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Regular decision processes (RDPs) are a subclass of non- Markovian decision processes where the transition and reward functions are guarded by some regular property of the past (a lookback). While RDPs enable intuitive and succinct rep- resentation of non-Markovian decision processes, their ex- pressive power coincides with finite-state Markov decision processes (MDPs). We introduce omega-regular decision pro- cesses (ODPs) where the non-Markovian aspect of the transi- tion and reward functions are extended to an ω-regular looka- head over the system evolution. Semantically, these looka- heads can be considered as promises made by the decision maker or the learning agent about her future behavior. In par- ticular, we assume that if the promised lookaheads are not fulfilled, then the decision maker receives a payoff of ⊥ (the least desirable payoff), overriding any rewards collected by the decision maker. We enable optimization and learning for ODPs under the discounted-reward objective by reducing them to lexicographic optimization and learning over finite MDPs. We present experimental results demonstrating the effectiveness of the proposed reduction. 

Reinforcement learning (RL) is a powerful approach for training agents to perform tasks, but designing an appropriate re- ward mechanism is critical to its success. However, in many cases, the complexity of the learning objectives goes beyond the capabili- ties of the Markovian assumption, necessitating a more sophisticated reward mechanism. Reward machines and ω-regular languages are two formalisms used to express non-Markovian rewards for quantita- tive and qualitative objectives, respectively. This paper introduces ω- regular reward machines, which integrate reward machines with ω- regular languages to enable an expressive and effective reward mech- anism for RL. We present a model-free RL algorithm to compute ε-optimal strategies against ω-regular reward machines and evaluate the effectiveness of the proposed algorithm through experiments. 

We characterize the class of nondeterministic 𝜔-automata that can be used for the analysis of finite Markov decision processes (MDPs). We call these automata ‘good-for-MDPs’ (GFM). We show that GFM automata are closed under classic simulation as well as under more powerful simulation relations that leverage properties of optimal control strategies for MDPs. This closure enables us to exploit state-space reduction techniques, such as those based on direct and delayed simulation, that guarantee simulation equivalence. We demonstrate the promise of GFM automata by defining a new class of automata with favorable properties—they are Büchi automata with low branching degree obtained through a simple construction—and show that going beyond limit-deterministic automata may significantly benefit reinforcement learning.