Formal Verification

Formal verification guarantees proof validity but not formalization faithfulness. For natural-language logical reasoning, where models construct axiom systems from scratch without library constraints, this gap between valid proofs and faithful translations is especially acute. We investigate whether frontier models exploit this gap when generating Lean 4 proofs, a behavior we term formalization gaming. We evaluate GPT-5 and DeepSeek-R1 on 303 first-order logic problems (203 from FOLIO, 100 from Multi-LogiEval), comparing unified generation against a two-stage pipeline that separates formalization from proving. Despite compilation rates of 87-99%, we find no evidence of systematic gaming in unified generation: models prefer reporting failure over forcing proofs, even under prompting designed to encourage it. However, unfaithfulness that evades our detection signals may still occur. The two-stage pipeline reveals two distinct modes of unfaithfulness: GPT-5 fabricates axioms during proof generation, a reactive fallback detectable via cross-stage comparison, while DeepSeek-R1 mistranslates premises during formalization, producing internally consistent outputs that evade detection entirely. These findings show that high compilation rates or accuracies should not be equated with faithful reasoning. Code and data are available at https://github.com/koreankiwi99/formalization-gaming.

Half-truths, claims that are factually correct yet misleading due to omitted context, remain a blind spot for fact verification systems focused on explicit falsehoods. Addressing such omission-based manipulation requires reasoning not only about what is said, but also about what is left unsaid. We propose RADAR, a role-anchored multi-agent debate framework for omission-aware fact verification under realistic, noisy retrieval. RADAR assigns complementary roles to a Politician and a Scientist, who reason adversarially over shared retrieved evidence, moderated by a neutral Judge. A dual-threshold early termination controller adaptively decides when sufficient reasoning has been reached to issue a verdict. Experiments show that RADAR consistently outperforms strong single- and multi-agent baselines across datasets and backbones, improving omission detection accuracy while reducing reasoning cost. These results demonstrate that role-anchored, retrieval-grounded debate with adaptive control is an effective and scalable framework for uncovering missing context in fact verification. The code is available at https://github.com/tangyixuan/RADAR.

Statement autoformalization acts as a critical bridge between human mathematics and formal mathematics by translating natural language problems into formal language. While prior works have focused on data synthesis and diverse training paradigms to optimize end-to-end Large Language Models (LLMs), they typically treat formal code as flat sequences, neglecting the hierarchical logic inherent in mathematical statements. In this work, we introduce Decompose, Structure, and Repair (DSR), a neuro-symbolic framework that restructures autoformalization into a modular pipeline. DSR decomposes statements into logical components and maps them to structured operator trees, leveraging this topological blueprint to precisely localize and repair errors via sub-tree refinement. Furthermore, we introduce PRIME, a benchmark of 156 undergraduate and graduate-level theorems selected from canonical textbooks and expertly annotated in Lean 4. Experimental results demonstrate that DSR establishes a new state-of-the-art, consistently outperforming baselines under equivalent computational budgets. The datasets, model, and code are available at https://github.com/XiaoyangLiu-sjtu/DSR.

We present a formally verified framework for patent analysis as a hybrid AI + Lean 4 pipeline. The DAG-coverage core (Algorithm 1b) is fully machine-verified once bounded match scores are fixed. Freedom-to-operate, claim-construction sensitivity, cross-claim consistency, and doctrine-of-equivalents analyses are formalized at the specification level with kernel-checked candidate certificates. Existing patent-analysis approaches rely on manual expert analysis (slow, non-scalable) or ML/NLP methods (probabilistic, opaque, non-compositional). To our knowledge, this is the first framework that applies interactive theorem proving based on dependent type theory to intellectual property analysis. Claims are encoded as DAGs in Lean 4, match strengths as elements of a verified complete lattice, and confidence scores propagate through dependencies via proven-correct monotone functions. We formalize five IP use cases (patent-to-product mapping, freedom-to-operate, claim construction sensitivity, cross-claim consistency, doctrine of equivalents) via six algorithms. Structural lemmas, the coverage-core generator, and the closed-path identity coverage = W_cov are machine-verified in Lean 4. Higher-level theorems for the other use cases remain informal proof sketches, and their proof-generation functions are architecturally mitigated (untrusted generators whose outputs are kernel-checked and sorry-free axiom-audited). Guarantees are conditional on the ML layer: they certify mathematical correctness of computations downstream of ML scores, not the accuracy of the scores themselves. A case study on a synthetic memory-module claim demonstrates weighted coverage and construction-sensitivity analysis. Validation against adjudicated cases is future work.

