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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Bibliographic Details
Published in:Theoretical computer science Vol. 333; no. 1; pp. 265 - 296
Main Authors: Santocanale, Luigi, Arnold, André
Format: Journal Article Conference Proceeding
Language:English
Published: Amsterdam Elsevier B.V 01.03.2005
Elsevier
Series:Foundations of Software Science and Computation Structures
Subjects:
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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Summary: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