Daily Papers

One way to increase confidence in the outputs of Large Language Models (LLMs) is to support them with reasoning that is clear and easy to check -- a property we call legibility. We study legibility in the context of solving grade-school math problems and show that optimizing chain-of-thought solutions only for answer correctness can make them less legible. To mitigate the loss in legibility, we propose a training algorithm inspired by Prover-Verifier Game from Anil et al. (2021). Our algorithm iteratively trains small verifiers to predict solution correctness, "helpful" provers to produce correct solutions that the verifier accepts, and "sneaky" provers to produce incorrect solutions that fool the verifier. We find that the helpful prover's accuracy and the verifier's robustness to adversarial attacks increase over the course of training. Furthermore, we show that legibility training transfers to time-constrained humans tasked with verifying solution correctness. Over course of LLM training human accuracy increases when checking the helpful prover's solutions, and decreases when checking the sneaky prover's solutions. Hence, training for checkability by small verifiers is a plausible technique for increasing output legibility. Our results suggest legibility training against small verifiers as a practical avenue for increasing legibility of large LLMs to humans, and thus could help with alignment of superhuman models.

We consider the problem of how a trusted, but computationally bounded agent (a 'verifier') can learn to interact with one or more powerful but untrusted agents ('provers') in order to solve a given task. More specifically, we study the case in which agents are represented using neural networks and refer to solutions of this problem as neural interactive proofs. First we introduce a unifying framework based on prover-verifier games, which generalises previously proposed interaction protocols. We then describe several new protocols for generating neural interactive proofs, and provide a theoretical comparison of both new and existing approaches. Finally, we support this theory with experiments in two domains: a toy graph isomorphism problem that illustrates the key ideas, and a code validation task using large language models. In so doing, we aim to create a foundation for future work on neural interactive proofs and their application in building safer AI systems.

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).

Solving math problems through verifiable languages such as Lean has significantly impacted both the mathematics and computer science communities. Current state-of-the-art models are often trained with expensive online Reinforcement Learning (RL) or expert iteration. However, these approaches rely on fixed problem sets, which causes inefficient training and limits the model to tackle complex problems. To overcome these limitations, we propose GAR: Generative Adversarial Reinforcement learning, a comprehensive RL training framework that jointly trains the problem composer and solver in an adversarial loop. GAR introduces an implicit curriculum learning mechanism, which aligns task difficulty with the prover's evolving capability. It thereby improves the training efficiency and enables stronger performance of proving advanced theorems. Experiments show that with GAR training, Goedel-Prover-V2-8B and DeepSeek-Prover-V2-7B achieve an average relative improvement in pass@32 of 4.20% on MiniF2F-Test benchmark, while DeepSeek-Prover-V2's pass@32 on ProofNet-Test increases from 22.58% to 25.81%. Beyond formal proving, GAR establishes a general RL paradigm for co-evolution of problem generation and solving under verifiable environments.

Deploying machine learning models in production may allow adversaries to infer sensitive information about training data. There is a vast literature analyzing different types of inference risks, ranging from membership inference to reconstruction attacks. Inspired by the success of games (i.e., probabilistic experiments) to study security properties in cryptography, some authors describe privacy inference risks in machine learning using a similar game-based style. However, adversary capabilities and goals are often stated in subtly different ways from one presentation to the other, which makes it hard to relate and compose results. In this paper, we present a game-based framework to systematize the body of knowledge on privacy inference risks in machine learning. We use this framework to (1) provide a unifying structure for definitions of inference risks, (2) formally establish known relations among definitions, and (3) to uncover hitherto unknown relations that would have been difficult to spot otherwise.

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.

Large language models have made significant progress in mathematical reasoning, which serves as an important testbed for AI and could impact scientific research if further advanced. By scaling reasoning with reinforcement learning that rewards correct final answers, LLMs have improved from poor performance to saturating quantitative reasoning competitions like AIME and HMMT in one year. However, this approach faces fundamental limitations. Pursuing higher final answer accuracy doesn't address a key issue: correct answers don't guarantee correct reasoning. Moreover, many mathematical tasks like theorem proving require rigorous step-by-step derivation rather than numerical answers, making final answer rewards inapplicable. To push the limits of deep reasoning, we believe it is necessary to verify the comprehensiveness and rigor of mathematical reasoning. Self-verification is particularly important for scaling test-time compute, especially for open problems without known solutions. Towards self-verifiable mathematical reasoning, we investigate how to train an accurate and faithful LLM-based verifier for theorem proving. We then train a proof generator using the verifier as the reward model, and incentivize the generator to identify and resolve as many issues as possible in their own proofs before finalizing them. To maintain the generation-verification gap as the generator becomes stronger, we propose to scale verification compute to automatically label new hard-to-verify proofs, creating training data to further improve the verifier. Our resulting model, DeepSeekMath-V2, demonstrates strong theorem-proving capabilities, achieving gold-level scores on IMO 2025 and CMO 2024 and a near-perfect 118/120 on Putnam 2024 with scaled test-time compute.

Training models through self-play alone (without any human data) has been a longstanding goal in AI, but its effectiveness for training large language models remains unclear, particularly in code generation where rewards based on unit tests are brittle and prone to error propagation. We study self-play in the verified code generation setting, where formal verification provides reliable correctness signals. We introduce Propose, Solve, Verify (PSV) a simple self-play framework where formal verification signals are used to create a proposer capable of generating challenging synthetic problems and a solver trained via expert iteration. We use PSV to train PSV-Verus, which across three benchmarks improves pass@1 by up to 9.6x over inference-only and expert-iteration baselines. We show that performance scales with the number of generated questions and training iterations, and through ablations identify formal verification and difficulty-aware proposal as essential ingredients for successful self-play.

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.

Nowadays, formal theorem provers have made monumental progress on high-school and competition-level mathematics, but few of them generalize to more advanced mathematics. In this paper, we present REAL-Prover, a new open-source stepwise theorem prover for Lean 4 to push this boundary. This prover, based on our fine-tuned large language model (REAL-Prover-v1) and integrated with a retrieval system (Leansearch-PS), notably boosts performance on solving college-level mathematics problems. To train REAL-Prover-v1, we developed HERALD-AF, a data extraction pipeline that converts natural language math problems into formal statements, and a new open-source Lean 4 interactive environment (Jixia-interactive) to facilitate synthesis data collection. In our experiments, our prover using only supervised fine-tune achieves competitive results with a 23.7% success rate (Pass@64) on the ProofNet dataset-comparable to state-of-the-art (SOTA) models. To further evaluate our approach, we introduce FATE-M, a new benchmark focused on algebraic problems, where our prover achieves a SOTA success rate of 56.7% (Pass@64).

We introduce our Leanabell-Prover-V2, a 7B large language models (LLMs) that can produce formal theorem proofs in Lean 4, with verifier-integrated Long Chain-of-Thoughts (CoT). Following our previous work Leanabell-Prover-V1, we continual to choose to posttrain existing strong prover models for further performance improvement. In our V2 version, we mainly upgrade the Reinforcement Learning (RL) with feedback provided by the Lean 4 verifier. Crucially, verifier feedback, such as indicating success or detailing specific errors, allows the LLM to become ``self-aware'' of the correctness of its own reasoning process and learn to reflexively correct errors. Leanabell-Prover-V2 directly optimizes LLM reasoning trajectories with multi-turn verifier interactions, together with feedback token masking for stable RL training and a simple reward strategy. Experiments show that Leanabell-Prover-V2 improves performance by 3.2% (pass@128) with Kimina-Prover-Preview-Distill-7B and 2.0% (pass@128) with DeepSeek-Prover-V2-7B on the MiniF2F test set. The source codes, curated data and models are available at: https://github.com/Leanabell-LM/Leanabell-Prover-V2.

Trustworthy verifiers are essential for the success of reinforcement learning with verifiable reward (RLVR), which is the core methodology behind various large reasoning models such as DeepSeek-R1. In complex domains like mathematical reasoning, rule-based verifiers have been widely adopted in previous works to train strong reasoning models. However, the reliability of these verifiers and their impact on the RL training process remain poorly understood. In this work, we take mathematical reasoning as a case study and conduct a comprehensive analysis of various verifiers in both static evaluation and RL training scenarios. First, we find that current open-source rule-based verifiers often fail to recognize equivalent answers presented in different formats across multiple commonly used mathematical datasets, resulting in non-negligible false negative rates. This limitation adversely affects RL training performance and becomes more pronounced as the policy model gets stronger. Subsequently, we investigate model-based verifiers as a potential solution to address these limitations. While the static evaluation shows that model-based verifiers achieve significantly higher verification accuracy, further analysis and RL training results imply that they are highly susceptible to hacking, where they misclassify certain patterns in responses as correct (i.e., false positives). This vulnerability is exploited during policy model optimization, leading to artificially inflated rewards. Our findings underscore the unique risks inherent to both rule-based and model-based verifiers, aiming to offer valuable insights to develop more robust reward systems in reinforcement learning.

The combination of verifiable languages and LLMs has significantly influenced both the mathematical and computer science communities because it provides a rigorous foundation for theorem proving. Recent advancements in the field provide foundation models and sophisticated agentic systems pushing the boundaries of formal mathematical reasoning to approach the natural language capability of LLMs. However, little attention has been given to the formal physics reasoning, which also heavily relies on similar problem-solving and theorem-proving frameworks. To solve this problem, this paper presents, to the best of our knowledge, the first approach to enhance formal theorem proving in the physics domain. We compose a dedicated dataset PhysLeanData for the task. It is composed of theorems sampled from PhysLean and data generated by a conjecture-based formal data generation pipeline. In the training pipeline, we leverage DeepSeek-Prover-V2-7B, a strong open-source mathematical theorem prover, and apply Reinforcement Learning with Verifiable Rewards (RLVR) to train our model PhysProver. Comprehensive experiments demonstrate that, using only sim5K training samples, PhysProver achieves an overall 2.4\% improvement in multiple sub-domains. Furthermore, after formal physics training, we observe 1.3\% gains on the MiniF2F-Test benchmark, which indicates non-trivial generalization beyond physics domains and enhancement for formal math capability as well. The results highlight the effectiveness and efficiency of our approach, which provides a paradigm for extending formal provers outside mathematical domains. To foster further research, we will release both our dataset and model to the community.

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.

As Large Language Models (LLMs) are increasingly applied in high-stakes domains, their ability to reason strategically under uncertainty becomes critical. Poker provides a rigorous testbed, requiring not only strong actions but also principled, game-theoretic reasoning. In this paper, we conduct a systematic study of LLMs in multiple realistic poker tasks, evaluating both gameplay outcomes and reasoning traces. Our analysis reveals LLMs fail to compete against traditional algorithms and identifies three recurring flaws: reliance on heuristics, factual misunderstandings, and a "knowing-doing" gap where actions diverge from reasoning. An initial attempt with behavior cloning and step-level reinforcement learning improves reasoning style but remains insufficient for accurate game-theoretic play. Motivated by these limitations, we propose ToolPoker, a tool-integrated reasoning framework that combines external solvers for GTO-consistent actions with more precise professional-style explanations. Experiments demonstrate that ToolPoker achieves state-of-the-art gameplay while producing reasoning traces that closely reflect game-theoretic principles.

Answer verification methods are widely employed in language model training pipelines spanning data curation, evaluation, and reinforcement learning with verifiable rewards (RLVR). While prior work focus on developing unified verifiers applicable across multiple reasoning scenarios, significant challenges remain in computation-oriented scientific domains, such as algebraic equivalence checking and physical constant substitution. In this paper, we introduce \model, a tool-augmented verifier that leverages external executors to perform precise computations and symbolic simplifications. \model enables robust verification that goes beyond simple semantic matching. We propose a novel two-stage pipeline, which begin with cold-start fine-tuning and followed by multi-turn reinforcement learning with tool integration. Extensive experiments conducted on STEM subjects, general QA, and long-form reasoning tasks demonstrates strong generalization of \model. The results shows that the \model achieves state-of-the-art performance on VerifyBench-Hard and SCI-Bench. And we also employ our \model in RLVR as a reward model, the results show that it consistently outperforms both rubric-based and model-based verifiers on AIME'24 and AIME'25, demonstrating strong potential to enhance reasoning capabilities of LLM. Our model is released at https://huggingface.co/Nanbeige/CoSineVerifier-Tool-4B{https://huggingface.co/Nanbeige/CoSineVerifier-Tool-4B}.

