Well Quasiorders and Hierarchy Theory

We discuss some applications of WQOs to several fields were hierarchies and reducibilities are the principal classification tools, notably to Descriptive Set Theory, Computability theory and Automata Theory. While the classical hierarchies of sets usually degenerate to structures very close to ordinals, the extension of them to functions requires more complicated WQOs, and the same applies to reducibilities. We survey some results obtained so far and discuss open problems and possible research directions.

[1]  K. Weihrauch.  The Degrees of Discontinuity of some Translators Between Representations of the Real Numbers , 1992 .

[2]  A. J. M. van Engelen,et al.  Homogeneous zero-dimensional absolute Borel sets , 1986 .

[3]  Victor L. Selivanov,et al.  Towards a descriptive set theory for domain-like structures , 2006, Theor. Comput. Sci..

[4]  Dominique Perrin,et al.  Finite Automata , 1958, Philosophy.

[5]  F. Stephan,et al.  Set theory , 2018, Mathematical Statistics with Applications in R.

[6]  Anna Papst.  Finiteness And Regularity In Semigroups And Formal Languages , 2016 .

[7]  Christian Glaßer,et al.  Efficient algorithms for membership in boolean hierarchies of regular languages , 2008, Theor. Comput. Sci..

[8]  Victor L. Selivanov.  Extending Wadge Theory to k-Partitions , 2017, CiE.

[9]  David Haussler,et al.  On Regularity of Context-Free Languages , 1983, Theor. Comput. Sci..

[10]  Victor L. Selivanov,et al.  A reducibility for the dot-depth hierarchy , 2005, Theor. Comput. Sci..

[11]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[12]  Wilfrid Hodges,et al.  Model Theory: The existential case , 1993 .

[13]  Victor L. Selivanov.  A Logical Approach to Decidability of Hierarchies of Regular Star-Free Languages , 2001, STACS.

[14]  Philippe Schnoebelen,et al.  The Height of Piecewise-Testable Languages with Applications in Logical Complexity , 2015, CSL.

[15]  Howard Becker,et al.  A characterization of jump operators , 1988, Journal of Symbolic Logic.

[16]  Philipp Schlicht,et al.  Continuous reducibility for the real line , 2012 .

[17]  Richard Laver,et al.  On Fraisse's order type conjecture , 1971 .

[18]  Victor L. Selivanov,et al.  A Gandy Theorem for Abstract Structures and Applications to First-Order Definability , 2009, CiE.

[19]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[20]  Dick H. J. Jongh,et al.  Well-partial orderings and hierarchies , 1977 .

[21]  Ludwig Staiger,et al.  Ω-languages , 1997 .

[22]  Victor L. Selivanov,et al.  Fine hierarchies via Priestley duality , 2012, Ann. Pure Appl. Log..

[23]  Nitakshi Goyal,et al.  General Topology-I , 2017 .

[24]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[25]  Victor L. Selivanov,et al.  Total Representations , 2013, Log. Methods Comput. Sci..

[26]  C. Kuratowski.  Sur une généralisation de la notion d'homéomorphie , 1934 .

[27]  Nick Bezhanishvili,et al.  Operations on a Wadge-Type Hierarchy of Ordinal-Valued Functions , 2014 .

[28]  Victor L. Selivanov,et al.  Hierarchies and reducibilities on regular languages related to modulo counting , 2009, RAIRO Theor. Informatics Appl..

[29]  Takayuki Kihara,et al.  The uniform Martin’s conjecture for many-one degrees , 2016, Transactions of the American Mathematical Society.

[30]  Paul D. Seymour,et al.  Graph Minors. XX. Wagner's conjecture , 2004, J. Comb. Theory B.

[31]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[32]  Victor L. Selivanov,et al.  Definability in the h-quasiorder of labeled forests , 2009, Ann. Pure Appl. Log..

[33]  Mizuhito Ogawa.  Well-quasi-orders and Regular ω-languages , 2018 .

