Daily Papers

Logical reasoning is a fundamental capability of large language models. However, existing studies often overlook the interaction between logical complexity and semantic complexity, leading to systems that struggle with abstract propositions, ambiguous contexts, and conflicting stances that are central to human reasoning. We propose LogicAgent, a semiotic-square-guided framework that jointly addresses these two axes of difficulty. The semiotic square provides a principled structure for multi-perspective semantic analysis, and LogicAgent integrates automated deduction with reflective verification to manage logical complexity across deeper reasoning chains. To support evaluation under these conditions, we introduce RepublicQA, a benchmark that couples semantic complexity with logical depth. RepublicQA reaches college-level semantic difficulty (FKGL 11.94), contains philosophically grounded abstract propositions with systematically constructed contrary and contradictory forms, and offers a semantically rich setting for assessing logical reasoning in large language models. Experiments show that LogicAgent achieves state-of-the-art performance on RepublicQA with a 6.25 percent average improvement over strong baselines, and generalizes effectively to mainstream logical reasoning benchmarks including ProntoQA, ProofWriter, FOLIO, and ProverQA, achieving an additional 7.05 percent average gain. These results demonstrate the effectiveness of semiotic-grounded multi-perspective reasoning in enhancing logical performance.

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.

Formal, automated theorem proving has long been viewed as a challenge to artificial intelligence. We introduce here a new approach to computer theorem proving, one that employs specialized language models for Lean4 proof generation combined with recursive decomposition of difficult theorems into simpler entailing propositions. These models are coordinated through a multi-agent architecture that orchestrates autoformalization (if required), proof generation, decomposition of difficult theorems into simpler entailing propositions, and recursive proof (and/or decomposition) of these propositions. Without decomposition, we achieve a 90.4% pass rate on miniF2F. With decomposition, this is significantly improved. A key technical contribution lies in our extension of the Kimina Lean Server with abstract syntax tree (AST) parsing capabilities to facilitate automated, recursive proof decomposition. The system is made available on PyPI as goedels-poetry (at https://pypi.org/project/goedels-poetry ), and the open-source implementation KellyJDavis/goedels-poetry (at https://github.com/KellyJDavis/goedels-poetry ) facilitates both adaptation to alternative language models and extension with custom functionality.

Formal mathematical reasoning remains a critical challenge for artificial intelligence, hindered by limitations of existing benchmarks in scope and scale. To address this, we present FormalMATH, a large-scale Lean4 benchmark comprising 5,560 formally verified problems spanning from high-school Olympiad challenges to undergraduate-level theorems across diverse domains (e.g., algebra, applied mathematics, calculus, number theory, and discrete mathematics). To mitigate the inefficiency of manual formalization, we introduce a novel human-in-the-loop autoformalization pipeline that integrates: (1) specialized large language models (LLMs) for statement autoformalization, (2) multi-LLM semantic verification, and (3) negation-based disproof filtering strategies using off-the-shelf LLM-based provers. This approach reduces expert annotation costs by retaining 72.09% of statements before manual verification while ensuring fidelity to the original natural-language problems. Our evaluation of state-of-the-art LLM-based theorem provers reveals significant limitations: even the strongest models achieve only 16.46% success rate under practical sampling budgets, exhibiting pronounced domain bias (e.g., excelling in algebra but failing in calculus) and over-reliance on simplified automation tactics. Notably, we identify a counterintuitive inverse relationship between natural-language solution guidance and proof success in chain-of-thought reasoning scenarios, suggesting that human-written informal reasoning introduces noise rather than clarity in the formal reasoning settings. We believe that FormalMATH provides a robust benchmark for benchmarking formal mathematical reasoning.

We study a synthetic corpus based approach for language models (LMs) to acquire logical deductive reasoning ability. The previous studies generated deduction examples using specific sets of deduction rules. However, these rules were limited or otherwise arbitrary, limiting the generalizability of acquired reasoning ability. We rethink this and adopt a well-grounded set of deduction rules based on formal logic theory, which can derive any other deduction rules when combined in a multistep way. Then, using the proposed corpora, which we name FLD (Formal Logic Deduction), we first evaluate and analyze the logical reasoning ability of the latest LLMs. Even GPT-4 can solve only half of the problems, suggesting that pure logical reasoning isolated from knowledge is still challenging for the LLMs, and additional training specialized in logical reasoning is indeed essential. We next empirically verify that LMs trained on FLD corpora acquire more generalizable reasoning ability. Furthermore, we identify the aspects of reasoning ability on which deduction corpora can enhance LMs and those on which they cannot, and discuss future directions on each aspect. The released corpora serve both as learning resources and as challenging benchmarks.

Autoformalization addresses the scarcity of data for Automated Theorem Proving (ATP) by translating mathematical problems from natural language into formal statements. Efforts in recent work shift from directly prompting large language models to training an end-to-end formalizer model from scratch, achieving remarkable advancements. However, existing formalizer still struggles to consistently generate valid statements that meet syntactic validity and semantic consistency. To address this issue, we propose the Autoformalizer with Tool Feedback (ATF), a novel approach that incorporates syntactic and consistency information as tools into the formalization process. By integrating Lean 4 compilers for syntax corrections and employing a multi-LLMs-as-judge approach for consistency validation, the model is able to adaptively refine generated statements according to the tool feedback, enhancing both syntactic validity and semantic consistency. The training of ATF involves a cold-start phase on synthetic tool-calling data, an expert iteration phase to improve formalization capabilities, and Direct Preference Optimization to alleviate ineffective revisions. Experimental results show that ATF markedly outperforms a range of baseline formalizer models, with its superior performance further validated by human evaluations. Subsequent analysis reveals that ATF demonstrates excellent inference scaling properties. Moreover, we open-source Numina-ATF, a dataset containing 750K synthetic formal statements to facilitate advancements in autoformalization and ATP research.

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.

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/ .

Large Language Models (LLMs) significantly benefit from Chain-of-Thought (CoT) prompting in performing various reasoning tasks. While CoT allows models to produce more comprehensive reasoning processes, its emphasis on intermediate reasoning steps can inadvertently introduce hallucinations and accumulated errors, thereby limiting models' ability to solve complex reasoning tasks. Inspired by how humans engage in careful and meticulous deductive logical reasoning processes to solve tasks, we seek to enable language models to perform explicit and rigorous deductive reasoning, and also ensure the trustworthiness of their reasoning process through self-verification. However, directly verifying the validity of an entire deductive reasoning process is challenging, even with advanced models like ChatGPT. In light of this, we propose to decompose a reasoning verification process into a series of step-by-step subprocesses, each only receiving their necessary context and premises. To facilitate this procedure, we propose Natural Program, a natural language-based deductive reasoning format. Our approach enables models to generate precise reasoning steps where subsequent steps are more rigorously grounded on prior steps. It also empowers language models to carry out reasoning self-verification in a step-by-step manner. By integrating this verification process into each deductive reasoning stage, we significantly enhance the rigor and trustfulness of generated reasoning steps. Along this process, we also improve the answer correctness on complex reasoning tasks. Code will be released at https://github.com/lz1oceani/verify_cot.

Large language models (LLMs) are capable of solving a wide range of tasks, yet they have struggled with reasoning. To address this, we propose Additional Logic Training (ALT), which aims to enhance LLMs' reasoning capabilities by program-generated logical reasoning samples. We first establish principles for designing high-quality samples by integrating symbolic logic theory and previous empirical insights. Then, based on these principles, we construct a synthetic corpus named Formal Logic Deduction Diverse (FLD^{times 2}), comprising numerous samples of multi-step deduction with unknown facts, diverse reasoning rules, diverse linguistic expressions, and challenging distractors. Finally, we empirically show that ALT on FLD^{times2} substantially enhances the reasoning capabilities of state-of-the-art LLMs, including LLaMA-3.1-70B. Improvements include gains of up to 30 points on logical reasoning benchmarks, up to 10 points on math and coding benchmarks, and 5 points on the benchmark suite BBH.

We use automated theorem provers to significantly shorten a formal development in higher order set theory. The development includes many standard theorems such as the fundamental theorem of arithmetic and irrationality of square root of two. Higher order automated theorem provers are particularly useful here, since the underlying framework of higher order set theory coincides with the classical extensional higher order logic of (most) higher order automated theorem provers, so no significant translation or encoding is required. Additionally, many subgoals are first order and so first order automated provers often suffice. We compare the performance of different provers on the subgoals generated from the development. We also discuss possibilities for proof reconstruction, i.e., obtaining formal proof terms when an automated theorem prover claims to have proven the subgoal.

The demonstrated code-understanding capability of LLMs raises the question of whether they can be used for automated program verification, a task that often demands high-level abstract reasoning about program properties, which is challenging for verification tools. We propose a general methodology to combine the power of LLMs and automated reasoners for automated program verification. We formally describe this methodology as a set of derivation rules and prove its soundness. We instantiate the calculus as a sound automated verification procedure, which led to practical improvements on a set of synthetic and competition benchmarks.

Exploiting large language models (LLMs) to tackle deductive reasoning has garnered growing attention. It still remains highly challenging to achieve satisfactory results in complex deductive problems, characterized by plenty of premises (i.e., facts or rules) entailing intricate relationships among entities and requiring multi-hop reasoning. One intuitive solution is to decompose the original task into smaller sub-tasks, and then chain the multiple casual reasoning steps together in a forward (e.g., Selection-Inference) or backward (e.g., LAMBADA) direction. However, these techniques inevitably necessitate a large number of overall stages, leading to computationally expensive operations and a higher possibility of making misleading steps. In addition to stage-by-stage decomposition, we draw inspiration from another aspect of human problem-solving. Humans tend to distill the most relevant information and organize their thoughts systematically (e.g., creating mind maps), which assists them in answering questions or drawing conclusions precisely and quickly. In light of this, we propose a novel reasoning approach named Concise and Organized Perception (COP). COP carefully analyzes the given statements to efficiently identify the most pertinent information while eliminating redundancy. It then prompts the LLMs in a more organized form that adapts to the model's inference process. By perceiving concise and organized proofs, the deductive reasoning abilities of LLMs can be better elicited, and the risk of acquiring errors caused by excessive reasoning stages is mitigated. Furthermore, our approach can be combined with the aforementioned ones to further boost their performance. Extensive experimental results on three popular deductive benchmarks (i.e., ProofWriter, PrOntoQA and PrOntoQA-OOD) show that COP significantly outperforms previous state-of-the-art methods.

Remarkable progress has been made on automated reasoning with natural text, by using Language Models (LMs) and methods such as Chain-of-Thought and Selection-Inference. These techniques search for proofs in the forward direction from axioms to the conclusion, which suffers from a combinatorial explosion of the search space, and thus high failure rates for problems requiring longer chains of reasoning. The classical automated reasoning literature has shown that reasoning in the backward direction (i.e. from the intended conclusion to supporting axioms) is significantly more efficient at proof-finding. Importing this intuition into the LM setting, we develop a Backward Chaining algorithm, called LAMBADA, that decomposes reasoning into four sub-modules. These sub-modules are simply implemented by few-shot prompted LM inference. We show that LAMBADA achieves sizable accuracy boosts over state-of-the-art forward reasoning methods on challenging logical reasoning datasets, particularly when deep and accurate proof chains are required.