We introduce Goedel-Prover, an open-source large language model (LLM) that achieves the state-of-the-art (SOTA) performance in automated formal proof generation for mathematical problems. The key challenge in this field is the scarcity of formalized math statements and proofs, which we tackle in the following ways. We train statement formalizers to translate the natural language math problems from Numina into formal language (Lean 4), creating a dataset of 1.64 million formal statements. LLMs are used to check that the formal statements accurately preserve the content of the original natural language problems. We then iteratively build a large dataset of formal proofs by training a series of provers. Each prover succeeds in proving many statements that the previous ones could not, and these new proofs are added to the training set for the next prover. The final prover outperforms all existing open-source models in whole-proof generation. On the miniF2F benchmark, it achieves a 57.6% success rate (Pass@32), exceeding the previous best open-source model by 7.6%. On PutnamBench, Goedel-Prover successfully solves 7 problems (Pass@512), ranking first on the leaderboard. Furthermore, it generates 29.7K formal proofs for Lean Workbook problems, nearly doubling the 15.7K produced by earlier works.

Reinforcement Learning with Verifiable Rewards (RLVR) trains policies against automated verifiers to avoid costly human labeling. To reduce vulnerability to verifier hacking, many RLVR systems collapse rewards to binary {0,1} during training. This choice carries a cost: it introduces false negatives (rejecting correct answers, FNs) and false positives (accepting incorrect ones, FPs). For instance, a rule-based checker may mark the correct fraction 12{36} as wrong when compared against the canonical 1{3} due to brittle parsing/equivalence rules (FN), while a large language model (LLM) judges can be gamed by superficial cues or even a single adversarial token, yielding inflated correctness for wrong solutions (FP). We formalize verifier unreliability by modeling the verifier as a stochastic reward channel with asymmetric noise rates. From this abstraction, we derive two correction algorithms for verifier errors. The first is a backward correction that de-biases the observed binary reward to recover an unbiased estimator of the clean policy gradient. The second is a forward correction that reweights score-function terms so that the expected update direction aligns with the clean gradient; notably, it requires only the FN rate. We implement both as lightweight hooks in a group relative policy optimization (GRPO)-based RLVR pipeline and evaluate them on math-reasoning models and benchmarks. Across models and datasets, both corrections improve over uncorrected training; the forward variant converges faster and remains stable under heavier noise. Finally, we show a practical appeal mechanism in which a lightweight LLM verifier estimates the FN rate online by rechecking rule-based negatives, obtaining outperformance compared with other state-of-the-art contenders.

The recent paradigm shift towards training large language models (LLMs) using DeepSeek-R1-Zero-style reinforcement learning (RL) on verifiable rewards has led to impressive advancements in code and mathematical reasoning. However, this methodology is limited to tasks where rule-based answer verification is possible and does not naturally extend to real-world domains such as chemistry, healthcare, engineering, law, biology, business, and economics. Current practical workarounds use an additional LLM as a model-based verifier; however, this introduces issues such as reliance on a strong verifier LLM, susceptibility to reward hacking, and the practical burden of maintaining the verifier model in memory during training. To address this and extend DeepSeek-R1-Zero-style training to general reasoning domains, we propose a verifier-free method (VeriFree) that bypasses answer verification and instead uses RL to directly maximize the probability of generating the reference answer. We compare VeriFree with verifier-based methods and demonstrate that, in addition to its significant practical benefits and reduced compute requirements, VeriFree matches and even surpasses verifier-based methods on extensive evaluations across MMLU-Pro, GPQA, SuperGPQA, and math-related benchmarks. Moreover, we provide insights into this method from multiple perspectives: as an elegant integration of training both the policy and implicit verifier in a unified model, and as a variational optimization approach. Code is available at https://github.com/sail-sg/VeriFree.

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.

Post-training for reasoning of large language models (LLMs) increasingly relies on verifiable rewards: deterministic checkers that provide 0-1 correctness signals. While reliable, such binary feedback is brittle--many tasks admit partially correct or alternative answers that verifiers under-credit, and the resulting all-or-nothing supervision limits learning. Reward models offer richer, continuous feedback, which can serve as a complementary supervisory signal to verifiers. We introduce HERO (Hybrid Ensemble Reward Optimization), a reinforcement learning framework that integrates verifier signals with reward-model scores in a structured way. HERO employs stratified normalization to bound reward-model scores within verifier-defined groups, preserving correctness while refining quality distinctions, and variance-aware weighting to emphasize challenging prompts where dense signals matter most. Across diverse mathematical reasoning benchmarks, HERO consistently outperforms RM-only and verifier-only baselines, with strong gains on both verifiable and hard-to-verify tasks. Our results show that hybrid reward design retains the stability of verifiers while leveraging the nuance of reward models to advance reasoning.

Reinforcement Learning with Verifiable Rewards (RLVR) demonstrates promising potential in advancing the reasoning capabilities of LLMs. However, its success remains largely confined to mathematical and code domains. This primary limitation stems from the heavy reliance on domain-specific verifiers, which results in prohibitive complexity and limited scalability. To address the challenge, our key observation is that LLM's intrinsic probability of generating a correct free-form answer directly indicates its own evaluation of the reasoning reward (i.e., how well the reasoning process leads to the correct answer). Building on this insight, we propose RLPR, a simple verifier-free framework that extrapolates RLVR to broader general domains. RLPR uses the LLM's own token probability scores for reference answers as the reward signal and maximizes the expected reward during training. We find that addressing the high variance of this noisy probability reward is crucial to make it work, and propose prob-to-reward and stabilizing methods to ensure a precise and stable reward from LLM intrinsic probabilities. Comprehensive experiments in four general-domain benchmarks and three mathematical benchmarks show that RLPR consistently improves reasoning capabilities in both areas for Gemma, Llama, and Qwen based models. Notably, RLPR outperforms concurrent VeriFree by 7.6 points on TheoremQA and 7.5 points on Minerva, and even surpasses strong verifier-model-dependent approaches General-Reasoner by 1.6 average points across seven benchmarks.

We present Prover Agent, a novel AI agent for automated theorem proving that integrates large language models (LLMs) with a formal proof assistant, Lean. Prover Agent coordinates an informal reasoning LLM, a formal prover model, and feedback from Lean while also generating auxiliary lemmas. These auxiliary lemmas are not limited to subgoals in the formal proof but can also include special cases or potentially useful facts derived from the assumptions, which help in discovering a viable proof strategy. It achieves an 88.1% success rate on the MiniF2F benchmark, establishing a new state-of-the-art among methods using small language models (SLMs) with a much lower sample budget than previous approaches. We also present theoretical analyses and case studies that illustrate how these generated lemmas contribute to solving challenging problems. Our code is publicly available at: https://github.com/kAIto47802/Prover-Agent.

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.

Mathematical reasoning is a central challenge for large language models (LLMs), requiring not only correct answers but also faithful reasoning processes. Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a promising approach for enhancing such capabilities; however, its ability to foster genuine reasoning remains unclear. We investigate RLVR on two combinatorial problems with fully verifiable solutions: Activity Scheduling and the Longest Increasing Subsequence, using carefully curated datasets with unique optima. Across multiple reward designs, we find that RLVR improves evaluation metrics but often by reinforcing superficial heuristics rather than acquiring new reasoning strategies. These findings highlight the limits of RLVR generalization, emphasizing the importance of benchmarks that disentangle genuine mathematical reasoning from shortcut exploitation and provide faithful measures of progress. Code available at https://github.com/xashru/rlvr-seq-generalization.

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.

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.

The breakthrough of OpenAI o1 highlights the potential of enhancing reasoning to improve LLM. Yet, most research in reasoning has focused on mathematical tasks, leaving domains like medicine underexplored. The medical domain, though distinct from mathematics, also demands robust reasoning to provide reliable answers, given the high standards of healthcare. However, verifying medical reasoning is challenging, unlike those in mathematics. To address this, we propose verifiable medical problems with a medical verifier to check the correctness of model outputs. This verifiable nature enables advancements in medical reasoning through a two-stage approach: (1) using the verifier to guide the search for a complex reasoning trajectory for fine-tuning LLMs, (2) applying reinforcement learning (RL) with verifier-based rewards to enhance complex reasoning further. Finally, we introduce HuatuoGPT-o1, a medical LLM capable of complex reasoning, which outperforms general and medical-specific baselines using only 40K verifiable problems. Experiments show complex reasoning improves medical problem-solving and benefits more from RL. We hope our approach inspires advancements in reasoning across medical and other specialized domains.

The rapid advancement of large language models (LLMs) has accelerated their application in reasoning, with strategic reasoning drawing increasing attention. To evaluate LLMs' strategic reasoning capabilities, game theory, with its concise structure, has become a preferred approach. However, current research focuses on a limited selection of games, resulting in low coverage. Classic game scenarios risk data leakage, and existing benchmarks often lack extensibility, making them inadequate for evaluating state-of-the-art models. To address these challenges, we propose TMGBench, a benchmark with comprehensive game type coverage, novel scenarios, and flexible organization. Specifically, we incorporate all 144 game types summarized by the Robinson-Goforth topology of 2x2 games, constructed as classic games. We also employ synthetic data generation to create diverse, higher-quality scenarios through topic guidance and human inspection, referred to as story-based games. Lastly, we provide a sustainable framework for increasingly powerful LLMs by treating these games as atomic units and organizing them into more complex forms via sequential, parallel, and nested structures. Our comprehensive evaluation of mainstream LLMs covers tests on rational reasoning, robustness, Theory-of-Mind (ToM), and reasoning in complex forms. Results reveal flaws in accuracy, consistency, and varying mastery of ToM. Additionally, o1-mini, OpenAI's latest reasoning model, achieved accuracy rates of 66.6%, 60.0%, and 70.0% on sequential, parallel, and nested games, highlighting TMGBench's challenges.

Large reasoning models (LRMs) have demonstrated impressive reasoning capabilities across a broad range of tasks including Olympiad-level mathematical problems, indicating evidence of their complex reasoning abilities. While many reasoning benchmarks focus on the STEM domain, the ability of LRMs to reason correctly in broader task domains remains underexplored. In this work, we introduce TTT-Bench, a new benchmark that is designed to evaluate basic strategic, spatial, and logical reasoning abilities in LRMs through a suite of four two-player Tic-Tac-Toe-style games that humans can effortlessly solve from a young age. We propose a simple yet scalable programmatic approach for generating verifiable two-player game problems for TTT-Bench. Although these games are trivial for humans, they require reasoning about the intentions of the opponent, as well as the game board's spatial configurations, to ensure a win. We evaluate a diverse set of state-of-the-art LRMs, and discover that the models that excel at hard math problems frequently fail at these simple reasoning games. Further testing reveals that our evaluated reasoning models score on average downarrow 41\% \& downarrow 5\% lower on TTT-Bench compared to MATH 500 \& AIME 2024 respectively, with larger models achieving higher performance using shorter reasoning traces, where most of the models struggle on long-term strategic reasoning situations on simple and new TTT-Bench tasks.

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.

We introduce Goedel-Prover-V2, a series of open-source language models that set a new state-of-the-art in automated theorem proving. Built on the standard expert iteration and reinforcement learning pipeline, our approach incorporates three key innovations: (1) Scaffolded data synthesis: We generate synthetic tasks of increasing difficulty to train the model to master increasingly complex theorems; (2) Verifier-guided self-correction: We enable the model to iteratively revise its proofs by leveraging feedback from the Lean compiler; (3) Model averaging: We merge model checkpoints to mitigate the decrease in model output diversity in later stages of training. Our small model, Goedel-Prover-V2-8B, reaches 84.6% pass@32 on MiniF2F and outperforms DeepSeek-Prover-V2-671B under the same metric, despite being 80X smaller. Our flagship model, Goedel-Prover-V2-32B, achieves 88.1% on MiniF2F at pass@32 in standard mode and 90.4% in self-correction mode, outperforming prior SOTA by a large margin. Additionally, our flagship model solves 86 problems on PutnamBench at pass@184, securing the first place among open-source models on the leaderboard, surpassing DeepSeek-Prover-V2-671B's record of solving 47 problems by pass@1024 with a significantly smaller model size and compute budget. At the time of its release (July-August 2025), Goedel-Prover-V2 achieves the strongest overall performance among all open-source theorem provers. It also ranks among the top-performing models--including closed-source systems with publicly reported performance--under a constrained test-time compute budget. Our models, code, and data are released at https://github.com/Goedel-LM/Goedel-Prover-V2.

This paper considers the development of an AI-based provably-correct mathematical proof tutor. While Large Language Models (LLMs) allow seamless communication in natural language, they are error prone. Theorem provers such as Lean allow for provable-correctness, but these are hard for students to learn. We present a proof-of-concept system (LeanTutor) by combining the complementary strengths of LLMs and theorem provers. LeanTutor is composed of three modules: (i) an autoformalizer/proof-checker, (ii) a next-step generator, and (iii) a natural language feedback generator. To evaluate the system, we introduce PeanoBench, a dataset of 371 Peano Arithmetic proofs in human-written natural language and formal language, derived from the Natural Numbers Game.

