Hierarchies and reducibilities on regular languages related to modulo counting

We discuss some known and introduce some new hierarchies and reducibilities on regular languages, with the emphasis on the quantifier-alternation and difference hierarchies of the quasi-aperiodic languages. The non-collapse of these hierarchies and decidability of some levels are established. Complete sets in the levels of the hierarchies under the polylogtime and some quantifier-free reducibilities are found. Some facts about the corresponding degree structures are established. As an application, we characterize the regular languages whose balanced leaf-language classes are contained in the polynomial hierarchy. For any discussed reducibility we try to give motivations and open questions, in a hope to convince the reader that the study of these reducibilities is interesting for automata theory and computational complexity.

[1]  Heribert Vollmer,et al.  The Chain Method to Separate Counting Classes , 1998, Theory of Computing Systems.

[2]  Howard Straubing,et al.  Actions, wreath products of C-varieties and concatenation product , 2006, Theor. Comput. Sci..

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

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

[5]  Thomas Wilke,et al.  Classifying Discrete Temporal Properties , 1999, STACS.

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

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

[8]  Howard Straubing,et al.  On Logical Descriptions of Regular Languages , 2002, LATIN.

[9]  Victor L. Selivanov.  Relating Automata-Theoretic Hierarchies to Complexity-Theoretic Hierarchies , 2001, FCT.

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

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

[12]  Bernd Borchert.  On the Acceptance Power of Regular Languages , 1994, STACS.

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

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

[15]  Janusz A. Brzozowski,et al.  Dot-Depth of Star-Free Events , 1971, Journal of computer and system sciences (Print).

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

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

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

[19]  Howard Straubing,et al.  regular Languages Defined with Generalized Quantifiers , 1988, ICALP.

[20]  Christian Glaßer,et al.  Polylog-Time Reductions Decrease Dot-Depth , 2005, STACS.

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

[22]  Joseph R. Shoenfield,et al.  Mathematical logic , 1967 .

[23]  Wolfgang Thomas.  An application of the Ehrenfeucht-Fraisse game in formal language theory , 1984 .

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

[25]  Robert McNaughton,et al.  Algebraic decision procedures for local testability , 1974, Mathematical systems theory.

[26]  Kim G. Larsen,et al.  Regular languages definable by Lindström quantifiers , 2003, RAIRO Theor. Informatics Appl..

[27]  Gerd Wechsung,et al.  A survey on counting classes , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[28]  Gerd Wechsung,et al.  Counting classes with finite acceptance types , 1987 .

[29]  Berndt Farwer,et al.  ω-automata , 2002 .

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

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

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

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

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

[35]  Marcel Paul Schützenberger,et al.  On Finite Monoids Having Only Trivial Subgroups , 1965, Inf. Control..

[36]  Pascal Weil,et al.  Polynomial closure and unambiguous product , 1995, Theory of Computing Systems.

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

[38]  Howard Straubing,et al.  Regular Languages in NC¹ , 1992, J. Comput. Syst. Sci..

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

[40]  Jean-Éric Pin,et al.  Syntactic Semigroups , 1997, Handbook of Formal Languages.