Abstract: 
One source of complexity in the μcalculus 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 μcalculus, 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 alternationfree fragment of the μcalculus, where no alternations between least and greatest fixed points are allowed. Our results imply that efforts to find a polynomialtime modelchecking algorithm for the μcalculus can be replaced by efforts to find such an algorithm for the universal or existential fragment.
