Ambiguous classes in μ -calculi hierarchies

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 Σ 2 ∩ Π 2 and Comp ( Σ 1 , Π...

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Vydáno v:	Theoretical computer science Ročník 333; číslo 1; s. 265 - 296
Hlavní autoři:	Santocanale, Luigi, Arnold, André
Médium:	Journal Article Konferenční příspěvek
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier B.V 01.03.2005 Elsevier
Edice:	Foundations of Software Science and Computation Structures
Témata:	Algorithmics. Computability. Computer arithmetics Applied sciences Automata. Abstract machines. Turing machines Combinatorics Combinatorics. Ordered structures Computer Science Computer science; control theory; systems Distributed, Parallel, and Cluster Computing Exact sciences and technology General logic Graph theory Logic and foundations Mathematical logic, foundations, set theory Mathematics Sciences and techniques of general use Theoretical computing Complexity class Hierarchy Second order logic Fix point Separation Computer theory Propositional calculus Mu calculus Parity Tree automaton Monadic logic Depth
ISSN:	0304-3975, 1879-2294
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Popis
Shrnutí:	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 Σ 2 ∩ Π 2 and Comp ( Σ 1 , Π 1 ) 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 Σ n + 1 ∩ Π n + 1 and the composition classes Comp ( Σ n , Π n ) 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 Σ n + 1 and to a game in Π n + 1 is also equivalent to a game obtained by composing games in Σ n and Π n . 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 Π n + 1 automata, then there exists a third language recognized by an alternating automaton in Comp ( Σ n , Π n ) 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.
ISSN:	0304-3975 1879-2294
DOI:	10.1016/j.tcs.2004.10.024