On the universal and existential fragments of the mu-calculus
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 p-calculus can be replaced by efforts to find such an algorithm for the universal or existential fragment. (c) 2005 Elsevier B.V. All rights reserved.
mu-calculus satisfiability model checking automata and logic