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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; Jeremie.Cabessa@unil.ch
Jacques Duparc
Affiliation:
University of Lausanne, Faculty of Business and Economics, HEC - ISI, 1015 Lausanne, Switzerland; Jeremie.Cabessa@unil.ch
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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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