Ambiguous classes in µ-calculi hierarchies

• Authors: Luigi Santocanale;André Arnold
• Venue: Theoretical Computer Science - Foundations of software science and computation structures
• Year: 2005

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.