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Well-Quasi Orders in Computation, Logic, Language and Reasoning: A Unifying Concept of Proof Theory, Automata Theory, Formal Languages and Descriptive Set Theory [1st ed. 2020]
 978-3-030-30228-3, 978-3-030-30229-0

Table of contents :
Front Matter ....Pages i-x
Well, Better and In-Between (Raphaël Carroy, Yann Pequignot)....Pages 1-27
On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders (Mirna Džamonja, Sylvain Schmitz, Philippe Schnoebelen)....Pages 29-54
The Ideal Approach to Computing Closed Subsets in Well-Quasi-orderings (Jean Goubault-Larrecq, Simon Halfon, Prateek Karandikar, K. Narayan Kumar, Philippe Schnoebelen)....Pages 55-105
Strong WQO Tree Theorems (Lev Gordeev)....Pages 107-125
Well Quasi-orderings and Roots of Polynomials in a Hahn Field (Julia F. Knight, Karen Lange)....Pages 127-144
Upper Bounds on the Graph Minor Theorem (Martin Krombholz, Michael Rathjen)....Pages 145-159
Recent Progress on Well-Quasi-ordering Graphs (Chun-Hung Liu)....Pages 161-188
The Reverse Mathematics of wqos and bqos (Alberto Marcone)....Pages 189-219
Well Quasi-orders and the Functional Interpretation (Thomas Powell)....Pages 221-269
Well-Quasi Orders and Hierarchy Theory (Victor Selivanov)....Pages 271-319
A Combinatorial Bound for a Restricted Form of the Termination Theorem (Silvia Steila)....Pages 321-338
A Mechanized Proof of Higman’s Lemma by Open Induction (Christian Sternagel)....Pages 339-350
Well-Partial Orderings and their Maximal Order Types ( Diana Schmidt)....Pages 351-391

Citation preview

Trends in Logic 53

Peter M. Schuster Monika Seisenberger Andreas Weiermann   Editors

Well-Quasi Orders in Computation, Logic, Language and Reasoning A Unifying Concept of Proof Theory, Automata Theory, Formal Languages and Descriptive Set Theory

Trends in Logic Volume 53

TRENDS IN LOGIC Studia Logica Library VOLUME 53 Editor-in-Chief Heinrich Wansing, Department of Philosophy, Ruhr University Bochum, Bochum, Germany Editorial Board Arnon Avron, Department of Computer Science, University of Tel Aviv, Tel Aviv, Israel Katalin Bimbó, Department of Philosophy, University of Alberta, Edmonton, AB, Canada Giovanna Corsi, Department of Philosophy, University of Bologna, Bologna, Italy Janusz Czelakowski, Institute of Mathematics and Informatics, University of Opole, Opole, Poland Roberto Giuntini, Department of Philosophy, University of Cagliari, Cagliari, Italy Rajeev Goré, Australian National University, Canberra, ACT, Australia Andreas Herzig, IRIT, University of Toulouse, Toulouse, France Wesley Holliday, UC Berkeley, Lafayette, CA, USA Andrzej Indrzejczak, Department of Logic, University of Lódz, Lódz, Poland Daniele Mundici, Mathematics and Computer Science, University of Florence, Firenze, Italy Sergei Odintsov, Sobolev Institute of Mathematics, Novosibirsk, Russia Ewa Orlowska, Institute of Telecommunications, Warsaw, Poland Peter Schroeder-Heister, Wilhelm-Schickard-Institut, Universität Tübingen, Tübingen, Baden-Württemberg, Germany Yde Venema, ILLC, Universiteit van Amsterdam, Amsterdam, Noord-Holland, The Netherlands Andreas Weiermann, Vakgroep Zuivere Wiskunde en Computeralgebra, University of Ghent, Ghent, Belgium Frank Wolter, Department of Computing, University of Liverpool, Liverpool, UK Ming Xu, Department of Philosophy, Wuhan University, Wuhan, China Jacek Malinowski, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warszawa, Poland Assistant Editor Daniel Skurt, Ruhr-University Bochum, Bochum, Germany Founding Editor Ryszard Wojcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland The book series Trends in Logic covers essentially the same areas as the journal Studia Logica, that is, contemporary formal logic and its applications and relations to other disciplines. The series aims at publishing monographs and thematically coherent volumes dealing with important developments in logic and presenting significant contributions to logical research. Volumes of Trends in Logic may range from highly focused studies to presentations that make a subject accessible to a broader scientific community or offer new perspectives for research. The series is open to contributions devoted to topics ranging from algebraic logic, model theory, proof theory, philosophical logic, non-classical logic, and logic in computer science to mathematical linguistics and formal epistemology. This thematic spectrum is also reflected in the editorial board of Trends in Logic. Volumes may be devoted to specific logical systems, particular methods and techniques, fundamental concepts, challenging open problems, different approaches to logical consequence, combinations of logics, classes of algebras or other structures, or interconnections between various logic-related domains. Authors interested in proposing a completed book or a manuscript in progress or in conception can contact either [email protected] or one of the Editors of the Series. More information about this series at http://www.springer.com/series/6645