Large Language Models (LLMs) demonstrate impressive mathematical reasoning abilities, but their solutions frequently contain errors that cannot be automatically verified. Formal theorem proving systems such as Lean 4 offer automated verification with complete accuracy, motivating recent efforts to build specialized prover LLMs that generate verifiable proofs in formal languages. However, a significant gap remains: current prover LLMs solve substantially fewer problems than general-purpose LLMs operating in natural language. We introduce Hilbert, an agentic framework that bridges this gap by combining the complementary strengths of informal reasoning and formal verification. Our system orchestrates four components: an informal LLM that excels at mathematical reasoning, a specialized prover LLM optimized for Lean 4 tactics, a formal verifier, and a semantic theorem retriever. Given a problem that the prover is unable to solve, Hilbert employs recursive decomposition to split the problem into subgoals that it solves with the prover or reasoner LLM. It leverages verifier feedback to refine incorrect proofs as necessary. Experimental results demonstrate that Hilbert substantially outperforms existing approaches on key benchmarks, achieving 99.2% on miniF2F, 6.6% points above the best publicly available method. Hilbert achieves the best known result on PutnamBench. It solves 462/660 problems (70.0%), outperforming proprietary approaches like SeedProver (50.4%) and achieving a 422% improvement over the best publicly available baseline. Thus, Hilbert effectively narrows the gap between informal reasoning and formal proof generation.

Large language models (LLMs) increasingly rely on reinforcement learning (RL) to enhance their reasoning capabilities through feedback. A critical challenge is verifying the consistency of model-generated responses and reference answers, since these responses are often lengthy, diverse, and nuanced. Rule-based verifiers struggle with complexity, prompting the use of model-based verifiers. However, specialized verifiers lack flexibility, while general LLM judges can be inconsistent. Existing research primarily focuses on building better verifiers, yet a systematic evaluation of different types of verifiers' performance across domains remains lacking, severely constraining the reliable development of Reinforcement Learning with Verifiable Reward (RLVR). To address this, we propose VerifyBench--a cross-domain comprehensive benchmark for systematically evaluating verifiers. We construct 4,000 expert-level questions covering mathematics, physics, chemistry, and biology. Each question is equipped with reference answers and diverse responses. The reliability of the evaluation is ensured through a rigorous annotation process conducted by a multidisciplinary expert team. We design a four-dimensional experimental framework to comprehensively compare the performance boundaries of specialized verifiers and general LLMs under combined conditions of extracted answers vs. complete responses, and short vs. long outputs. Our evaluation uncovers fundamental trade-offs in verifiers: while specialized verifiers achieve leading accuracy, they exhibit deficiencies in recall; general models show stronger inclusivity but unstable precision. More importantly, we discover verifiers' high sensitivity to input structure and inherent limitations in cross-domain generalization, providing critical insights into the bottlenecks of current verifier technology.

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.

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.

State-of-the-art neural theorem provers like DeepSeek-Prover-V1.5 combine large language models with reinforcement learning, achieving impressive results through sophisticated training. We ask: do these highly-trained models still benefit from simple structural guidance at inference time? We evaluate a lightweight intervention -- a fixed prompt schedule over 15 common tactic skeletons -- on the miniF2F benchmark. This simple approach yields 21.7% pass@16 compared to 15.2% for standard sampling from the same model, a 43% relative improvement using the same number of samples (k=16) and same maximum generation length (1024 tokens). Our results suggest that even capable RL-trained provers underutilize structural priors available in the tactic language, and that simple inference-time guidance remains a cheap, complementary boost.

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.

Large language models (LLMs) have shown promise in proving formal theorems using proof assistants such as Lean. However, existing methods are difficult to reproduce or build on, due to private code, data, and large compute requirements. This has created substantial barriers to research on machine learning methods for theorem proving. This paper removes these barriers by introducing LeanDojo: an open-source Lean playground consisting of toolkits, data, models, and benchmarks. LeanDojo extracts data from Lean and enables interaction with the proof environment programmatically. It contains fine-grained annotations of premises in proofs, providing valuable data for premise selection: a key bottleneck in theorem proving. Using this data, we develop ReProver (Retrieval-Augmented Prover): the first LLM-based prover that is augmented with retrieval for selecting premises from a vast math library. It is inexpensive and needs only one GPU week of training. Our retriever leverages LeanDojo's program analysis capability to identify accessible premises and hard negative examples, which makes retrieval much more effective. Furthermore, we construct a new benchmark consisting of 96,962 theorems and proofs extracted from Lean's math library. It features challenging data split requiring the prover to generalize to theorems relying on novel premises that are never used in training. We use this benchmark for training and evaluation, and experimental results demonstrate the effectiveness of ReProver over non-retrieval baselines and GPT-4. We thus provide the first set of open-source LLM-based theorem provers without any proprietary datasets and release it under a permissive MIT license to facilitate further research.

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%).

Unlike perfect information games, where all elements are known to every player, imperfect information games emulate the real-world complexities of decision-making under uncertain or incomplete information. GPT-4, the recent breakthrough in large language models (LLMs) trained on massive passive data, is notable for its knowledge retrieval and reasoning abilities. This paper delves into the applicability of GPT-4's learned knowledge for imperfect information games. To achieve this, we introduce Suspicion-Agent, an innovative agent that leverages GPT-4's capabilities for performing in imperfect information games. With proper prompt engineering to achieve different functions, Suspicion-Agent based on GPT-4 demonstrates remarkable adaptability across a range of imperfect information card games. Importantly, GPT-4 displays a strong high-order theory of mind (ToM) capacity, meaning it can understand others and intentionally impact others' behavior. Leveraging this, we design a planning strategy that enables GPT-4 to competently play against different opponents, adapting its gameplay style as needed, while requiring only the game rules and descriptions of observations as input. In the experiments, we qualitatively showcase the capabilities of Suspicion-Agent across three different imperfect information games and then quantitatively evaluate it in Leduc Hold'em. The results show that Suspicion-Agent can potentially outperform traditional algorithms designed for imperfect information games, without any specialized training or examples. In order to encourage and foster deeper insights within the community, we make our game-related data publicly available.

Explanations for AI models in high-stakes domains like medicine often lack verifiability, which can hinder trust. To address this, we propose an interactive agent that produces explanations through an auditable sequence of actions. The agent learns a policy to strategically seek external visual evidence to support its diagnostic reasoning. This policy is optimized using reinforcement learning, resulting in a model that is both efficient and generalizable. Our experiments show that this action-based reasoning process significantly improves calibrated accuracy, reducing the Brier score by 18\% compared to a non-interactive baseline. To validate the faithfulness of the agent's explanations, we introduce a causal intervention method. By masking the visual evidence the agent chooses to use, we observe a measurable degradation in its performance (DeltaBrier=+0.029), confirming that the evidence is integral to its decision-making process. Our work provides a practical framework for building AI systems with verifiable and faithful reasoning capabilities.

Mathematical reasoning lies at the heart of artificial intelligence, underpinning applications in education, program verification, and research-level mathematical discovery. Mathematical competitions, in particular, present two challenging problem types: theorem proving, which requires rigorous proofs of stated conclusions, and answer construction, which involves hypothesizing and formally verifying mathematical objects. Large Language Models (LLMs) effectively generate creative candidate answers but struggle with formal verification, while symbolic provers ensure rigor but cannot efficiently handle creative conjecture generation. We introduce the Enumerate-Conjecture-Prove (ECP) framework, a modular neuro-symbolic method integrating LLM-based enumeration and pattern-driven conjecturing with formal theorem proving. We present ConstructiveBench, a dataset of 3,431 answer-construction problems in various math competitions with verified Lean formalizations. On the ConstructiveBench dataset, ECP improves the accuracy of answer construction from a Chain-of-Thought (CoT) baseline of 14.54% to 45.06% with the gpt-4.1-mini model. Moreover, combined with ECP's constructed answers, the state-of-the-art DeepSeek-Prover-V2-7B model generates correct proofs for 858 of the 3,431 constructive problems in Lean, achieving 25.01% accuracy compared to 9.86% for symbolic-only baselines. Our code and dataset are publicly available at https://github.com/JackSun200312/ECP.

Reinforcement Learning with Verifiable Rewards (RLVR) offers a robust mechanism for enhancing mathematical reasoning in large models. However, we identify a systematic lack of emphasis on more challenging questions in existing methods from both algorithmic and data perspectives, despite their importance for refining underdeveloped capabilities. Algorithmically, widely used Group Relative Policy Optimization (GRPO) suffers from an implicit imbalance where the magnitude of policy updates is lower for harder questions. Data-wise, augmentation approaches primarily rephrase questions to enhance diversity without systematically increasing intrinsic difficulty. To address these issues, we propose a two-dual MathForge framework to improve mathematical reasoning by targeting harder questions from both perspectives, which comprises a Difficulty-Aware Group Policy Optimization (DGPO) algorithm and a Multi-Aspect Question Reformulation (MQR) strategy. Specifically, DGPO first rectifies the implicit imbalance in GRPO via difficulty-balanced group advantage estimation, and further prioritizes harder questions by difficulty-aware question-level weighting. Meanwhile, MQR reformulates questions across multiple aspects to increase difficulty while maintaining the original gold answer. Overall, MathForge forms a synergistic loop: MQR expands the data frontier, and DGPO effectively learns from the augmented data. Extensive experiments show that MathForge significantly outperforms existing methods on various mathematical reasoning tasks. The code and augmented data are all available at https://github.com/AMAP-ML/MathForge.

We introduce Reasoning Gym (RG), a library of reasoning environments for reinforcement learning with verifiable rewards. It provides over 100 data generators and verifiers spanning multiple domains including algebra, arithmetic, computation, cognition, geometry, graph theory, logic, and various common games. Its key innovation is the ability to generate virtually infinite training data with adjustable complexity, unlike most previous reasoning datasets, which are typically fixed. This procedural generation approach allows for continuous evaluation across varying difficulty levels. Our experimental results demonstrate the efficacy of RG in both evaluating and reinforcement learning of reasoning models.

Language models have become increasingly powerful tools for formal mathematical reasoning. However, most existing approaches rely exclusively on either large general-purpose models or smaller specialized models, each with distinct limitations, while training specialized large models still requires significant computational resources. This paper introduces ProofCompass, a novel hybrid methodology that achieves remarkable computational efficiency by strategically guiding existing specialized prover methods, such as DeepSeek-Prover-v1.5-RL (DSP-v1.5) with a Large Language Model (LLM) without requiring additional model training. The LLM provides natural language proof strategies and analyzes failed attempts to select intermediate lemmas, enabling effective problem decomposition. On the miniF2F benchmark, ProofCompass demonstrates substantial resource efficiency: it outperforms DSP-v1.5 (54.9% rightarrow 55.3%) while using 25x fewer attempts (3200 rightarrow 128). Our synergistic approach paves the way for simultaneously improving computational efficiency and accuracy in formal theorem proving.

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.

Large Language Models (LLMs) excel in reasoning and generation across domains, but still struggle with identifying and diagnosing complex errors. This stems mainly from training objectives that prioritize correct answers, limiting exposure to and learning from errors. While recent studies have begun to address this by introducing error signals, most rely on shallow, static errors, restricting improvement in deep diagnostic ability. To overcome this, we propose Hide and Seek Game (HSG), a dynamic adversarial framework for error generation and diagnosis, and evaluate it on mathematical problem-solving. HSG involves two adversarial roles: Sneaky, which "hides" by generating subtle, deceptive reasoning errors, and Diagnosis, which "seeks" to accurately detect them. Through adversarial co-evolution, both error stealth and diagnostic precision are enhanced. Experiments on several math reasoning tasks show that HSG significantly boosts error diagnosis, achieving 16.8\%--31.4\% higher accuracy than baselines like GPT-4o. We also release a challenging dataset of deceptive errors and diagnostic annotations as a benchmark for future research.

Game theory is a powerful framework for reasoning about strategic interactions, with applications in domains ranging from day-to-day life to international politics. However, applying formal reasoning tools in such contexts is challenging, as these scenarios are often expressed in natural language. To address this, we introduce a framework for the autoformalization of game-theoretic scenarios, which translates natural language descriptions into formal logic representations suitable for formal solvers. Our approach utilizes one-shot prompting and a solver that provides feedback on syntactic correctness to allow LLMs to refine the code. We evaluate the framework using GPT-4o and a dataset of natural language problem descriptions, achieving 98% syntactic correctness and 88% semantic correctness. These results show the potential of LLMs to bridge the gap between real-life strategic interactions and formal reasoning.

