 → δ512 → δ512 → δ512 → δ512 (i.e., 100011 sees that δ512 100 → 110100000 → 111011101 → 111010101 → 111110010 → 101111010 → 100111100 → 100011100 corresponding to proteins Rb, E2F, CycE, CycA, p27, Cdc20, Cdh1, UbcH10, CycB) is the unique attractor of the BN obtained from BCN (3.22) by setting u(t) ≡ δ21 (i.e., presence of protein CycD).

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3 Invertibility and Nonsingularity of Boolean Control Networks

Now we show how these sequences alternate states between the two attractors under constant inputs. At the start of cell cycle, Cdc20 must be inactive. It is because a certain amount of RNA and proteins must be synthesized before cell divide. On the 236 236 → δ512 , in other hand, when u(t) ≡ δ22 , the BCN (3.22) has a unique attractor δ512 2 2 which Cdc20 is in state δ2 , i.e., inactive. So when u(t) ≡ δ2 , no matter which state the network is in, Cdc20 will become inactive after a finite number of iterations of the network and then remains invariant. Therefore we might as well set the prefix of the target input sequence as δ22 . . . δ22 , and then the prefix of the corresponding 236 236 . . . δ512 . This stable period corresponds to the G0 phase. Since state trajectory is δ512 236 δ512 is a fixed point when u(t) ≡ δ22 , Cdc20 cannot be active until CycD becomes active. When the amount of RNA and proteins is enough for the division of cells, CycD is activated, and then Cdc20 will also be active. Hence we set a consecutive δ21 segment of CycD after the former consecutive δ22 segment of CycD. Although we know consecutive δ21 CycD sequence can lead the state into the unique attractor of length 7, we do not know whether there will be alternative δ21 and δ22 Cdc20 before the state trajectory enters the attractor of length 7, which we do not hope will appear. Once this happens, cells will divide without enough RNA and proteins. Luckily, this does not happen, i.e., before the state trajectory enters the attractor of length 7, Cdc20 is always inactive. And even more luckily, during the period of CycD being active, 378 , a state of the attractor of length 7, if one sets a consecutive when the state is in δ512 2 δ2 segment of CycD, then the state trajectory enters the attractor of length 1, and during this period, Cdc20 is always inactive. That is, we have found a series of periodic Cdc20 sequences. From Table 3.2, one sees the sequences seem consistent with alternating the two attractors found in Fauré et al. (2006) for the synchronous simulation: a stable state corresponding to the G0 phase, with inactive Cdc20 for CycD = 0 and a cyclic activation of Cdc20 corresponding to cell division for CycD = 1. From the process of looking for the series of periodic Cdc20 sequences, we can obtain the following conclusions: • Adding times with inactive CycD increases the number of states where Cdc20 is inactive. • When the system is in the quiescent state, Cdc20 cannot be activated until CycD has been activated. • The times in the sequence where CycD can be either active or inactive show that CycD is not required during the whole cell cycle but only at the beginning and the end of the cell cycle. Based on the above discussion, we obtain that one can control the period of the mammalian cell cycle to a large extent, since we have found a large class of (partially) periodically Cdc20 sequences (as shown in Table 3.2). Practically, in order to generate the above meaningful (partially) periodically Cdc20 sequences, one should in advance synthesize the corresponding CycD sequences, which should depend on biological techniques, which is far beyond the scope of this book.

3.5 Application to the Mammalian Cell Cycle

85

Notes The invertibility problem has been extensively studied for linear systems (cf. Brockett and Mesarovi 1965; Silverman 1969; Sain and Massey 1969; Morse and Wonham 1971; Moylan 1977, etc.), nonlinear systems (cf. Hirschorn 1979; Singh 1982; Nijmeijer 1982, etc.), and switched systems (cf. Vu and Liberzon 2008; Tanwani and Liberzon 2010, etc.). Invertibility of linear and nonlinear systems is dealt with by using a frequency domain method, a linear-space method, or a differential geometrical method. For BCNs, the updating functions of nodes are essentially polynomials defined on linear spaces (D, Dn ) with module-2 addition and multiplication, and hence are nonlinear. Hence the linear-space method cannot be used to deal with invertibility of BCNs. Also, BCNs do not have frequency domain structure. Hence the frequency domain method does not apply to BCNs. The state spaces of BCNs are not manifolds, since they are totally disconnected. Hence the differential geometrical method does not apply to invertibility of BCNs either. Similar to invertibility, one will see that nonsingularity of BCN (3.2) also remains invariant under logical coordinate transformations. Reconsider the BCN (3.19). We have shown that the BCN is nonsingular in p Example 3.2. We have also shown that (H L)δ4 is not injective for any p ∈ Z+ in the 4 example. Hence BCN (3.19) has no “inverse” BCN. When does a nonsingular BCN have an “inverse” BCN just like invertible BCNs? This is an interesting question. Extensions for invertibility and nonsingularity of BCNs can be found in Zhao et al. (2016), Yu et al. (2018).

References Brockett RW, Mesarovi MD (1965) The reproducibility of multivariable systems. J Math Anal Appl 11:548–563 Cheng D, Li Z, Qi H (2010) Realization of Boolean control networks. Automatica 46(1):62–69 Fauré A et al (2006) Dynamical analysis of a generic Boolean model for the control of the mammalian cell cycle. Bioinformatics 22(14):e124 Hirschorn R (1979) Invertibility of multivariable nonlinear control systems. IEEE Trans Autom Control 24(6):855–865 Jungnickel D (2013) Graphs, networks and algorithms, 4th edn. Springer Publishing Company, Incorporated, Berlin Morse A, Wonham W (1971) Status of noninteracting control. IEEE Trans Autom Control 16(6):568–581 Moylan P (1977) Stable inversion of linear systems. IEEE Trans Autom Control 22(1):74–78 Nijmeijer H (1982) Invertibility of affine nonlinear control systems: a geometric approach. Syst Control Lett 2(3):163–168 Sain M, Massey J (1969) Invertibility of linear time-invariant dynamical systems. IEEE Trans Autom Control 14(2):141–149 Silverman L (1969) Inversion of multivariable linear systems. IEEE Trans Autom Control 14(3):270–276

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Singh S (1982) Invertibility of observable multivariable nonlinear systems. IEEE Trans Autom Control 27(2):487–489 Tanwani A, Liberzon D (2010) Invertibility of switched nonlinear systems. Automatica 46(12):1962–1973 Vu L, Liberzon D (2008) Invertibility of switched linear systems. Automatica 44(4):949–958 Yu Y, Wang B, Feng J (2018) Input observability of Boolean control networks. Neurocomputing Zhang K, Zhang L, Mou S (2017) An application of invertibility of Boolean control networks to the control of the mammalian cell cycle. IEEE/ACM Trans Comput Biol Bioinform 14(1):225–229 Zhang K, Zhang L, Xie L (2015) Invertibility and nonsingularity of Boolean control networks. Automatica 60:155–164 Zhao G, Wang Y, Li H (2016) Invertibility of higher order k-valued logical control networks and its application in trajectory control. J Frankl Inst 353(17):4667–4679

Chapter 4

Observability of Boolean Control Networks

Given a dynamical system, as the system evolves, a state trajectory is generated. Generally speaking, a quantitative analysis of the system closely depends on states of a trajectory. Particularly for a deterministic system, if the initial state has been determined, then the corresponding trajectory will be naturally determined by using an input sequence. That is, the initial state will help understand the whole information of the corresponding trajectory for deterministic systems. Of course, if states of a trajectory can be directly observed, then it is feasible to do quantitative analysis for the system. However, if states are only partially observed, which can be represented as an (observation) output map, then it is meaningful to study whether the initial state can be obtained by using an input sequence and the corresponding output sequence, resulting in the concept of observability (Kalman 1963; Kalman et al. 1969; Wonham 1985). Intuitively, observability implies that one could use an input sequence and the corresponding output sequence to determine the initial state. For different uses, observability can be formulated in different ways. If it is formulated as a strong definition, then it is easy to recover the initial state by using the definition, but fewer systems satisfy such a definition. However, if it is formulated as a weak definition, it is relatively difficult to recover the initial state, but more systems satisfy such a definition. For classical linear time-invariant control systems, since the sum of two solutions is still solution, the notion of observability does not depend on inputs or initial states. Hence most notions of observability are equivalent. However, for Boolean control networks (BCNs), there exist nonequivalent notions of observability. Later on,1 we introduce four notions of observability for BCNs and their verification methods step by step.

