Table Of ContentTrends 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,
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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
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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