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

Software testing plays a critical role in ensuring that systems behave as intended. However, existing automated testing approaches struggle to match the capabilities of human engineers due to key limitations such as test locality, lack of general reliability, and business logic blindness. In this work, we propose a novel framework that leverages functional programming and type systems to translate Scala backend code into formal Lean representations. Our pipeline automatically generates theorems that specify the intended behavior of APIs and database operations, and uses LLM-based provers to verify them. When a theorem is proved, the corresponding logic is guaranteed to be correct and no further testing is needed. If the negation of a theorem is proved instead, it confirms a bug. In cases where neither can be proved, human intervention is required. We evaluate our method on realistic backend systems and find that it can formally verify over 50% of the test requirements, which suggests that half of a testing engineer's workload can be automated. Additionally, with an average cost of only $2.19 per API, LLM-based verification is significantly more cost-effective than manual testing and can be scaled easily through parallel execution. Our results indicate a promising direction for scalable, AI-powered software testing, with the potential to greatly improve engineering productivity as models continue to advance.

Recent advancements have significantly augmented the reasoning capabilities of Large Language Models (LLMs) through various methodologies, especially chain-of-thought (CoT) reasoning. However, previous methods fail to address reasoning errors in intermediate steps, leading to accumulative errors. In this paper, we propose Deductive Beam Search (DBS), which seamlessly integrates CoT and deductive reasoning with step-wise beam search for LLMs. Our approach deploys a verifier, verifying the deducibility of a reasoning step and its premises, thus alleviating the error accumulation. Furthermore, we introduce a scalable and labor-free data construction method to amplify our model's verification capabilities. Extensive experiments demonstrate that our approach significantly enhances the base performance of LLMs of various scales (7B, 13B, 70B, and ChatGPT) across 8 reasoning datasets from 3 diverse reasoning genres, including arithmetic, commonsense, and symbolic. Moreover, our analysis proves DBS's capability of detecting diverse and subtle reasoning errors and robustness on different model scales.

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

Humans prove theorems by relying on substantial high-level reasoning and problem-specific insights. Proof assistants offer a formalism that resembles human mathematical reasoning, representing theorems in higher-order logic and proofs as high-level tactics. However, human experts have to construct proofs manually by entering tactics into the proof assistant. In this paper, we study the problem of using machine learning to automate the interaction with proof assistants. We construct CoqGym, a large-scale dataset and learning environment containing 71K human-written proofs from 123 projects developed with the Coq proof assistant. We develop ASTactic, a deep learning-based model that generates tactics as programs in the form of abstract syntax trees (ASTs). Experiments show that ASTactic trained on CoqGym can generate effective tactics and can be used to prove new theorems not previously provable by automated methods. Code is available at https://github.com/princeton-vl/CoqGym.

Although contemporary large language models (LMs) demonstrate impressive question-answering capabilities, their answers are typically the product of a single call to the model. This entails an unwelcome degree of opacity and compromises performance, especially on problems that are inherently multi-step. To address these limitations, we show how LMs can be made to perform faithful multi-step reasoning via a process whose causal structure mirrors the underlying logical structure of the problem. Our approach works by chaining together reasoning steps, where each step results from calls to two fine-tuned LMs, one for selection and one for inference, to produce a valid reasoning trace. Our method carries out a beam search through the space of reasoning traces to improve reasoning quality. We demonstrate the effectiveness of our model on multi-step logical deduction and scientific question-answering, showing that it outperforms baselines on final answer accuracy, and generates humanly interpretable reasoning traces whose validity can be checked by the user.

Automated theorem proving (ATP) has become an appealing domain for exploring the reasoning ability of the recent successful generative language models. However, current ATP benchmarks mainly focus on symbolic inference, but rarely involve the understanding of complex number combination reasoning. In this work, we propose TRIGO, an ATP benchmark that not only requires a model to reduce a trigonometric expression with step-by-step proofs but also evaluates a generative LM's reasoning ability on formulas and its capability to manipulate, group, and factor number terms. We gather trigonometric expressions and their reduced forms from the web, annotate the simplification process manually, and translate it into the Lean formal language system. We then automatically generate additional examples from the annotated samples to expand the dataset. Furthermore, we develop an automatic generator based on Lean-Gym to create dataset splits of varying difficulties and distributions in order to thoroughly analyze the model's generalization ability. Our extensive experiments show our proposed TRIGO poses a new challenge for advanced generative LM's including GPT-4 which is pre-trained on a considerable amount of open-source formal theorem-proving language data, and provide a new tool to study the generative LM's ability on both formal and mathematical reasoning.

Proof automation is crucial to large-scale formal mathematics and software/hardware verification projects in ITPs. Sophisticated tools called hammers have been developed to provide general-purpose proof automation in ITPs such as Coq and Isabelle, leveraging the power of ATPs. An important component of a hammer is the translation algorithm from the ITP's logical system to the ATP's logical system. In this paper, we propose a novel translation algorithm for ITPs based on dependent type theory. The algorithm is implemented in Lean 4 under the name Lean-auto. When combined with ATPs, Lean-auto provides general-purpose, ATP-based proof automation in Lean 4 for the first time. Soundness of the main translation procedure is guaranteed, and experimental results suggest that our algorithm is sufficiently complete to automate the proof of many problems that arise in practical uses of Lean 4. We also find that Lean-auto solves more problems than existing tools on Lean 4's math library Mathlib4.

Robustness of reasoning remains a significant challenge for large language models, and addressing it is essential for the practical applicability of AI-driven reasoning systems. We introduce Semantic Self-Verification (SSV), a novel approach that addresses the key challenge in combining language models with the rigor of logical solvers: to accurately formulate the reasoning problem from natural language to the formal language of the solver. SSV uses a consistency-based approach to produce strong abstract formalizations of problems using concrete instantiations that are generated by the model and verified by the solver. In addition to significantly advancing the overall reasoning accuracy over the state-of-the-art, a key novelty that this approach presents is a feature of verification that has near-perfect precision over a significant coverage of cases, as we demonstrate on open reasoning benchmarks. We propose such *near-certain reasoning* as a new approach to reduce the need for manual verification in many cases, taking us closer to more dependable and autonomous AI reasoning systems.

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

Deductive reasoning is the process of deriving conclusions strictly from the given premises, without relying on external knowledge. We define honesty in this setting as a model's ability to respond only when the conclusion is logically entailed by the premises, and to abstain otherwise. However, current language models often fail to reason honestly, producing unwarranted answers when the input is insufficient. To study this challenge, we formulate honest deductive reasoning as multi-step tasks where models must either derive the correct conclusion or abstain. We curate two datasets from graph structures, one for linear algebra and one for logical inference, and introduce unanswerable cases by randomly perturbing an edge in half of the instances. We find that prompting and existing training methods, including GRPO with or without supervised fine-tuning initialization, struggle on these tasks. In particular, GRPO optimize only for final task outcomes, leaving models vulnerable to collapse when negative rewards dominate early training. To address this, we propose ACNCHOR, a reinforcement learning method that injects ground truth trajectories into rollouts, preventing early training collapse. Our results demonstrate that this method stabilizes learning and significantly improves the overall reasoning performance, underscoring the importance of training dynamics for enabling honest deductive reasoning in language models.

Large language models (LLMs) have shown remarkable reasoning capabilities given chain-of-thought prompts (examples with intermediate reasoning steps). Existing benchmarks measure reasoning ability indirectly, by evaluating accuracy on downstream tasks such as mathematical reasoning. However, it is unclear how these models obtain the answers and whether they rely on simple heuristics rather than the generated chain-of-thought. To enable systematic exploration of the reasoning ability of LLMs, we present a new synthetic question-answering dataset called PrOntoQA, where each example is generated from a synthetic world model represented in first-order logic. This allows us to parse the generated chain-of-thought into symbolic proofs for formal analysis. Our analysis on InstructGPT and GPT-3 shows that LLMs are quite capable of making correct individual deduction steps, and so are generally capable of reasoning, even in fictional contexts. However, they have difficulty with proof planning: When multiple valid deduction steps are available, they are not able to systematically explore the different options.

Logical reasoning remains a challenge for natural language processing, but it can be improved by training language models to mimic theorem provers on procedurally generated problems. Previous work used domain-specific proof generation algorithms, which biases reasoning toward specific proof traces and limits auditability and extensibility. We present a simpler and more general declarative framework with flexible context-sensitive rules binding multiple languages (specifically, simplified English and the TPTP theorem-proving language). We construct first-order logic problems by selecting up to 32 premises and one hypothesis. We demonstrate that using semantic constraints during generation and careful English verbalization of predicates enhances logical reasoning without hurting natural English tasks. We use relatively small DeBERTa-v3 models to achieve state-of-the-art accuracy on the FOLIO human-authored logic dataset, surpassing GPT-4 in accuracy with or without an external solver by 12%.

Premise selection is a fundamental problem of automated theorem proving. Previous works often use intricate symbolic methods, rely on domain knowledge, and require significant engineering effort to solve this task. In this work, we show that Magnushammer, a neural transformer-based approach, can outperform traditional symbolic systems by a large margin. Tested on the PISA benchmark, Magnushammer achieves 59.5% proof rate compared to a 38.3% proof rate of Sledgehammer, the most mature and popular symbolic-based solver. Furthermore, by combining Magnushammer with a neural formal prover based on a language model, we significantly improve the previous state-of-the-art proof rate from 57.0% to 71.0%.

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.

Efficient and accurate autoformalization methods, which leverage large-scale datasets of extensive natural language mathematical problems to construct formal language datasets, are key to advancing formal mathematical reasoning. In this paper, we propose an autoformalization pipeline based on large language models with error feedback, achieving a fully automatic and training-free formalization approach. Using this pipeline, we curate an Olympiad-level dataset aligning natural language problems with Lean formalizations. The dataset comprises 3,922 mathematical problems in natural language and 9,787 in Lean, of which 64.46% were assessed as at least above-average quality, making it suitable as a benchmark for automated theorem provers. Additionally, we investigate the formalization and reasoning capabilities of various LLMs and empirically demonstrate that few-shot learning, error feedback, and increasing sampling numbers enhance the autoformalization process. Experiments of three automated theorem provers on the \dataset\ dataset also highlight its challenging nature and its value as a benchmark for formal reasoning tasks.

Large language models (LLMs) can perform complex reasoning in few- and zero-shot settings by generating intermediate chain of thought (CoT) reasoning steps. Further, each reasoning step can rely on external tools to support computation beyond the core LLM capabilities (e.g. search/running code). Prior work on CoT prompting and tool use typically requires hand-crafting task-specific demonstrations and carefully scripted interleaving of model generations with tool use. We introduce Automatic Reasoning and Tool-use (ART), a framework that uses frozen LLMs to automatically generate intermediate reasoning steps as a program. Given a new task to solve, ART selects demonstrations of multi-step reasoning and tool use from a task library. At test time, ART seamlessly pauses generation whenever external tools are called, and integrates their output before resuming generation. ART achieves a substantial improvement over few-shot prompting and automatic CoT on unseen tasks in the BigBench and MMLU benchmarks, and matches performance of hand-crafted CoT prompts on a majority of these tasks. ART is also extensible, and makes it easy for humans to improve performance by correcting errors in task-specific programs or incorporating new tools, which we demonstrate by drastically improving performance on select tasks with minimal human intervention.

Recent progress in large language models (LLMs) has shown promise in formal theorem proving, yet existing benchmarks remain limited to isolated, static proof tasks, failing to capture the iterative, engineering-intensive workflows of real-world formal mathematics libraries. Motivated by analogous advances in software engineering, we introduce the paradigm of Automated Proof Engineering (APE), which aims to automate proof engineering tasks such as feature addition, proof refactoring, and bug fixing using LLMs. To facilitate research in this direction, we present APE-Bench I, the first realistic benchmark built from real-world commit histories of Mathlib4, featuring diverse file-level tasks described in natural language and verified via a hybrid approach combining the Lean compiler and LLM-as-a-Judge. We further develop Eleanstic, a scalable parallel verification infrastructure optimized for proof checking across multiple versions of Mathlib. Empirical results on state-of-the-art LLMs demonstrate strong performance on localized edits but substantial degradation on handling complex proof engineering. This work lays the foundation for developing agentic workflows in proof engineering, with future benchmarks targeting multi-file coordination, project-scale verification, and autonomous agents capable of planning, editing, and repairing formal libraries.

