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

Jérémie Cabessa; Jacques Duparc

RAIRO - Theoretical Informatics and Applications (2009)

  • Volume: 43, Issue: 3, page 443-461
  • ISSN: 0988-3754

Abstract

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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.

How to cite

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Cabessa, Jérémie, and Duparc, Jacques. "A game theoretical approach to the algebraic counterpart of the Wagner hierarchy : Part I." RAIRO - Theoretical Informatics and Applications 43.3 (2009): 443-461. <http://eudml.org/doc/250579>.

@article{Cabessa2009,
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. },
author = {Cabessa, Jérémie, Duparc, Jacques},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {ω-automata; ω-rational languages; ω-semigroups; infinite games; hierarchical games; Wadge game; Wadge hierarchy; Wagner hierarchy.; -automata; -rational languages; -semigroups; Wagner hierarchy},
language = {eng},
month = {3},
number = {3},
pages = {443-461},
publisher = {EDP Sciences},
title = {A game theoretical approach to the algebraic counterpart of the Wagner hierarchy : Part I},
url = {http://eudml.org/doc/250579},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Cabessa, Jérémie
AU - Duparc, Jacques
TI - A game theoretical approach to the algebraic counterpart of the Wagner hierarchy : Part I
JO - RAIRO - Theoretical Informatics and Applications
DA - 2009/3//
PB - EDP Sciences
VL - 43
IS - 3
SP - 443
EP - 461
AB - 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.
LA - eng
KW - ω-automata; ω-rational languages; ω-semigroups; infinite games; hierarchical games; Wadge game; Wadge hierarchy; Wagner hierarchy.; -automata; -rational languages; -semigroups; Wagner hierarchy
UR - http://eudml.org/doc/250579
ER -

References

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