# Unpredictable Planning Under Partial Observability

## TL;DR

This paper develops a method to synthesize controllers that maximize unpredictability in partially observable Markov decision processes while ensuring task completion, using entropy maximization and nonlinear optimization.

## Contribution

It introduces a novel approach to maximize entropy in POMDPs via finite-state controllers and recasts the problem as parameter synthesis for parametric Markov chains.

## Key findings

- The proposed algorithm effectively maximizes entropy in motion planning scenarios.
- A lower bound on POMDP entropy is established through pMC analysis.
- The method guarantees task completion while enhancing unpredictability.

## Abstract

We study the problem of synthesizing a controller that maximizes the entropy of a partially observable Markov decision process (POMDP) subject to a constraint on the expected total reward. Such a controller minimizes the predictability of a decision-maker's trajectories while guaranteeing the completion of a task expressed by a reward function. First, we prove that a decision-maker with perfect observations can randomize its paths at least as well as a decision-maker with partial observations. Then, focusing on finite-state controllers, we recast the entropy maximization problem as a so-called parameter synthesis problem for a parametric Markov chain (pMC). We show that the maximum entropy of a POMDP is lower bounded by the maximum entropy of this pMC. Finally, we present an algorithm, based on a nonlinear optimization problem, to synthesize an FSC that locally maximizes the entropy of a POMDP over FSCs with the same number of memory states. In numerical examples, we demonstrate the proposed algorithm on motion planning scenarios.