We explore whether Large Language Models (LLMs) are capable of logical reasoning with distorted facts, which we call Deduction under Perturbed Evidence (DUPE). DUPE presents a unique challenge to LLMs since they typically rely on their parameters, which encode mostly accurate information, to reason and make inferences. However, in DUPE, LLMs must reason over manipulated or falsified evidence present in their prompts, which can result in false conclusions that are valid only under the manipulated evidence. Our goal with DUPE is to determine whether LLMs can arrive at these false conclusions and identify whether the dominant factor influencing the deduction process is the encoded data in the parameters or the manipulated evidence in the prompts. To evaluate the DUPE capabilities of LLMs, we create a DUPEd version of the StrategyQA dataset, where facts are manipulated to reverse the answer to the question. Our findings show that even the most advanced GPT models struggle to reason on manipulated facts - showcasing poor DUPE skills - with accuracy dropping by 45% compared to the original dataset. We also investigate prompt settings inspired from student simulation models, which mitigate the accuracy drop to some extent. Our findings have practical implications for understanding the performance of LLMs in real-world applications such as student simulation models that involve reasoning over inaccurate information.

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

We propose an online training procedure for a transformer-based automated theorem prover. Our approach leverages a new search algorithm, HyperTree Proof Search (HTPS), inspired by the recent success of AlphaZero. Our model learns from previous proof searches through online training, allowing it to generalize to domains far from the training distribution. We report detailed ablations of our pipeline's main components by studying performance on three environments of increasing complexity. In particular, we show that with HTPS alone, a model trained on annotated proofs manages to prove 65.4% of a held-out set of Metamath theorems, significantly outperforming the previous state of the art of 56.5% by GPT-f. Online training on these unproved theorems increases accuracy to 82.6%. With a similar computational budget, we improve the state of the art on the Lean-based miniF2F-curriculum dataset from 31% to 42% proving accuracy.

In this work, we propose a novel method named Automated Process-Supervised Verifier (\textsc{AutoPSV}) to enhance the reasoning capabilities of large language models (LLMs) by automatically annotating the reasoning steps. AutoPSV begins by training a verification model on the correctness of final answers, enabling it to generate automatic process annotations. This verification model assigns a confidence score to each reasoning step, indicating the probability of arriving at the correct final answer from that point onward. We detect relative changes in the verification's confidence scores across reasoning steps to automatically annotate the reasoning process, enabling error detection even in scenarios where ground truth answers are unavailable. This alleviates the need for numerous manual annotations or the high computational costs associated with model-induced annotation approaches. We experimentally validate that the step-level confidence changes learned by the verification model trained on the final answer correctness can effectively identify errors in the reasoning steps. We demonstrate that the verification model, when trained on process annotations generated by AutoPSV, exhibits improved performance in selecting correct answers from multiple LLM-generated outputs. Notably, we achieve substantial improvements across five datasets in mathematics and commonsense reasoning. The source code of AutoPSV is available at https://github.com/rookie-joe/AutoPSV.

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.

We explore the application of transformer-based language models to automated theorem proving. This work is motivated by the possibility that a major limitation of automated theorem provers compared to humans -- the generation of original mathematical terms -- might be addressable via generation from language models. We present an automated prover and proof assistant, GPT-f, for the Metamath formalization language, and analyze its performance. GPT-f found new short proofs that were accepted into the main Metamath library, which is to our knowledge, the first time a deep-learning based system has contributed proofs that were adopted by a formal mathematics community.

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.

Transformers have been shown to emulate logical deduction over natural language theories (logical rules expressed in natural language), reliably assigning true/false labels to candidate implications. However, their ability to generate implications of a theory has not yet been demonstrated, and methods for reconstructing proofs of answers are imperfect. In this work we show that a generative model, called ProofWriter, can reliably generate both implications of a theory and the natural language proof(s) that support them. In particular, iterating a 1-step implication generator results in proofs that are highly reliable, and represent actual model decisions (rather than post-hoc rationalizations). On the RuleTaker dataset, the accuracy of ProofWriter's proofs exceed previous methods by +9% absolute, and in a way that generalizes to proof depths unseen in training and on out-of-domain problems. We also show that generative techniques can perform a type of abduction with high precision: Given a theory and an unprovable conclusion, identify a missing fact that allows the conclusion to be proved, along with a proof. These results significantly improve the viability of neural methods for systematically reasoning over natural language.

Autoformalization is the process of automatically translating from natural language mathematics to formal specifications and proofs. A successful autoformalization system could advance the fields of formal verification, program synthesis, and artificial intelligence. While the long-term goal of autoformalization seemed elusive for a long time, we show large language models provide new prospects towards this goal. We make the surprising observation that LLMs can correctly translate a significant portion (25.3%) of mathematical competition problems perfectly to formal specifications in Isabelle/HOL. We demonstrate the usefulness of this process by improving a previously introduced neural theorem prover via training on these autoformalized theorems. Our methodology results in a new state-of-the-art result on the MiniF2F theorem proving benchmark, improving the proof rate from 29.6% to 35.2%.

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.

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.

Logical reasoning, i.e., deductively inferring the truth value of a conclusion from a set of premises, is an important task for artificial intelligence with wide potential impacts on science, mathematics, and society. While many prompting-based strategies have been proposed to enable Large Language Models (LLMs) to do such reasoning more effectively, they still appear unsatisfactory, often failing in subtle and unpredictable ways. In this work, we investigate the validity of instead reformulating such tasks as modular neurosymbolic programming, which we call LINC: Logical Inference via Neurosymbolic Computation. In LINC, the LLM acts as a semantic parser, translating premises and conclusions from natural language to expressions in first-order logic. These expressions are then offloaded to an external theorem prover, which symbolically performs deductive inference. Leveraging this approach, we observe significant performance gains on FOLIO and a balanced subset of ProofWriter for three different models in nearly all experimental conditions we evaluate. On ProofWriter, augmenting the comparatively small open-source StarCoder+ (15.5B parameters) with LINC even outperforms GPT-3.5 and GPT-4 with Chain-of-Thought (CoT) prompting by an absolute 38% and 10%, respectively. When used with GPT-4, LINC scores 26% higher than CoT on ProofWriter while performing comparatively on FOLIO. Further analysis reveals that although both methods on average succeed roughly equally often on this dataset, they exhibit distinct and complementary failure modes. We thus provide promising evidence for how logical reasoning over natural language can be tackled through jointly leveraging LLMs alongside symbolic provers. All corresponding code is publicly available at https://github.com/benlipkin/linc

Translating natural language mathematical statements into formal, executable code is a fundamental challenge in automated theorem proving. While prior work has focused on generation and compilation success, little attention has been paid to the critic phase-the evaluation of whether generated formalizations truly capture the semantic intent of the original problem. In this paper, we introduce CriticLean, a novel critic-guided reinforcement learning framework that elevates the role of the critic from a passive validator to an active learning component. Specifically, first, we propose the CriticLeanGPT, trained via supervised fine-tuning and reinforcement learning, to rigorously assess the semantic fidelity of Lean 4 formalizations. Then, we introduce CriticLeanBench, a benchmark designed to measure models' ability to distinguish semantically correct from incorrect formalizations, and demonstrate that our trained CriticLeanGPT models can significantly outperform strong open- and closed-source baselines. Building on the CriticLean framework, we construct FineLeanCorpus, a dataset comprising over 285K problems that exhibits rich domain diversity, broad difficulty coverage, and high correctness based on human evaluation. Overall, our findings highlight that optimizing the critic phase is essential for producing reliable formalizations, and we hope our CriticLean will provide valuable insights for future advances in formal mathematical reasoning.

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.

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

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.

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.

Interactive theorem provers (ITPs) require manual formalization, which is labor-intensive and demands expert knowledge. While automated formalization offers a potential solution, it faces two major challenges: model hallucination (e.g., undefined predicates, symbol misuse, and version incompatibility) and the semantic gap caused by ambiguous or missing premises in natural language descriptions. To address these issues, we propose CRAMF, a Concept-driven Retrieval-Augmented Mathematical Formalization framework. CRAMF enhances LLM-based autoformalization by retrieving formal definitions of core mathematical concepts, providing contextual grounding during code generation. However, applying retrieval-augmented generation (RAG) in this setting is non-trivial due to the lack of structured knowledge bases, the polymorphic nature of mathematical concepts, and the high precision required in formal retrieval. We introduce a framework for automatically constructing a concept-definition knowledge base from Mathlib4, the standard mathematical library for the Lean 4 theorem prover, indexing over 26,000 formal definitions and 1,000+ core mathematical concepts. To address conceptual polymorphism, we propose contextual query augmentation with domain- and application-level signals. In addition, we design a dual-channel hybrid retrieval strategy with reranking to ensure accurate and relevant definition retrieval. Experiments on miniF2F, ProofNet, and our newly proposed AdvancedMath benchmark show that CRAMF can be seamlessly integrated into LLM-based autoformalizers, yielding consistent improvements in translation accuracy, achieving up to 62.1% and an average of 29.9% relative improvement.

We present Generative Logic (GL), a deterministic architecture that begins from user-supplied axiomatic definitions -- written in a minimalist Mathematical Programming Language (MPL) -- and systematically explores their deductive neighborhood. Definitions are compiled into a distributed grid of simple Logic Blocks (LBs) that exchange messages; any time several expressions unify under an inference rule, a new fact is emitted with full provenance to its sources, yielding replayable, auditable proof graphs. A prototype software implementation instantiates the workflow on first-order Peano arithmetic. Starting only from the Peano axioms, GL enumerates candidate implications, applies normalization and type filters, and automatically reconstructs machine-checkable proofs of foundational arithmetic laws including associativity and commutativity of addition, associativity and commutativity of multiplication, and distributivity. Generated proofs export to navigable HTML so that every inference step can be inspected independently. We outline a hardware-software co-design path toward massively parallel realizations and describe prospective integration with probabilistic models (e.g., Large Language Models (LLMs)) for autoformalization and conjecture seeding. The Python and MPL code to reproduce the Peano experiments, along with the full HTML proof graphs, are available in the project's GitHub repository at https://github.com/Generative-Logic/GL/tree/35a111ea9ba53afe051703d6050be0c3923e9724 and are permanently archived at https://doi.org/10.5281/zenodo.16408441. We invite community feedback and collaboration.

Claim verification is essential in combating misinformation, and large language models (LLMs) have recently emerged in this area as powerful tools for assessing the veracity of claims using external knowledge. Existing LLM-based methods for claim verification typically adopt a Decompose-Then-Verify paradigm, which involves decomposing complex claims into several independent sub-claims and verifying each sub-claim separately. However, this paradigm often introduces errors during the claim decomposition process. To mitigate these errors, we propose to develop the Chain-of-Thought (CoT)-Verify paradigm, which leverages LLM reasoning methods to generate CoT-verification paths for the original complex claim without requiring decompositions into sub-claims and separate verification stages. The CoT-Verify paradigm allows us to propose a natural fine-tuning method called Reasoning-CV to enhance the verification capabilities in LLMs. Reasoning-CV includes a supervised fine-tuning (SFT) stage and a self-improvement direct preference optimization (DPO) stage. Utilizing only an 8B pre-trained LLM, Reasoning-CV demonstrates superior knowledge-assisted claim verification performances compared to existing Decompose-Then-Verify methods, as well as powerful black-box LLMs such as GPT-4o+CoT and o1-preview. Our code is available.

