Probability-Raising Causality for Uncertain Parametric Markov Decision Processes with PAC Guarantees

Abstract: Recent decision-making systems are increasingly complicated, making it crucial to verify and understand their behavior for a given specification. A promising approach is to comprehensively explain undesired behavior in the systems modeled by Markov decision processes (MDPs) through formal verification and causal reasoning. However, the reliable explanation using model-based probabilistic causal analysis has not been explored when the MDP's transition probabilities are uncertain. This paper proposes a method to identify potential causes of undesired behaviors in an uncertain parametric MDP (upMDP) using parameter sampling, model checking, and a set covering for the samples. A cause is defined as a subset of states based on a probability-raising principle. We show that the probability of each identified subset being a cause exceeds a specified threshold. Further, a lower bound of the probability that the undesired paths visit the subsets is maximized as much as possible while satisfying a nonredundancy condition. While computing these probabilities is complicated, this study derives probabilistically approximately correct lower bounds of both probabilities by the sampling. We demonstrate the effectiveness of the proposed method through a path-planning scenario.