GUI agents that control desktop computers via screenshot-and-click loops introduce a new class of vulnerability: the observation-to-action gap (mean 6.51 s on real OSWorld workloads) creates a Time-Of-Check, Time-Of-Use (TOCTOU) window during which an unprivileged attacker can manipulate the UI state. We formalize this as a Visual Atomicity Violation and characterize three concrete attack primitives: (A) Notification Overlay Hijack, (B) Window Focus Manipulation, and (C) Web DOM Injection. Primitive B, the closest desktop analog to Android Action Rebinding, achieves 100% action-redirection success rate with zero visual evidence at the observation time. We propose Pre-execution UI State Verification (PUSV), a lightweight three-layer defense that re-verifies the UI state immediately before each action dispatch: masked pixel SSIM at the click target (L1), global screenshot diff (L2a), and X Window snapshot diff (L2b). PUSV achieves 100% Action Interception Rate across 180 adversarial trials (135 Primitive A + 45 Primitive B) with zero false positives and < 0.1 s overhead. Against Primitive C (zero-visual-footprint DOM injection), PUSV reveals a structural blind spot (~0% AIR), motivating future OS+DOM defense-in-depth architectures. No single PUSV layer alone achieves full coverage; different primitives require different detection signals, validating the layered design.

Many neural network (NN) verification systems represent the network's input-output relation as a constraint program. Sound and complete, representations involve integer constraints, for simulating the activations. Recent works convexly relax the integer constraints, improving performance, at the cost of soundness. Convex relaxations consider outputs that are unreachable by the original network. We study the worst case divergence between the original network and its convex relaxations; both qualitatively and quantitatively. The relaxations' space forms a lattice, where the top element corresponds to a full relaxation, with every neuron linearized. The bottom element corresponds to the original network. We provide analytical upper and lower bounds for the $\ell_\infty$-distance between the fully relaxed and original outputs. This distance grows exponentially, w.r.t. the network's depth, and linearly w.r.t. the input's radius. The misclassification probability exhibits a step-like behavior, w.r.t. input radius. Our results are supported by experiments on MNIST, Fashion MNIST and random networks.

Verification of model outputs is rapidly emerging as a key primitive for both training and real-world deployment of large language models (LLMs). In practice, this often involves using imperfect LLM judges and reward models since ground truth acquisition can be time-consuming and expensive. We introduce Fully Unsupervised Score Ensembling (FUSE), a method for improving verification quality by ensembling verifiers without access to ground truth correctness labels. The key idea behind FUSE is to control conditional dependencies between verifiers in a manner that improves the unsupervised performance of a class of spectral algorithms from the ensembling literature. Despite requiring zero ground truth labels, FUSE typically matches or improves upon semi-supervised alternatives in test-time scaling experiments with diverse sets of generator models, verifiers, and benchmarks. In particular, we validate our method on both conventional academic benchmarks such as GPQA Diamond and on frontier, unsaturated benchmarks such as Humanity's Last Exam and IMO Shortlist questions.

As reinforcement learning continues to scale the training of large language model-based agents, reliably verifying agent behaviors in complex environments has become increasingly challenging. Existing approaches rely on rule-based verifiers or LLM-as-a-Judge models, which struggle to generalize beyond narrow domains. Agent-as-a-Judge addresses this limitation by actively interacting with environments and tools to acquire verifiable evidence, yet its capabilities remain underexplored. We introduce a benchmark AJ-Bench to systematically evaluate Agent-as-a-Judge across three domains-search, data systems, and graphical user interfaces-comprising 155 tasks and 516 annotated trajectories. The benchmark comprehensively assesses judge agents' abilities in information acquisition, state verification, and process verification. Experiments demonstrate consistent performance gains over LLM-as-a-Judge baselines, while also revealing substantial open challenges in agent-based verification. Our data and code are available at https://aj-bench.github.io/.