Peter M. Schuster Monika Seisenberger Andreas Weiermann •



Editors

Well-Quasi Orders in Computation, Logic, Language and Reasoning A Unifying Concept of Proof Theory, Automata Theory, Formal Languages and Descriptive Set Theory

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Editors Peter M. Schuster Dipartimento di Informatica Università degli Studi di Verona Verona, Italy

Monika Seisenberger Department of Computer Science Swansea University Swansea, UK

Andreas Weiermann Vakgroep Wiskunde Ghent University Ghent, Belgium

ISSN 1572-6126 ISSN 2212-7313 (electronic) Trends in Logic ISBN 978-3-030-30228-3 ISBN 978-3-030-30229-0 (eBook) https://doi.org/10.1007/978-3-030-30229-0 © 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

The theory of well-quasi orders, also known as wqos, is a highly active branch of combinatorics deeply rooted in and between many fields of mathematics and logic, among which there are proof theory, commutative algebra, braid groups, graph theory, analytic combinatorics, theory of relations, reverse mathematics and subrecursive hierarchies. As a unifying concept for slick finiteness or termination proofs, wqos have been rediscovered in diverse contexts, and turned out utmost useful in computer science. With this volume we intend to display the many facets of and recent developments about wqos, through chapters written by scholars from different areas. Last but not least we thus wish to transfer knowledge between different areas of logic, mathematics and computer science. A special highlight of the present volume is Diana Schmidt’s habilitation thesis ‘Well-partial ordering and the maximal order type’ at the University of Heidelberg from 1979. Since publication this thesis has been extremely influential but never published, not even in parts. This volume grew out of the following two meetings: the minisymposium ‘Well-quasi orders: from theory to applications’ organised by Peter Schuster, Monika Seisenberger and Andreas Weiermann within the ‘Jahrestagung 2015 der Deutschen Mathematiker-Vereinigung (DMV)’ from 21 to 25 September 2015 in Hamburg, and the Dagstuhl Seminar 16031 ‘Well Quasi-Orders in Computer Science’ organised by Jean Goubault-Larrecq, Monika Seisenberger, Victor Selivanov and Andreas Weiermann from 17 to 22 January 2016 in Schloss Dagstuhl. The related financial support by the ‘Deutsche Vereinigung für Mathematische Logik und für Grundlagenforschung der exakten Wissenschaften (DVMLG)’ and by ‘Schloss Dagstuhl: Leibniz Zentrum für Informatik’ is gratefully acknowledged. Parts of the editing process took place during Schuster’s participation in the John Templeton Foundation’s project ‘A New Dawn of Intuitionism: Mathematical and Philosophical Advances’ (ID: 60842). The opinions expressed in this paper are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

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The editors wish to thank the speakers and further participants of the aforementioned meetings, and the authors and referees of the chapters of the present volume. Verona, Italy Swansea, UK Ghent, Belgium June 2019

Peter M. Schuster Monika Seisenberger Andreas Weiermann

Contents

Well, Better and In-Between . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Raphaël Carroy and Yann Pequignot On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mirna Džamonja, Sylvain Schmitz and Philippe Schnoebelen The Ideal Approach to Computing Closed Subsets in Well-Quasi-orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean Goubault-Larrecq, Simon Halfon, Prateek Karandikar, K. Narayan Kumar and Philippe Schnoebelen

