Probabilistic temporal logics for finite and bounded models

We present two (closely-related) propositional probabilistic temporal logics based on temporal logics of branching time as introduced by Ben-Ari, Pnueli and Manna and by Clarke and Emerson. The first logic, <italic>PTL<subscrpt>f</subscrpt></italic>, is interpreted over finite models, while the second logic, <italic>PTL<subscrpt>b</subscrpt></italic>, which is an extension of the first one, is interpreted over infinite models with transition probabilities bounded away from 0. The logic <italic>PTL<subscrpt>f</subscrpt></italic> allows us to reason about finite-state sequential probabilistic programs, and the logic <italic>PTL<subscrpt>b</subscrpt></italic> 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 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 <italic>PTL<subscrpt>b</subscrpt></italic>, and the connection between satisfiable formulae of <italic>PTL<subscrpt>b</subscrpt></italic> and finite state concurrent probabilistic programs, are also discussed.