On the Complexity of Semantic Self-minimization

Partial Kripke structures model only parts of a state space and so enable aggressive abstraction of systems prior to verifying them with respect to a formula of temporal logic. This partiality of models means that verifications may reply with true (all refinements satisfy the formula under check), false (no refinement satisfies the formula under check) or don't know. Generalized model checking is the most precise verification for such models (all don't know answers imply that some refinements satisfy the formula, some don't), but computationally expensive. A compositional model-checking algorithm for partial Kripke structures is efficient, sound (all answers true and false are truthful), but may lose precision by answering don't know instead of a factual true or false. Recent work has shown that such a loss of precision does not occur for this compositional algorithm for most practically relevant patterns of temporal logic formulas. Formulas that never lose precision in this manner are called semantically self-minimizing. In this paper we provide a systematic study of the complexity of deciding whether a formula of propositional logic, propositional modal logic or the propositional modal mu-calculus is semantically self-minimizing