Title: Accelerated Learning with Linear Temporal Logic using Differentiable Simulation
Venue: NeurIPS 2025
Accepted: no
Keywords: reinforcement learning

temporal logic

differentiable simulation
Abstract: To ensure learned controllers comply with safety and reliability requirements for reinforcement learning in real-world settings remains challenging. Traditional safety assurance approaches, such as state avoidance and constrained Markov decision processes, often inadequately capture trajectory requirements or may result in overly conservative behaviors. To address these limitations, recent studies advocate the use of formal specification languages such as linear temporal logic (LTL), enabling the derivation of correct-by-construction learning objectives from the specified requirements. However, the sparse rewards associated with LTL specifications make learning extremely difficult, whereas dense heuristic-based rewards risk compromising correctness.
In this work, we propose the first method, to our knowledge, that integrates LTL with differentiable simulators, facilitating efficient gradient-based learning directly from LTL specifications by coupling with differentiable paradigms. Our approach introduces soft labeling to achieve differentiable rewards and states, effectively mitigating the sparse-reward issue intrinsic to LTL without compromising objective correctness. We validate the efficacy of our method through experiments, demonstrating significant improvements in both reward attainment and training time compared to the discrete methods.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nLogical formalisms enable the specification of agents’ behaviors. For instance, they serve as building blocks for formally verifying agents and synthesizing agents’ controllers that guarantee satisfying the specification. In RL, the user needs to engineer a reward signal to align its intention with the induced agent’s behaviors, which is, in practice, very hard. Alternatively, one can use that logic to automatically derive a reward scheme for the agent. In particular, the authors consider LTL; in a nutshell, a return is assigned to each agent’s trajectory depending on its satisfaction of the LTL formula. However, traditional methods lead to a sparse reward scheme, which is not suitable for allowing the agent to learn to satisfy the user specification. The authors propose to use differentiable environments and labeling functions to allow the policy’s gradients to flow through both the environment and specification dynamics. The method relies on considering the labeling function as a probability distribution over the atomic proposition space instead of a deterministic function. Then, doing the product of the automaton translated from the formula with the environment leads to a differentiable environment that can be exploited to derive smooth, low-variance gradients of the policy.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n- In the tasks considered in the paper, the proposed approach empirically confirms the claims of the authors that differentiable LTL allows for avoiding high-variance gradient updates and alleviates the sparse rewards problem\n- The idea of introducing soft labeling to allow differentiating the LTL formula (through its automata construction and generated rewards) is very promising. \n- The experimental section is quite convincing and very detailed.\n# Weaknesses \n- The paper lacks accessibility; in my opinion, Section 3 lacks crucial details for understading the contribution. There is no real introduction to differentiable RL. Also, the formal logic specification part is extremely compact. For instance, for a noephyte, it is unclear what an LDBA actually does (what are \\epsilon-actions, what do they represent, what does a path in the LDBA correspond to, how does the reward scheme of Theorem 1 work, what is beta, and how to choose it?). All those notions are essential to understanding the main contribution and would benefit from additional details and explanations. \n- While being experimentally convincing, the method seems purely empirical and lacks theoretical support. For instance, how does the resulting probability signal relate to Theorem 1? Why is the sigmoid function a good fit for learning the rewards?\n- Since the whole method relies on differentiable environments, to compute the gradients of the policy, I would have liked to see in Algorithm 1, called ‘differentiable RL with LTL’, how and where the gradients are updated. Currently, the algorithm is very generic and does not help understand that point. Note that I’m raising that issue because differentiable RL hasn’t been formally introduced anywhere in the paper.