1 Parts of Sects. 4.2 and 4.3 were reproduced from Zhang and Zhang (2014) with permission @ 2016

IEEE. Some of the material in Sect. 4.4 were reproduced from Laschov et al. (2013) with permission @ 2013 Elsevier Ltd. © Springer Nature Switzerland AG 2020 K. Zhang et al., Discrete-Time and Discrete-Space Dynamical Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-25972-3_4

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4.1 Notions of Observability Recall a BCN (2.4) as follows: x(t + 1) = f (x(t), u(t)), y(t) = h(x(t)),

(4.1)

where t = 0, 1, 2, . . . ; x(t) ∈ Dn , u(t) ∈ Dm , y(t) ∈ Dq ; f : Dn+m → Dn and h : Dn → Dq . Recall 2m = M, 2n = N , 2q = Q, and the algebraic form of a BCN as follows: x(t + 1) = Lu(t)x(t), y(t) = H x(t),

(4.2)

where t = 0, 1, . . . ; x(t) ∈  N , u(t) ∈  M , and y(t) ∈  Q ; L ∈ L N ×(N M) ; H ∈ L Q×N . Now we give the first definition for observability of BCNs, which is actually the same as the widely used notion of observability for nonlinear control systems (Sontag 1979) (cf. (C) of p. 142). It is not known whether this notion is decidable p for nonlinear control systems to the best of our knowledge. Recall that (H L)x0 (U ) N (resp. (H L)x0 (U )) means the corresponding output sequence (excluding y0 = H x0 ) of (4.2) generated by a given initial state x0 and a given p-length input sequence U (resp. an infinite-length input sequence U ), where p ∈ Z+ . Definition 4.1 A BCN (4.2) is called multiple-experiment observable if for very two different initial states x0 , x0 ∈  N , there is an input sequence U ∈ ( M ) p for p p some p ∈ Z+ such that H x0 = H x0 implies (H L)x0 (U ) = (H L)x  (U ). Such an 0 input sequence U is called a distinguishing input sequence of x0 and x0 . If a BCN is multiple-experiment observable, then every two different initial states have a distinguishing input sequence, and then the initial state can be determined by doing a finite number of experiments. In the first experiment, assume two different initial states as candidates, and feed one of their distinguishing input sequences into the BCN, then at least one of the obtained output sequences differs from the real output sequence (generated by the real initial state and the input sequence), hence at least one of the two candidates is not the real initial state, and mark each candidate that is not real. In the second experiment, repeat the first experiment by assuming two initial states as candidates from the unmarked initial states. Repeat the experiment until there is only one unmarked initial state left. Then the only state is the real initial state. This notion for BCNs is first studied in Zhao et al. (2010), in which a sufficient but not necessary condition is given. Later on, equivalent conditions for this notion are given in Li et al. (2015), Zhang and Zhang (2014, 2016), Zhu et al. (2018), Cheng et al. (2018), Cheng et al. (2016), etc. The results in Zhang and Zhang (2014, 2016) are mainly based on graph theory and finite automata, those in Cheng et al. (2018),

4.1 Notions of Observability

89

Cheng et al. (2016) are based on the semitensor product (STP) of matrices, and the one in Zhu et al. (2018) is based on the STP and graph theory. All these results are focused on states of BCNs. However, differently, the result in Li et al. (2015) is focused on mappings of BCNs and obtained by using results in computational algebra. The methods in Zhang and Zhang (2014, 2016), Cheng et al. (2016) are mathematically equivalent. The methods in Cheng et al. (2018), Zhu et al. (2018) are mathematically equivalent, but the verification method in Zhu et al. (2018) is more effective than that in Cheng et al. (2018). The method in Zhu et al. (2018) is an improved version of the one in Zhang and Zhang (2014, 2016) based on the structure proposed in Zhang and Zhang (2014, 2016). To be more specific, consider a BCN (4.2), in Zhang and Zhang (2014, 2016), an observability graph with complexity O(22n+m−1 ) is constructed, then observability is verified by checking at most 22n−1 deterministic finite automata (DFAs) each with size at most the graph. The method in Zhang and Zhang (2014, 2016) runs in time O(24n+m−2 ). However, the verification in Zhu et al. (2018) is done directly on the observability graph, and in time O(22n+m−1 ). The efficiency of the methods in Zhang and Zhang (2014, 2016), Zhu et al. (2018), Cheng et al. (2018), Cheng et al. (2016) are quite similar, and generally higher than the one in Li et al. (2015) in the worst case. However, for sparse BCNs, the efficiency of the method in Li et al. (2015) is generally higher than the ones in Zhang and Zhang (2014, 2016), Zhu et al. (2018), Cheng et al. (2018), Cheng et al. (2016), since for sparse BCNs, mappings depend on very few Boolean variables. Let us look at the second notion of observability. Definition 4.2 A BCN (4.2) is called strongly multiple-experiment observable if for every initial state x0 ∈  N , there exists an input sequence U ∈ ( M ) p for some p ∈ Z+ such that for each initial state x0 ∈  N different from x0 , H x0 = H x0 implies p p (H L)x0 (U ) = (H L)x  (U ). Such an input sequence U is called a distinguishing input 0 sequence of x0 . If a BCN is strongly multiple-experiment observable, then every initial state has a distinguishing input sequence, and then the initial state can be determined also by doing a finite number of experiments (sometimes one experiment is enough). In the first experiment, assume one initial state x0 as a candidate, and feed one of its distinguishing input sequences into the BCN, then x0 is the real initial state x0 if and only if the output sequence generated by x0 and U and the real output sequence (generated by x0 and U ) are the same. With any luck, after the first experiment, the real initial state can be found. With no luck, after repeating this experiment for finitely many times for different candidates, the real initial state will be determined. Intuitively, one sees that Definition 4.2 is stronger than Definition 4.1. The second notion for BCNs is first studied in Cheng and Qi (2009), where an equivalent condition for this notion for controllable BCNs2 is given. Later on, an equivalent condition for BCNs is given in Zhang and Zhang (2014, 2016). We now state the third notion of observability, which is actually the “singleexperiment observability” for nonlinear control systems (Sontag 1979) (cf. (A) of 2 Actually

a sufficient but not necessary condition for BCNs.

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p. 142). It is also not known whether this notion is decidable for nonlinear control systems. Definition 4.3 A BCN (4.2) is called single-experiment observable if there exists an input sequence U ∈ ( M ) p for some p ∈ Z+ such that for every two different p p initial states x0 , x0 ∈  N , H x0 = H x0 implies (H L)x0 (U ) = (H L)x  (U ). Such an 0 input sequence U is called a distinguishing input sequence (of (4.2)). If a BCN is single-experiment observable, then the initial state can be determined by every distinguishing input sequence. Hence, in order to determine the initial state, one only needs to do one experiment. Definition 4.3 is stronger than Definition 4.2. The third notion for BCNs is first studied in Cheng and Zhao (2011), where an equivalent condition for this notion for controllable BCNs3 is given. Later on, equivalent conditions for BCNs are given in Li et al. (2014), Zhang and Zhang (2014, 2016). The method in Zhang and Zhang (2014, 2016) is more effective than that in Li et al. (2014), since in the former only necessary input sequences with a length upper bound are needed to test, while in the latter all input sequences with the length upper bound are needed to test. Finally we give the fourth notion of observability, which is essentially the notion of observability for linear control systems. There are several verification methods for linear control systems, e.g., a matrix rank criterion (Kalman 1963) and a linearspace characterization (Wonham 1985). However, these methods do not apply to BCNs because BCNs are not linear. Definition 4.4 A BCN (4.2) is called arbitrary-experiment observable if for every two different initial states x0 , x0 ∈  N , for every input sequence U ∈ ( M )N , H x0 = H x0 implies (H L)Nx0 (U ) = (H L)Nx0 (U ). One directly sees that a BCN (4.2) is arbitrary-experiment observable if and only if for every two different initial states x0 , x0 ∈  N , for every input sequence p U ∈ ( M ) p , where p is sufficiently large, H x0 = H x0 implies (H L)x0 (U ) = p (H L)x0 (U ). If a BCN is arbitrary-experiment observable, then the initial state can be determined by every sufficiently long input sequence. Hence, in order to determine the initial state, one also only needs to do one experiment. Definition 4.4 is stronger than Definition 4.3. The fourth notion for BCNs is first studied in Fornasini and Valcher (2013). Equivalent conditions for this notion for BCNs are given in Fornasini and Valcher (2013), Zhang and Zhang (2014). The methods used in these two papers are based on different ideas, but almost have the same computational complexity in general. However, the method in Zhang and Zhang (2014) is easier to follow, because only a simple weighted directed graph (called observability graph) is needed to construct. In Zhang and Zhang (2014), a notion of observability graph is proposed. By using the graph, a given BCN is transformed into different DFAs to verify the above four 3 Also

actually a sufficient but not necessary condition for BCNs.

4.1 Notions of Observability

91

notions of observability, resulting in a unified verification method. The computationally algebraic method used in Li et al. (2015) applies to Definition 4.1, and should also apply to Definition 4.2, but does not apply to Definition 4.3 or Definition 4.4, because for the latter two definitions, no algebraic variety can be defined. The method used in Fornasini and Valcher (2013) only applies to Definition 4.4, because it assumes that the notion of observability does not depend on input sequences. The method used in Li et al. (2014) only applies to Definition 4.3. The method used in Cheng et al. (2018) is transforming verification of observability into verification of a notion of set controllability, and applies only to Definition 4.1.

