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

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

Received: 10 April 2008
Accepted: 18 December 2008

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 ω^ω.