\almost Always" and \deenitely Sometime" Are Not Enough: Probabilistic Quantiiers and Probabilistic Model-checking 1

Speciications for probabilistic programs often use the notion of almost always and deenitely sometime to capture the probabilistic information. But there are a number of instances (eg. network protocols) where probabilistic information needs to be explicitly speciied. In this paper we present PCTL , a probabilistic version of the branching time logic CTL , where the quantiiers for universality (almost always) and existence (deenitely sometime) are replaced by a single probabilistic quantiier. We argue that this logic is more general than previous logics. We show how to model-check probabilistic programs (given as markov chains) against speciications in PCTL. We also prove our algorithm correct and provide an analysis of its computational complexity.