4.2 Observability Graphs In order to give a unified verification method for the above four notions of observability, we introduce the notion of observability graph. Definition 4.5 Consider a BCN (4.1). A weighted directed graph Go = (V, E, W) m (with weight set 2D ) is called the observability graph of (4.1) if the vertex set V  n is {{x, x } ∈ D × Dn |h(x) = h(x  )}, the edge set E is {({x1 , x1 }, {x2 , x2 }) ∈ V × V|(∃u ∈ Dm )[( f (x1 , u) = x2 ∧ f (x1 , u) = x2 )∨( f (x1 , u) = x2 ∧ f (x1 , u) = x2 )]} ⊂ m V×V, and the weight function W : E → 2D maps each edge ({x1 , x1 }, {x2 , x2 }) ∈  m E to a set {u ∈ D |( f (x1 , u) = x2 ∧ f (x1 , u) = x2 ) ∨ ( f (x1 , u) = x2 ∧ f (x1 , u) = x2 )} of inputs. A vertex {x, x  } is called diagonal if x = x  , and non-diagonal otherwise. Note that for all vertices {x, x  } ∈ V, we have {x, x  } = {x  , x}. In each observability graph, the children of a diagonal vertex are always diagonal. In addition, each diagonal vertex will go into a cycle consisting of diagonal vertices, since there are totally finitely many vertices. We then use symbol to denote the subgraph of an observability graph generated by all diagonal vertices, and call diagonal subgraph. The subgraph of an observability graph generated by all non-diagonal vertices is called the non-diagonal subgraph. Intuitively, an observability graph collects all state trajectory pairs over the same input sequences and generating the same outputs. For a BCN (4.2), the computational cost of constructing its observability graph is at most (1 + 2n (2n − 1)/2) + (1 + 2n (2n − 1)/2)2m = (1 + 2n (2n − 1)/2) + 2m + 22n+m−1 − 2n+m−1 , where 1 + 2n (2n − 1)/2 and (1 + 2n (2n − 1)/2)2m are upper bounds of the number of vertices and the number of edges, respectively. Hence the computational complexity is O(22n+m−1 ). Example 4.1 Consider BCN x(t + 1) = δ4 [1, 1, 2, 2, 2, 4, 1, 1]x(t)u(t), y(t) = δ2 [1, 2, 2, 2]x(t),

(4.3)

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4 Observability of Boolean Control Networks

Fig. 4.1 Observability graph of BCN (4.3), where the number i j in each circle denotes state pair j {δ4i , δ4 }, number k beside each edge denotes input δ2k , denotes the diagonal subgraph

Fig. 4.2 Sketch of using the notion of observability graph and finite automaton to verify observability of BCNs

where t ∈ N, x(t) ∈ 4 , u(t), y(t) ∈ . The observability graph of (4.3) is depicted in Fig. 4.1.

4.3 Verifying Different Notions of Observability With the observability graph, we start to verify the above four notions of observability for BCNs one by one. The idea is simple but fundamental: (1) We are given a BCN and construct its observability graph; (2) use the graph to construct a DFA according to a specific notion of observability; and (3) use the DFA to verify observability (see Fig. 4.2). With respect to different notions of observability, different DFAs will be constructed. In order to realize the scenario shown in Fig. 4.2, we show a proposition on DFAs and the opposite statement for notions of observability. Proposition 4.1 Consider a DFA A = (S, , δ, s0 , S). Assume that each state is reachable from s0 . Then L(A) =  ∗ if and only if A is complete. Proof “if”: Since each state is final, if A is complete, then  ∈ L(A) and every nonempty word w ∈  ∗ also belongs to L(A), i.e., L(A) =  ∗ . “only if”: Assume that A is not complete. Choose an s ∈ S such that δ is not well defined at (s, a) for some a ∈ . Choose word w ∈  ∗ such that δ(s0 , w) = s, then wa ∈ / L(A) because A is deterministic. That is, L(A)   ∗ .  Proposition 4.2 A BCN (4.2) is not multiple-experiment observable (in the sense of Definition 4.1) if and only if there exist two different initial states x0 , x0 ∈  N such that for each p ∈ Z+ and each input sequence U ∈ ( M ) p , we have H x0 = H x0 p p and (H L)x0 (U ) = (H L)x  (U ). 0

Proposition 4.3 A BCN (4.2) is not strongly multiple-experiment observable (in the sense of Definition 4.2) if and only if there exists an initial state x0 ∈  N such that for each p ∈ Z+ and each input sequence U ∈ ( M ) p , there exists an initial state p p x0 ∈  N different from x0 such that H x0 = H x0 and (H L)x0 (U ) = (H L)x  (U ). 0

4.3 Verifying Different Notions of Observability

93

Proposition 4.4 A BCN (4.2) is not single-experiment observable (in the sense of Definition 4.3) if and only if for each p ∈ Z+ and each input sequence U ∈ ( M ) p , there exist two different initial states x0 , x0 ∈  N such that H x0 = H x0 and p p (H L)x0 (U ) = (H L)x  (U ). 0

Proposition 4.5 A BCN (4.2) is not arbitrary-experiment observable (in the sense of Definition 4.4) if and only if there exist two different initial states x0 , x0 ∈  N and an input sequence U ∈ ( M )N such that H x0 = H x0 and (H L)Nx0 (U ) = (H L)Nx0 (U ).

4.3.1 Verifying Multiple-Experiment Observability In order to verify Definition 4.1, we design the following Algorithm 4.1. Algorithm 4.1 Require: The observability graph Go of a BCN (4.2) and {v}, where v is a non-diagonal vertex of Go Ensure: A DFA Av 1: Put Go and {v} into Algorithm 1.1 to obtain DFA Av

Theorem 4.1 A BCN (4.2) is not multiple-experiment observable if and only if there exist two different initial states x0 and x0 both in  N such that H x0 = H x0 and the DFA A{x0 ,x0 } returned by Algorithm 4.1 recognizes language ( M )∗ , i.e., A{x0 ,x0 } is complete.4 Proof For every two different initial states x0 , x0 ∈  N satisfying H x0 = H x0 , by Proposition 4.2 and the structure of DFA A{x0 ,x0 } , a word U ∈ ( M )∗ is accepted |U | |U | by A{x0 ,x0 } if and only if (H L)x0 (U ) = (H L)x  (U ). Hence, all words that are 0 not accepted by A{x0 ,x0 } can distinguish x0 and x0 . Consequently the conclusion holds.  For a BCN (4.2), ones sees that for every two different initial states x0 and x0 both in  N , the size of the DFA A{x0 ,x0 } returned by Algorithm 4.1 is no greater than that of the observability graph of (4.2). Since at most 2n (2n − 1)/2 DFAs need to be checked and the size of the graph is O(22n+m−1 ), the computational complexity of using Theorem 4.1 to verify multiple-experiment observability is O(24n+m−2 ). Let us further consider Definition 4.1 and the observability graph Go of a BCN (4.2). After a bit more careful thinking, the notion of observability can be verified directly by using the graph. The new idea is based on the observability graph and quite intuitive. order to verify whether L(A{x0 ,x0 } ) = ( M )∗ holds, it is a very intuitive way to verify the completeness of A{x0 ,x0 } by Proposition 4.1.

4 In

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Theorem 4.2 A BCN (4.2) is multiple-experiment observable if and only if in its observability graph, for each non-diagonal vertex v that has outdegree M, there is a path from v to a vertex that has outdegree less than M. Proof “if”: By assumption, since a BCN is deterministic, one has that for each nondiagonal vertex v, the DFA Av returned by Algorithm 4.1 is not complete. Hence the BCN is multiple-experiment observable by Theorem 4.1. “only if”: If there is a non-diagonal vertex v of outdegree M such that there is no path from v to any vertex with outdegree less than M, then the DFA Av returned by Algorithm 4.1 is complete. Hence the BCN is not multiple-experiment observable by Theorem 4.1.  The condition in Theorem 4.2 can be verified in time linear of the size of the observability graph Go = (V, E, W) as follows: (1) Find the set V i, M j − Mi is a term, where Mi and M j are marking variables. c. T1 + T2 and T1 − T2 are terms if T1 and T2 are terms.

10.1 Preliminaries

199

3. Atomic Predicates. There are two types of atomic predicates, namely, transition predicates and marking predicates. a. Transition predicates. • y  (si ) < c, y  (si ) = c, and y  (si ) > c are predicates, where i >  denotes the inner product (i.e., 1, constant y ∈ ZT , constant c ∈ N, and |T | (a1 , . . . , a|T | )  (b1 , . . . , b|T | ) = i=1 ak bk ). • (s1 )(t) ≤ c and (s1 )(t) ≥ c are predicates, where constant c ∈ N, t ∈ T . b. Marking predicates. • Type 1. M( p) ≥ c and M( p) > c are predicates, where M is a marking variable and c ∈ Z is constant. • Type 2. T1 (i) = T2 ( j), T1 (i) < T2 ( j), and T1 (i) > T2 ( j) are predicates, where T1 , T2 are terms and i, j ∈ T . 4. F1 ∨ F2 and F1 ∧ F2 are predicates if F1 and F2 are predicates. A Yen’s path formula f is of the following form (with respect to Petri net (N , M0 ), where N = (P, T, Pr e, Post)): (∃M1 , . . . , Mn ∈ N P )(∃s1 , . . . , sn ∈ T ∗ )[(M0 [s1 M1 [s2 · · · [sn Mn ) ∧ F(M1 , . . . , Mn , s1 , . . . , sn )],

(10.1)

where F(M1 , . . . , Mn , s1 , . . . , sn ) is a predicate. Given a Petri net G and a Yen’s path formula f , we use G |= f to denote that f is true in G. The satisfiability problem is the problem of determining, given a Petri net G and a Yen’s path formula f , whether G |= f . A Yen’s path formula (10.1) is called increasing if F does not contain transition predicates and implies Mn ≥ M1 . When n = 1, it naturally holds Mn ≥ M1 , then in this case an increasing Yen’s path formula is (∃M1 )(∃s1 )[(M0 [s1 M1 ) ∧ F(M1 )]. The unboundedness problem can be formulated as the satisfiability of the increasing Yen’s path formula (∃M1 , M2 )(∃s1 , s2 )[(M0 [s1 M1 [s2 M2 ) ∧ (M2 > M1 )]. The coverability problem can be formulated as the satisfiability of the increasing Yen’s path formula (∃M1 )(∃s1 )[(M0 [s1 M1 ) ∧ (M1 ≥ M)], where M is the destination marking.