\n- In Figure 3 (or in an additional figure), I think it could also be relevant to also plot the probability of the formula’s satisfaction with respect to the original rewards, i.e., the reward function provided in the considered environment. I think this is important to highlight the advantage of learning from LTL instead of classic reward schemes.\n- The authors restrict themselves to deterministic MDPs and deterministic, stationary policies, and then claim that the goal of the agent is to optimize the expected return over trajectories. If everything is deterministic, we don’t need any expectation/probabilistic measures.\n- Some limitations (listed below) are not acknowledged by the authors.\n\nQUESTIONS:\n- How does the method compare to RL with usual reward schemes (e.g., those proposed in the original environments or reward machines; see e.g. [1])?\n- Could you comment on the concerns raised about the function $g$ (see limitations)?\n- Does the method work on (subclasses of) stochastic environments?\n\n[1] LTL and Beyond: Formal Languages for Reward Function Specification in Reinforcement Learning. Alberto Camacho, Rodrigo Toro Icarte, Toryn Q. Klassen, Richard Valenzano, Sheila A. McIlraith, IJCAI 2019.\n\nLIMITATIONS:\n- The method heavily relies on signal functions $g_a \\colon S \\to \\mathbb{R}$ and thresholds such that the atomic proposition $a$ is part of the label iff $g_a(s) > 0$. On the other hand, the ‘intensity’ of this signal is used (heuristically) to define the probability of the atomic proposition being included in the label. I’m convinced (e.g., through the braking example) that this heuristic is of interest in some robotics applications. However, it is not clear to me how it is useful/sound in general, in particular for safety applications. For instance, hard thresholds might be relevant for safety, and it might be very difficult to design this function. One could, e.g., design a region that the robot is strictly forbidden to visit, whereas it is free to move within the rest of the state space.\n- The authors restrict themselves to deterministic MDPs. If I am not mistaken, several stochastic transition functions (e.g., reparameterizable distributions) are differentiable. So, why not take such functions into account? The proposed method does not seem to require the MDP to be deterministic, as the resulting product is stochastic by introducing soft labeling.\n- As acknowledged by the authors, the proposed methods shift from tuning a reward signal to tuning an LTL specification to make the agent learn, while requiring extra hyperparameter tuning. It would be great to include an analysis of how much this LTL formula tuning influences the results (in the appendix if space doesn’t permit it).", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-17"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel approach that integrates LTL specifications with differentiable physics simulators to enable efficient gradient based policy optimization directly from the logical objectives. The authors propose \"soft labeling\" technique that makes the normal discrete satisfaction conditions of LTL differentiable, generating comparatively smooth reward signals and automaton state transitions as functions of the agent's actions. Empirical results in simulated robotics show significantly faster convergence and higher attainment compared to traditional LTL-based methods.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n**1**, Literature survey is comprehensive and well written.\\\n**2**, Background introduction on labels, LTL, LDBA, and Control Synthesis are concise and easy to interpret for readers without model checking prior knowledge. \\\n**3**, The parking example helps reviewer conceptually better understand the rate of change of LTL satisfaction probability.\\\n**4**, The idea of integrating LTL with differential simulation and enable differentiability of discrete LTL objective/reward is novel.\\\n**5**, The differentiable rewards can provide more guidance in more rewarded direction, enabling faster convergence. \n\n\n# Weakness\n**1**, citation missing: LTL RL{[1,3,4],}; BF RL{[2]}.\\\n**2**, the explanation of how discounted function $\\beta$ is defined is unclear to the reviewer and same for the discounting function $\\mathfrak{D}$.\\\n**3**, The parking example is intuitive, but from figure 1 reviewer cannot tell the advantage in terms of low-variance.\\\n**4**, The reviewer think some other discrete LTL reward-based RL baselines are needed for comparisons. \\\n**5**, The reviewer think it is also necessary to demonstrate the safety rate/probability of satisfaction for both discrete and differentiable cases in more direct way, since intuitively the differentiability might introduce some slackness to the safety constraints.\\\n\n\n\n[1]Jackermeier, Mathias, and Alessandro Abate. \"Deepltl: Learning to efficiently satisfy complex ltl specifications for multi-task rl.