Daily Papers

Recent advancements in large language models (LLMs) have sparked considerable interest in automated theorem proving and a prominent line of research integrates stepwise LLM-based provers into tree search. In this paper, we introduce a novel proof-state exploration approach for training data synthesis, designed to produce diverse tactics across a wide range of intermediate proof states, thereby facilitating effective one-shot fine-tuning of LLM as the policy model. We also propose an adaptive beam size strategy, which effectively takes advantage of our data synthesis method and achieves a trade-off between exploration and exploitation during tree search. Evaluations on the MiniF2F and ProofNet benchmarks demonstrate that our method outperforms strong baselines under the stringent Pass@1 metric, attaining an average pass rate of 60.74% on MiniF2F and 21.18% on ProofNet. These results underscore the impact of large-scale synthetic data in advancing automated theorem proving.

Large language models (LLMs) can generate plausible code but offer limited guarantees of correctness. Formally verifying that implementations satisfy specifications requires constructing machine-checkable proofs, a task that remains beyond current automation. We propose a hierarchical proof search framework for automated code verification in Lean~4 that decomposes complex verification goals into structurally simpler subgoals before attempting tactic-level proving. Central to our approach is a principled decomposition score that combines constructive justification with structural effectiveness. Crucially, this score serves as both the training reward and the inference-time ranking criterion, ensuring strict alignment between optimization and deployment. We train Goedel-Code-Prover-8B, a single unified policy for both decomposition and completion, via supervised initialization followed by hybrid reinforcement learning, where a continuous decomposition reward drives planning exploration while supervised replay stabilizes proof generation. On three Lean-based code verification benchmarks comprising 427 tasks, our 8B-parameter model achieves a 62.0\% prove success rate, a 2.6times improvement over the strongest baseline, surpassing neural provers up to 84times larger. We further observe consistent inference-time scaling: success rates improve monotonically with search iterations and sampling budget, with our trained model achieving greater efficiency than frontier off-the-shelf models of comparable scale.

Large Language Models (LLMs) have shown remarkable abilities in structured reasoning and symbolic tasks, with coding emerging as a particular area of strength. This success has sparked growing interest in applying LLMs to mathematics, both in informal problem-solving and formal theorem proving. However, progress in formal mathematics has proven to be significantly more difficult, despite surface-level similarities between programming and proof construction. This discrepancy raises important questions about how LLMs ``reason'', how they are supervised, and whether they internally track a notion of computational or deductive state. In this article, we address the state-of-the-art of the discipline, focusing on recent models and benchmarks, and explore three central issues at the intersection of machine learning and mathematical cognition: (i) the trade-offs between formal and informal mathematics as training domains; (ii) the deeper reasons why proof generation remains more brittle than code synthesis; (iii) and the question of whether LLMs represent, or merely mimic, a notion of evolving logical state. Our goal is not to draw hard boundaries, but to identify where the current limits lie, and how they might be extended.

Agentic theorem provers combine a reasoning model, retrieval, search, and a proof assistant verifier, yet it remains unclear which components actually improve finite-budget proof success and why they help on real mathematical workloads. We study this question through statistical provability: the probability of reaching a verified proof within a budget on a specified stream of theorem instances. We model formal proof search as a finite-horizon reachability MDP with deterministic verifier dynamics, and show that under a faithful state abstraction the optimal success probability coincides with ordinary syntactic provability. We then analyze a simple but practically important pipeline: depth-wise offline action-value regression followed by greedy test-time proving. Our main theorem bounds the provability gap between the learned prover and the optimal prover by an occupancy-weighted sum of uniform action-value errors; in the common uniform-error reading, the leading complexity multiplier is the learned prover's average truncated proof length. The error decomposes into approximation error, geometric coverage of the training distribution, and Monte Carlo label noise, and improves to a fast rate under an action-gap margin condition. The result gives a component-sensitive account of why verifier feedback, retrieval, representation geometry, and proof-shortening mechanisms help on biased theorem workloads, without contradicting classical worst-case hardness.

Large language model based agents often fail in unfamiliar environments due to premature exploitation: a tendency to act on prior knowledge before acquiring sufficient environment-specific information. We identify autonomous exploration as a critical yet underexplored capability for building adaptive agents. To formalize and quantify this capability, we introduce Exploration Checkpoint Coverage, a verifiable metric that measures how broadly an agent discovers key states, objects, and affordances. Our systematic evaluation reveals that agents trained with standard task-oriented reinforcement learning consistently exhibit narrow and repetitive behaviors that impede downstream performance. To address this limitation, we develop a training strategy that interleaves task-execution rollouts and exploration rollouts, with each type of rollout optimized by its corresponding verifiable reward. Building on this training strategy, we propose the Explore-then-Act paradigm, which decouples information-gathering from task execution: agents first utilize an interaction budget to acquire grounded environmental knowledge, then leverage it for task resolution. Our results demonstrate that learning to systematically explore is imperative for building generalizable and real-world-ready agents.

Achieving effective test-time scaling requires models to engage in In-Context Exploration -- the intrinsic ability to generate, verify, and refine multiple reasoning hypotheses within a single continuous context. Grounded in State Coverage theory, our analysis identifies a critical bottleneck to enabling this capability: while broader state coverage requires longer reasoning trajectories, the probability of sampling such sequences decays exponentially during autoregressive generation, a phenomenon we term the ``Shallow Exploration Trap''. To bridge this gap, we propose Length-Incentivized Exploration(\method). This simple yet effective recipe explicitly encourages models to explore more via a length-based reward coupled with a redundancy penalty, thereby maximizing state coverage in two-step manner. Comprehensive experiments across different models (Qwen3, Llama) demonstrate that \method effectively incentivize in-context exploration. As a result, our method achieves an average improvement of 4.4\% on in-domain tasks and a 2.7\% gain on out-of-domain benchmarks.

Automated Theorem Proving (ATP) in formal languages remains a formidable challenge in AI, demanding rigorous logical deduction and navigating vast search spaces. While large language models (LLMs) have shown promising performance, existing stepwise provers often suffer from biased search guidance, leading to inefficiencies and suboptimal proof strategies. This paper introduces the Multi-Perspective Search Prover (MPS-Prover), a novel stepwise ATP system designed to overcome these limitations. MPS-Prover incorporates two key innovations: a highly effective post-training data curation strategy that prunes approximately 40% of redundant training data without sacrificing performance, and a multi-perspective tree search mechanism. This search integrates a learned critic model with strategically designed heuristic rules to diversify tactic selection, prevent getting trapped in unproductive states, and enhance search robustness. Extensive evaluations demonstrate that MPS-Prover achieves state-of-the-art performance on multiple challenging benchmarks, including miniF2F and ProofNet, outperforming prior 7B parameter models. Furthermore, our analyses reveal that MPS-Prover generates significantly shorter and more diverse proofs compared to existing stepwise and whole-proof methods, highlighting its efficiency and efficacy. Our work advances the capabilities of LLM-based formal reasoning and offers a robust framework and a comprehensive analysis for developing more powerful theorem provers.

Go-Explore is a powerful family of algorithms designed to solve hard-exploration problems, built on the principle of archiving discovered states, and iteratively returning to and exploring from the most promising states. This approach has led to superhuman performance across a wide variety of challenging problems including Atari games and robotic control, but requires manually designing heuristics to guide exploration, which is time-consuming and infeasible in general. To resolve this, we propose Intelligent Go-Explore (IGE) which greatly extends the scope of the original Go-Explore by replacing these heuristics with the intelligence and internalized human notions of interestingness captured by giant foundation models (FMs). This provides IGE with a human-like ability to instinctively identify how interesting or promising any new state is (e.g. discovering new objects, locations, or behaviors), even in complex environments where heuristics are hard to define. Moreover, IGE offers the exciting and previously impossible opportunity to recognize and capitalize on serendipitous discoveries that cannot be predicted ahead of time. We evaluate IGE on a range of language-based tasks that require search and exploration. In Game of 24, a multistep mathematical reasoning problem, IGE reaches 100% success rate 70.8% faster than the best classic graph search baseline. Next, in BabyAI-Text, a challenging partially observable gridworld, IGE exceeds the previous SOTA with orders of magnitude fewer online samples. Finally, in TextWorld, we show the unique ability of IGE to succeed in settings requiring long-horizon exploration where prior SOTA FM agents like Reflexion completely fail. Overall, IGE combines the tremendous strengths of FMs and the powerful Go-Explore algorithm, opening up a new frontier of research into creating more generally capable agents with impressive exploration capabilities.

Theorem proving is a fundamental task in mathematics. With the advent of large language models (LLMs) and interactive theorem provers (ITPs) like Lean, there has been growing interest in integrating LLMs and ITPs to automate theorem proving. In this approach, the LLM generates proof steps (tactics), and the ITP checks the applicability of the tactics at the current goal. The two systems work together to complete the proof. In this paper, we introduce DS-Prover, a novel dynamic sampling method for theorem proving. This method dynamically determines the number of tactics to apply to expand the current goal, taking into account the remaining time compared to the total allocated time for proving a theorem. This makes the proof search process more efficient by adjusting the balance between exploration and exploitation as time passes. We also augment the training dataset by decomposing simplification and rewrite tactics with multiple premises into tactics with single premises. This gives the model more examples to learn from and helps it to predict the tactics with premises more accurately. We perform our experiments using the Mathlib dataset of the Lean theorem prover and report the performance on two standard datasets, MiniF2F and ProofNet. Our methods achieve significant performance gains on both datasets. We achieved a state-of-the-art performance (Pass@1) of 14.2% on the ProofNet dataset and a performance of 29.8% on MiniF2F, slightly surpassing the best-reported Pass@1 of 29.6% using Lean.

We introduce ProofGrid, a benchmark suite for evaluating LLM reasoning through machine-checkable proofs rather than final answers alone. ProofGrid contains 15 tasks spanning proof writing, proof checking, proof masking, and proof gap-filling. Tasks are expressed in minimal formal notation, especially NDL, a compact natural-deduction language that fits in short prompts and supports precise, auditable verification. This yields mechanical, reproducible, and fine-grained evaluation rather than judgments by humans or LLMs. ProofGrid covers a calibrated difficulty spectrum, from foundational reasoning tests to structurally rich challenge tasks that no current model solves, while minimizing reliance on domain knowledge, solver delegation, and long-context artifacts. We also develop a comparative framework for reasoning benchmarks and use it to situate ProofGrid relative to existing work in terms of representation, verification guarantees, and reasoning depth. Methodologically, we introduce an instrumented proof-checking pipeline that tolerates minor surface deviations while locating the first substantive reasoning failure, improving measurement resolution and separating proof planning from low-level execution noise. Using this pipeline, we evaluate a broad range of open and proprietary models. Results show rapid progress but substantial remaining limits: frontier models perform well on several foundational tasks, yet difficult tasks, especially those requiring global combinatorial reasoning or low-level proof synthesis, remain far from solved. We also identify epistemic instability, where models generate flawed proofs yet correctly reject those local inferences in isolation, and formalize this with an Epistemic Stability Index. Finally, we complement accuracy with 2PL IRT analyses, Wright maps, and a normalized task-discrimination measure based on Fisher information.

Exploration is a fundamental aspect of Reinforcement Learning, typically implemented using stochastic action-selection. Exploration, however, can be more efficient if directed toward gaining new world knowledge. Visit-counters have been proven useful both in practice and in theory for directed exploration. However, a major limitation of counters is their locality. While there are a few model-based solutions to this shortcoming, a model-free approach is still missing. We propose E-values, a generalization of counters that can be used to evaluate the propagating exploratory value over state-action trajectories. We compare our approach to commonly used RL techniques, and show that using E-values improves learning and performance over traditional counters. We also show how our method can be implemented with function approximation to efficiently learn continuous MDPs. We demonstrate this by showing that our approach surpasses state of the art performance in the Freeway Atari 2600 game.

A promising approach for improving reasoning in large language models is to use process reward models (PRMs). PRMs provide feedback at each step of a multi-step reasoning trace, potentially improving credit assignment over outcome reward models (ORMs) that only provide feedback at the final step. However, collecting dense, per-step human labels is not scalable, and training PRMs from automatically-labeled data has thus far led to limited gains. To improve a base policy by running search against a PRM or using it as dense rewards for reinforcement learning (RL), we ask: "How should we design process rewards?". Our key insight is that, to be effective, the process reward for a step should measure progress: a change in the likelihood of producing a correct response in the future, before and after taking the step, corresponding to the notion of step-level advantages in RL. Crucially, this progress should be measured under a prover policy distinct from the base policy. We theoretically characterize the set of good provers and our results show that optimizing process rewards from such provers improves exploration during test-time search and online RL. In fact, our characterization shows that weak prover policies can substantially improve a stronger base policy, which we also observe empirically. We validate our claims by training process advantage verifiers (PAVs) to predict progress under such provers, and show that compared to ORMs, test-time search against PAVs is >8% more accurate, and 1.5-5times more compute-efficient. Online RL with dense rewards from PAVs enables one of the first results with 5-6times gain in sample efficiency, and >6% gain in accuracy, over ORMs.

One of the challenges in online reinforcement learning (RL) is that the agent needs to trade off the exploration of the environment and the exploitation of the samples to optimize its behavior. Whether we optimize for regret, sample complexity, state-space coverage or model estimation, we need to strike a different exploration-exploitation trade-off. In this paper, we propose to tackle the exploration-exploitation problem following a decoupled approach composed of: 1) An "objective-specific" algorithm that (adaptively) prescribes how many samples to collect at which states, as if it has access to a generative model (i.e., a simulator of the environment); 2) An "objective-agnostic" sample collection exploration strategy responsible for generating the prescribed samples as fast as possible. Building on recent methods for exploration in the stochastic shortest path problem, we first provide an algorithm that, given as input the number of samples b(s,a) needed in each state-action pair, requires O(B D + D^{3/2} S^2 A) time steps to collect the B=sum_{s,a} b(s,a) desired samples, in any unknown communicating MDP with S states, A actions and diameter D. Then we show how this general-purpose exploration algorithm can be paired with "objective-specific" strategies that prescribe the sample requirements to tackle a variety of settings -- e.g., model estimation, sparse reward discovery, goal-free cost-free exploration in communicating MDPs -- for which we obtain improved or novel sample complexity guarantees.

We investigate the exploration of an unknown environment when no reward function is provided. Building on the incremental exploration setting introduced by Lim and Auer [1], we define the objective of learning the set of ε-optimal goal-conditioned policies attaining all states that are incrementally reachable within L steps (in expectation) from a reference state s_0. In this paper, we introduce a novel model-based approach that interleaves discovering new states from s_0 and improving the accuracy of a model estimate that is used to compute goal-conditioned policies to reach newly discovered states. The resulting algorithm, DisCo, achieves a sample complexity scaling as O(L^5 S_{L+ε} Γ_{L+ε} A ε^{-2}), where A is the number of actions, S_{L+ε} is the number of states that are incrementally reachable from s_0 in L+ε steps, and Γ_{L+ε} is the branching factor of the dynamics over such states. This improves over the algorithm proposed in [1] in both ε and L at the cost of an extra Γ_{L+ε} factor, which is small in most environments of interest. Furthermore, DisCo is the first algorithm that can return an ε/c_{min}-optimal policy for any cost-sensitive shortest-path problem defined on the L-reachable states with minimum cost c_{min}. Finally, we report preliminary empirical results confirming our theoretical findings.

