Daily Papers

Private BitTorrent trackers enforce upload-to-download ratios to prevent free-riding, but suffer from three critical weaknesses: reputation cannot move between trackers, centralized servers create single points of failure, and upload statistics are self-reported and unverifiable. When a tracker shuts down, users lose their contribution history and cannot prove their standing to new communities. We address these problems by storing reputation in smart contracts and replacing self-reports with cryptographic attestations. Peers sign receipts for received pieces; the tracker aggregates them via BLS signatures and updates reputation. If a tracker is unavailable, peers fall back to an authenticated distributed hash table (DHT): stored reputation acts as a public key infrastructure (PKI), preserving access control without the tracker. Reputation is portable across tracker failures through single-hop migration in factory-deployed contracts. We also address the privacy implications of publishing public keys and reputations tied to private trackers on a public ledger: we propose ephemeral session keys to prevent linking peer identities, zero-knowledge membership proofs for anonymous DHT participation, and confidential reputation using homomorphic commitments. We formalize the security requirements, prove four security properties under standard cryptographic assumptions, and evaluate a prototype. Measurements show that transfer receipts add less than 5\% end-to-end overhead with typical piece sizes. To minimize signing overhead, we adopt a hybrid signature scheme: ECDSA signs individual piece receipts at transfer time for low per-operation latency, while BLS serves as the overarching scheme, enabling compact aggregation of many receipts into a single proof at report time. This design reduces client-side signing cost by an order of magnitude compared to using BLS throughout.

Zero-Knowledge Proofs (ZKPs) are a cryptographic primitive that allows a prover to demonstrate knowledge of a secret value to a verifier without revealing anything about the secret itself. ZKPs have shown to be an extremely powerful tool, as evidenced in both industry and academic settings. In recent years, the utilization of user data in practical applications has necessitated the rapid development of privacy-preserving techniques, including ZKPs. This has led to the creation of several robust open-source ZKP frameworks. However, there remains a significant gap in understanding the capabilities and real-world applications of these frameworks. Furthermore, identifying the most suitable frameworks for the developers' specific applications and settings is a challenge, given the variety of options available. The primary goal of our work is to lower the barrier to entry for understanding and building applications with open-source ZKP frameworks. In this work, we survey and evaluate 25 general-purpose, prominent ZKP frameworks. Recognizing that ZKPs have various constructions and underlying arithmetic schemes, our survey aims to provide a comprehensive overview of the ZKP landscape. These systems are assessed based on their usability and performance in SHA-256 and matrix multiplication experiments. Acknowledging that setting up a functional development environment can be challenging for these frameworks, we offer a fully open-source collection of Docker containers. These containers include a working development environment and are accompanied by documented code from our experiments. We conclude our work with a thorough analysis of the practical applications of ZKPs, recommendations for ZKP settings in different application scenarios, and a discussion on the future development of ZKP frameworks.

Recent advances in artificial intelligence (AI), particularly deep learning, have led to widespread adoption across various applications. Yet, a fundamental challenge persists: how can we verify the correctness of AI model inference when model owners cannot (or will not) reveal their parameters? These parameters represent enormous training costs and valuable intellectual property, making transparent verification difficult. In this paper, we introduce a zero-knowledge framework capable of verifying deep learning inference without exposing model internal parameters. Built on recursively composed zero-knowledge proofs and requiring no trusted setup, our framework supports both linear and nonlinear neural network layers, including matrix multiplication, normalization, softmax, and SiLU. Leveraging the Fiat-Shamir heuristic, we obtain a succinct non-interactive argument of knowledge (zkSNARK) with constant-size proofs. To demonstrate the practicality of our approach, we translate the DeepSeek model into a fully SNARK-verifiable version named ZK-DeepSeek and show experimentally that our framework delivers both efficiency and flexibility in real-world AI verification workloads.

The recent advancements in deep learning have brought about significant changes in various aspects of people's lives. Meanwhile, these rapid developments have raised concerns about the legitimacy of the training process of deep neural networks. To protect the intellectual properties of AI developers, directly examining the training process by accessing the model parameters and training data is often prohibited for verifiers. In response to this challenge, we present zero-knowledge deep learning (zkDL), an efficient zero-knowledge proof for deep learning training. To address the long-standing challenge of verifiable computations of non-linearities in deep learning training, we introduce zkReLU, a specialized proof for the ReLU activation and its backpropagation. zkReLU turns the disadvantage of non-arithmetic relations into an advantage, leading to the creation of FAC4DNN, our specialized arithmetic circuit design for modelling neural networks. This design aggregates the proofs over different layers and training steps, without being constrained by their sequential order in the training process. With our new CUDA implementation that achieves full compatibility with the tensor structures and the aggregated proof design, zkDL enables the generation of complete and sound proofs in less than a second per batch update for an 8-layer neural network with 10M parameters and a batch size of 64, while provably ensuring the privacy of data and model parameters. To our best knowledge, we are not aware of any existing work on zero-knowledge proof of deep learning training that is scalable to million-size networks.

Federated Learning (FL) has emerged as a promising paradigm in distributed machine learning, enabling collaborative model training while preserving data privacy. However, despite its many advantages, FL still contends with significant challenges -- most notably regarding security and trust. Zero-Knowledge Proofs (ZKPs) offer a potential solution by establishing trust and enhancing system integrity throughout the FL process. Although several studies have explored ZKP-based FL (ZK-FL), a systematic framework and comprehensive analysis are still lacking. This article makes two key contributions. First, we propose a structured ZK-FL framework that categorizes and analyzes the technical roles of ZKPs across various FL stages and tasks. Second, we introduce a novel algorithm, Verifiable Client Selection FL (Veri-CS-FL), which employs ZKPs to refine the client selection process. In Veri-CS-FL, participating clients generate verifiable proofs for the performance metrics of their local models and submit these concise proofs to the server for efficient verification. The server then selects clients with high-quality local models for uploading, subsequently aggregating the contributions from these selected clients. By integrating ZKPs, Veri-CS-FL not only ensures the accuracy of performance metrics but also fortifies trust among participants while enhancing the overall efficiency and security of FL systems.

Machine learning models are increasingly used in societal applications, yet legal and privacy concerns demand that they very often be kept confidential. Consequently, there is a growing distrust about the fairness properties of these models in the minds of consumers, who are often at the receiving end of model predictions. To this end, we propose \name -- a system that uses Zero-Knowledge Proofs (a cryptographic primitive) to publicly verify the fairness of a model, while maintaining confidentiality. We also propose a fairness certification algorithm for fully-connected neural networks which is befitting to ZKPs and is used in this system. We implement \name in Gnark and demonstrate empirically that our system is practically feasible. Code is available at https://github.com/infinite-pursuits/FairProof.

Machine learning models trained with differentially-private (DP) algorithms such as DP-SGD enjoy resilience against a wide range of privacy attacks. Although it is possible to derive bounds for some attacks based solely on an (varepsilon,delta)-DP guarantee, meaningful bounds require a small enough privacy budget (i.e., injecting a large amount of noise), which results in a large loss in utility. This paper presents a new approach to evaluate the privacy of machine learning models against specific record-level threats, such as membership and attribute inference, without the indirection through DP. We focus on the popular DP-SGD algorithm, and derive simple closed-form bounds. Our proofs model DP-SGD as an information theoretic channel whose inputs are the secrets that an attacker wants to infer (e.g., membership of a data record) and whose outputs are the intermediate model parameters produced by iterative optimization. We obtain bounds for membership inference that match state-of-the-art techniques, whilst being orders of magnitude faster to compute. Additionally, we present a novel data-dependent bound against attribute inference. Our results provide a direct, interpretable, and practical way to evaluate the privacy of trained models against specific inference threats without sacrificing utility.

Low-Rank Adaptation (LoRA) is a widely adopted method for customizing large-scale language models. In distributed, untrusted training environments, an open source base model user may want to use LoRA weights created by an external contributor, leading to two requirements: (1) the base model user must confirm that the LoRA weights are effective when paired with the intended base model, and (2) the LoRA contributor must keep their proprietary weights private until compensation is assured. We present ZKLoRA, a zero-knowledge verification protocol that relies on succinct proofs and our novel Multi-Party Inference procedure to verify LoRA-base model compatibility without exposing LoRA weights. ZKLoRA produces deterministic correctness guarantees and validates each LoRA module in only 1-2 seconds on state-of-the-art large language models. This low-latency approach enables nearly real-time verification and promotes secure collaboration among geographically decentralized teams and contract-based training pipelines. The protocol ensures that the delivered LoRA module works as claimed, safeguarding the contributor's intellectual property while providing the base model user with verification of compatibility and lineage.

The rapid advancement of ML models in critical sectors such as healthcare, finance, and security has intensified the need for robust data security, model integrity, and reliable outputs. Large multimodal foundational models, while crucial for complex tasks, present challenges in scalability, reliability, and potential misuse. Decentralized systems offer a solution by distributing workload and mitigating central points of failure, but they introduce risks of unauthorized access to sensitive data across nodes. We address these challenges with a comprehensive framework designed for responsible AI development. Our approach incorporates: 1) Zero-knowledge proofs for secure model verification, enhancing trust without compromising privacy. 2) Consensus-based verification checks to ensure consistent outputs across nodes, mitigating hallucinations and maintaining model integrity. 3) Split Learning techniques that segment models across different nodes, preserving data privacy by preventing full data access at any point. 4) Hardware-based security through trusted execution environments (TEEs) to protect data and computations. This framework aims to enhance security and privacy and improve the reliability and fairness of multimodal AI systems. Promoting efficient resource utilization contributes to more sustainable AI development. Our state-of-the-art proofs and principles demonstrate the framework's effectiveness in responsibly democratizing artificial intelligence, offering a promising approach for building secure and private foundational models.