Evaluating the capabilities of Large Language Models (LLMs) has traditionally relied on static benchmark datasets, human assessments, or model-based evaluations - methods that often suffer from overfitting, high costs, and biases. ZeroSumEval is a novel competition-based evaluation protocol that leverages zero-sum games to assess LLMs with dynamic benchmarks that resist saturation. ZeroSumEval encompasses a diverse suite of games, including security challenges (PyJail), classic games (Chess, Liar's Dice, Poker), knowledge tests (MathQuiz), and persuasion challenges (Gandalf, Debate). These games are designed to evaluate a range of AI capabilities such as strategic reasoning, planning, knowledge application, and creativity. Building upon recent studies that highlight the effectiveness of game-based evaluations for LLMs, ZeroSumEval enhances these approaches by providing a standardized and extensible framework. To demonstrate this, we conduct extensive experiments with >7000 simulations across 7 games and 13 models. Our results show that while frontier models from the GPT and Claude families can play common games and answer questions, they struggle to play games that require creating novel and challenging questions. We also observe that models cannot reliably jailbreak each other and fail generally at tasks requiring creativity. We release our code at https://github.com/facebookresearch/ZeroSumEval.

Large language models demonstrate remarkable reasoning capabilities but often produce unreliable or incorrect responses. Existing verification methods are typically model-specific or domain-restricted, requiring significant computational resources and lacking scalability across diverse reasoning tasks. To address these limitations, we propose VerifiAgent, a unified verification agent that integrates two levels of verification: meta-verification, which assesses completeness and consistency in model responses, and tool-based adaptive verification, where VerifiAgent autonomously selects appropriate verification tools based on the reasoning type, including mathematical, logical, or commonsense reasoning. This adaptive approach ensures both efficiency and robustness across different verification scenarios. Experimental results show that VerifiAgent outperforms baseline verification methods (e.g., deductive verifier, backward verifier) among all reasoning tasks. Additionally, it can further enhance reasoning accuracy by leveraging feedback from verification results. VerifiAgent can also be effectively applied to inference scaling, achieving better results with fewer generated samples and costs compared to existing process reward models in the mathematical reasoning domain. Code is available at https://github.com/Jiuzhouh/VerifiAgent

Post-training of reasoning LLMs is a holistic process that typically consists of an offline SFT stage followed by an online reinforcement learning (RL) stage. However, SFT is often optimized in isolation to maximize SFT performance alone. We show that, after identical RL training, models initialized from stronger SFT checkpoints can significantly underperform those initialized from weaker ones. We attribute this to a mismatch typical in current SFT-RL pipelines: the distribution that generates the offline SFT data can differ substantially from the policy optimized during online RL, which learns from its own rollouts. We propose PEAR (Policy Evaluation-inspired Algorithm for Offline Learning Loss Re-weighting), an SFT-stage method that corrects this mismatch and better prepares the model for RL. PEAR uses importance sampling to reweight the SFT loss, with three variants operating at the token, block, and sequence levels. It can be used to augment standard SFT objectives and incurs little additional training overhead once probabilities for the offline data are collected. We conduct controlled experiments on verifiable reasoning games and mathematical reasoning tasks on Qwen 2.5 and 3 and DeepSeek-distilled models. PEAR consistently improves post-RL performance over canonical SFT, with pass at 8 gains up to a 14.6 percent on AIME2025. Our results suggest that PEAR is an effective step toward more holistic LLM post-training by designing and evaluating SFT with downstream RL in mind rather than in isolation.

Large reasoning models such as OpenAI o1 and DeepSeek-R1 have achieved remarkable performance in the domain of reasoning. A key component of their training is the incorporation of verifiable rewards within reinforcement learning (RL). However, existing reward benchmarks do not evaluate reference-based reward systems, leaving researchers with limited understanding of the accuracy of verifiers used in RL. In this paper, we introduce two benchmarks, VerifyBench and VerifyBench-Hard, designed to assess the performance of reference-based reward systems. These benchmarks are constructed through meticulous data collection and curation, followed by careful human annotation to ensure high quality. Current models still show considerable room for improvement on both VerifyBench and VerifyBench-Hard, especially smaller-scale models. Furthermore, we conduct a thorough and comprehensive analysis of evaluation results, offering insights for understanding and developing reference-based reward systems. Our proposed benchmarks serve as effective tools for guiding the development of verifier accuracy and the reasoning capabilities of models trained via RL in reasoning tasks.

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.

As a seemingly self-explanatory task, problem-solving has been a significant component of science and engineering. However, a general yet concrete formulation of problem-solving itself is missing. With the recent development of AI-based problem-solving agents, the demand for process-level verifiability is rapidly increasing yet underexplored. To fill these gaps, we present a principled formulation of problem-solving as a deterministic Markov decision process; a novel framework, FPS (Formal Problem-Solving), which utilizes existing FTP (formal theorem proving) environments to perform process-verified problem-solving; and D-FPS (Deductive FPS), decoupling solving and answer verification for better human-alignment. The expressiveness, soundness and completeness of the frameworks are proven. We construct three benchmarks on problem-solving: FormalMath500, a formalization of a subset of the MATH500 benchmark; MiniF2F-Solving and PutnamBench-Solving, adaptations of FTP benchmarks MiniF2F and PutnamBench. For faithful, interpretable, and human-aligned evaluation, we propose RPE (Restricted Propositional Equivalence), a symbolic approach to determine the correctness of answers by formal verification. We evaluate four prevalent FTP models and two prompting methods as baselines, solving at most 23.77% of FormalMath500, 27.47% of MiniF2F-Solving, and 0.31% of PutnamBench-Solving.

Machine learning models can leak private information about their training data. The standard methods to measure this privacy risk, based on membership inference attacks (MIAs), only check if a given data point exactly matches a training point, neglecting the potential of similar or partially overlapping memorized data revealing the same private information. To address this issue, we introduce the class of range membership inference attacks (RaMIAs), testing if the model was trained on any data in a specified range (defined based on the semantics of privacy). We formulate the RaMIAs game and design a principled statistical test for its composite hypotheses. We show that RaMIAs can capture privacy loss more accurately and comprehensively than MIAs on various types of data, such as tabular, image, and language. RaMIA paves the way for more comprehensive and meaningful privacy auditing of machine learning algorithms.

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.

Reinforcement Learning with Verifiable Rewards (RLVR) has demonstrated great potential in enhancing the reasoning capabilities of large language models (LLMs), achieving remarkable progress in domains such as mathematics and programming where standard answers are available. However, for open-ended tasks lacking ground-truth solutions (e.g., creative writing and instruction following), existing studies typically regard them as non-reasoning scenarios, thereby overlooking the latent value of reasoning capabilities. This raises a key question: Can strengthening reasoning improve performance in open-ended tasks? To address this, we explore the transfer of the RLVR paradigm to the open domain. Yet, since RLVR fundamentally relies on verifiers that presuppose the existence of standard answers, it cannot be directly applied to open-ended tasks. To overcome this challenge, we introduce Verifiable Multiple-Choice Reformulation (VMR), a novel training strategy that restructures open-ended data into verifiable multiple-choice formats, enabling effective training even in the absence of explicit ground truth. Experimental results on multiple benchmarks validate the effectiveness of our method in improving LLM performance on open-ended tasks. Notably, across eight open-ended benchmarks, our VMR-based training delivers an average gain of 5.99 points over the baseline. Code will be released upon acceptance to facilitate reproducibility.

Large Language Models (LLMs) show great promise in complex reasoning, with Reinforcement Learning with Verifiable Rewards (RLVR) being a key enhancement strategy. However, a prevalent issue is ``superficial self-reflection'', where models fail to robustly verify their own outputs. We introduce RISE (Reinforcing Reasoning with Self-Verification), a novel online RL framework designed to tackle this. RISE explicitly and simultaneously trains an LLM to improve both its problem-solving and self-verification abilities within a single, integrated RL process. The core mechanism involves leveraging verifiable rewards from an outcome verifier to provide on-the-fly feedback for both solution generation and self-verification tasks. In each iteration, the model generates solutions, then critiques its own on-policy generated solutions, with both trajectories contributing to the policy update. Extensive experiments on diverse mathematical reasoning benchmarks show that RISE consistently improves model's problem-solving accuracy while concurrently fostering strong self-verification skills. Our analyses highlight the advantages of online verification and the benefits of increased verification compute. Additionally, RISE models exhibit more frequent and accurate self-verification behaviors during reasoning. These advantages reinforce RISE as a flexible and effective path towards developing more robust and self-aware reasoners.

Verifiers or reward models are often used to enhance the reasoning performance of large language models (LLMs). A common approach is the Best-of-N method, where N candidate solutions generated by the LLM are ranked by a verifier, and the best one is selected. While LLM-based verifiers are typically trained as discriminative classifiers to score solutions, they do not utilize the text generation capabilities of pretrained LLMs. To overcome this limitation, we instead propose training verifiers using the ubiquitous next-token prediction objective, jointly on verification and solution generation. Compared to standard verifiers, such generative verifiers (GenRM) can benefit from several advantages of LLMs: they integrate seamlessly with instruction tuning, enable chain-of-thought reasoning, and can utilize additional inference-time compute via majority voting for better verification. We demonstrate that when using Gemma-based verifiers on algorithmic and grade-school math reasoning tasks, GenRM outperforms discriminative verifiers and LLM-as-a-Judge, showing a 16-64% improvement in the percentage of problems solved with Best-of-N. Furthermore, we show that GenRM scales favorably across dataset size, model capacity, and inference-time compute.

Large language models (LLMs) have shown increasing competence in solving mathematical reasoning problems. However, many open-source LLMs still struggle with errors in calculation and semantic understanding during intermediate reasoning steps. In this work, we introduce Prove, a simple yet effective framework that leverages translated programs derived from natural language solutions as a verification mechanism to filter out potentially incorrect reasoning paths before aggregating final answers. Unlike vanilla majority voting, our approach filters out solutions whose corresponding program output is inconsistent with the generated solution, aggregating only those that pass verification. We conducted extensive experiments using 13 open-source LLMs from various model families and sizes, ranging from 0.5B to 13B parameters, across eight mathematical benchmarks. Our results show that Prove consistently outperforms vanilla majority voting as a heuristic for solving mathematical reasoning tasks across all model sizes and datasets, achieving improvements of up to 18% on GSM8K and 8% on MATH-500. Our codes are available at https://github.com/declare-lab/prove.

The integration of machine learning (ML) systems into critical industries such as healthcare, finance, and cybersecurity has transformed decision-making processes, but it also brings new challenges around trust, security, and accountability. As AI systems become more ubiquitous, ensuring the transparency and correctness of AI-driven decisions is crucial, especially when they have direct consequences on privacy, security, or fairness. Verifiable AI, powered by Zero-Knowledge Machine Learning (zkML), offers a robust solution to these challenges. zkML enables the verification of AI model inferences without exposing sensitive data, providing an essential layer of trust and privacy. However, traditional zkML systems typically require deep cryptographic expertise, placing them beyond the reach of most ML engineers. In this paper, we introduce JSTprove, a specialized zkML toolkit, built on Polyhedra Network's Expander backend, to enable AI developers and ML engineers to generate and verify proofs of AI inference. JSTprove provides an end-to-end verifiable AI inference pipeline that hides cryptographic complexity behind a simple command-line interface while exposing auditable artifacts for reproducibility. We present the design, innovations, and real-world use cases of JSTprove as well as our blueprints and tooling to encourage community review and extension. JSTprove therefore serves both as a usable zkML product for current engineering needs and as a reproducible foundation for future research and production deployments of verifiable AI.

Recent advances in large reasoning models highlight Reinforcement Learning with Verifiable Rewards (RLVR) as a promising method for enhancing AI's capabilities, particularly in solving complex logical tasks. However, it remains unclear whether RLVR truly expands a model's reasoning boundary or merely amplifies high-reward outputs that the base model already knows for improved precision. This study presents a theoretical and empirical investigation that provides fresh insights into the potential limits of RLVR. First, we offer a new theoretical perspective that RLVR is constrained by the base model's support-unable to sample solutions with zero initial probability-and operates as a conservative reweighting mechanism that may restrict the discovery of entirely original solutions. We also identify an entropy-reward tradeoff: while RLVR reliably enhances precision, it may progressively narrow exploration and potentially overlook correct yet underrepresented solutions. Extensive empirical experiments validate that while RLVR consistently improves pass@1, the shrinkage of empirical support generally outweighs the expansion of empirical support under larger sampling budgets, failing to recover correct answers that were previously accessible to the base model. Interestingly, we also observe that while RLVR sometimes increases token-level entropy, resulting in greater uncertainty at each generation step, answer-level entropy declines, indicating that these seemingly more uncertain paths ultimately converge onto a smaller set of distinct answers. Taken together, these findings reveal potential limits of RLVR in extending reasoning horizons. Breaking this invisible leash may require future algorithmic innovations such as explicit exploration mechanisms or hybrid strategies that seed probability mass into underrepresented solution regions.

