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Hierarchies and reducibilities on regular languages related to modulo counting

Published online by Cambridge University Press:  15 January 2008

Victor L. Selivanov
Victor L. Selivanov*
Affiliation:
A.P. Ershov Institute of Informatics Systems, Siberian Division of the Russian Academy of Sciences; vseliv@nspu.ru
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Abstract

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.

Keywords

Regular languagequasi-aperiodic regular languagequantifier-alternation hierarchydifference hierarchypolylogtime reducibilityquantifier-free reducibilityforbidden pattern.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 43 , Issue 1 , January 2009 , pp. 95 - 132
Copyright
© EDP Sciences, 2008

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References

Borchert, B., On the acceptance power of regular languages. Theor. Comput. Sci. 148 (1995) 207–225. CrossRef
Borchert, B., Kuske, D. and Stephan, F., On existentially first-order definable languages and their relation to NP. RAIRO-Theor. Inf. Appl. 33 (1999) 259–269. CrossRef
Bovet, D.P., Crescenzi, P. and Silvestri, R., A uniform approach to define complexity classes. Theor. Comput. Sci. 104 (1992) 263–283. CrossRef
Büchi, J.R., Weak second-order arithmetic and finite automata. Z. Math. Logic Grundl. Math. 6 (1960) 66–92. CrossRef
Barrington, D.A.M., Compton, K., Straubing, H. and Thérien, D., Regular languages in NC 1. J. Comput. System Sci. 44 (1992) 478–499. CrossRef
Cronauer, K., Hertrampf, U., Vollmer, H. and Wagner, K.W., The chain method to separate counting classes. Theor. Comput. Syst. 31 (1998) 93–108. CrossRef
Cohen, R.S. and Brzozowski, J.A., Dot-depth of star-free events. J. Comput. System Sci. 5 (1971) 1–16. CrossRef
Chaubard, L., Pin, J.-E. and Straubing, H., Actions, wreath products of C-varieties and concatenation product. Theor. Comput. Sci. 356 (2006) 73–89. CrossRef
S. Eilenberg, Automata, Languages and Machines v. A and B. Academic Press (1974 and 1976).
Esik, Z. and Ito, M., Temporal logic with cyclic counting and the degree of aperiodicity of finite automata. Acta Cybern. 16 (2003) 1–28.
Esik, Z. and Larsen, K.G., Regular languages definable by Lindström quantifiers. RAIRO-Theor. Inf. Appl. 37 (2003) 179–241. CrossRef
C. Glaßer, Polylogtime-reductions decrease dot-depth, in Proc. of STACS-2005. Lect. Notes Comput. Sci. 3404 (2005).
Glaßer, C., Languages Polylog-Time Reducible to Dot-Depth 1/2. J. Comput. System Sci. 73 (2007) 36–56. CrossRef
Glaßer, C. and Schmitz, H., The Boolean Structure of Dot-Depth One. J. Autom. Lang. Comb. 6 (2001) 437–452.
Gundermann, T. and Wechsung, G., Counting classes of finite accepting types. Computers and Artificial Intelligence 6 (1987) 395–409.
T. Gundermann, N.A. Nasser and G. Wechsung, A survey on counting classes, in Proc. of Structures in Complexity Theory (1990) 140–153.
U. Hertrampf, C. Lautemann, T. Schwentick, H. Vollmer and K.W. Wagner, On the power of polynomial time bit-reductions, Proc. 8th Structure in Complexity Theory (1993) 200–207.
A.S. Kechris, Classical Descriptive Set Theory. Springer, New York (1994).
McNaughton, R., Algebraic decision procedures for local testability. Math. Syst. Theor. 8 (1974) 60–76. CrossRef
R. McNaughton and S. Papert, Counter-Free Automata. MIT Press, Cambridge, Massachussets (1971).