AI agents that execute tasks via tool calls frequently hallucinate results - fabricating tool executions, misstating output counts, or presenting inferences as facts. Recent approaches to verifiable AI inference rely on zero-knowledge proofs, which provide cryptographic guarantees but impose minutes of proving time per query, making them impractical for interactive agents. We propose NabaOS, a lightweight verification framework inspired by Indian epistemology (Nyaya Shastra), which classifies every claim in an LLM response by its epistemic source (pramana): direct tool output (pratyaksha), inference (anumana), external testimony (shabda), absence (abhava), or ungrounded opinion. Our runtime generates HMAC-signed tool execution receipts that the LLM cannot forge, then cross-references claims against these receipts to detect hallucinations in real time. We evaluate on NyayaVerifyBench, a new benchmark of 1,800 agent response scenarios across four languages with injected hallucinations of six types. NabaOS detects 94.2% of fabricated tool references, 87.6% of count misstatements, and 91.3% of false absence claims, with <15ms verification overhead per response. For deep delegation (agents performing multi-step web tasks), our cross-checking protocol catches 78.4% of URL fabrications via independent re-fetching. We compare against five approaches: zkLLM (cryptographic proofs, 180s/query), TOPLOC (locality-sensitive hashing), SPEX (sampling-based proof of execution), tensor commitments, and self-consistency checking. NabaOS achieves the best cost-latency-coverage trade-off for interactive agents: 94.2% coverage at <15ms versus zkLLM's near-perfect coverage at 180,000ms. For interactive agents, practical receipt-based verification provides better cost-benefit than cryptographic proofs, and epistemic classification gives users actionable trust signals rather than binary judgments.

Privacy-preserving federated learning allows multiple users to jointly train a model with coordination of a central server. The server only learns the final aggregation result, thus the users' (private) training data is not leaked from the individual model updates. However, keeping the individual updates private allows malicious users to perform Byzantine attacks and degrade the accuracy without being detected. Best existing defenses against Byzantine workers rely on robust rank-based statistics, e.g., median, to find malicious updates. However, implementing privacy-preserving rank-based statistics is nontrivial and not scalable in the secure domain, as it requires sorting all individual updates. We establish the first private robustness check that uses high break point rank-based statistics on aggregated model updates. By exploiting randomized clustering, we significantly improve the scalability of our defense without compromising privacy. We leverage our statistical bounds in zero-knowledge proofs to detect and remove malicious updates without revealing the private user updates. Our novel framework, zPROBE, enables Byzantine resilient and secure federated learning. Empirical evaluations demonstrate that zPROBE provides a low overhead solution to defend against state-of-the-art Byzantine attacks while preserving privacy.

Membership inference attacks serves as useful tool for fair use of language models, such as detecting potential copyright infringement and auditing data leakage. However, many current state-of-the-art attacks require access to models' hidden states or probability distribution, which prevents investigation into more widely-used, API-access only models like GPT-4. In this work, we introduce N-Gram Coverage Attack, a membership inference attack that relies solely on text outputs from the target model, enabling attacks on completely black-box models. We leverage the observation that models are more likely to memorize and subsequently generate text patterns that were commonly observed in their training data. Specifically, to make a prediction on a candidate member, N-Gram Coverage Attack first obtains multiple model generations conditioned on a prefix of the candidate. It then uses n-gram overlap metrics to compute and aggregate the similarities of these outputs with the ground truth suffix; high similarities indicate likely membership. We first demonstrate on a diverse set of existing benchmarks that N-Gram Coverage Attack outperforms other black-box methods while also impressively achieving comparable or even better performance to state-of-the-art white-box attacks - despite having access to only text outputs. Interestingly, we find that the success rate of our method scales with the attack compute budget - as we increase the number of sequences generated from the target model conditioned on the prefix, attack performance tends to improve. Having verified the accuracy of our method, we use it to investigate previously unstudied closed OpenAI models on multiple domains. We find that more recent models, such as GPT-4o, exhibit increased robustness to membership inference, suggesting an evolving trend toward improved privacy protections.

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

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

Zero-knowledge proof (ZKP) provers remain costly because multi-scalar multiplication (MSM) and number-theoretic transforms (NTTs) dominate runtime as they need significant computation. AI ASICs such as TPUs provide massive matrix throughput and SotA energy efficiency. We present MORPH, the first framework that reformulates ZKP kernels to match AI-ASIC execution. We introduce Big-T complexity, a hardware-aware complexity model that exposes heterogeneous bottlenecks and layout-transformation costs ignored by Big-O. Guided by this analysis, (1) at arithmetic level, MORPH develops an MXU-centric extended-RNS lazy reduction that converts high-precision modular arithmetic into dense low-precision GEMMs, eliminating all carry chains, and (2) at dataflow level, MORPH constructs a unified-sharding layout-stationary TPU Pippenger MSM and optimized 3/5-step NTT that avoid on-TPU shuffles to minimize costly memory reorganization. Implemented in JAX, MORPH enables TPUv6e8 to achieve up-to 10x higher throughput on NTT and comparable throughput on MSM than GZKP. Our code: https://github.com/EfficientPPML/MORPH.

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

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

Membership inference attacks aim to infer whether an individual record was used to train a model, serving as a key tool for disclosure auditing. While such evaluations are useful to demonstrate risk, they are computationally expensive and often make strong assumptions about potential adversaries' access to models and training environments, and thus do not provide very tight bounds on leakage from potential attacks. We show how prior claims around black-box access being sufficient for optimal membership inference do not hold for most useful settings such as stochastic gradient descent, and that optimal membership inference indeed requires white-box access. We validate our findings with a new white-box inference attack IHA (Inverse Hessian Attack) that explicitly uses model parameters by taking advantage of computing inverse-Hessian vector products. Our results show that both audits and adversaries may be able to benefit from access to model parameters, and we advocate for further research into white-box methods for membership privacy auditing.

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

As industrial applications are increasingly automated by machine learning models, enforcing personal data ownership and intellectual property rights requires tracing training data back to their rightful owners. Membership inference algorithms approach this problem by using statistical techniques to discern whether a target sample was included in a model's training set. However, existing methods only utilize the unaltered target sample or simple augmentations of the target to compute statistics. Such a sparse sampling of the model's behavior carries little information, leading to poor inference capabilities. In this work, we use adversarial tools to directly optimize for queries that are discriminative and diverse. Our improvements achieve significantly more accurate membership inference than existing methods, especially in offline scenarios and in the low false-positive regime which is critical in legal settings. Code is available at https://github.com/YuxinWenRick/canary-in-a-coalmine.

The integration of machine learning (ML) in numerous critical applications introduces a range of privacy concerns for individuals who provide their datasets for model training. One such privacy risk is Membership Inference (MI), in which an attacker seeks to determine whether a particular data sample was included in the training dataset of a model. Current state-of-the-art MI attacks capitalize on access to the model's predicted confidence scores to successfully perform membership inference, and employ data poisoning to further enhance their effectiveness. In this work, we focus on the less explored and more realistic label-only setting, where the model provides only the predicted label on a queried sample. We show that existing label-only MI attacks are ineffective at inferring membership in the low False Positive Rate (FPR) regime. To address this challenge, we propose a new attack Chameleon that leverages a novel adaptive data poisoning strategy and an efficient query selection method to achieve significantly more accurate membership inference than existing label-only attacks, especially at low FPRs.

Membership Inference Attacks (MIAs) have emerged as a valuable framework for evaluating privacy leakage by machine learning models. Score-based MIAs are distinguished, in particular, by their ability to exploit the confidence scores that the model generates for particular inputs. Existing score-based MIAs implicitly assume that the adversary has access to the target model's hyperparameters, which can be used to train the shadow models for the attack. In this work, we demonstrate that the knowledge of target hyperparameters is not a prerequisite for MIA in the transfer learning setting. Based on this, we propose a novel approach to select the hyperparameters for training the shadow models for MIA when the attacker has no prior knowledge about them by matching the output distributions of target and shadow models. We demonstrate that using the new approach yields hyperparameters that lead to an attack near indistinguishable in performance from an attack that uses target hyperparameters to train the shadow models. Furthermore, we study the empirical privacy risk of unaccounted use of training data for hyperparameter optimization (HPO) in differentially private (DP) transfer learning. We find no statistically significant evidence that performing HPO using training data would increase vulnerability to MIA.

Adapting Large Language Models (LLMs) to specific tasks introduces concerns about computational efficiency, prompting an exploration of efficient methods such as In-Context Learning (ICL). However, the vulnerability of ICL to privacy attacks under realistic assumptions remains largely unexplored. In this work, we present the first membership inference attack tailored for ICL, relying solely on generated texts without their associated probabilities. We propose four attack strategies tailored to various constrained scenarios and conduct extensive experiments on four popular large language models. Empirical results show that our attacks can accurately determine membership status in most cases, e.g., 95\% accuracy advantage against LLaMA, indicating that the associated risks are much higher than those shown by existing probability-based attacks. Additionally, we propose a hybrid attack that synthesizes the strengths of the aforementioned strategies, achieving an accuracy advantage of over 95\% in most cases. Furthermore, we investigate three potential defenses targeting data, instruction, and output. Results demonstrate combining defenses from orthogonal dimensions significantly reduces privacy leakage and offers enhanced privacy assurances.

Blockchain-based Attribute-Based Access Control (BC-ABAC) offers a decentralized paradigm for secure data governance but faces two inherent challenges: the transparency of blockchain ledgers threatens user privacy by enabling reidentification attacks through attribute analysis, while the computational complexity of policy matching clashes with blockchain's performance constraints. Existing solutions, such as those employing Zero-Knowledge Proofs (ZKPs), often incur high overhead and lack measurable anonymity guarantees, while efficiency optimizations frequently ignore privacy implications. To address these dual challenges, this paper proposes QAEBAC (Quantifiable Anonymity and Efficiency in Blockchain-Based Access Control with Attribute). QAE-BAC introduces a formal (r, t)-anonymity model to dynamically quantify the re-identification risk of users based on their access attributes and history. Furthermore, it features an Entropy-Weighted Path Tree (EWPT) that optimizes policy structure based on realtime anonymity metrics, drastically reducing policy matching complexity. Implemented and evaluated on Hyperledger Fabric, QAE-BAC demonstrates a superior balance between privacy and performance. Experimental results show that it effectively mitigates re-identification risks and outperforms state-of-the-art baselines, achieving up to an 11x improvement in throughput and an 87% reduction in latency, proving its practicality for privacy-sensitive decentralized applications.

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

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

Small language models (SLMs) are increasingly valued for their efficiency and deployability in resource-constrained environments, making them useful for on-device, privacy-sensitive, and edge computing applications. On the other hand, membership inference attacks (MIAs), which aim to determine whether a given sample was used in a model's training, are an important threat with serious privacy and intellectual property implications. In this paper, we study MIAs on SLMs. Although MIAs were shown to be effective on large language models (LLMs), they are relatively less studied on emerging SLMs, and furthermore, their effectiveness decreases as models get smaller. Motivated by this finding, we propose a new MIA called win-k, which builds on top of a state-of-the-art attack (min-k). We experimentally evaluate win-k by comparing it with five existing MIAs using three datasets and eight SLMs. Results show that win-k outperforms existing MIAs in terms of AUROC, TPR @ 1% FPR, and FPR @ 99% TPR metrics, especially on smaller models.

