Fine hierarchies and m-reducibilities in theoretical computer science

This is a survey of results about versions of fine hierarchies and many-one reducibilities that appear in different parts of theoretical computer science. These notions and related techniques play a crucial role in understanding complexity of finite and infinite computations. We try not only to present the corresponding notions and facts from the particular fields but also to identify the unifying notions, techniques and ideas.

[1]  Edith Hemaspaandra,et al.  What's up with downward collapse: using the easy-hard technique to link Boolean and polynomial hierarchy collapses , 1998, SIGA.

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

[3]  Klaus W. Wagner Leaf Language Classes , 2004, MCU.

[4]  Eric Allender,et al.  Reducibility and Completeness , 2010, Algorithms and Theory of Computation Handbook.

[5]  Olivier Finkel,et al.  An Effective Extension of the Wagner Hierarchy to Blind Counter Automata , 2001, CSL.

[6]  Victor L. Selivanov,et al.  A Useful Undecidable Theory , 2007, CiE.

[7]  Thomas Wilke,et al.  Computing the Wadge Degree, the Lifschitz Degree, and the Rabin Index of a Regular Language of Infinite Words in Polynomial Time , 1995, TAPSOFT.

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

[9]  Eric Allender,et al.  Complexity , 2007, Scholarpedia.

[10]  V. L. Selivanov Fine hierarchy and definable index sets , 1991 .

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

[12]  Melven R. Krom Separation Principles in the Hierarchy Theory of Pure First-Order Logic , 1963, J. Symb. Log..

[13]  José L. Balcázar,et al.  Structural Complexity I , 1995, Texts in Theoretical Computer Science An EATCS Series.

[14]  Klaus W. Wagner Eine topologische Charakterisierung einiger Klassen regulärer Folgenmengen , 1977, J. Inf. Process. Cybern..

[15]  Wolfgang Thomas,et al.  Classifying Regular Events in Symbolic Logic , 1982, J. Comput. Syst. Sci..

[16]  Klaus W. Wagner,et al.  The Boolean Hierarchy over Level 1/2 of the Straubing-Therien Hierarchy , 1998, ArXiv.

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

[18]  P. Odifreddi Classical recursion theory , 1989 .

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

[20]  S. Lempp Hyperarithmetical index sets in recursion theory , 1987 .

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

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

[23]  Klaus Weihrauch,et al.  Computable Analysis: An Introduction , 2014, Texts in Theoretical Computer Science. An EATCS Series.

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

[25]  J. R. Büchi,et al.  The monadic second order theory of all countable ordinals , 1973 .

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

[27]  Heribert Vollmer,et al.  On Balanced vs . Unbalanced Computation Trees , 2007 .

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

[29]  V. L. Selivanov Structures of the degrees of unsolvability of index sets , 1979 .

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

[31]  N. Vereshchagin RELATIVIZABLE AND NONRELATIVIZABLE THEOREMS IN THE POLYNOMIAL THEORY OF ALGORITHMS , 1994 .

[32]  O. H. Lowry Academic press. , 1972, Analytical chemistry.

[33]  Olivier Finkel,et al.  Borel ranks and Wadge degrees of context free $\omega$-languages , 2006, Mathematical Structures in Computer Science.

[34]  Victor L. Selivanov Fine Hierarchy of Regular omega-Languages , 1995, TAPSOFT.

[35]  Dung T. Huynh,et al.  Finite-Automaton Aperiodicity is PSPACE-Complete , 1991, Theor. Comput. Sci..

[36]  Joseph B. Kruskal,et al.  The Theory of Well-Quasi-Ordering: A Frequently Discovered Concept , 1972, J. Comb. Theory A.

[37]  D. C. Cooper,et al.  Theory of Recursive Functions and Effective Computability , 1969, The Mathematical Gazette.

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

[39]  J. Richard Büchi,et al.  The monadic second order theory of ω1 , 1973 .

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

[41]  Victor L. Selivanov,et al.  Fine Hierarchy of Regular Aperiodic omega -Languages , 2007, Developments in Language Theory.

[42]  A. I. Mal'cev Algorithms and Recursive Functions , 1970 .

[43]  Olivier Finkel,et al.  Topology and Ambiguity in Omega Context Free Languages , 2008, ArXiv.

[44]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

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

[46]  Victor L. Selivanov,et al.  Wadge degrees of ω-languages of deterministic Turing machines , 2003 .

[47]  Peter Hertling,et al.  Topological Complexity with Continuous Operations , 1996, J. Complex..

[48]  Gerd Wechsung,et al.  On the Boolean closure of NP , 1985, FCT.

[49]  Bernd Borchert,et al.  On the Acceptance Power of Regular Languages , 1994, Theor. Comput. Sci..

[50]  R. Epstein,et al.  Hierarchies of sets and degrees below 0 , 1981 .

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

[52]  Heribert Vollmer,et al.  Lindström Quantifiers and Leaf Language Definability , 1996, Int. J. Found. Comput. Sci..

[53]  A. Tang Chain Properties in P omega , 1979, Theor. Comput. Sci..

