Scenario-Based Verification of Uncertain Parametric MDPs

We consider parametric Markov decision processes (pMDPs) that are augmented with unknown probability distributions over parameter values. The problem is to compute the probability to satisfy a temporal logic specification within any concrete MDP that corresponds to a sample from these distributions. As this problem is infeasible to solve precisely, we resort to sampling techniques that exploit the so-called scenario approach. Based on a finite number of samples of the parameters, the proposed method yields high-confidence bounds on the probability of satisfying the specification. The number of samples required to obtain a high confidence on these bounds is independent of the number of states and the number of random parameters. Experiments on a large set of benchmarks show that several thousand samples suffice to obtain tight and high-confidence lower and upper bounds on the satisfaction probability.