\" The Thirteenth International Conference on Learning Representations. 2025.\n[2]Wang, Yixuan, et al. \"Joint differentiable optimization and verification for certified reinforcement learning.\" Proceedings of the ACM/IEEE 14th International Conference on Cyber-Physical Systems (with CPS-IoT Week 2023). 2023.\n[3]Camacho, Alberto, et al. \"LTL and beyond: Formal languages for reward function specification in reinforcement learning.\" IJCAI. Vol. 19. 2019.\n[4]Yalcinkaya, Beyazit, et al. \"Compositional automata embeddings for goal-conditioned reinforcement learning.\" Advances in Neural Information Processing Systems 37 (2024): 72933-72963.\n\nQUESTIONS:\n**1**, The rewards seem to be correlated with the probability of reaching accepting states in the LDBA. Depending on the reward design, will it possibly induce some reward hacking scenarios(eg. unexpected behaviors with high reward returns)? It would be interesting to provide some discussion in this direction.  \n**2**, If  we use the probabilistic/soft labels introduced in line 209 to 219, which also induced probabilistic automaton transitions, are we still dealing with LDBA? And also does Theorem 1 still holds under this setting?  \n**3**, What is the definition of $\\psi$ defined in line 249? Is it controller parameters?  \n**4**, The motivation of ablation study conducting in simplified LTL is still unclear to the reviewer. Since here authors are using induced LTL formula not different set of formula how it can isolate the impact of differentiability of LTL rewards form inherent environment.\n\nLIMITATIONS:\nThis methods are limited for high dimensional continuous tasks due to the automaton generation process, but in general all LTL-based RL methods all have these kind of limitations. In general, this paper is technically solid with clear motivation and novelty. *", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-22"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the problem of how to use differentiable simulators/differentiable reinforcement learning to learn in linear temporal logic problems. The reason this is hard to do is that formulating an LTL spec as an RL objective produces sparse discrete rewards which only appear in states that satisfy the spec. The paper produces smooth, differentiable rewards by calculating labels using a distance measure then turning this distance measure into a probability using a sigmoid function. The probability for a spec being satisfied is the product of the label probabilities. This approach is evaluated using two state of the art differentiable RL algorithms and compared against two standard RL algorithms on some basic control problems in simulation. The results show that the use of the differentiable rewards speed up learning and also find better policies i.e. policies that are more likely to satisfy the specification.\n\nSTRENGTHS AND WEAKNESSES:\nThe key strength of the paper is proposing a dense reward structure that is correctly aligned with the LTL spec objective. This goes beyond existing work which generates dense rewards but without guarantees of creating optimal policies for the specification.  Whilst the idea of generating dense rewards is not original, this approach for propagating satisfaction probabilities for a dense reward is new, at least as far as I am aware. This is also quite significant since looks to allow scaling of RL for LTL specs, which is desirable to the community. \n\nThere are a few weaknesses of the paper. \n\nI find there's not enough time spent comparing to or discussing existing work. Whilst there are a huge number of references, no individual reference is really explored in detail. Also there are no competing methods included in the evaluation as baselines. I was surprised not to see reward machines [37] compared to, since they seem a direct competitor to this method. \n\nAlso, given the way predicate satisfaction is formulated, I would have liked to see a deeper comparison to STL, since this appears equivalent to the taken approach to labelling. \n\nI also think the evaluation domains don't really align with the robotics/long-horizon pitch of the paper. Cartpole and Deepmind Control are good starter control from state observation domains, but are reasonably easy for modern continuous control RL approaches. These are not long-horizon problems in robotics, even if specifying a sequence of objectives (move then stop) does go beyond a typical control horizon. They are also deterministic problems, and the specifications are over times which smoothly vary in state space anyway (i.e. velocity) which I assume makes it easier to learn probability maximising policies. In summary, I'd like to see the work evaluated in Richard domains as well as over more meaningful temporal specifications, particularly where those temporal variations may be more discrete than changes in velocity.