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

This definitive research memoria presents a comprehensive, mathematically verified paradigm for neural communication with Bitcoin mining Application-Specific Integrated Circuits (ASICs), integrating five complementary frameworks: thermodynamic reservoir computing, hierarchical number system theory, algorithmic analysis, network latency optimization, and machine-checked mathematical formalization. We establish that obsolete cryptocurrency mining hardware exhibits emergent computational properties enabling bidirectional information exchange between AI systems and silicon substrates. The research program demonstrates: (1) reservoir computing with NARMA-10 Normalized Root Mean Square Error (NRMSE) of 0.8661; (2) the Thermodynamic Probability Filter (TPF) achieving 92.19% theoretical energy reduction; (3) the Virtual Block Manager achieving +25% effective hashrate; and (4) hardware universality across multiple ASIC families including Antminer S9, Lucky Miner LV06, and Goldshell LB-Box. A significant contribution is the machine-checked mathematical formalization using Lean 4 and Mathlib, providing unambiguous definitions, machine-verified theorems, and reviewer-proof claims. Key theorems proven include: independence implies zero leakage, predictor beats baseline implies non-independence (the logical core of TPF), energy savings theoretical maximum, and Physical Unclonable Function (PUF) distinguishability witnesses. Vladimir Veselov's hierarchical number system theory explains why early-round information contains predictive power. This work establishes a new paradigm: treating ASICs not as passive computational substrates but as active conversational partners whose thermodynamic state encodes exploitable computational information.

We revisit the longstanding open problem of implementing Nakamoto's proof-of-work (PoW) consensus based on a real-world computational task T(x) (as opposed to artificial random hashing), in a truly permissionless setting where the miner itself chooses the input x. The challenge in designing such a Proof-of-Useful-Work (PoUW) protocol, is using the native computation of T(x) to produce a PoW certificate with prescribed hardness and with negligible computational overhead over the worst-case complexity of T(cdot) -- This ensures malicious miners cannot ``game the system" by fooling the verifier to accept with higher probability compared to honest miners (while using similar computational resources). Indeed, obtaining a PoUW with O(1)-factor overhead is trivial for any task T, but also useless. Our main result is a PoUW for the task of Matrix Multiplication MatMul(A,B) of arbitrary matrices with 1+o(1) multiplicative overhead compared to naive MatMul (even in the presence of Fast Matrix Multiplication-style algorithms, which are currently impractical). We conjecture that our protocol has optimal security in the sense that a malicious prover cannot obtain any significant advantage over an honest prover. This conjecture is based on reducing hardness of our protocol to the task of solving a batch of low-rank random linear equations which is of independent interest. Since MatMuls are the bottleneck of AI compute as well as countless industry-scale applications, this primitive suggests a concrete design of a new L1 base-layer protocol, which nearly eliminates the energy-waste of Bitcoin mining -- allowing GPU consumers to reduce their AI training and inference costs by ``re-using" it for blockchain consensus, in exchange for block rewards (2-for-1). This blockchain is currently under construction.

Federated learning (FL) has been widely adopted in various fields of study and business. Traditional centralized FL systems suffer from serious issues. To address these concerns, decentralized federated learning (DFL) systems have been introduced in recent years. With the help of blockchains, they attempt to achieve more integrity and efficiency. However, privacy preservation remains an uncovered aspect of these systems. To tackle this, as well as to scale the blockchain-based computations, we propose a zero-knowledge proof (ZKP)-based aggregator (zkDFL). This allows clients to share their large-scale model parameters with a trusted centralized server without revealing their individual data to other clients. We utilize blockchain technology to manage the aggregation algorithm via smart contracts. The server performs a ZKP algorithm to prove to the clients that the aggregation is done according to the accepted algorithm. Additionally, the server can prove that all inputs from clients have been used. We evaluate our approach using a public dataset related to the wearable Internet of Things. As demonstrated by numerical evaluations, zkDFL introduces verifiability of the correctness of the aggregation process and enhances the privacy protection and scalability of DFL systems, while the gas cost has significantly declined.

Human verification under adversarial information flow operates as a cost-bounded decision procedure constrained by working memory limits and cognitive biases. We introduce the Verification Cost Asymmetry (VCA) coefficient, formalizing it as the ratio of expected verification work between populations under identical claim distributions. Drawing on probabilistically checkable proofs (PCP) and parameterized complexity theory, we construct dissemination protocols that reduce verification for trusted audiences to constant human effort while imposing superlinear costs on adversarial populations lacking cryptographic infrastructure. We prove theoretical guarantees for this asymmetry, validate the framework through controlled user studies measuring verification effort with and without spot-checkable provenance, and demonstrate practical encoding of real-world information campaigns. The results establish complexity-theoretic foundations for engineering democratic advantage in cognitive warfare, with immediate applications to content authentication, platform governance, and information operations doctrine.

We often interact with untrusted parties. Prioritization of privacy can limit the effectiveness of these interactions, as achieving certain goals necessitates sharing private data. Traditionally, addressing this challenge has involved either seeking trusted intermediaries or constructing cryptographic protocols that restrict how much data is revealed, such as multi-party computations or zero-knowledge proofs. While significant advances have been made in scaling cryptographic approaches, they remain limited in terms of the size and complexity of applications they can be used for. In this paper, we argue that capable machine learning models can fulfill the role of a trusted third party, thus enabling secure computations for applications that were previously infeasible. In particular, we describe Trusted Capable Model Environments (TCMEs) as an alternative approach for scaling secure computation, where capable machine learning model(s) interact under input/output constraints, with explicit information flow control and explicit statelessness. This approach aims to achieve a balance between privacy and computational efficiency, enabling private inference where classical cryptographic solutions are currently infeasible. We describe a number of use cases that are enabled by TCME, and show that even some simple classic cryptographic problems can already be solved with TCME. Finally, we outline current limitations and discuss the path forward in implementing them.

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.

A Membership Inference Attack (MIA) assesses how much a trained machine learning model reveals about its training data by determining whether specific query instances were included in the dataset. We classify existing MIAs into adaptive or non-adaptive, depending on whether the adversary is allowed to train shadow models on membership queries. In the adaptive setting, where the adversary can train shadow models after accessing query instances, we highlight the importance of exploiting membership dependencies between instances and propose an attack-agnostic framework called Cascading Membership Inference Attack (CMIA), which incorporates membership dependencies via conditional shadow training to boost membership inference performance. In the non-adaptive setting, where the adversary is restricted to training shadow models before obtaining membership queries, we introduce Proxy Membership Inference Attack (PMIA). PMIA employs a proxy selection strategy that identifies samples with similar behaviors to the query instance and uses their behaviors in shadow models to perform a membership posterior odds test for membership inference. We provide theoretical analyses for both attacks, and extensive experimental results demonstrate that CMIA and PMIA substantially outperform existing MIAs in both settings, particularly in the low false-positive regime, which is crucial for evaluating privacy risks.

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

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

In recent years, generative artificial intelligence (GenAI) has demonstrated remarkable capabilities in high-stakes domains such as molecular science. However, challenges related to the verifiability and structural privacy of its outputs remain largely unresolved. This paper focuses on the task of molecular toxicity repair. It proposes a structure-private verification framework - ToxiEval-ZKP - which, for the first time, introduces zero-knowledge proof (ZKP) mechanisms into the evaluation process of this task. The system enables model developers to demonstrate to external verifiers that the generated molecules meet multidimensional toxicity repair criteria, without revealing the molecular structures themselves. To this end, we design a general-purpose circuit compatible with both classification and regression tasks, incorporating evaluation logic, Poseidon-based commitment hashing, and a nullifier-based replay prevention mechanism to build a complete end-to-end ZK verification system. Experimental results demonstrate that ToxiEval-ZKP facilitates adequate validation under complete structural invisibility, offering strong circuit efficiency, security, and adaptability, thereby opening up a novel paradigm for trustworthy evaluation in generative scientific tasks.

A membership inference attack allows an adversary to query a trained machine learning model to predict whether or not a particular example was contained in the model's training dataset. These attacks are currently evaluated using average-case "accuracy" metrics that fail to characterize whether the attack can confidently identify any members of the training set. We argue that attacks should instead be evaluated by computing their true-positive rate at low (e.g., <0.1%) false-positive rates, and find most prior attacks perform poorly when evaluated in this way. To address this we develop a Likelihood Ratio Attack (LiRA) that carefully combines multiple ideas from the literature. Our attack is 10x more powerful at low false-positive rates, and also strictly dominates prior attacks on existing metrics.

Membership inference attacks (MIAs) pose a critical privacy threat to fine-tuned large language models (LLMs), especially when models are adapted to domain-specific tasks using sensitive data. While prior black-box MIA techniques rely on confidence scores or token likelihoods, these signals are often entangled with a sample's intrinsic properties - such as content difficulty or rarity - leading to poor generalization and low signal-to-noise ratios. In this paper, we propose ICP-MIA, a novel MIA framework grounded in the theory of training dynamics, particularly the phenomenon of diminishing returns during optimization. We introduce the Optimization Gap as a fundamental signal of membership: at convergence, member samples exhibit minimal remaining loss-reduction potential, while non-members retain significant potential for further optimization. To estimate this gap in a black-box setting, we propose In-Context Probing (ICP), a training-free method that simulates fine-tuning-like behavior via strategically constructed input contexts. We propose two probing strategies: reference-data-based (using semantically similar public samples) and self-perturbation (via masking or generation). Experiments on three tasks and multiple LLMs show that ICP-MIA significantly outperforms prior black-box MIAs, particularly at low false positive rates. We further analyze how reference data alignment, model type, PEFT configurations, and training schedules affect attack effectiveness. Our findings establish ICP-MIA as a practical and theoretically grounded framework for auditing privacy risks in deployed LLMs.

As large language models (LLMs) are trained on increasingly opaque corpora, membership inference attacks (MIAs) have been proposed to audit whether copyrighted texts were used during training, despite growing concerns about their reliability under realistic conditions. We ask whether MIAs can serve as admissible evidence in adversarial copyright disputes where an accused model developer may obfuscate training data while preserving semantic content, and formalize this setting through a judge-prosecutor-accused communication protocol. To test robustness under this protocol, we introduce SAGE (Structure-Aware SAE-Guided Extraction), a paraphrasing framework guided by Sparse Autoencoders (SAEs) that rewrites training data to alter lexical structure while preserving semantic content and downstream utility. Our experiments show that state-of-the-art MIAs degrade when models are fine-tuned on SAGE-generated paraphrases, indicating that their signals are not robust to semantics-preserving transformations. While some leakage remains in certain fine-tuning regimes, these results suggest that MIAs are brittle in adversarial settings and insufficient, on their own, as a standalone mechanism for copyright auditing of LLMs.