[34]  William W. Wadge,et al.  Reducibility and Determinateness on the Baire Space , 1982 .

[35]  Denis R. Hirschfeldt.  Slicing the Truth - On the Computable and Reverse Mathematics of Combinatorial Principles , 2014, Lecture Notes Series / Institute for Mathematical Sciences / National University of Singapore.

[36]  V. L. Selivanov.  Hierearchies of hyperarithmetical sets and functions , 1983 .

[37]  Alexander S. Kechris,et al.  Linear algebraic groups and countable Borel equivalence relations , 2000 .

[38]  A. Wires.  Definability in the Substructure Ordering of Simple Graphs , 2016 .

[39]  Su Gao.  Invariant Descriptive Set Theory , 2008 .

[40]  C. St. J. A. Nash-Williams,et al.  On well-quasi-ordering transfinite sequences , 1965, Mathematical Proceedings of the Cambridge Philosophical Society.

[41]  V. L. Selivanov,et al.  Structure of powers of generalized index sets , 1982 .

[42]  Boris A. Trakhtenbrot,et al.  Finite automata : behavior and synthesis , 1973 .

[43]  Peter Hertling,et al.  Unstetigkeitsgrade von Funktionen in der effektiven Analysis , 1996 .

[44]  Y. Moschovakis.  Descriptive Set Theory , 1980 .

[45]  B. Mohar,et al.  Graph Minors , 2009 .

[46]  V. L. Selivanov.  Algorithmic complexity of algebraic systems , 1988 .

[47]  Luca Motto Ros,et al.  Borel-amenable reducibilities for sets of reals , 2008, The Journal of Symbolic Logic.

[48]  Victor L. Selivanov.  On the Wadge Reducibility of k-Partitions , 2008, Electron. Notes Theor. Comput. Sci..

[49]  Daniela Kühn,et al.  On well-quasi-ordering infinite trees – Nash–Williams's theorem revisited , 2001, Mathematical Proceedings of the Cambridge Philosophical Society.

[50]  Victor L. Selivanov,et al.  Definability in the Infix Order on Words , 2009, Developments in Language Theory.

[51]  Victor L. Selivanov,et al.  Fine hierarchies and m-reducibilities in theoretical computer science , 2008, Theor. Comput. Sci..

[52]  Matthew de Brecht.  Quasi-Polish spaces , 2011, Ann. Pure Appl. Log..

[53]  Christian Glaßer,et al.  The Boolean Structure of Dot-Depth One , 2001, J. Autom. Lang. Comb..

[54]  Diana Schmidt,et al.  Well-Partial Orderings and their Maximal Order Types , 2020 .

[55]  J. R. Büchi,et al.  Solving sequential conditions by finite-state strategies , 1969 .

[56]  Klaus Weihrauch,et al.  Levels of Degeneracy and Exact Lower Complexity Bounds for Geometric Algorithms , 1994, CCCG.

[57]  A. Kechris.  Classical descriptive set theory , 1987 .

[58]  Philippe Schnoebelen,et al.  Decidability in the Logic of Subsequences and Supersequences , 2015, FSTTCS.

[59]  Julia A. Knight,et al.  Computable structures and the hyperarithmetical hierarchy , 2000 .

[60]  Victor L. Selivanov,et al.  Hierarchies of Δ02‐measurable k ‐partitions , 2007, Math. Log. Q..

[61]  Chi Tat Chong,et al.  7 Classification of jump operators , 2015 .

[62]  Victor L. Selivanov,et al.  The quotient algebra of labeled forests modulo h-equivalence , 2007 .

[63]  Philipp Schlicht,et al.  Continuous reducibility and dimension of metric spaces , 2017, Arch. Math. Log..

[64]  John R. Steel.  A Classification of Jump Operator , 1982, J. Symb. Log..

[65]  J. W. Addison,et al.  Separation principles in the hierarchies of classical and effective descriptive set theory , 1958 .

