Characterizing finite Kripke structures in propositional temporal logic

Abstract

We show that if two finite Kripke structures can be distinguished by some CTL ∗ formula that contains both branching-time and linear-time operators, then the structures can be distinguished by a CTL formula that contains only branching-time operators. Our proof involves showing that, for any finite Kripke structure M, it is possible to construct a CTL formula F M that uniquely characterizes M. Since one Kripke structure may be a trivial unrolling of another, we use a notion of equivalence between Kripke structures that is similar to the notion of bisimulation studied by Milner [15]. Our first construction of F M requires the use of the nexttime operator. We also consider the case in which the nexttime operator is disallowed in CTL formulas. The proof, in this case, requires another notion of equivalence— equivalence with respect to stuttering and is much more difficult since it is possible for two inequivalent states to have exactly the same finite behaviors (modulo stuttering), but different infinite behaviors. We also give a polynomial algorithm for determining if two structures are stuttering equivalent and discuss the relevance of our results for temporal logic model checking and synthesis procedures.