Markov Decision Processes with Functional Rewards

Markov decision processes (MDP) have become one of the standard models for decision-theoretic planning problems under uncertainty. In its standard form, rewards are assumed to be numerical additive scalars. In this paper, we propose a generalization of this model allowing rewards to be functional. The value of a history is recursively computed by composing the reward functions. We show that several variants of MDPs presented in the literature can be instantiated in this setting. We then identify sufficient conditions on these reward functions for dynamic programming to be valid. In order to show the potential of our framework, we conclude the paper by presenting several illustrative examples.