Value Functions for Temporal Logic: Optimal Policies and Safety Filters
For researchers in reinforcement learning and formal methods, this work resolves a subtle optimality issue in undiscounted infinite-horizon temporal logic tasks and provides a principled approach to safety filtering for complex specifications.
The paper addresses the pathology where greedy maximization of Q-functions for temporal logic tasks leads to policies that defer task completion indefinitely. It constructs non-Markovian policies based on state history that achieve optimality for nested Until, Globally, and Globally-Until specifications, and extends Q-function-based safety filters to complex temporal logic specifications.
While Bellman equations for basic reach, avoid, and reach-avoid problems are well studied, the relationship between value optimality and policy optimality becomes subtle in the undiscounted infinite-horizon setting, particularly for more complicated tasks. Greedily maximizing the Q-function can produce policies that indefinitely defer task completion for reach-avoid problems, or equivalently, Until specifications, even when the value function is optimal. Building upon recent results decomposing the value function for temporal logic (TL) into a graph of constituent value functions, we construct non-Markovian policies based on state history that avoid this pathology and prove their optimality with respect to the quantitative robustness score for nested Until, Globally, and Globally-Until specifications. We further show how the Q function can serve as a safety filter for complex TL specifications, extending prior results beyond simple avoid or reach-avoid tasks.