[54]  Victor L. Selivanov,et al.  Complexity of Topological Properties of Regular omega-Languages , 2008, Fundam. Informaticae.

[55]  Albert R. Meyer,et al.  The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space , 1972, SWAT.

[56]  Lane A. Hemaspaandra,et al.  Query Order , 1998, SIAM J. Comput..

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

[58]  V. L. Selivanov Hierarchy of limiting computations , 1984 .

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

[60]  Hilary Putnam,et al.  Trial and error predicates and the solution to a problem of Mostowski , 1965, Journal of Symbolic Logic.

[61]  Jean-Éric Pin,et al.  Logic on Words , 2001, Bull. EATCS.

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

[63]  Paul Gastin,et al.  First-order definable languages , 2008, Logic and Automata.

[64]  Victor L. Selivanov Two Refinements of the Polynomial Hierarcht , 1994, STACS.

[65]  Victor L. Selivanov,et al.  On the Wadge Reducibility of k-Partitions , 2008, CCA.

[66]  Heribert Vollmer,et al.  On the power of number-theoretic operations with respect to counting , 1995, Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference.

[67]  Raymond E. Miller,et al.  Varieties of Formal Languages , 1986 .

[68]  Frank Stephan,et al.  The dot-depth and the polynomial hierarchies correspond on the delta levels , 2005, Int. J. Found. Comput. Sci..

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

[70]  V. L. Selivanov Refining the polynomial hierarchy , 1999 .

[71]  Thomas Schwentick,et al.  On the power of polynomial time bit-reductions , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[72]  Jean Saint Raymond,et al.  Les propriétés de réduction et de norme pour les classes de Boréliens , 1988 .

[73]  Christian Glaßer,et al.  Languages polylog-time reducible to dot-depth 1/2 , 2007, J. Comput. Syst. Sci..

[74]  Ludwig Staiger,et al.  Automatentheoretische und automatenfreie Charakterisierungen topologischer Klassen regulärer Folgenmengen , 1974, J. Inf. Process. Cybern..

[75]  R. McNaughton,et al.  Counter-Free Automata , 1971 .

[76]  Max Crochemore,et al.  Algorithms and Theory of Computation Handbook , 2010 .

[77]  Wolfgang Thomas,et al.  Languages, Automata, and Logic , 1997, Handbook of Formal Languages.

[78]  Erkko Lehtonen Labeled posets are universal , 2008, Eur. J. Comb..

[79]  José L. Balcázar,et al.  Structural Complexity II , 2012, EATCS.

[80]  Jeanleah Mohrherr Kleene Index Sets and Functional m-Degrees , 1983, J. Symb. Log..

[81]  Olivier Carton,et al.  Chains and Superchains for ω-Rational Sets, Automata and Semigroups , 1997, Int. J. Algebra Comput..

[82]  Pierluigi Crescenzi,et al.  A Uniform Approach to Define Complexity Classes , 1992, Theor. Comput. Sci..

[83]  Victor L. Selivanov,et al.  Hierarchies in φ‐spaces and applications , 2005, Math. Log. Q..

[84]  Tom Linton,et al.  Countable structures, Ehrenfeucht strategies, and Wadge reductions , 1991, Journal of Symbolic Logic.

[85]  Filip Murlak The Wadge Hierarchy of Deterministic Tree Languages , 2008, Log. Methods Comput. Sci..

[86]  Olivier Finkel,et al.  Wadge hierarchy of omega context-free languages , 2001, Theor. Comput. Sci..

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

[88]  K. Hofmann,et al.  Continuous Lattices and Domains , 2003 .

[89]  Victor L. Selivanov Relating Automata-theoretic Hierarchies to Complexity-theoretic Hierarchies , 2002, RAIRO Theor. Informatics Appl..

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

[91]  Victor L. Selivanov Wadge Degrees of [omega]-Languages of Deterministic Turing Machines , 2003, RAIRO Theor. Informatics Appl..

[92]  V. L. Selivanov Refined hierarchy of formulas , 1991 .

[93]  Stefan Friedrich,et al.  Topology , 2019, Arch. Formal Proofs.

[94]  Yuri L. Ershov,et al.  Theory of Numberings , 1999, Handbook of Computability Theory.

[95]  V. L. Selivanov Index sets of classes of hyper-hypersimple sets , 1990 .

[96]  Edith Hemaspaandra,et al.  A Downward Collapse within the Polynomial Hierarchy , 1999, SIAM J. Comput..

[97]  Armin Hemmerling,et al.  Hierarchies of Function Classes Defined by the First-Value Operator: (Extended Abstract) , 2005, CCA.

[98]  Victor Selivanov Fine hierarchy and definability in the Lindenbaum algebra , 1996 .

[99]  Christian Glaßer,et al.  The Shrinking Property for NP and coNP , 2008, CiE.

[100]  Victor L. Selivanov Classifying omega-regular partitions , 2007, LATA.

[101]  S. S. Goncharov,et al.  Computability and models - perspectives east and west , 2003, The University series in mathematics.