10.2 Notions of Detectability The same as finite-state automata (Sect. 9.1), we formulate the notions of strong detectability and weak detectability for labeled Petri nets. Definition 10.1 A labeled Petri net G is called strongly detectable if there exists a positive integer k such that for each label sequence σ ∈ Lω (G), |M(G, σ  )| = 1 for every prefix σ  of σ satisfying |σ  | > k.

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10 Detectability of Labeled Petri Nets

Definition 10.2 A labeled Petri net G is called weakly detectable if there exists a label sequence σ ∈ Lω (G) such that for some positive integer k, |M(G, σ  )| = 1 for every prefix σ  of σ satisfying |σ  | > k. Also, the same as finite-state automata, we state the following usual Assumption 3, which guarantees that a labeled Petri net will always run and always generate an infinitely long label sequence. Assumption 3 A labeled Petri net G = (N , M0 , , ) satisfies that (1) it is deadlock-free, i.e., for each M ∈ R(N , M0 ), there exists t ∈ T such that M[t ; (2) it is prompt, i.e., there exists no firing sequence M0 [s1 M1 [s2 M2 such that M1 ≤ M2 , s2 ∈ T + , and (s2 ) = . Verifying promptness of labeled Petri nets belongs to EXPSPACE (Atig and Habermehl 2009). Note that the promptness assumption is equivalent to all infinite firing sequences labeled with infinitely long label sequences. If there is a firing sequence M0 [s1 M1 [s2 M2 such that M1 ≤ M2 , s2 ∈ T + , and (s2 ) = , then this sequence can be extended to an infinite firing sequence with label (s1 ) of finite length, since M1 [s2 M2 is repetitive. On the other hand, if there is an infinite firing sequence M0 [s1 such that (s1 ) ∈  ∗ , then there exist s11 ∈ T ∗ and s12 ∈ T ω such that s1 = s11 s12 , M0 [s11 M1 [s12 , (s11 ) = (s1 ), and (s12 ) = ; by Dickson’s lemma, there exist s121 ∈ T ∗ , s122 ∈ T + , and s123 ∈ T ω such that s12 = s121 s122 s123 , M1 [s121 M2 [s122 M3 [s123 , and M2 ≤ M3 , i.e., the net is not prompt. Next we prove that it is decidable and EXPSPACE-hard to verify strong detectability under the promptness assumption, and it is undecidable to verify weak detectability for labeled Petri nets. We refer the reader to Masopust and Yin (2019) for further reading. In addition, there is an earlier undecidable result of weak detectability for labeled Petri nets with inhibitor arcs, and an undecidable result of weak approximate detectability for labeled Petri nets which are proved in Zhang and Giua (2018) by reducing the undecidable language equivalence problem (Hack 1976, Theorem 8.2) to negation of the above two problems, where weak approximate detectability is a natural generalization for weak detectability, which means that there is an infinite label sequence σ generated by a labeled Petri net such that after a time period every prefix of σ allows determining to which partition cell (of a given partition of the set of reachable markings) the current marking belongs. The result on weak detectability proved in Masopust and Yin (2019) strengthens the counterpart in Zhang and Giua (2018), as labeled Petri nets with inhibitor arcs are more general than labeled Petri nets. It is an open question whether there is a reduction from weak detectability of a labeled Petri net to weak approximate detectability of the net with respect to a given partition of the set of its reachable markings.

10.3 Decidability and Complexity of Strong Detectability

201

10.3 Decidability and Complexity of Strong Detectability In this section, we characterize strong detectability of labeled Petri nets under Assumption 3. In order to characterize strong detectability for labeled Petri nets, we introduce the concurrent composition of a labeled Petri net. Given a labeled Petri net G = (N = (P, T, Pr e, Post), M0 , , ), we construct its concurrent composition as a Petri net (10.2) G  = (N  = (P  , T  , Pr e , Post  ), M0 ) which aggregates every pair of firing sequences of G producing the same label sequence. Denote P = { p˘ 1 , . . . , p˘ |P| } and T = {t˘1 , . . . , t˘|T | }, duplicate them to Pi = i i ˘1 ˘2 ˘ } and Ti = {t˘1i , . . . , t˘|T { p˘ 1i , . . . , p˘ |P| | }, i = 1, 2, where we let (ti ) = (ti ) = (ti )  for all i in [1, |T |]. Then we specify G as follows: 1. P  = P1 ∪ P2 ; 2. T  = To ∪ T , where To = {(t˘i1 , t˘2j ) ∈ T1 × T2 |i, j ∈ [1, |T |], (t˘i1 ) = (t˘2j ) ∈ }, T = {(t˘1 , )|t˘1 ∈ T1 , (t˘1 ) = } ∪ {(, t˘2 )|t˘2 ∈ T2 , (t˘2 ) = }; 3. for all k ∈ [1, 2], all l ∈ [1, |P|], and all i, j ∈ [1, |T |] such that (t˘i1 ) = (t˘2j ) ∈ , Pr e Post





( p˘lk , (t˘i1 , t˘2j ))



( p˘lk , (t˘i1 , t˘2j ))

=  =

Pr e( p˘lk , t˘i1 ) if k = 1, Pr e( p˘lk , t˘2j ) if k = 2, Post ( p˘lk , t˘i1 ) if k = 1, Post ( p˘lk , t˘2j ) if k = 2;

4. for all l ∈ [1, |P|], all i ∈ [1, |T |] such that (t˘i1 ) = (t˘i2 ) = , Pr e ( p˘l1 , (t˘i1 , )) = Pr e( p˘l1 , t˘i1 ), Pr e ( p˘l2 , (, t˘i2 )) = Pr e( p˘l2 , t˘i2 ), Post  ( p˘l1 , (t˘i1 , )) = Post ( p˘l1 , t˘i1 ), Post  ( p˘l2 , (, t˘i2 )) = Post ( p˘l2 , t˘i2 );

5. M0 ( p˘lk ) = M0 ( p˘l ) for any k in [1, 2] and any l in [1, |P|]. A labeled Petri net and its concurrent composition are shown in Figs. 10.1 and 10.2, respectively. Assume that there exists a label sequence σ ∈ L(G) such that |M(G, σ )| > 1, then there exist transitions tμ1 , . . . , tμn , tω1 , . . . , tωn ∈ T ∪ {}, where n ≥ 1, such that (tμi ) = (tωi ) for all i ∈ [1, n], (tμ1 . . . tμn ) = (tω1 . . . tωn ) = σ , M0 [tμ1 . . . tμn M1 and M0 [tω1 . . . tωn M2 for different M1 and M2 both in N P . Then for G  , we have

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10 Detectability of Labeled Petri Nets

Fig. 10.1 A labeled Petri net G, where event a is unobservable, but b can be directly observed

Fig. 10.2 Concurrent composition of the net in Fig. 10.1

M0 [(tμ1 1 , tω21 ) . . . (tμ1 n , tω2n ) M  , where M  ( p˘lk ) = Mk ( p˘l ), k ∈ [1, 2], l ∈ [1, |P|], and M  ( p˘l1 ) = M  ( p˘l2 ) for some l  ∈ [1, |P|] (briefly denoted by M  | P1 = M  | P2 ). Assume that for each label sequence σ ∈ L(G), we have |M(G, σ )| = 1. Then for all M  ∈ R(N  , M0 ), M  ( p˘l1 ) = M  ( p˘l2 ) for each l in [1, |P|] (briefly denoted by M  | P1 = M  | P2 ). Add a new set Tφ = Tφ1 ∪ Tφ2 of transitions into G  , where φ ∈ / T1 ∪ T2 , Tφ1 = {(t˘1 , φ)|t˘1 ∈ T1 }, Tφ2 = {(φ, t˘2 )|t˘2 ∈ T2 }. Append the following rules into Pr e and Post  : for all l ∈ [1, |P|], all i ∈ [1, |T |]: Pr e ( p˘l1 , (t˘i1 , φ)) = Pr e( p˘l1 , t˘i1 ), Pr e ( p˘l2 , (φ, t˘i2 )) = Pr e( p˘l2 , t˘i2 ), Post  ( p˘l1 , (t˘i1 , φ)) = Post ( p˘l1 , t˘i1 ), Post  ( p˘l2 , (φ, t˘i2 )) = Post ( p˘l2 , t˘i2 ). The newly obtained extended concurrent composition is denoted by G  = (N  = (P  , T  , Pr e , Post  ), M0 ),

(10.3)

