Minimal counterexamples for linear-time probabilistic verification

Counterexamples for property violations have a number of important applications like supporting thedebugging of erroneous systems and verifying large systems via counterexample-guided abstraction refinement.In this paper, we propose the usage of minimal critical subsystems of discrete-time Markov chains and Markovdecision processes as counterexamples for violated ω-regular properties. Minimality can thereby be defined interms of the number of states or transitions. This problem is known to be NP-complete for Markov decisionprocesses. We show how to compute such subsystems using mixed integer linear programming and evaluatethe practical applicability in a number of experiments. They show that our method yields substantiallysmaller counterexample than using existing techniques.