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) PDF
Preview 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)
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 HeinrichWansing,DepartmentofPhilosophy,RuhrUniversityBochum, Bochum,Germany EditorialBoard ArnonAvron,DepartmentofComputerScience,UniversityofTelAviv,TelAviv,Israel KatalinBimbó,DepartmentofPhilosophy,UniversityofAlberta,Edmonton,AB,Canada GiovannaCorsi,DepartmentofPhilosophy,UniversityofBologna,Bologna,Italy JanuszCzelakowski,InstituteofMathematicsandInformatics,UniversityofOpole, Opole,Poland RobertoGiuntini,DepartmentofPhilosophy,UniversityofCagliari,Cagliari,Italy RajeevGoré,AustralianNationalUniversity,Canberra,ACT,Australia AndreasHerzig,IRIT,UniversityofToulouse,Toulouse,France WesleyHolliday,UCBerkeley,Lafayette,CA,USA AndrzejIndrzejczak,DepartmentofLogic,UniversityofLódz,Lódz,Poland DanieleMundici,MathematicsandComputerScience,UniversityofFlorence,Firenze,Italy SergeiOdintsov,SobolevInstituteofMathematics,Novosibirsk,Russia EwaOrlowska,InstituteofTelecommunications,Warsaw,Poland PeterSchroeder-Heister,Wilhelm-Schickard-Institut,UniversitätTübingen,Tübingen, Baden-Württemberg,Germany YdeVenema,ILLC,UniversiteitvanAmsterdam, Amsterdam,Noord-Holland,TheNetherlands AndreasWeiermann,VakgroepZuivereWiskundeenComputeralgebra,UniversityofGhent, Ghent,Belgium FrankWolter,DepartmentofComputing,UniversityofLiverpool,Liverpool,UK MingXu,DepartmentofPhilosophy,WuhanUniversity,Wuhan,China JacekMalinowski,InstituteofPhilosophyandSociology,PolishAcademyofSciences, Warszawa,Poland AssistantEditor DanielSkurt,Ruhr-UniversityBochum,Bochum,Germany FoundingEditor RyszardWojcicki,InstituteofPhilosophyandSociology,PolishAcademyofSciences, Warsaw,Poland The book series Trends in Logic covers essentially the same areas as the journal Studia Logica, that is, contemporaryformallogicanditsapplicationsandrelationstootherdisciplines.Theseriesaimsatpublishing monographsandthematicallycoherentvolumesdealingwithimportantdevelopmentsinlogicandpresenting significantcontributionstologicalresearch. VolumesofTrendsinLogicmayrangefromhighlyfocusedstudiestopresentationsthatmakeasubject accessible to a broader scientific community or offer new perspectives for research. The series is open to contributionsdevotedtotopicsrangingfromalgebraiclogic,modeltheory,prooftheory,philosophicallogic, non-classicallogic,andlogicincomputersciencetomathematicallinguisticsandformalepistemology.This thematicspectrumisalsoreflectedintheeditorialboardofTrendsinLogic.Volumesmaybedevotedtospecific logicalsystems,particularmethodsandtechniques,fundamentalconcepts,challengingopenproblems,different approaches to logical consequence, combinations of logics, classes of algebras or other structures, or interconnectionsbetweenvariouslogic-relateddomains.Authorsinterestedinproposingacompletedbookora manuscriptinprogressorinconceptioncancontacteitherchristi.lue@springer.comoroneoftheEditorsofthe Series. Moreinformationaboutthisseriesathttp://www.springer.com/series/6645 Peter M. Schuster Monika Seisenberger (cid:129) (cid:129) 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 123 Editors PeterM. Schuster Monika Seisenberger Dipartimento di Informatica Department ofComputer Science Universitàdegli Studi diVerona SwanseaUniversity Verona,Italy Swansea, UK Andreas Weiermann Vakgroep Wiskunde GhentUniversity Ghent, Belgium ISSN 1572-6126 ISSN 2212-7313 (electronic) Trends inLogic ISBN978-3-030-30228-3 ISBN978-3-030-30229-0 (eBook) https://doi.org/10.1007/978-3-030-30229-0 ©SpringerNatureSwitzerlandAG2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart 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 orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. 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 hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface The theory of well-quasi orders, also known as wqos, is a highly active branch of combinatoricsdeeplyrootedinandbetweenmanyfieldsofmathematicsandlogic, among which there are proof theory, commutative algebra, braid groups, graph theory, analytic combinatorics, theory of relations, reverse mathematics and sub- recursive 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 develop- ments 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-partialordering and themaximal order type’attheUniversity ofHeidelberg 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 SelivanovandAndreasWeiermannfrom17to22January2016inSchlossDagstuhl. The related financial support by the ‘Deutsche Vereinigung für Mathematische LogikundfürGrundlagenforschungderexaktenWissenschaften(DVMLG)’andby ‘SchlossDagstuhl: Leibniz Zentrum fürInformatik’is gratefullyacknowledged. PartsoftheeditingprocesstookplaceduringSchuster’sparticipationintheJohn Templeton Foundation’s project ‘A New Dawn of Intuitionism: Mathematical and Philosophical Advances’ (ID: 60842). The opinions expressed in this paper are thoseoftheauthorsanddonotnecessarilyreflecttheviewsoftheJohnTempleton Foundation. v vi Preface The editors wish to thank the speakers and further participants of the afore- mentioned meetings, and the authors and referees of the chapters of the present volume. Verona, Italy Peter M. Schuster Swansea, UK Monika Seisenberger Ghent, Belgium Andreas Weiermann June 2019 Contents Well, Better and In-Between . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Raphaël Carroy and Yann Pequignot On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Mirna Džamonja, Sylvain Schmitz and Philippe Schnoebelen The Ideal Approach to Computing Closed Subsets in Well-Quasi-orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Jean Goubault-Larrecq, Simon Halfon, Prateek Karandikar, K. Narayan Kumar and Philippe Schnoebelen 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 viii 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 PeterM.Schuster isAssociateProfessorforMathematicalLogicattheUniversity ofVerona.AfterbothdoctorateandhabilitationinmathematicsattheUniversityof Munich,hewasLecturerattheUniversityofLeedsandmemberoftheLeedsLogic Group. Apart from constructive mathematics at large, his principal research inter- ests 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’attheUniversityofMunichshetookupapositionasResearchAssistantat Swansea University, and subsequently as Lecturer and Programme Director. Her research focuses on Logic as well as on theorem proving and verification. AndreasWeiermann isFullProfessorforMathematicsatGhentUniversity.After doctorate and habilitation in mathematics at the University of Münster, he held postdoctoralpositionsinMünsterandUtrechtandbecamefirstAssociateProfessor andsomewhatlaterFullProfessorinGhent.Hisresearchinterestsareprooftheory, 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 ix
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