Formal Methods for RL

In many real-world scenarios we need to consider formal requirements, such as safety and constraint compliance, to deploy Reinforcement Learning (RL) agents. In this paper we introduce RecReg, a safe RL framework capable of enforcing non-Markovian regular safety properties via recovery-based shielding. RecReg is a unified approach that supports different levels of a priori knowledge and safety guarantees. The paper primarily focuses on building the theoretical foundations for RecReg in the tabular setting, providing rigorous safety and convergence guarantees that are not present in related work. We provide an extensive set of empirical results using the MASA-Safe-RL library, and we outline an extension of RecReg to deep RL domains using SOTA model-based RL. Our empirical results demonstrate that RecReg achieves efficient learning and matches the theory in this paper.

Enhancing the agentic capabilities of Large Language Models (LLMs) to tackle complex software engineering (SWE) and general terminal-usage problems is a pressing challenge. Current state-of-the-art LLMs often go through large-scale agentic Reinforcement Learning (RL). However, they mostly rely on high-level outcome rewards, with minimal consideration of low-level toolcall and mid-level workflow-stage rewards, which makes the training less effective. To overcome this limitation, we propose Lean4AgenticRL, which, to the best of our knowledge, is the first framework that grounds multi-level process rewards in the unified Lean4 formalization schema for open-ended agentic RL. Lean4AgenticRL first builds a problem-specific workflow system that balances the agent's own capability and controllability during execution, laying the foundation for further verification. Subsequently, our workflow formalization and multi-level reward annotation methods enable agent behavior verification within a general harness system. It provides multi-level rewards to support more effective agentic RL with the workflow-stage-normalized GRPO algorithm. Using Lean4AgenticRL, we train LAR-9B/27B, a family of models with enhanced agentic capability. Extensive experiments indicate that our model outperforms the leading model of similar size. LAR-27B achieves 77.3% on TerminalBench-2.1. On average, our models outperform the base model by 9.89% and surpass the second-best model of comparable size we know by 5.07%. These results indicate the potential of integrating formal methods in enhancing open-ended agentic RL.

Formal specification is a powerful tool to guide the learning process and provides significant advantages over reward shaping: (1) mathematical rigor; (2) expressiveness to specify objectives and constraints, and (3) the ability to define tactics to achieve objectives. However, these benefits remain largely unexplored in the context of Multi-Agent Reinforcement Learning (MARL). This paper introduces HyPOLE, a novel framework for MARL under partial observability, where learning is guided by the expressive power of the so-called hyperproperties and, in particular, the temporal logic HyperLTL. We integrate Centralized Training for Decentralized Execution (CTDE) techniques with HyPOLE to synthesize decentralized policies, and our evaluation on SMAC, MessySMAC, and WildFire benchmark demonstrates clear advantages over baselines.

Multi-agent reinforcement learning (MARL) enables agents to develop coordination strategies through emergent communication, but neural policies lack the formal safety guarantees required for safety-critical robotic deployment in drone swarms and autonomous vehicle fleets. We present the first end-to-end framework for safety verification of learned multi-agent communication policies through policy abstraction: neural policies are distilled into interpretable decision trees, then formally verified, with empirical validation confirming that verified safety properties transfer to original networks. Our four-stage pipeline consists of domain-specific feature extraction from agent observations, decision tree distillation achieving 97.9% +/- 1.2% fidelity to neural policies, automated translation to PRISM probabilistic model checker specifications with complete feature-to-state-variable correspondence, and compositional verification of Probabilistic Computation Tree Logic (PCTL) properties via pairwise decomposition with union-bound aggregation and empirical neighbor modeling. Evaluating Vector-Quantized Variational Information Bottleneck (VQ-VIB) policies for multi-drone coordination with 5-7 agents, we verify 18 temporal logic properties across safety, liveness, and cooperation, achieving 88.9% property satisfaction with all five safety thresholds satisfied (0.3% collision probability vs. 1% threshold). Monte Carlo validation of original neural policies confirms that verified safety properties transfer with <=0.6 percentage-point deviation (95% CI). Discrete VQ-VIB messages provide +11.6 to +13.6 percentage-point fidelity advantages over continuous methods, enabling 3-4x faster verification. Our framework provides empirically validated safety verification for distilled policy abstractions, serving as a practical bridge between deep MARL and formal safety workflows for multi-robot deployment.

