Computing the Reachability Value of Posterior-Deterministic POMDPs

TL;DR

This paper introduces posterior-deterministic POMDPs, a new class where the reachability probability can be approximated, contrasting with the intractability in general POMDPs.

Contribution

The paper defines posterior-deterministic POMDPs and proves that their reachability probabilities can be approximated arbitrarily closely, expanding computable POMDP classes.

Findings

Posterior-deterministic POMDPs include all MDPs and classical examples like the Tiger POMDP.

Reachability probabilities in these POMDPs can be approximated up to arbitrary precision.

This class is one of the largest where such approximation is possible.

Abstract

Partially observable Markov decision processes (POMDPs) are a fundamental model for sequential decision-making under uncertainty. However, many verification and synthesis problems for POMDPs are undecidable or intractable. Most prominently, the seminal result of Madani et al. (2003) states that there is no algorithm that, given a POMDP and a set of target states, can compute the maximal probability of reaching the target states, or even approximate it up to a non-trivial constant. This is in stark contrast to fully observable Markov decision processes (MDPs), where the reachability value can be computed in polynomial time. In this work, we introduce posterior-deterministic POMDPs, a novel class of POMDPs. Our main technical contribution is to show that for posterior-deterministic POMDPs, the maximal probability of reaching a given set of states can be approximated up to arbitrary precision. A POMDP is posterior-deterministic if the next state can be uniquely determined by the current state, the action taken, and the observation received. While the actual state is generally uncertain in POMDPs, the posterior-deterministic property tells us that once the true state is known it remains known forever. This simple and natural definition includes all MDPs and captures classical non-trivial examples such as the Tiger POMDP (Kaelbling et al. 1998), making it one of the largest known classes of POMDPs for which the reachability value can be approximated.