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

Generative diffusion models, including Stable Diffusion and Midjourney, can generate visually appealing, diverse, and high-resolution images for various applications. These models are trained on billions of internet-sourced images, raising significant concerns about the potential unauthorized use of copyright-protected images. In this paper, we examine whether it is possible to determine if a specific image was used in the training set, a problem known in the cybersecurity community and referred to as a membership inference attack. Our focus is on Stable Diffusion, and we address the challenge of designing a fair evaluation framework to answer this membership question. We propose a methodology to establish a fair evaluation setup and apply it to Stable Diffusion, enabling potential extensions to other generative models. Utilizing this evaluation setup, we execute membership attacks (both known and newly introduced). Our research reveals that previously proposed evaluation setups do not provide a full understanding of the effectiveness of membership inference attacks. We conclude that the membership inference attack remains a significant challenge for large diffusion models (often deployed as black-box systems), indicating that related privacy and copyright issues will persist in the foreseeable future.

Automated negotiations in insurance and business-to-business (B2B) commerce encounter substantial challenges. Current systems force a trade-off between convenience and privacy by routing sensitive financial data through centralized servers, increasing security risks, and diminishing user trust. This study introduces a device-native autonomous Agentic AI system for privacy-preserving negotiations. The proposed system operates exclusively on user hardware, enabling real-time bargaining while maintaining sensitive constraints locally. It integrates zero-knowledge proofs to ensure privacy and employs distilled world models to support advanced on-device reasoning. The architecture incorporates six technical components within an Agentic AI workflow. Agents autonomously plan negotiation strategies, conduct secure multi-party bargaining, and generate cryptographic audit trails without exposing user data to external servers. The system is evaluated in insurance and B2B procurement scenarios across diverse device configurations. Results show an average success rate of 87 %, a 2.4x reduction in latency relative to cloud baselines, and strong privacy preservation through zero-knowledge proofs. User studies show 27 % higher trust scores when decision trails are available. These findings establish a foundation for trustworthy autonomous agents in privacy-sensitive financial domains.

Understanding what generative models retain from training data remains challenging, with implications for copyright and privacy. Beyond verbatim reproduction, models can encode subtler traces of their training data that never surface in their outputs yet remain exploitable. We study this regime for Rectified Flows, which are increasingly used in deployed generative systems. We analyse the interpolation path X_λ= (1-λ)X_0 + λX_1 that defines the Rectified Flow training. We show that a gap exists between the reconstruction of train and test data that follows a bell-shaped curve over λ, wich accumulates during training, while the validation metrics remain stable. The signal has a maximum whose location we derive in closed form under Gaussian assumptions. We validate these predictions on both audio and images and show that the bell-shaped structure is universal, while the peak prediction holds when our assumptions are satisfied. As a proof of concept, we exploit this specific λ-resolved structure to perform a Membership Inference Attack, distinguishing members of the training set from non-members.

Machine unlearning aims to remove the influence of specific data points from a trained model to satisfy privacy, copyright, and safety requirements. In real deployments, providers distribute a global model to many edge devices, where each client personalizes the model using private data. When a deletion request is issued, clients may ignore it or falsely claim compliance, and providers cannot check their parameters or data. This makes verification difficult, especially because personalized models must forget the targeted samples while preserving local utility, and verification must remain lightweight on edge devices. We introduce ZK APEX, a zero-shot personalized unlearning method that operates directly on the personalized model without retraining. ZK APEX combines sparse masking on the provider side with a small Group OBS compensation step on the client side, using a blockwise empirical Fisher matrix to create a curvature-aware update designed for low overhead. Paired with Halo2 zero-knowledge proofs, it enables the provider to verify that the correct unlearning transformation was applied without revealing any private data or personalized parameters. On Vision Transformer classification tasks, ZK APEX recovers nearly all personalization accuracy while effectively removing the targeted information. Applied to the OPT125M generative model trained on code data, it recovers around seventy percent of the original accuracy. Proof generation for the ViT case completes in about two hours, more than ten million times faster than retraining-based checks, with less than one gigabyte of memory use and proof sizes around four hundred megabytes. These results show the first practical framework for verifiable personalized unlearning on edge devices.

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

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.

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.

Membership inference attacks (MIAs) pose a significant threat to the privacy of machine learning models and are widely used as tools for privacy assessment, auditing, and machine unlearning. While prior MIA research has primarily focused on performance metrics such as AUC, accuracy, and TPR@low FPR - either by developing new methods to enhance these metrics or using them to evaluate privacy solutions - we found that it overlooks the disparities among different attacks. These disparities, both between distinct attack methods and between multiple instantiations of the same method, have crucial implications for the reliability and completeness of MIAs as privacy evaluation tools. In this paper, we systematically investigate these disparities through a novel framework based on coverage and stability analysis. Extensive experiments reveal significant disparities in MIAs, their potential causes, and their broader implications for privacy evaluation. To address these challenges, we propose an ensemble framework with three distinct strategies to harness the strengths of state-of-the-art MIAs while accounting for their disparities. This framework not only enables the construction of more powerful attacks but also provides a more robust and comprehensive methodology for privacy evaluation.

Large Language Models (LLMs) demonstrate strong capabilities for solving scientific and mathematical problems, yet they struggle to produce valid, challenging, and novel problems - an essential component for advancing LLM training and enabling autonomous scientific research. Existing problem generation approaches either depend on expensive human expert involvement or adopt naive self-play paradigms, which frequently yield invalid problems due to reward hacking. This work introduces VHG, a verifier-enhanced hard problem generation framework built upon three-party self-play. By integrating an independent verifier into the conventional setter-solver duality, our design constrains the setter's reward to be jointly determined by problem validity (evaluated by the verifier) and difficulty (assessed by the solver). We instantiate two verifier variants: a Hard symbolic verifier and a Soft LLM-based verifier, with evaluations conducted on indefinite integral tasks and general mathematical reasoning tasks. Experimental results show that VHG substantially outperforms all baseline methods by a clear margin.

The vulnerability of machine learning models to membership inference attacks has received much attention in recent years. However, existing attacks mostly remain impractical due to having high false positive rates, where non-member samples are often erroneously predicted as members. This type of error makes the predicted membership signal unreliable, especially since most samples are non-members in real world applications. In this work, we argue that membership inference attacks can benefit drastically from difficulty calibration, where an attack's predicted membership score is adjusted to the difficulty of correctly classifying the target sample. We show that difficulty calibration can significantly reduce the false positive rate of a variety of existing attacks without a loss in accuracy.

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

Recently, large language models (LLMs) have been gaining a lot of interest due to their adaptability and extensibility in emerging applications, including communication networks. It is anticipated that ZSM networks will be able to support LLMs as a service, as they provide ultra reliable low-latency communications and closed loop massive connectivity. However, LLMs are vulnerable to data and model privacy issues that affect the trustworthiness of LLMs to be deployed for user-based services. In this paper, we explore the security vulnerabilities associated with fine-tuning LLMs in ZSM networks, in particular the membership inference attack. We define the characteristics of an attack network that can perform a membership inference attack if the attacker has access to the fine-tuned model for the downstream task. We show that the membership inference attacks are effective for any downstream task, which can lead to a personal data breach when using LLM as a service. The experimental results show that the attack success rate of maximum 92% can be achieved on named entity recognition task. Based on the experimental analysis, we discuss possible defense mechanisms and present possible research directions to make the LLMs more trustworthy in the context of ZSM networks.

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

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.

This whitepaper seeks to elucidate implications that the capabilities of developing quantum architectures have on blockchain vulnerabilities and mitigation strategies. First, we provide new resource estimates for breaking the 256-bit Elliptic Curve Discrete Logarithm Problem, the core of modern blockchain cryptography. We demonstrate that Shor's algorithm for this problem can execute with either <1200 logical qubits and <90 million Toffoli gates or <1450 logical qubits and <70 million Toffoli gates. In the interest of responsible disclosure, we use a zero-knowledge proof to validate these results without disclosing attack vectors. On superconducting architectures with 1e-3 physical error rates and planar connectivity, those circuits can execute in minutes using fewer than half a million physical qubits. We introduce a critical distinction between fast-clock (such as superconducting and photonic) and slow-clock (such as neutral atom and ion trap) architectures. Our analysis reveals that the first fast-clock CRQCs would enable on-spend attacks on public mempool transactions of some cryptocurrencies. We survey major cryptocurrency vulnerabilities through this lens, identifying systemic risks associated with advanced features in some blockchains such as smart contracts, Proof-of-Stake consensus, and Data Availability Sampling, as well as the enduring concern of abandoned assets. We argue that technical solutions would benefit from accompanying public policy and discuss various frameworks of digital salvage to regulate the recovery or destruction of dormant assets while preventing adversarial seizure. We also discuss implications for other digital assets and tokenization as well as challenges and successful examples of the ongoing transition to Post-Quantum Cryptography (PQC). Finally, we urge all vulnerable cryptocurrency communities to join the ongoing migration to PQC without delay.

The Bernays-Sch\"onfinkel first-order logic fragment over simple linear real arithmetic constraints BS(SLR) is known to be decidable. We prove that BS(SLR) clause sets with both universally and existentially quantified verification conditions (conjectures) can be translated into BS(SLR) clause sets over a finite set of first-order constants. For the Horn case, we provide a Datalog hammer preserving validity and satisfiability. A toolchain from the BS(LRA) prover SPASS-SPL to the Datalog reasoner VLog establishes an effective way of deciding verification conditions in the Horn fragment. This is exemplified by the verification of supervisor code for a lane change assistant in a car and of an electronic control unit for a supercharged combustion engine.