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Strong WQO Tree Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Lev Gordeev Well Quasi-orderings and Roots of Polynomials in a Hahn Field . . . . . . 127 Julia F. Knight and Karen Lange Upper Bounds on the Graph Minor Theorem . . . . . . . . . . . . . . . . . . . . 145 Martin Krombholz and Michael Rathjen Recent Progress on Well-Quasi-ordering Graphs . . . . . . . . . . . . . . . . . . 161 Chun-Hung Liu The Reverse Mathematics of wqos and bqos . . . . . . . . . . . . . . . . . . . . . 189 Alberto Marcone Well Quasi-orders and the Functional Interpretation . . . . . . . . . . . . . . . 221 Thomas Powell Well-Quasi Orders and Hierarchy Theory . . . . . . . . . . . . . . . . . . . . . . . 271 Victor Selivanov A Combinatorial Bound for a Restricted Form of the Termination Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Silvia Steila vii

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Contents

A Mechanized Proof of Higman’s Lemma by Open Induction . . . . . . . . 339 Christian Sternagel Well-Partial Orderings and their Maximal Order Types . . . . . . . . . . . . 351 Diana Schmidt

Editors and Contributors

About the Editors Peter M. Schuster is Associate Professor for Mathematical Logic at the University of Verona. After both doctorate and habilitation in mathematics at the University of Munich, he was Lecturer at the University of Leeds and member of the Leeds Logic Group. Apart from constructive mathematics at large, his principal research interests are about the computational content of classical proofs in abstract algebra and related fields in which maximum or minimum principles are invoked. Monika Seisenberger is Associate Professor in Computer Science at Swansea University. After her doctorate in the Graduate Programme ‘Logic in Computer Science’ at the University of Munich she took up a position as Research Assistant at Swansea University, and subsequently as Lecturer and Programme Director. Her research focuses on Logic as well as on theorem proving and verification. Andreas Weiermann is Full Professor for Mathematics at Ghent University. After doctorate and habilitation in mathematics at the University of Münster, he held postdoctoral positions in Münster and Utrecht and became first Associate Professor and somewhat later Full Professor in Ghent. His research interests are proof theory, theoretical computer science and discrete mathematics.

Contributors Raphaël Carroy University of Vienna, Wien, Austria Mirna Džamonja University of East Anglia, Norwich, UK Lev Gordeev Universität Tübingen, Ghent University, Ghent, Belgium

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Editors and Contributors

Jean Goubault-Larrecq LSV, ENS Paris-Saclay, Université Paris-Saclay, Cachan, France Simon Halfon LSV, ENS Paris-Saclay, Université Paris-Saclay, Cachan, France Prateek Karandikar LSV, ENS Paris-Saclay, Université Paris-Saclay, Cachan, France; Chennai Mathematical Institute, Chennai, India; CNRS UMI ReLaX, Chennai, India Julia F. Knight Department of Mathematics, University of Notre Dame, Notre Dame, IN, USA Martin Krombholz School of Mathematics, University of Leeds, Leeds, UK Karen Lange Department of Mathematics, Wellesley College, Wellesley, MA, USA Chun-Hung Liu Department of Mathematics, Texas A&M University, College Station, TX, USA; Department of Mathematics, Princeton University, Princeton, NJ, USA Alberto Marcone Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, Udine, Italy K. Narayan Kumar Chennai Mathematical Institute, Chennai, India; CNRS UMI ReLaX, Chennai, India Yann Pequignot University of Vienna, Wien, Austria Thomas Powell Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany Michael Rathjen School of Mathematics, University of Leeds, Leeds, UK Diana Schmidt Emeritus, Hochschule Heilbronn, Heilbronn, Germany Sylvain Schmitz IRIF, Université de Paris & CNRS, Paris, France Philippe Schnoebelen LSV, CNRS, ENS Paris-Saclay, Université Paris-Saclay, Cachan, France; CNRS UMI ReLaX, Chennai, India Victor Selivanov A.P. Ershov Institute of Informatics Systems SB RAS and Kazan Federal University, Novosibirsk, Russia Silvia Steila Institut für Informatik, Universität Bern, Bern, Switzerland Christian Sternagel University of Innsbruck, Innsbruck, Austria

Well, Better and In-Between Raphaël Carroy and Yann Pequignot

Abstract Starting from well-quasi-orders (wqos), we motivate step by step the introduction of the complicated notion of better-quasi-order (bqo). We then discuss the equivalence between the two main approaches to defining bqo and state several essential results of bqo theory. After recalling the rôle played by the ideals of a wqo in its bqoness, we give a new presentation of known examples of wqos which fail to be bqo. We also provide new forbidden pattern conditions ensuring that a quasi-order is a better quasi-order.