Recent advances in large language models have been driven by reinforcement learning (RL)-style post-training, which improves reasoning by optimizing model outputs based on reward or preference signals. GRPO-style approaches implement this by using self-generated samples labeled by an outcome-based verifier. However, these methods depend heavily on the model's initial ability to produce positive samples. They primarily refine what the model already knows (distribution sharpening) rather than enabling the model to solve problems where it initially fails. This limitation is especially problematic in early-stage RL training and on challenging reasoning tasks, where positive samples are unlikely to be generated. To unlock reasoning ability in such settings, the model must explore new reasoning trajectories beyond its current output distribution. Such exploration requires access to sufficiently good positive samples to guide the learning. While expert demonstrations seem like a natural solution, we find that they are often ineffective in RL post-training. Instead, we identify two key properties of effective positive samples: they should (1) be likely under the current policy, and (2) increase the model's likelihood of predicting the correct answer. Based on these insights, we propose Self-Explanation Policy Optimization (ExPO)-a simple and modular framework that generates such samples by conditioning on the ground-truth answer. ExPO enables efficient exploration and guides the model to produce reasoning trajectories more aligned with its policy than expert-written CoTs, while ensuring higher quality than its own (incorrect) samples. Experiments show that ExPO improves both learning efficiency and final performance on reasoning benchmarks, surpassing expert-demonstration-based methods in challenging settings such as MATH level-5, where the model initially struggles the most.

Language Model (LM) agents are increasingly used in complex open-ended decision-making tasks, from AI coding to physical AI. A core requirement in these settings is the ability to both explore the problem space and exploit acquired knowledge effectively. However, systematically distinguishing and quantifying exploration and exploitation from observed actions without access to the agent's internal policy remains challenging. To address this, we design controllable environments inspired by practical embodied AI scenarios. Each environment consists of a partially observable 2D grid map and an unknown task Directed Acyclic Graph (DAG). The map generation can be programmatically adjusted to emphasize exploration or exploitation difficulty. To enable policy-agnostic evaluation, we design a metric to quantify exploration and exploitation errors from agent's actions. We evaluate a variety of frontier LM agents and find that even state-of-the-art models struggle on our task, with different models exhibiting distinct failure modes. We further observe that reasoning models solve the task more effectively and show both exploration and exploitation can be significantly improved through minimal harness engineering. We release our code https://github.com/jjj-madison/measurable-explore-exploit{here}.

A fundamental challenge in formal theorem proving by LLMs is the lack of high-quality training data. Although reinforcement learning or expert iteration partially mitigates this issue by alternating between LLM generating proofs and finetuning them on correctly generated ones, performance quickly plateaus due to the scarcity of correct proofs (sparse rewards). To keep improving the models with limited data, we draw inspiration from mathematicians, who continuously develop new results, partly by proposing novel conjectures or exercises (which are often variants of known results) and attempting to solve them. We design the Self-play Theorem Prover (STP) that simultaneously takes on two roles, conjecturer and prover, each providing training signals to the other. The conjecturer is trained iteratively on previously generated conjectures that are barely provable by the current prover, which incentivizes it to generate increasingly challenging conjectures over time. The prover attempts to prove the conjectures with standard expert iteration. We evaluate STP with both Lean and Isabelle formal versifiers. With 19.8 billion tokens generated during the training in Lean, STP proves 26.3% of the statements in the LeanWorkbook dataset, doubling the previous best result of 13.2% achieved through expert iteration. The final model achieves state-of-the-art performance among whole-proof generation methods on miniF2F-test (61.7%, pass@3200), Proofnet-test (23.1%, pass@3200) and PutnamBench (8/644, pass@3200).

We study the autonomous exploration (AX) problem proposed by Lim & Auer (2012). In this setting, the objective is to discover a set of epsilon-optimal policies reaching a set S_L^{rightarrow} of incrementally L-controllable states. We introduce a novel layered decomposition of the set of incrementally L-controllable states that is based on the iterative application of a state-expansion operator. We leverage these results to design Layered Autonomous Exploration (LAE), a novel algorithm for AX that attains a sample complexity of mathcal{O}(LS^{rightarrow}_{L(1+epsilon)}Gamma_{L(1+epsilon)} A ln^{12}(S^{rightarrow}_{L(1+epsilon)})/epsilon^2), where S^{rightarrow}_{L(1+epsilon)} is the number of states that are incrementally L(1+epsilon)-controllable, A is the number of actions, and Gamma_{L(1+epsilon)} is the branching factor of the transitions over such states. LAE improves over the algorithm of Tarbouriech et al. (2020a) by a factor of L^2 and it is the first algorithm for AX that works in a countably-infinite state space. Moreover, we show that, under a certain identifiability assumption, LAE achieves minimax-optimal sample complexity of mathcal{O}(LS^{rightarrow}_{L}Aln^{12}(S^{rightarrow}_{L})/epsilon^2), outperforming existing algorithms and matching for the first time the lower bound proved by Cai et al. (2022) up to logarithmic factors.

We propose a new method for count-based exploration in high-dimensional state spaces. Unlike previous work which relies on density models, we show that counts can be derived by averaging samples from the Rademacher distribution (or coin flips). This insight is used to set up a simple supervised learning objective which, when optimized, yields a state's visitation count. We show that our method is significantly more effective at deducing ground-truth visitation counts than previous work; when used as an exploration bonus for a model-free reinforcement learning algorithm, it outperforms existing approaches on most of 9 challenging exploration tasks, including the Atari game Montezuma's Revenge.

Recent advancements in large language models (LLMs) have spurred growing interest in automatic theorem proving using Lean4, where effective tree search methods are crucial for navigating proof search spaces. While the existing approaches primarily rely on value functions and Monte Carlo Tree Search (MCTS), the potential of simpler methods like Best-First Search (BFS) remains underexplored. This paper investigates whether BFS can achieve competitive performance in large-scale theorem proving tasks. We present BFS-Prover, a scalable expert iteration framework, featuring three key innovations. First, we implement strategic data filtering at each expert iteration round, excluding problems solvable via beam search node expansion to focus on harder cases. Second, we improve the sample efficiency of BFS through Direct Preference Optimization (DPO) applied to state-tactic pairs automatically annotated with compiler error feedback, refining the LLM's policy to prioritize productive expansions. Third, we employ length normalization in BFS to encourage exploration of deeper proof paths. BFS-Prover achieves a score of 71.31 on the MiniF2F test set and therefore challenges the perceived necessity of complex tree search methods, demonstrating that BFS can achieve competitive performance when properly scaled.

Deep research requires agents to find answers that jointly satisfy multiple constraints. Discovering such answers is costly, whereas verifying a candidate can often be decomposed into tractable constraint-wise checks. This discovery--verification asymmetry suggests that a research agent should do more than simply search longer: it should recursively improve its current answer by verifying intermediate results and using the partially verified state to guide subsequent refinement. We introduce AREX, a family of Recursively Self-Improving (RSI) deep research agents. AREX alternates between an inner research loop that gathers evidence and constructs a provisional answer, and an outer self-improvement loop that audits the answer constraint-wise, identifies unresolved claims, and launches targeted follow-up research. To sustain RSI over long horizons, AREX learns an autonomous context-update tool that compresses growing interaction history into a compact improvement state preserving verified evidence and unresolved constraints, without relying on an external model. We train AREX on verified synthetic tasks and high-quality trajectories through agentic mid-training and long-horizon reinforcement learning. To mitigate sparse final rewards during long horizon learning, we emphasize key steps where decisive evidence is acquired or erroneous research directions are corrected. We instantiate a dense 4B model and a 122B-A10B Mixture-of-Experts model. Across BrowseComp, WideSearch, DeepSearchQA, Humanity's Last Exam (HLE), and other reasoning and tool-use benchmarks, AREX substantially outperforms comparable-scale baselines and remains competitive with models using substantially more activated parameters.

We explore a central question in AI for mathematics: can AI systems produce original, nontrivial proofs for open research problems? Despite strong benchmark performance, producing genuinely novel proofs remains an outstanding challenge for LLMs. Through systematic experiments with frontier LLMs on research-level proof tasks, we identify seven failure modes that prevent reliable proof generation, including context contamination, citation hallucination, hand-waving on key steps and misallocation of proof effort, unstable proof plans, unfocused verification, problem modification and single-model bottleneck. We argue that the gap between benchmark success and research-level proving is primarily one of system design, due to those failure modes. We present QED, an open-source multi-agent proof system in which each architectural decision directly addresses a specific failure mode. Evaluated on five open problems in applied analysis and PDEs contributed by domain experts, QED produces correct proofs for three problems, each verified by the contributing experts as original and nontrivial. QED is released as open-source software at https://github.com/proofQED/QED.

Realistic environments often provide agents with very limited feedback. When the environment is initially unknown, the feedback, in the beginning, can be completely absent, and the agents may first choose to devote all their effort on exploring efficiently. The exploration remains a challenge while it has been addressed with many hand-tuned heuristics with different levels of generality on one side, and a few theoretically-backed exploration strategies on the other. Many of them are incarnated by intrinsic motivation and in particular explorations bonuses. A common rule of thumb for exploration bonuses is to use 1/n bonus that is added to the empirical estimates of the reward, where n is a number of times this particular state (or a state-action pair) was visited. We show that, surprisingly, for a pure-exploration objective of reward-free exploration, bonuses that scale with 1/n bring faster learning rates, improving the known upper bounds with respect to the dependence on the horizon H. Furthermore, we show that with an improved analysis of the stopping time, we can improve by a factor H the sample complexity in the best-policy identification setting, which is another pure-exploration objective, where the environment provides rewards but the agent is not penalized for its behavior during the exploration phase.

Reasoning requires going beyond pattern matching or memorization of solutions to identify and implement "algorithmic procedures" that can be used to deduce answers to hard problems. Doing so requires realizing the most relevant primitives, intermediate results, or shared procedures, and building upon them. While RL post-training on long chains of thought ultimately aims to uncover this kind of algorithmic behavior, most reasoning traces learned by large models fail to consistently capture or reuse procedures, instead drifting into verbose and degenerate exploration. To address more effective reasoning, we introduce reasoning abstractions: concise natural language descriptions of procedural and factual knowledge that guide the model toward learning successful reasoning. We train models to be capable of proposing multiple abstractions given a problem, followed by RL that incentivizes building a solution while using the information provided by these abstractions. This results in a two-player RL training paradigm, abbreviated as RLAD, that jointly trains an abstraction generator and a solution generator. This setup effectively enables structured exploration, decouples learning signals of abstraction proposal and solution generation, and improves generalization to harder problems. We also show that allocating more test-time compute to generating abstractions is more beneficial for performance than generating more solutions at large test budgets, illustrating the role of abstractions in guiding meaningful exploration.

We introduce DeepSeek-Prover-V1.5, an open-source language model designed for theorem proving in Lean 4, which enhances DeepSeek-Prover-V1 by optimizing both training and inference processes. Pre-trained on DeepSeekMath-Base with specialization in formal mathematical languages, the model undergoes supervised fine-tuning using an enhanced formal theorem proving dataset derived from DeepSeek-Prover-V1. Further refinement is achieved through reinforcement learning from proof assistant feedback (RLPAF). Beyond the single-pass whole-proof generation approach of DeepSeek-Prover-V1, we propose RMaxTS, a variant of Monte-Carlo tree search that employs an intrinsic-reward-driven exploration strategy to generate diverse proof paths. DeepSeek-Prover-V1.5 demonstrates significant improvements over DeepSeek-Prover-V1, achieving new state-of-the-art results on the test set of the high school level miniF2F benchmark (63.5%) and the undergraduate level ProofNet benchmark (25.3%).

Large language models (LLMs) have demonstrated significant potential in formal theorem proving, yet state-of-the-art performance often necessitates prohibitive test-time compute via massive roll-outs or extended context windows. In this work, we address this scalability bottleneck by exploiting an informative structure in formal verification: the observation that compilers map a vast space of diverse proof attempts to a compact set of structured failure modes. We introduce a learning-to-refine framework that leverages this compression to perform efficient learning and proof exploration. We perform tree search that corrects errors locally conditioned on explicit verifier feedback, thereby circumventing the costs associated with accumulating a long history of proof attempts. Extensive evaluations show that our method consistently amplifies the reasoning capabilities of base provers across varying scales. Notably, our approach achieves state-of-the-art performance on PutnamBench among publicly reported sim8B and sim32B parameter models under comparable test-time budgets, offering a scalable paradigm for next-generation verifier-guided reasoning.

We explore the use of expert iteration in the context of language modeling applied to formal mathematics. We show that at same compute budget, expert iteration, by which we mean proof search interleaved with learning, dramatically outperforms proof search only. We also observe that when applied to a collection of formal statements of sufficiently varied difficulty, expert iteration is capable of finding and solving a curriculum of increasingly difficult problems, without the need for associated ground-truth proofs. Finally, by applying this expert iteration to a manually curated set of problem statements, we achieve state-of-the-art on the miniF2F benchmark, automatically solving multiple challenging problems drawn from high school olympiads.

Large Language Models (LLMs) have shown significant promise in automated theorem proving, yet progress is often constrained by the scarcity of diverse and high-quality formal language data. To address this issue, we introduce Spark-Prover-X1, a 7B parameter model trained via an three-stage framework designed to unlock the reasoning potential of more accessible and moderately-sized LLMs. The first stage infuses deep knowledge through continuous pre-training on a broad mathematical corpus, enhanced by a suite of novel data tasks. Key innovation is a "CoT-augmented state prediction" task to achieve fine-grained reasoning. The second stage employs Supervised Fine-tuning (SFT) within an expert iteration loop to specialize both the Spark-Prover-X1-7B and Spark-Formalizer-X1-7B models. Finally, a targeted round of Group Relative Policy Optimization (GRPO) is applied to sharpen the prover's capabilities on the most challenging problems. To facilitate robust evaluation, particularly on problems from real-world examinations, we also introduce ExamFormal-Bench, a new benchmark dataset of 402 formal problems. Experimental results demonstrate that Spark-Prover achieves state-of-the-art performance among similarly-sized open-source models within the "Whole-Proof Generation" paradigm. It shows exceptional performance on difficult competition benchmarks, notably solving 27 problems on PutnamBench (pass@32) and achieving 24.0\% on CombiBench (pass@32). Our work validates that this diverse training data and progressively refined training pipeline provides an effective path for enhancing the formal reasoning capabilities of lightweight LLMs. We will release both Spark-Prover-X1-7B and Spark-Formalizer-X1-7B, along with the ExamFormal-Bench dataset, in the near future.

Despite their remarkable capabilities, large language models often struggle with tasks requiring complex reasoning and planning. While existing approaches like Chain-of-Thought prompting and tree search techniques show promise, they are limited by their reliance on predefined heuristics and computationally expensive exploration strategies. We propose Policy-Guided Tree Search (PGTS), a framework that combines reinforcement learning with structured tree exploration to efficiently navigate reasoning paths. Our key innovation is a learned policy that dynamically decides between expanding, branching, backtracking, or terminating exploration, eliminating the need for manual heuristics or exhaustive search. Experiments across mathematical reasoning, logical deduction, and planning benchmarks demonstrate that PGTS achieves superior reasoning performance while significantly reducing computational costs compared to existing methods. These results establish PGTS as a scalable and effective solution for tackling complex reasoning tasks with LLMs.

Reinforcement learning with verifiable rewards (RLVR) has proven effective in enhancing the reasoning of large language models (LLMs). Monte Carlo Tree Search (MCTS)-based extensions improve upon vanilla RLVR (e.g., GRPO) by providing tree-based reasoning rollouts that enable fine-grained and segment-level credit assignment. However, existing methods still suffer from limited exploration diversity and inefficient reasoning. To address the above challenges, we propose reinforced efficient reasoning via semantically diverse explorations, i.e., ROSE, for LLMs. To encourage more diverse reasoning exploration, our method incorporates a semantic-entropy-based branching strategy and an varepsilon-exploration mechanism. The former operates on already sampled reasoning rollouts to capture semantic uncertainty and select branching points with high semantic divergence to generate new successive reasoning paths, whereas the latter stochastically initiates reasoning rollouts from the root, preventing the search process from becoming overly local. To improve efficiency, we design a length-aware segment-level advantage estimator that rewards concise and correct reasoning while penalizing unnecessarily long reasoning chains. Extensive experiments on various mathematical reasoning benchmarks with Qwen and Llama models validate the effectiveness and efficiency of ROSE. Codes are available at https://github.com/ZiqiZhao1/ROSE-rl.

