Probabilistic verification and approximation

We study the existence of efficient approximation methods to verify quantitative specifications of probabilistic systems. Models of such systems are labelled discrete time Markov chains and checking specifications consists of computing satisfaction probabilities of linear temporal logic formulas. We prove that, in general, there is no polynomial time randomized approximation scheme with relative error for probabilistic verification. However, in many applications, specifications can be expressed by monotone formulas or negation of monotone formulas and randomized approximation schemes with absolute error are sufficient. We show how some simple randomized approximation algorithms can improve the efficiency of verification of such probabilistic specifications and can be implemented in a probabilistic model checker.