It is the variety of these applications, rather than any depth in the results obtained, that suggests that the theorems may be interesting. Graham Higman [13]

While studying a generalization of the partial order of divisibility on the natural numbers for an abstract algebra, Higman [13] identified the following desirable property for a quasi-order (qo). A qo has the finite basis property if every upwards closed subset is the upward closure of a finite subset. He notices that this property is equivalent to that defining a well-quasi-order (wqo): being well-founded and having no infinite antichains. Higman proves the following essential fact: in order to be wqo it suffices to be generated by means of finitary operations from a wqo. He then proceeds to apply his theorem to solve a problem posed by Erd˝os, to provide a new proof of a theorem on power-series ring and also to the study of fully invariant subgroups of a free group. These were only the first instances of a long series of applications of this result that became known as Higman’s Theorem. Pouzet [28] later commented on the possibilities and the limitations of that fruitful approach:

Raphaël Carroy was supported by FWF Grants P28153 and P29999. Yann Pequignot gratefully acknowledges the support of the Swiss National Science Foundation (SNF) through grant P2L A P2_164904. R. Carroy (B) · Y. Pequignot University of Vienna, Oskar-Morgenstern-Platz 1,1090 Wien, Austria e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. M. Schuster et al. (eds.), Well-Quasi Orders in Computation, Logic, Language and Reasoning, Trends in Logic 53, https://doi.org/10.1007/978-3-030-30229-0_1

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R. Carroy and Y. Pequignot In order to show that a certain class of posets (finite or infinite) is wqo, one tries first to see if the class can be constructed from some simpler class by means of some operations. If these operations are finitary, then it is possible that Higman’s theorem can be applied. However, for infinite posets, these operations may very well also be infinitary and then there is no possibility of applying Higman’s theorem since the obvious generalization of this is false for infinitary operations.

Pouzet here refers to the fact that well-founded quasi-orders, as well as wqos, lack closure under certain infinitary operations as first proven by Rado [31]. This is explained in detail in Sect. 1 where we point out how it opens the way to the definition by Alvah Nash-Williams [1] of the concept of better-quasi-order (bqo): a stronger property than wqo which allows for an infinitary analogue to Higman’s Theorem. We first provide a gentle introduction to the original definition of Nash-Williams, before presenting the more concise definition introduced by Simpson [33]. We use some new terminology with the hope that it makes it easier for the unacquainted reader to appreciate the respective advantages of these two complementary approaches to defining bqo. To offer a few more words of introduction about this intriguing concept, we briefly comment on the emblematic case of the quasi-order LINℵ0 of countable linear orders, equipped with the relation of embeddability. Fraïssé [10] conjectured that LINℵ0 was well-founded, but the statement that became known as Fraïssé’s Conjecture (FRA) is that LINℵ0 is wqo; it follows from the famous theorem of Laver [16] that LINℵ0 is in fact bqo. The reason for the use of the concept of bqo in Laver’s proof of FRA was already alluded to in the above quote by Pouzet. While using Hausdorff’s analysis of scattered linear orders and proceeding by induction is a very reasonable way to tackle FRA, the operations underlying this analysis are infinitary and this is a main obstacle when working with wqos alone. This masterly use by Laver of the concept of bqo introduced by Nash-Williams inspired many other delightful results. But however successful this story is, it raises at least two questions. Firstly, one may ask if the use of the concept of bqo in the proof of FRA is in a sense necessary. In the framework of Reverse Mathematics, one can formalize this question by asking for the exact proof-theoretic strength of FRA. The answer is still unknown despite many efforts, but important results have already been obtained (see [21] and more recently [23]). Secondly, one may ask if this strategy for proving wqoness always works. On the one hand, many other quasi-orders were proved to be bqo in the subsequent years attesting to the effectiveness of this concept (see Sect. 3.3). On the other hand, there does exist a large range of examples of wqo that are not bqo (see Sect. 4). Nevertheless, these examples appear to have a somehow artificial flavor since as Kruskal [14, p. 302] observed in his very nice historical introduction to wqo: “all ‘naturally occurring’ wqo sets which are known are bqo”.1 In a quest towards a deeper understanding of the discrepancy between wqo and bqo, we mention a result of ours on the role played by the ideals of a wqo in it being minor relations on finite graphs, proved to be wqo by Robertson and Seymour [32], is to our knowledge the only naturally occurring wqo which is not yet known to be bqo.