Network change validation remains a critical yet predominantly manual, time-consuming, and error-prone process in modern network operations. While formal network verification has made substantial progress in proving correctness properties, it is typically applied in offline, pre-deployment settings and faces challenges in accommodating continuous changes and validating live production behavior. Current operational approaches typically involve scattered testing tools, resulting in partial coverage and errors that surface only after deployment. In this paper, we present Aether, a novel approach that integrates Generative Agentic AI with a multi-functional Network Digital Twin to automate and streamline network change validation workflows. It features an agentic architecture with five specialized Network Operations AI agents that collaboratively handle the change validation lifecycle from intent analysis to network verification and testing. Aether agents use a unified Network Digital Twin integrating modeling, simulation, and emulation to maintain a consistent, up-to-date network view for verification and testing. By orchestrating agent collaboration atop this digital twin, Aether enables automated, rapid network change validation while reducing manual effort, minimizing errors, and improving operational agility and cost-effectiveness. We evaluate Aether over synthetic network change scenarios covering main classes of network changes and on past incidents from a major ISP operational network, demonstrating promising results in error detection (100%), diagnostic coverage (92-96%), and speed (6-7 minutes) over traditional methods.

State-of-the-art code generation frameworks rely on mental simulation, where LLMs internally trace execution to verify correctness. We expose a fundamental limitation: the Mental-Reality Gap -- where models hallucinate execution traces and confidently validate buggy code. This gap manifests along two orthogonal dimensions: the Specification Gap (overlooking edge cases during planning) and the Verification Gap (hallucinating correct behavior for flawed code). We propose SolidCoder with a simple principle: don't imagine -- execute. The S.O.L.I.D. architecture addresses both dimensions by forcing edge-case awareness before algorithm design and replacing imagined traces with sandboxed execution using property-based oracles. With GPT-4o, SolidCoder achieves state-of-the-art pass@1 performance: 95.7% on HumanEval (+0.6%p), 77.0% on CodeContests (+4.3%p), and 26.7% on APPS (+3.4%p). Ablation reveals that edge-case awareness provides the largest individual gain, while execution grounding catches categorically different errors that specification improvements cannot address. These gains generalize to RL post-trained models, validating that bridging both gap dimensions is essential for robust code synthesis. We release our code and framework to facilitate future research.

State-of-the-art reasoning models utilize long chain-of-thought (CoT) to solve increasingly complex problems using more test-time computation. In this work, we explore a long CoT setting where the model makes up to K successive attempts at solving a problem, in which each attempt is allowed to build on earlier ones after the model receives a hard verifier feedback. This motivates RL methods that can harness per-attempt rewards by carefully weighting individual attempts. We study optimizing the Verification@K reward (the model succeeds by the K-th attempt) and show that naively weighing the attempts by their pass/fail results in biased gradients. We introduce Calibrated Attempt-Level (CAL) GRPO by devising a weighing strategy to obtain unbiased gradients while maintaining small variance. Our theory reveals how incorporating per-attempt rewards influence the training and the eventual Verification@K performance. Experiments, baselines, and ablations on synthetic and real data corroborate our theory and the benefits of CAL-GRPO over vanilla GRPO as well as naive weighting.

General-purpose LLMs pose misinformation risks for development and policy experts, lacking epistemic humility for verifiable outputs. We present AVA (AI + Verified Analysis), a GenAI platform built on a curated library of over 4,000 World Bank Reports with multilingual capabilities. AVA's multi-agent pipeline enables users to query and receive evidence-based syntheses. It operationalizes epistemic humility through two mechanisms: citation verifiability (tracing claims to sources) and reasoned abstention (declining unsupported queries with justification and redirection). We conducted an in-the-wild evaluation with over 2,200 individuals from heterogeneous organisations and roles in 116 countries, via log analysis, surveys, and 20 interviews. Difference-in-Differences estimates associate sustained engagement with 2.4-3.9 hours saved weekly. Qualitatively, participants used AVA as a specialized "evidence engine"; reasoned abstention clarified scope boundaries, and trust was calibrated through institutional provenance and page-anchored citations. We contribute design guidelines for specialized AI and articulate a vision for "ecosystem-aware" Humble AI.

