Boolean functions and their representation through logics, circuits, machine learning classifiers, or binary decision diagrams (BDDs) play a central role in the design and analysis of computing systems. Quantifying the relative impact of variables on the truth value by means of importance values can provide useful insights to steer system design and debugging. In this paper, we introduce a uniform framework for reasoning about such values, relying on a generic notion of importance value functions (IVFs). The class of IVFs is defined by axioms motivated from several notions of importance values introduced in the literature, including Ben-Or and Linial’s influence and Chockler, Halpern, and Kupferman’s notion of responsibility and blame. We establish a connection between IVFs and game-theoretic concepts such as Shapley and Banzhaf values, both of which measure the impact of players on outcomes in cooperative games. Exploiting BDD-based symbolic methods and projected model counting, we devise and evaluate practical computation schemes for IVFs.

Witnessing subsystems for probabilistic reachability thresholds in discrete Markovian models are an important concept both as diagnostic information on why a property holds, and as input to refinement algorithms.We present SWITSS, a tool for the computation of Small WITnessing SubSystems.SWITSS implements exact and heuristic approaches based on reducing the problem to (mixed integer) linear programming.Returned subsystems can automatically be rendered graphically and are accompanied with a certificate which proves that the subsystem is indeed a witness.