Safe exploration is a key challenge in Reinforcement Learning (RL) that aims to prevent agents from making harmful decisions while exploring their environment. Safe exploration is a key challenge in Reinforcement Learning (RL) that aims to prevent agents from making harmful decisions while exploring their environment. Shielding is one such technique that assumes domain knowledge in the form of an environment model to decide upon action safety. Although well-established, shielding has seen limited adoption in RL due to the lack of accessible end-to-end infrastructure connecting formal shield synthesis with standard RL frameworks. Applying shielding typically requires expertise in formal methods and substantial engineering effort, keeping it outside the typical RL workflow. We address this by extending our shield synthesis tool Tempest into a practical backend for safe RL. Our core contribution is tempestpy, a Python library that integrates Tempest-based shield synthesis directly into the Gymnasium API, allowing shields to be synthesized and deployed within existing RL pipelines. This lowers the barrier to entry for shielding and turns formal safe-exploration methods into a usable component for RL practitioners. We also extend Tempest's algorithmic support to compute sound shields for stochastic multiplayer games, preserving formal safety guarantees. We demonstrate the resulting workflow end to end and evaluate shielded and unshielded RL across multiple environments. To facilitate modeling, we provide symbolic models for MiniGrid and introduce MiniGridSafe, a collection of playground environments designed to make shielding easily accessible and experimentally transparent. MiniGridSafe extends MiniGrid with safety-oriented scenarios featuring probabilistic transitions and additional agents, enabling the study of challenging safety aspects in a simple and intuitive setting.

This work presents a logic-driven framework to evaluate the performance of reinforcement learning (RL) algorithms in their ability to generalize to unseen tasks. Our framework defines a family of inductive reach-avoid tasks, characterized by structural similarities in task dynamics, enabling evaluation of generalization capabilities. We introduce a neural certificate function that validates trajectories generated by RL algorithms by enforcing key conditions, thereby serving as a litmus test for RL generalization. We empirically demonstrate our method's capability in certifying generalization for several state-of-the-art generalizable RL algorithms on challenging continuous environments. Our results show that a lower percentage of certificate function violations correlates with a higher number of test tasks successfully solved, highlighting the effectiveness of our framework in evaluating and distinguishing generalization capabilities of RL algorithms. This work provides a principled approach for benchmarking RL generalization.

Certification methods for stochastic systems provide sufficient proof rules, based on real-valued supermartingale certificates, to determine the almost-sure satisfaction of $ω$-regular properties (and therefore of linear temporal logic) over general state spaces, encompassing both countably infinite and continuous state spaces. Conversely, reinforcement learning (RL) methods for $ω$-regular tasks have received considerable attention, but they typically lack formal guarantees that the learned policy satisfies the specification, except possibly for finite state and action spaces. We bridge these two lines of research by establishing a novel theoretical connection: under an appropriate reward, the value function associated to a policy that almost surely satisfies an $ω$-regular property encodes a Streett supermartingale certificate for that specification. Our results, validated experimentally on finite Markov decision processes, hold for finite, countably infinite, and continuous state spaces, suggesting a principled route to certificate synthesis via RL.

Discounted reinforcement learning is usually presented through Bellman equations on closed Markov decision processes. This paper develops a compositional view: a one-step decision process is treated as an open stochastic component, and infinite-horizon policy evaluation is obtained by closing a contractive feedback loop. The resulting semantics assigns typed Bellman transformers to open components, interprets series and parallel wiring as composition and tensoring of transformers, and interprets feedback as an admissible guarded Banach trace realized by a unique fixed point. This perspective yields three theoretical consequences. First, approximate component equivalence is a contextual congruence for admitted well-typed guarded one-hole contexts: local operator error remains controlled after plugging the component into a surrounding circuit that uses the hole once and whose feedback nodes have certified uniform guardedness. Second, exact and approximate state abstractions become commuting or near-commuting coalgebraic diagrams, giving value-preservation and explicit sup-norm distortion bounds. Third, under monotone $ω$-continuous contract-transformer semantics, safety, risk, and resource specifications can be represented as quantale-valued contracts, where local inductive bounds lift through wiring and feedback by least-fixed-point reasoning. Its central claim is not that all RL morphisms form a global traced monoidal category, but that discounted Bellman evaluation admits a contractive feedback semantics on the admissible class of guarded circuits.

Aging bridge networks require proactive, verifiable, and interpretable maintenance strategies, yet reinforcement learning (RL) policies trained solely on reward signals provide no formal safety guarantees and remain opaque to infrastructure managers. We demonstrate COOL-MC as a tool for verifying and explaining RL policies for multi-bridge network maintenance, building on a single-bridge Markov decision process (MDP) from the literature and extending it to a parallel network of three heterogeneous bridges with a shared periodic budget constraint, encoded in the PRISM modeling language. We train an RL agent on this MDP and apply probabilistic model checking and explainability methods to the induced discrete-time Markov chain (DTMC) that arises from the interaction between the learned policy and the underlying MDP. Probabilistic model checking reveals that the trained policy has a safety-violation probability of 3.5% over the planning horizon, being slightly above the theoretical minimum of 0% and indicating the suboptimality of the learned policy, noting that these results are based on artificially constructed transition probabilities and deterioration rates rather than real-world data, so absolute performance figures should be interpreted with caution. The explainability analysis further reveals, for instance, a systematic bias in the trained policy toward the state of bridge 1 over the remaining bridges in the network. These results demonstrate COOL-MC's ability to provide formal, interpretable, and practical analysis of RL maintenance policies.

