Reasoning with time and chance

Abstract

The temporal propositional logic of linear time is generalized to an uncertain world, in which random events may occur. The formulas do not mention probabilities explicitly, i.e., the only probability appearing explicitly in formulas is probability one. This logic is claimed to be useful for stating and proving properties of probabilistic programs. It is convenient for proving those properties that do not depend on the specific distribution of probabilities used in the program's random draws. The formulas describe properties of execution sequences. The models are stochastic systems, with state transition probabilities. Three different axiomatic systems are proposed and shown complete for general models, finite models, and models with bounded transition probabilities, respectively. All three systems are decidable, by the results of Rabin ( Trans. Amer. Math. Soc. 141 (1969), 1–35).