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A Game Theoretical Approach to The Algebraic Counterpart of The Wagner Hierarchy : Part II

Published online by Cambridge University Press:  12 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 counterpart of the Wagner hierarchy consists of a well-founded and decidable classification of finite pointed ω-semigroups of width 2 and height ωω. 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.

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. 463 - 515
Copyright
© EDP Sciences, 2009

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