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

GossipSub is a new peer-to-peer communication protocol designed to counter attacks from misbehaving peers by controlling what information is sent and to whom, via a score function computed by each peer that captures positive and negative behaviors of its neighbors. The score function depends on several parameters (weights, caps, thresholds) that can be configured by applications using GossipSub. The specification for GossipSub is written in English and its resilience to attacks from misbehaving peers is supported empirically by emulation testing using an implementation in Golang. In this work we take a foundational approach to understanding the resilience of GossipSub to attacks from misbehaving peers. We build the first formal model of GossipSub, using the ACL2s theorem prover. Our model is officially endorsed by the GossipSub developers. It can simulate GossipSub networks of arbitrary size and topology, with arbitrarily configured peers, and can be used to prove and disprove theorems about the protocol. We formalize fundamental security properties stating that the score function is fair, penalizes bad behavior, and rewards good behavior. We prove that the score function is always fair, but can be configured in ways that either penalize good behavior or ignore bad behavior. Using our model, we run GossipSub with the specific configurations for two popular real-world applications: the FileCoin and Eth2.0 blockchains. We show that all properties hold for FileCoin. However, given any Eth2.0 network (of any topology and size) with any number of potentially misbehaving peers, we can synthesize attacks where these peers are able to continuously misbehave by never forwarding topic messages, while maintaining positive scores so that they are never pruned from the network by GossipSub.

We present MaxProof, a population-level test-time scaling framework for competition-level mathematical proof in the MiniMax-M3 series. M3 first trains three proof-oriented capabilities -- proof generation, proof verification, and critique-conditioned proof repair -- using a defense-in-depth generative verifier engineered for low false-positive rate. These capabilities are merged into a single released M3 model. At test time, MaxProof treats the model as a generator, verifier, refiner, and ranker, searches over a population of candidate proofs, and returns one final proof through tournament selection. With MaxProof test-time scaling, the M3 model reaches 35/42 on IMO 2025 and 36/42 on USAMO 2026, exceeding the human gold-medal threshold on both.

Privacy amplification exploits randomness in data selection to provide tighter differential privacy (DP) guarantees. This analysis is key to DP-SGD's success in machine learning, but, is not readily applicable to the newer state-of-the-art algorithms. This is because these algorithms, known as DP-FTRL, use the matrix mechanism to add correlated noise instead of independent noise as in DP-SGD. In this paper, we propose "MMCC", the first algorithm to analyze privacy amplification via sampling for any generic matrix mechanism. MMCC is nearly tight in that it approaches a lower bound as epsilonto0. To analyze correlated outputs in MMCC, we prove that they can be analyzed as if they were independent, by conditioning them on prior outputs. Our "conditional composition theorem" has broad utility: we use it to show that the noise added to binary-tree-DP-FTRL can asymptotically match the noise added to DP-SGD with amplification. Our amplification algorithm also has practical empirical utility: we show it leads to significant improvement in the privacy-utility trade-offs for DP-FTRL algorithms on standard benchmarks.

Automated formal verification remains challenging for large language models because data for proof assistants and verification-aware languages is scarce, and correctness depends on satisfying precise machine-checkable specifications rather than producing plausible code. This thesis studies how verifier environments can improve LLM generation of verified programs and proofs through reinforcement learning from verifiable rewards (RLVR) and verifier-guided inference-time search. First, we train open-source models in Dafny with RLVR using Group Relative Policy Optimization (GRPO) and related variants, assembling generated candidates into complete programs and scoring them with compiler and verifier outcomes. Initial experiments on an APPS-derived Dafny dataset increased verified reward from 2.2% to 58.1%, but revealed specification hacking, where models exploit weak formal specifications instead of implementing the intended solutions. After filtering underspecified and vulnerable tasks, multi-turn RLVR on the refined benchmark improves the verified pass rate from 9.7% to 31.1%. Second, we develop a verifier-guided inference scaffold in Lean that treats proof generation as structured search over decomposed subgoals, verifier feedback, diagnostics, and repair. With a fixed base model, the full scaffold with proof reviser improves pass rate on an initial VeriCoding pilot set from 46.2% under direct repair to 69.2%. On the larger VERINA dataset, whole-task decomposition plus proof reviser solves 7 of 42 previously unsolved tasks. We also introduce Dalek-Bench, a repository-scale Lean benchmark derived from the Rust curve25519-dalek verification project; preliminary results remain weak, indicating that stronger progress evaluation and task-specific tool-use policies are still needed.

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

Membership Inference Attacks (MIAs) expose privacy risks by determining whether a specific sample was part of a model's training set. These threats are especially serious in sensitive domains such as healthcare and finance. Traditional mitigation techniques, such as static differential privacy, rely on injecting a fixed amount of noise during training or inference. However, this often leads to a detrimental trade-off: the noise may be insufficient to counter sophisticated attacks or, when increased, can substantially degrade model accuracy. To address this limitation, we propose DynaNoise, an adaptive inference-time defense that modulates injected noise based on per-query sensitivity. DynaNoise estimates risk using measures such as Shannon entropy and scales the noise variance accordingly, followed by a smoothing step that re-normalizes the perturbed outputs to preserve predictive utility. We further introduce MIDPUT (Membership Inference Defense Privacy-Utility Trade-off), a scalar metric that captures both privacy gains and accuracy retention. Our evaluation on several benchmark datasets demonstrates that DynaNoise substantially lowers attack success rates while maintaining competitive accuracy, achieving strong overall MIDPUT scores compared to state-of-the-art defenses.

Large language models (LLMs) have proven to be very capable, but access to the best models currently rely on inference providers which introduces trust challenges -- how can we be sure that the provider is using the model configuration they claim? We propose TOPLOC, a novel method for verifiable inference that addresses this problem. TOPLOC leverages a compact locality sensitive hashing mechanism for intermediate activations which can detect unauthorized modifications to models, prompts, or precision with 100% accuracy, achieving no false positives or negatives in our empirical evaluations. Our approach is robust across diverse hardware configurations, GPU types, and algebraic reorderings, which allows for validation speeds significantly faster than the original inference. By introducing a polynomial encoding scheme, TOPLOC minimizes memory overhead of the generated commits by 1000times, requiring only 258 bytes of storage per 32 new tokens compared to the 262KB requirement of storing the token embeddings directly for Llama-3.1-8B-Instruct. Our method empowers users to verify LLM inference computations efficiently, fostering greater trust and transparency in open ecosystems and lays a foundation for decentralized and verifiable AI services.

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 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.

Membership Inference Attacks (MIAs) act as a crucial auditing tool for the opaque training data of Large Language Models (LLMs). However, existing techniques predominantly rely on inaccessible model internals (e.g., logits) or suffer from poor generalization across domains in strict black-box settings where only generated text is available. In this work, we propose SimMIA, a robust MIA framework tailored for this text-only regime by leveraging an advanced sampling strategy and scoring mechanism. Furthermore, we present WikiMIA-25, a new benchmark curated to evaluate MIA performance on modern proprietary LLMs. Experiments demonstrate that SimMIA achieves state-of-the-art results in the black-box setting, rivaling baselines that exploit internal model information.

As post-training techniques evolve, large language models (LLMs) are increasingly augmented with structured multi-step reasoning abilities, often optimized through reinforcement learning. These reasoning-enhanced models outperform standard LLMs on complex tasks and now underpin many commercial LLM APIs. However, to protect proprietary behavior and reduce verbosity, providers typically conceal the reasoning traces while returning only the final answer. This opacity introduces a critical transparency gap: users are billed for invisible reasoning tokens, which often account for the majority of the cost, yet have no means to verify their authenticity. This opens the door to token count inflation, where providers may overreport token usage or inject synthetic, low-effort tokens to inflate charges. To address this issue, we propose CoIn, a verification framework that audits both the quantity and semantic validity of hidden tokens. CoIn constructs a verifiable hash tree from token embedding fingerprints to check token counts, and uses embedding-based relevance matching to detect fabricated reasoning content. Experiments demonstrate that CoIn, when deployed as a trusted third-party auditor, can effectively detect token count inflation with a success rate reaching up to 94.7%, showing the strong ability to restore billing transparency in opaque LLM services. The dataset and code are available at https://github.com/CASE-Lab-UMD/LLM-Auditing-CoIn.

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

Membership inference attacks (MIAs) attempt to predict whether a particular datapoint is a member of a target model's training data. Despite extensive research on traditional machine learning models, there has been limited work studying MIA on the pre-training data of large language models (LLMs). We perform a large-scale evaluation of MIAs over a suite of language models (LMs) trained on the Pile, ranging from 160M to 12B parameters. We find that MIAs barely outperform random guessing for most settings across varying LLM sizes and domains. Our further analyses reveal that this poor performance can be attributed to (1) the combination of a large dataset and few training iterations, and (2) an inherently fuzzy boundary between members and non-members. We identify specific settings where LLMs have been shown to be vulnerable to membership inference and show that the apparent success in such settings can be attributed to a distribution shift, such as when members and non-members are drawn from the seemingly identical domain but with different temporal ranges. We release our code and data as a unified benchmark package that includes all existing MIAs, supporting future work.

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.

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/ .

Modern Lean theorem provers achieve strong performance only with substantial training and inference compute, driven in part by scarce verified proof data and the long reasoning traces of formal proof search, making both supervised fine-tuning (SFT) and sampling expensive. We introduce Pythagoras-Prover, a compute-efficient open-source family of Lean theorem provers built for practical compute budgets. The family spans two generation paradigms: autoregressive models at 4B and 32B parameters, and a first proof-of-concept diffusion-based prover (4B) that iteratively refines Lean proofs at inference time. For training efficiency, we build a Lean-verified corpus stratified into easy, medium, and hard problems for curriculum SFT, so models acquire proof skills progressively from shorter, simpler proofs to longer, harder ones. During SFT, a dynamic proof-reasoning filtering scheme preserves informative proof traces while keeping each instance within an 8k-token context budget. We also introduce Augmented Lean Formalisation (ALF), which expands scarce verified corpora into variants of formal statements, populated via self-distillation for extra training signal without formally verifying every mutated instance. By perturbing known problems while preserving their formal character, ALF reduces reliance on any statement's surface form. Empirically, Pythagoras-Prover-4B surpasses DeepSeek-Prover-V2-671B at pass@32 on MiniF2F-Test (86.1% vs 82.4%) with ~167x fewer parameters, while Pythagoras-Prover-32B sets the open-source state of the art at 93.0% on MiniF2F-Test and solves 93 of 672 PutnamBench problems. We release MiniF2F-ALF, an ALF-mutated contamination-sensitive benchmark on which every evaluated model loses accuracy; here our 32B remains strongest and our 4B matches the prior state of the art, Goedel-Prover-V2-32B.

