Reinforcement Learning with $\omega$-Regular Objectives and Constraints

Reinforcement learning (RL) commonly relies on scalar rewards with limited ability to express temporal, conditional, or safety-critical goals, and can lead to reward hacking. Temporal logic expressible via the more general class of $\omega$-regular objectives addresses this by precisely specifying rich behavioural properties. Even still, measuring performance by a single scalar (be it reward or satisfaction probability) masks safety-performance trade-offs that arise in settings with a tolerable level of risk. We address both limitations simultaneously by combining $\omega$-regular objectives with explicit constraints, allowing safety requirements and optimisation targets to be treated separately. We develop a model-based RL algorithm based on linear programming, which in the limit produces a policy maximising the probability of satisfying an $\omega$-regular objective while also adhering to $\omega$-regular constraints within specified thresholds. Furthermore, we establish a translation to constrained limit-average problems with optimality-preserving guarantees.