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

Recent advances have shown that scaling test-time computation enables large language models (LLMs) to solve increasingly complex problems across diverse domains. One effective paradigm for test-time scaling (TTS) involves LLM generators producing multiple solution candidates, with LLM verifiers assessing the correctness of these candidates without reference answers. In this paper, we study generative verifiers, which perform verification by generating chain-of-thought (CoT) reasoning followed by a binary verdict. We systematically analyze verification dynamics across three dimensions - problem difficulty, generator capability, and verifier generation capability - with empirical studies on 12 benchmarks across mathematical reasoning, knowledge, and natural language reasoning tasks using 14 open-source models (2B to 72B parameter range) and GPT-4o. Our experiments reveal three key findings about verification effectiveness: (1) Easy problems allow verifiers to more reliably certify correct responses; (2) Weak generators produce errors that are easier to detect than strong generators; (3) Verification ability is generally correlated with the verifier's own problem-solving capability, but this relationship varies with problem difficulty. These findings reveal opportunities to optimize basic verification strategies in TTS applications. First, given the same verifier, some weak generators can nearly match stronger ones in post-verification TTS performance (e.g., the Gemma2-9B to Gemma2-27B performance gap shrinks by 75.5%). Second, we identify cases where strong verifiers offer limited advantage over weak ones, as both fail to provide meaningful verification gains, suggesting that verifier scaling alone cannot overcome fundamental verification challenges.

Reinforcement learning (RL) has become a standard paradigm for refining large language models (LLMs) beyond pre-training and instruction tuning. A prominent line of work is RL with verifiable rewards (RLVR), which leverages automatically verifiable outcomes (e.g., correctness or executability) to generate reward signals. While efficient, this framework faces two key limitations: First, its binary feedback is too sparse to capture the quality of the reasoning process. Second, its coarse-grained rewards potentially lead to vanishing gradients. Inspired by observations from human learning, we introduce a RL technique that integrates verifiable outcomes with the model's own confidence estimates. This joint design enriches the reward signal, providing finer-grained feedback and implicitly supervising the reasoning process. Experimental results demonstrate that our proposed method enhances RL performance across multiple datasets and reduces token consumption during inference, while incurring negligible additional training cost. Moreover, it can be used as a plug-in module to enhance other state-of-the-art RL methods.

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.

Recent advances in large language models (LLMs) have improved their performance on coding benchmarks. However, improvement is plateauing due to the exhaustion of readily available high-quality data. Prior work has shown the potential of synthetic self-instruct data, but naively training on a model's own outputs can cause error accumulation, especially in coding tasks, where generalization may collapse due to overly simple or erroneous training data, highlighting the need for rigorous quality checks on synthetic data. In this work, we explore an effective approach whereby the model itself verifies the correctness of its own data. We thus propose Sol-Ver, a self-play solver-verifier framework that jointly improves a single model's code and test generation capacity. By iteratively refining code (LLM-as-a-solver) and tests (LLM-as-a-verifier) together, we boost both capabilities without relying on human annotations or larger teacher models. Experiments with the Llama 3.1 8B model demonstrate substantial performance enhancements, achieving average relative improvements of 19.63% in code generation and 17.49% in test generation on MBPP and LiveCodeBench.

Reinforcement learning with verifiable rewards (RLVR) is a practical and scalable approach to enhancing large language models in areas such as math, code, and other structured tasks. Two questions motivate this paper: how much of the reported gains survive under strictly parity-controlled evaluation, and whether RLVR is cost-free or exacts a measurable tax. We argue that progress is real, but gains are often overstated due to three forces - an RLVR tax, evaluation pitfalls, and data contamination. Using a partial-prompt contamination audit and matched-budget reproductions across base and RL models, we show that several headline gaps shrink or vanish under clean, parity-controlled evaluation. We then propose a tax-aware training and evaluation protocol that co-optimizes accuracy, grounding, and calibrated abstention and standardizes budgeting and provenance checks. Applied to recent RLVR setups, this protocol yields more reliable estimates of reasoning gains and, in several cases, revises prior conclusions. Our position is constructive: RLVR is valuable and industry-ready; we advocate keeping its practical benefits while prioritizing reliability, safety, and measurement.

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.

Large language model (LLM)-based reasoning systems have recently achieved gold medal-level performance in the IMO 2025 competition, writing mathematical proofs where, to receive full credit, each step must be not only correct but also sufficiently supported. To train LLM-based reasoners in such challenging, open-ended settings, strong verifiers capable of catching step-level mistakes are necessary prerequisites. We introduce Hard2Verify, a human-annotated, step-level verification benchmark produced with over 500 hours of human labor. Hard2Verify is designed to rigorously assess step-level verifiers at the frontier: Verifiers must provide step-level annotations or identify the first error in responses generated by frontier LLMs for very recent, challenging, and open-ended math questions. We evaluate 29 generative critics and process reward models, demonstrating that, beyond a few standouts, open-source verifiers lag closed source models. We subsequently analyze what drives poor performance in step-level verification, the impacts of scaling verifier compute, as well as fundamental questions such as self-verification and verification-generation dynamics.

Recent advances such as DeepSeek R1-Zero highlight the effectiveness of incentive training, a reinforcement learning paradigm that computes rewards solely based on the final answer part of a language model's output, thereby encouraging the generation of intermediate reasoning steps. However, these methods fundamentally rely on external verifiers, which limits their applicability to domains like mathematics and coding where such verifiers are readily available. Although reward models can serve as verifiers, they require high-quality annotated data and are costly to train. In this work, we propose NOVER, NO-VERifier Reinforcement Learning, a general reinforcement learning framework that requires only standard supervised fine-tuning data with no need for an external verifier. NOVER enables incentive training across a wide range of text-to-text tasks and outperforms the model of the same size distilled from large reasoning models such as DeepSeek R1 671B by 7.7 percent. Moreover, the flexibility of NOVER enables new possibilities for optimizing large language models, such as inverse incentive training.

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 with Verifiable Rewards (RLVR) has emerged as a key method for improving Large Language Models' reasoning capabilities, yet recent evidence suggests it may paradoxically shrink the reasoning boundary rather than expand it. This paper investigates the shrinkage issue of RLVR by analyzing its learning dynamics and reveals two critical phenomena that explain this failure. First, we expose negative interference in RLVR, where learning to solve certain training problems actively reduces the likelihood of correct solutions for others, leading to the decline of Pass@k performance, or the probability of generating a correct solution within k attempts. Second, we uncover the winner-take-all phenomenon: RLVR disproportionately reinforces problems with high likelihood, correct solutions, under the base model, while suppressing other initially low-likelihood ones. Through extensive theoretical and empirical analysis on multiple mathematical reasoning benchmarks, we show that this effect arises from the inherent on-policy sampling in standard RL objectives, causing the model to converge toward narrow solution strategies. Based on these insights, we propose a simple yet effective data curation algorithm that focuses RLVR learning on low-likelihood problems, achieving notable improvement in Pass@k performance. Our code is available at https://github.com/mail-research/SELF-llm-interference.

Mathematical verfier achieves success in mathematical reasoning tasks by validating the correctness of solutions. However, existing verifiers are trained with binary classification labels, which are not informative enough for the model to accurately assess the solutions. To mitigate the aforementioned insufficiency of binary labels, we introduce step-wise natural language feedbacks as rationale labels (i.e., the correctness of the current step and the explanations). In this paper, we propose Math-Minos, a natural language feedback enhanced verifier by constructing automatically-generated training data and a two-stage training paradigm for effective training and efficient inference. Our experiments reveal that a small set (30k) of natural language feedbacks can significantly boost the performance of the verifier by the accuracy of 1.6\% (86.6\% rightarrow 88.2\%) on GSM8K and 0.8\% (37.8\% rightarrow 38.6\%) on MATH. We have released our code and data for further exploration.

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.

Reinforcement learning (RL) has recently emerged as a compelling approach for enhancing the reasoning capabilities of large language models (LLMs), where an LLM generator serves as a policy guided by a verifier (reward model). However, current RL post-training methods for LLMs typically use verifiers that are fixed (rule-based or frozen pretrained) or trained discriminatively via supervised fine-tuning (SFT). Such designs are susceptible to reward hacking and generalize poorly beyond their training distributions. To overcome these limitations, we propose Tango, a novel framework that uses RL to concurrently train both an LLM generator and a verifier in an interleaved manner. A central innovation of Tango is its generative, process-level LLM verifier, which is trained via RL and co-evolves with the generator. Importantly, the verifier is trained solely based on outcome-level verification correctness rewards without requiring explicit process-level annotations. This generative RL-trained verifier exhibits improved robustness and superior generalization compared to deterministic or SFT-trained verifiers, fostering effective mutual reinforcement with the generator. Extensive experiments demonstrate that both components of Tango achieve state-of-the-art results among 7B/8B-scale models: the generator attains best-in-class performance across five competition-level math benchmarks and four challenging out-of-domain reasoning tasks, while the verifier leads on the ProcessBench dataset. Remarkably, both components exhibit particularly substantial improvements on the most difficult mathematical reasoning problems. Code is at: https://github.com/kaiwenzha/rl-tango.

As Large Language Models (LLMs) are integrated into critical real-world applications, their strategic and logical reasoning abilities are increasingly crucial. This paper evaluates LLMs' reasoning abilities in competitive environments through game-theoretic tasks, e.g., board and card games that require pure logic and strategic reasoning to compete with opponents. We first propose GTBench, a language-driven environment composing 10 widely-recognized tasks, across a comprehensive game taxonomy: complete versus incomplete information, dynamic versus static, and probabilistic versus deterministic scenarios. Then, we investigate two key problems: (1) Characterizing game-theoretic reasoning of LLMs; (2) LLM-vs-LLM competitions as reasoning evaluation. We observe that (1) LLMs have distinct behaviors regarding various gaming scenarios; for example, LLMs fail in complete and deterministic games yet they are competitive in probabilistic gaming scenarios; (2) Open-source LLMs, e.g., CodeLlama-34b-Instruct, are less competitive than commercial LLMs, e.g., GPT-4, in complex games. In addition, code-pretraining greatly benefits strategic reasoning, while advanced reasoning methods such as Chain-of-Thought (CoT) and Tree-of-Thought (ToT) do not always help. Detailed error profiles are also provided for a better understanding of LLMs' behavior.

This paper presents a quantitative program verification infrastructure for discrete probabilistic programs. Our infrastructure can be viewed as the probabilistic analogue of Boogie: its central components are an intermediate verification language (IVL) together with a real-valued logic. Our IVL provides a programming-language-style for expressing verification conditions whose validity implies the correctness of a program under investigation. As our focus is on verifying quantitative properties such as bounds on expected outcomes, expected run-times, or termination probabilities, off-the-shelf IVLs based on Boolean first-order logic do not suffice. Instead, a paradigm shift from the standard Boolean to a real-valued domain is required. Our IVL features quantitative generalizations of standard verification constructs such as assume- and assert-statements. Verification conditions are generated by a weakest-precondition-style semantics, based on our real-valued logic. We show that our verification infrastructure supports natural encodings of numerous verification techniques from the literature. With our SMT-based implementation, we automatically verify a variety of benchmarks. To the best of our knowledge, this establishes the first deductive verification infrastructure for expectation-based reasoning about probabilistic programs.

Large Language Models (LLMs), such as OpenAI's o1 and DeepSeek's R1, excel at advanced reasoning tasks like math and coding via Reinforcement Learning with Verifiable Rewards (RLVR), but still struggle with puzzles solvable by humans without domain knowledge. We introduce Enigmata, the first comprehensive suite tailored for improving LLMs with puzzle reasoning skills. It includes 36 tasks across seven categories, each with 1) a generator that produces unlimited examples with controllable difficulty and 2) a rule-based verifier for automatic evaluation. This generator-verifier design supports scalable, multi-task RL training, fine-grained analysis, and seamless RLVR integration. We further propose Enigmata-Eval, a rigorous benchmark, and develop optimized multi-task RLVR strategies. Our trained model, Qwen2.5-32B-Enigmata, consistently surpasses o3-mini-high and o1 on the puzzle reasoning benchmarks like Enigmata-Eval, ARC-AGI (32.8%), and ARC-AGI 2 (0.6%). It also generalizes well to out-of-domain puzzle benchmarks and mathematical reasoning, with little multi-tasking trade-off. When trained on larger models like Seed1.5-Thinking (20B activated parameters and 200B total parameters), puzzle data from Enigmata further boosts SoTA performance on advanced math and STEM reasoning tasks such as AIME (2024-2025), BeyondAIME and GPQA (Diamond), showing nice generalization benefits of Enigmata. This work offers a unified, controllable framework for advancing logical reasoning in LLMs. Resources of this work can be found at https://seed-enigmata.github.io.