10.3 Decidability and Complexity of Strong Detectability

203

Fig. 10.3 Extended concurrent composition of the net in Fig. 10.1

where P  = P  , T  = T  ∪ Tφ , M0 = M0 . For a transition sequence s  ∈ (T  )∗ , we use s  (L) and s  (R) to denote its left and right components, respectively. For example, the extended concurrent composition of the net in Fig. 10.1 is shown in Fig. 10.3. Theorem 10.1 1. It is decidable to verify if a prompt labeled Petri net G is strongly detectable. 2. It is EXPSPACE-hard to verify if a labeled Petri net G is strongly detectable. Proof (1) Consider a prompt labeled Petri net G = (N = (P, T, Pr e, Post), M0 , , ), its concurrent composition G  = (N  = (P  , T  , Pr e , Post  ), M0 ) defined by (10.2), and its extended concurrent composition G  = (N  = (P  , T  , Pr e , Post  ), M0 ) defined by (10.3). We claim that G is not strongly detectable if and only if in G  , there exists a firing sequence M0 [s1 M1 [s2 M2 [s3 M3 [s4 M4 [s5 M5

(10.4a)

such that s1 , s2 , s3 ∈ (T  )∗ ; s4 , s5 ∈ (T  ∪ Tφ1 )∗ ; |s2 | > 0; |s5 | > 0;

(10.4b)

M1

(10.4c)

≤

M2 ;

M3 | P1

=

M3 | P2 ;

M4 | P1

≤

M5 | P1 .

(if:) Assume that (10.4) holds and arbitrarily choose a firing sequence (10.4a). Then for every sufficiently large n ∈ Z+ , we have a firing sequence 



M0 | P1 [s1 (L) M1 | P1 [(s2 (L))n M 2 | P1 [s3 (L) M 3 | P1 



[s4 (L) M 4 | P1 [s5 (L) M 5 | P1

204

10 Detectability of Labeled Petri Nets

of G such that 

M i | P1 = Mi | P1 + (n − 1)(M2 − M1 )| P1 , i = 2, 3, 4, 5, |M(G, (s1 (L)(s2 (L))n s3 (L)))| > 1,

(10.5a) (10.5b)

|(s2 (L))| > 0, |(s5 (L))| > 0,

(10.5c)

where (10.5c) holds because of the promptness assumption. Note that the repetitive   firing sequence M 4 | P1 [s5 (L) M 5 | P1 can be fired for infinitely many times, then net G is not strongly detectable. (only if:) Suppose that net G is not strongly detectable. Then for every n ∈ Z+ there exists a firing sequence (10.6) M0 [s1 M1 [s2 such that |(s1 )| > n, |M(G, (s1 ))| > 1, s2 ∈ T ω , and hence (s2 ) ∈  ω by the promptness assumption. Then there exists a firing sequence M0 [s1 M1 of G  such that s1 (L) = s1 , M1 | P1 = M1 = M1 | P2 . Collect all such M0 [s1 M1 , we obtain a locally  finite, infinite tree. By König’s lemma, there is an infinite path M0 [s 1 M 1 [s 2 · · ·   in the tree. By Dickson’s lemma, there exist 0 < i < j such that M i ≤ M j (hence 

 . . . s j )| > 0 by the promptness assumption). One also has that M0 [· · · M j |(s i+1 



is a path of the above tree, then there exist firing sequences M j [s  M in G  and  ˆ s M˜ in G such that M  | P1 = M  | P2 , s  ∈ (T  )∗ , Mˆ ≤ M, ˜ s˜ ∈ T + , and M | P1 [ˆs M[˜ hence |(˜s )| > 0 also by the promptness assumption. Thus, (10.4) holds. Observe that all conditions in (10.4b) and (10.4c) but M3 | P1 = M3 | P2 are predicates. Hence, (10.4) is not a Yen’s path formula. Finally, we transform the satisfiability of (10.4) to the satisfiability of a Yen’s path formula of a new Petri net. Then by Proposition 10.1, the satisfiability of (10.4) is decidable. Add two new places p0 and p1 into G  , where initially p0 contains exactly 1 token, but p1 contains no token; add one new transition r1 and arcs p0 → r1 → p1 , both with weight 1. Also, for each transition t in G  , add arcs p1 → t → p1 , both with weight 1. Then we obtain a new Petri net G  = (N  = (P  , T  , Pr e , Post  ), M0 ). We then have for G  , (10.4) holds if and only if G  satisfies the Yen’s path formula (∃M1 , M2 , M3 , M4 , M5 , M6 )(∃s1 , s2 , s3 , s4 , s5 , s6 ) [(M0 [s1 M1 [s2 M2 [s3 M3 [s4 M4 [s5 M5 [s6 M6 ) ∧ (s1 = r1 )∧ (s2 , s3 , s4 ∈ (T  )∗ ) ∧ (s5 , s6 ∈ (T  ∪ Tφ1 )∗ ) ∧ (|s3 | > 0) ∧ (|s6 | > 0) (M2 ≤ M3 ) ∧ ((M4 − M1 )| P1 = (M4 − M1 )| P2 ) ∧ (M5 | P1 ≤ M6 | P1 )]. (10.7) (2) Next we prove the hardness result by reducing the coverability problem to the non-strong detectability problem in polynomial time.

10.3 Decidability and Complexity of Strong Detectability

205

Fig. 10.4 Sketch for the reduction in the hardness proof of Theorem 10.1

We are given a Petri net G = (N = (P, T, Pr e, Post), M0 ) and a destination marking M ∈ N P , and construct a labeled Petri net G  = (N  = (P  , T  , Pr e , Post  ), M0 , T ∪ {σG }, )

(10.8)

as follows (see Fig. 10.4 as a sketch): 1. Add three places p0 , p1 , p2 , where initially p0 contains exactly one token, but p1 and p2 contain no token; 2. Add three transitions t0 , t1 , t2 , and arcs p0 → t0 → p0 , t1 → p1 , t2 → p2 , all with weight 1; for every p ∈ P, add arcs p → t1 and p → t2 , both with weight M( p); / T ∪ {t0 , t1 , t2 }, (t) = t for each t ∈ T ∪ {t0 }, (t) = σG for 3. Add label σG ∈ each t ∈ {t1 , t2 }. It is clear that if M is not covered by G then G  shown in (10.8) is strongly detectable. If M is covered by G, then there exists a firing sequence M0 [σ1 M1 with M1 ≥ M. Furthermore, there exist two infinite firing sequences M0 [(t0 )n M0 [σ1 M1 [t1 M2 [t0 M2 [t0 · · · , M0 [(t0 )n M0 [σ1 M1 [t2 M2 [t0 M2 [t0 · · · for every n ∈ Z+ , where M2 = M2 since M2 ( p1 ) > 0, M2 ( p2 ) = 0, M2 ( p2 ) > 0, M2 ( p1 ) = 0; in both sequences, after t1 , all firing transitions are t0 . Also by (t1 ) = (t2 ), we have G  is not strongly detectable. This reduction runs in time linear of the number of places of G and the number of tokens of the destination marking M. Since the coverability problem is EXPSPACE-hard in the number of transitions of G, deciding non-strong detectability is EXPSPACE-hard in the numbers of places and transitions of G  and the number of tokens of M, hence deciding strong detectability is also EXPSPACE-hard, which completes the proof. 

206

10 Detectability of Labeled Petri Nets

10.4 Decidability of Weak Detectability In this section, we characterize weak detectability of labeled Petri nets. Theorem 10.2 It is undecidable to verify if a labeled Petri net G is weakly detectable. Proof We reduce the undecidable language inclusion problem to the non-weak detectability problem (see Fig. 10.5 as a sketch). Given an -free labeled Petri net G 1 and two copies of another -free labeled Petri net G 2 , where G 1 and G 2 share the same alphabet, we effectively construct another labeled Petri net G by adding places, transitions, labels, and arcs (all with weight 1) into the three nets as follows: 1. Add place p0 with exactly 1 token; add places p1 , . . . , p6 , p4 , . . . , p6 ; 2. Add three labels x, a, b; 3. Add transitions tx1 , . . . , tx6 all with label x, add transitions ta1 , ta2 , ta3 all with label a, and add transitions tb1 , . . . , tb6 all with labels b; 4. Add arcs p0 → tx1 → p1 → tx4 → p2 → ta1 → p2 → tb1 → p3 → tb2 → p3 , p0 → tx2 → p4 → tx5 → p5 → ta2 → p5 → tb3 → p6 → tb4 → p6 , p0 → tx3 → p4 → tx6 → p5 → ta3 → p5 → tb5 → p6 → tb6 → p6 ; 5. For all transitions t of G 1 add arcs t → p1 → t, for all places p of G 1 add arcs p → ta1 , and for all transitions t of the first (resp. second) copy of G 2 add arcs t → p4 → t (resp. t → p4 → t). For net G, initially only one of the transitions tx1 , tx2 , tx3 can fire. If tx1 fires, then G 1 can run. After tx4 fires, G 1 will stop and never run again. If tx2 fires, then the first copy of G 2 can run. After tx5 fires, the first copy of G 2 will stop and never run again. The functionality of tx3 is similar to tx2 but induces the second copy of G 2 to run. Hence, one has Lω (G) ={xσ |σ ∈ Lω (G 1 ) ∪ Lω (G 2 )} n ω

