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A game theoretical approach to the algebraic counterpart of the Wagner hierarchy : Part I

Published online by Cambridge University Press:  06 March 2009

Jérémie Cabessa  and
Jacques Duparc
Jérémie Cabessa
Affiliation:
University of Lausanne, Faculty of Business and Economics, HEC - ISI, 1015 Lausanne, Switzerland; [email protected]
Jacques Duparc
Affiliation:
University of Lausanne, Faculty of Business and Economics, HEC - ISI, 1015 Lausanne, Switzerland; [email protected]
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Abstract

The algebraic study of formal languages shows that ω-rational sets correspond precisely to the ω-languages recognizable by finite ω-semigroups. Within this framework, we provide a construction of the algebraic counterpart of the Wagner hierarchy. We adopt a hierarchical game approach, by translating the Wadge theory from the ω-rational language to the ω-semigroup context. More precisely, we first show that the Wagner degree is indeed a syntactic invariant. We then define a reduction relation on finite pointed ω-semigroups by means of a Wadge-like infinite two-player game. The collection of these algebraic structures ordered by this reduction is then proven to be isomorphic to the Wagner hierarchy, namely a well-founded and decidable partial ordering of width 2 and height ωω.

Keywords

ω-automataω-rational languagesω-semigroupsinfinite gameshierarchical gamesWadge gameWadge hierarchyWagner hierarchy.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 43 , Issue 3 , July 2009 , pp. 443 - 461
Copyright
© EDP Sciences, 2009

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References

Andretta, A., Equivalence between Wadge and Lipschitz determinacy. Ann. Pure Appl. Logic 123 (2003) 163–192. CrossRef
J. Cabessa and J. Duparc, An infinite game over ω-semigroups. In Foundations of the Formal Sciences V, Infinite Games, edited by S. Bold, B. Löwe, T. Räsch, J. van Benthem, Studies in Logic 11, College Publications, London (2007) 63–78.
Carton, O. and Perrin, D., Chains and superchains for ω-rational sets, automata and semigroups. Int. J. Algebra Comput. 7 (1997) 673–695. CrossRef
Carton, O. and Perrin, D., The Wadge-Wagner hierarchy of ω-rational sets. In Automata, languages and programming (Bologna, 1997). Lect. Notes Comput. Sci. 1256 (1997) 17–35. CrossRef
Carton, O. and Perrin, D., The Wagner hierarchy. Int. J. Algebra Comput. 9 (1999) 597–620. CrossRef
Duparc, J., Wadge hierarchy and Veblen hierarchy. I. Borel sets of finite rank. J. Symbolic Logic 66 (2001) 56–86. CrossRef
Duparc, J. and Riss, M., The missing link for ω-rational sets, automata, and semigroups. Int. J. Algebra Comput. 16 (2006) 161–185. CrossRef
Ladner, R.E., Application of model theoretic games to discrete linear orders and finite automata. Inform. Control 33 (1977) 281–303. CrossRef
Martin, D.A., Borel determinacy. Ann. Math. 102 (1975) 363–371. CrossRef
R.t McNaughton and S.A. Papert, Counter-Free Automata (M.I.T. research monograph No. 65). The MIT Press (1971).
Perrin, D. and Pin, J.-E., First-order logic and star-free sets. J. Comput. System Sci. 32 (1986) 393–406. CrossRef
D. Perrin and J.-E. Pin, Semigroups and automata on infinite words. In Semigroups, formal languages and groups (York, 1993). Kluwer Acad. Publ., Dordrecht (1995) 49–72.
D. Perrin and J.-E. Pin, Infinite words. Pure and Applied Mathematics 141, Elsevier (2004).
Pin, J.-E., Logic, semigroups and automata on words. Ann. Math. Artif. Intell. 16 (1996) 343–384. CrossRef
Pin, J.-E., Positive varieties and infinite words. in edited by Latin'98, edited by C.L. Lucchesi and A.V. Moura. Lect. Notes Comput. Sci. 1380 (1998) 76–87. CrossRef
Sakarovitch, J., Monoïdes pointés. Semigroup Forum 18 (1979) 235–264. CrossRef
Schützenberger, M.P., On finite monoids having only trivial subgroups. Inform. Control 8 (1965) 190–194. CrossRef
Selivanov, V., Fine hierarchy of regular ω-languages. Theoret. Comput. Sci. 191 (1998) 37–59. CrossRef
Thomas, W., Star-free regular sets of ω-sequences. Inform. Control 42 (1979) 148–156. CrossRef
W.W. Wadge, Degrees of complexity of subsets of the baire space. Notice A.M.S. (1972) A714–A715.
W.W. Wadge, Reducibility and determinateness on the Baire space. Ph.D. thesis, University of California, Berkeley (1983).
Wagner, K., On ω-regular sets. Inform. Control 43 (1979) 123–177. CrossRef
Wilke, T., Eilenberg, An theorem for ∞-languages. In Automata, languages and programming (Madrid, 1991). Lect. Notes Comput. Sci. 510 (1991) 588–599. CrossRef
Wilke, T. and Yoo, H., Computing the Wadge degree, the Lifshitz degree, and the Rabin index of a regular language of infinite words in polynomial time, in TAPSOFT '95: Theory and Practive of Software Development, edited by P.D. Mosses, M. Nielsen and M.I. Schwartzbach. Lect. Notes Comput. Sci. 915 (1995) 288–302. CrossRef