Approximate Model Checking of PCTL Involving Unbounded Path Properties

We study the problem of applying statistical methods for approximate model checking of probabilistic systems against properties encoded as PCTL formulas. Such approximate methods have been proposed primarily to deal with state-space explosion that makes the exact model checking by numerical methods practically infeasible for large systems. However, the existing statistical methods either consider a restricted subset of PCTL, specifically, the subset that can only express bounded until properties; or rely on user-specified finite bound on the sample path length. We propose a new method that does not have such restrictions and can be effectively used to reason about unbounded until properties. We approximate probabilistic characteristics of an unbounded until property by that of a bounded until property for a suitably chosen value of the bound. In essence, our method is a two-phase process: (a) the first phase is concerned with identifying the bound k0; (b) the second phase computes the probability of satisfying the k0-bounded until property as an estimate for the probability of satisfying the corresponding unbounded until property. In both phases, it is sufficient to verify bounded until properties which can be effectively done using existing statistical techniques. We prove the correctness of our technique and present its prototype implementations. We empirically show the practical applicability of our method by considering different case studies including a simple infinite-state model, and large finite-state models such as IPv4 zeroconf protocol and dining philosopher protocol modeled as Discrete Time Markov chains.