Verifiers have been demonstrated to enhance LLM reasoning via test-time scaling (TTS). Yet, they face significant challenges in complex domains. Error propagation from incorrect intermediate reasoning can lead to false positives for seemingly plausible solutions, while lacking external grounding makes verifiers unreliable on computation or knowledge-intensive tasks. To address these challenges, we propose Agentic Verifier, a framework that transforms reward modeling into a multi-turn, tool-augmented deliberative process. We introduce complementary forward and backward agents: one traces solutions from premises to conclusions, while the other re-checks conclusions against their underlying premises. This bidirectional process enables a comprehensive, reliable, and interpretable assessment of solutions. To facilitate practical deployment, we propose AgentV-RL. Through proactive exploration and reinforcement learning, the verifier autonomously interleaves tool-use with internal reasoning. Extensive experiments show that Agentic Verifier yields consistent performance gains under both parallel and sequential TTS. Notably, our 4B variant surpasses state-of-the-art ORMs by 25.2%, positioning it as a promising paradigm for agentic reward modeling.

The capability of autonomous exploration in complex, unknown environments is important in many robotic applications. While recent research on autonomous exploration have achieved much progress, there are still limitations, e.g., existing methods relying on greedy heuristics or optimal path planning are often hindered by repetitive paths and high computational demands. To address such limitations, we propose a novel exploration framework that utilizes the global topology information of observed environment to improve exploration efficiency while reducing computational overhead. Specifically, global information is utilized based on a skeletal topological graph representation of the environment geometry. We first propose an incremental skeleton extraction method based on wavefront propagation, based on which we then design an approach to generate a lightweight topological graph that can effectively capture the environment's structural characteristics. Building upon this, we introduce a finite state machine that leverages the topological structure to efficiently plan coverage paths, which can substantially mitigate the back-and-forth maneuvers (BFMs) problem. Experimental results demonstrate the superiority of our method in comparison with state-of-the-art methods. The source code will be made publicly available at: https://github.com/Haochen-Niu/STGPlanner.

Large language models have recently made significant progress to generate rigorous mathematical proofs. In contrast, utilizing LLMs for theorem proving in formal languages (such as Lean) remains challenging and computationally expensive, particularly when addressing problems at the undergraduate level and beyond. In this work, we present Seed-Prover 1.5, a formal theorem-proving model trained via large-scale agentic reinforcement learning, alongside an efficient test-time scaling (TTS) workflow. Through extensive interactions with Lean and other tools, the model continuously accumulates experience during the RL process, substantially enhancing the capability and efficiency of formal theorem proving. Furthermore, leveraging recent advancements in natural language proving, our TTS workflow efficiently bridges the gap between natural and formal languages. Compared to state-of-the-art methods, Seed-Prover 1.5 achieves superior performance with a smaller compute budget. It solves 88\% of PutnamBench (undergraduate-level), 80\% of Fate-H (graduate-level), and 33\% of Fate-X (PhD-level) problems. Notably, using our system, we solved 11 out of 12 problems from Putnam 2025 within 9 hours. Our findings suggest that scaling learning from experience, driven by high-quality formal feedback, holds immense potential for the future of formal mathematical reasoning.

We systematically investigate all 25 public ARC-AGI-3 games and find that every one is reachable through non-intelligent strategies: 10 in a single blind step, 5 after one probing action, 1 via repeated ACTION1 presses, 1 via diverse exploration, and 8 via single repeated actions with sufficient budget (50-200 steps). A library-level null-coordinate vulnerability additionally bypasses 18 games in 1 step. This benchmark critique implies the public evaluation set cannot discriminate intelligent exploration from trivial heuristics - the private 55-game evaluation is the only genuine intelligence test. Against this backdrop, we present AERA (Adaptive Epistemic Reasoning Agent), a three-phase (EXPLORE / VERIFY / PLAN) agent achieving RHAE=0.2116 (4/25 solved) on these 25 games with Qwen2.5-0.5B, while random and no-explore baselines score 0.0000. We formalise AERA through a Speed--Depth trade-off framework: under a convexity assumption (proved for a class of environments in the Appendix), RHAE's quadratic form emerges as a second-order penalty for deviating from the Pareto frontier between action efficiency and information gain. Contributions: (i) a benchmark validity analysis showing that current interactive reasoning benchmarks fail to measure the exploration they claim to require, and (ii) the EXPLORE-before-PLAN framework and model-capability x exploration interaction. The linked code track entry achieves RHAE=0.30 on the full 55-game private evaluation. Code: CC0.

Exploration is essential for solving complex Reinforcement Learning (RL) tasks. Maximum State-Visitation Entropy (MSVE) formulates the exploration problem as a well-defined policy optimization problem whose solution aims at visiting all states as uniformly as possible. This is in contrast to standard uncertainty-based approaches where exploration is transient and eventually vanishes. However, existing approaches to MSVE are theoretically justified only for discrete state-spaces as they are oblivious to the geometry of continuous domains. We address this challenge by introducing Geometric Entropy Maximisation (GEM), a new algorithm that maximises the geometry-aware Shannon entropy of state-visits in both discrete and continuous domains. Our key theoretical contribution is casting geometry-aware MSVE exploration as a tractable problem of optimising a simple and novel noise-contrastive objective function. In our experiments, we show the efficiency of GEM in solving several RL problems with sparse rewards, compared against other deep RL exploration approaches.

We introduce DeepSeek-Prover-V2, an open-source large language model designed for formal theorem proving in Lean 4, with initialization data collected through a recursive theorem proving pipeline powered by DeepSeek-V3. The cold-start training procedure begins by prompting DeepSeek-V3 to decompose complex problems into a series of subgoals. The proofs of resolved subgoals are synthesized into a chain-of-thought process, combined with DeepSeek-V3's step-by-step reasoning, to create an initial cold start for reinforcement learning. This process enables us to integrate both informal and formal mathematical reasoning into a unified model. The resulting model, DeepSeek-Prover-V2-671B, achieves state-of-the-art performance in neural theorem proving, reaching 88.9% pass ratio on the MiniF2F-test and solving 49 out of 658 problems from PutnamBench. In addition to standard benchmarks, we introduce ProverBench, a collection of 325 formalized problems, to enrich our evaluation, including 15 selected problems from the recent AIME competitions (years 24-25). Further evaluation on these 15 AIME problems shows that the model successfully solves 6 of them. In comparison, DeepSeek-V3 solves 8 of these problems using majority voting, highlighting that the gap between formal and informal mathematical reasoning in large language models is substantially narrowing.

Reinforcement learning (RL) promises to expand the capabilities of language models, but it is unclear if current RL techniques promote the discovery of novel behaviors, or simply sharpen those already present in the base model. In this paper, we investigate the value of deliberate exploration -- explicitly incentivizing the model to discover novel and diverse behaviors -- and aim to understand how the knowledge in pre-trained models can guide this search. Our main finding is that exploration with a simple, principled, representation-based bonus derived from the pre-trained language model's hidden states significantly improves diversity and pass@k rates -- both for post-training, and in a novel inference-time scaling setting we introduce. For inference-time, exploration with representation-based diversity improves efficiency, consistently improving pass@k rates across a variety of models and reasoning tasks. For example, for Qwen-2.5-14b-Instruct we obtain over 50% improvement in verifier efficiency on almost all tasks. For post-training, we show that integrating this exploration strategy into an RL pipeline improves reasoning performance over that of the initial model and over standard RL post-training. For example, on AIME 2024, our post-trained Qwen-2.5-7b-Instruct's pass@80 matches the pass@256 of GRPO on the same model, demonstrating a 3x improvement in test-time sample efficiency. Overall, our findings suggest that deliberate exploration -- with the right notion of diversity -- is a practical path toward discovery of new behaviors beyond sharpening.

Machine-assisted theorem proving refers to the process of conducting structured reasoning to automatically generate proofs for mathematical theorems. Recently, there has been a surge of interest in using machine learning models in conjunction with proof assistants to perform this task. In this paper, we introduce Pantograph, a tool that provides a versatile interface to the Lean 4 proof assistant and enables efficient proof search via powerful search algorithms such as Monte Carlo Tree Search. In addition, Pantograph enables high-level reasoning by enabling a more robust handling of Lean 4's inference steps. We provide an overview of Pantograph's architecture and features. We also report on an illustrative use case: using machine learning models and proof sketches to prove Lean 4 theorems. Pantograph's innovative features pave the way for more advanced machine learning models to perform complex proof searches and high-level reasoning, equipping future researchers to design more versatile and powerful theorem provers.

The standard post-training recipe for large reasoning models, supervised fine-tuning followed by reinforcement learning (SFT-then-RL), may limit the benefits of the RL stage: while SFT imitates expert demonstrations, it often causes overconfidence and reduces generation diversity, leaving RL with a narrowed solution space to explore. Adding entropy regularization during SFT is not a cure-all; it tends to flatten token distributions toward uniformity, increasing entropy without improving meaningful exploration capability. In this paper, we propose CurioSFT, an entropy-preserving SFT method designed to enhance exploration capabilities through intrinsic curiosity. It consists of (a) Self-Exploratory Distillation, which distills the model toward a self-generated, temperature-scaled teacher to encourage exploration within its capability; and (b) Entropy-Guided Temperature Selection, which adaptively adjusts distillation strength to mitigate knowledge forgetting by amplifying exploration at reasoning tokens while stabilizing factual tokens. Extensive experiments on mathematical reasoning tasks demonstrate that, in SFT stage, CurioSFT outperforms the vanilla SFT by 2.5 points on in-distribution tasks and 2.9 points on out-of-distribution tasks. We also verify that exploration capabilities preserved during SFT successfully translate into concrete gains in RL stage, yielding an average improvement of 5.0 points.

In this paper, we introduce Latent Go-Explore (LGE), a simple and general approach based on the Go-Explore paradigm for exploration in reinforcement learning (RL). Go-Explore was initially introduced with a strong domain knowledge constraint for partitioning the state space into cells. However, in most real-world scenarios, drawing domain knowledge from raw observations is complex and tedious. If the cell partitioning is not informative enough, Go-Explore can completely fail to explore the environment. We argue that the Go-Explore approach can be generalized to any environment without domain knowledge and without cells by exploiting a learned latent representation. Thus, we show that LGE can be flexibly combined with any strategy for learning a latent representation. Our results indicate that LGE, although simpler than Go-Explore, is more robust and outperforms state-of-the-art algorithms in terms of pure exploration on multiple hard-exploration environments including Montezuma's Revenge. The LGE implementation is available as open-source at https://github.com/qgallouedec/lge.

Given the intractably large size of the space of proofs, any model that is capable of general deductive reasoning must generalize to proofs of greater complexity. Recent studies have shown that large language models (LLMs) possess some abstract deductive reasoning ability given chain-of-thought prompts. However, they have primarily been tested on proofs using modus ponens or of a specific size, and from the same distribution as the in-context examples. To measure the general deductive reasoning ability of LLMs, we test on a broad set of deduction rules and measure their ability to generalize to more complex proofs from simpler demonstrations from multiple angles: depth-, width-, and compositional generalization. To facilitate systematic exploration, we construct a new synthetic and programmable reasoning dataset that enables control over deduction rules and proof complexity. Our experiments on four LLMs of various sizes and training objectives show that they are able to generalize to longer and compositional proofs. However, they require explicit demonstrations to produce hypothetical subproofs, specifically in proof by cases and proof by contradiction.

Reasoning remains a challenging task for large language models (LLMs), especially within the logically constrained environment of automated theorem proving (ATP), due to sparse rewards and the vast scale of proofs. These challenges are amplified in benchmarks like PutnamBench, which contains university-level problems requiring complex, multi-step reasoning. To address this, we introduce self-generated goal-conditioned MDPs (sG-MDPs), a new framework in which agents generate and pursue their subgoals based on the evolving proof state. Given this more structured generation of goals, the resulting problem becomes more amenable to search. We then apply Monte Carlo Tree Search (MCTS)-like algorithms to solve the sG-MDP, instantiating our approach in Bourbaki (7B), a modular system that can ensemble multiple 7B LLMs for subgoal generation and tactic synthesis. On PutnamBench, Bourbaki (7B) solves 26 problems, achieving new state-of-the-art results with models at this scale.

Recent advances in automated theorem proving (ATP) through LLMs have highlighted the potential of formal reasoning with Lean 4 codes. However, ATP has not yet be revolutionized by the recent posttraining scaling as demonstrated by Open AI O1/O3 and Deepseek R1. In this work, we investigate the entire posttraining of ATP, aiming to align it with breakthroughs in reasoning models in natural languages.To begin, we continual train current ATP models with a hybrid dataset, which consists of numerous statement-proof pairs, and additional data aimed at incorporating cognitive behaviors that emulate human reasoning and hypothesis refinement. Next, we explore reinforcement learning with the use of outcome reward returned by Lean 4 compiler. Through our designed continual training and reinforcement learning processes, we have successfully improved existing formal provers, including both DeepSeek-Prover-v1.5 and Goedel-Prover, achieving state-of-the-art performance in the field of whole-proof generation. For example, we achieve a 59.8% pass rate (pass@32) on MiniF2F. This is an on-going project and we will progressively update our findings, release our data and training details.

Large Language Models (LLMs) have achieved significant advances in reasoning tasks. A key approach is tree-based search with verifiers, which expand candidate reasoning paths and use reward models to guide pruning and selection. Although effective in improving accuracy, these methods are not optimal in terms of efficiency: they perform simple decomposition on the reasoning process, but ignore the planning-execution nature of tasks such as math reasoning or code generation. This results in inefficient exploration of reasoning process. To address this, we propose a dual-phase test-time scaling framework that explicitly separates reasoning into planning and execution, and performs search over the two phases individually. Specifically, we decompose reasoning trajectories and develop reward models for each phase, enabling the search to explore and prune plans and executions separately. We further introduce a dynamic budget allocation mechanism that adaptively redistributes sampling effort based on reward feedback, allowing early stopping on confident steps and reallocation of computation to more challenging parts of the reasoning process. Experiments on both mathematical reasoning and code generation benchmarks demonstrate that our approach consistently improves accuracy while reducing redundant computation.

Reinforcement Learning with Verifiable Rewards (RLVR) is a powerful paradigm for enhancing the reasoning ability of Large Language Models (LLMs). Yet current RLVR methods often explore poorly, leading to premature convergence and entropy collapse. To address this challenge, we introduce Curiosity-Driven Exploration (CDE), a framework that leverages the model's own intrinsic sense of curiosity to guide exploration. We formalize curiosity with signals from both the actor and the critic: for the actor, we use perplexity over its generated response, and for the critic, we use the variance of value estimates from a multi-head architecture. Both signals serve as an exploration bonus within the RLVR framework to guide the model. Our theoretical analysis shows that the actor-wise bonus inherently penalizes overconfident errors and promotes diversity among correct responses; moreover, we connect the critic-wise bonus to the well-established count-based exploration bonus in RL. Empirically, our method achieves an approximate +3 point improvement over standard RLVR using GRPO/PPO on AIME benchmarks. Further analysis identifies a calibration collapse mechanism within RLVR, shedding light on common LLM failure modes.

Large language models have achieved remarkable success on final-answer mathematical problems, largely due to the ease of applying reinforcement learning with verifiable rewards. However, the reasoning underlying these solutions is often flawed. Advancing to rigorous proof-based mathematics requires reliable proof verification capabilities. We begin by analyzing multiple evaluation setups and show that focusing on a single benchmark can lead to brittle or misleading conclusions. To address this, we evaluate both proof-based and final-answer reasoning to obtain a more reliable measure of model performance. We then scale two major generative verification methods (GenSelect and LLM-as-a-Judge) to millions of tokens and identify their combination as the most effective framework for solution verification and selection. We further show that the choice of prompt for LLM-as-a-Judge significantly affects the model's performance, but reinforcement learning can reduce this sensitivity. However, despite improving proof-level metrics, reinforcement learning does not enhance final-answer precision, indicating that current models often reward stylistic or procedural correctness rather than mathematical validity. Our results establish practical guidelines for designing and evaluating scalable proof-verification and selection systems.