∪ {xσ xa b |σ ∈ L(G 1 ), n ∈ (G 1 , σ )} ∪ {xσ xa n bω , xσ xa ω |σ ∈ L(G 2 ), n ∈ N},

(10.9a) (10.9b) (10.9c)

 where (G 1 , σ ) denotes { p∈P M( p)|M ∈ M(G 1 , σ )}, i.e., the set of sums of numbers of tokens of all places of all reachable markings of G 1 after finite label sequence σ has been observed, where P is the set of places of G 1 . Note that G 1 is -free, then (G 1 , σ ) is a finite set. Note also that only one of G 1 and the two copies of G 2 can run. Next we prove that L(G 1 ) ⊂ L(G 2 ) if and only if G is not weakly detectable. Then the weak detectability of -free labeled Petri nets is undecidable by Proposition 10.1. Assume L(G 1 ) ⊂ L(G 2 ). Consider xσ in (10.9a). If σ ∈ Lω (G 1 ), then after xσ  has been observed, where σ   σ , one has either p1 contains 1 token (G 1 runs), or p4 contains 1 token (the first copy of G 2 runs), or p4 contains 1 token (the second copy of G 2 runs). If σ ∈ Lω (G 2 ) \ Lω (G 1 ), then after xσ  has been observed, where

10.4 Decidability of Weak Detectability

207

Fig. 10.5 Sketch for the reduction in the undecidability proof of Theorem 10.2, where for each newly added transition, its label is just its subscript, reproduced from Masopust and Yin (2019) with permission @ 2019 Elsevier Ltd.

σ   σ , one has either p4 contains 1 token (the first copy of G 2 runs) or p4 contains 1 token (the second copy of G 2 runs). Consider xσ xa n bω in (10.9b) or (10.9b). If σ ∈ L(G 1 ), then after xσ xa n bm has been observed, where m ∈ Z+ , one has either p3 contains 1 token (G 1 runs), or p6 contains 1 token (the first copy of G 2 runs), or p6 contains 1 token (the second copy of G 2 runs). If σ ∈ L(G 2 ) \ L(G 1 ), then after xσ xa n bm has been observed, where m ∈ Z+ , one has either p6 contains 1 token (the first copy of G 2 runs) or p6 contains 1 token (the second copy of G 2 runs). Consider xσ xa ω in (10.9c). Note that here σ cannot belong to L(G 1 ) since G 1 is -free. After xσ xa n has been observed, where n ∈ Z+ , one has either p5 contains 1 token (the first copy of G 2 runs) or p5 contains 1 token (the second copy of G 2 runs). Hence, G is not weakly detectable. Assume L(G 1 ) ⊂ L(G 2 ). Arbitrarily choose σ ∈ L(G 1 ) \ L(G 2 ), and choose xσ xa k bω in (10.9a), where k = max{(G 1 , σ )}. Then after xσ xa k bm has been observed, where m ∈ Z+ , one has p3 contains 1 token, all places of G 1 contain no token, the two copies of G 2 are in their initial marking, and all other newly added places contain no token. Hence, G is weakly detectable. 

208

10 Detectability of Labeled Petri Nets

Notes The decidability of strong detectability for labeled Petri nets without the promptness assumption is an interesting open problem. In Zhang and Giua (2018), the weak detectability of labeled Petri nets with inhibitor arcs was proved to be undecidable by reducing the undecidable language equivalence problem (Proposition 10.1) to negation of the weak detectability problem. The reduction is as follows: given two labeled Petri nets G 1 and G 2 with the same alphabet, effectively compute a labeled Petri net G with inhibitor arcs. It was shown that if L(G 1 ) = L(G 2 ) then G is not weakly detectable. When L(G 1 ) = L(G 2 ), without loss of generality there exists σ ∈ L(G 1 ) \ L(G 2 ), an infinitely long label sequence σ  generated by G having σ as a subsequence was found such that after observing a special prefix of σ  containing σ , all places of G 1 became empty, resulting in that G is weakly detectable. It is not known whether such a reduction could be found for labeled Petri nets. However, later on in Masopust and Yin (2019), when the language inclusion problem was chosen instead of the language equivalence problem, the above idea of clearing all places of G 1 was implemented for labeled Petri nets (see the proof of Theorem 10.2) by a similar novel reduction, resulting in the stronger result (i.e., Theorem 10.2). Results on other interesting variants of notions of strong detectability and weak detectability for labeled Petri nets can be found in the manuscript (Zhang and Giua 2018).

References Atig MF, Habermehl P (2009) On Yen’s path logic for Petri nets. In: Bournez O, Potapov I (eds) Reachability problems. Springer, Berlin, pp 51–63 Cassandras CG, Lafortune S (2010) Introduction to discrete event systems, 2nd edn. Springer Publishing Company Dickson LE (1913) Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. Am J Math 35(4):413–422 Hack M (1976) Petri net languages. Technical report Cambridge, MA, USA Li Z, Zhou M (2009) Deadlock resolution in automated manufacturing systems: a novel petri net approach, 1st edn. Springer Publishing Company Lipton RJ (1976) The reachability problem requires exponential space. Yale University. Department of computer science. Research report. Department of computer science, Yale University Masopust T, Yin X (2019) Deciding detectability for labeled petri nets. Automatica 104:238–241 Mazaré L (2004) Using unification for opacity properties. Verimag Tech Rep Rackoff C (1978) The covering and boundedness problems for vector addition systems. Theor Comput Sci 6(2):223–231 Reutenauer C (1990) The mathematics of petri nets. Upper Saddle River, Prentice-Hall Inc, NJ, USA Saboori A, Hadjicostis CN (2013) Verification of initial-state opacity in security applications of discrete event systems. Inf Sci 246:115–132

References

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Seatzu C, Silva M, van Schuppen JH (eds) (2013) Control of discrete-event systems: automata and petri-net perspectives. Lecture notes in control and information sciences, vol 433. Springer, London, p 478 Tong Y et al (2017) Decidability of opacity verification problems in labeled Petri net systems. Automatica 80:48–53 Yen HC (1992) A unified approach for deciding the existence of certain petri net paths. Inf Comput 96(1):119–137 Zhang K, Giua A (2018) On detectability of labeled Petri nets and finite automata. https://arxiv.org/ abs/1802.07551 Zhang K, Giua A (2018) Weak (approximate) detectability of labeled petri net systems with inhibitor arcs. IFAC-PapersOnLine 51(7):167–171. 14th IFAC workshop on discrete event systems WODES

Part VI

Cellular Automata

Chapter 11

Generalized Reversibility of Cellular Automata

Reversibility is a fundamental property of microscopic physical systems, implied by the laws of quantum mechanics, which seems to be at odds with the Second Law of Thermodynamics (Schiff 2008; Toffoli and Margolus 1990). Nonreversibility always implies energy dissipation, in practice, in the form of heat. Using reversible cellular automata (CAs) to simulate such systems has caused wide attention since the early days of the investigation of CAs (Toffoli and Margolus 1990; Kari 2005). On the other hand, if a CA is not reversible but reversible over an invariant closed subset, e.g., the limit set (Taaki 2007), it can also be used to describe physical systems locally. In this chapter,1 we present a formal definition to represent this class of generalized reversible CAs, and investigate some of their topological properties. We refer the reader to Zhang and Zhang (2015), Taaki (2007) for further reading. Other variants of generalized reversibility can be found in Castillo-Ramirez and Gadouleau (2017). A CA is a dynamical system which consists of a regular network of finite-state automata (cells) that change their states synchronously depending on the states of their neighbors, according to a local update rule. The update rule is quite simple, but CAs show complicated behavior (Schiff 2008; Wolfram 2002).

11.1 Cellular Automata Let Z and d be the set of integers and a given positive integer, respectively. Zd denotes a d-dimensional cellular space, and elements of Zd are called cells. A finite set S that has at least two elements denotes the state set. A map c : Zd → S that specifies d the states of all cells is called a configuration (point). The symbol S Z denotes the d d set of all configurations, which is an uncountable set. A function τv : S Z → S Z 1 Theorems

11.1, 11.2, and 11.4 were reproduced from Zhang and Zhang (2015) with permission @ 2015 Old City Publishing Inc. Theorems 11.3 and 11.5 were reproduced from Taaki (2007) with permission @ 2007 Old City Publishing Inc.

