Interval Markov Decision Processes with Multiple Objectives: From Robust Strategies to Pareto Curves

Accurate Modelling of a real-world system with probabilistic behaviour is a difficult task. Sensor noise and statistical estimations, among other imprecisions, make the exact probability values impossible to obtain. In this article, we consider Interval Markov decision processes (IMDPs), which generalise classical MDPs by having interval-valued transition probabilities. They provide a powerful modelling tool for probabilistic systems with an additional variation or uncertainty that prevents the knowledge of the exact transition probabilities. We investigate the problem of robust multi-objective synthesis for IMDPs and Pareto curve analysis of multi-objective queries on IMDPs. We study how to find a robust (randomised) strategy that satisfies multiple objectives involving rewards, reachability, and more general ω-regular properties against all possible resolutions of the transition probability uncertainties, as well as to generate an approximate Pareto curve providing an explicit view of the trade-offs between multiple objectives. We show that the multi-objective synthesis problem is PSPACE-hard and provide a value iteration-based decision algorithm to approximate the Pareto set of achievable points. We finally demonstrate the practical effectiveness of our proposed approaches by applying them on several case studies using a prototype tool.