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

Cabessa, Jérémie; Duparc, Jacques

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 3, pp. 443-461.

The algebraic study of formal languages shows that $\omega$-rational sets correspond precisely to the $\omega$-languages recognizable by finite $\omega$-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 $\omega$-rational language to the $\omega$-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 $\omega$-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 ${\omega }^{\omega }$.

Classification : O3D55, 20M35, 68Q70, 91A65
Mots clés : $\omega $-automata, $\omega $-rational languages, $\omega $-semigroups, infinite games, hierarchical games, Wadge game, Wadge hierarchy, Wagner hierarchy

[1] A. Andretta, Equivalence between Wadge and Lipschitz determinacy. Ann. Pure Appl. Logic 123 (2003) 163-192. | MR | Zbl

[2] J. Cabessa and J. Duparc, An infinite game over $\omega$-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. | MR | Zbl

[3] O. Carton and D. Perrin, Chains and superchains for $\omega$-rational sets, automata and semigroups. Int. J. Algebra Comput. 7 (1997) 673-695. | MR | Zbl

[4] O. Carton and D. Perrin, The Wadge-Wagner hierarchy of $\omega$-rational sets. In Automata, languages and programming (Bologna, 1997). Lect. Notes Comput. Sci. 1256 (1997) 17-35. | MR

[5] O. Carton and D. Perrin, The Wagner hierarchy. Int. J. Algebra Comput. 9 (1999) 597-620. | MR | Zbl

[6] J. Duparc, Wadge hierarchy and Veblen hierarchy. I. Borel sets of finite rank. J. Symbolic Logic 66 (2001) 56-86. | MR | Zbl

[7] J. Duparc and M. Riss, The missing link for $\omega$-rational sets, automata, and semigroups. Int. J. Algebra Comput. 16 (2006) 161-185. | MR | Zbl

[8] R.E. Ladner, Application of model theoretic games to discrete linear orders and finite automata. Inform. Control 33 (1977) 281-303. | MR | Zbl

[9] D.A. Martin, Borel determinacy. Ann. Math. 102 (1975) 363-371. | MR | Zbl

[10] R.t McNaughton and S.A. Papert, Counter-Free Automata (M.I.T. research monograph No. 65). The MIT Press (1971). | MR | Zbl

[11] D. Perrin and J.-E. Pin, First-order logic and star-free sets. J. Comput. System Sci. 32 (1986) 393-406. | MR | Zbl

[12] 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. | MR | Zbl

[13] D. Perrin and J.-E. Pin, Infinite words. Pure and Applied Mathematics 141, Elsevier (2004). | Zbl

[14] J.-E. Pin, Logic, semigroups and automata on words. Ann. Math. Artif. Intell. 16 (1996) 343-384. | MR | Zbl

[15] J.-E. Pin, 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. | MR | Zbl

[16] J. Sakarovitch, Monoïdes pointés. Semigroup Forum 18 (1979) 235-264. | Zbl

[17] M.P. Schützenberger, On finite monoids having only trivial subgroups. Inform. Control 8 (1965) 190-194. | MR | Zbl

[18] V. Selivanov, Fine hierarchy of regular $\omega$-languages. Theoret. Comput. Sci. 191 (1998) 37-59. | MR | Zbl

[19] W. Thomas, Star-free regular sets of $\omega$-sequences. Inform. Control 42 (1979) 148-156. | MR | Zbl

[20] W.W. Wadge, Degrees of complexity of subsets of the baire space. Notice A.M.S. (1972) A714-A715.

[21] W.W. Wadge, Reducibility and determinateness on the Baire space. Ph.D. thesis, University of California, Berkeley (1983).

[22] K. Wagner, On $\omega$-regular sets. Inform. Control 43 (1979) 123-177. | MR | Zbl

[23] T. Wilke, An Eilenberg theorem for $\infty$-languages. In Automata, languages and programming (Madrid, 1991). Lect. Notes Comput. Sci. 510 (1991) 588-599. | MR | Zbl

[24] T. Wilke and H. Yoo, 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.