© Springer Nature Switzerland AG 2020 K. Zhang et al., Discrete-Time and Discrete-Space Dynamical Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-25972-3_11

213

214

11 Generalized Reversibility of Cellular Automata

is called a translation determined by vector v ∈ Zd if for all c ∈ S Z , all u ∈ Zd , d τv (c)(u) = c(u + v). A configuration c ∈ S Z is said to be uniform if all its cells are Zd in the same state. A configuration c ∈ S is called spatially periodic if there exist linearly independent vectors v1 , . . . , vd ∈ Zd such that τv1 (c) = c, …, τvd (c) = c. It d can be seen that each open subset of S Z contains a spatially periodic configuration (this can be obtained by the topological base consisting of cylinders, see Sect. 2.5), d then the set of all spatially periodic configurations is dense in S Z . d Endowed with the following metric: for all configurations c, e ∈ S Z , d

 d(e, c) =

0 if c = e, − min{|v||v∈Zd ,c(v)=e(v)} if c = e, 2

where | · | denotes the Euclidean norm or the max-norm, which induce the same d topology, the whole configuration space S Z forms a compact, totally disconnected (that is, any two distinct points can be separated by two disjoint clopen sets) and perfect (that is, there is no isolated point) space (Proposition 1.27). A function G : d d S Z → S Z is called a CA function if G is continuous and commutes with every d d translation. In this case, (S Z , G) is called a CA. For any two CAs (S Z , G) and Zd (S , H ), the composition H ◦ G (or briefly H G) is also a CA function. It is known d d that (Proposition 2.2) G : S Z → S Z is a CA function if and only if there is a finite d ordered set N = {n 1 , . . . , n m } ⊂ Z of m distinct vectors, called neighborhood, and d a local rule g : S m → S such that for all c ∈ S Z , all n ∈ Zd , G(c)(n) = g(c(n + n 1 ), . . . , c(n + n m )). In a CA G, every initial configuration evolves under iterations of the global function G. A CA G is said to be injective (surjective), if its global function G is injective (surjective). It is said to be reversible if its global function G is bijective and the inverse function G −1 is also a CA function. All injective CAs are reversible (Proposition 2.3). For a reversible CA, every configuration has one and only one predecessor. Intuitively, in a reversible CA, each configuration can evolve both forward and backward uniquely at any time step. In Amoroso and Patt (1972), an algorithm for determining whether a given one-dimensional CA is injective or surjective was given. However, in Kari (1994), it was proved that it is undecidable whether a given two- or higher dimensional CA is injective or surjective. Hence, designing a reversible CA with certain behavior remains far from trivial.  d n Zd For a CA (S Z , G), its limit set G := ∞ n=0 G (S ) (Definition 2.4) is nonempty, closed, contains a uniform configuration, and satisfies G(G ) = G (Proposition 2.4). That is to say, the restriction of each CA on its limit set is surjective. If the limit set of a CA is finite, then it contains only one configuration (Proposition 2.4). Such CAs are called nilpotent. The nilpotency of a CA is undecidable (Kari 1992; Culik et al. 1989). The limit set G is called reached in finite time d d if G = G n (S Z ) for some n ∈ Z+ . Later on, usually we simply denote S Z =: X .

11.2 Drazin Inverses

215

11.2 Drazin Inverses The notion of Drazin inverse has applications in many areas, such as in special matrix theory, singular differential and difference equations, finite Markov chains, and graph theory (Meyer 1975; Bu et al. 2011; Zhang and Bu 2012; Wang et al. 2004). In Drazin (1958), the concept of Drazin inverse was first proposed. Definition 11.1 (Drazin 1958) Consider an associative ring (or a semigroup). Given G, H , two elements of the ring (or semigroup), H is called a Drazin inverse of G, written H = G D , if Gk H G = Gk , (11.1) G H = H G, HGH = H for some k ∈ N. The least nonnegative integer k for which these equations hold is called the index, written Ind(G), of G. By definition, if G is invertible, G D = G −1 . Note that if an element G in a semigroup S has a Drazin inverse H ∈ S, then H is a Drazin inverse of G also in every semigroup containing S, and in any subsemigroup of S that contains G and H . It is well known (proved in Drazin 1958) that any element G of any associative ring (or semigroup) has at most one Drazin inverse, and if G does have a Drazin inverse, G D commutes with any element that commutes with G. Since we will use these properties, next we write a short proof. Assume that H1 and H2 are both Drazin inverses of G. Denote k = ind(G), then one has H1 = H12 G = H1k+1 G k = H1k+1 G k+1 H2 = H1k G k H2 . Furthermore, one has H1 = H1k G k H2 = H1k−1 G k H22 = · · · = H1 G k H2k = G k H2k+1 = H2 . Let C be an element such that C G = GC, next we show C H = H C. By definition, one has H C = H k+1 G k C = H k+1 C G k = H k+1 C G k+1 H = H k+1 G k+1 C H = H k G k C H . Furthermore, H C = H k G k C H = H k−1 G k C H 2 = · · · = G k C H k+1 = C G k H k+1 = C H .

11.3 Generalized Reversibility In this section, we characterize generalized reversibility of CAs. Theorem 11.1 Consider a CA (X, F). If there exists a function H : X → X such that H is a Drazin inverse of F, then (i) F is reversible over  F , (ii)  F is reached in finite time, (iii) (X, H ) is also a CA, and (iv)  F =  H = H (X ). Proof Denote the index of F by t. By (11.1), we have F t (X ) = (F t+1 H )(X ) ⊂ F t+1 (X ) ⊂ F t (X ).

216

11 Generalized Reversibility of Cellular Automata

That is,  F = F t (X ) =: , which shows that  is reached in finite time. It is easy to verify that  is a compact subspace of X by the continuity of F and the compactness of X . For every point c ∈ , there exists a point e ∈ X such that F t (e) = c. Then we have (H | F| )(c) = (H | F| F t )(e) = (H F F t )(e) = F t (e) = c, (F| H | )(c) = (F| H | F t )(e) = (F H F t )(e) = F t (e) = c. Then F is reversible over , and H | = (F| )−1 . Since F| is continuous, by the compactness of , H | is also continuous. Furthermore, H = H F H = H (F H )t = H t+1 F t = (H | )t+1 F t is continuous. That is, (X, H ) is also a CA. Since H | is reversible, we have H (X ) = (H t+1 F t )(X ) = H t+1 () = , H 2 (X ) = H () = . Hence  H =  =  F = H (X ).



Next based on Theorem 11.1, we give an equivalent algebraic characterization for reversibility of a CA over its limit set. Theorem 11.2 Consider a CA (X, F). The following three expressions are equivalent: 1.  F is reached in finite time, and F| F is injective. 2. There exists a CA (X, H ) such that H = F D and  F =  H = H (X ). 3. There exists a function H : X → X such that H = F D . Proof (1)⇒(2): Since  F is reached in finite time, we set min{k|k ≥ 0,  F = F k (X )} =: t, and have F( F ) = F(F t (X )) = F t+1 (X ) = F t (X ) =  F . Also F| F is injective and F| F is bijective. Define H = (F| F )−(t+1) F t , then H = F D , since (F| F )−(t+1) F t F = (F| F )−(t+1) F| F F t = F| F (F| F )−(t+1) F t = F(F| F )−(t+1) F t , (F| F )−(t+1) F t F(F| F )−(t+1) F t = (F| F )

−(t+1)

F ((F| F ) t

(F| F ) F| F (F| F )

−(t+1)

t

−(t+1)

(11.2) F = (F| F ) t

−(t+1)

F, t

F )F = F . t

t

Then from Theorem 11.1, (X, H ) is a CA and  H =  F = H (X ). (2)⇒(3): Obvious. (3)⇒(1): This implication holds by Theorem 11.1.



Note that in both cases (2) and (3) of Theorem 11.2, H is unique. Finally, based on Theorems 11.1 and 11.2, we give the definition of generalized inverse CA.

11.3 Generalized Reversibility

217

Definition 11.2 Consider a CA (X, F). If F has a Drazin inverse H , then CA (X, H ) is called the generalized inverse CA of CA (X, F). Next we prove an implication relation between injectivity of F and  F being reached in finite time. Actually, this result is a generalization of one result in Culik et al. (1989): for a nilpotent CA, the limit set can be reached in finite time (the limit set for such a CA is a singleton, and hence the CA is injective over its limit set). Theorem 11.3 If a CA (X, G) is injective (i.e., bijective) over its limit set G , then G is reached in finite time. Proof One sees (by G( X ) = x , Proposition 2.4)  X ⊂ G −1 ( X ) ⊂ G −2 (G ) ⊂ · · · ⊂ X. If for some k, G −k ( X ) = G −(k+1) ( X ), then for all m > k we have G −m ( X ) = G −k ( X ). In this case, we claim that G −k ( X ) = X . Suppose on the contrary that X \ G −k ( X ) =: Y = ∅, then none of configurations in Y will go into  X . However, since Y is open, there is c ∈ Y that is spatially periodic (since the set of all spatially periodically configurations is dense). Note that such c is eventually periodic, and hence will go into  X , a contradiction. Hence, either G m (X ) =  X for some m ∈ N, or for every m ∈ N there is a configuration c−m ∈ X that enters  X after exactly m steps. Now consider a CA such that G| X is injective, we show that  X is reached in finite time. Suppose on the contrary that  X is not reached in finite time. Then for (− j) (−( j−1)) , each i ∈ N, there exist configurations ci(−i) , . . . , ci(0) such that G(ci ) = ci (− j) (0) (0) ∈ /  X for all 0 < j ≤ i. For all i ∈ N, since ci ∈  X , there exist ci ∈  X , but ci ei ∈  X such that G(ei ) = ci(0) (by Proposition 2.4), then ei = ci(−1) , and ei (0) = ci(−1) (0) without loss of generality (by using proper translations). Consider sequences {ei }i≥0 , {ci(0) }i≥0 , {ci(−1) }i≥1 , {ci(−2) }i≥2 , . . . . Since X is compact, there is a strictly increasing function μ : N → N such that all sequences (0) (−1) }i≥0 , {cμ(i) }i≥0 {eμ(i) }i≥0 , {cμ(i) converge. Denote their limits by e, c(0) , and c(−1) , respectively. Then by the continuity of G (preserving limits), we have G(e) = c(0) = G(c(−1) ). Furthermore, e ∈  X because  X is closed (by Proposition 2.4). Apparently e = c(−1) . Similarly, we can find a converging subsequence of {ci(−2) }i≥2 such that the subsequence converges to c(−2) and G(c(−2) ) = c(−1) . Repeating this procedure, we can find a sequence . . . , c(−2) , c(−1) such that G(c−( j+1) ) = c(− j) for all j ∈ Z+ . Hence, c(−1) ∈  X (again by Proposition 2.4), and G| X is not injective, a contradiction. 

