Ambiguous classes in mu-calculi hierarchies

نویسندگان

  • Luigi Santocanale
  • André Arnold
چکیده

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 -calculi, this theorem asserts the equality between the complexity classes 2 ∩ 2 and Comp( 1, 1) of the fixed-point alternation-depth hierarchy of the -calculus of tree languages. It is natural to ask whether at higher levels of the hierarchy the ambiguous classes n+1 ∩ n+1 and the composition classes Comp( n, n) are equal, and for which -calculi. The first result of this paper is that the alternation-depth hierarchy of the games -calculus—whose canonical interpretation is the class of all complete lattices—enjoys this property. More explicitly, every parity gamewhich is equivalent both to a game in n+1 and to a game in n+1 is also equivalent to a game obtained by composing games in n and n. The second result is that the alternation-depth hierarchy of the -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 n+1 automata, then there exists a third language recognized by an alternating automaton inComp( n, n) containing one and disjoint from the other. Finally, we lift the results obtained for the -calculus of tree languages to the propositional modal -calculus: ambiguous classes do not coincide with composition classes, but a separation theorem is established for disjunctive formulas. © 2004 Elsevier B.V. All rights reserved. Research supported by the European Commission through a Marie Curie Individual Fellowship. ∗ Corresponding author. E-mail addresses: [email protected] (L. Santocanale), [email protected] (A. Arnold). 0304-3975/$ see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.tcs.2004.10.024 266 L. Santocanale, A. Arnold / Theoretical Computer Science 333 (2005) 265–296

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عنوان ژورنال:
  • Theor. Comput. Sci.

دوره 333  شماره 

صفحات  -

تاریخ انتشار 2005