On the Universal and Existential Fragments of the Mu-Calculus

Thomas A. Henzinger, Orna Kupferman, and Rupak Majumdar

One source of complexity in the mu-calculus is its ability to specify an unbounded number of switches between universal (AX) and existential (EX) branching modes. We therefore study the problems of satisfiability, validity, model checking, and implication for the universal and existential fragments of the mu-calculus, in which only one branching mode is allowed. The universal fragment is rich enough to express most specifications of interest, and therefore improved algorithms are of practical importance. We show that while the satisfiability and validity problems become indeed simpler for the existential and universal fragments, this is, unfortunately, not the case for model checking and implication. We also show the corresponding results for the alternation-free fragment of the mu-calculus, where no alternations between least and greatest fixed points are allowed. Our results imply that efforts to find a polynomial-time model-checking algorithm for the mu-calculus can be replaced by efforts to find such an algorithm for the universal or existential fragment.

Theoretical Computer Science 354:173-186, 2006. A preliminary version appeared in the Proceedings of the Ninth International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS), Lecture Notes in Computer Science 2619, Springer, 2003, pp. 49-64.

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