Stochastic Fairness and Language-Theoretic Fairness in Planning on Nondeterministic Domains

TL;DR

This paper compares two fairness notions in nondeterministic planning, clarifies their differences for extended goals, and introduces an optimal algorithm for state-action fair planning with LTL goals.

Contribution

It distinguishes stochastic and language-theoretic fairness, corrects misconceptions, and provides an optimal algorithm for state-action fair planning with LTL/LTLf goals.

Findings

Stochastic fairness is more well-behaved than state-action fairness.

The paper corrects previous algorithmic assumptions for state-action fairness.

Provides a lower bound proof for goal complexity in fair planning.

Abstract

We address two central notions of fairness in the literature of planning on nondeterministic fully observable domains. The first, which we call stochastic fairness, is classical, and assumes an environment which operates probabilistically using possibly unknown probabilities. The second, which is language-theoretic, assumes that if an action is taken from a given state infinitely often then all its possible outcomes should appear infinitely often (we call this state-action fairness). While the two notions coincide for standard reachability goals, they diverge for temporally extended goals. This important difference has been overlooked in the planning literature, and we argue has led to confusion in a number of published algorithms which use reductions that were stated for state-action fairness, for which they are incorrect, while being correct for stochastic fairness. We remedy this and provide an optimal sound and complete algorithm for solving state-action fair planning for LTL/LTLf goals, as well as a correct proof of the lower bound of the goal-complexity (our proof is general enough that it provides new proofs also for the no-fairness and stochastic-fairness cases). Overall, we show that stochastic fairness is better behaved than state-action fairness.