Autoformalization, which translates natural language mathematics into machine-verifiable formal statements, is critical for using formal mathematical reasoning to solve math problems stated in natural language. While Large Language Models can generate syntactically correct formal statements, they often fail to preserve the original problem's semantic intent. This limitation arises from the LLM approaches' treating autoformalization as a simplistic translation task which lacks mechanisms for self-reflection and iterative refinement that human experts naturally employ. To address these issues, we propose ReForm, a Reflective Autoformalization method that tightly integrates semantic consistency evaluation into the autoformalization process. This enables the model to iteratively generate formal statements, assess its semantic fidelity, and self-correct identified errors through progressive refinement. To effectively train this reflective model, we introduce Prospective Bounded Sequence Optimization (PBSO), which employs different rewards at different sequence positions to ensure that the model develops both accurate autoformalization and correct semantic validations, preventing superficial critiques that would undermine the purpose of reflection. Extensive experiments across four autoformalization benchmarks demonstrate that ReForm achieves an average improvement of 17.2 percentage points over the strongest baselines. To further ensure evaluation reliability, we introduce ConsistencyCheck, a benchmark of 859 expert-annotated items that not only validates LLMs as judges but also reveals that autoformalization is inherently difficult: even human experts produce semantic errors in up to 38.5% of cases.

When prompted with a few examples and intermediate steps, large language models (LLMs) have demonstrated impressive performance in various reasoning tasks. However, prompting methods that rely on implicit knowledge in an LLM often generate incorrect answers when the implicit knowledge is wrong or inconsistent with the task. To tackle this problem, we present Hypotheses-to-Theories (HtT), a framework that learns a rule library for reasoning with LLMs. HtT contains two stages, an induction stage and a deduction stage. In the induction stage, an LLM is first asked to generate and verify rules over a set of training examples. Rules that appear and lead to correct answers sufficiently often are collected to form a rule library. In the deduction stage, the LLM is then prompted to employ the learned rule library to perform reasoning to answer test questions. Experiments on relational reasoning, numerical reasoning and concept learning problems show that HtT improves existing prompting methods, with an absolute gain of 10-30% in accuracy. The learned rules are also transferable to different models and to different forms of the same problem.

As a vital stage of automated rule checking (ARC), rule interpretation of regulatory texts requires considerable effort. However, interpreting regulatory clauses with implicit properties or complex computational logic is still challenging due to the lack of domain knowledge and limited expressibility of conventional logic representations. Thus, LLM-FuncMapper, an approach to identifying predefined functions needed to interpret various regulatory clauses based on the large language model (LLM), is proposed. First, by systematically analysis of building codes, a series of atomic functions are defined to capture shared computational logics of implicit properties and complex constraints, creating a database of common blocks for interpreting regulatory clauses. Then, a prompt template with the chain of thought is developed and further enhanced with a classification-based tuning strategy, to enable common LLMs for effective function identification. Finally, the proposed approach is validated with statistical analysis, experiments, and proof of concept. Statistical analysis reveals a long-tail distribution and high expressibility of the developed function database, with which almost 100% of computer-processible clauses can be interpreted and represented as computer-executable codes. Experiments show that LLM-FuncMapper achieve promising results in identifying relevant predefined functions for rule interpretation. Further proof of concept in automated rule interpretation also demonstrates the possibility of LLM-FuncMapper in interpreting complex regulatory clauses. To the best of our knowledge, this study is the first attempt to introduce LLM for understanding and interpreting complex regulatory clauses, which may shed light on further adoption of LLM in the construction domain.

Legal reasoning requires both precise interpretation of statutory language and consistent application of complex rules, presenting significant challenges for AI systems. This paper introduces a modular multi-agent framework that decomposes legal reasoning into distinct knowledge acquisition and application stages. In the first stage, specialized agents extract legal concepts and formalize rules to create verifiable intermediate representations of statutes. The second stage applies this knowledge to specific cases through three steps: analyzing queries to map case facts onto the ontology schema, performing symbolic inference to derive logically entailed conclusions, and generating final answers using a programmatic implementation that operationalizes the ontological knowledge. This bridging of natural language understanding with symbolic reasoning provides explicit and verifiable inspection points, significantly enhancing transparency compared to end-to-end approaches. Evaluation on statutory tax calculation tasks demonstrates substantial improvements, with foundational models achieving 76.4\% accuracy compared to 18.8\% baseline performance, effectively narrowing the performance gap between reasoning and foundational models. These findings suggest that modular architectures with formalized knowledge representations can make sophisticated legal reasoning more accessible through computationally efficient models while enhancing consistency and explainability in AI legal reasoning, establishing a foundation for future research into more transparent, trustworthy, and effective AI systems for legal domain.

Transformers have been shown to be able to perform deductive reasoning on a logical rulebase containing rules and statements written in English natural language. While the progress is promising, it is currently unclear if these models indeed perform logical reasoning by understanding the underlying logical semantics in the language. To this end, we propose RobustLR, a suite of evaluation datasets that evaluate the robustness of these models to minimal logical edits in rulebases and some standard logical equivalence conditions. In our experiments with RoBERTa and T5, we find that the models trained in prior works do not perform consistently on the different perturbations in RobustLR, thus showing that the models are not robust to the proposed logical perturbations. Further, we find that the models find it especially hard to learn logical negation and disjunction operators. Overall, using our evaluation sets, we demonstrate some shortcomings of the deductive reasoning-based language models, which can eventually help towards designing better models for logical reasoning over natural language. All the datasets and code base have been made publicly available.

The demand for synthetic data in mathematical reasoning has increased due to its potential to enhance the mathematical capabilities of large language models (LLMs). However, ensuring the validity of intermediate reasoning steps remains a significant challenge, affecting data quality. While formal verification via theorem provers effectively validates LLM reasoning, the autoformalisation of mathematical proofs remains error-prone. In response, we introduce iterative autoformalisation, an approach that iteratively refines theorem prover formalisation to mitigate errors, thereby increasing the execution rate on the Lean prover from 60% to 87%. Building upon that, we introduce Theorem Prover as a Judge (TP-as-a-Judge), a method that employs theorem prover formalisation to rigorously assess LLM intermediate reasoning, effectively integrating autoformalisation with synthetic data generation. Finally, we present Reinforcement Learning from Theorem Prover Feedback (RLTPF), a framework that replaces human annotation with theorem prover feedback in Reinforcement Learning from Human Feedback (RLHF). Across multiple LLMs, applying TP-as-a-Judge and RLTPF improves benchmarks with only 3,508 samples, achieving 5.56% accuracy gain on Mistral-7B for MultiArith, 6.00% on Llama-2-7B for SVAMP, and 3.55% on Llama-3.1-8B for AQUA.

Translating human-written mathematical theorems and proofs from natural language (NL) into formal languages (FLs) like Lean 4 has long been a significant challenge for AI. Most state-of-the-art methods address this separately, first translating theorems and then generating proofs, creating a fundamental disconnect vis-a-vis true proof auto-formalization. This two-step process and its limitations were evident even in AlphaProof's silver-medal performance at the 2024 IMO, where problem statements needed manual translation before automated proof synthesis. We present ProofBridge, a unified framework for automatically translating entire NL theorems and proofs into Lean 4. At its core is a joint embedding model that aligns NL and FL (NL-FL) theorem-proof pairs in a shared semantic space, enabling cross-modal retrieval of semantically relevant FL examples to guide translation. Our training ensures that NL-FL theorems (and their proofs) are mapped close together in this space if and only if the NL-FL pairs are semantically equivalent. ProofBridge integrates retrieval-augmented fine-tuning with iterative proof repair, leveraging Lean's type checker and semantic equivalence feedback to ensure both syntactic correctness and semantic fidelity. Experiments show substantial improvements in proof auto-formalization over strong baselines (including GPT-5, Gemini-2.5, Kimina-Prover, DeepSeek-Prover), with our retrieval-augmented approach yielding significant gains in semantic correctness (SC, via proving bi-directional equivalence) and type correctness (TC, via type-checking theorem+proof) across pass@k metrics on miniF2F-Test-PF, a dataset we curated. In particular, ProofBridge improves cross-modal retrieval quality by up to 3.28x Recall@1 over all-MiniLM-L6-v2, and achieves +31.14% SC and +1.64% TC (pass@32) compared to the baseline Kimina-Prover-RL-1.7B.

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

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

We present GenesisGeo, an automated theorem prover in Euclidean geometry. We have open-sourced a large-scale geometry dataset of 21.8 million geometric problems, over 3 million of which contain auxiliary constructions. Specially, we significantly accelerate the symbolic deduction engine DDARN by 120x through theorem matching, combined with a C++ implementation of its core components. Furthermore, we build our neuro-symbolic prover, GenesisGeo, upon Qwen3-0.6B-Base, which solves 24 of 30 problems (IMO silver medal level) in the IMO-AG-30 benchmark using a single model, and achieves 26 problems (IMO gold medal level) with a dual-model ensemble.

We study the effectiveness of neural sequence models for premise selection in automated theorem proving, one of the main bottlenecks in the formalization of mathematics. We propose a two stage approach for this task that yields good results for the premise selection task on the Mizar corpus while avoiding the hand-engineered features of existing state-of-the-art models. To our knowledge, this is the first time deep learning has been applied to theorem proving on a large scale.

Formal proofs are challenging to write even for experienced experts. Recent progress in Neural Theorem Proving (NTP) shows promise in expediting this process. However, the formal corpora available on the Internet are limited compared to the general text, posing a significant data scarcity challenge for NTP. To address this issue, this work proposes Alchemy, a general framework for data synthesis that constructs formal theorems through symbolic mutation. Specifically, for each candidate theorem in Mathlib, we identify all invocable theorems that can be used to rewrite or apply to it. Subsequently, we mutate the candidate theorem by replacing the corresponding term in the statement with its equivalent form or antecedent. As a result, our method increases the number of theorems in Mathlib by an order of magnitude, from 110k to 6M. Furthermore, we perform continual pretraining and supervised finetuning on this augmented corpus for large language models. Experimental results demonstrate the effectiveness of our approach, achieving a 5% absolute performance improvement on Leandojo benchmark. Additionally, our synthetic data achieve a 2.5% absolute performance gain on the out-of-distribution miniF2F benchmark. To provide further insights, we conduct a comprehensive analysis of synthetic data composition and the training paradigm, offering valuable guidance for developing a strong theorem prover.

Current long chain-of-thought (long-CoT) models excel at mathematical reasoning but rely on slow and error-prone natural language traces. Tool-augmented agents address arithmetic via code execution, but often falter on complex logical tasks. We introduce a fine-tuning framework, DualDistill, that distills complementary reasoning strategies from multiple teachers into a unified student model. Using this approach, we train Agentic-R1, which dynamically selects the optimal strategy for each query, invoking tools for arithmetic and algorithmic problems, and using text-based reasoning for abstract ones. Our method improves accuracy across a range of tasks, including both computation-intensive and standard benchmarks, demonstrating the effectiveness of multi-strategy distillation in achieving robust and efficient reasoning. Our project is available at https://github.com/StigLidu/DualDistill

Inductive reasoning is a core problem-solving capacity: humans can identify underlying principles from a few examples, which can then be robustly generalized to novel scenarios. Recent work has evaluated large language models (LLMs) on inductive reasoning tasks by directly prompting them yielding "in context learning." This can work well for straightforward inductive tasks, but performs very poorly on more complex tasks such as the Abstraction and Reasoning Corpus (ARC). In this work, we propose to improve the inductive reasoning ability of LLMs by generating explicit hypotheses at multiple levels of abstraction: we prompt the LLM to propose multiple abstract hypotheses about the problem, in natural language, then implement the natural language hypotheses as concrete Python programs. These programs can be directly verified by running on the observed examples and generalized to novel inputs. Because of the prohibitive cost of generation with state-of-the-art LLMs, we consider a middle step to filter the set of hypotheses that will be implemented into programs: we either ask the LLM to summarize into a smaller set of hypotheses, or ask human annotators to select a subset of the hypotheses. We verify our pipeline's effectiveness on the ARC visual inductive reasoning benchmark, its variant 1D-ARC, and string transformation dataset SyGuS. On a random 40-problem subset of ARC, our automated pipeline using LLM summaries achieves 27.5% accuracy, significantly outperforming the direct prompting baseline (accuracy of 12.5%). With the minimal human input of selecting from LLM-generated candidates, the performance is boosted to 37.5%. (And we argue this is a lower bound on the performance of our approach without filtering.) Our ablation studies show that abstract hypothesis generation and concrete program representations are both beneficial for LLMs to perform inductive reasoning tasks.

