Minimal counterexamples for linear-time probabilistic verification

Counterexamples for property violations have a number of important applications like supporting the debugging 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 Markov decision processes as counterexamples for violated ω-regular properties. Minimality can thereby be defined in terms of the number of states or transitions. This problem is known to be NP-complete for Markov decision processes. We show how to compute such subsystems using mixed integer linear programming and evaluate the practical applicability in a number of experiments. They show that our method yields substantially smaller counterexample than using existing techniques.