Ambiguous classes in μ-calculi hierarchies

Abstract : A classical result by Rabin states that if a set of trees and its complement are both Büchi definable in the monadic second order logic, then these sets are weakly definable. In the language of μ-calculi, this theorem asserts the equality between the complexity classes Σ2∩Π2 and Comp(Σ1,Π1) of the fixed-point alternation-depth hierarchy of the μ-calculus of tree languages. It is natural to ask whether at higher levels of the hierarchy the ambiguous classes Σn+1∩Πn+1 and the composition classes Comp(Σn,Πn) are equal, and for which μ-calculi. The first result of this paper is that the alternation-depth hierarchy of the games μ-calculus—whose canonical interpretation is the class of all complete lattices—enjoys this property. More explicitly, every parity game which is equivalent both to a game in Σn+1 and to a game in Πn+1 is also equivalent to a game obtained by composing games in Σn and Πn. The second result is that the alternation-depth hierarchy of the μ-calculus of tree languages does not enjoy the property. Taking into account that any Büchi definable set is recognized by a nondeterministic Büchi automaton, we generalize Rabin's result in terms of the following separation theorem: if two disjoint languages are recognized by nondeterministic Πn+1 automata, then there exists a third language recognized by an alternating automaton in Comp(Σn,Πn) containing one and disjoint from the other. Finally, we lift the results obtained for the μ-calculus of tree languages to the propositional modal μ-calculus: ambiguous classes do not coincide with composition classes, but a separation theorem is established for disjunctive formulas.
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Contributor : Luigi Santocanale <>
Submitted on : Saturday, January 23, 2016 - 11:55:26 AM
Last modification on : Friday, April 12, 2019 - 10:18:09 AM

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André Arnold, Luigi Santocanale. Ambiguous classes in μ-calculi hierarchies. Theoretical Computer Science, Elsevier, 2005, Foundations of Software Science and Computation Structures, 333 (1--2), pp.265-296. ⟨10.1016/j.tcs.2004.10.024⟩. ⟨hal-01261055⟩

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