Probabilistic propositional temporal logics

Abstract

We present two (closely-related) propositional probabilistic temporal logics based on temporal logics of branching time as introduced by Ben-Ari, Pnueli, and Manna (Acta Inform. 20 (1983), 207–226), Emerson and Halpern (“Proceedings, 14th ACM Sympos. Theory of Comput.,” 1982, pp. 169–179, and Emerson and Clarke (Sci. Comput. Program. 2 (1982), 241–266). The first logic, PTL_f, is interpreted over finite models, while the second logic, PTL_b, which is an extension of the first one, is interpreted over infinite models with transition probabilities bounded away from 0. The logic PTL_f allows us to reason about finite-state sequential probabilistic programs, and the logic PTL_b allows us to reason about (finite-state) concurrent probabilistic programs, without any explicit reference to the actual values of their state-transition probabilities. A generalization of the tableau method yields deterministic single-exponential time decision procedures for our logics, and complete axiomatizations of them are given. Several meta-results, including the absence of a finite-model property for PTL_b, and the connection between satisfiable formulae of PTL_b and finite state concurrent probabilistic programs, are also discussed.