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a crucial paradigm for incentivizing reasoning capabilities in Large Language Models (LLMs). Due to vast state-action spaces and reward sparsity in reasoning tasks, existing methods often struggle with deficient exploration and poor sample efficiency. In the paper, we propose DIVER (Diversity-Incentivized Exploration for VersatilE Reasoning), an innovative framework that highlights the pivotal role of global sequence-level diversity to incentivize deep exploration for versatile reasoning. We first conduct a primary empirical study to reveal a strong positive correlation between global diversity and reasoning capacity. Building on this insight, we introduce global diversity incentives as an intrinsic reward to promote deep exploration in a semantically structured space. Incorporating the intrinsic reward, we develop a potential-based reward shaping mechanism to preserve optimal policy invariance and design simple heuristics to mitigate possible reward hacking. Experimental results show that DIVER outperforms competitive RLVR baselines with various exploration strategies on both in-domain and out-of-domain tasks, excelling in both Pass@1 and Pass@k evaluations. Our code is available at https://github.com/NJU-RL/DIVER.

We present BYOL-Explore, a conceptually simple yet general approach for curiosity-driven exploration in visually-complex environments. BYOL-Explore learns a world representation, the world dynamics, and an exploration policy all-together by optimizing a single prediction loss in the latent space with no additional auxiliary objective. We show that BYOL-Explore is effective in DM-HARD-8, a challenging partially-observable continuous-action hard-exploration benchmark with visually-rich 3-D environments. On this benchmark, we solve the majority of the tasks purely through augmenting the extrinsic reward with BYOL-Explore s intrinsic reward, whereas prior work could only get off the ground with human demonstrations. As further evidence of the generality of BYOL-Explore, we show that it achieves superhuman performance on the ten hardest exploration games in Atari while having a much simpler design than other competitive agents.

Balancing exploration and exploitation is a central goal in reinforcement learning (RL). Despite recent advances in enhancing language model (LM) reasoning, most methods lean toward exploitation, and increasingly encounter performance plateaus. In this work, we revisit entropy -- a signal of exploration in RL -- and examine its relationship to exploratory reasoning in LMs. Through empirical analysis, we uncover strong positive correlations between high-entropy regions and three types of exploratory reasoning actions: (1) pivotal tokens that determine or connect logical steps, (2) reflective actions such as self-verification and correction, and (3) rare behaviors under-explored by the base LMs. Motivated by this, we introduce a minimal modification to standard RL with only one line of code: augmenting the advantage function with an entropy-based term. Unlike traditional maximum-entropy methods which encourage exploration by promoting uncertainty, we encourage exploration by promoting longer and deeper reasoning chains. Notably, our method achieves significant gains on the Pass@K metric -- an upper-bound estimator of LM reasoning capabilities -- even when evaluated with extremely large K values, pushing the boundaries of LM reasoning.

The formalization of existing mathematical proofs is a notoriously difficult process. Despite decades of research on automation and proof assistants, writing formal proofs remains arduous and only accessible to a few experts. While previous studies to automate formalization focused on powerful search algorithms, no attempts were made to take advantage of available informal proofs. In this work, we introduce Draft, Sketch, and Prove (DSP), a method that maps informal proofs to formal proof sketches, and uses the sketches to guide an automated prover by directing its search to easier sub-problems. We investigate two relevant setups where informal proofs are either written by humans or generated by a language model. Our experiments and ablation studies show that large language models are able to produce well-structured formal sketches that follow the same reasoning steps as the informal proofs. Guiding an automated prover with these sketches enhances its performance from 20.9% to 39.3% on a collection of mathematical competition problems.

We present Ax-Prover, a multi-agent system for automated theorem proving in Lean that can solve problems across diverse scientific domains and operate either autonomously or collaboratively with human experts. To achieve this, Ax-Prover approaches scientific problem solving through formal proof generation, a process that demands both creative reasoning and strict syntactic rigor. Ax-Prover meets this challenge by equipping Large Language Models (LLMs), which provide knowledge and reasoning, with Lean tools via the Model Context Protocol (MCP), which ensure formal correctness. To evaluate its performance as an autonomous prover, we benchmark our approach against frontier LLMs and specialized prover models on two public math benchmarks and on two Lean benchmarks we introduce in the fields of abstract algebra and quantum theory. On public datasets, Ax-Prover is competitive with state-of-the-art provers, while it largely outperforms them on the new benchmarks. This shows that, unlike specialized systems that struggle to generalize, our tool-based agentic theorem prover approach offers a generalizable methodology for formal verification across diverse scientific domains. Furthermore, we demonstrate Ax-Prover's assistant capabilities in a practical use case, showing how it enabled an expert mathematician to formalize the proof of a complex cryptography theorem.

Recent advances in Reinforcement Learning with Verifiable Rewards (RLVR) for Large Language Model (LLM) reasoning have been hindered by a persistent challenge: exploration collapse. The semantic homogeneity of random rollouts often traps models in narrow, over-optimized behaviors. While existing methods leverage policy entropy to encourage exploration, they face inherent limitations. Global entropy regularization is susceptible to reward hacking, which can induce meaningless verbosity, whereas local token-selective updates struggle with the strong inductive bias of pre-trained models. To address this, we propose Latent Policy Optimization via Iterative Information Bottleneck (IIB-LPO), a novel approach that shifts exploration from statistical perturbation of token distributions to topological branching of reasoning trajectories. IIB-LPO triggers latent branching at high-entropy states to diversify reasoning paths and employs the Information Bottleneck principle both as a trajectory filter and a self-reward mechanism, ensuring concise and informative exploration. Empirical results across four mathematical reasoning benchmarks demonstrate that IIB-LPO achieves state-of-the-art performance, surpassing prior methods by margins of up to 5.3% in accuracy and 7.4% in diversity metrics.

Reinforcement learning with verifiable rewards (RLVR) is a powerful paradigm for enhancing the reasoning capabilities of large language models (LLMs), yet its success hinges on effective exploration. An ideal exploration strategy must navigate two fundamental challenges: it must preserve sample quality while also ensuring training stability. While standard fixed-temperature sampling is simple, it struggles to balance these competing demands, as high temperatures degrade sample quality and low temperatures limit discovery. In this work, we propose a simpler and more effective strategy, Exploratory Annealed Decoding (EAD), grounded in the insight that exploration is most impactful on early tokens which define a sequence's semantic direction. EAD implements an intuitive **explore-at-the-beginning, exploit-at-the-end** strategy by annealing the sampling temperature from high to low during generation. This dynamic schedule encourages meaningful, high-level diversity at the start, then gradually lowers the temperature to preserve sample quality and keep the sampling distribution close to the target policy, which is essential for stable training. We demonstrate that EAD is a lightweight, plug-and-play method that significantly improves sample efficiency, consistently outperforming fixed-temperature sampling across various RLVR algorithms and model sizes. Our work suggests that aligning exploration with the natural dynamics of sequential generation offers a robust path to improving LLM reasoning.

Recent advancements, such as DeepSeek-Prover-V2-671B and Kimina-Prover-Preview-72B, demonstrate a prevailing trend in leveraging reinforcement learning (RL)-based large-scale training for automated theorem proving. Surprisingly, we discover that even without any training, careful neuro-symbolic coordination of existing off-the-shelf reasoning models and tactic step provers can achieve comparable performance. This paper introduces DSP+, an improved version of the Draft, Sketch, and Prove framework, featuring a fine-grained and integrated neuro-symbolic enhancement for each phase: (1) In the draft phase, we prompt reasoning models to generate concise natural-language subgoals to benefit the sketch phase, removing thinking tokens and references to human-written proofs; (2) In the sketch phase, subgoals are autoformalized with hypotheses to benefit the proving phase, and sketch lines containing syntactic errors are masked according to predefined rules; (3) In the proving phase, we tightly integrate symbolic search methods like Aesop with step provers to establish proofs for the sketch subgoals. Experimental results show that, without any additional model training or fine-tuning, DSP+ solves 80.7\%, 32.8\%, and 24 out of 644 problems from miniF2F, ProofNet, and PutnamBench, respectively, while requiring fewer budgets compared to state-of-the-arts. DSP+ proves imo\_2019\_p1, an IMO problem in miniF2F that is not solved by any prior work. Additionally, DSP+ generates proof patterns comprehensible by human experts, facilitating the identification of formalization errors; For example, eight wrongly formalized statements in miniF2F are discovered. Our results highlight the potential of classical reasoning patterns besides the RL-based training. All components will be open-sourced.

Large language models (LLMs) have recently demonstrated remarkable progress in formal theorem proving. Yet their ability to serve as practical assistants for mathematicians, filling in missing steps within complex proofs, remains underexplored. We identify this challenge as the task of subgoal completion, where an LLM must discharge short but nontrivial proof obligations left unresolved in a human-provided sketch. To study this problem, we introduce FormalML, a Lean 4 benchmark built from foundational theories of machine learning. Using a translation tactic that converts procedural proofs into declarative form, we extract 4937 problems spanning optimization and probability inequalities, with varying levels of difficulty. FormalML is the first subgoal completion benchmark to combine premise retrieval and complex research-level contexts. Evaluation of state-of-the-art provers highlights persistent limitations in accuracy and efficiency, underscoring the need for more capable LLM-based theorem provers for effective subgoal completion,

We introduce a generic strategy for provably efficient multi-goal exploration. It relies on AdaGoal, a novel goal selection scheme that leverages a measure of uncertainty in reaching states to adaptively target goals that are neither too difficult nor too easy. We show how AdaGoal can be used to tackle the objective of learning an ε-optimal goal-conditioned policy for the (initially unknown) set of goal states that are reachable within L steps in expectation from a reference state s_0 in a reward-free Markov decision process. In the tabular case with S states and A actions, our algorithm requires O(L^3 S A ε^{-2}) exploration steps, which is nearly minimax optimal. We also readily instantiate AdaGoal in linear mixture Markov decision processes, yielding the first goal-oriented PAC guarantee with linear function approximation. Beyond its strong theoretical guarantees, we anchor AdaGoal in goal-conditioned deep reinforcement learning, both conceptually and empirically, by connecting its idea of selecting "uncertain" goals to maximizing value ensemble disagreement.

Research Agents enable models to gather information from the web using tools to answer user queries, requiring them to dynamically interleave internal reasoning with tool use. While such capabilities can in principle be learned via reinforcement learning with verifiable rewards (RLVR), we observe that agents often exhibit poor exploration behaviors, including premature termination and biased tool usage. As a result, RLVR alone yields limited improvements. We propose SynPlanResearch-R1, a framework that synthesizes tool-use trajectories that encourage deeper exploration to shape exploration during cold-start supervised fine-tuning, providing a strong initialization for subsequent RL. Across seven multi-hop and open-web benchmarks, \framework improves performance by up to 6.0% on Qwen3-8B and 5.8% on Qwen3-4B backbones respectively compared to SOTA baselines. Further analyses of tool-use patterns and training dynamics compared to baselines shed light on the factors underlying these gains. Our code is publicly available at https://github.com/HansiZeng/syn-plan-research.

LLMs have demonstrated strong mathematical reasoning abilities by leveraging reinforcement learning with long chain-of-thought, yet they continue to struggle with theorem proving due to the lack of clear supervision signals when solely using natural language. Dedicated domain-specific languages like Lean provide clear supervision via formal verification of proofs, enabling effective training through reinforcement learning. In this work, we propose Seed-Prover, a lemma-style whole-proof reasoning model. Seed-Prover can iteratively refine its proof based on Lean feedback, proved lemmas, and self-summarization. To solve IMO-level contest problems, we design three test-time inference strategies that enable both deep and broad reasoning. Seed-Prover proves 78.1% of formalized past IMO problems, saturates MiniF2F, and achieves over 50\% on PutnamBench, outperforming the previous state-of-the-art by a large margin. To address the lack of geometry support in Lean, we introduce a geometry reasoning engine Seed-Geometry, which outperforms previous formal geometry engines. We use these two systems to participate in IMO 2025 and fully prove 5 out of 6 problems. This work represents a significant advancement in automated mathematical reasoning, demonstrating the effectiveness of formal verification with long chain-of-thought reasoning.

Large language models (LLMs) increasingly excel at mathematical reasoning, but their unreliability limits their utility in mathematics research. A mitigation is using LLMs to generate formal proofs in languages like Lean. We perform the first large-scale evaluation of this method's ability to solve open problems. Our most capable agent autonomously resolved 9 of 353 open Erdős problems at the per-problem cost of a few hundred dollars, proved 44/492 OEIS conjectures, and is being deployed in combinatorics, optimization, graph theory, algebraic geometry, and quantum optics research. A basic agent alternating LLM-based generation with Lean-based verification replicated the Erdős successes but proved costlier on the hardest problems. These findings demonstrate the power of AI-aided formal proof search and shed light on the agent designs that enable it.

The use of LLMs for code generation has naturally extended to code testing and evaluation. As codebases grow in size and complexity, so does the need for automated test generation. Current approaches for LLM-based test generation rely on strategies that maximize immediate coverage gain, a greedy approach that plateaus on code where reaching deep branches requires setup steps that individually yield zero new coverage. Drawing on principles of Bayesian exploration, we treat the program's branch structure as an unknown environment, and an evolving coverage map as a proxy probabilistic posterior representing what the LLM has discovered so far. Our method, CovQValue, feeds the coverage map back to the LLM, generates diverse candidate plans in parallel, and selects the most informative plan by LLM-estimated Q-values, seeking actions that balance immediate branch discovery with future reachability. Our method outperforms greedy selection on TestGenEval Lite, achieving 51-77% higher branch coverage across three popular LLMs and winning on 77-84% of targets. In addition, we build a benchmark for iterative test generation, RepoExploreBench, where they achieve 40-74%. These results show the potential of curiosity-driven planning methods for LLM-based exploration, enabling more effective discovery of program behavior through sequential interaction

A grand challenge in reinforcement learning is intelligent exploration, especially when rewards are sparse or deceptive. Two Atari games serve as benchmarks for such hard-exploration domains: Montezuma's Revenge and Pitfall. On both games, current RL algorithms perform poorly, even those with intrinsic motivation, which is the dominant method to improve performance on hard-exploration domains. To address this shortfall, we introduce a new algorithm called Go-Explore. It exploits the following principles: (1) remember previously visited states, (2) first return to a promising state (without exploration), then explore from it, and (3) solve simulated environments through any available means (including by introducing determinism), then robustify via imitation learning. The combined effect of these principles is a dramatic performance improvement on hard-exploration problems. On Montezuma's Revenge, Go-Explore scores a mean of over 43k points, almost 4 times the previous state of the art. Go-Explore can also harness human-provided domain knowledge and, when augmented with it, scores a mean of over 650k points on Montezuma's Revenge. Its max performance of nearly 18 million surpasses the human world record, meeting even the strictest definition of "superhuman" performance. On Pitfall, Go-Explore with domain knowledge is the first algorithm to score above zero. Its mean score of almost 60k points exceeds expert human performance. Because Go-Explore produces high-performing demonstrations automatically and cheaply, it also outperforms imitation learning work where humans provide solution demonstrations. Go-Explore opens up many new research directions into improving it and weaving its insights into current RL algorithms. It may also enable progress on previously unsolvable hard-exploration problems in many domains, especially those that harness a simulator during training (e.g. robotics).