J.-E. Pin, Varieties of Formal Languages. North Oxford Academic (1986).
J.-E. Pin, Syntactic semigroups, Chap. 10 in Handbook of language theory, Vol. I, edited by G. Rozenberg and A. Salomaa. Springer Verlag (1997) 679–746.
Perrin, D. and Pin, J.-E., First-order logic and star-free sets. J. Comput. System Sci. 32 (1986) 393–496. CrossRef
Pin, J.-E. and Weil, P., Polynomial closure and unambiguous product. Theor. Comput. Syst. 30 (1997) 383–422. CrossRef
Schützenberger, M.P., On finite monoids having only trivial subgroups. Inform. Control 8 (1965) 190–194. CrossRef
Selivanov, V.L., A logical approach to decidability of hierarchies of regular star-free languages, in Proc. of STACS-2001. Lect. Notes Comput. Sci. 2010 (2001) 539–550. CrossRef
Selivanov, V.L., Relating automata-theoretic hierarchies to complexity-theoretic hierarchies. RAIRO-Theor. Inf. Appl. 36 (2002) 29–42. CrossRef
V.L. Selivanov, Some hierarchies and reducibilities on regular languages. University of Würzburg, Technical Report 349 (2004).
Selivanov, V.L., Some reducibilities on regular sets, in Proc. of CIE-2005. Lect. Notes Comput. Sci. 3526 (2005) 430–440. CrossRef
Selivanov, V.L., Fine hierarchy of regular aperiodic ω-languages, in Proc. of DLT-2007, edited by T. Harju, J. Karhumäki and A. Lepistö. Lect. Notes Comput. Sci. 4588 (2007) 399–410. CrossRef
J. Shoenfield, Mathematical Logic. Addison Wesley, Massachussets (1967).
A.G. Shukin, Difference hierarchies of regular languages. Comput. Systems, Novosibirsk 161 (1998) 141–155 (in Russian).
V.L. Selivanov and A.G. Shukin, On hierarchies of regular star-free languages (in Russian). Preprint 69 of A.P. Ershov Institute of Informatics Systems (2000) 28.
Stern, J., Characterizations of some classes of regular events. Theor. Comput. Sci. 35 (1985) 163–176. CrossRef
H. Straubing, Finite automata, formal logic and circuit complexity. Birkhäuser, Boston (1994).
Straubing, H., On logical description of regular languages, in Proc. of LATIN-2002. Lect. Notes Comput. Sci. 2286 (2002) 528–538. CrossRef
Straubing, H., Thérien, D. and Thomas, W., Regular languages defined with generalized quantifiers. Inform. Comput. 118 (1995) 289–301. CrossRef
Selivanov, V.L. and Wagner, K.W., A reducibility for the dot-depth hierarchy. Proc. 29th Int. Symp. on Mathematical Foundations of Computer Science. Lect. Notes Comput. Sci. 3153 (2004) 783–793. CrossRef
Selivanov, V.L. and Wagner, K.W., A reducibility for the dot-depth hierarchy. Theor. Comput. Sci. 345 (2005) 448–472. CrossRef
Thomas, W., Classifying regular events in symbolic logic. J. Comput. System Sci. 25 (1982) 360–376. CrossRef
Thomas, W., An application of the Ehrenteucht-Fraïssé game in formal language theory. Mém. Soc. Math. France Ser. 2 16 (1984) 11–21. CrossRef
Trakhtenbrot, B.A., Synthesis of logic networks whose operators are described by means of single-placed predicate calculus. Doklady Akad. Nauk SSSR 118 (1958) 646–649.
Vereshchagin, N.K., Relativizable and non-relativizable theorems in the polynomial theory of algorithms. Izvestiya Rossiiskoi Akademii Nauk 57 (1993) 51–90 (in Russian).
Wagner, K.W., On ω-regular sets. Inform. Control 43 (1979) 123–177. CrossRef
Wagner, K.W., Leaf language classes. MCU-2004. Lect. Notes Comput. Sci. 3354 (2005) 60–81.
Wilke, T., Classifying discrete temporal properties, in Proc. STACS-99. Lect. Notes Comput. Sci. 1563 (1999) 32–46. CrossRef