Approximate Dynamic Programming with Probabilistic Temporal Logic Constraints

In this paper, we develop approximate dynamic programming methods for stochastic systems modeled as Markov Decision Processes, given both soft performance criteria and hard constraints in a class of probabilistic temporal logic called Probabilistic Computation Tree Logic (PCTL). Our approach consists of two steps: First, we show how to transform a class of PCTL formulas into chance constraints that can be enforced during planning in stochastic systems. Second, by integrating randomized optimization and dynamic programming with softmax Bellman operator, we devise a novel trajectory sampling-based approximate value iteration method to iteratively solve for an upper bound on the value function while ensuring the constraints that PCTL specifications are satisfied. Particularly, we show that by the on-policy sampling of the trajectories, a tight bound can be achieved between the upper bound given by the approximation and the true value function. The correctness and efficiency of the method are demonstrated using robotic motion planning examples.