Reinforcement Learning with Verifiable Rewards (RLVR) has catalyzed significant advances in the reasoning capabilities of Large Language Models (LLMs). However, effectively managing the exploration and exploitation trade-off remains a critical challenge. In this paper, we fully analyze the exploration and exploitation dilemma of extremely hard and easy samples during the training and propose a new fine-grained trade-off mechanism. Concretely, we introduce a perplexity space disentangling strategy that divides the sample space into distinct exploration (high perplexity) and exploitation (low perplexity) subspaces, thereby mining fine-grained samples requiring exploration-exploitation trade-off. Subsequently, we propose a bidirectional reward allocation mechanism with a minimum impact on verification rewards to implement perplexity-guided exploration and exploitation, enabling more stable policy optimization. Finally, we have evaluated our method on two mainstream tasks: mathematical reasoning and function calling, and experimental results demonstrate the superiority of the proposed method, confirming its effectiveness in enhancing LLM performance by fine-grained exploration-exploitation trade-off.

Parallel thinking has emerged as a new paradigm for large reasoning models (LRMs) in tackling complex problems. Recent methods leverage Reinforcement Learning (RL) to enhance parallel thinking, aiming to address the limitations in computational resources and effectiveness encountered with supervised fine-tuning. However, most existing studies primarily focus on optimizing the aggregation phase, with limited attention to the path exploration stage. In this paper, we theoretically analyze the optimization of parallel thinking under the Reinforcement Learning with Verifiable Rewards (RLVR) setting, and identify that the mutual information bottleneck among exploration paths fundamentally restricts overall performance. To address this, we propose Outline-Guided Path Exploration (OPE), which explicitly partitions the solution space by generating diverse reasoning outlines prior to parallel path reasoning, thereby reducing information redundancy and improving the diversity of information captured across exploration paths. We implement OPE with an iterative RL strategy that optimizes outline planning and outline-guided reasoning independently. Extensive experiments across multiple challenging mathematical benchmarks demonstrate that OPE effectively improves reasoning performance in different aggregation strategies, enabling LRMs to more reliably discover correct solutions.

We introduce Kimina-Prover Preview, a large language model that pioneers a novel reasoning-driven exploration paradigm for formal theorem proving, as showcased in this preview release. Trained with a large-scale reinforcement learning pipeline from Qwen2.5-72B, Kimina-Prover demonstrates strong performance in Lean 4 proof generation by employing a structured reasoning pattern we term formal reasoning pattern. This approach allows the model to emulate human problem-solving strategies in Lean, iteratively generating and refining proof steps. Kimina-Prover sets a new state-of-the-art on the miniF2F benchmark, reaching 80.7% with pass@8192. Beyond improved benchmark performance, our work yields several key insights: (1) Kimina-Prover exhibits high sample efficiency, delivering strong results even with minimal sampling (pass@1) and scaling effectively with computational budget, stemming from its unique reasoning pattern and RL training; (2) we demonstrate clear performance scaling with model size, a trend previously unobserved for neural theorem provers in formal mathematics; (3) the learned reasoning style, distinct from traditional search algorithms, shows potential to bridge the gap between formal verification and informal mathematical intuition. We open source distilled versions with 1.5B and 7B parameters of Kimina-Prover

Reinforcement Learning (RL) has empowered Large Language Models (LLMs) to master autonomous search for complex question answering. However, particularly within multi-turn search scenarios, this interaction introduces a critical challenge: search results often suffer from high redundancy and low signal-to-noise ratios. Consequently, agents easily fall into "Tunnel Vision," where the forced interpretation of early noisy retrievals leads to irreversible error accumulation. To address these challenges, we propose SIGHT, a framework that enhances search-based reasoning through Self-Evidence Support (SES) and Information-Gain Driven Diverse Branching. SIGHT distills search results into high-fidelity evidence via SES and calculates an Information Gain score to pinpoint pivotal states where observations maximally reduce uncertainty. This score guides Dynamic Prompting Interventions - including de-duplication, reflection, or adaptive branching - to spawn new branches with SES. Finally, by integrating SES and correctness rewards via Group Relative Policy Optimization, SIGHT internalizes robust exploration strategies without external verifiers. Experiments on single-hop and multi-hop QA benchmarks demonstrate that SIGHT significantly outperforms existing approaches, particularly in complex reasoning scenarios, using fewer search steps.

Reasoning is a fundamental substrate for solving novel and complex problems. Deliberate efforts in learning and developing frameworks around System 2 reasoning have made great strides, yet problems of sufficient complexity remain largely out of reach for open models. To address this gap, we examine the potential of Generative Flow Networks as a fine-tuning method for LLMs to unlock advanced reasoning capabilities. In this paper, we present a proof of concept in the domain of formal reasoning, specifically in the Neural Theorem Proving (NTP) setting, where proofs specified in a formal language such as Lean can be deterministically and objectively verified. Unlike classical reward-maximization reinforcement learning, which frequently over-exploits high-reward actions and fails to effectively explore the state space, GFlowNets have emerged as a promising approach for sampling compositional objects, improving generalization, and enabling models to maintain diverse hypotheses. Our early results demonstrate GFlowNet fine-tuning's potential for enhancing model performance in a search setting, which is especially relevant given the paradigm shift towards inference time compute scaling and "thinking slowly."

Formal reasoning and automated theorem proving constitute a challenging subfield of machine learning, in which machines are tasked with proving mathematical theorems using formal languages like Lean. A formal verification system can check whether a formal proof is correct or not almost instantaneously, but generating a completely correct formal proof with large language models (LLMs) remains a formidable task. The usual approach in the literature is to prompt the LLM many times (up to several thousands) until one of the generated proofs passes the verification system. In this work, we present APOLLO (Automated PrOof repair via LLM and Lean cOllaboration), a modular, model-agnostic pipeline that combines the strengths of the Lean compiler with an LLM's reasoning abilities to achieve better proof-generation results at a low sampling budget. Apollo directs a fully automated process in which the LLM generates proofs for theorems, a set of agents analyze the proofs, fix the syntax errors, identify the mistakes in the proofs using Lean, isolate failing sub-lemmas, utilize automated solvers, and invoke an LLM on each remaining goal with a low top-K budget. The repaired sub-proofs are recombined and reverified, iterating up to a user-controlled maximum number of attempts. On the miniF2F benchmark, we establish a new state-of-the-art accuracy of 75.0% among 7B-parameter models while keeping the sampling budget below one thousand. Moreover, Apollo raises the state-of-the-art accuracy for Goedel-Prover-SFT to 65.6% while cutting sample complexity from 25,600 to a few hundred. General-purpose models (o3-mini, o4-mini) jump from 3-7% to over 40% accuracy. Our results demonstrate that targeted, compiler-guided repair of LLM outputs yields dramatic gains in both efficiency and correctness, suggesting a general paradigm for scalable automated theorem proving.

Diffusion large language models (dLLMs) theoretically permit token decoding in arbitrary order, a flexibility that could enable richer exploration of reasoning paths than autoregressive (AR) LLMs. In practice, however, random-order decoding often hurts generation quality. To mitigate this, low-confidence remasking improves single-sample quality (e.g., Pass@1) by prioritizing confident tokens, but it also suppresses exploration and limits multi-sample gains (e.g., Pass@k), creating a fundamental quality--exploration dilemma. In this paper, we provide a unified explanation of this dilemma. We show that low-confidence remasking improves a myopic proxy for quality while provably constraining the entropy of the induced sequence distribution. To overcome this limitation, we characterize the optimal distribution that explicitly balances quality and exploration, and develop a simple Independent Metropolis--Hastings sampler that approximately targets this distribution during decoding. Experiments across a range of reasoning benchmarks including MATH500, AIME24/25, HumanEval, and MBPP show that our approach yields better exploration-quality tradeoff than both random and low-confidence remasking.

Formal theorem proving with TLA+ provides rigorous guarantees for system specifications, but constructing proofs requires substantial expertise and effort. While large language models have shown promise in automating proofs for tactic-based theorem provers like Lean, applying these approaches directly to TLA+ faces significant challenges due to the hierarchical proof structure of the TLA+ proof system. We present a prompt-based approach that leverages LLMs to guide hierarchical decomposition of complex proof obligations into simpler sub-claims, while relying on symbolic provers for verification. Our key insight is to constrain LLMs to generate normalized claim decompositions rather than complete proofs, significantly reducing syntax errors. We also introduce a benchmark suite of 119 theorems adapted from (1) established mathematical collections and (2) inductive proofs of distributed protocols. Our approach consistently outperforms baseline methods across the benchmark suite.

Automatically generated code is gaining traction recently, owing to the prevalence of Large Language Models (LLMs). Further, the AlphaProof initiative has demonstrated the possibility of using AI for general mathematical reasoning. Reasoning about computer programs (software) can be accomplished via general mathematical reasoning; however, it tends to be more structured and richer in contexts. This forms an attractive proposition, since then AI agents can be used to reason about voluminous code that gets generated by AI. In this work, we present a first LLM agent, AutoRocq, for conducting program verification. Unlike past works, which rely on extensive training of LLMs on proof examples, our agent learns on-the-fly and improves the proof via an iterative refinement loop. The iterative improvement of the proof is achieved by the proof agent communicating with the Rocq (formerly Coq) theorem prover to get additional context and feedback. The final result of the iteration is a proof derivation checked by the Rocq theorem prover. In this way, our proof construction involves autonomous collaboration between the proof agent and the theorem prover. This autonomy facilitates the search for proofs and decision-making in deciding on the structure of the proof tree. Experimental evaluation on SV-COMP benchmarks and on Linux kernel modules shows promising efficacy in achieving automated program verification. As automation in code generation becomes more widespread, we posit that our proof agent can be potentially integrated with AI coding agents to achieve a generate and validate loop, thus moving closer to the vision of trusted automatic programming.

Large Reasoning Models (LRMs) have recently achieved strong mathematical and code reasoning performance through Reinforcement Learning (RL) post-training. However, we show that modern reasoning post-training induces an unintended exploration collapse: temperature-based sampling no longer increases pass@n accuracy. Empirically, the final-layer posterior of post-trained LRMs exhibit sharply reduced entropy, while the entropy of intermediate layers remains relatively high. Motivated by this entropy asymmetry, we propose Latent Exploration Decoding (LED), a depth-conditioned decoding strategy. LED aggregates intermediate posteriors via cumulative sum and selects depth configurations with maximal entropy as exploration candidates. Without additional training or parameters, LED consistently improves pass@1 and pass@16 accuracy by 0.61 and 1.03 percentage points across multiple reasoning benchmarks and models. Project page: https://GitHub.com/Xiaomi-Research/LED.

Reinforcement learning with verifiable rewards (RLVR) has emerged as a powerful paradigm for enhancing the reasoning capabilities of large language models (LLMs). Unlike traditional RL approaches, RLVR leverages rule-based feedback to guide LLMs in generating and refining complex reasoning chains -- a process critically dependent on effective exploration strategies. While prior work has demonstrated RLVR's empirical success, the fundamental mechanisms governing LLMs' exploration behaviors remain underexplored. This technical report presents a systematic investigation of exploration capacities in RLVR, covering four main aspects: (1) exploration space shaping, where we develop quantitative metrics to characterize LLMs' capability boundaries; (2) entropy-performance exchange, analyzed across training stages, individual instances, and token-level patterns; and (3) RL performance optimization, examining methods to effectively translate exploration gains into measurable improvements. By unifying previously identified insights with new empirical evidence, this work aims to provide a foundational framework for advancing RLVR systems.

Exploration remains a critical challenge in online reinforcement learning, as an agent must effectively explore unknown environments to achieve high returns. Currently, the main exploration algorithms are primarily count-based methods and curiosity-based methods, with prediction-error methods being a prominent example. In this paper, we propose a novel method called Random Distribution Distillation (RDD), which samples the output of a target network from a normal distribution. RDD facilitates a more extensive exploration by explicitly treating the difference between the prediction network and the target network as an intrinsic reward. Furthermore, by introducing randomness into the output of the target network for a given state and modeling it as a sample from a normal distribution, intrinsic rewards are bounded by two key components: a pseudo-count term ensuring proper exploration decay and a discrepancy term accounting for predictor convergence. We demonstrate that RDD effectively unifies both count-based and prediction-error approaches. It retains the advantages of prediction-error methods in high-dimensional spaces, while also implementing an intrinsic reward decay mode akin to the pseudo-count method. In the experimental section, RDD is compared with more advanced methods in a series of environments. Both theoretical analysis and experimental results confirm the effectiveness of our approach in improving online exploration for reinforcement learning tasks.

Automated Theorem Proving (ATP) in formal languages is a foundational challenge for AI. While Large Language Models (LLMs) have driven remarkable progress, a significant gap remains between their powerful informal reasoning capabilities and their weak formal proving performance. Recent studies show that the informal accuracy exceeds 80% while formal success remains below 8% on benchmarks like PutnamBench. We argue this gap persists because current state-of-the-art provers, by tightly coupling reasoning and proving, are trained with paradigms that inadvertently punish deep reasoning in favor of shallow, tactic-based strategies. To bridge this fundamental gap, we propose a novel framework that decouples high-level reasoning from low-level proof generation. Our approach utilizes two distinct, specialized models: a powerful, general-purpose Reasoner to generate diverse, strategic subgoal lemmas, and an efficient Prover to rigorously verify them. This modular design liberates the model's full reasoning potential and bypasses the pitfalls of end-to-end training. We evaluate our method on a challenging set of post-2000 IMO problems, a problem set on which no prior open-source prover has reported success. Our decoupled framework successfully solves 5 of these problems, demonstrating a significant step towards automated reasoning on exceptionally difficult mathematical challenges. To foster future research, we release our full dataset of generated and verified lemmas for a wide range of IMO problems, available at https://tencent-imo.github.io/ .

Reinforcement learning (RL) agents improve through trial-and-error, but when reward is sparse and the agent cannot discover successful action sequences, learning stagnates. This has been a notable problem in training deep RL agents to perform web-based tasks, such as booking flights or replying to emails, where a single mistake can ruin the entire sequence of actions. A common remedy is to "warm-start" the agent by pre-training it to mimic expert demonstrations, but this is prone to overfitting. Instead, we propose to constrain exploration using demonstrations. From each demonstration, we induce high-level "workflows" which constrain the allowable actions at each time step to be similar to those in the demonstration (e.g., "Step 1: click on a textbox; Step 2: enter some text"). Our exploration policy then learns to identify successful workflows and samples actions that satisfy these workflows. Workflows prune out bad exploration directions and accelerate the agent's ability to discover rewards. We use our approach to train a novel neural policy designed to handle the semi-structured nature of websites, and evaluate on a suite of web tasks, including the recent World of Bits benchmark. We achieve new state-of-the-art results, and show that workflow-guided exploration improves sample efficiency over behavioral cloning by more than 100x.

Reinforcement Learning with Verifiable Rewards (RLVR) has propelled Large Language Models in complex reasoning, yet its scalability is often hindered by a training bottleneck where performance plateaus as policy entropy collapses, signaling a loss of exploration. Previous methods typically address this by maintaining high policy entropy, yet the precise mechanisms that govern meaningful exploration have remained underexplored. Our analysis suggests that an unselective focus on entropy risks amplifying irrelevant tokens and destabilizing training. This paper investigates the exploration dynamics within RLVR and identifies a key issue: the gradual elimination of valuable low-probability exploratory tokens, which we term \textit{reasoning sparks}. We find that while abundant in pre-trained models, these sparks are systematically extinguished during RLVR due to over-penalization, leading to a degeneracy in exploration. To address this, we introduce Low-probability Regularization (Lp-Reg). Its core mechanism regularizes the policy towards a heuristic proxy distribution. This proxy is constructed by filtering out presumed noise tokens and re-normalizing the distribution over the remaining candidates. The result is a less-noisy proxy where the probability of reasoning sparks is amplified, which then serves as a soft regularization target to shield these valuable tokens from elimination via KL divergence. Experiments show that Lp-Reg enables stable on-policy training for around 1,000 steps, a regime where baseline entropy-control methods collapse. This sustained exploration leads to state-of-the-art performance, achieving a 60.17% average accuracy on five math benchmarks, an improvement of 2.66% over prior methods. Code is available at https://github.com/CarlanLark/Lp-Reg.