1 The

Well, Better and In-Between

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bqo (see Theorem 3.23). We show the relevance of this rather singular theorem by giving two applications of it. First, we use it in Sect. 4 to give a new presentation of examples of wqos that, while failing to be bqos, still enjoy stronger and stronger properties. Finally, we use this theorem to give in Sect. 5 some new conditions under which the two notions of wqo and bqo coincide.

1 Well Is Not Good Enough In the sequel, (Q, ≤ Q ) always stands for a quasi-order, qo for short, i.e. a reflexive and transitive relation ≤ Q on a non-empty set Q. An antisymmetric qo is a partial order, or po. A sequence (qn )n∈ω in Q is bad if and only if for all integers m and n such that m < n we have qm  Q qn . The strict quasi-order associated to ≤ Q is defined by p < Q q if and only if p ≤ Q q and q  Q p. We say that Q is well-founded if there is no infinite descending chain in Q, i.e. no sequence (qn )n such that qn+1 < Q qn for every n. An antichain in Q is a subset A of Q consisting of pairwise ≤ Q -incomparable elements, i.e. p = q implies p  Q q for every p, q ∈ A. A subset D of Q is a called a downset, if q ∈ D and p ≤ Q q implies p ∈ D. For any S ⊆ Q, we write ↓ S for the downset generated by S in Q, i.e. the set {q ∈ Q | ∃ p ∈ S q ≤ Q p}. We also write ↓ p for ↓{ p}. Finally we denote by D(Q) the po of downsets of Q under inclusion. We start by proving the equivalence between three of the main characterizations of a wqo. Proposition 1.1 A quasi-order (Q, ≤ Q ) is a wqo if and only if one of the following equivalent conditions is fulfilled: 1. there is no bad sequence in (Q, ≤ Q ), 2. (Q, ≤ Q ) is well-founded and contains no infinite antichain, 3. (D(Q), ⊆) is well-founded. Proof Item 2 ↔ Item 1 Notice that an infinite descending chain and a countably infinite antichain are both special cases of a bad sequence. Conversely if (qn )n is a bad sequence in Q, then using Ramsey’s theorem we obtain either an infinite descending chain or an infinite antichain. Item 1 → Item 3 By contraposition, suppose that (Dn )n∈ω is an infinite descending chain inside (D(Q), ⊆). Then for each n ∈ ω we can pick some qn ∈ Dn \Dn+1 . Then n → qn is a bad sequence in Q. To see this, suppose towards a contradiction that for m < n we have qm ≤ qn . As qn ∈ Dn and Dn is a downset, we have qm ∈ Dn . But since Dn ⊂ Dm+1 , we have qm ∈ Dm+1 , a contradiction with the choice of qm .

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Item 3 → Item 1 By contraposition, suppose that (qn )n∈ω is a bad sequence in Q. Set Dn = ↓{qk | n ≤ k}, then (Dn )n is a descending chain in D(Q). Indeed for every n we clearly have Dn+1 ⊆ Dn and since the sequence is bad, k ≥ n + 1 implies qn  qk and so qn ∈ Dn while qn ∈ / Dn+1 .  After Proposition 1.1, it is natural to ask if being well-founded and being wqo is actually equivalent for the partial order of downsets of any quasi-order. The answer is negative and the first example of a wqo with an antichain of downsets was identified by Richard Rado. Example 1.2 ([31]) Rado’s partial order R is the set [ω]2 of pairs of natural numbers, partially ordered by (cf. Fig. 1): {m, n} ≤R {m  , n  }

 m = m  and n ≤ n  , or n < m.

←→

where by convention a pair {m, n} of natural numbers is always assumed to be written in increasing order (m < n). The po R is wqo. To see this, consider any map f : ω → [ω]2 and let f (n) = { f 0 (n), f 1 (n)} for all n ∈ ω. Now if f 0 is unbounded, then there exists n > 0 with f 1 (0) < f 0 (n) and so f (0) ≤R f (n) by the second clause. So f is good in this case. Next if f 0 is bounded, then by going to a subsequence we can assume that f 0 is constantly equal to some k. But then the restriction of R to the pairs {k, n 1 } is simply ω which is wqo, so f must be good in this case too. Fig. 1 Rado’s poset R

.. .

.. .

.. .

.. .

.. .

.. .