Researchers have made significant progress in automating the software development process in the past decades. Recent progress in Large Language Models (LLMs) has significantly impacted the development process, where developers can use LLM-based programming assistants to achieve automated coding. Nevertheless, software engineering involves the process of program improvement apart from coding, specifically to enable software maintenance (e.g. bug fixing) and software evolution (e.g. feature additions). In this paper, we propose an automated approach for solving GitHub issues to autonomously achieve program improvement. In our approach called AutoCodeRover, LLMs are combined with sophisticated code search capabilities, ultimately leading to a program modification or patch. In contrast to recent LLM agent approaches from AI researchers and practitioners, our outlook is more software engineering oriented. We work on a program representation (abstract syntax tree) as opposed to viewing a software project as a mere collection of files. Our code search exploits the program structure in the form of classes/methods to enhance LLM's understanding of the issue's root cause, and effectively retrieve a context via iterative search. The use of spectrum-based fault localization using tests, further sharpens the context, as long as a test-suite is available. Experiments on SWE-bench-lite (300 real-life GitHub issues) show increased efficacy in solving GitHub issues (19% on SWE-bench-lite), which is higher than the efficacy of the recently reported SWE-agent. In addition, AutoCodeRover achieved this efficacy with significantly lower cost (on average, $0.43 USD), compared to other baselines. We posit that our workflow enables autonomous software engineering, where, in future, auto-generated code from LLMs can be autonomously improved.

Autoformalization aims to translate natural-language mathematical statements into a formal language. While LLMs have accelerated progress in this area, existing methods still suffer from low accuracy. We identify two key abilities for effective autoformalization: comprehensive mastery of formal-language domain knowledge, and reasoning capability of natural language problem understanding and informal-formal alignment. Without the former, a model cannot identify the correct formal objects; without the latter, it struggles to interpret real-world contexts and map them precisely into formal expressions. To address these gaps, we introduce ThinkingF, a data synthesis and training pipeline that improves both abilities. First, we construct two datasets: one by distilling and selecting large-scale examples rich in formal knowledge, and another by generating informal-to-formal reasoning trajectories guided by expert-designed templates. We then apply SFT and RLVR with these datasets to further fuse and refine the two abilities. The resulting 7B and 32B models exhibit both comprehensive formal knowledge and strong informal-to-formal reasoning. Notably, StepFun-Formalizer-32B achieves SOTA BEq@1 scores of 40.5% on FormalMATH-Lite and 26.7% on ProverBench, surpassing all prior general-purpose and specialized models.

Writing competitive programming problems is exacting. Authors must: set constraints, input distributions, and edge cases that rule out shortcuts; target specific algorithms (e.g., max-flow, dynamic programming, data structures); and calibrate complexity beyond the reach of most competitors. We argue that this makes for an ideal test of general large language model capabilities and study whether they can do this reliably. We introduce AutoCode, which uses multiple rounds of validation to yield competition-grade problem statements and test cases. On held-out problems, AutoCode test suites approach 99% consistency with official judgments, a significant improvement over current state-of-the-art methods like HardTests, which achieve less than 81%. Furthermore, starting with a random seed problem, AutoCode can create novel variants with reference and brute-force solutions. By cross-verifying these generated solutions against test cases, we can further filter out malformed problems. Our system ensures high correctness, as verified by human experts. AutoCode successfully produces novel problems judged by Grandmaster-level (top 0.3%) competitive programmers to be of contest quality.

We present semantic correctness proofs of Automatic Differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming, based on diffeological spaces. We show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is, in essence, a logical relations argument. Finally, we sketch how the analysis extends to other AD methods by considering a continuation-based method.

This paper investigates the mathematical reasoning capabilities of large language models (LLMs) using 50 newly constructed high-school-level word problems. Unlike prior studies that focus solely on answer correctness, we rigorously analyze both final answers and solution steps to identify reasoning failures. Evaluating eight state-of-the-art models - including Mixtral, Llama, Gemini, GPT-4o, and OpenAI's o1 variants - we find that while newer models (e.g., o3-mini, deepseek-r1) achieve higher accuracy, all models exhibit errors in spatial reasoning, strategic planning, and arithmetic, sometimes producing correct answers through flawed logic. Common failure modes include unwarranted assumptions, over-reliance on numerical patterns, and difficulty translating physical intuition into mathematical steps. Manual analysis reveals that models struggle with problems requiring multi-step deduction or real-world knowledge, despite possessing broad mathematical knowledge. Our results underscore the importance of evaluating reasoning processes, not just answers, and caution against overestimating LLMs' problem-solving proficiency. The study highlights persistent gaps in LLMs' generalization abilities, emphasizing the need for targeted improvements in structured reasoning and constraint handling.

We present semantic correctness proofs of automatic differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming, based on diffeological spaces. We show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is, in essence, a logical relations argument. Throughout, we show how the analysis extends to AD methods for computing higher order derivatives using a Taylor approximation.

Logical reasoning is a critical benchmark for evaluating the capabilities of large language models (LLMs), as it reflects their ability to derive valid conclusions from given premises. While the combination of test-time scaling with dedicated outcome or process reward models has opened up new avenues to enhance LLMs performance in complex reasoning tasks, this space is under-explored in deductive logical reasoning. We present a set of Outcome Reward Models (ORMs) for deductive reasoning. To train the ORMs we mainly generate data using Chain-of-Thought (CoT) with single and multiple samples. Additionally, we propose a novel tactic to further expand the type of errors covered in the training dataset of the ORM. In particular, we propose an echo generation technique that leverages LLMs' tendency to reflect incorrect assumptions made in prompts to extract additional training data, covering previously unexplored error types. While a standard CoT chain may contain errors likely to be made by the reasoner, the echo strategy deliberately steers the model toward incorrect reasoning. We show that ORMs trained on CoT and echo-augmented data demonstrate improved performance on the FOLIO, JustLogic, and ProverQA datasets across four different LLMs.

We introduce ProofNet, a benchmark for autoformalization and formal proving of undergraduate-level mathematics. The ProofNet benchmarks consists of 371 examples, each consisting of a formal theorem statement in Lean 3, a natural language theorem statement, and a natural language proof. The problems are primarily drawn from popular undergraduate pure mathematics textbooks and cover topics such as real and complex analysis, linear algebra, abstract algebra, and topology. We intend for ProofNet to be a challenging benchmark that will drive progress in autoformalization and automatic theorem proving. We report baseline results on statement autoformalization via in-context learning. Moreover, we introduce two novel statement autoformalization methods: prompt retrieval and distilled backtranslation.

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.

Recent advancements in large language models (LLMs) have substantially enhanced their mathematical reasoning abilities. However, these models still struggle with complex problems that require multiple reasoning steps, frequently leading to logical or numerical errors. While numerical mistakes can be largely addressed by integrating a code interpreter, identifying logical errors within intermediate steps is more challenging. Moreover, manually annotating these steps for training is not only expensive but also labor-intensive, requiring the expertise of professional annotators. In our study, we introduce an innovative approach that bypasses the need for process annotations (from human or GPTs) by utilizing the Monte Carlo Tree Search (MCTS) framework. This technique automatically generates both the process supervision and the step-level evaluation signals. Our method iteratively trains the policy and value models, leveraging the capabilities of a well-pretrained LLM to progressively enhance its mathematical reasoning skills. Furthermore, we propose an efficient inference strategy-step-level beam search, where the value model is crafted to assist the policy model (i.e., LLM) in navigating more effective reasoning paths, rather than solely relying on prior probabilities. The experimental results on both in-domain and out-of-domain datasets demonstrate that even without GPT-4 or human-annotated process supervision, our AlphaMath framework achieves comparable or superior results to previous state-of-the-art methods.

The Automated Model Evaluation (AutoEval) framework entertains the possibility of evaluating a trained machine learning model without resorting to a labeled testing set. Despite the promise and some decent results, the existing AutoEval methods heavily rely on computing distribution shifts between the unlabelled testing set and the training set. We believe this reliance on the training set becomes another obstacle in shipping this technology to real-world ML development. In this work, we propose Contrastive Automatic Model Evaluation (CAME), a novel AutoEval framework that is rid of involving training set in the loop. The core idea of CAME bases on a theoretical analysis which bonds the model performance with a contrastive loss. Further, with extensive empirical validation, we manage to set up a predictable relationship between the two, simply by deducing on the unlabeled/unseen testing set. The resulting framework CAME establishes a new SOTA results for AutoEval by surpassing prior work significantly.

The advent of Large Language Models (LLMs) has profoundly transformed our lives, revolutionizing interactions with AI and lowering the barrier to AI usage. While LLMs are primarily designed for natural language interaction, the extensive embedded knowledge empowers them to comprehend digital sensor data. This capability enables LLMs to engage with the physical world through IoT sensors and actuators, performing a myriad of AIoT tasks. Consequently, this evolution triggers a paradigm shift in conventional AIoT application development, democratizing its accessibility to all by facilitating the design and development of AIoT applications via natural language. However, some limitations need to be addressed to unlock the full potential of LLMs in AIoT application development. First, existing solutions often require transferring raw sensor data to LLM servers, which raises privacy concerns, incurs high query fees, and is limited by token size. Moreover, the reasoning processes of LLMs are opaque to users, making it difficult to verify the robustness and correctness of inference results. This paper introduces AutoIOT, an LLM-based automated program generator for AIoT applications. AutoIOT enables users to specify their requirements using natural language (input) and automatically synthesizes interpretable programs with documentation (output). AutoIOT automates the iterative optimization to enhance the quality of generated code with minimum user involvement. AutoIOT not only makes the execution of AIoT tasks more explainable but also mitigates privacy concerns and reduces token costs with local execution of synthesized programs. Extensive experiments and user studies demonstrate AutoIOT's remarkable capability in program synthesis for various AIoT tasks. The synthesized programs can match and even outperform some representative baselines.

Deceptive patterns in digital interfaces manipulate users into making unintended decisions, exploiting cognitive biases and psychological vulnerabilities. These patterns have become ubiquitous on various digital platforms. While efforts to mitigate deceptive patterns have emerged from legal and technical perspectives, a significant gap remains in creating usable and scalable solutions. We introduce our AutoBot framework to address this gap and help web stakeholders navigate and mitigate online deceptive patterns. AutoBot accurately identifies and localizes deceptive patterns from a screenshot of a website without relying on the underlying HTML code. AutoBot employs a two-stage pipeline that leverages the capabilities of specialized vision models to analyze website screenshots, identify interactive elements, and extract textual features. Next, using a large language model, AutoBot understands the context surrounding these elements to determine the presence of deceptive patterns. We also use AutoBot, to create a synthetic dataset to distill knowledge from 'teacher' LLMs to smaller language models. Through extensive evaluation, we demonstrate AutoBot's effectiveness in detecting deceptive patterns on the web, achieving an F1-score of 0.93 when detecting deceptive patterns, underscoring its potential as an essential tool for mitigating online deceptive patterns. We implement AutoBot, across three downstream applications targeting different web stakeholders: (1) a local browser extension providing users with real-time feedback, (2) a Lighthouse audit to inform developers of potential deceptive patterns on their sites, and (3) as a measurement tool designed for researchers and regulators.

