Small Decision Trees for MDPs with Deductive Synthesis

Markov decision processes (MDPs) describe decision making subject to probabilistic uncertainty. A classical problem on MDPs is to compute a policy, selecting actions in every state, that maximizes the probability of reaching a dedicated set of target states. Computing such policies in tabular form is efficiently possible via standard algorithms. However, for further processing by either humans or machines, policies should be represented concisely, e.g., as a decision tree. This paper considers finding (almost) optimal decision trees of minimal depth and contributes a deductive synthesis approach. Technically, we combine pruning the space of concise policies with an abstraction-refinement loop with an SMT-encoding that maps candidate policies into decision trees. Our experiments show that this approach beats the state-of-the-art solver using an MILP encoding by orders of magnitude. The approach also pairs well with heuristic approaches that map a fixed policy into a decision tree: for an MDP with 1.5M states, our approach reduces the size of the given tree by 90%, while sacrificing only 1% of the optimal performance.