Symbolic Dynamic Programming for Discrete and Continuous State MDPs

Abstract

Many real-world decision-theoretic planning problems can be naturally modeled with discrete and continuous state Markov decision processes (DC-MDPs). While previous work has addressed automated decision-theoretic planning for DC-MDPs, optimal solutions have only been defined so far for limited settings, e.g., DC-MDPs having hyper-rectangular piecewise linear value functions. In this work, we extend symbolic dynamic programming (SDP) techniques to provide optimal solutions for a vastly expanded class of DC-MDPs. To address the inherent combinatorial aspects of SDP, we introduce the XADD — a continuous variable extension of the algebraic decision diagram (ADD) — that maintains compact representations of the exact value function. Empirically, we demonstrate an implementation of SDP with XADDs on various DC-MDPs, showing the first optimal automated solutions to DC-MDPs with linear and nonlinear piecewise partitioned value functions and showing the advantages of constraint-based pruning for XADDs.