Formal verification (FV) has witnessed growing significance with current emerging program synthesis by the evolving large language models (LLMs). However, current formal verification mainly resorts to symbolic verifiers or hand-craft rules, resulting in limitations for extensive and flexible verification. On the other hand, formal languages for automated theorem proving, such as Isabelle, as another line of rigorous verification, are maintained with comprehensive rules and theorems. In this paper, we propose FVEL, an interactive Formal Verification Environment with LLMs. Specifically, FVEL transforms a given code to be verified into Isabelle, and then conducts verification via neural automated theorem proving with an LLM. The joined paradigm leverages the rigorous yet abundant formulated and organized rules in Isabelle and is also convenient for introducing and adjusting cutting-edge LLMs. To achieve this goal, we extract a large-scale FVELER3. The FVELER dataset includes code dependencies and verification processes that are formulated in Isabelle, containing 758 theories, 29,125 lemmas, and 200,646 proof steps in total with in-depth dependencies. We benchmark FVELER in the FVEL environment by first fine-tuning LLMs with FVELER and then evaluating them on Code2Inv and SV-COMP. The results show that FVEL with FVELER fine-tuned Llama3- 8B solves 17.39% (69 -> 81) more problems, and Mistral-7B 12% (75 -> 84) more problems in SV-COMP. And the proportion of proof errors is reduced. Project page: https://fveler.github.io/.

Conducting contamination-free evaluation of mathematical capabilities can be difficult for two reasons: models may memorize a test set once it is made public, and current mathematical benchmarks are prone to overfitting due to having limited diversity of symbols and rules, coupled with closed-ended answers. This paper proposes a method to leverage these shortcomings as useful features to a construct dynamic, counterfactual benchmark, which can be used to both reveal overfitting and measure true reasoning. We demonstrate this via MatheMagic, which generates math test instances with the interpretations of numbers and operators altered, yet has automatically verifiable answers. Test instances are randomly seeded and constructed at test time to evaluate a model's induction or deduction capability, offering stability, extensibility, comparability, and robustness to overfitting. Our experiments find that models solve deduction more easily than induction, but they revert to standard math. Further analysis reveals that math-adapted models fail to exhibit a general "skill" of reasoning, and fine-tuning on induction tasks generalizes poorly.

Automated machine learning (AutoML) accelerates AI development by automating tasks in the development pipeline, such as optimal model search and hyperparameter tuning. Existing AutoML systems often require technical expertise to set up complex tools, which is in general time-consuming and requires a large amount of human effort. Therefore, recent works have started exploiting large language models (LLM) to lessen such burden and increase the usability of AutoML frameworks via a natural language interface, allowing non-expert users to build their data-driven solutions. These methods, however, are usually designed only for a particular process in the AI development pipeline and do not efficiently use the inherent capacity of the LLMs. This paper proposes AutoML-Agent, a novel multi-agent framework tailored for full-pipeline AutoML, i.e., from data retrieval to model deployment. AutoML-Agent takes user's task descriptions, facilitates collaboration between specialized LLM agents, and delivers deployment-ready models. Unlike existing work, instead of devising a single plan, we introduce a retrieval-augmented planning strategy to enhance exploration to search for more optimal plans. We also decompose each plan into sub-tasks (e.g., data preprocessing and neural network design) each of which is solved by a specialized agent we build via prompting executing in parallel, making the search process more efficient. Moreover, we propose a multi-stage verification to verify executed results and guide the code generation LLM in implementing successful solutions. Extensive experiments on seven downstream tasks using fourteen datasets show that AutoML-Agent achieves a higher success rate in automating the full AutoML process, yielding systems with good performance throughout the diverse domains.

Automated Theorem Proving (ATP) faces challenges due to its complexity and computational demands. Recent work has explored using Large Language Models (LLMs) for ATP action selection, but these methods can be resource-intensive. This study introduces FEAS, an agent that enhances the COPRA in-context learning framework within Lean. FEAS refines prompt generation, response parsing, and incorporates domain-specific heuristics for functional equations. It introduces FunEq, a curated dataset of functional equation problems with varying difficulty. FEAS outperforms baselines on FunEq, particularly with the integration of domain-specific heuristics. The results demonstrate FEAS's effectiveness in generating and formalizing high-level proof strategies into Lean proofs, showcasing the potential of tailored approaches for specific ATP challenges.

We automate deep step-by step reasoning in an LLM dialog thread by recursively exploring alternatives (OR-nodes) and expanding details (AND-nodes) up to a given depth. Starting from a single succinct task-specific initiator we steer the automated dialog thread to stay focussed on the task by synthesizing a prompt that summarizes the depth-first steps taken so far. Our algorithm is derived from a simple recursive descent implementation of a Horn Clause interpreter, except that we accommodate our logic engine to fit the natural language reasoning patterns LLMs have been trained on. Semantic similarity to ground-truth facts or oracle advice from another LLM instance is used to restrict the search space and validate the traces of justification steps returned as answers. At the end, the unique minimal model of a generated Horn Clause program collects the results of the reasoning process. As applications, we sketch implementations of consequence predictions, causal explanations, recommendation systems and topic-focussed exploration of scientific literature.

Scientists often infer abstract procedures from specific instances of problems and use the abstractions to generate new, related instances. For example, programs encoding the formal rules and properties of a system have been useful in fields ranging from RL (procedural environments) to physics (simulation engines). These programs can be seen as functions which execute to different outputs based on their parameterizations (e.g., gridworld configuration or initial physical conditions). We introduce the term EFA (Executable Functional Abstraction) to denote such programs for math problems. EFA-like constructs have been shown to be useful for math reasoning as problem generators for stress-testing models. However, prior work has been limited to abstractions for grade-school math (whose simple rules are easy to encode in programs), while generating EFAs for advanced math has thus far required human engineering. We explore the automatic construction of EFAs for advanced math problems. We operationalize the task of automatically constructing EFAs as a program synthesis task, and develop EFAGen, which conditions an LLM on a seed math problem and its step-by-step solution to generate candidate EFA programs that are faithful to the generalized problem and solution class underlying the seed problem. Furthermore, we formalize properties any valid EFA must possess in terms of executable unit tests, and show how the tests can be used as verifiable rewards to train LLMs to become better writers of EFAs. We demonstrate that EFAs constructed by EFAGen behave rationally by remaining faithful to seed problems, produce learnable problem variations, and that EFAGen can infer EFAs across multiple diverse sources of competition-level math problems. Finally, we show downstream uses of model-written EFAs e.g. finding problem variations that are harder or easier for a learner to solve, as well as data generation.

While Large Reasoning Models (LRMs) have demonstrated success in complex reasoning tasks through long chain-of-thought (CoT) reasoning, their inference often involves excessively verbose reasoning traces, resulting in substantial inefficiency. To address this, we propose Distilled Reasoning Pruning (DRP), a hybrid framework that combines inference-time pruning with tuning-based distillation, two widely used strategies for efficient reasoning. DRP uses a teacher model to perform skill-aware step decomposition and content pruning, and then distills the pruned reasoning paths into a student model, enabling it to reason both efficiently and accurately. Across several challenging mathematical reasoning datasets, we find that models trained with DRP achieve substantial improvements in token efficiency without sacrificing accuracy. Specifically, DRP reduces average token usage on GSM8K from 917 to 328 while improving accuracy from 91.7% to 94.1%, and achieves a 43% token reduction on AIME with no performance drop. Further analysis shows that aligning the reasoning structure of training CoTs with the student's reasoning capacity is critical for effective knowledge transfer and performance gains.

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.

Algorithmic reasoning refers to the ability to understand the complex patterns behind the problem and decompose them into a sequence of reasoning steps towards the solution. Such nature of algorithmic reasoning makes it a challenge for large language models (LLMs), even though they have demonstrated promising performance in other reasoning tasks. Within this context, some recent studies use programming languages (e.g., Python) to express the necessary logic for solving a given instance/question (e.g., Program-of-Thought) as inspired by their strict and precise syntaxes. However, it is non-trivial to write an executable code that expresses the correct logic on the fly within a single inference call. Also, the code generated specifically for an instance cannot be reused for others, even if they are from the same task and might require identical logic to solve. This paper presents Think-and-Execute, a novel framework that decomposes the reasoning process of language models into two steps. (1) In Think, we discover a task-level logic that is shared across all instances for solving a given task and then express the logic with pseudocode; (2) In Execute, we further tailor the generated pseudocode to each instance and simulate the execution of the code. With extensive experiments on seven algorithmic reasoning tasks, we demonstrate the effectiveness of Think-and-Execute. Our approach better improves LMs' reasoning compared to several strong baselines performing instance-specific reasoning (e.g., CoT and PoT), suggesting the helpfulness of discovering task-level logic. Also, we show that compared to natural language, pseudocode can better guide the reasoning of LMs, even though they are trained to follow natural language instructions.

Large language models show promise for autoformalization, the task of automatically translating natural language into formal languages. However, current autoformalization methods remain limited. The last reported state-of-the-art performance on the ProofNet formalization benchmark for the Lean proof assistant, achieved using Codex for Lean 3, only showed successful formalization of 16.1% of informal statements. Similarly, our evaluation of GPT-4o for Lean 4 only produces successful translations 34.9% of the time. Our analysis shows that the performance of these models is largely limited by their inability to generate formal statements that successfully type-check (i.e., are syntactically correct and consistent with types) - with a whopping 86.6% of GPT-4o errors starting from a type-check failure. In this work, we propose a method to fix this issue through decoding with type-check filtering, where we initially sample a diverse set of candidate formalizations for an informal statement, then use the Lean proof assistant to filter out candidates that do not type-check. Using GPT-4o as a base model, and combining our method with self-consistency, we obtain a +18.3% absolute increase in formalization accuracy, and achieve a new state-of-the-art of 53.2% on ProofNet with Lean 4.

We propose utilizing background operators for mathematical reasoning in large language models (LLMs). To achieve this, we define a set of fundamental mathematical predicates as the basic building blocks. For each mathematical problem, we develop a Prolog solution that includes problem-specific predicates and intermediate predicates derived from these background operators, ensuring that each solution adheres to the defined operator set. We introduce the MATH-Prolog corpus, which is derived from the counting and probability categories of the MATH corpus. For efficient data augmentation, we apply K-fold cross-validated self-training. This method incrementally generates new Prolog solutions for each fold, incorporating those verified as correct into the training set throughout the model training process. Our experimental results demonstrate that 5-fold crossvalidated self-training effectively identifies new, accurate Prolog solutions, achieving an accuracy of 84.6% on the cross-validated set, and 84.8% on the test set during fine-tuning the Meta-Llama-3.1-8B-Instruct model. This approach successfully uncovers new solutions with fully computable inference steps for previously unseen problems. Additionally, incorporating the background mathematical predicates into the prompt enhances solution coverage.