Chain-of-thought (CoT) reasoning enables large language models (LLMs) to break down complex problems into interpretable intermediate steps, significantly enhancing model transparency and performance in reasoning tasks. However, conventional CoT methods rely on heuristic sampling without structured modeling of reasoning transitions, constraining their ability to systematically explore and discover diverse and effective reasoning trajectories. In this work, we introduce CTRLS, a framework that formulates CoT reasoning as a Markov decision process (MDP) with latent state transitions, enabling principled and state-aware exploration via distributional reinforcement learning. By modelling reasoning actions as explicit probability distributions in latent space, our approach explicitly models epistemic uncertainty, facilitating robust exploration of the reasoning space. As part of our framework, we introduce an on-policy reinforcement learning strategy incorporating epsilon-greedy exploration and entropy-based regularization to iteratively refine latent state transitions without requiring additional fine-tuning of the underlying LLM. Theoretical analyses provide evidence lower bounds (ELBO), theoretically grounding our transition-aware modeling of latent reasoning dynamics. Further experiments demonstrate improvements in reasoning accuracy, diversity, and exploration efficiency across benchmark reasoning tasks.

Reinforcement learning algorithms typically struggle in the absence of a dense, well-shaped reward function. Intrinsically motivated exploration methods address this limitation by rewarding agents for visiting novel states or transitions, but these methods offer limited benefits in large environments where most discovered novelty is irrelevant for downstream tasks. We describe a method that uses background knowledge from text corpora to shape exploration. This method, called ELLM (Exploring with LLMs) rewards an agent for achieving goals suggested by a language model prompted with a description of the agent's current state. By leveraging large-scale language model pretraining, ELLM guides agents toward human-meaningful and plausibly useful behaviors without requiring a human in the loop. We evaluate ELLM in the Crafter game environment and the Housekeep robotic simulator, showing that ELLM-trained agents have better coverage of common-sense behaviors during pretraining and usually match or improve performance on a range of downstream tasks.

LLM-based formal proof assistants (e.g., in Lean) hold great promise for automating mathematical discovery. But beyond syntactic correctness, do these systems truly understand mathematical structure as humans do? We investigate this question through the lens of mathematical inequalities -- a fundamental tool across many domains. While modern provers can solve basic inequalities, we probe their ability to handle human-intuitive compositionality. We introduce Ineq-Comp, a benchmark built from elementary inequalities through systematic transformations, including variable duplication, algebraic rewriting, and multi-step composition. Although these problems remain easy for humans, we find that most provers -- including Goedel, STP, and Kimina-7B -- struggle significantly. DeepSeek-Prover-V2-7B shows relative robustness -- possibly because it is trained to decompose the problems into sub-problems -- but still suffers a 20\% performance drop (pass@32). Strikingly, performance remains poor for all models even when formal proofs of the constituent parts are provided in context, revealing that the source of weakness is indeed in compositional reasoning. Our results expose a persisting gap between the generalization behavior of current AI provers and human mathematical intuition.

In this paper, we present empirical evidence of skills and directed exploration emerging from a simple RL algorithm long before any successful trials are observed. For example, in a manipulation task, the agent is given a single observation of the goal state and learns skills, first for moving its end-effector, then for pushing the block, and finally for picking up and placing the block. These skills emerge before the agent has ever successfully placed the block at the goal location and without the aid of any reward functions, demonstrations, or manually-specified distance metrics. Once the agent has learned to reach the goal state reliably, exploration is reduced. Implementing our method involves a simple modification of prior work and does not require density estimates, ensembles, or any additional hyperparameters. Intuitively, the proposed method seems like it should be terrible at exploration, and we lack a clear theoretical understanding of why it works so effectively, though our experiments provide some hints.

Language environments such as web browsers, code terminals, and interactive simulations emit raw text rather than states, and provide none of the runtime structure that MDP analysis requires. No explicit state space, no observation-to-state mapping, no certified transitions, and no termination criterion. We introduce the State-Centric Decision Process (SDP), a runtime framework that constructs these missing inputs by having the agent build them, predicate by predicate, as it acts. At each step the agent commits to a natural-language predicate describing how the world should look, takes an action to make it true, and checks the observation against it. Predicates that pass become certified states, and the resulting trajectory carries the four objects language environments do not provide, namely a task-induced state space, an observation-to-state mapping, certified transitions, and a termination criterion. We evaluate SDP on five benchmarks spanning planning, scientific exploration, web reasoning, and multi-hop question answering. SDP achieves the best training-free results on all five, with the advantage widening as the horizon grows. The certified trajectories additionally support analyses unavailable to reactive agents, including per-predicate credit assignment, failure localization, partial-progress measurement, and modular operator replacement.

We are at the cusp of a transition from "learning from data" to "learning what data to learn from" as a central focus of artificial intelligence (AI) research. While the first-order learning problem is not completely solved, large models under unified architectures, such as transformers, have shifted the learning bottleneck from how to effectively train our models to how to effectively acquire and use task-relevant data. This problem, which we frame as exploration, is a universal aspect of learning in open-ended domains, such as the real world. Although the study of exploration in AI is largely limited to the field of reinforcement learning, we argue that exploration is essential to all learning systems, including supervised learning. We propose the problem of generalized exploration to conceptually unify exploration-driven learning between supervised learning and reinforcement learning, allowing us to highlight key similarities across learning settings and open research challenges. Importantly, generalized exploration serves as a necessary objective for maintaining open-ended learning processes, which in continually learning to discover and solve new problems, provides a promising path to more general intelligence.

The scarcity of high-quality, logically sound data is a critical bottleneck for advancing the mathematical reasoning of Large Language Models (LLMs). Our work confronts this challenge by turning decades of automated theorem proving research into a scalable data engine. Rather than relying on error-prone LLMs or complex proof-assistant syntax like Lean and Isabelle, our framework leverages E-prover's saturation capabilities on the vast TPTP axiom library to derive a massive, guaranteed-valid corpus of theorems. Our pipeline is principled and simple: saturate axioms, filter for "interesting" theorems, and generate tasks. With no LLMs in the loop, we eliminate factual errors by construction. This purely symbolic data is then transformed into three difficulty-controlled challenges: entailment verification, premise selection, and proof reconstruction. Our zero-shot experiments on frontier models reveal a clear weakness: performance collapses on tasks requiring deep, structural reasoning. Our framework provides both the diagnostic tool to measure this gap and a scalable source of symbolic training data to address it. We make the code and data publicly available. https://github.com/sileod/reasoning_core https://hf.co/datasets/reasoning-core/rc1

Existing LMs struggle with proof-oriented programming due to data scarcity, which manifest in two key ways: (1) a lack of sufficient corpora for proof-oriented programming languages such as F*, and (2) the absence of large-scale, project-level proof-oriented implementations that can teach the model the intricate reasoning process when performing proof-oriented programming. We present the first on synthetic data augmentation for project level proof oriented programming for both generation and repair. Our method addresses data scarcity by synthesizing basic proof-oriented programming problems for proficiency in that language; incorporating diverse coding data for reasoning capability elicitation and creating new proofs and repair data within existing repositories. This approach enables language models to both synthesize and repair proofs for function- and repository-level code. We show that our fine-tuned 14B parameter model, PoPilot, can exceed the performance of the models that outperforms GPT-4o in project-level proof-oriented programming by 64% relative margin, and can improve GPT-4o's performance by 54% by repairing its outputs over GPT-4o's self-repair.

Exploration bonuses in reinforcement learning guide long-horizon exploration by defining custom intrinsic objectives. Several exploration objectives like count-based bonuses, pseudo-counts, and state-entropy maximization are non-stationary and hence are difficult to optimize for the agent. While this issue is generally known, it is usually omitted and solutions remain under-explored. The key contribution of our work lies in transforming the original non-stationary rewards into stationary rewards through an augmented state representation. For this purpose, we introduce the Stationary Objectives For Exploration (SOFE) framework. SOFE requires identifying sufficient statistics for different exploration bonuses and finding an efficient encoding of these statistics to use as input to a deep network. SOFE is based on proposing state augmentations that expand the state space but hold the promise of simplifying the optimization of the agent's objective. We show that SOFE improves the performance of several exploration objectives, including count-based bonuses, pseudo-counts, and state-entropy maximization. Moreover, SOFE outperforms prior methods that attempt to stabilize the optimization of intrinsic objectives. We demonstrate the efficacy of SOFE in hard-exploration problems, including sparse-reward tasks, pixel-based observations, 3D navigation, and procedurally generated environments.

Recent advances in Reinforcement Learning with Verified Reward (RLVR) have driven the emergence of more sophisticated cognitive behaviors in large language models (LLMs), thereby enhancing their reasoning capabilities. However, in prior RLVR pipelines, the repeated use of static initial-state sampling drawn exactly from the dataset distribution during each sampling phase produced overly deterministic, low diversity model behavior, which manifested as rapid entropy collapse and hindered sustained performance gains during prolonged training. To address this issue, we introduce CURE (Critical-token-gUided Re concatenation for Entropy-collapse prevention), a two-stage framework that balances exploration and exploitation. Specifically, in the first stage, to deliberately steer the model toward novel yet coherent contexts, we re-generate at high-entropy critical tokens and jointly optimize the original and the branched trajectories. The further comparison with vanilla DAPO shows that the regeneration process achieves a better performance on math reasoning tasks while sustaining a high-level entropy degree for exploration. In the second stage, we continue training with static initial-state sampling by DAPO, intentionally placing the model in a familiar state to gradually strengthen exploitation. Extensive experiments on Qwen-2.5-Math-7B show that, compared to other RLVR methods, CURE achieves a 5% performance gain across six math benchmarks, establishing state-of-the-art performance in both entropy and accuracy. A series of experiments further validate the effectiveness of our approach. Code is available at https://github.com/bytedance/CURE.

Scaling test-time compute with multi-path chain-of-thought improves reasoning accuracy, but its effectiveness depends critically on the exploration-exploitation trade-off. Existing approaches address this trade-off in rigid ways: tree-structured search hard-codes exploration through brittle expansion rules that interfere with post-trained reasoning, while parallel reasoning over-explores redundant hypothesis paths and relies on weak answer selection. Motivated by the observation that the optimal balance is phase-dependent and that correct and incorrect reasoning paths often diverge only at late stages, we reformulate test-time scaling as a dynamic expand-reduce control problem over a pool of hypotheses. We propose HyPER, a training-free online control policy for multi-path decoding in mixture-of-experts models that reallocates computation under a fixed budget using lightweight path statistics. HyPER consists of an online controller that transitions from exploration to exploitation as the hypothesis pool evolves, a token-level refinement mechanism that enables efficient generation-time exploitation without full-path resampling, and a length- and confidence-aware aggregation strategy for reliable answer-time exploitation. Experiments on four mixture-of-experts language models across diverse reasoning benchmarks show that HyPER consistently achieves a superior accuracy-compute trade-off, improving accuracy by 8 to 10 percent while reducing token usage by 25 to 40 percent.

Reinforcement learning with verifiable rewards has significantly advanced the reasoning capabilities of large language models, yet how to explicitly steer training toward exploration or exploitation remains an open problem. We introduce Token Hidden Reward (THR), a token-level metric that quantifies each token's influence on the likelihood of correct responses under Group Relative Policy Optimization (GRPO). We find that training dynamics are dominated by a small subset of tokens with high absolute THR values. Most interestingly, tokens with positive THR strengthen confidence in correct outputs, thus favoring exploitation, while tokens with negative THR preserve probability mass for alternative outputs, enabling exploration. This insight suggests a natural intervention: a THR-guided reweighting algorithm that modulates GRPO's learning signals to explicitly bias training toward exploitation or exploration. We validate the efficacy of this algorithm on diverse math reasoning benchmarks. By amplifying tokens with positive THR value and weakening negative ones, our algorithm improves greedy-decoding accuracy, favoring exploitation. The reverse strategy yields consistent gains in Pass@K accuracy, favoring exploration. We further demonstrate that our algorithm integrates seamlessly with other RL objectives such as GSPO and generalizes across architectures including Llama. These findings establish THR as a principled and fine-grained mechanism for dynamically controlling exploration and exploitation in RL-tuned LLMs, providing new tools for targeted fine-tuning in reasoning-intensive applications.

Autonomous coding agents increasingly execute multi-step software work, but lifecycle states such as reviewed, tested, DONE, and ready-to-merge remain claims unless supported by current evidence. We present Proof-or-Stop Lifecycle Control, a method that permits lifecycle transitions only when fresh, tracked-source-state-bound, mechanically verifiable evidence satisfies the relevant gate. The method treats agent outputs as claims rather than lifecycle state, and uses proof operationally to mean gate-admissible evidence under a stated trust model, not semantic program correctness. We evaluate an open-source implementation through mechanism tests, a powered control-policy ablation, and operated self-application evidence. The unattended-loop engine passed 10 of 10 scenarios with zero false-DONE, and local-key receipt bundles rejected 18 tamper classes with zero false accepts. In a 9,240-cell ablation, the pre-registered A4 versus A2-prime comparison reduced visible-pass/hidden-fail amplification from 31 of 1,800 injected cells under a compute-budgeted naive loop to 2 of 1,800 under the gated loop, a 1.6 percentage-point improvement in not-amplified rate with a 95 percent confidence interval of [0.8, 2.5]. A near-compute A3 versus A4 comparison, 14 of 1,800 versus 2 of 1,800, indicates that the gain is associated with enforcing review as a lifecycle gate rather than merely adding a reviewer. The self-application corpus contains 565 stories and 1,007 review findings, with 94.8 percent resolved, plus a 68-row high/critical cross-vendor exhibit. These results support Proof-or-Stop as a model-agnostic, host-neutral control layer for deciding which autonomous-agent claims a lifecycle may act on. The evaluation is limited to one model family, 24 ablation tasks, and a self-hosted corpus.

The integration of Large Language Models (LLMs) into automated theorem proving has shown immense promise, yet is fundamentally constrained by challenges in scaling up both training-time reinforcement learning (RL) and inference-time compute. This paper introduces BFS-Prover-V2, a system designed to address this dual scaling problem. We present two primary innovations. The first is a novel multi-turn off-policy RL framework for continually improving the performance of LLM step-prover at training time. This framework, inspired by the principles of AlphaZero, utilizes a multi-stage expert iteration pipeline featuring adaptive tactic-level data filtering and periodic retraining to surmount the performance plateaus that typically curtail long-term RL in LLM-based agents. The second innovation is a planner-enhanced multi-agent search architecture that scales reasoning capabilities at inference time. This architecture employs a general reasoning model as a high-level planner to iteratively decompose complex theorems into a sequence of simpler subgoals. This hierarchical approach substantially reduces the search space, enabling a team of parallel prover agents to collaborate efficiently by leveraging a shared proof cache. We demonstrate that this dual approach to scaling yields state-of-the-art results on established formal mathematics benchmarks. BFS-Prover-V2 achieves 95.08\% and 41.4\% on the MiniF2F and ProofNet test sets respectively. While demonstrated in the domain of formal mathematics, the RL and inference techniques presented in this work are of broader interest and may be applied to other domains requiring long-horizon multi-turn reasoning and complex search.

We study the design of sample-efficient algorithms for reinforcement learning in the presence of rich, high-dimensional observations, formalized via the Block MDP problem. Existing algorithms suffer from either 1) computational intractability, 2) strong statistical assumptions that are not necessarily satisfied in practice, or 3) suboptimal sample complexity. We address these issues by providing the first computationally efficient algorithm that attains rate-optimal sample complexity with respect to the desired accuracy level, with minimal statistical assumptions. Our algorithm, MusIK, combines systematic exploration with representation learning based on multi-step inverse kinematics, a learning objective in which the aim is to predict the learner's own action from the current observation and observations in the (potentially distant) future. MusIK is simple and flexible, and can efficiently take advantage of general-purpose function approximation. Our analysis leverages several new techniques tailored to non-optimistic exploration algorithms, which we anticipate will find broader use.

