A game theoretical approach to the algebraic counterpart of the Wagner hierarchy : part II
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 3, pp. 463-515.

The algebraic counterpart of the Wagner hierarchy consists of a well-founded and decidable classification of finite pointed ω-semigroups of width 2 and height ω ω . This paper completes the description of this algebraic hierarchy. We first give a purely algebraic decidability procedure of this partial ordering by introducing a graph representation of finite pointed ω-semigroups allowing to compute their precise Wagner degrees. The Wagner degree of any ω-rational language can therefore be computed directly on its syntactic image. We then show how to build a finite pointed ω-semigroup of any given Wagner degree. We finally describe the algebraic invariants characterizing every degree of this hierarchy.

DOI : 10.1051/ita/2009007
Classification : O3D55, 20M35, 68Q70, 91A65
Mots clés : $\omega $-automata, $\omega $-rational languages, $\omega $-semigroups, infinite games, hierarchical games, Wadge game, Wadge hierarchy, Wagner hierarchy
@article{ITA_2009__43_3_463_0,
     author = {Cabessa, J\'er\'emie and Duparc, Jacques},
     title = {A game theoretical approach to the algebraic counterpart of the {Wagner} hierarchy : part {II}},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {463--515},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {3},
     year = {2009},
     doi = {10.1051/ita/2009007},
     mrnumber = {2541208},
     zbl = {1175.03022},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ita/2009007/}
}
TY  - JOUR
AU  - Cabessa, Jérémie
AU  - Duparc, Jacques
TI  - A game theoretical approach to the algebraic counterpart of the Wagner hierarchy : part II
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2009
SP  - 463
EP  - 515
VL  - 43
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ita/2009007/
DO  - 10.1051/ita/2009007
LA  - en
ID  - ITA_2009__43_3_463_0
ER  - 
%0 Journal Article
%A Cabessa, Jérémie
%A Duparc, Jacques
%T A game theoretical approach to the algebraic counterpart of the Wagner hierarchy : part II
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2009
%P 463-515
%V 43
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ita/2009007/
%R 10.1051/ita/2009007
%G en
%F ITA_2009__43_3_463_0
Cabessa, Jérémie; Duparc, Jacques. A game theoretical approach to the algebraic counterpart of the Wagner hierarchy : part II. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 3, pp. 463-515. doi : 10.1051/ita/2009007. http://archive.numdam.org/articles/10.1051/ita/2009007/

[1] 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. | MR | Zbl

[2] O. Carton and D. Perrin, Chains and superchains in ω-semigroups, edited by Almeida Jorge et al., Semigroups, automata and languages. Papers from the conference, Porto, Portugal (1994) June 20-24. World Scientific, Singapore (1996) 17-28. | MR | Zbl

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

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

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

[6] J. Duparc, A hierarchy of deterministic context-free ω-languages. Theoret. Comput. Sci. 290 (2003) 1253-1300. | MR | Zbl

[7] J. Duparc, Wadge hierarchy and Veblen hierarchy. Part II: Borel sets of infinite rank (to appear). | Zbl

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

[9] O. Finkel, An effective extension of the Wagner hierarchy to blind counter automata. In Computer Science Logic (Paris, 2001); Lect. Notes Comput. Sci. 2142 (2001) 369-383. | Zbl

[10] O. Finkel, Borel ranks and Wadge degrees of context free omega languages. In New Computational Paradigms, First Conference on Computability in Europe, CiE. Lect. Notes Comput. Sci. 2142 (2005) 129-138. | Zbl

[11] A.S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics 156. Springer-Verlag, New York (1995). | Zbl

[12] K. Kunen, Set theory. An introduction to independence proofs. 2nd print. Studies in Logic and the Foundations of Mathematics 102. North-Holland (1983) 313. | Zbl

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

[14] Y.N. Moschovakis, Descriptive set theory. Studies in Logic and the Foundations of Mathematics 100. North-Holland Publishing Company (1980) 637. | MR | Zbl

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

[16] D. Perrin and J.-Éric Pin, Infinite words. Pure Appl. Mathematics 141. Elsevier (2004). | Zbl

[17] J.-E. Pin, Varieties of formal languages. North Oxford, London and Plenum, New-York (1986). | MR | Zbl

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

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

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

[21] K. Wagner, On ω-regular sets. Inform. Control 43 (1979) 123-177. | MR | Zbl

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

[23] 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 Peter D. Mosses, M. Nielsen, M.I. Schwartzbach. Lect. Notes Comput. Sci. 915 (1995) 288-302.

Cité par Sources :