Reinforcement Learning with Verifiable Rewards (RLVR) has become a standard paradigm for reasoning in Large Language Models. However, optimizing solely for final-answer correctness often drives models into aimless, verbose exploration, where they rely on exhaustive trial-and-error tactics rather than structured planning to reach solutions. While heuristic constraints like length penalties can reduce verbosity, they often truncate essential reasoning steps, creating a difficult trade-off between efficiency and verification. In this paper, we argue that discriminative capability is a prerequisite for efficient generation: by learning to distinguish valid solutions, a model can internalize a guidance signal that prunes the search space. We propose JudgeRLVR, a two-stage judge-then-generate paradigm. In the first stage, we train the model to judge solution responses with verifiable answers. In the second stage, we fine-tune the same model with vanilla generating RLVR initialized from the judge. Compared to Vanilla RLVR using the same math-domain training data, JudgeRLVR achieves a better quality--efficiency trade-off for Qwen3-30B-A3B: on in-domain math, it delivers about +3.7 points average accuracy gain with -42\% average generation length; on out-of-domain benchmarks, it delivers about +4.5 points average accuracy improvement, demonstrating enhanced generalization.

Common self-improvement approaches for large language models (LLMs), such as STaR (Zelikman et al., 2022), iteratively fine-tune LLMs on self-generated solutions to improve their problem-solving ability. However, these approaches discard the large amounts of incorrect solutions generated during this process, potentially neglecting valuable information in such solutions. To address this shortcoming, we propose V-STaR that utilizes both the correct and incorrect solutions generated during the self-improvement process to train a verifier using DPO that judges correctness of model-generated solutions. This verifier is used at inference time to select one solution among many candidate solutions. Running V-STaR for multiple iterations results in progressively better reasoners and verifiers, delivering a 4% to 17% test accuracy improvement over existing self-improvement and verification approaches on common code generation and math reasoning benchmarks with LLaMA2 models.

Reliable verifiable data has become a key driver of capability gains in modern language models, enabling stable reinforcement learning with verifiable rewards and effective distillation that transfers competence across math, coding, and agentic tasks. Yet constructing generalizable synthetic verifiable data remains difficult due to hallucination-prone generation, and weak or trivial verification artifacts that fail to separate strong from weak solutions. Existing approaches often rely on task-specific heuristics or post-hoc filters that do not transfer across domains and lack a principled, universal evaluator of verifiability. In this work, we introduce an evolutionary, task-agnostic, strategy-guided, executably-checkable data synthesis framework that, from minimal seed supervision, jointly synthesizes problems, diverse candidate solutions, and verification artifacts, and iteratively discovers strategies via a consistency-based evaluator that enforces agreement between human-annotated and strategy-induced checks. This pipeline upgrades filtering into principled synthesis: it reliably assembles coherent, verifiable training instances and generalizes without domain-specific rules. Our experiments demonstrate the effectiveness of the proposed approach under both RLVR and model distillation training paradigms. The results show that training with our synthesized data yields significant improvements on both the LiveCodeBench and AgentBench-OS tasks, highlighting the robust generalization of our framework.

Fine-tuning large language models (LLMs) on reasoning benchmarks via reinforcement learning requires a specific reward function, often binary, for each benchmark. This comes with two potential limitations: the need to design the reward, and the potentially sparse nature of binary rewards. Here, we systematically investigate rewards derived from the probability or log-probability of emitting the reference answer (or any other prompt continuation present in the data), which have the advantage of not relying on specific verifiers and being available at scale. Several recent works have advocated for the use of similar rewards (e.g., VeriFree, JEPO, RLPR, NOVER). We systematically compare variants of likelihood-based rewards with standard baselines, testing performance both on standard mathematical reasoning benchmarks, and on long-form answers where no external verifier is available. We find that using the log-probability of the reference answer as the reward for chain-of-thought (CoT) learning is the only option that performs well in all setups. This reward is also consistent with the next-token log-likelihood loss used during pretraining. In verifiable settings, log-probability rewards bring comparable or better success rates than reinforcing with standard binary rewards, and yield much better perplexity. In non-verifiable settings, they perform on par with SFT. On the other hand, methods based on probability, such as VeriFree, flatline on non-verifiable settings due to vanishing probabilities of getting the correct answer. Overall, this establishes log-probability rewards as a viable method for CoT fine-tuning, bridging the short, verifiable and long, non-verifiable answer settings.

Verifiers play a crucial role in large language model (LLM) reasoning, needed by post-training techniques such as reinforcement learning. However, reliable verifiers are hard to get for difficult coding problems, because a well-disguised wrong solution may only be detected by carefully human-written edge cases that are difficult to synthesize. To address this issue, we propose HARDTESTGEN, a pipeline for high-quality test synthesis using LLMs. With this pipeline, we curate a comprehensive competitive programming dataset HARDTESTS with 47k problems and synthetic high-quality tests. Compared with existing tests, HARDTESTGEN tests demonstrate precision that is 11.3 percentage points higher and recall that is 17.5 percentage points higher when evaluating LLM-generated code. For harder problems, the improvement in precision can be as large as 40 points. HARDTESTS also proves to be more effective for model training, measured by downstream code generation performance. We will open-source our dataset and synthesis pipeline at https://leililab.github.io/HardTests/.

DSperse is a modular framework for distributed machine learning inference with strategic cryptographic verification. Operating within the emerging paradigm of distributed zero-knowledge machine learning, DSperse avoids the high cost and rigidity of full-model circuitization by enabling targeted verification of strategically chosen subcomputations. These verifiable segments, or "slices", may cover part or all of the inference pipeline, with global consistency enforced through audit, replication, or economic incentives. This architecture supports a pragmatic form of trust minimization, localizing zero-knowledge proofs to the components where they provide the greatest value. We evaluate DSperse using multiple proving systems and report empirical results on memory usage, runtime, and circuit behavior under sliced and unsliced configurations. By allowing proof boundaries to align flexibly with the model's logical structure, DSperse supports scalable, targeted verification strategies suited to diverse deployment needs.

Recent advances in synergizing large reasoning models (LRMs) with retrieval-augmented generation (RAG) have shown promising results, yet two critical challenges remain: (1) reasoning models typically operate from a single, unchallenged perspective, limiting their ability to conduct deep, self-correcting reasoning over external documents, and (2) existing training paradigms rely excessively on outcome-oriented rewards, which provide insufficient signal for shaping the complex, multi-step reasoning process. To address these issues, we propose an Reasoner-Verifier framework named Adversarial Reasoning RAG (ARR). The Reasoner and Verifier engage in reasoning on retrieved evidence and critiquing each other's logic while being guided by process-aware advantage that requires no external scoring model. This reward combines explicit observational signals with internal model uncertainty to jointly optimize reasoning fidelity and verification rigor. Experiments on multiple benchmarks demonstrate the effectiveness of our method.

Large language models are increasingly post-trained with reinforcement learning in verifiable domains such as code and math. Yet, current methods for reinforcement learning with verifiable rewards (RLVR) learn only from a scalar outcome reward per attempt, creating a severe credit-assignment bottleneck. Many verifiable environments actually provide rich textual feedback, such as runtime errors or judge evaluations, that explain why an attempt failed. We formalize this setting as reinforcement learning with rich feedback and introduce Self-Distillation Policy Optimization (SDPO), which converts tokenized feedback into a dense learning signal without any external teacher or explicit reward model. SDPO treats the current model conditioned on feedback as a self-teacher and distills its feedback-informed next-token predictions back into the policy. In this way, SDPO leverages the model's ability to retrospectively identify its own mistakes in-context. Across scientific reasoning, tool use, and competitive programming on LiveCodeBench v6, SDPO improves sample efficiency and final accuracy over strong RLVR baselines. Notably, SDPO also outperforms baselines in standard RLVR environments that only return scalar feedback by using successful rollouts as implicit feedback for failed attempts. Finally, applying SDPO to individual questions at test time accelerates discovery on difficult binary-reward tasks, achieving the same discovery probability as best-of-k sampling or multi-turn conversations with 3x fewer attempts.

We investigate the effectiveness of reinforcement learning methods for finetuning large language models when transitioning from offline to semi-online to fully online regimes for both verifiable and non-verifiable tasks. Our experiments cover training on verifiable math as well as non-verifiable instruction following with a set of benchmark evaluations for both. Across these settings, we extensively compare online and semi-online Direct Preference Optimization and Group Reward Policy Optimization objectives, and surprisingly find similar performance and convergence between these variants, which all strongly outperform offline methods. We provide a detailed analysis of the training dynamics and hyperparameter selection strategies to achieve optimal results. Finally, we show that multi-tasking with verifiable and non-verifiable rewards jointly yields improved performance across both task types.

Agentic reinforcement learning (RL) holds great promise for the development of autonomous agents under complex GUI tasks, but its scalability remains severely hampered by the verification of task completion. Existing task verification is treated as a passive, post-hoc process: a verifier (i.e., rule-based scoring script, reward or critic model, and LLM-as-a-Judge) analyzes the agent's entire interaction trajectory to determine if the agent succeeds. Such processing of verbose context that contains irrelevant, noisy history poses challenges to the verification protocols and therefore leads to prohibitive cost and low reliability. To overcome this bottleneck, we propose SmartSnap, a paradigm shift from this passive, post-hoc verification to proactive, in-situ self-verification by the agent itself. We introduce the Self-Verifying Agent, a new type of agent designed with dual missions: to not only complete a task but also to prove its accomplishment with curated snapshot evidences. Guided by our proposed 3C Principles (Completeness, Conciseness, and Creativity), the agent leverages its accessibility to the online environment to perform self-verification on a minimal, decisive set of snapshots. Such evidences are provided as the sole materials for a general LLM-as-a-Judge verifier to determine their validity and relevance. Experiments on mobile tasks across model families and scales demonstrate that our SmartSnap paradigm allows training LLM-driven agents in a scalable manner, bringing performance gains up to 26.08% and 16.66% respectively to 8B and 30B models. The synergizing between solution finding and evidence seeking facilitates the cultivation of efficient, self-verifying agents with competitive performance against DeepSeek V3.1 and Qwen3-235B-A22B.

Reinforcement learning with verifiable rewards (RLVR) has become a key technique for enhancing large language models (LLMs), with verification engineering playing a central role. However, best practices for RL in instruction following remain underexplored. In this work, we explore the verification challenge in RL for instruction following and propose VerIF, a verification method that combines rule-based code verification with LLM-based verification from a large reasoning model (e.g., QwQ-32B). To support this approach, we construct a high-quality instruction-following dataset, VerInstruct, containing approximately 22,000 instances with associated verification signals. We apply RL training with VerIF to two models, achieving significant improvements across several representative instruction-following benchmarks. The trained models reach state-of-the-art performance among models of comparable size and generalize well to unseen constraints. We further observe that their general capabilities remain unaffected, suggesting that RL with VerIF can be integrated into existing RL recipes to enhance overall model performance. We have released our datasets, codes, and models to facilitate future research at https://github.com/THU-KEG/VerIF.

Reinforcement learning with verifiable rewards (RLVR) succeeds in reasoning tasks (e.g., math and code) by checking the final verifiable answer (i.e., a verifiable dot signal). However, extending this paradigm to open-ended generation is challenging because there is no unambiguous ground truth. Relying on single-dot supervision often leads to inefficiency and reward hacking. To address these issues, we propose reinforcement learning with verifiable reference-based rewards (RLVRR). Instead of checking the final answer, RLVRR extracts an ordered linguistic signal from high-quality references (i.e, reward chain). Specifically, RLVRR decomposes rewards into two dimensions: content, which preserves deterministic core concepts (e.g., keywords), and style, which evaluates adherence to stylistic properties through LLM-based verification. In this way, RLVRR combines the exploratory strength of RL with the efficiency and reliability of supervised fine-tuning (SFT). Extensive experiments on more than 10 benchmarks with Qwen and Llama models confirm the advantages of our approach. RLVRR (1) substantially outperforms SFT trained with ten times more data and advanced reward models, (2) unifies the training of structured reasoning and open-ended generation, and (3) generalizes more effectively while preserving output diversity. These results establish RLVRR as a principled and efficient path toward verifiable reinforcement learning for general-purpose LLM alignment. We release our code and data at https://github.com/YJiangcm/RLVRR.

Reinforcement Learning with Verifiable Rewards (RLVR) has proven effective for training large language models (LLMs) on complex reasoning tasks, such as mathematical problem solving. A prerequisite for the scalability of RLVR is a high-quality problem set with precise and verifiable answers. However, the scarcity of well-crafted human-labeled math problems and limited-verification answers in existing distillation-oriented synthetic datasets limit their effectiveness in RL. Additionally, most problem synthesis strategies indiscriminately expand the problem set without considering the model's capabilities, leading to low efficiency in generating useful questions. To mitigate this issue, we introduce a Self-aware Weakness-driven problem Synthesis framework (SwS) that systematically identifies model deficiencies and leverages them for problem augmentation. Specifically, we define weaknesses as questions that the model consistently fails to learn through its iterative sampling during RL training. We then extract the core concepts from these failure cases and synthesize new problems to strengthen the model's weak areas in subsequent augmented training, enabling it to focus on and gradually overcome its weaknesses. Without relying on external knowledge distillation, our framework enables robust generalization byempowering the model to self-identify and address its weaknesses in RL, yielding average performance gains of 10.0% and 7.7% on 7B and 32B models across eight mainstream reasoning benchmarks.