Large language models (LLMs) have shown promising first-order logic (FOL) reasoning capabilities with applications in various areas. However, their effectiveness in complex mathematical reasoning involving multi-step FOL deductions is still under-researched. While LLMs perform competitively on established mathematical reasoning benchmarks, they struggle with multi-step FOL tasks, as demonstrated by Deepseek-Prover-V2-7B's low accuracy (4.2%) on our proposed theorem proving dataset. This issue arises from the limited exploration of diverse proof strategies and the potential for early reasoning mistakes to undermine entire proofs. To address these issues, we propose DREAM, a self-adaptive solution that enhances the Diversity and REAsonability of LLMs' generation strategies. DREAM incorporates an Axiom-Driven Strategy Diversification mechanism to promote varied strategic outcomes and a Sub-Proposition Error Feedback to help LLMs reflect on and correct their proofs. Our contributions include pioneering advancements in LLMs' mathematical reasoning through FOL theorem proving, introducing a novel inference stage solution that improves performance by 0.6% to 6.4%, and providing a curated dataset of 447 mathematical theorems in Lean 4 format for evaluation.

Although RLVR has become an essential component for developing advanced reasoning skills in LLMs, contemporary studies have documented training plateaus that emerge following thousands of optimization steps, demonstrating notable decreases in performance gains despite increased computational investment. This limitation stems from the sparse exploration patterns inherent in current RLVR practices, where models rely on limited rollouts that often miss critical reasoning paths and fail to provide systematic coverage of the solution space. We present DeepSearch, a framework that integrates Monte Carlo Tree Search directly into RLVR training. In contrast to existing methods that rely on tree search only at inference, DeepSearch embeds structured search into the training loop, enabling systematic exploration and fine-grained credit assignment across reasoning steps. Through training-time exploration, DeepSearch addresses the fundamental bottleneck of insufficient exploration, which leads to diminishing performance improvements over prolonged training steps. Our contributions include: (1) a global frontier selection strategy that prioritizes promising nodes across the search tree, (2) selection with entropy-based guidance that identifies confident paths for supervision, and (3) adaptive replay buffer training with solution caching for efficiency. Experiments on mathematical reasoning benchmarks show that DeepSearch achieves 62.95% average accuracy and establishes a new state-of-the-art for 1.5B reasoning models - using 5.7x fewer GPU hours than extended training approaches. These results highlight the importance of strategic exploration over brute-force scaling and demonstrate the promise of algorithmic innovation for advancing RLVR methodologies. DeepSearch establishes a new direction for scaling reasoning capabilities through systematic search rather than prolonged computation.

We evaluate language models on their ability to explore interactive environments under a limited interaction budget. We introduce three parametric tasks with controllable exploration difficulty, spanning continuous and discrete environments. Across state-of-the-art models, we find systematic under-exploration and suboptimal solutions, with performance often significantly worse than simple explore--exploit heuristic baselines and scaling weakly as the budget increases. Finally, we study two lightweight interventions: splitting a fixed budget into parallel executions, which surprisingly improves performance despite a no-gain theoretical result for our tasks, and periodically summarizing the interaction history, which preserves key discoveries and further improves exploration.

Recent advancements have shown that reinforcement learning (RL) can substantially improve the reasoning abilities of large language models (LLMs). The effectiveness of such RL training, however, depends critically on the exploration space defined by the pre-trained model's token-output distribution. In this paper, we revisit the standard cross-entropy loss, interpreting it as a specific instance of policy gradient optimization applied within a single-step episode. To systematically study how the pre-trained distribution shapes the exploration potential for subsequent RL, we propose a generalized pre-training objective that adapts on-policy RL principles to supervised learning. By framing next-token prediction as a stochastic decision process, we introduce a reward-shaping strategy that explicitly balances diversity and precision. Our method employs a positive reward scaling factor to control probability concentration on ground-truth tokens and a rank-aware mechanism that treats high-ranking and low-ranking negative tokens asymmetrically. This allows us to reshape the pre-trained token-output distribution and investigate how to provide a more favorable exploration space for RL, ultimately enhancing end-to-end reasoning performance. Contrary to the intuition that higher distribution entropy facilitates effective exploration, we find that imposing a precision-oriented prior yields a superior exploration space for RL.

Reinforcement learning has empowered large language models to act as intelligent agents, yet training them for long-horizon tasks remains challenging due to the scarcity of high-quality trajectories, especially under limited resources. Existing methods typically scale up rollout sizes and indiscriminately allocate computational resources among intermediate steps. Such attempts inherently waste substantial computation budget on trivial steps while failing to guarantee sample quality. To address this, we propose Spark (Strategic Policy-Aware exploRation via Key-state dynamic branching), a novel framework that selectively branches at critical decision states for resource-efficient exploration. Our key insight is to activate adaptive branching exploration at critical decision points to probe promising trajectories, thereby achieving precise resource allocation that prioritizes sampling quality over blind coverage. This design leverages the agent's intrinsic decision-making signals to reduce dependence on human priors, enabling the agent to autonomously expand exploration and achieve stronger generalization. Experiments across diverse tasks (e.g., embodied planning), demonstrate that Spark achieves superior success rates with significantly fewer training samples, exhibiting robust generalization even in unseen scenarios.

Logical reasoning encompasses deduction, induction, and abduction. However, while Large Language Models (LLMs) have effectively mastered the former two, abductive reasoning remains significantly underexplored. Existing frameworks, predominantly designed for static deductive tasks, fail to generalize to abductive reasoning due to unstructured state representation and lack of explicit state control. Consequently, they are inevitably prone to Evidence Fabrication, Context Drift, Failed Backtracking, and Early Stopping. To bridge this gap, we introduce Graph of States (GoS), a general-purpose neuro-symbolic framework tailored for abductive tasks. GoS grounds multi-agent collaboration in a structured belief states, utilizing a causal graph to explicitly encode logical dependencies and a state machine to govern the valid transitions of the reasoning process. By dynamically aligning the reasoning focus with these symbolic constraints, our approach transforms aimless, unconstrained exploration into a convergent, directed search. Extensive evaluations on two real-world datasets demonstrate that GoS significantly outperforms all baselines, providing a robust solution for complex abductive tasks. Code repo and all prompts: https://anonymous.4open.science/r/Graph-of-States-5B4E.

The rapid progress of large language models (LLMs) has sparked growing interest in building Artificial General Intelligence (AGI) within Graphical User Interface (GUI) environments. However, existing GUI agents based on LLMs or vision-language models (VLMs) often fail to generalize to novel environments and rely heavily on manually curated, diverse datasets. To overcome these limitations, we introduce ScreenExplorer, a VLM trained via Group Relative Policy Optimization(GRPO) in real, dynamic, and open-ended GUI environments. Innovatively, we introduced a world-model-based curiosity reward function to help the agent overcome the cold-start phase of exploration. Additionally, distilling experience streams further enhances the model's exploration capabilities. Our training framework enhances model exploration in open GUI environments, with trained models showing better environmental adaptation and sustained exploration compared to static deployment models. Our findings offer a scalable pathway toward AGI systems with self-improving capabilities in complex interactive settings.

Reinforcement learning (RL) has emerged as a powerful method for improving the reasoning abilities of large language models (LLMs). Outcome-based RL, which rewards policies solely for the correctness of the final answer, yields substantial accuracy gains but also induces a systematic loss in generation diversity. This collapse undermines real-world performance, where diversity is critical for test-time scaling. We analyze this phenomenon by viewing RL post-training as a sampling process and show that, strikingly, RL can reduce effective diversity even on the training set relative to the base model. Our study highlights two central findings: (i) a transfer of diversity degradation, where reduced diversity on solved problems propagates to unsolved ones, and (ii) the tractability of the outcome space, since reasoning tasks admit only a limited set of distinct answers. Motivated by these insights, we propose outcome-based exploration, which assigns exploration bonuses according to final outcomes. We introduce two complementary algorithms: historical exploration, which encourages rarely observed answers via UCB-style bonuses, and batch exploration, which penalizes within-batch repetition to promote test-time diversity. Experiments on standard competition math with Llama and Qwen models demonstrate that both methods improve accuracy while mitigating diversity collapse. On the theoretical side, we formalize the benefit of outcome-based exploration through a new model of outcome-based bandits. Together, these contributions chart a practical path toward RL methods that enhance reasoning without sacrificing the diversity essential for scalable deployment.

To operate robustly in open-world environments, autonomous agents should be able to infer the behavior of unfamiliar systems through interaction alone, even in the absence of documentation. However, existing tool-use benchmarks expose semantic tool schemas in static environments, allowing agents to rely on prior knowledge rather than autonomous discovery. To address this limitation, we introduce ScrambleToolBench, an interactive terminal benchmark designed to isolate behavioral reasoning. By removing semantic cues and enforcing a continuous task curriculum, the benchmark requires agents to uncover hidden tool behaviors entirely through trial-and-error interaction. The benchmark further introduces dynamic challenges, including mapping drift, stochastic action failures, and temporal execution windows, to evaluate whether agents can revise and adapt their hypotheses as the environment changes. Our evaluation of state-of-the-art language models reveals that successful initial discovery does not translate into robust adaptation. When faced with structural changes such as mapping drift, agents fail to use deductive strategies such as cycle tracing, and instead exhibit belief inertia or fall back to exhaustive search. Increasing test-time reasoning only amplifies this expensive brute-force search rather than enabling deductive recovery. While equipping agents with persistent memory reduces compounding errors, they remain unable to efficiently infer structural changes, highlighting a gap in current agent reasoning.

Reinforcement learning is emerging as a primary driver for improving language model reasoning capabilities. A fundamental question is whether current reinforcement learning algorithms -- such as Group Relative Policy Optimization (GRPO), the de facto standard algorithm used to improve language model reasoning -- merely sharpen the base model's distribution around problems it can already solve. We investigate this question in the context of formal theorem proving, which has access to a perfect verifier. We identify a degenerate rank bias in GRPO in which highly probable trajectories are reinforced and rare ones are neglected. This results in distribution sharpening: the model can solve some problems with fewer samples, but underperforms simply sampling more solutions from the original model. To overcome GRPO's rank bias we introduce unlikeliness reward, a simple method for explicitly up-weighting rare but correct solutions. We show that unlikeliness reward mitigates rank bias and improves pass@N across a large range of N in both synthetic and real theorem proving settings. We also uncover an unexpected link between rank bias and a seemingly mundane hyperparameter -- the number of updates per batch -- that leads to a second, complementary mitigation. We combine our insights into a revised GRPO training recipe for formal theorem proving, yielding an open pipeline that achieves competitive performance to DeepSeek-Prover-V1.5-RL on the miniF2F-test benchmark. We release our implementation at https://github.com/AndreHe02/rewarding-unlikely-release

As automated reasoning systems advance rapidly, there is a growing need for research-level formal mathematical problems to accurately evaluate their capabilities. To address this, we present Formal Conjectures, an evolving benchmark of currently 2615 mathematical problem statements formalized in Lean 4. Sourced from areas of active mathematical research, the dataset features 1029 open research conjectures providing a zero-contamination benchmark for mathematical proof discovery, and 836 solved problems for proof autoformalization. Notably, the repository provides a structured interface connecting mathematicians who formalize and clarify problems with the AI systems and humans attempting to solve them. Demonstrating its immediate utility, the benchmark has already been leveraged to make new mathematical discoveries, including the resolution of open research conjectures. We describe our approach to ensuring the correctness of these formalizations in a collaborative open-source project where contributions stem from an active community. In this framework, AI-generated proofs and disproofs serve as a valuable auditing mechanism to iteratively improve the fidelity of the benchmark. Finally, we provide a standardized evaluation setup and report baseline results on frozen evaluation subsets, demonstrating a climbable signal that measures the current frontier of automated reasoning on research-level mathematics.

Recent advancements in agentic test-time scaling allow models to gather environmental feedback before committing to final actions. A key limitation of existing methods is that they typically employ undifferentiated exploration strategies, lacking the ability to adaptively distinguish when exploration is truly required. In this paper, we propose an exploration-aware reinforcement learning framework that enables LLM agents to adaptively explore only when uncertainty is high. Our method introduces a fine-grained reward function via variational inference that explicitly evaluates exploratory actions by estimating their potential to improve future decision-making, together with an exploration-aware grouping mechanism that separates exploratory actions from task-completion actions during optimization. By targeting informational gaps, this design allows agents to explore selectively and transition to execution as soon as the task context is clear. Empirically, we demonstrate that our approach achieves consistent improvements across a range of challenging text-based and GUI-based agent benchmarks. Code is available at https://github.com/HansenHua/EAPO-ICML26 and models are available at https://huggingface.co/hansenhua/EAPO-ICML26.

Theorem proving serves as a major testbed for evaluating complex reasoning abilities in large language models (LLMs). However, traditional automated theorem proving (ATP) approaches rely heavily on formal proof systems that poorly align with LLMs' strength derived from informal, natural language knowledge acquired during pre-training. In this work, we propose DeepTheorem, a comprehensive informal theorem-proving framework exploiting natural language to enhance LLM mathematical reasoning. DeepTheorem includes a large-scale benchmark dataset consisting of 121K high-quality IMO-level informal theorems and proofs spanning diverse mathematical domains, rigorously annotated for correctness, difficulty, and topic categories, accompanied by systematically constructed verifiable theorem variants. We devise a novel reinforcement learning strategy (RL-Zero) explicitly tailored to informal theorem proving, leveraging the verified theorem variants to incentivize robust mathematical inference. Additionally, we propose comprehensive outcome and process evaluation metrics examining proof correctness and the quality of reasoning steps. Extensive experimental analyses demonstrate DeepTheorem significantly improves LLM theorem-proving performance compared to existing datasets and supervised fine-tuning protocols, achieving state-of-the-art accuracy and reasoning quality. Our findings highlight DeepTheorem's potential to fundamentally advance automated informal theorem proving and mathematical exploration.

Large Language Models (LLMs) present an intriguing avenue for exploration in the field of formal theorem proving. Nevertheless, their full potential, particularly concerning the mitigation of hallucinations and refinement through prover error messages, remains an area that has yet to be thoroughly investigated. To enhance the effectiveness of LLMs in the field, we introduce the Lyra, a new framework that employs two distinct correction mechanisms: Tool Correction (TC) and Conjecture Correction (CC). To implement Tool Correction in the post-processing of formal proofs, we leverage prior knowledge to utilize predefined prover tools (e.g., Sledgehammer) for guiding the replacement of incorrect tools. Tool Correction significantly contributes to mitigating hallucinations, thereby improving the overall accuracy of the proof. In addition, we introduce Conjecture Correction, an error feedback mechanism designed to interact with prover to refine formal proof conjectures with prover error messages. Compared to the previous refinement framework, the proposed Conjecture Correction refines generation with instruction but does not collect paired (generation, error & refinement) prompts. Our method has achieved state-of-the-art (SOTA) performance on both miniF2F validation (48.0% -> 55.3%) and test (45.5% -> 51.2%). We also present 3 IMO problems solved by Lyra. We believe Tool Correction (post-process for hallucination mitigation) and Conjecture Correction (subgoal adjustment from interaction with environment) could provide a promising avenue for future research in this field.

