Omega-Regular Reward Machines

Updated 18 July 2025

Omega-regular reward machines are a framework integrating omega-regular automata with reinforcement learning to enforce long-run qualitative and quantitative task objectives.
They combine temporal logic specifications with product MDP constructions to enable strategy synthesis that satisfies infinite-trace acceptance conditions.
Applications include safe robotics, adaptive control, and formal verification, where policies must adhere to both liveness constraints and performance metrics.

An omega-regular reward machine is a mathematical and algorithmic framework that combines the formal expressiveness of omega-regular languages (used to specify properties of infinite sequences, such as those expressible by Linear Temporal Logic or Büchi automata) with the operational mechanism of reward machines commonly used in reinforcement learning (RL) to provide guidance for agents on complex, temporally extended tasks. This construction facilitates the specification, reward assignment, and automated synthesis of strategies or policies that must satisfy both quantitative performance criteria and long-run qualitative behavioral constraints in stochastic environments.

1. Formal Definition and Theoretical Foundations

An omega-regular reward machine (ω-RM) is formally defined as a structure combining an automaton with omega-regular acceptance (typically Büchi or parity conditions) and a reward function mapped onto its transitions. Specifically, a typical formalization is:

$\mathcal{R} = (\Sigma, U, u_0, \delta, \rho, F)$

where:

$\Sigma$ is the input alphabet (often $2^{AP}$ for atomic propositions),
$U$ is a finite set of states (nodes of the automaton),
$u_0 \in U$ is the initial state,
$\delta: U \times \Sigma \to 2^U$ is the (possibly nondeterministic) transition function,
$\rho: U \times \Sigma \times U \to \mathbb{R}$ is the scalar reward function on transitions,
$F \subseteq U \times \Sigma \times U$ is the set of accepting (Büchi) transitions, which encode the omega-regular constraint.

A run $w = w_0w_1\ldots$ is accepted if the set of transitions in $F$ is visited infinitely often (the Büchi condition). The reward along a path is accumulated according to $\rho$ . This dual structure enables specification of both qualitative (correctness, fairness, liveness) and quantitative (cost, mean-payoff, discounted) objectives (Hahn et al., 2023).

2. Algorithmic and Learning Frameworks

Omega-regular reward machines serve as the backbone for algorithms that synthesize or learn strategies in Markov Decision Processes (MDPs) or their extensions. The standard workflow is:

Specification: The desired behavior is written as an omega-regular property, often in LTL or as a Büchi automaton. Quantitative aspects are added via transition rewards.
Product Construction: Form the product between an MDP $\mathcal{M} = (S, s_0, A, T, AP, L)$ and the ω-RM $\mathcal{R}$ , yielding an extended MDP whose states are pairs $(s, u)$ and whose transitions are guided by both $T$ and $\delta$ .
Reduction to RL: The resulting product MDP is used as the basis for applying reinforcement learning algorithms (e.g., Q-learning, UCBVI, Differential Q-learning), with the reward function inherited from $\rho$ and the acceptance structure guiding episodic/mean/discounted reward assignment.
Optimization Objective: A lexicographic optimization principle is often employed—first, maximize the probability of satisfying the omega-regular (accepting) condition, and second, among all maximally satisfying policies, select one maximizing expected (discounted or mean) reward (Hahn et al., 2023, Křetínský et al., 2018, Le et al., 16 Oct 2024).

A central technical nuance is that, unlike Markovian reward functions, optimization over ω-RM requires incorporating infinite-memory automaton states and enforcing acceptance conditions that cannot be checked using local or memoryless mechanisms (Hahn et al., 2023, Le et al., 16 Oct 2024, Kazemi et al., 21 May 2025).

3. Expressiveness, Applications, and Variants

Omega-regular reward machines strictly subsume regular reward machines, as they can encode specifications over infinite traces, including fairness and liveness constraints. Notable research extensions include:

Hierarchies of Reward Machines: Nested compositions of (omega-)RMs to modularize long-horizon or sparse reward tasks, decomposing them into reusable subprocedures, with exponentially more compact representations than flat automata (Furelos-Blanco et al., 2022).
Robustness Quantification: Quantitative refinement of ω-regular languages for robustness, associating each infinite trace with a robustness value via semantic ranking in the automaton's strongly connected component hierarchy; this can be used to design reward machines that grade not just acceptance but degrees of satisfaction (Fisman et al., 16 Mar 2025).
Mean-Payoff and Lexicographic Rewards: Integration with average or mean-payoff objectives, often in continuing (non-episodic) environments. For instance, model-free RL via Differential Q-learning or Q-learning in the average-reward criterion directly supports infinite-horizon tasks such as those specified by absolute liveness properties (Kazemi et al., 21 May 2025).
Noisy or Uncertain Symbolic Environments: Handling noisy evaluations of propositions (e.g., in POMDP settings), where the reward machine's grounding is probabilistic and state inference is performed by history-dependent (possibly learned) observation models (Li et al., 31 May 2024).