We provide here a dataset for tasks related to natural language understanding and natural language inference. The dataset contains logical puzzles in natural language from three domains: comparing puzzles, knighs and knaves, and zebra puzzles. Each puzzle is associated with the entire set of atomic questions that can be generated based on the relations and individuals occurring in the text. For each question we provide the correct answer: entailment, contradiction or ambiguity. The answer's correctness is verified against theorem provers. Good puzzles have two properties: (i) each piece of information is necessary and (ii) no unnecessary information is provided. These properties make puzzles interesting candidates for machine comprehension tasks.

Formally verifying properties of software code has been a highly desirable task, especially with the emergence of LLM-generated code. In the same vein, they provide an interesting avenue for the exploration of formal verification and mechanistic interpretability. Since the introduction of code-specific models, despite their successes in generating code in Lean4 and Isabelle, the task of generalized theorem proving still remains far from being fully solved and will be a benchmark for reasoning capability in LLMs. In this work, we introduce a framework that generates whole proofs in a formal language to be used within systems that utilize the power of built-in tactics and off-the-shelf automated theorem provers. Our framework includes 3 components: generating natural language statements of the code to be verified, an LLM that generates formal proofs for the given statement, and a module employing heuristics for building the final proof. To train the LLM, we employ a 2-stage fine-tuning process, where we first use SFT-based training to enable the model to generate syntactically correct Isabelle code and then RL-based training that encourages the model to generate proofs verified by a theorem prover. We validate our framework using the miniF2F-test benchmark and the Isabelle proof assistant and design a use case to verify the correctness of the AWS S3 bucket access policy code. We also curate a dataset based on the FVEL\textnormal{ER} dataset for future training tasks.

Poker, also known as Texas Hold'em, has always been a typical research target within imperfect information games (IIGs). IIGs have long served as a measure of artificial intelligence (AI) development. Representative prior works, such as DeepStack and Libratus heavily rely on counterfactual regret minimization (CFR) to tackle heads-up no-limit Poker. However, it is challenging for subsequent researchers to learn CFR from previous models and apply it to other real-world applications due to the expensive computational cost of CFR iterations. Additionally, CFR is difficult to apply to multi-player games due to the exponential growth of the game tree size. In this work, we introduce PokerGPT, an end-to-end solver for playing Texas Hold'em with arbitrary number of players and gaining high win rates, established on a lightweight large language model (LLM). PokerGPT only requires simple textual information of Poker games for generating decision-making advice, thus guaranteeing the convenient interaction between AI and humans. We mainly transform a set of textual records acquired from real games into prompts, and use them to fine-tune a lightweight pre-trained LLM using reinforcement learning human feedback technique. To improve fine-tuning performance, we conduct prompt engineering on raw data, including filtering useful information, selecting behaviors of players with high win rates, and further processing them into textual instruction using multiple prompt engineering techniques. Through the experiments, we demonstrate that PokerGPT outperforms previous approaches in terms of win rate, model size, training time, and response speed, indicating the great potential of LLMs in solving IIGs.

The widespread adoption of deep neural networks (DNNs) requires efficient techniques for verifying their safety. DNN verifiers are complex tools, which might contain bugs that could compromise their soundness and undermine the reliability of the verification process. This concern can be mitigated using proofs: artifacts that are checkable by an external and reliable proof checker, and which attest to the correctness of the verification process. However, such proofs tend to be extremely large, limiting their use in many scenarios. In this work, we address this problem by minimizing proofs of unsatisfiability produced by DNN verifiers. We present algorithms that remove facts which were learned during the verification process, but which are unnecessary for the proof itself. Conceptually, our method analyzes the dependencies among facts used to deduce UNSAT, and removes facts that did not contribute. We then further minimize the proof by eliminating remaining unnecessary dependencies, using two alternative procedures. We implemented our algorithms on top of a proof producing DNN verifier, and evaluated them across several benchmarks. Our results show that our best-performing algorithm reduces proof size by 37%-82% and proof checking time by 30%-88%, while introducing a runtime overhead of 7%-20% to the verification process itself.

Formal theorem proving (FTP) has emerged as a critical foundation for evaluating the reasoning capabilities of large language models, enabling automated verification of mathematical proofs at scale. However, progress has been constrained by limited datasets due to the high cost of manual curation and the scarcity of challenging problems with verified formal-informal correspondences. We propose leveraging theoretical computer science (TCS) as a scalable source of rigorous proof problems, where algorithmic definitions enable automated generation of arbitrarily many challenging theorem-proof pairs. We demonstrate this approach on two TCS domains: Busy Beaver problems, which involve proving bounds on Turing machine halting behavior, and Mixed Boolean Arithmetic problems, which combine logical and arithmetic reasoning. Our framework automatically synthesizes problems with parallel formal (Lean4) and informal (Markdown) specifications, creating a scalable pipeline for generating verified proof challenges. Evaluation on frontier models reveals substantial gaps in automated theorem proving: while DeepSeekProver-V2-671B achieves 57.5\% success on Busy Beaver problems, it manages only 12\% on Mixed Boolean Arithmetic problems. These results highlight the difficulty of long-form proof generation even for problems that are computationally easy to verify, demonstrating the value of TCS domains for advancing automated reasoning research.

Current research on the Decompose-Then-Verify paradigm for evaluating the factuality of long-form text typically treats decomposition and verification in isolation, overlooking their interactions and potential misalignment. We find that existing decomposition policies, typically hand-crafted demonstrations, do not align well with downstream verifiers in terms of atomicity -- a novel metric quantifying information density -- leading to suboptimal verification results. We formulate finding the optimal decomposition policy for optimal verification as a bilevel optimization problem. To approximate a solution for this strongly NP-hard problem, we propose dynamic decomposition, a reinforcement learning framework that leverages verifier feedback to learn a policy for dynamically decomposing claims to verifier-preferred atomicity. Experimental results show that dynamic decomposition outperforms existing decomposition policies, improving verification confidence by 0.07 and accuracy by 0.12 (on a 0-1 scale) on average across varying verifiers, datasets, and atomcities of input claims.

Despite significant advancements in the general capability of large language models (LLMs), they continue to struggle with consistent and accurate reasoning, especially in complex tasks such as mathematical and code reasoning. One key limitation is that LLMs are trained primarily on correct solutions, reducing their ability to detect and learn from errors, which hampers their ability to reliably verify and rank outputs. To address this, we scale up the inference-time computation by generating multiple reasoning paths and employing verifiers to assess and rank the generated outputs by correctness. To facilitate this, we introduce a comprehensive dataset consisting of correct and incorrect solutions for math and code tasks, generated by multiple LLMs. This diverse set of solutions enables verifiers to more effectively distinguish and rank correct answers from erroneous outputs. The training methods for building verifiers were selected based on an extensive comparison of existing approaches. Moreover, to leverage the unique strengths of different reasoning strategies, we propose a novel collaborative method integrating Chain-of-Thought (CoT) and Program-of-Thought (PoT) solutions for verification. CoT provides a clear, step-by-step reasoning process that enhances interpretability, while PoT, being executable, offers a precise and error-sensitive validation mechanism. By taking both of their strengths, our approach significantly improves the accuracy and reliability of reasoning verification. Our verifiers, Math-Rev and Code-Rev, demonstrate substantial performance gains to existing LLMs, achieving state-of-the-art results on benchmarks such as GSM8k and MATH and even outperforming GPT-4o with Qwen-72B-Instruct as the reasoner.

While Large Language Model (LLM) agents are often approached from the angle of action planning/generation to accomplish a goal (e.g., given by language descriptions), their abilities to collaborate with each other to achieve a joint goal are not well explored. To address this limitation, this paper studies LLM agents in task collaboration, particularly under the condition of information asymmetry, where agents have disparities in their knowledge and skills and need to work together to complete a shared task. We extend Einstein Puzzles, a classical symbolic puzzle, to a table-top game. In this game, two LLM agents must reason, communicate, and act to satisfy spatial and relational constraints required to solve the puzzle. We apply a fine-tuning-plus-verifier framework in which LLM agents are equipped with various communication strategies and verification signals from the environment. Empirical results highlight the critical importance of aligned communication, especially when agents possess both information-seeking and -providing capabilities. Interestingly, agents without communication can still achieve high task performance; however, further analysis reveals a lack of true rule understanding and lower trust from human evaluators. Instead, by integrating an environment-based verifier, we enhance agents' ability to comprehend task rules and complete tasks, promoting both safer and more interpretable collaboration in AI systems. https://github.com/Roihn/EinsteinPuzzles

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.

We study privacy leakage in the reasoning traces of large reasoning models used as personal agents. Unlike final outputs, reasoning traces are often assumed to be internal and safe. We challenge this assumption by showing that reasoning traces frequently contain sensitive user data, which can be extracted via prompt injections or accidentally leak into outputs. Through probing and agentic evaluations, we demonstrate that test-time compute approaches, particularly increased reasoning steps, amplify such leakage. While increasing the budget of those test-time compute approaches makes models more cautious in their final answers, it also leads them to reason more verbosely and leak more in their own thinking. This reveals a core tension: reasoning improves utility but enlarges the privacy attack surface. We argue that safety efforts must extend to the model's internal thinking, not just its outputs.

Zero Reinforcement Learning (Zero-RL) has proven to be an effective approach for enhancing the reasoning capabilities of large language models (LLMs) by directly applying reinforcement learning with verifiable rewards on pretrained models, without the need for a supervised fine-tuning phase. However, current research on zero-RL primarily focuses on domains with easily verifiable reward signals, such as mathematics, programming, and other reasoning tasks. The challenge of eliciting reasoning abilities in more diverse scenarios, where verification is not straightforward, remains underexplored. To address this gap, we propose a novel zero-RL paradigm designed to improve a model's reasoning ability across both verifiable and non-verifiable domains. By combining verifiable rewards with a generative reward model, we conduct multi-task zero-RL training across both domains, facilitating the transfer of reasoning capabilities between them. Furthermore, to mitigate reward hacking in the generative reward model, we design a smooth length penalty that encourages the generation of more comprehensive thinking tokens in general domains. Experimental results on Qwen3-8B-Base and Qwen3-14B-Base demonstrate that our approach achieves superior reasoning performance, not only on tasks requiring extensive reasoning but also on more general tasks.

Recent advances in synthetic data generation (SDG) have been hailed as a solution to the difficult problem of sharing sensitive data while protecting privacy. SDG aims to learn statistical properties of real data in order to generate "artificial" data that are structurally and statistically similar to sensitive data. However, prior research suggests that inference attacks on synthetic data can undermine privacy, but only for specific outlier records. In this work, we introduce a new attribute inference attack against synthetic data. The attack is based on linear reconstruction methods for aggregate statistics, which target all records in the dataset, not only outliers. We evaluate our attack on state-of-the-art SDG algorithms, including Probabilistic Graphical Models, Generative Adversarial Networks, and recent differentially private SDG mechanisms. By defining a formal privacy game, we show that our attack can be highly accurate even on arbitrary records, and that this is the result of individual information leakage (as opposed to population-level inference). We then systematically evaluate the tradeoff between protecting privacy and preserving statistical utility. Our findings suggest that current SDG methods cannot consistently provide sufficient privacy protection against inference attacks while retaining reasonable utility. The best method evaluated, a differentially private SDG mechanism, can provide both protection against inference attacks and reasonable utility, but only in very specific settings. Lastly, we show that releasing a larger number of synthetic records can improve utility but at the cost of making attacks far more effective.

Recent advancements in reasoning-focused language models such as OpenAI's O1 and DeepSeek-R1 have shown that scaling test-time computation-through chain-of-thought reasoning and iterative exploration-can yield substantial improvements on complex tasks like mathematics and code generation. These breakthroughs have been driven by large-scale reinforcement learning (RL), particularly when combined with verifiable reward signals that provide objective and grounded supervision. In this report, we investigate the effects of prolonged reinforcement learning on a small language model across a diverse set of reasoning domains. Our work identifies several key ingredients for effective training, including the use of verifiable reward tasks, enhancements to Group Relative Policy Optimization (GRPO), and practical techniques to improve training stability and generalization. We introduce controlled KL regularization, clipping ratio, and periodic reference policy resets as critical components for unlocking long-term performance gains. Our model achieves significant improvements over strong baselines, including +14.7% on math, +13.9% on coding, and +54.8% on logic puzzle tasks. To facilitate continued research, we release our model publicly.