[66]  Jacques Stern,et al.  Characterizations of Some Classes of Regular Events , 1985, Theor. Comput. Sci..

[67]  A. Louveau,et al.  Some results in the wadge hierarchy of borel sets , 1983 .

[68]  Theodore A. Slaman,et al.  Definable functions on degrees , 1988 .

[69]  Vasco Brattka,et al.  Effective Choice and Boundedness Principles in Computable Analysis , 2009, The Bulletin of Symbolic Logic.

[70]  Philipp Schlicht,et al.  Wadge-like reducibilities on arbitrary quasi-Polish spaces , 2012, Mathematical Structures in Computer Science.

[71]  Jacques Duparc,et al.  Wadge hierarchy and Veblen hierarchy Part I: Borel sets of finite rank , 2001, Journal of Symbolic Logic.

[72]  Howard Straubing.  Finite Automata, Formal Logic, and Circuit Complexity , 1994, Progress in Theoretical Computer Science.

[73]  Andrew Marks,et al.  The universality of polynomial time Turing equivalence , 2016, Mathematical Structures in Computer Science.

[74]  Friedhelm Meyer auf der Heide,et al.  Hierarchies , 2013, Encyclopedia of GIS.

[75]  John R. Steel,et al.  Determinateness and the separation property , 1981, Journal of Symbolic Logic.

[76]  Takayuki Kihara,et al.  On the structure of the Wadge degrees of bqo-valued Borel functions , 2019, Transactions of the American Mathematical Society.

[77]  Robert van Wesep,et al.  Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II: Wadge degrees and descriptive set theory , 1978 .

[78]  李幼升,et al.  Ph , 1989 .

[79]  Alberto Marcone,et al.  Foundations of BQO theory , 1994 .

[80]  Daisuke Ikegami.  Games in set theory and logic , 2010 .

[81]  R. McKenzie,et al.  Definability in substructure orderings, I: finite semilattices , 2009 .

[82]  R. Soare.  Recursively enumerable sets and degrees , 1987 .

[83]  Samson Abramsky,et al.  Domain theory , 1995, LICS 1995.

[84]  J. Hatzenbuhler,et al.  DIMENSION THEORY , 1997 .

[85]  J. Ježek,et al.  Definability in substructure orderings, IV: Finite lattices , 2009 .

[86]  Victor L. Selivanov.  A Fine Hierarchy of ω-Regular k-Partitions , 2011, CiE.

[87]  David Abend,et al.  Recursive Aspects Of Descriptive Set Theory , 2016 .

[88]  Dietrich Kuske,et al.  Theories of orders on the set of words , 2006, RAIRO Theor. Informatics Appl..

[89]  Philippe Schnoebelen,et al.  On the state complexity of closures and interiors of regular languages with subwords and superwords , 2014, Theor. Comput. Sci..

[90]  Victor L. Selivanov,et al.  Complexity Issues for Preorders on Finite Labeled Forests , 2011, CiE.

[91]  Andrew S. Marks,et al.  Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations , 2011, 1109.1875.

[92]  Donald A. Martin,et al.  Borel-Wadge degrees , 2003 .

[93]  C. St. J. A. Nash-Williams,et al.  On better-quasi-ordering transfinite sequences , 1968, Mathematical Proceedings of the Cambridge Philosophical Society.

[94]  Математика.  Well-quasi-ordering , 2010 .

[95]  Victor L. Selivanov,et al.  Fine hierarchies and Boolean terms , 1995, Journal of Symbolic Logic.

[96]  Victor L. Selivanov,et al.  Undecidability in the Homomorphic Quasiorder of Finite Labeled Forests , 2006, CiE.

[97]  Ramaswamy Ramanujam,et al.  Definability in First Order Theories of Graph Orderings , 2016, LFCS.

[98]  J. W. Addison.  THE METHOD OF ALTERNATING CHAINS , 2014 .