Common applications are found in formal verification and synthesis (correct-by-construction RL), safe robotics, adaptive control subject to high-level mission constraints, and automated repair/synthesis of controllers that must satisfy temporal logic requirements in uncertain environments.

4. Reduction Techniques and Learning Guarantees

A central insight of contemporary research is that RL problems with $\omega$ -regular objectives can be optimally reduced to average-reward, mean-payoff, or discounted-reward RL tasks in an explicit, structure-preserving fashion using omega-regular reward machines. Key results include:

Optimality-Preserving Reductions: There exist reward machine constructions such that the limit-average reward under any policy matches the satisfaction probability of the target $\omega$ -regular objective; this is crucially achieved only using reward machines with memory, not simple per-step reward functions (Le et al., 16 Oct 2024).
Average-Reward RL for Omega-Regular Objectives: By assigning rewards only on transitions within specific accepting end components of the product MDP, average reward maximization yields policies that are (asymptotically) optimal with respect to the original $\omega$ -regular objective. Model-free learning is supported via algorithms such as Differential Q-learning, which suit continuing tasks without episodic resetting (Kazemi et al., 21 May 2025).
Sequence of Discounted Problems: It is shown that optimal policies for the average-reward problem can be approximated (Blackwell optimality) by solving a sequence of classic discounted RL problems with discount factor $\gamma \to 1$ (Le et al., 16 Oct 2024).

Moreover, structural properties and tight bounds have been established for the sample and memory complexity required—for example, finite-memory strategies yield high-probability (not almost-sure) mean-payoff guarantees under omega-regular constraints (Křetínský et al., 2018).

5. Implications, Expressiveness, and Explainability

The main implications and advantages of omega-regular reward machines include:

Unified Specification and Learning: They reconcile the need for high-level, interpretable, temporal logic specifications (e.g., “infinitely often service each request before a deadline”) with conventional RL approaches that require scalar, local rewards.
Expressive Modularization: Their automata-based structure supports loops, sequencing, branching, and hierarchy, enabling efficient transfer (e.g., via hierarchy or abstraction (Furelos-Blanco et al., 2022, Azran et al., 2023)), curriculum generation (Koprulu et al., 2023), and robustification (Fisman et al., 16 Mar 2025).
Correctness and Verification: Learning agents can be synthesized with formal correctness certificates against non-Markovian specifications; post-hoc verification via model checking on the product automaton is directly supported (Hahn et al., 2021).
Explainability: Policies derived using omega-regular reward machines inherit the transparently specified behavioral goals of the underlying temporal logic, supporting interpretability and explanation in safety-critical and autonomous systems (Le et al., 16 Oct 2024).

A limitation lies in potential state-space explosion in the full product MDP, though techniques such as on-the-fly construction, abstraction, or state-reduction alleviate these challenges in practice (Hahn et al., 2023).

6. Extensions: Synthesis, Inference, and Robustness

Recent advances include methods for learning the structure of reward machines from partially observed policies (using prefix tree policies and SAT-based synthesis), extending their practical applicability to inverse RL and real-world robotics (Shehab et al., 6 Feb 2025). Mechanisms for quantifying robustness of satisfaction for omega-regular properties have yielded semantic, preference-based grading that can be embedded in the design of reward machines to distinguish not just correctness but degrees of resilience or promptness (Fisman et al., 16 Mar 2025).

For continuous-time systems (CTMDPs), translation of omega-regular specifications into scalar rewards for RL extends the classical theory to domains with dense-time, enabling average and expected residence-time semantics (Falah et al., 2023).

7. Practical Considerations and Empirical Results

Empirical studies indicate that omega-regular reward machines, when integrated with model-free RL algorithms such as Q-learning, Differential Q-learning, or sample-efficient exploration strategies, are effective in producing verifiably correct and often more sample-efficient policies in both tabular and deep RL settings, robust to specification structure and environment variation (Hahn et al., 2023, Kazemi et al., 21 May 2025, Lin et al., 19 Aug 2024).

The average-reward criterion, as exemplified by large-scale experiments, is particularly suited to infinite-horizon, continuing RL problems and addresses shortcomings in discount-based approaches that rely on episodic resets, which mismatch the semantics of omega-regular objectives (Kazemi et al., 21 May 2025).

Omega-regular reward machines constitute a principled, expressive, and implementable bridge between the formal world of temporal logic specifications and the practical requirements of reinforcement learning. They support both the specification and learning of strategies for complex, temporally extended objectives, equipped with both theoretical guarantees and empirical effectiveness.