Large language models (LLMs) struggle with multi-step reasoning, where inference-time scaling has emerged as a promising strategy for performance improvement. Verifier-guided search outperforms repeated sampling when sample size is limited by selecting and prioritizing valid reasoning paths. However, we identify a critical limitation: scaling flaws, prevalent across different models (Mistral 7B and DeepSeekMath 7B), benchmarks (GSM8K and MATH), and verifiers (outcome value models and process reward models). As sample size increases, verifier-guided search exhibits diminishing advantages and eventually underperforms repeated sampling. Our analysis attributes this to verifier failures, where imperfect verifiers misrank candidates and erroneously prune all valid paths. These issues are further exacerbated in challenging and out-of-distribution problems, restricting search effectiveness. To mitigate verifier failures, we explore reducing reliance on verifiers and conduct preliminary investigations using two simple methods. Our findings reveal fundamental limitations in verifier-guided search and suggest future directions.

Reinforcement Learning (RL) has become a powerful tool for enhancing the reasoning abilities of large language models (LLMs) by optimizing their policies with reward signals. Yet, RL's success relies on the reliability of rewards, which are provided by verifiers. In this paper, we expose and analyze a widespread problem--false negatives--where verifiers wrongly reject correct model outputs. Our in-depth study of the Big-Math-RL-Verified dataset reveals that over 38% of model-generated responses suffer from false negatives, where the verifier fails to recognize correct answers. We show, both empirically and theoretically, that these false negatives severely impair RL training by depriving the model of informative gradient signals and slowing convergence. To mitigate this, we propose tinyV, a lightweight LLM-based verifier that augments existing rule-based methods, which dynamically identifies potential false negatives and recovers valid responses to produce more accurate reward estimates. Across multiple math-reasoning benchmarks, integrating TinyV boosts pass rates by up to 10% and accelerates convergence relative to the baseline. Our findings highlight the critical importance of addressing verifier false negatives and offer a practical approach to improve RL-based fine-tuning of LLMs. Our code is available at https://github.com/uw-nsl/TinyV.

Large language models (LLMs) often struggle with maintaining accuracy across a sequence of intermediate reasoning steps in mathematical reasoning, leading to error propagation that undermines the final result. The current methodology to mitigate this issue primarily involves using a verifier model to assess the correctness of generated solution candidates, focusing either on the overall reasoning path or on an incomplete reasoning path. By rethinking this approach, we argue that assessing potentials of incomplete reasoning paths could be more advantageous as it guides towards correct final answers, transforming the task into a planning problem. Our proposed verifier, the Outcome-supervision Value Model (OVM), employs outcome supervision for training, offering an efficient and intuitive method for planning by prioritizing steps that lead to accurate conclusions over mere per-step correctness. Furthermore, the OVM eschews the need for labor-intensive annotations on step-level correctness, enhancing its scalability. Our experiments on two multi-step mathematical reasoning datasets, GSM8K and Game of 24, demonstrate the superior performance of the OVM model. Notably, in GSM8K, our OVM-7B model achieves state-of-the-art results among LLMs up to 13B parameters; especially it does not utilize GPT-4 or code execution. These findings offer a novel perspective on the role of outcome supervision in training verifiers for multi-step reasoning tasks and provide theoretical justification for its advantage in value estimation for planning.

Large Language Model (LLM) reasoning for complex tasks inherently involves a trade-off between solution accuracy and computational efficiency. The subsequent step of verification, while intended to improve performance, further complicates this landscape by introducing its own challenging trade-off: sophisticated Generative Reward Models (GenRMs) can be computationally prohibitive if naively integrated with LLMs at test-time, while simpler, faster methods may lack reliability. To overcome these challenges, we introduce FlexiVe, a novel generative verifier that flexibly balances computational resources between rapid, reliable fast thinking and meticulous slow thinking using a Flexible Allocation of Verification Budget strategy. We further propose the Solve-Detect-Verify pipeline, an efficient inference-time scaling framework that intelligently integrates FlexiVe, proactively identifying solution completion points to trigger targeted verification and provide focused solver feedback. Experiments show FlexiVe achieves superior accuracy in pinpointing errors within reasoning traces on ProcessBench. Furthermore, on challenging mathematical reasoning benchmarks (AIME 2024, AIME 2025, and CNMO), our full approach outperforms baselines like self-consistency in reasoning accuracy and inference efficiency. Our system offers a scalable and effective solution to enhance LLM reasoning at test time.

Can large language models improve without external data -- by generating their own questions and answers? We hypothesize that a pre-trained language model can improve its reasoning skills given only a single prompt specifying the topic (e.g., algebra word problems) and asking the model to generate its own questions. To do this, we propose Self-Questioning Language Models (SQLM): an asymmetric self-play framework where a proposer is given the topic and generates a question for a solver, who tries to answer it. Both the proposer and solver are trained via reinforcement learning. The proposer receives a reward if the problem is not too easy or too difficult, and the solver receives a reward based on majority voting, a proxy for correctness in the absence of ground-truth answers. For coding, the proposer can instead generate unit tests which are used for verification. We study this asymmetric self-play framework on three benchmarks: three-digit multiplication, algebra problems from the OMEGA benchmark, and programming problems from Codeforces. By continually generating more interesting problems and attempting to solve them, language models can improve on downstream benchmarks without access to any curated training datasets.

Traditional language model-based theorem proving assumes that by training on a sufficient amount of formal proof data, a model will learn to prove theorems. Our key observation is that a wealth of informal information that is not present in formal proofs can be useful for learning to prove theorems. For instance, humans think through steps of a proof, but this thought process is not visible in the resulting code. We present Lean-STaR, a framework for training language models to produce informal thoughts prior to each step of a proof, thereby boosting the model's theorem-proving capabilities. Lean-STaR uses retrospective ground-truth tactics to generate synthetic thoughts for training the language model. At inference time, the trained model directly generates the thoughts prior to the prediction of the tactics in each proof step. Building on the self-taught reasoner framework, we then apply expert iteration to further fine-tune the model on the correct proofs it samples and verifies using the Lean solver. Lean-STaR achieves state-of-the-art results on the miniF2F-test benchmark within the Lean theorem proving environment, significantly outperforming base models (43.4% rightarrow 46.3%, Pass@64). We also analyze the impact of the augmented thoughts on various aspects of the theorem proving process, providing insights into their effectiveness.

Formal methods is pivotal for verifying the reliability of critical systems through rigorous mathematical proofs. However, its adoption is hindered by labor-intensive manual proofs and the expertise required to use theorem provers. Recent advancements in large language models (LLMs) offer new opportunities for automated theorem proving. Two promising approaches are generating tactics step by step and generating a whole proof directly with an LLM. However, existing work makes no attempt to combine the two approaches. In this work, we introduce HybridProver, a dual-model proof synthesis framework that combines tactic-based generation and whole-proof synthesis to harness the benefits of both approaches. HybridProver generates whole proof candidates for evaluation directly, then extracts proof sketches from those candidates. It then uses a tactic-based generation model that integrates automated tools to complete the sketches via stepwise refinement. We implement HybridProver for the Isabelle theorem prover and fine-tune LLMs on our optimized Isabelle datasets. Evaluation on the miniF2F dataset illustrates HybridProver's effectiveness. We achieve a 59.4% success rate on miniF2F, where the previous SOTA is 56.1%. Our ablation studies show that this SOTA result is attributable to combining whole-proof and tactic-based generation. Additionally, we show how the dataset quality, training parameters, and sampling diversity affect the final result during automated theorem proving with LLMs. All of our code, datasets, and LLMs are open source.

Recent advances in large language models show strong promise for formal reasoning. However, most LLM-based theorem provers have long been constrained by the need for expert-written formal statements as inputs, limiting their applicability to real-world problems expressed in natural language. We tackle this gap with Mathesis, the first end-to-end theorem proving pipeline processing informal problem statements. It contributes Mathesis-Autoformalizer, the first autoformalizer using reinforcement learning to enhance the formalization ability of natural language problems, aided by our novel LeanScorer framework for nuanced formalization quality assessment. It also proposes a Mathesis-Prover, which generates formal proofs from the formalized statements. To evaluate the real-world applicability of end-to-end formal theorem proving, we introduce Gaokao-Formal, a benchmark of 488 complex problems from China's national college entrance exam. Our approach is carefully designed, with a thorough study of each component. Experiments demonstrate Mathesis's effectiveness, with the autoformalizer outperforming the best baseline by 22% in pass-rate on Gaokao-Formal. The full system surpasses other model combinations, achieving 64% accuracy on MiniF2F with pass@32 and a state-of-the-art 18% on Gaokao-Formal.

We explore the self-play training procedure of large language models (LLMs) in a two-player adversarial language game called Adversarial Taboo. In this game, an attacker and a defender communicate around a target word only visible to the attacker. The attacker aims to induce the defender to speak the target word unconsciously, while the defender tries to infer the target word from the attacker's utterances. To win the game, both players should have sufficient knowledge about the target word and high-level reasoning ability to infer and express in this information-reserved conversation. Hence, we are curious about whether LLMs' reasoning ability can be further enhanced by self-play in this adversarial language game (SPAG). With this goal, we select several open-source LLMs and let each act as the attacker and play with a copy of itself as the defender on an extensive range of target words. Through reinforcement learning on the game outcomes, we observe that the LLMs' performances uniformly improve on a broad range of reasoning benchmarks. Furthermore, iteratively adopting this self-play process can continuously promote LLMs' reasoning abilities. The code is at https://github.com/Linear95/SPAG.

First-order logic (FOL) reasoning, which involves sequential deduction, is pivotal for intelligent systems and serves as a valuable task for evaluating reasoning capabilities, particularly in chain-of-thought (CoT) contexts. Existing benchmarks often rely on extensive human annotation or handcrafted templates, making it difficult to achieve the necessary complexity, scalability, and diversity for robust evaluation. To address these limitations, we propose a novel framework called ProverGen that synergizes the generative strengths of Large Language Models (LLMs) with the rigor and precision of symbolic provers, enabling the creation of a scalable, diverse, and high-quality FOL reasoning dataset, ProverQA. ProverQA is also distinguished by its inclusion of accessible and logically coherent intermediate reasoning steps for each problem. Our evaluation shows that state-of-the-art LLMs struggle to solve ProverQA problems, even with CoT prompting, highlighting the dataset's challenging nature. We also finetune Llama3.1-8B-Instruct on a separate training set generated by our framework. The finetuned model demonstrates consistent improvements on both in-distribution and out-of-distribution test sets, suggesting the value of our proposed data generation framework. Code available at: https://github.com/opendatalab/ProverGen

Social reasoning - inferring unobservable beliefs and intentions from partial observations of other agents - remains a challenging task for large language models (LLMs). We evaluate the limits of current reasoning language models in the social deduction game Avalon and find that while the largest models demonstrate strong performance, they require extensive test-time inference and degrade sharply when distilled to smaller, real-time-capable variants. To address this, we introduce a hybrid reasoning framework that externalizes belief inference to a structured probabilistic model, while using an LLM for language understanding and interaction. Our approach achieves competitive performance with much larger models in Agent-Agent play and, notably, is the first language agent to defeat human players in a controlled study - achieving a 67% win rate and receiving higher qualitative ratings than both reasoning baselines and human teammates. We release code, models, and a dataset to support future work on social reasoning in LLM agents, which can be found at https://camp-lab-purdue.github.io/bayesian-social-deduction/

Chain-of-Thought (CoT) prompting has become the de facto method to elicit reasoning capabilities from large language models (LLMs). However, to mitigate hallucinations in CoT that are notoriously difficult to detect, current methods such as process reward models (PRMs) or self-consistency operate as opaque boxes and do not provide checkable evidence for their judgments, possibly limiting their effectiveness. To address this issue, we draw inspiration from the idea that "the gold standard for supporting a mathematical claim is to provide a proof". We propose a retrospective, step-aware formal verification framework Safe. Rather than assigning arbitrary scores, we strive to articulate mathematical claims in formal mathematical language Lean 4 at each reasoning step and provide formal proofs to identify hallucinations. We evaluate our framework Safe across multiple language models and various mathematical datasets, demonstrating a significant performance improvement while offering interpretable and verifiable evidence. We also propose FormalStep as a benchmark for step correctness theorem proving with 30,809 formal statements. To the best of our knowledge, our work represents the first endeavor to utilize formal mathematical language Lean 4 for verifying natural language content generated by LLMs, aligning with the reason why formal mathematical languages were created in the first place: to provide a robust foundation for hallucination-prone human-written proofs.

Users like sharing personal photos with others through social media. At the same time, they might want to make automatic identification in such photos difficult or even impossible. Classic obfuscation methods such as blurring are not only unpleasant but also not as effective as one would expect. Recent studies on adversarial image perturbations (AIP) suggest that it is possible to confuse recognition systems effectively without unpleasant artifacts. However, in the presence of counter measures against AIPs, it is unclear how effective AIP would be in particular when the choice of counter measure is unknown. Game theory provides tools for studying the interaction between agents with uncertainties in the strategies. We introduce a general game theoretical framework for the user-recogniser dynamics, and present a case study that involves current state of the art AIP and person recognition techniques. We derive the optimal strategy for the user that assures an upper bound on the recognition rate independent of the recogniser's counter measure. Code is available at https://goo.gl/hgvbNK.