Well-Quasi Orders in Computation, Logic, Language and Reasoning

1. Front Matter

   Pages i-x

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2. Well, Better and In-Between

   • Raphaël Carroy, Yann Pequignot

   Pages 1-27

3. On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders

   • Mirna Džamonja, Sylvain Schmitz, Philippe Schnoebelen

   Pages 29-54

4. 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

5. Strong WQO Tree Theorems

   • Lev Gordeev

   Pages 107-125

6. Well Quasi-orderings and Roots of Polynomials in a Hahn Field

   • Julia F. Knight, Karen Lange

   Pages 127-144

7. Upper Bounds on the Graph Minor Theorem

   • Martin Krombholz, Michael Rathjen

   Pages 145-159

8. Recent Progress on Well-Quasi-ordering Graphs

   • Chun-Hung Liu

   Pages 161-188

9. The Reverse Mathematics of wqos and bqos

   • Alberto Marcone

   Pages 189-219

10. Well Quasi-orders and the Functional Interpretation

    • Thomas Powell

    Pages 221-269

11. Well-Quasi Orders and Hierarchy Theory

    • Victor Selivanov

    Pages 271-319

12. A Combinatorial Bound for a Restricted Form of the Termination Theorem

    • Silvia Steila

    Pages 321-338

13. A Mechanized Proof of Higman’s Lemma by Open Induction

    • Christian Sternagel

    Pages 339-350

14. Well-Partial Orderings and their Maximal Order Types

    • Diana Schmidt

    Pages 351-391

This book bridges the gaps between logic, mathematics and computer science by delving into the theory of well-quasi orders, also known as wqos. This highly active branch of combinatorics is deeply rooted in and between many fields of mathematics and logic, including 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 proven to be extremely useful in computer science. 

The book introduces readers to the many facets of, and recent developments in, wqos through chapters contributed by scholars from various fields. As such, it offers a valuable asset for logicians, mathematicians and computer scientists, as well as scholars and students.

• Well Quasi-order
• Combinatorics
• Graph Theory
• Proof Theory
• Descriptive Set Theory
• Maximal Order Type
• Ordinal Notation System
• Reverse Mathematics
• Graph-minor Theorem
• Termination Proofs
• constructive mathematics
• computational content of classical proofs
• Theorem Proving and Verification
• discrete mathematics
• commutative algebra
• braid groups
• analytic combinatorics
• subrecursive hierarchies
• theory of relations
• Kriz's Theorem

• Dipartimento di Informatica, Università degli Studi di Verona, Verona, Italy

  Peter M. Schuster

• Department of Computer Science, Swansea University, Swansea, UK

  Monika Seisenberger

• Vakgroep Wiskunde, Ghent University, Ghent, Belgium

  Andreas Weiermann

Peter Schuster is an Associate Professor of Mathematical Logic at the University of Verona. After completing both his doctorate and habilitation in mathematics at the University of Munich, he was a Lecturer at the University of Leeds and member of the Leeds Logic Group. Apart from constructive mathematics in general, his principal research interests are in the computational content of classical proofs in abstract algebra and related fields, in which maximum or minimum principles are invoked.


Monika Seisenberger is an Associate Professor of Computer Science at Swansea University. After completing a PhD in the Graduate Programme “Logic in Computer Science” at the LMU Munich she took up a position as research assistant at Swansea University, where she was subsequently appointed lecturer and later programme director. Her research focuses on logic, and on theorem proving and verification.


Andreas Weiermann is a FullProfessor of Mathematics at Ghent University. After completing both his doctorate and habilitation in mathematics at the University of Münster, he held postdoctoral positions in Münster and Utrecht and became first an Associate Professor and later Full Professor in Ghent. His research interests include proof theory, theoretical computer science and discrete mathematics.

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