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

Multi-agent LLM systems share state through memory stores, vector indices, and tool registries. We model such sharing as long-running read-generate-write operations under deterministic-generation semantics -- the regime durable-execution engines enforce by deterministic replay -- and formalize four concurrency anomalies in TLA+: stale-generation, phantom-tool, causal-cascade, and tool-effect reordering, structural analogues of classical isolation anomalies, each with a TLC counter-example. The exclusion lattice over these anomalies is trivial; the contribution is the mechanically verified realizability and strict separation of one maximal chain within it, L_0 subsetneq cdots subsetneq L_4, to our knowledge the first machine-checked consistency hierarchy for such runtimes. A development of 274 Verus obligations (zero assume, zero admit; trust base: two structural axioms and a mutex correspondence) proves the detectors sound and complete against the specifications and each runtime its avoidance set. Three deployed Rust runtimes realize L0-L1 (pessimistic locking, serializable snapshot isolation, default-SI), each verified against stale-generation and refined to its state machine; L2-L4 are exec-mode-verified with dependency-free prevention twins (A3, A6, A2: 0/1000 versus 1000/1000), and L2 is run live across three model families (A3 prevented in all 120 retracted sessions). We reproduce a silent lost update in ByteDance's deer-flow, formalizing its fix as a verified L_0 to L_1 refinement, and exhibit tool-effect reordering in LangGraph's ToolNode on unmodified output, removed by an L3 commit-order sequencer. The verified detector, refinements, and realizability artifacts are the contribution; the phenomena and lattice are classical.

Text-to-image diffusion models have achieved tremendous success in the field of controllable image generation, while also coming along with issues of privacy leakage and data copyrights. Membership inference arises in these contexts as a potential auditing method for detecting unauthorized data usage. While some efforts have been made on diffusion models, they are not applicable to text-to-image diffusion models due to the high computation overhead and enhanced generalization capabilities. In this paper, we first identify a conditional overfitting phenomenon in text-to-image diffusion models, indicating that these models tend to overfit the conditional distribution of images given the corresponding text rather than the marginal distribution of images only. Based on this observation, we derive an analytical indicator, namely Conditional Likelihood Discrepancy (CLiD), to perform membership inference, which reduces the stochasticity in estimating memorization of individual samples. Experimental results demonstrate that our method significantly outperforms previous methods across various data distributions and dataset scales. Additionally, our method shows superior resistance to overfitting mitigation strategies, such as early stopping and data augmentation.

Membership Inference Attacks (MIAs) serve as a fundamental auditing tool for evaluating training data leakage in machine learning models. However, existing methodologies predominantly rely on static, handcrafted heuristics that lack adaptability, often leading to suboptimal performance when transferred across different large models. In this work, we propose AutoMIA, an agentic framework that reformulates membership inference as an automated process of self-exploration and strategy evolution. Given high-level scenario specifications, AutoMIA self-explores the attack space by generating executable logits-level strategies and progressively refining them through closed-loop evaluation feedback. By decoupling abstract strategy reasoning from low-level execution, our framework enables a systematic, model-agnostic traversal of the attack search space. Extensive experiments demonstrate that AutoMIA consistently matches or outperforms state-of-the-art baselines while eliminating the need for manual feature engineering.

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

As reinforcement Learning with Verifiable Rewards (RLVR) has become the dominant paradigm for scaling reasoning capabilities in LLMs, a new failure mode emerges: LLMs gaming verifiers. We study this phenomenon on inductive reasoning tasks, where models must induce and output logical rules. We find that RLVR-trained models systematically abandon rule induction. Instead of learning generalizable patterns (e.g., ``trains carrying red cars go east''), they enumerate instance-level labels, producing outputs that pass verifiers without capturing the relational patterns required by the task. We show that this behavior is not a failure of understanding but a form of reward hacking: imperfect verifiers that check only extensional correctness admit false positives. To detect such shortcuts, we introduce Isomorphic Perturbation Testing (IPT), which evaluates a single model output under both extensional and isomorphic verification, where the latter enforces invariance under logically isomorphic tasks. While genuine rule induction remains invariant, shortcut strategies fail. We find that shortcut behavior is specific to RLVR-trained reasoning models (e.g., GPT-5, Olmo3) and absent in non-RLVR models (e.g., GPT-4o, GPT-4.5, Ministral). Moreover, shortcut prevalence increases with task complexity and inference-time compute. In controlled training experiments, extensional verification directly induces shortcut strategies, while isomorphic verification eliminates them. These results show that RLVR can incentivize reward hacking not only through overt manipulation but also by exploiting what the verifier fails to enforce.

A membership inference attack (MIA) against a machine-learning model enables an attacker to determine whether a given data record was part of the model's training data or not. In this paper, we provide an in-depth study of the phenomenon of disparate vulnerability against MIAs: unequal success rate of MIAs against different population subgroups. We first establish necessary and sufficient conditions for MIAs to be prevented, both on average and for population subgroups, using a notion of distributional generalization. Second, we derive connections of disparate vulnerability to algorithmic fairness and to differential privacy. We show that fairness can only prevent disparate vulnerability against limited classes of adversaries. Differential privacy bounds disparate vulnerability but can significantly reduce the accuracy of the model. We show that estimating disparate vulnerability to MIAs by na\"ively applying existing attacks can lead to overestimation. We then establish which attacks are suitable for estimating disparate vulnerability, and provide a statistical framework for doing so reliably. We conduct experiments on synthetic and real-world data finding statistically significant evidence of disparate vulnerability in realistic settings. The code is available at https://github.com/spring-epfl/disparate-vulnerability

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.

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

In this paper we undertake a systematic study of privacy attacks against open source Large Language Models (LLMs), where an adversary has access to either the model weights, gradients, or losses, and tries to exploit them to learn something about the underlying training data. Our headline results are the first membership inference attacks (MIAs) against pre-trained LLMs that are able to simultaneously achieve high TPRs and low FPRs, and a pipeline showing that over 50% (!) of the fine-tuning dataset can be extracted from a fine-tuned LLM in natural settings. We consider varying degrees of access to the underlying model, customization of the language model, and resources available to the attacker. In the pre-trained setting, we propose three new white-box MIAs: an attack based on the gradient norm, a supervised neural network classifier, and a single step loss ratio attack. All outperform existing black-box baselines, and our supervised attack closes the gap between MIA attack success against LLMs and other types of models. In fine-tuning, we find that given access to the loss of the fine-tuned and base models, a fine-tuned loss ratio attack FLoRA is able to achieve near perfect MIA peformance. We then leverage these MIAs to extract fine-tuning data from fine-tuned language models. We find that the pipeline of generating from fine-tuned models prompted with a small snippet of the prefix of each training example, followed by using FLoRa to select the most likely training sample, succeeds the majority of the fine-tuning dataset after only 3 epochs of fine-tuning. Taken together, these findings show that highly effective MIAs are available in almost all LLM training settings, and highlight that great care must be taken before LLMs are fine-tuned on highly sensitive data and then deployed.

Machine learning programs, such as those performing inference, fine-tuning, and training of LLMs, are commonly delegated to untrusted compute providers. To provide correctness guarantees for the client, we propose adapting the cryptographic notion of refereed delegation to the machine learning setting. This approach enables a computationally limited client to delegate a program to multiple untrusted compute providers, with a guarantee of obtaining the correct result if at least one of them is honest. Refereed delegation of ML programs poses two technical hurdles: (1) an arbitration protocol to resolve disputes when compute providers disagree on the output, and (2) the ability to bitwise reproduce ML programs across different hardware setups, For (1), we design Verde, a dispute arbitration protocol that efficiently handles the large scale and graph-based computational model of modern ML programs. For (2), we build RepOps (Reproducible Operators), a library that eliminates hardware "non-determinism" by controlling the order of floating point operations performed on all hardware. Our implementation shows that refereed delegation achieves both strong guarantees for clients and practical overheads for compute providers.

It is important that consumers and regulators can verify the provenance of large neural models to evaluate their capabilities and risks. We introduce the concept of a "Proof-of-Training-Data": any protocol that allows a model trainer to convince a Verifier of the training data that produced a set of model weights. Such protocols could verify the amount and kind of data and compute used to train the model, including whether it was trained on specific harmful or beneficial data sources. We explore efficient verification strategies for Proof-of-Training-Data that are compatible with most current large-model training procedures. These include a method for the model-trainer to verifiably pre-commit to a random seed used in training, and a method that exploits models' tendency to temporarily overfit to training data in order to detect whether a given data-point was included in training. We show experimentally that our verification procedures can catch a wide variety of attacks, including all known attacks from the Proof-of-Learning literature.

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.

Neural Theorem Proving (NTP) employs LLMs to automate formal proofs in proof assistants. While LLMs have achieved relatively remarkable success in informal reasoning tasks using natural languages, the transition to mechanized formal theorem proving presents persistent challenges. Mechanized proof languages often contain many syntactic constructs and diverse, specialized proof tactics, which facilitate expert use but have no direct counterpart in informal mathematical proofs. These prover-specific idioms represent an additional burden for LLM-based NTPs that might be otherwise successful in generating informal proofs. Seeking to bridge this gap between formal proof construction and informal reasoning, in order to better facilitate NTP, this work approaches these challenges from a language design perspective. We look at common reasoning patterns in informal proofs and in existing mechanized proofs, and design Minilang -- a minimalist proof language that captures these reasoning patterns. In contrast to proof languages (informal and formal) that often feature a large collection of operations with unclear semantic boundaries, Minilang is deliberately kept minimalist -- its core design comprises only 10 operations, each with clear semantic distinctions. We further develop a rule-based translator from Isabelle's language (Isar) to Minilang, translating ~340K existing proofs with an ~85% success rate. Using this translated corpus, we finetune two LLMs to compare machine learning performance on Minilang versus the original Isar. Experiments show Minilang benefits the two LLMs by improving the pass@1 success rate on the PISA benchmark by up to 20/29 percentage points in comparison to the Isar-based LLMs w/wo Sledgehammer. The pass@1 rate reaches 69.1%, exceeding the prior work Baldur's pass@64 (65.7%); the pass@8 rate reaches 79.2%, exceeding the SOTA on PISA (71.0%) achieved by Magnushammer.

We present a complete formalization in Isabelle/HOL of the object part of an equivalence between L-mosaics and bounded join-semilattices, employing an AI-assisted methodology that integrates large language models as reasoning assistants throughout the proof development process. The equivalence was originally established by Cangiotti, Linzi, and Talotti in their study of hypercompositional structures related to orthomodular lattices and quantum logic. Our formalization rigorously verifies the main theoretical result and demonstrates the mutual inverse property of the transformations establishing this equivalence. The development showcases both the mathematical depth of multivalued algebraic operations and the potential for AI-enhanced interactive theorem proving in tackling complex formalization projects.

