On the Uniqueness of Solution for the Bellman Equation of LTL Objectives

TL;DR

This paper investigates the conditions under which the Bellman equation for LTL objectives has a unique solution, highlighting issues with multiple solutions when using certain discount factors, and proposing a condition for uniqueness.

Contribution

It identifies the non-uniqueness problem in Bellman equations with two discount factors and proposes a sufficient condition for ensuring a unique solution in LTL planning.

Findings

Multiple solutions occur when one discount factor is set to one.

A sufficient condition for uniqueness involves solutions inside rejecting BSCC being zero.

The proposed condition separates solutions for discounted and non-discounted states.

Abstract

Surrogate rewards for linear temporal logic (LTL) objectives are commonly utilized in planning problems for LTL objectives. In a widely-adopted surrogate reward approach, two discount factors are used to ensure that the expected return approximates the satisfaction probability of the LTL objective. The expected return then can be estimated by methods using the Bellman updates such as reinforcement learning. However, the uniqueness of the solution to the Bellman equation with two discount factors has not been explicitly discussed. We demonstrate with an example that when one of the discount factors is set to one, as allowed in many previous works, the Bellman equation may have multiple solutions, leading to inaccurate evaluation of the expected return. We then propose a condition for the Bellman equation to have the expected return as the unique solution, requiring the solutions for states inside a rejecting bottom strongly connected component (BSCC) to be 0. We prove this condition is sufficient by showing that the solutions for the states with discounting can be separated from those for the states without discounting under this condition