A game theoretical approach to the algebraic counterpart of the Wagner hierarchy : part II

Cabessa, Jérémie; Duparc, Jacques

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 3, pp. 463-515.

The algebraic counterpart of the Wagner hierarchy consists of a well-founded and decidable classification of finite pointed $\omega$-semigroups of width $2$ and height ${\omega }^{\omega }$. 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 $\omega$-semigroups allowing to compute their precise Wagner degrees. The Wagner degree of any $\omega$-rational language can therefore be computed directly on its syntactic image. We then show how to build a finite pointed $\omega$-semigroup of any given Wagner degree. We finally describe the algebraic invariants characterizing every degree of this hierarchy.

Classification : O3D55, 20M35, 68Q70, 91A65
Mots clés : $\omega $-automata, $\omega $-rational languages, $\omega $-semigroups, infinite games, hierarchical games, Wadge game, Wadge hierarchy, Wagner hierarchy

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