A decision algorithm for full propositional temporal logic

Abstract

The paper presents an efficient algorithm for checking the satisfiability of a propositional linear time temporal logic formula, which may have past as well as future operators. This algorithm can be used to check validity of such formulas over all models as well as over computations of a finite-state program (model checking). Unlike previous theoretical presentations of a decision method for checking satisfiability or validity, whose first step is to construct the full set of all possible atoms of a tableau (satisfaction graph) and immediately pay the worst case exponential complexity price, the algorithm presented here builds the tableau incrementally. This means that the algorithm constructs only those atoms that are reachable from a possible initial atom, satisfying the formula to be checked.While incremental tableau construction for the future fragment of linear time temporal logic can be done in a single pass, the presence of past operators requires multiple passes that successively construct augmented versions of existing atoms, while still maintaining consistency and reachability.The proof of correctness of the algorithm is based on showing that any model of the considered formula is embedded as a path in the tableau at all the construction stages, and can be delineated when the construction terminates.The paper also describes an implementation of the algorithm with further attention to efficiency. This implementation is available as a support system for the book [8] under the name “temporal prover”. It has been used to verify all the propositional temporal formulas and to model-check all the finite-state programs appearing in the book.Keywordstemporal logicsatisfiability checkingvalidity checkingmodel checkingpast and future operatorsincremental tableauautomatic verification