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

Cabessa, Jérémie; Duparc, Jacques

doi:10.1051/ita/2009007

Cabessa, Jérémie ; Duparc, Jacques

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

Résumé

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 $ω^{ω}$ . 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.

EuDML MR Zbl

DOI : 10.1051/ita/2009007

Classification : O3D55, 20M35, 68Q70, 91A65
Mots-clés :

ω

-automata,

ω

-rational languages,

ω

-semigroups, infinite games, hierarchical games, Wadge game, Wadge hierarchy, Wagner hierarchy

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     title = {A game theoretical approach to the algebraic counterpart of the {Wagner} hierarchy : part {II}},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
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     publisher = {EDP-Sciences},
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Cabessa, Jérémie; Duparc, Jacques. A game theoretical approach to the algebraic counterpart of the Wagner hierarchy : part II. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 3, pp. 463-515. doi : 10.1051/ita/2009007. https://www.numdam.org/articles/10.1051/ita/2009007/

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