Some results on complexity of µ-calculus evaluation in the black-box model

We consider μ -calculus formulas in a normal form: after a prefix of fixed-point quantifiers follows a quantifier-free expression. We are interested in the problem of evaluating (model checking) such formulas in a powerset lattice. We assume that the quantifier-free part of the expression can be any monotone function given by a black-box – we may only ask for its value for given arguments. As a first result we prove that when the lattice is fixed, the problem becomes polynomial (the assumption about the quantifier-free part strengthens this result). As a second result we show that any algorithm solving the problem has to ask at least about n 2 (namely Ω(n 2 /log n )) queries to the function, even when the expression consists of one μ and one ν (the assumption about the quantifier-free part weakens this result).