A game theoretical approach to the algebraic counterpart of the Wagner hierarchy : part I
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 43 (2009) no. 3, pp. 443-461.

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 ω ω .

DOI: 10.1051/ita/2009004
Classification: O3D55,  20M35,  68Q70,  91A65
Keywords: ω-automata, ω-rational languages, ω-semigroups, infinite games, hierarchical games, Wadge game, Wadge hierarchy, Wagner hierarchy
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Cabessa, Jérémie; Duparc, Jacques. A game theoretical approach to the algebraic counterpart of the Wagner hierarchy : part I. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 43 (2009) no. 3, pp. 443-461. doi : 10.1051/ita/2009004. http://www.numdam.org/articles/10.1051/ita/2009004/

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