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), p. 443-461 / Harvested from Numdam

Résumé

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 $.

Détail

Publié le : 2009-01-01

MR 2541207 | Zbl 1175.03021

DOI : https://doi.org/10.1051/ita/2009004
Classification:  O3D55,  20M35,  68Q70,  91A65

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