Deductive reasoning is a crucial logical capability that assists us in solving complex problems based on existing knowledge. Although augmented by Chain-of-Thought prompts, Large Language Models (LLMs) might not follow the correct reasoning paths. Enhancing the deductive reasoning abilities of LLMs, and leveraging their extensive built-in knowledge for various reasoning tasks, remains an open question. Attempting to mimic the human deductive reasoning paradigm, we propose a multi-stage Syllogistic-Reasoning Framework of Thought (SR-FoT) that enables LLMs to perform syllogistic deductive reasoning to handle complex knowledge-based reasoning tasks. Our SR-FoT begins by interpreting the question and then uses the interpretation and the original question to propose a suitable major premise. It proceeds by generating and answering minor premise questions in two stages to match the minor premises. Finally, it guides LLMs to use the previously generated major and minor premises to perform syllogistic deductive reasoning to derive the answer to the original question. Extensive and thorough experiments on knowledge-based reasoning tasks have demonstrated the effectiveness and advantages of our SR-FoT.

Claim verification can be a challenging task. In this paper, we present a method to enhance the robustness and reasoning capabilities of automated claim verification through the extraction of short facts from evidence. Our novel approach, FactDetect, leverages Large Language Models (LLMs) to generate concise factual statements from evidence and label these facts based on their semantic relevance to the claim and evidence. The generated facts are then combined with the claim and evidence. To train a lightweight supervised model, we incorporate a fact-detection task into the claim verification process as a multitasking approach to improve both performance and explainability. We also show that augmenting FactDetect in the claim verification prompt enhances performance in zero-shot claim verification using LLMs. Our method demonstrates competitive results in the supervised claim verification model by 15% on the F1 score when evaluated for challenging scientific claim verification datasets. We also demonstrate that FactDetect can be augmented with claim and evidence for zero-shot prompting (AugFactDetect) in LLMs for verdict prediction. We show that AugFactDetect outperforms the baseline with statistical significance on three challenging scientific claim verification datasets with an average of 17.3% performance gain compared to the best performing baselines.

Legislation can be viewed as a body of prescriptive rules expressed in natural language. The application of legislation to facts of a case we refer to as statutory reasoning, where those facts are also expressed in natural language. Computational statutory reasoning is distinct from most existing work in machine reading, in that much of the information needed for deciding a case is declared exactly once (a law), while the information needed in much of machine reading tends to be learned through distributional language statistics. To investigate the performance of natural language understanding approaches on statutory reasoning, we introduce a dataset, together with a legal-domain text corpus. Straightforward application of machine reading models exhibits low out-of-the-box performance on our questions, whether or not they have been fine-tuned to the legal domain. We contrast this with a hand-constructed Prolog-based system, designed to fully solve the task. These experiments support a discussion of the challenges facing statutory reasoning moving forward, which we argue is an interesting real-world task that can motivate the development of models able to utilize prescriptive rules specified in natural language.

Reasoning capability is pivotal for Large Language Models (LLMs) to solve complex tasks, yet achieving reliable and scalable reasoning remains challenging. While Chain-of-Thought (CoT) prompting has become a mainstream approach, existing methods often suffer from uncontrolled generation, insufficient quality, and limited diversity in reasoning paths. Recent efforts leverage code to enhance CoT by grounding reasoning in executable steps, but such methods are typically constrained to predefined mathematical problems, hindering scalability and generalizability. In this work, we propose Caco (Code-Assisted Chain-of-ThOught), a novel framework that automates the synthesis of high-quality, verifiable, and diverse instruction-CoT reasoning data through code-driven augmentation. Unlike prior work, Caco first fine-tunes a code-based CoT generator on existing math and programming solutions in a unified code format, then scales the data generation to a large amount of diverse reasoning traces. Crucially, we introduce automated validation via code execution and rule-based filtering to ensure logical correctness and structural diversity, followed by reverse-engineering filtered outputs into natural language instructions and language CoTs to enrich task adaptability. This closed-loop process enables fully automated, scalable synthesis of reasoning data with guaranteed executability. Experiments on our created Caco-1.3M dataset demonstrate that Caco-trained models achieve strong competitive performance on mathematical reasoning benchmarks, outperforming existing strong baselines. Further analysis reveals that Caco's code-anchored verification and instruction diversity contribute to superior generalization across unseen tasks. Our work establishes a paradigm for building self-sustaining, trustworthy reasoning systems without human intervention.

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.

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

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

Large language models (LLMs) are increasingly evaluated on formal tasks, where strong reasoning abilities define the state of the art. However, their ability to generalize to out-of-distribution problems remains limited. In this paper, we investigate how LLMs can achieve a systematic understanding of deductive rules. Our focus is on the task of identifying the appropriate subset of premises within a knowledge base needed to derive a given hypothesis. To tackle this challenge, we propose Meta-learning for In-context Deduction (MIND), a novel few-shot meta-learning fine-tuning approach. The goal of MIND is to enable models to generalize more effectively to unseen knowledge bases and to systematically apply inference rules. Our results show that MIND significantly improves generalization in small LMs ranging from 1.5B to 7B parameters. The benefits are especially pronounced in smaller models and low-data settings. Remarkably, small models fine-tuned with MIND outperform state-of-the-art LLMs, such as GPT-4o and o3-mini, on this task.

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.

Large language models (LLMs) demonstrate remarkable reasoning capabilities in tasks such as algorithmic coding and mathematical problem-solving. Recent methods have improved reasoning through expanded corpus and multistage training combining reinforcement learning and supervised fine-tuning. Although some methods suggest that small but targeted dataset can incentivize reasoning via only distillation, a reasoning scaling laws is still taking shape, increasing computational costs. To address this, we propose a data-efficient distillation framework (DED) that optimizes the Pareto frontier of reasoning distillation. Inspired by the on-policy learning and diverse roll-out strategies of reinforcement learning, the key idea of our approach is threefold: (1) We identify that benchmark scores alone do not determine an effective teacher model. Through comprehensive comparisons of leading reasoning LLMs, we develop a method to select an optimal teacher model. (2) While scaling distillation can enhance reasoning, it often degrades out-of-domain performance. A carefully curated, smaller corpus achieves a balanced trade-off between in-domain and out-of-domain capabilities. (3) Diverse reasoning trajectories encourage the student model to develop robust reasoning skills. We validate our method through evaluations on mathematical reasoning (AIME 2024/2025, MATH-500) and code generation (LiveCodeBench), achieving state-of-the-art results with only 0.8k carefully curated examples, bypassing the need for extensive scaling. Our systematic analysis demonstrates that DED outperforms existing methods by considering factors beyond superficial hardness, token length, or teacher model capability. This work offers a practical and efficient pathway to advanced reasoning while preserving general capabilities.

Solving math word problems requires deductive reasoning over the quantities in the text. Various recent research efforts mostly relied on sequence-to-sequence or sequence-to-tree models to generate mathematical expressions without explicitly performing relational reasoning between quantities in the given context. While empirically effective, such approaches typically do not provide explanations for the generated expressions. In this work, we view the task as a complex relation extraction problem, proposing a novel approach that presents explainable deductive reasoning steps to iteratively construct target expressions, where each step involves a primitive operation over two quantities defining their relation. Through extensive experiments on four benchmark datasets, we show that the proposed model significantly outperforms existing strong baselines. We further demonstrate that the deductive procedure not only presents more explainable steps but also enables us to make more accurate predictions on questions that require more complex reasoning.

Premise selection is a crucial yet challenging step in mathematical formalization, especially for users with limited experience. Due to the lack of available formalization projects, existing approaches that leverage language models often suffer from data scarcity. In this work, we introduce an innovative method for training a premise retriever to support the formalization of mathematics. Our approach employs a BERT model to embed proof states and premises into a shared latent space. The retrieval model is trained within a contrastive learning framework and incorporates a domain-specific tokenizer along with a fine-grained similarity computation method. Experimental results show that our model is highly competitive compared to existing baselines, achieving strong performance while requiring fewer computational resources. Performance is further enhanced through the integration of a re-ranking module. To streamline the formalization process, we will release a search engine that enables users to query Mathlib theorems directly using proof states, significantly improving accessibility and efficiency. Codes are available at https://github.com/ruc-ai4math/Premise-Retrieval.

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.

Logical reasoning is a fundamental task in natural language processing that presents significant challenges to Large Language Models (LLMs). The inherent characteristics of logical reasoning makes it well-suited for symbolic representations such as first-order logic (FOL). Research in symbolic logical reasoning explored FOL generation using state-of-the-art LLMs (i.e., GPT-4) to produce FOL translations of natural language (NL) statements, but errors in translation are usually not the focus. We address this by categorizing the translation errors in FOL statements generated by LLMs. To make progress towards improving the quality of FOL translations for smaller language models such as LLaMA-2 13B and Mistral 7B, we create ProofFOL, a high-quality FOL-annotated subset of ProofWriter dataset using GPT-4o. The models fine-tuned on this silver standard data achieve a significant gain in performance when compared to larger language models such as LLaMA-2 70B. In addition to improving the model using large data, we also tackle the issue of data scarcity and introduce an incremental framework encompassing of data augmentation and verification steps. In the augmentation process, a single pair of (premises, conclusion) is split into multiple new instances based on the predicates and FOLs. This data is used for fine-tuning, and the inference on this model generates FOLs with fewer errors over the model trained on the original data. Our investigation on the translation errors leads to generation of a perturbation dataset, which is used to train a verifier that corrects potential syntactic and semantic FOL translation errors. We demonstrate an efficient method for making the most of a limited existing human-annotated dataset. Our results show state-of-the-art performance for ProofWriter and ProntoQA datasets using ProofFOL on LLaMA-2 and Mistral models.

While most social norms are informal, they are often formalized by companies in contracts to regulate trades of goods and services. When poorly written, contracts may contain normative conflicts resulting from opposing deontic meanings or contradict specifications. As contracts tend to be long and contain many norms, manually identifying such conflicts requires human-effort, which is time-consuming and error-prone. Automating such task benefits contract makers increasing productivity and making conflict identification more reliable. To address this problem, we introduce an approach to detect and classify norm conflicts in contracts by converting them into latent representations that preserve both syntactic and semantic information and training a model to classify norm conflicts in four conflict types. Our results reach the new state of the art when compared to a previous approach.

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.

We present a novel reasoning approach called Flow-of-Options (FoO), designed to address intrinsic biases in Large Language Models (LLMs). FoO enables LLMs to systematically explore a diverse range of possibilities in their reasoning, as demonstrated by an FoO-based agentic system for autonomously solving Machine Learning tasks (AutoML). Our framework outperforms state-of-the-art baselines, achieving improvements of 38.2% - 69.2% on standard data science tasks, and 37.4% - 47.9% on therapeutic chemistry tasks. With an overall operation cost under $1 per task, our framework is well-suited for cost-sensitive applications. Beyond classification and regression, we illustrate the broader applicability of our FoO-based agentic system to tasks such as reinforcement learning and image generation. Our framework presents significant advancements compared to current state-of-the-art agentic systems for AutoML, due to the benefits of FoO in enforcing diversity in LLM solutions through compressed, explainable representations that also support long-term memory when combined with case-based reasoning.

Large reasoning models (LRMs) have recently achieved significant progress in complex reasoning tasks, aided by reinforcement learning with verifiable rewards. However, LRMs often suffer from overthinking, expending excessive computation on simple problems and reducing efficiency. Existing efficient reasoning methods typically require accurate task assessment to preset token budgets or select reasoning modes, which limits their flexibility and reliability. In this work, we revisit the essence of overthinking and identify that encouraging effective steps while penalizing ineffective ones is key to its solution. To this end, we propose a novel rule-based verifiable stepwise reward mechanism (VSRM), which assigns rewards based on the performance of intermediate states in the reasoning trajectory. This approach is intuitive and naturally fits the step-by-step nature of reasoning tasks. We conduct extensive experiments on standard mathematical reasoning benchmarks, including AIME24 and AIME25, by integrating VSRM with PPO and Reinforce++. Results show that our method achieves substantial output length reduction while maintaining original reasoning performance, striking an optimal balance between efficiency and accuracy. Further analysis of overthinking frequency and pass@k score before and after training demonstrates that our approach in deed effectively suppresses ineffective steps and encourages effective reasoning, fundamentally alleviating the overthinking problem. All code will be released upon acceptance.