While distributed device-edge speculative decoding enhances resource utilization across heterogeneous nodes, its performance is often bottlenecked by conventional token-level verification strategies. Such rigid alignment leads to excessive rejections, significantly diminishing the accepted sequence length and increasing interaction rounds under fluctuating wireless conditions. In this paper, we propose WISV (Wireless-Informed Semantic Verification), a novel distributed speculative decoding framework that goes beyond strict token-level matching via a channel-aware semantic acceptance policy. WISV integrates a lightweight decision head into the edge-side target LLM to dynamically evaluate speculative tokens by synthesizing high-dimensional hidden representations with instantaneous channel state information (CSI). To optimize the trade-off between verification fidelity and communication overhead, we further design two tailored communication protocols: full-hidden upload and mismatch-first selective-hidden upload. Extensive simulations using a 1B drafter and an 8B target model demonstrate that WISV achieves up to a 60.8% increase in accepted length, a 37.3% reduction in interaction rounds, and a 31.4% improvement in end-to-end latency compared to vanilla speculative decoding across tested settings, while maintaining a negligible task accuracy drop (<1%). Finally, we validate WISV on a hardware testbed comprising an NVIDIA Jetson AGX Orin and an A40-equipped server, confirming its real-world efficacy in accelerating edge-deployed LLM inference.

Retrieval-Augmented Generation (RAG) systems depend on the geometric properties of vector representations to retrieve contextually appropriate evidence. When source documents interleave multiple topics within contiguous text, standard vectorization produces embedding spaces in which semantically distinct content occupies overlapping neighborhoods. We term this condition semantic entanglement. We formalize entanglement as a model-relative measure of cross-topic overlap in embedding space and define an Entanglement Index (EI) as a quantitative proxy. We argue that higher EI constrains attainable Top-K retrieval precision under cosine similarity retrieval. To address this, we introduce the Semantic Disentanglement Pipeline (SDP), a four-stage preprocessing framework that restructures documents prior to embedding. We further propose context-conditioned preprocessing, in which document structure is shaped by patterns of operational use, and a continuous feedback mechanism that adapts document structure based on agent performance. We evaluate SDP on a real-world enterprise healthcare knowledge base comprising over 2,000 documents across approximately 25 sub-domains. Top-K retrieval precision improves from approximately 32% under fixed-token chunking to approximately 82% under SDP, while mean EI decreases from 0.71 to 0.14. We do not claim that entanglement fully explains RAG failure, but that it captures a distinct preprocessing failure mode that downstream optimization cannot reliably correct once encoded into the vector space.

Peer review is central to scientific publishing, yet reviewers frequently include claims that are subjective, rhetorical, or misaligned with the submitted work. Assessing whether review statements are factual and verifiable is crucial for fairness and accountability. At the scale of modern conferences and journals, manually inspecting the grounding of such claims is infeasible. We present Peerispect, an interactive system that operationalizes claim-level verification in peer reviews by extracting check-worthy claims from peer reviews, retrieving relevant evidence from the manuscript, and verifying the claims through natural language inference. Results are presented through a visual interface that highlights evidence directly in the paper, enabling rapid inspection and interpretation. Peerispect is designed as a modular Information Retrieval (IR) pipeline, supporting alternative retrievers, rerankers, and verifiers, and is intended for use by reviewers, authors, and program committees. We demonstrate Peerispect through a live, publicly available demo (https://app.reviewer.ly/app/peerispect) and API services (https://github.com/Reviewerly-Inc/Peerispect), accompanied by a video tutorial (https://www.youtube.com/watch?v=pc9RkvkUh14).

We introduce causal-temporal event graphs (CTEGs) as a formal model for fully resolved recursive agent execution records under single-parenthood causal semantics. We formalise direct event emissions and recursive subagent invocations as extension procedures on generic typed temporal graphs and show that the recursive closure $\mathscr{E}_\infty$ of the induced maximal dynamics starting from single causal roots consists entirely of finite sequences of CTEGs. A CTEG is a rooted arborescence whose nodes carry timestamps and event types, subject to the constraint that timestamps be strictly increasing along causal paths. We realise $\mathscr{E}_\infty$ as the increasing union of a recursive hierarchy $\mathscr{E}_0 \subseteq \mathscr{E}_1 \subseteq \cdots$ of agent execution levels parametrised by recursion depth, which is recognised as the ascending Kleene chain of a monotone operator $\varphi$ admitting $\mathscr{E}_\infty$ as its least fixed point. Although the introduction of the full hierarchy is natural, stabilisation occurs already at $\mathscr{E}_1$ if one insists that the internal construction of a subagent execution trace be a delegated and opaque computational unit. The CTEG formalism supports compositional construction of globally well-formed execution traces from local agent behaviour without centralised coordination, preserves well-formedness under partial execution failure, and admits a natural relational database encoding. The arborescent structure of CTEGs is further compatible with cryptographic Merkle tree commitments for tamper-evident session verification.