Safety-critical task planning in robotic systems remains challenging: classical planners suffer from poor scalability, Reinforcement Learning (RL)-based methods generalize poorly, and base Large Language Models (LLMs) cannot guarantee safety. To address this gap, we propose safety-generalizable large language models, named SafeGen-LLM. SafeGen-LLM can not only enhance the safety satisfaction of task plans but also generalize well to novel safety properties in various domains. We first construct a multi-domain Planning Domain Definition Language 3 (PDDL3) benchmark with explicit safety constraints. Then, we introduce a two-stage post-training framework: Supervised Fine-Tuning (SFT) on a constraint-compliant planning dataset to learn planning syntax and semantics, and Group Relative Policy Optimization (GRPO) guided by fine-grained reward machines derived from formal verification to enforce safety alignment and by curriculum learning to better handle complex tasks. Extensive experiments show that SafeGen-LLM achieves strong safety generalization and outperforms frontier proprietary baselines across multi-domain planning tasks and multiple input formats (e.g., PDDLs and natural language).

The next generation of autonomous agents must not only learn efficiently but also act reliably and adapt their behavior in open worlds. Standard approaches typically assume fixed tasks and environments with little or no novelty, which limits world models' ability to support agents that must evolve their policies as conditions change. This paper outlines a vision for foundation world models: persistent, compositional representations that unify reinforcement learning, reactive/program synthesis, and abstraction mechanisms. We propose an agenda built around four components: (i) learnable reward models from specifications to support optimization with clear objectives; (ii) adaptive formal verification integrated throughout learning; (iii) online abstraction calibration to quantify the reliability of the model's predictions; and (iv) test-time synthesis and world-model generation guided by verifiers. Together, these components enable agents to synthesize verifiable programs, derive new policies from a small number of interactions, and maintain correctness while adapting to novelty. The resulting framework positions foundation world models as a substrate for learning, reasoning, and adaptation, laying the groundwork for agents that not only act well but can explain and justify the behavior they adopt.

Safe and interpretable sequential decision-making is critical in healthcare, yet reinforcement learning (RL) policies for sepsis treatment optimization remain opaque and difficult to verify. Standard probabilistic model checkers operate on the full state space, which becomes infeasible for larger MDPs, and cannot explain why a learned policy makes particular decisions. COOL-MC wraps the model checker Storm but adds three key capabilities: it constructs only the reachable state space induced by a trained policy, yielding a smaller discrete-time Markov chain amenable to verification even when full-MDP analysis is intractable; it automatically labels states with clinically meaningful atomic propositions; and it integrates explainability methods with probabilistic computation tree logic (PCTL) queries to reveal which features drive decisions across treatment trajectories. We demonstrate COOL-MC's capabilities on the ICU-Sepsis MDP, a benchmark derived from approximately 17,000 sepsis patient records, which serves as a case study for applying COOL-MC to the formal analysis of sepsis treatment policies. Our analysis establishes hard bounds via full MDP verification, trains a safe RL policy that achieves optimal survival probability, and analyzes its behavior via PCTL verification and explainability on the induced DTMC. This reveals, for instance, that our trained policy relies predominantly on prior dosing history rather than the patient's evolving condition, a weakness that is invisible to standard evaluation but is exposed by COOL-MC's integration of formal verification and explainability. Our results illustrate how COOL-MC could serve as a tool for clinicians to investigate and debug sepsis treatment policies before deployment.

Zero-shot reinforcement learning (RL) has emerged as a setting for developing general agents, capable of solving downstream tasks without additional training or planning at test-time. While conventional RL optimizes policies for fixed rewards, zero-shot RL requires learning representations that enable immediate adaptation to arbitrary reward functions. As the field matures, the growing diversity of approaches demands a foundational framework reconciling different perspectives under a common unifying structure. In this work, we introduce a formal, unified framework for zero-shot RL, allowing for rigorous comparisons across methods. We propose a taxonomy organizing the algorithmic landscape along two levels: representation, distinguishing between compositional and direct methods based on their exploitation of action-value function decompositions; and learning paradigm, differentiating between reward-free and pseudo reward-free training. Additionally, we propose a unified view of existing error bounds, decomposing the total error into three primary contributing components: inference, reward, and approximation, serving as a foundation for more grounded comparisons of zero-shot methods.