Recent advancements in tree search algorithms guided by verifiers have significantly enhanced the reasoning capabilities of large language models (LLMs), but at the cost of increased computational resources. In this work, we identify two key challenges contributing to this inefficiency: over-exploration due to redundant states with semantically equivalent content, and under-exploration caused by high variance in verifier scoring leading to frequent trajectory switching. To address these issues, we propose FETCH, an efficient tree search framework, which is a flexible, plug-and-play system compatible with various tree search algorithms. Our framework mitigates over-exploration by merging semantically similar states using agglomerative clustering of text embeddings obtained from a fine-tuned SimCSE model. To tackle under-exploration, we enhance verifiers by incorporating temporal difference learning with adjusted lambda-returns during training to reduce variance, and employing a verifier ensemble to aggregate scores during inference. Experiments on GSM8K, GSM-Plus, and MATH datasets demonstrate that our methods significantly improve reasoning accuracy and computational efficiency across four different tree search algorithms, paving the way for more practical applications of LLM-based reasoning. The code is available at https://github.com/Soistesimmer/Fetch.

Tree search has become as a representative framework for test-time reasoning with large language models (LLMs), exemplified by methods such as Tree-of-Thought and Monte Carlo Tree Search that explore multiple reasoning paths. However, it remains difficult to provide instant and reliable quantitative assessments of intermediate reasoning step quality, and extensive path exploration is computationally costly. To address this, we propose Mutual Information Tree Search (MITS), a novel framework that guides reasoning with information-theoretic principles. MITS introduces an effective scoring function based on pointwise mutual information (PMI), which enables step-wise evaluation of reasoning paths and search tree expansion via beam search without expensive look-ahead simulations, achieving superior reasoning performances while maintaining computational efficiency. The framework is complemented by an entropy-based dynamic sampling strategy that adaptively allocates computational resources to uncertain reasoning steps where exploration is most beneficial. For final prediction, MITS employs a weighted voting scheme that combines PMI scores with prediction consensus. Through comprehensive experiments on diverse reasoning benchmarks, MITS consistently surpasses baseline methods, establishing a principled and efficient framework for LLM reasoning.

Combinatorics is central to Olympiad-level mathematical problem solving, requiring deep discrete reasoning, creative constructions, and rigorous structural insight. Recent evidence suggests that even today's strongest frontier models remain uneven on Olympiad combinatorics, revealing a gap in creative mathematical reasoning. We introduce ComBench, an Olympiad-level combinatorics benchmark for evaluating and diagnosing the combinatorial reasoning capabilities of large language models. ComBench contains 100 human-annotated competition-level problems organized around two complementary settings: analysis-centric problems, which primarily require rigorous mathematical arguments, and construction-centric problems, which require explicit constructions in addition to correctness justifications. The evaluation protocol combines rubric-guided proof grading with deterministic construction verification, exposing cases where proof quality and construction validity diverge. Experiments on frontier open- and closed-source models show that ComBench is far from saturated: the strongest model reaches 65.4% overall Avg. and 75.3% overall Best@4. We further find that Rigorous Proof Reasoning and Constructive Realization are distinct capabilities: Kimi-K2.6 trails GPT-5.5 on analysis-centric proof grading but surpasses it on construction-centric Best@4, while Existence and Construction problems remain consistently hardest across representative frontier models.

Exploration is a prerequisite for learning useful behaviors in sparse-reward, long-horizon tasks, particularly within 3D environments. Curiosity-driven reinforcement learning addresses this via intrinsic rewards derived from the mismatch between the agent's predictive model of the world and reality. However, translating this intrinsic motivation to complex, photorealistic environments remains difficult, as agents can become trapped in local loops and receive fresh rewards for revisiting forgotten states. In this work, we demonstrate that this failure stems from a lack of spatial persistence and episodic context. We show that effective curiosity requires a model of the world that is persistent and continuously updated, paired with an agent that maintains an episodic trajectory history to navigate toward novel regions. We achieve this using an online 3D reconstruction as a persistent model of the world, while the agent policy is parameterized as a sequence model over RGB observations to maintain episodic context. This design enables effective exploration during training while allowing the agent to navigate using solely RGB frames at deployment. Trained purely via curiosity on HM3D, our agent outperforms RL-based active mapping baselines and generalizes zero-shot to Gibson and AI-generated worlds. Our end-to-end policy enables efficient adaptation to downstream tasks, such as apple picking and image-goal navigation, outperforming from-scratch baselines. Please see video results at https://recuriosity.github.io/.

We propose MDP-GapE, a new trajectory-based Monte-Carlo Tree Search algorithm for planning in a Markov Decision Process in which transitions have a finite support. We prove an upper bound on the number of calls to the generative models needed for MDP-GapE to identify a near-optimal action with high probability. This problem-dependent sample complexity result is expressed in terms of the sub-optimality gaps of the state-action pairs that are visited during exploration. Our experiments reveal that MDP-GapE is also effective in practice, in contrast with other algorithms with sample complexity guarantees in the fixed-confidence setting, that are mostly theoretical.

Recent progress in reasoning models has substantially advanced long-horizon mathematical and scientific problem solving, with several systems now reaching gold-medal-level performance on International Mathematical Olympiad (IMO) and International Physics Olympiad (IPhO) problems. In this paper, we introduce a simple and unified recipe for converting a post-trained reasoning backbone into a rigorous olympiad-level solver. The recipe first uses a reverse-perplexity curriculum for SFT to instill rigorous proof-search and self-checking behaviors, then scales these behaviors through a two-stage RL pipeline that progresses from RL with verifiable rewards to more delicate proof-level RL, and finally boosts solving performance with test-time scaling. Applying this recipe, we train a 30B-A3B backbone with SFT on around 340K sub-8K-token trajectories followed by 200 RL steps. The resulting model, SU-01, supports stable reasoning on difficult problems with trajectories exceeding 100K tokens, while achieving gold-medal-level performance on mathematical and physical olympiad competitions, including IMO 2025/USAMO 2026 and IPhO 2024/2025. It also demonstrates strong generalization of scientific reasoning to domains beyond mathematics and physics.

Many challenging reasoning tasks require not just rapid, intuitive responses, but a more deliberate, multi-step approach. Recent progress in large language models (LLMs) highlights an important shift from the "System 1" way of quick reactions to the "System 2" style of reflection-and-correction problem solving. However, current benchmarks heavily rely on the final-answer accuracy, leaving much of a model's intermediate reasoning steps unexamined. This fails to assess the model's ability to reflect and rectify mistakes within the reasoning process. To bridge this gap, we introduce FINEREASON, a logic-puzzle benchmark for fine-grained evaluation of LLMs' reasoning capabilities. Each puzzle can be decomposed into atomic steps, making it ideal for rigorous validation of intermediate correctness. Building on this, we introduce two tasks: state checking, and state transition, for a comprehensive evaluation of how models assess the current situation and plan the next move. To support broader research, we also provide a puzzle training set aimed at enhancing performance on general mathematical tasks. We show that models trained on our state checking and transition data demonstrate gains in math reasoning by up to 5.1% on GSM8K.

A prevailing view in Reinforcement Learning for Verifiable Rewards (RLVR) interprets recent progress through the lens of an exploration-exploitation trade-off, a perspective largely shaped by token-level metrics. We re-examine this perspective, proposing that this perceived trade-off may not be a fundamental constraint but rather an artifact of the measurement level. To investigate this, we shift the analysis to the semantically rich hidden-state space, adopting Effective Rank (ER) to quantify exploration and proposing its novel first- and second-order derivatives, named Effective Rank Velocity (ERV) and Effective Rank Acceleration (ERA), to capture exploitation dynamics. Our analysis reveals that at the hidden-state level, exploration and exploitation could be decoupled (Sec. 4). This finding reveals an opportunity to enhance both capacities simultaneously. This insight motivates our method, Velocity-Exploiting Rank-Learning (VERL), the first to operationalize the principle of synergistic exploration-exploitation enhancement by directly shaping the RL advantage function. The key innovation is leveraging the theoretically stable ERA as a predictive meta-controller to create a synergistic, dual-channel incentive structure. Instead of forcing a trade-off, VERL prospectively amplifies rewards for exploration to preempt overconfidence and reinforces exploitative gains to consolidate reasoning. Experiments across diverse LLMs and reasoning benchmarks show consistent gains, including up to 21.4% absolute accuracy improvement on the challenging Gaokao 2024 dataset.

Recent advances in formal theorem proving have focused on Olympiad-level mathematics, leaving undergraduate domains largely unexplored. Optimization, fundamental to machine learning, operations research, and scientific computing, remains underserved by existing provers. Its reliance on domain-specific formalisms (convexity, optimality conditions, and algorithmic analysis) creates significant distribution shift, making naive domain transfer ineffective. We present OptProver, a trained model that achieves robust transfer from Olympiad to undergraduate optimization. Starting from a strong Olympiad-level prover, our pipeline mitigates distribution shift through two key innovations. First, we employ large-scale optimization-focused data curation via expert iteration. Second, we introduce a specialized preference learning objective that integrates perplexity-weighted optimization with a mechanism to penalize valid but non-progressing proof steps. This not only addresses distribution shifts but also guides the search toward efficient trajectories. To enable rigorous evaluation, we construct a novel benchmark in Lean 4 focused on optimization. On this benchmark, OptProver achieves state-of-the-art Pass@1 and Pass@32 among comparably sized models while maintaining competitive performance on general theorem-proving tasks, demonstrating effective domain transfer without catastrophic forgetting.

Large Language Model (LLM) agents are deployed in complex environments -- such as massive codebases, enterprise databases, and conversational histories -- where the relevant state far exceeds their context windows. To navigate these spaces, an agent must iteratively explore the environment to find relevant information. However, without explicit infrastructure, an agent's working memory can degrade into lossy representations of the search state, resulting in redundant work (e.g. repetitive looping) and premature stopping. In this work, we formalize this challenge as the Context Gathering Decision Process (CGDP), a specialized Partially Observable Markov Decision Process, where an agent's objective is to adaptively refine its belief state to isolate the necessary information for a task. We model an LLM's behavior as approximate Thompson Sampling within this CGDP, and introduce a predicate-based method that decomposes an LLM's implicit search into explicit and modular operations. We then derive two plug-and-play interventions for iterative LLM agents: a persistent, predicate-based belief state that bounds context while preserving multi-hop reasoning, and a programmatic exhaustion gate that halts unproductive search without premature stopping. Across four methods and three question-answering domains, we empirically validate that replacing an LLM's implicit state with our CGDP-motivated belief state improves multi-hop reasoning by up to 11.4%; while the modular programmatic exhaustion detection saves up to 39% of tokens without any degradation in agent performance. Ultimately, we argue that framing the LLM agent loop as a CGDP can guide the design of modular, non-interfering improvements to agentic search harnesses.

Large Language Models (LLMs) augmented with external tools have demonstrated remarkable capabilities in complex reasoning tasks. However, existing frameworks rely heavily on natural language reasoning to determine when tools can be invoked and whether their results should be committed, lacking formal guarantees for logical safety and verifiability. We present ToolGate, a forward execution framework that provides logical safety guarantees and verifiable state evolution for LLM tool calling. ToolGate maintains an explicit symbolic state space as a typed key-value mapping representing trusted world information throughout the reasoning process. Each tool is formalized as a Hoare-style contract consisting of a precondition and a postcondition, where the precondition gates tool invocation by checking whether the current state satisfies the required conditions, and the postcondition determines whether the tool's result can be committed to update the state through runtime verification. Our approach guarantees that the symbolic state evolves only through verified tool executions, preventing invalid or hallucinated results from corrupting the world representation. Experimental validation demonstrates that ToolGate significantly improves the reliability and verifiability of tool-augmented LLM systems while maintaining competitive performance on complex multi-step reasoning tasks. This work establishes a foundation for building more trustworthy and debuggable AI systems that integrate language models with external tools.

Proof-oriented programs mix computational content with proofs of program correctness. However, the human effort involved in programming and proving is still substantial, despite the use of Satisfiability Modulo Theories (SMT) solvers to automate proofs in languages such as F*. Seeking to spur research on using AI to automate the construction of proof-oriented programs, we curate a dataset of 600K lines of open-source F* programs and proofs, including software used in production systems ranging from Windows and Linux, to Python and Firefox. Our dataset includes around 32K top-level F* definitions, each representing a type-directed program and proof synthesis problem -- producing a definition given a formal specification expressed as an F* type. We provide a program-fragment checker that queries F* to check the correctness of candidate solutions. We believe this is the largest corpus of SMT-assisted program proofs coupled with a reproducible program-fragment checker. Grounded in this dataset, we investigate the use of AI to synthesize programs and their proofs in F*, with promising results. Our main finding in that the performance of fine-tuned smaller language models (such as Phi-2 or StarCoder) compare favorably with large language models (such as GPT-4), at a much lower computational cost. We also identify various type-based retrieval augmentation techniques and find that they boost performance significantly. With detailed error analysis and case studies, we identify potential strengths and weaknesses of models and techniques and suggest directions for future improvements.

We introduce LongCat-Flash-Prover, a flagship 560-billion-parameter open-source Mixture-of- Experts (MoE) model that advances Native Formal Reasoning in Lean4 through agentic tool-integrated reasoning (TIR). We decompose the native formal reasoning task into three independent formal capabilities, i.e., auto-formalization, sketching, and proving. To facilitate these capabilities, we propose a Hybrid-Experts Iteration Framework to expand high-quality task trajectories, including generating a formal statement based on a given informal problem, producing a whole-proof directly from the statement, or a lemma-style sketch. During agentic RL, we present a Hierarchical Importance Sampling Policy Optimization (HisPO) algorithm, which aims to stabilize the MoE model training on such long-horizon tasks. It employs a gradient masking strategy that accounts for the policy staleness and the inherent train-inference engine discrepancies at both sequence and token levels. Additionally, we also incorporate theorem consistency and legality detection mechanisms to eliminate reward hacking issues. Extensive evaluations show that our LongCat-Flash-Prover sets a new state-of-the-art for open-weights models in both auto-formalization and theorem proving. Demonstrating remarkable sample efficiency, it achieves a 97.1% pass rate on MiniF2F-Test using only 72 inference budget per problem. On more challenging benchmarks, it solves 70.8% of ProverBench and 41.5% of PutnamBench with no more than 220 attempts per problem, significantly outperforming existing open-weights baselines.

LLM-based agents have demonstrated strong capabilities in data-intensive analytical tasks, yet their outputs are rarely verifiable: a reliance on linear text trajectories makes their reasoning difficult to audit. In particular, deterministic computations over raw data and semantic deductions over natural-language claims are often entangled in an unstructured stream, leaving numerical conclusions hard to reproduce and qualitative judgments hard to inspect. To address this, we propose VeriGraph, a traceable neuro-symbolic reasoning framework that enables agents to construct an explicit heterogeneous evidence directed acyclic graph (DAG) during execution. VeriGraph introduces three evidence-expansion primitives, namely computational, grounding, and derivational expansion, to connect raw data, interpreter variables, computed results, and natural-language claims in a unified graph. Under this formulation, structural traceability is reduced to graph reachability from raw data sources to terminal claims, while semantic support is measured by claim-level evidence evaluation. To improve graph construction, we further design a graph-based policy optimization strategy with a composite reward that jointly supervises answer correctness, computational integrity, and derivational coherence. Experiments on four benchmarks show that VeriGraph-8B achieves the highest overall score among all baselines. More importantly, VeriGraph produces auditable evidence graphs with substantially stronger claim grounding, achieving a 87.61\% Grounding Rate under our claim-level evidence support evaluation. These results suggest that explicit evidence-graph construction is a promising path toward verifiable data-analytic agents. Our code is available at https://github.com/ignorejjj/VeriGraph.

Reasoning methods such as chain-of-thought prompting and self-consistency have shown immense potential to improve the accuracy of large language models across various reasoning tasks. However such methods involve generation of lengthy reasoning chains, which substantially increases token consumption, computational cost, and latency. To address this inefficiency, we propose ProofSketch, a verification-guided reasoning framework that integrates symbolic closure computation, lexicographic verification and adaptive sketch generation. Our experiments show that ProofSketch consistently reduces token usage while improving accuracy, demonstrating that this approach offers a promising path for efficient and trustworthy reasoning.