.. .













































(0, 3)

• (1, 3) • (2, 3) •

(0, 2)

• (1, 2) •

(0, 1)



D3

..

.

Well, Better and In-Between

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However the map n → Dn = ↓{{n, l} | n < l} is a bad sequence (in fact an infinite antichain) inside D(R). Indeed whenever m < n we have {m, n} ∈ Dm while {m, n} ∈ / Dn , and so Dm  Dn . Suppose we want to make sure that D(Q) is wqo. What condition on Q could ensure this? In other words, what phenomenon are we to exclude inside Q in order to rule out the existence of antichains inside D(Q)? Forbidding bad sequences in Q is certainly not enough, as shown by the existence of Rado’s example. But here is what we can do. Suppose that (Pn )n∈ω is a bad sequence in D(Q). Fix some m ∈ ω. Then whenever m < n we have Pm  Pn and we can choose a witness q ∈ Pm \Pn . In general, no single q ∈ Pm can witness that Pm  Pn for all n > m, so we have to pick a whole sequence f m : ω/m → Q, n → qmn of witnesses2 : / Pn , n ∈ ω/m. qmn ∈ Pm and qmn ∈ In this way we get a sequence f 0 , f 1 , . . . of sequences which is advantageously viewed as single map from [ω]2 : f : [ω]2 −→ Q {m, n} −→ f m (n) = qmn . By our choices this sequence of sequences satisfies the following condition: ∀m, n, l ∈ ω m < n < l → qmn  qnl . To see this, suppose towards a contradiction that for m < n < l we have qmn ≤ qnl . Since qnl ∈ Pn which is a downset, we would have qmn ∈ Pn , but we chose qmn such / Pn . that qmn ∈ Let us say that a sequence of sequences f : [ω]2 → Q is bad if for every m, n, l ∈ ω, m < n < l implies f ({m, n})  f ({n, l}). We have found the desired condition. Proposition 1.3 Let Q be a qo. Then D(Q) is wqo if and only if there is no bad sequence of sequences in Q. Proof As we have seen in the preceding discussion, if D(Q) is not wqo then from a bad sequence in D(Q) we can define a bad sequence of sequences in Q. Conversely, if f : [ω]2 → Q is a bad sequence of sequences, then for each m ∈ ω we can consider the set Pm = { f ({m, n}) | n ∈ ω/m} consisting of the image of the m th sequence. Then the sequence m → ↓ Pm in D(Q) is a bad sequence. Indeed / ↓ Pn , since otherwise every time m < n we have f ({m, n}) ∈ Pm while f ({m, n}) ∈ there would exist l > n with f ({m, n}) ≤ f ({n, l}), a contradiction with the fact that f is a bad sequence of sequences.  2 where

ω/m denotes the set {n ∈ ω | m < n}.

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Notice that in the case of Rado’s partial order R, the identity map itself is a bad sequence of sequences witnessing that D(R) is not wqo, since every time m < n < l then {m, n} R {n, l}. This example is actually minimal in the following sense; if Q is wqo but D(Q) is not wqo, then R embeds into Q, as proved by Laver [17]. For now, let us just say that a better-quasi-order is a quasi-order Q such that D(Q) is well-founded, D(D(Q)) is well founded, D(D(D(Q))) is well-founded, so on and so forth into the transfinite. While this idea can be formalized, we can already see that it cannot serve as a convenient definition.3 In the next Section, we introduce the super-sequences and the multi-sequences which are two equivalent way of generalizing the idea of sequence of sequences into the transfinite. This allows us to define better-quasi-orders in Sect. 3.

2 Super-Sequences Versus Multi-sequences As the preceding section suggests, we are going to define a better-quasi-order as a quasi-order with no bad sequence, with no bad sequence of sequences, no bad sequence of sequences of sequences, so on and so forth, into the transfinite. In order to formalize this idea, we need a convenient notion of “index set” for a sequence of sequences of … of sequences, in short a super-sequence. We first describe the original combinatorial approach of Nash-Williams, before presenting the more condensed topological definition due to Simpson. Let us first fix some notations. We adopt the set-theoretic convention that m ∈ n for all natural numbers m < n. A sequence is a map from an initial segment of ω to some non-empty set, a finite sequence s has domain an integer n, also called the length of s and denoted by |s|. When i < |s|, s(i) stands for the i-th element of the sequence s. If A is a non-empty set, A