\n\nQUESTIONS:\n1. How would this work in high dimensional settings?\n2. Would you expect similar performance in stochastic domains?\n3. How does this vary in practice from learning for for an STL spec?\n4. Can you compare to reward machines?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the challenge of sparse rewards in reinforcement learning with linear temporal logic (LTL) specifications by introducing a novel differentiable framework. Traditional methods translate LTL into automata and provide sparse, discrete rewards, which are difficult to optimize. The proposed method transforms this process into a differentiable one by introducing probabilistic “soft” labels for atomic propositions and modeling transitions through a probabilistic automaton. This allows the construction of a product MDP with fully differentiable state transitions and reward structures, enabling the use of first-order gradient-based optimization algorithms.\n\nThe method is implemented using differentiable simulators and evaluated using both traditional and differentiable RL algorithms across standard benchmarks like CartPole, Hopper, Cheetah, and Ant. The experiments show that differentiable RL methods using this reward structure outperform traditional methods in convergence speed and policy quality, especially for complex LTL specifications.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n- The idea of integrating differentiable reinforcement learning with LTL-based specifications is, to the best of my knowledge, novel and timely. With increasing interest in structured reward functions and formal methods in RL, this paper contributes a natural progression by introducing a differentiable framework grounded in formal logic other than STL.\n- The approach leverages soft labeling and probabilistic automaton state representations to construct end-to-end differentiable reward structures, addressing the longstanding challenge of reward sparsity in LTL-guided learning.\n- The experiments are extensive, covering both low- and high-dimensional robotic control tasks. The results convincingly demonstrate the advantage of differentiable LTL rewards in terms of convergence and final policy quality.\n- The ablation study strengthens the paper by isolating the contribution of differentiability from environmental factors.\n\nWeaknesses\n- There are multiple issues with mathematical clarity and presentation:\n1. The definition of policies as \\pi : S \\to A implies memoryless policies, yet the paper later discusses finite-memory policies without offering sufficient background or formal definitions.\n2. The notation M used in line 109 (transition dynamics) is not properly defined.\n3.\t\\sigma is defined as a path, but inconsistently used — e.g., line 112 uses it as a single instance, and expressions like \\sigma[i], \\sigma[t:] are used without explanation.\n4.\tThe set of atomic propositions (AP) is denoted in both bold and normal font without clarification (e.g., line 127).\n5.\tThe state q_{\\epsilon} is introduced in Equation (1) but is never defined or intuitively explained, leaving its role ambiguous.\n6.\tThe function f_L (Equation 5) is described in dense notation without sufficient elaboration, making its mechanics difficult to follow.\n7.\tLine 74 (“growing increasing”) seems to have typos and is not readable.\n8.\tThe theoretical results provided (e.g., Theorem 1) only apply to the discrete reward formulation, and no similar guarantees are provided for the differentiable LTL rewards introduced later in the paper. This weakens the formal justification of the core contribution.\n\nQUESTIONS:\n1.\tCould you provide a theoretical guarantee or approximation result for the differentiable LTL rewards?\n Theoretical guarantees (e.g., Theorem 1) only apply to the discrete formulation. Does a similar result hold in expectation for the differentiable case? \n2.\tPlease define or explain key notations more clearly (e.g., q_\\epsilon, \\sigma[i], \\sigma[t:], and f_L).\n Several notations are introduced without adequate definitions or intuition, especially in Section 4. A short paragraph walking through the probabilistic automaton update step (Equation 5) would be helpful.\n\n3.\tCan you explain whether and how the gradient-based learning framework respects the Büchi acceptance condition probabilistically?\n While differentiable rewards are defined, it’s unclear how they preserve the infinite visitation condition (\\always \\eventually B) in practice or theory. Do the smooth reward functions approximate this in some probabilistic sense over finite horizons?\n\n*The authors have answered my questions satisfactorily.*\n\nLIMITATIONS:\n- Have the authors acknowledged limitations and potential negative societal impact?\n     Yes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}