[99]  Christian Glaßer,et al.  Efficient algorithms for membership in boolean hierarchies of regular languages , 2016, Theor. Comput. Sci..

[100]  Antonio Montalbán,et al.  Computable Linearizations of Well-partial-orderings , 2007, Order.

[101]  Philippe Schnoebelen,et al.  Generalized Post Embedding Problems , 2014, Theory of Computing Systems.

[102]  Victor L. Selivanov,et al.  Towards a descriptive theory of cb0-spaces , 2014, Mathematical Structures in Computer Science.

[103]  J. Kruskal.  Well-quasi-ordering, the Tree Theorem, and Vazsonyi’s conjecture , 1960 .

[104]  Alessandro Andretta.  More on Wadge determinacy , 2006, Ann. Pure Appl. Log..

[105]  Victor L. Selivanov.  Hierarchies of [ ... ] º 2-measurable k -partitions , 2007 .

[106]  Klaus Weihrauch,et al.  Computable Analysis , 2005, CiE.

[107]  Victor L. Selivanov,et al.  Definability in the Homomorphic Quasiorder of Finite Labeled Forests , 2007, CiE.

[108]  Ádám Kunos.  Definability in the Embeddability Ordering of Finite Directed Graphs , 2015, Order.

[109]  V. Selivanov.  Boolean Hierarchies of Partitions over a Reducible Base , 2004 .

[110]  Mizuhito Ogawa.  Well-quasi-orders and regular omega-languages , 2004, Theor. Comput. Sci..

[111]  Raphaël Carroy.  A quasi-order on continuous functions , 2013, J. Symb. Log..

[112]  Y. Ershov.  A hierarchy of sets. I , 1968 .

[113]  Graham Higman,et al.  Ordering by Divisibility in Abstract Algebras , 1952 .

[114]  Victor L. Selivanov,et al.  Variations on Wadge Reducibility Extended Abstract , 2005, CCA.

[115]  Jaroslav Jezek,et al.  Definability in Substructure Orderings, II: Finite Ordered Sets , 2010, Order.

[116]  Yann Pequignot,et al.  A Wadge hierarchy for second countable spaces , 2015, Arch. Math. Log..

[117]  A N Skinner,et al.  THE UNITED STATES NAVAL OBSERVATORY. , 1899, Science.

[118]  Douglas A. Cenzer,et al.  Polynomial-Time versus Recursive Models , 1991, Ann. Pure Appl. Log..

[119]  Volker Diekert,et al.  Fragments of First-Order Logic over Infinite Words , 2009, Theory of Computing Systems.

[120]  Robert I. Soare,et al.  Recursively enumerable sets and degrees - a study of computable functions and computability generated sets , 1987, Perspectives in mathematical logic.

[121]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[122]  Parosh Aziz Abdulla,et al.  Using Forward Reachability Analysis for Verification of Lossy Channel Systems , 2004, Formal Methods Syst. Des..

[123]  Arnold W. Miller,et al.  Rigid Borel sets and better quasi-order theory , 1985 .

[124]  V. Brattka,et al.  Weihrauch Degrees, Omniscience Principles and Weak Computability , 2009 .

[125]  Wolfgang Thomas,et al.  Automata on Infinite Objects , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[126]  Jean Saint Raymond.  Preservation of the Borel class under countable-compact-covering mappings , 2007 .

[127]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[128]  Victor L. Selivanov,et al.  Undecidability in the Homomorphic Quasiorder of Finite Labelled Forests , 2007, J. Log. Comput..

[129]  Victor Selivanov,et al.  Fine Hierarchy of Regular Omega-Languages , 1995, Theor. Comput. Sci..

[130]  Ralph McKenzie,et al.  Definability in substructure orderings, III: Finite distributive lattices , 2009 .

[131]  Victor L. Selivanov,et al.  Definability in the Subword Order , 2010, CiE.

[132]  Philipp Schlicht,et al.  Borel subsets of the real line and continuous reducibility , 2019, Fundamenta Mathematicae.