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

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

Membership inference attacks (MIA) attempt to verify the membership of a given data sample in the training set for a model. MIA has become relevant in recent years, following the rapid development of large language models (LLM). Many are concerned about the usage of copyrighted materials for training them and call for methods for detecting such usage. However, recent research has largely concluded that current MIA methods do not work on LLMs. Even when they seem to work, it is usually because of the ill-designed experimental setup where other shortcut features enable "cheating." In this work, we argue that MIA still works on LLMs, but only when multiple documents are presented for testing. We construct new benchmarks that measure the MIA performances at a continuous scale of data samples, from sentences (n-grams) to a collection of documents (multiple chunks of tokens). To validate the efficacy of current MIA approaches at greater scales, we adapt a recent work on Dataset Inference (DI) for the task of binary membership detection that aggregates paragraph-level MIA features to enable MIA at document and collection of documents level. This baseline achieves the first successful MIA on pre-trained and fine-tuned LLMs.

Membership inference attacks aim to detect if a particular data point was used in training a model. We design a novel statistical test to perform robust membership inference attacks (RMIA) with low computational overhead. We achieve this by a fine-grained modeling of the null hypothesis in our likelihood ratio tests, and effectively leveraging both reference models and reference population data samples. RMIA has superior test power compared with prior methods, throughout the TPR-FPR curve (even at extremely low FPR, as low as 0). Under computational constraints, where only a limited number of pre-trained reference models (as few as 1) are available, and also when we vary other elements of the attack (e.g., data distribution), our method performs exceptionally well, unlike prior attacks that approach random guessing. RMIA lays the groundwork for practical yet accurate data privacy risk assessment in machine learning.

Reliable verification of proofs remains a bottleneck for training and evaluating AI systems on hard mathematical reasoning. Fully formal proofs, in languages like Lean, are easy to verify because they are unambiguous and modular. Most proofs, particularly those written by AI systems, have neither property, and translating them into formal languages remains challenging in many frontier math settings. We propose Pseudo-Formalization (PF), a proof format that captures the modularity and precision of formal proofs while retaining the flexibility of natural language. A Pseudo-Formal proof is decomposed into self-contained modules, each stating its premises, conclusion, and proof in natural language. To verify the correctness of a regular natural language proof, an LLM translates it to Pseudo-Formal and then verifies each module independently, an algorithm we call Block Verification (BV). We evaluate PF+BV on two benchmarks spanning olympiad and research-level mathematics, where it pareto-dominates LLM-as-judge baselines on error-finding precision and recall. To support future work, we release our research-level proof verification benchmark ArxivMathGradingBench.

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.

Diffusion-based generative models have shown great potential for image synthesis, but there is a lack of research on the security and privacy risks they may pose. In this paper, we investigate the vulnerability of diffusion models to Membership Inference Attacks (MIAs), a common privacy concern. Our results indicate that existing MIAs designed for GANs or VAE are largely ineffective on diffusion models, either due to inapplicable scenarios (e.g., requiring the discriminator of GANs) or inappropriate assumptions (e.g., closer distances between synthetic samples and member samples). To address this gap, we propose Step-wise Error Comparing Membership Inference (SecMI), a query-based MIA that infers memberships by assessing the matching of forward process posterior estimation at each timestep. SecMI follows the common overfitting assumption in MIA where member samples normally have smaller estimation errors, compared with hold-out samples. We consider both the standard diffusion models, e.g., DDPM, and the text-to-image diffusion models, e.g., Latent Diffusion Models and Stable Diffusion. Experimental results demonstrate that our methods precisely infer the membership with high confidence on both of the two scenarios across multiple different datasets. Code is available at https://github.com/jinhaoduan/SecMI.

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

Blockchains have seen growing traction with cryptocurrencies reaching a market cap of over 1 trillion dollars, major institution investors taking interests, and global impacts on governments, businesses, and individuals. Also growing significantly is the heterogeneity of the ecosystem where a variety of blockchains co-exist. Cross-chain bridge is a necessary building block in this multi-chain ecosystem. Existing solutions, however, either suffer from performance issues or rely on trust assumptions of committees that significantly lower the security. Recurring attacks against bridges have cost users more than 1.5 billion USD. In this paper, we introduce zkBridge, an efficient cross-chain bridge that guarantees strong security without external trust assumptions. With succinct proofs, zkBridge not only guarantees correctness, but also significantly reduces on-chain verification cost. We propose novel succinct proof protocols that are orders-of-magnitude faster than existing solutions for workload in zkBridge. With a modular design, zkBridge enables a broad spectrum of use cases and capabilities, including message passing, token transferring, and other computational logic operating on state changes from different chains. To demonstrate the practicality of zkBridge, we implemented a prototype bridge from Cosmos to Ethereum, a particularly challenging direction that involves large proof circuits that existing systems cannot efficiently handle. Our evaluation shows that zkBridge achieves practical performance: proof generation takes less than 20 seconds, while verifying proofs on-chain costs less than 230K gas. For completeness, we also implemented and evaluated the direction from Ethereum to other EVM-compatible chains (such as BSC) which involves smaller circuits and incurs much less overhead.

This article shows yet another proof of NP=CoNP$. In a previous article, we proved that NP=PSPACE and from it we can conclude that NP=CoNP immediately. The former proof shows how to obtain polynomial and, polynomial in time checkable Dag-like proofs for all purely implicational Minimal logic tautologies. From the fact that Minimal implicational logic is PSPACE-complete we get the proof that NP=PSPACE. This first proof of NP=CoNP uses Hudelmaier linear upper-bound on the height of Sequent Calculus minimal implicational logic proofs. In an addendum to the proof of NP=PSPACE, we observe that we do not need to use Hudelmaier upper-bound since any proof of non-hamiltonicity for any graph is linear upper-bounded. By the CoNP-completeness of non-hamiltonicity, we obtain NP=CoNP as a corollary of the first proof. In this article we show the third proof of CoNP=NP, also providing polynomial size and polynomial verifiable certificates that are Dags. They are generated from normal Natural Deduction proofs, linear height upper-bounded too, by removing redundancy, i.e., repeated parts. The existence of repeated parts is a consequence of the redundancy theorem for a family of super-polynomial proofs in the purely implicational Minimal logic. It is mandatory to read at least two previous articles to get the details of the proof presented here. The article that proves the redundancy theorem and the article that shows how to remove the repeated parts of a normal Natural Deduction proof to have a polynomial Dag certificate for minimal implicational logic tautologies.

When should an AI system's answer be trusted? Formal proof assistants offer certainty but cannot reach most of the problem distribution; scalar LLM judges offer coverage but produce opaque scores that cannot be audited after the fact and are subject to the same coherence issues as any LLM. We present Theoria, a verification architecture that closes this gap. A candidate solution is rewritten into a sequence of typed state transitions, each licensed by an explicit justification, whether that be a citation, computation, or problem-given fact, and every transition is independently auditable. The foundational invariant is completeness of change: every difference between consecutive proof states must be accounted for, so hidden premises surface as unlicensed mutations rather than passing silently. On HLE-Verified Gold (185 text-only expert problems), Theoria certifies 105 at 91.4% strict precision (Wilson 95% CI [84.5%, 95.4%]). Every certification produces a human readable proof trace in which each step can be independently challenged. Holistic LLM judges achieve comparable precision at matched coverage but fail on different problems (Jaccard 0.14-0.36), making the approaches complementary. On 95 adversarial poisoned proofs across 15 domains, structured judges catch 94.7% versus 83.2% for holistic judging (p= 0.0017). The overall 11.5 pp gap concentrates in hidden premises (90.6% vs. 62.5%, a 28 pp difference) and fabricated citations (100% vs. 90%), the error classes where the formal analysis predicts an advantage; performance is identical on arithmetic and theorem-misapplication errors, where no advantage is predicted. On GPQA Diamond (n= 65), certified precision is 97.1% (Wilson CI [85.1%, 99.5%]).

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.

Differential privacy (DP) has a wide range of applications for protecting data privacy, but designing and verifying DP algorithms requires expert-level reasoning, creating a high barrier for non-expert practitioners. Prior works either rely on specialized verification languages that demand substantial domain expertise or remain semi-automated and require human-in-the-loop guidance. In this work, we investigate whether large language models (LLMs) can automate DP reasoning. We introduce DPrivBench, a benchmark in which each instance asks whether a function or algorithm satisfies a stated DP guarantee under specified assumptions. The benchmark is carefully designed to cover a broad range of DP topics, span diverse difficulty levels, and resist shortcut reasoning through trivial pattern matching. Experiments show that while the strongest models handle textbook mechanisms well, all models struggle with advanced algorithms, revealing substantial gaps in current DP reasoning capabilities. Through further analytic study and failure-mode analysis, we identify several promising directions for improving automated DP reasoning. Our benchmark provides a solid foundation for developing and evaluating such methods, and complements existing benchmarks for mathematical reasoning.

We study the Stochastic Boolean Function Certification (SBFC) problem, where we are given n Bernoulli random variables {X_e: e in U} on a ground set U of n elements with joint distribution p, a Boolean function f: 2^U to {0, 1}, and an (unknown) scenario S = {e in U: X_e = 1} of active elements sampled from p. We seek to probe the elements one-at-a-time to reveal if they are active until we can certify f(S) = 1, while minimizing the expected number of probes. Unlike most previous results that assume independence, we study correlated distributions p and give approximation algorithms for several classes of functions f. When f(S) is the indicator function for whether S is the spanning set of a given matroid, our problem reduces to finding a basis of active elements of a matroid by probing elements. We give a non-adaptive O(log n)-approximation algorithm for arbitrary distributions p, and show that this is tight up to constants unless P = NP, even for partition matroids. For uniform matroids, we give constant factor 4.642-approximation ([BBFT20]) that can be further improved to a 2-approximation if additionally the random variables are negatively correlated for the case of 1-uniform matroid. We also give an adaptive O(log k)-approximation algorithm for SBFC for k-uniform matroids for the Graph Probing problem, where we seek to probe the edges of a graph one-at-a-time until we find k active edges. The underlying distribution on edges arises from (hidden) independent vertex random variables, with an edge being active if at least one of its endpoints is active. This significantly improves over the information-theoretic lower bound on Ω(poly(n)) ([JGM19]) for adaptive algorithms for k-uniform matroids with arbitrary distributions.