AI for software engineering has made remarkable progress recently, becoming a notable success within generative AI. Despite this, there are still many challenges that need to be addressed before automated software engineering reaches its full potential. It should be possible to reach high levels of automation where humans can focus on the critical decisions of what to build and how to balance difficult tradeoffs while most routine development effort is automated away. Reaching this level of automation will require substantial research and engineering efforts across academia and industry. In this paper, we aim to discuss progress towards this in a threefold manner. First, we provide a structured taxonomy of concrete tasks in AI for software engineering, emphasizing the many other tasks in software engineering beyond code generation and completion. Second, we outline several key bottlenecks that limit current approaches. Finally, we provide an opinionated list of promising research directions toward making progress on these bottlenecks, hoping to inspire future research in this rapidly maturing field.

Deductive coding is a widely used qualitative research method for determining the prevalence of themes across documents. While useful, deductive coding is often burdensome and time consuming since it requires researchers to read, interpret, and reliably categorize a large body of unstructured text documents. Large language models (LLMs), like ChatGPT, are a class of quickly evolving AI tools that can perform a range of natural language processing and reasoning tasks. In this study, we explore the use of LLMs to reduce the time it takes for deductive coding while retaining the flexibility of a traditional content analysis. We outline the proposed approach, called LLM-assisted content analysis (LACA), along with an in-depth case study using GPT-3.5 for LACA on a publicly available deductive coding data set. Additionally, we conduct an empirical benchmark using LACA on 4 publicly available data sets to assess the broader question of how well GPT-3.5 performs across a range of deductive coding tasks. Overall, we find that GPT-3.5 can often perform deductive coding at levels of agreement comparable to human coders. Additionally, we demonstrate that LACA can help refine prompts for deductive coding, identify codes for which an LLM is randomly guessing, and help assess when to use LLMs vs. human coders for deductive coding. We conclude with several implications for future practice of deductive coding and related research methods.

How capable are large language models (LLMs) in the domain of taxation? Although numerous studies have explored the legal domain in general, research dedicated to taxation remain scarce. Moreover, the datasets used in these studies are either simplified, failing to reflect the real-world complexities, or unavailable as open source. To address this gap, we introduce PLAT, a new benchmark designed to assess the ability of LLMs to predict the legitimacy of additional tax penalties. PLAT is constructed to evaluate LLMs' understanding of tax law, particularly in cases where resolving the issue requires more than just applying related statutes. Our experiments with six LLMs reveal that their baseline capabilities are limited, especially when dealing with conflicting issues that demand a comprehensive understanding. However, we found that enabling retrieval, self-reasoning, and discussion among multiple agents with specific role assignments, this limitation can be mitigated.

Neural theorem proving combines large language models (LLMs) with proof assistants such as Lean, where the correctness of formal proofs can be rigorously verified, leaving no room for hallucination. With existing neural theorem provers pretrained on a fixed collection of data and offering valuable suggestions at times, it is challenging for them to continually prove novel theorems in a fully autonomous mode, where human insights may be critical. In this paper, we explore LLMs as copilots that assist humans in proving theorems. We introduce Lean Copilot, a general framework for running LLM inference natively in Lean. It enables programmers to build various LLM-based proof automation tools that integrate seamlessly into the workflow of Lean users. Lean users can use our pretrained models or bring their own ones that run either locally (with or without GPUs) or on the cloud. Using Lean Copilot, we build LLM-based tools that suggest proof steps, complete proof goals, and select relevant premises. Experimental results on the Mathematics in Lean textbook demonstrate the effectiveness of our method compared to existing rule-based proof automation in Lean (aesop). When assisting humans, Lean Copilot requires only 2.08 manually-entered proof steps on average (3.86 required by aesop); when automating the theorem proving process, Lean Copilot automates 74.2% proof steps on average, 85% better than aesop (40.1%). We open source all code and artifacts under a permissive MIT license to facilitate further research.

Large reasoning models (LRMs) are proficient at generating explicit, step-by-step reasoning sequences before producing final answers. However, such detailed reasoning can introduce substantial computational overhead and latency, particularly for simple problems. To address this over-thinking problem, we explore how to equip LRMs with adaptive thinking capabilities: enabling them to dynamically decide whether or not to engage in explicit reasoning based on problem complexity. Building on R1-style distilled models, we observe that inserting a simple ellipsis ("...") into the prompt can stochastically trigger either a thinking or no-thinking mode, revealing a latent controllability in the reasoning behavior. Leveraging this property, we propose AutoThink, a multi-stage reinforcement learning (RL) framework that progressively optimizes reasoning policies via stage-wise reward shaping. AutoThink learns to invoke explicit reasoning only when necessary, while defaulting to succinct responses for simpler tasks. Experiments on five mainstream mathematical benchmarks demonstrate that AutoThink achieves favorable accuracy-efficiency trade-offs compared to recent prompting and RL-based pruning methods. It can be seamlessly integrated into any R1-style model, including both distilled and further fine-tuned variants. Notably, AutoThink improves relative accuracy by 6.4 percent while reducing token usage by 52 percent on DeepSeek-R1-Distill-Qwen-1.5B, establishing a scalable and adaptive reasoning paradigm for LRMs. Project Page: https://github.com/ScienceOne-AI/AutoThink.

We introduce a large language model (LLM) based approach to answer complex questions requiring multi-hop numerical reasoning over financial reports. While LLMs have exhibited remarkable performance on various natural language and reasoning tasks, complex reasoning problems often rely on few-shot prompts that require carefully crafted examples. In contrast, our approach uses novel zero-shot prompts that guide the LLM to encode the required reasoning into a Python program or a domain specific language. The generated program is then executed by a program interpreter, thus mitigating the limitations of LLM in performing accurate arithmetic calculations. We evaluate the proposed approach on three financial datasets using some of the recently developed generative pretrained transformer (GPT) models and perform comparisons with various zero-shot baselines. The experimental results demonstrate that our approach significantly improves the accuracy for all the LLMs over their respective baselines. We provide a detailed analysis of the results, generating insights to support our findings. The success of our approach demonstrates the enormous potential to extract complex domain specific numerical reasoning by designing zero-shot prompts to effectively exploit the knowledge embedded in LLMs.

Autoformalization, the conversion of natural language mathematics into formal languages, offers significant potential for advancing mathematical reasoning. However, existing efforts are limited to formal languages with substantial online corpora and struggle to keep pace with rapidly evolving languages like Lean 4. To bridge this gap, we propose a new benchmark Formalization for Lean~4 (\name) designed to evaluate the autoformalization capabilities of large language models (LLMs). This benchmark encompasses a comprehensive assessment of questions, answers, formal statements, and proofs. Additionally, we introduce a Process-Supervised Verifier (PSV) model that leverages the precise feedback from Lean 4 compilers to enhance autoformalization. Our experiments demonstrate that the PSV method improves autoformalization, enabling higher accuracy using less filtered training data. Furthermore, when fine-tuned with data containing detailed process information, PSV can leverage the data more effectively, leading to more significant improvements in autoformalization for Lean 4. Our dataset and code are available at https://github.com/rookie-joe/PDA.

Automated code generation is gaining significant importance in intelligent computer programming and system deployment. However, current approaches often face challenges in computational efficiency and lack robust mechanisms for code parsing and error correction. In this work, we propose a novel framework, PyCapsule, with a simple yet effective two-agent pipeline and efficient self-debugging modules for Python code generation. PyCapsule features sophisticated prompt inference, iterative error handling, and case testing, ensuring high generation stability, safety, and correctness. Empirically, PyCapsule achieves up to 5.7% improvement of success rate on HumanEval, 10.3% on HumanEval-ET, and 24.4% on BigCodeBench compared to the state-of-art methods. We also observe a decrease in normalized success rate given more self-debugging attempts, potentially affected by limited and noisy error feedback in retention. PyCapsule demonstrates broader impacts on advancing lightweight and efficient code generation for artificial intelligence systems.

We perform a thorough analysis of the formal and informal statements in the miniF2F benchmark from the perspective of an AI system that is tasked to participate in a math Olympiad consisting of the problems in miniF2F. In such setting, the model has to read and comprehend the problems in natural language, formalize them in Lean language, then proceed with proving the problems, and it will get credit for each problem if the formal proof corresponds to the original informal statement presented to the model. Our evaluation results reveal that the best accuracy of such pipeline can be about 36% using the SoTA models in the literature, considerably lower than the individual SoTA accuracies, 97% and 69% reported in the autoformalization and theorem proving literature. Analyzing the failure modes, we trace back a considerable portion of this drop to discrepancies between the formal and informal statements for more than half of the problems in miniF2F. We proceed with correcting all the errors, discrepancies and simplifications in formal and informal statements, and present the miniF2F-v2 with fully verified formal and informal statements and proofs. Evaluating the full theorem proving pipeline on miniF2F-v2 leads to the best accuracy of 70%, a significant improvement from the 40% on the original miniF2F, yet indicating considerable misalignment between the autoformalization models and theorem provers. Our deep analysis suggests that a higher quality benchmark can help the community better evaluate progress in the field of formal reasoning and also better diagnose the failure and success modes of autoformalization and theorem proving models. Our dataset is available at https://github.com/roozbeh-yz/miniF2F_v2.

We propose a framework that amortizes the cost of inference-time reasoning by converting transient critiques into retrievable guidelines, through a file-based memory system and agent-controlled tool calls. We evaluate this method on the Rubric Feedback Bench, a novel dataset for rubric-based learning. Experiments demonstrate that our augmented LLMs rapidly match the performance of test-time refinement pipelines while drastically reducing inference cost.

General-purpose Large Language Models (LLMs) have achieved remarkable success in intelligence, performing comparably to human experts on complex reasoning tasks such as coding and mathematical reasoning. However, generating formal proofs in specialized languages like Lean 4 remains a significant challenge for these models, limiting their application in complex theorem proving and automated verification. Current approaches typically require specializing models through fine-tuning on dedicated formal corpora, incurring high costs for data collection and training. In this work, we introduce Delta Prover, an agent-based framework that orchestrates the interaction between a general-purpose LLM and the Lean 4 proof environment. Delta Prover leverages the reflection and reasoning capabilities of general-purpose LLMs to interactively construct formal proofs in Lean 4, circumventing the need for model specialization. At its core, the agent integrates two novel, interdependent components: an algorithmic framework for reflective decomposition and iterative proof repair, and a custom Domain-Specific Language (DSL) built upon Lean 4 for streamlined subproblem management. Delta Prover achieves a state-of-the-art 95.9\% success rate on the miniF2F-test benchmark, surpassing all existing approaches, including those requiring model specialization. Furthermore, Delta Prover exhibits a significantly stronger test-time scaling law compared to standard Best-of-N proof strategies. Crucially, our findings demonstrate that general-purpose LLMs, when guided by an effective agentic structure, possess substantial untapped theorem-proving capabilities. This presents a computationally efficient alternative to specialized models for robust automated reasoning in formal environments.

We report on a new, simple, modular, and flexible approach for automated generation of illustrations for (readable) synthetic geometry proofs. The underlying proofs are generated using the Larus automated prover for coherent logic, and corresponding illustrations are generated in the GCLC language. Animated illustrations are also supported.