Project Yanasse presents a method for discovering new proofs of theorems in one area of mathematics by transferring proof strategy patterns (e.g., Lean 4 tactic invocation patterns) from a structurally distant area. The system extracts tactic usage distributions across 27 top-level areas of Mathlib (217,133 proof states), computes z-scores to identify tactics that are heavily used in a source area but rare or absent in a target area, matches source and target proof states via GPU-accelerated NP-hard analogy (running on a MacBook Air via Apple's MPS backend), and then asks an AI reasoning agent to semantically adapt--not symbol-substitute--the source tactics invocation pattern to the target theorem. In this first part of the study, the method is applied to the pair Probability -> Representation Theory, producing 4 Lean-verified new proofs out of 10 attempts (40%). The proofs compile with zero sorry declarations. The key finding is that tactic schemas decompose into a head (domain-gated, rarely transfers) and a modifier (domain-general, often transfers): filter upwards's head fails in representation theory (no Filter structure), but its [LIST] with ω modifier transfers cleanly as ext1 + simp [LIST] + rfl. Crucially, the underlying matching engine--deep vision lib.py--is entirely domain independent: the same optimization code for an NP-hard matching that matches chess positions by analogy matches Lean proof states by analogy, without knowing which domain it is processing. Only a relation extractor is domain-specific.

We present a novel approach for claim verification from tabular data documents. Recent LLM-based approaches either employ complex pretraining/fine-tuning or decompose verification into subtasks, often lacking comprehensive explanations and generalizability. To address these limitations, we propose a Multi-Agentic framework for Claim verification (MACE) consisting of three specialized agents: Planner, Executor, and Verifier. Instead of elaborate finetuning, each agent employs a zero-shot Chain-of-Thought setup to perform its tasks. MACE produces interpretable verification traces, with the Planner generating explicit reasoning strategies, the Executor providing detailed computation steps, and the Verifier validating the logic. Experiments demonstrate that MACE achieves state-of-the-art (SOTA) performance on two datasets and performs on par with the best models on two others, while achieving 80--100% of best performance with substantially smaller models: 27--92B parameters versus 235B. This combination of competitive performance, memory efficiency, and transparent reasoning highlights our framework's effectiveness.

We introduce a self-play framework for semantic equivalence in Haskell, utilizing formal verification to guide adversarial training between a generator and an evaluator. The framework leverages Liquid Haskell proofs for validating equivalence and execution-based counterexamples for inequivalence, organized via a difficulty-aware curriculum. To facilitate this, we release \textbf{OpInstruct-HSx}, a synthetic dataset of $\approx$28k validated Haskell programs. Empirical experiments show that our evaluator transfers effectively to downstream tasks, achieving up to 13.3pp accuracy gain on EquiBench and consistent gains on PySecDB. Ablation studies on the SEQ-SINQ regimes indicate that while inequivalence supervision provides data volume, equivalence proofs are uniquely responsible for the model's reasoning capabilities. The entire training pipeline and dataset are publicly released on GitHub and Hugging Face respectively.

We report new results on eight problems in mathematics and theoretical computer science, produced with the assistance of Bolzano, an open-source multi-agent LLM system. Bolzano orchestrates rounds of interaction between parallel prover agents and a verifier agent while maintaining a persistent knowledge base that is carried across rounds. Classified using the significance-autonomy taxonomy of Feng et al., six of the eight results reach the level of publishable research, and five of the eight were produced essentially autonomously by Bolzano. Our results provide evidence that LLMs can contribute meaningfully to mathematical research, complementing recent reports by Bubeck et al., Woodruff et al., and others.