[102]  J. Ersov Theorie der Numerierungen II , 1973 .

[103]  M Sidman,et al.  Equivalence relations. , 1997, Journal of the experimental analysis of behavior.

[104]  Jörg Flum,et al.  Mathematical logic , 1985, Undergraduate texts in mathematics.

[105]  Alexander S. Kechris,et al.  Π11 Borel sets , 1989, Journal of Symbolic Logic.

[106]  Erkko Lehtonen Descending Chains and Antichains of the Unary, Linear, and Monotone Subfunction Relations , 2006, Order.

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

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

[109]  J. Büchi Weak Second‐Order Arithmetic and Finite Automata , 1960 .

[110]  Robert Fleischer New Physics in B and K Decays , 2005 .

[111]  Frank Stephan,et al.  On Existentially First-Order Definable Languages and Their Relation to NP , 1998, ICALP.

[112]  Barbara F. Csima,et al.  Boolean Algebras, Tarski Invariants, and Index Sets , 2006, Notre Dame J. Formal Log..

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

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

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

[116]  Scott Aaronson,et al.  The Complexity Zoo , 2008 .

[117]  J. U. L. Ersov,et al.  Theorie der Numerierungen II , 1975, Math. Log. Q..

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

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

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

[121]  Alexander Moshe Rabinovich,et al.  Logical Refinements of Church's Problem , 2007, CSL.

[122]  Grzegorz Rozenberg,et al.  Handbook of Formal Languages , 1997, Springer Berlin Heidelberg.

[123]  Zoltán Ésik,et al.  Temporal Logic with Cyclic Counting and the Degree of Aperiodicity of Finite Automata , 2001, Acta Cybern..

[124]  José L. Balcázar,et al.  Structural complexity 1 , 1988 .

[125]  Victor L. Selivanov Some Reducibilities on Regular Sets , 2005, CiE.

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

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

[128]  Olivier Finkel,et al.  Borel ranks and Wadge degrees of context free ω-languages , 2005 .

[129]  Yuri Leonidovich Ershov,et al.  Theory of Domains and Nearby (Invited Paper) , 1993, Formal Methods in Programming and Their Applications.

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

[131]  Jin-Yi Cai,et al.  The Boolean Hierarchy: Hardware over NP , 1986, SCT.

[132]  Armin Hemmerling Characterizations of the class Deltata2 over Euclidean spaces , 2004, Math. Log. Q..

[133]  Filip Murlak,et al.  On the Topological Complexity of Weakly Recognizable Tree Languages , 2007, FCT.

[134]  Victor L. Selivanov Undecidability in Some Structures Related to Computation Theory , 2009, J. Log. Comput..

[135]  Klaus W. Wagner,et al.  The Difference and Truth-Table Hierarchies for NP , 1987, RAIRO Theor. Informatics Appl..

[136]  Olivier Finkel,et al.  Borel hierarchy and omega context free languages , 2003, Theor. Comput. Sci..

[137]  Klaus W. Wagner A Note on Parallel Queries and the Symmetric-Difference Hierarchy , 1998, Inf. Process. Lett..

[138]  Klaus W. Wagner,et al.  The boolean hierarchy of NP-partitions , 2008, Inf. Comput..

[139]  Jim Kadin The Polynomial Time Hierarchy Collapses if the Boolean Hierarchy Collapses , 1988, SIAM J. Comput..

[140]  William W. Wadge,et al.  Degrees of complexity of subsets of the baire space , 1972 .

[141]  Rajeev Alur,et al.  Visibly pushdown languages , 2004, STOC '04.

[142]  Dominique Perrin,et al.  First-Order Logic and Star-Free Sets , 1986, J. Comput. Syst. Sci..

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

[144]  Armin Hemmerling,et al.  The Hausdorff-Ershov Hierarchy in Euclidean Spaces , 2006, Arch. Math. Log..

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

[146]  W. Thomas Star-Free Regular Sets of ~o-Sequences , 2004 .

[147]  Louise Hay,et al.  A discrete chain of degrees of index sets , 1972, Journal of Symbolic Logic.

[148]  R. Vaught Invariant sets in topology and logic , 1974 .

[149]  Heribert Vollmer,et al.  On balanced versus unbalanced computation trees , 2005, Mathematical systems theory.

[150]  Frank Stephan,et al.  The Dot-Depth and the Polynomial Hierarchy Correspond on the Delta Levels , 2004, Developments in Language Theory.

[151]  José L. Balcázar,et al.  Structural complexity 2 , 1990 .

[152]  Andreas Blass,et al.  Equivalence Relations, Invariants, and Normal Forms , 1983, SIAM J. Comput..

[153]  Mikhail J. Atallah,et al.  Algorithms and Theory of Computation Handbook , 2009, Chapman & Hall/CRC Applied Algorithms and Data Structures series.

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

[155]  Victor L. Selivanov,et al.  Complexity of Aperiodicity for Topological Properties of Regular omega-Languages , 2008, CiE.

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

[157]  F. Hausdorff Grundzüge der Mengenlehre , 1914 .