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Remark 11.1 Note that G being reached in finite time does not imply the injectivity of G|G , since there are surjective CAs that are not injective, e.g., elementary CA 102, i.e., the CA specified by   {0, 1}Z , f, {−1, 0, 1} , where f (1, 1, 1) = 0, f (0, 1, 1) = 0,

f (1, 1, 0) = 1, f (0, 1, 0) = 1,

f (1, 0, 1) = 1, f (0, 0, 1) = 1,

f (1, 0, 0) = 0, f (0, 0, 0) = 0,

(01101110) is the binary representation of 102. For this CA, every configuration has exactly two predecessors. Based on Theorems 11.3 and 11.2, we can give an equivalent algebraic characterization for reversibility of a CA over its limit set. Theorem 11.4 Consider a CA (X, G). The following three expressions are equivalent: 1. G is injective over G . 2. There exists a CA (X, H ) such that H = G D and G =  H = H (X ). 3. There exists a function H : X → X such that H = G D . Proof (1) ⇒ (2): Assume that G|G is injective. From Theorem 11.3, G is reached in finite time. Then from Theorem 11.1, there is a function H : X → X such that H = G D , H is continuous, and G =  H = H (X ). Since G commutes with any translation, then H also commutes with any translation. Hence, (X, H ) is a cellular automaton. (2) ⇒ (3): Obvious. (3) ⇒ (1): This implication follows from Theorem 11.2.  On decidability of existence of a generalized inverse CA of a CA, the following result holds, which implies that it is far from trivial to design a CA that possesses a generalized inverse CA. Theorem 11.5 It is undecidable whether a d-dimensional CA G (d ≥ 1) has a generalized inverse CA, i.e., whether G is reversible over G . Proof The original proof of the undecidability of nilpotency in Kari (1992) (for d = 1, which implies undecidability for d > 1) can be used to prove the undecidability of reversibility of G over G . Consider the undecidable NW-deterministic tiling problem (proved in Kari 1992): given a partial function φ : S × S → S, determine where S is a finite set, whether

11.3 Generalized Reversibility

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the subshift φ := {c ∈ S Z |φ(x(i, j), x(i + 1, j)) = x(i, j + 1)∀i, j ∈ Z} defined by φ is empty. A partial function φ can be extended to a function 2

ψ : (S ∪ {q}) × (S ∪ {q}) → S ∪ {q} by ψ(a, b) = φ(a, b) if φ(a, b) is defined, and ψ(a, b) = q otherwise. Then, it can be seen that the one-dimensional CA G with local rule ψ is nilpotent if and only if φ is empty. If G is nilpotent, then its limit set G is a singleton (by Proposition 2.4), is reached in finite time (Culik et al. 1989), and consists of xq with xq (i) = q for all i ∈ Z, hence φ = ∅. If G is not nilpotent, then there is a configuration x in G that differs from xq . Without loss of generality, we assume that x(0) = q, then by arranging x with its predecessors, we obtain an upper half place of Z2 such that all cells in positions (i, j) ∈ Z2 with 0 ≤ i ≤ j are in states of S, i.e., it tiles a whole 1/8 plane Z2 . We then define a configuration sequence {τ(k,2k) (x)}k∈N , by 2 the compactness of S Z (by Proposition 1.27), one subsequence of {τ(k,2k) (x)}k∈N converges and hence the corresponding limit has all cells in states of S, i.e., φ = ∅. In addition, if the above CA G is nilpotent, then it is obviously reversible over its limit set. Otherwise, configuration xq ∈ G has at least two different preimages in G , one of which is xq , another of which is c0 that has all cells in state q but the cell 0. We can choose c0 (0) as follows: since φ = ∅, we find an arbitrary configuration x  ∈ φ , and set c0 (0) = x  (0, 0). Then one easily sees that there exist configurations . . . , c−2 , c−1 ∈ S Z such that G(c−( j+1) ) = c− j for all j ∈ N. Add in addition, for  each j ∈ N, c− j has all cells in state q but the cells 0, . . . , j. Hence c0 ∈ G .

Notes Results on other interesting variants of notions of generalized reversibility of CAs can be found in Castillo-Ramirez and Gadouleau (2017).

References Amoroso S, Patt YN (1972) Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures. J Comput Syst Sci 6(5):448–464 Bu C, Zhang K, Zhao J (2011) Representations of the Drazin inverse on solution of a class singular differential equations. Linear Multilinear Algebr 59(8):863–877 Castillo-Ramirez A, Gadouleau M (2017) Von Neumann regular cellular automata. In: Dennunzio Alberto et al (eds) Cellular automata and discrete complex systems. Springer International Publishing, Cham, pp 44–55 Culik K II, Pachl J, Yu S (1989) On the limit sets of cellular automata. SIAM J Comput 18(4):831– 842

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Drazin MP (1958) Pseudo-inverses in associative rings and semigroups. Am Math Mon 65(7):506– 514 Kari J (1992) The nilpotency problem of one-dimensional cellular automata. SIAM J Comput 21(3):571–586 Kari J (1994) Reversibility and surjectivity problems of cellular automata. J Comput Syst Sci 48(1):149–182 Kari J (2005) Theory of cellular automata: a survey. Theor Comput Sci 334(1):3–33 Meyer CD Jr (1975) The role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev 17(3):443–464 Schiff JL (2008) Cellular automata: a discrete view of the world, 1st edn. Wiley-Interscience Taaki S (2007) Cellular automata reversible over limit set. J Cell Autom 2:167–177 Toffoli T, Margolus NH (1990) Invertible cellular automata: a review. Phys D: Nonlinear Phenom 45(1):229–253 Wang G, Wei Y, Qiao S (2004) Generalized inverses: theory and computations. Science Press, Beijing/New York Wolfram S (2002) A new kind of science. Wolfram Media Zhang K, Bu C (2012) Group inverses of matrices over right Ore domains. Appl Math Comput 218(12):6942–6953 Zhang K, Zhang L (2015) Generalized reversibility of topological dynamical systems and cellular automata. J Cell Autom 10:425–434

Index

A Aggregated graph, 122 Alphabet, 10 Associated graph, 119

B Ball, 30 Boolean control network, 35 Boolean matrix, 9 Boolean network, 35

C Cantor set, 26 Cantor space, 29 ˇ Cech–Lebesgue covering dimension, 22 Cell, 46 Cellular automaton, 46 Clopen, 26 Compact, 19 Complexity, 16 Configuration, 26, 46 Controllability, 38 Cover, 18 Cycle, 4 Cylinder, 26

D Decidability, 16 Dependency graph, 119 Detectability, 106, 107, 167, 182, 199, 200 Detectability graph, 107 Deterministic finite automaton, 10 Deterministic Turing machine, 49

Dimension, 47 Directed graph, 4 Discrete-event system, 40 Discrete space, 3 Drazin inverse, 215 E Edge, 4 -free labeled Petri net, 44 -nondeterministic finite automaton, 14 Event, 10 F Finite-state automaton, 41 Fire, 43 Formal language, 10 Fullshift, 53 G Generalized inverse cellular automaton, 217 Genetic regulatory network, 35 H Homeomorphism, 29 Homing input sequence, 106, 112, 167 I Invertibility, 59, 65 K Khatri–Rao product, 9

© Springer Nature Switzerland AG 2020 K. Zhang et al., Discrete-Time and Discrete-Space Dynamical Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-25972-3

221

222 L Labeled Petri net, 44 Labeling function, 44 Lebesgue number, 24 Left shift, 46 Limit set, 52 Logical matrix, 9

M Marking, 43 Maximal run, 40 Mealy machine, 39 Moore machine, 39

N Node aggregation, 121 Nondeterministic finite automaton, 10 Nondeterministic finite-transition system, 39 Nonsingularity, 60, 65 Nonsingularity graph, 74

O Observability, 88–90, 146, 147 Observability graph, 91 ω-language, 44 Order, 21

P Partition, 21 Path, 4 Pattern, 30 Perfect, 19 Petri net, 43 Place/transition net, 43

Index Post-incidence function, 43 Power-reducing matrix, 9 Pre-incidence function, 43

R Regular language, 10 Resulting subnetwork, 122 Reversibility, 52 Run, 40

S Self-loop, 4 Semitensor product, 6, 37 Strong connectedness, 4 Subshift, 52 Swap matrix, 8 Symbolic space, 26

T Topological dimension, 21 Totally disconnected, 19 Totally periodic, 52 Transition, 39, 43 Transition relation, 39, 145, 166 Translation, 46

V Vertex, 4

W Weak connectedness, 4 Weighted directed graph, 4 Word, 26