We quantitatively investigate how machine learning models leak information about the individual data records on which they were trained. We focus on the basic membership inference attack: given a data record and black-box access to a model, determine if the record was in the model's training dataset. To perform membership inference against a target model, we make adversarial use of machine learning and train our own inference model to recognize differences in the target model's predictions on the inputs that it trained on versus the inputs that it did not train on. We empirically evaluate our inference techniques on classification models trained by commercial "machine learning as a service" providers such as Google and Amazon. Using realistic datasets and classification tasks, including a hospital discharge dataset whose membership is sensitive from the privacy perspective, we show that these models can be vulnerable to membership inference attacks. We then investigate the factors that influence this leakage and evaluate mitigation strategies.

Attempts to render deep learning models interpretable, data-efficient, and robust have seen some success through hybridisation with rule-based systems, for example, in Neural Theorem Provers (NTPs). These neuro-symbolic models can induce interpretable rules and learn representations from data via back-propagation, while providing logical explanations for their predictions. However, they are restricted by their computational complexity, as they need to consider all possible proof paths for explaining a goal, thus rendering them unfit for large-scale applications. We present Conditional Theorem Provers (CTPs), an extension to NTPs that learns an optimal rule selection strategy via gradient-based optimisation. We show that CTPs are scalable and yield state-of-the-art results on the CLUTRR dataset, which tests systematic generalisation of neural models by learning to reason over smaller graphs and evaluating on larger ones. Finally, CTPs show better link prediction results on standard benchmarks in comparison with other neural-symbolic models, while being explainable. All source code and datasets are available online, at https://github.com/uclnlp/ctp.

Large-scale pre-training frameworks like CLIP have revolutionized multimodal learning, but their reliance on web-scraped datasets, frequently containing private user data, raises serious concerns about misuse. Unlearnable Examples (UEs) have emerged as a promising countermeasure against unauthorized model training, employing carefully crafted unlearnable noise to disrupt the learning of meaningful representations from protected data. Current approaches typically generate UEs by jointly optimizing unlearnable noise for both images and their associated text descriptions (or labels). However, this optimization process is often computationally prohibitive for on-device execution, forcing reliance on external third-party services. This creates a fundamental privacy paradox: users must initially expose their data to these very services to achieve protection, thereby compromising privacy in the process. Such a contradiction has severely hindered the development of practical, scalable data protection solutions. To resolve this paradox, we introduce Text-to-Unlearnable Example (T2UE), a novel framework that enables users to generate UEs using only text descriptions. T2UE circumvents the need for original image data by employing a text-to-image (T2I) model to map text descriptions into the image (noise) space, combined with an error-minimization framework to produce effective unlearnable noise. Extensive experiments show that T2UE-protected data substantially degrades performance in downstream tasks (e.g., cross-modal retrieval) for state-of-the-art models. Notably, the protective effect generalizes across diverse architectures and even to supervised learning settings. Our work demonstrates the feasibility of "zero-contact data protection", where personal data can be safeguarded based solely on their textual descriptions, eliminating the need for direct data exposure.

Distribution inference, sometimes called property inference, infers statistical properties about a training set from access to a model trained on that data. Distribution inference attacks can pose serious risks when models are trained on private data, but are difficult to distinguish from the intrinsic purpose of statistical machine learning -- namely, to produce models that capture statistical properties about a distribution. Motivated by Yeom et al.'s membership inference framework, we propose a formal definition of distribution inference attacks that is general enough to describe a broad class of attacks distinguishing between possible training distributions. We show how our definition captures previous ratio-based property inference attacks as well as new kinds of attack including revealing the average node degree or clustering coefficient of a training graph. To understand distribution inference risks, we introduce a metric that quantifies observed leakage by relating it to the leakage that would occur if samples from the training distribution were provided directly to the adversary. We report on a series of experiments across a range of different distributions using both novel black-box attacks and improved versions of the state-of-the-art white-box attacks. Our results show that inexpensive attacks are often as effective as expensive meta-classifier attacks, and that there are surprising asymmetries in the effectiveness of attacks. Code is available at https://github.com/iamgroot42/FormEstDistRisks

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

Modern machine learning models are sensitive to the manipulation of both the training data (poisoning attacks) and inference data (adversarial examples). Recognizing this issue, the community has developed many empirical defenses against both attacks and, more recently, certification methods with provable guarantees against inference-time attacks. However, such guarantees are still largely lacking for training-time attacks. In this work, we present FullCert, the first end-to-end certifier with sound, deterministic bounds, which proves robustness against both training-time and inference-time attacks. We first bound all possible perturbations an adversary can make to the training data under the considered threat model. Using these constraints, we bound the perturbations' influence on the model's parameters. Finally, we bound the impact of these parameter changes on the model's prediction, resulting in joint robustness guarantees against poisoning and adversarial examples. To facilitate this novel certification paradigm, we combine our theoretical work with a new open-source library BoundFlow, which enables model training on bounded datasets. We experimentally demonstrate FullCert's feasibility on two datasets.

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.

Synthetic data is emerging as one of the most promising solutions to share individual-level data while safeguarding privacy. While membership inference attacks (MIAs), based on shadow modeling, have become the standard to evaluate the privacy of synthetic data, they currently assume the attacker to have access to an auxiliary dataset sampled from a similar distribution as the training dataset. This is often seen as a very strong assumption in practice, especially as the proposed main use cases for synthetic tabular data (e.g. medical data, financial transactions) are very specific and don't have any reference datasets directly available. We here show how this assumption can be removed, allowing for MIAs to be performed using only the synthetic data. Specifically, we developed three different scenarios: (S1) Black-box access to the generator, (S2) only access to the released synthetic dataset and (S3) a theoretical setup as upper bound for the attack performance using only synthetic data. Our results show that MIAs are still successful, across two real-world datasets and two synthetic data generators. These results show how the strong hypothesis made when auditing synthetic data releases - access to an auxiliary dataset - can be relaxed, making the attacks more realistic in practice.

Formal verification offers the strongest guarantee of software correctness, but it does not scale: the proofs demanded by interactive theorem provers such as Coq require enormous expert effort. Large language models (LLMs) promise to generate these proofs automatically, yet existing approaches wire a fixed, human-designed proof strategy into the system and constrain the model to follow it (retrieving premises and predicting tactics one step at a time, or splitting goals by divide-and-conquer), and still prove only a fraction of their target theorems. We show that imposing such a strategy is unnecessary and limiting. Handing the whole lemma to a general LLM code agent (for example, Claude Code), free to choose its own approach, and wrapping it in a verification harness is both simpler and more effective, achieving full coverage: every targeted lemma proved, with no failures and no Coq expert intervention. The agent writes the proofs under feedback and hard constraints from the harness that keep each one sound (accepted only when the prover's kernel closes it), complete (no obligation left unproved or silently dropped), and terminating (no divergent tactics). We evaluate this harness plus code agent along three dimensions. (1) Core logic: on Iris, the state-of-the-art separation logic for concurrent and memory-manipulating programs, Aria proves all 4,257 lemmas of the four core modules and the 217 lemmas verifying Rust's standard libraries built on it, fully automatically. (2) Comparison with prior LLM provers: on reglang, where prior provers manage barely one in eight, Aria proves all 318. (3) Generality: on iris-lean, the unfinished Lean 4 port of Iris, it proves 72 not-yet-ported lemmas, showing the approach is not specific to Coq. A state-of-the-art model (Claude Opus 4.7) can write proofs for verified software development fully and automatically.

Recently, diffusion models have achieved remarkable success in generating tasks, including image and audio generation. However, like other generative models, diffusion models are prone to privacy issues. In this paper, we propose an efficient query-based membership inference attack (MIA), namely Proximal Initialization Attack (PIA), which utilizes groundtruth trajectory obtained by epsilon initialized in t=0 and predicted point to infer memberships. Experimental results indicate that the proposed method can achieve competitive performance with only two queries on both discrete-time and continuous-time diffusion models. Moreover, previous works on the privacy of diffusion models have focused on vision tasks without considering audio tasks. Therefore, we also explore the robustness of diffusion models to MIA in the text-to-speech (TTS) task, which is an audio generation task. To the best of our knowledge, this work is the first to study the robustness of diffusion models to MIA in the TTS task. Experimental results indicate that models with mel-spectrogram (image-like) output are vulnerable to MIA, while models with audio output are relatively robust to MIA. {Code is available at https://github.com/kong13661/PIA}.

Communicating state machines provide a formal foundation for distributed computation. Unfortunately, they are Turing-complete and, thus, challenging to analyse. In this paper, we classify restrictions on channels which have been proposed to work around the undecidability of verification questions. We compare half-duplex communication, existential B-boundedness, and k-synchronisability. These restrictions do not prevent the communication channels from growing arbitrarily large but still restrict the power of the model. Each restriction gives rise to a set of languages so, for every pair of restrictions, we check whether one subsumes the other or if they are incomparable. We investigate their relationship in two different contexts: first, the one of communicating state machines, and, second, the one of communication protocol specifications using high-level message sequence charts. Surprisingly, these two contexts yield different conclusions. In addition, we integrate multiparty session types, another approach to specify communication protocols, into our classification. We show that multiparty session type languages are half-duplex, existentially 1-bounded, and 1-synchronisable. To~show this result, we provide the first formal embedding of multiparty session types into high-level message sequence charts.

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

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

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

The proliferation of large language models (LLMs) in the real world has come with a rise in copyright cases against companies for training their models on unlicensed data from the internet. Recent works have presented methods to identify if individual text sequences were members of the model's training data, known as membership inference attacks (MIAs). We demonstrate that the apparent success of these MIAs is confounded by selecting non-members (text sequences not used for training) belonging to a different distribution from the members (e.g., temporally shifted recent Wikipedia articles compared with ones used to train the model). This distribution shift makes membership inference appear successful. However, most MIA methods perform no better than random guessing when discriminating between members and non-members from the same distribution (e.g., in this case, the same period of time). Even when MIAs work, we find that different MIAs succeed at inferring membership of samples from different distributions. Instead, we propose a new dataset inference method to accurately identify the datasets used to train large language models. This paradigm sits realistically in the modern-day copyright landscape, where authors claim that an LLM is trained over multiple documents (such as a book) written by them, rather than one particular paragraph. While dataset inference shares many of the challenges of membership inference, we solve it by selectively combining the MIAs that provide positive signal for a given distribution, and aggregating them to perform a statistical test on a given dataset. Our approach successfully distinguishes the train and test sets of different subsets of the Pile with statistically significant p-values < 0.1, without any false positives.