Although most of the automated theorem-proving approaches depend on formal proof systems, informal theorem proving can align better with large language models' (LLMs) strength in natural language processing. In this work, we identify a primary bottleneck in informal theorem proving as a lack of insight, namely the difficulty of recognizing the core techniques required to solve complex problems. To address this, we propose $\texttt{DeepInsight}$, a unified training framework designed to cultivate this essential reasoning skill and enable LLMs to perform insightful reasoning. Our framework consists of three components: (1) $\texttt{DeepInsightTheorem}$, a hierarchical dataset that structures informal proofs by explicitly extracting core techniques and proof sketches alongside the final proof; (2) a Progressive Multi-Stage SFT strategy that mimics the human learning process, teaching the model proof writing, planning, and insight identification; and (3) $\texttt{InsightPO}$, a policy optimization method that assigns structured rewards over this insight hierarchy. Our experiments on challenging mathematical benchmarks demonstrate that this insight-aware generation strategy significantly outperforms baselines. These results demonstrate that teaching models to identify and apply core techniques can substantially improve their mathematical reasoning.

Certified program synthesis (aka vericoding) is the process of automatically generating a program, its formal specification, and a machine-checkable proof of their alignment from a natural-language description. Two challenges make vericoding difficult. First, specifications synthesised from natural language are often either too weak to be meaningful or too strong to be implementable, yet existing approaches lack systematic means to detect such defects. Second, the landscape of program verifiers is fragmented: each tool supports a particular reasoning mode -- auto-active (e.g., Dafny, Verus) or interactive (e.g., Coq, Lean) -- with its own trade-off between automation and expressivity. This forces every synthesis methodology to be tailored to a single verification paradigm, limiting the class of tasks it can handle effectively. We overcome both challenges by structuring the certified synthesis workflow around a multi-modal verifier -- a single tool combining dynamic validation, automated proofs, and interactive proof scripting in one foundational framework. We realise this idea in LeetProof, an agentic pipeline built on Velvet, a multi-modal verifier embedded in Lean. Multi-modality enables LeetProof to validate generated specifications via randomised property-based testing before any code is synthesised, decompose the synthesis task into sub-problems guided by verification conditions, and delegate residual proof obligations to frontier AI provers specialised for Lean. We evaluate LeetProof on benchmarks derived from prior work on certified synthesis. Our specification validation uncovers defects in existing reference benchmarks, and LeetProof's staged pipeline achieves a significantly higher rate of fully certified solutions than a single-mode baseline at the same budget -- consistently across two frontier LLM backends.

While Diffusion Large Language Models (dLLMs) offer structural advantages for global planning, efficiently verifying that they arrive at correct answers via valid reasoning traces remains a critical challenge. In this work, we propose a geometric perspective: Reasoning on the Manifold. We hypothesize that valid generation trajectories reside as stable attractors on the high-density manifold of the learned distribution, whereas invalid paths exhibit off-manifold drift. To operationalize this, we introduce Bidirectional Manifold Consistency (BMC), a training-free, unsupervised metric that quantifies the stability of the generated sequence through a forward-masking and backward-reconstruction cycle. Empirically, we demonstrate BMC's versatility across the full reasoning lifecycle: (1) in Diagnosis, it serves as a robust discriminator of solution validity without ground truth answer; (2) in Inference, it enables rejection resampling to effectively concentrate computational resources on complex reasoning tasks; and (3) in Alignment, it functions as a dense geometric reward that transforms sparse outcome supervision into fine-grained guidance, empowering models to self-evolve beyond standard baselines. Our results establish intrinsic geometric stability as a robust indicator of correctness for dLLMs.

Most ATP benchmarks embed the final answer within the formal statement -- a convention we call "Easy Mode" -- a design that simplifies the task relative to what human competitors face and may lead to optimistic estimates of model capability. We call the stricter, more realistic setting "Hard Mode": the system must independently discover the answer before constructing a formal proof. To enable Hard Mode research, we make two contributions. First, we release MiniF2F-Hard and FIMO-Hard, expert-reannotated Hard Mode variants of two widely-used ATP benchmarks. Second, we introduce Discover And Prove (DAP), an agentic framework that uses LLM natural-language reasoning with explicit self-reflection to discover answers, then rewrites Hard Mode statements into Easy Mode ones for existing ATP provers. DAP sets the state of the art: on CombiBench it raises solved problems from 7 (previous SOTA, Pass@16) to 10; on PutnamBench it is the first system to formally prove 36 theorems in Hard Mode -- while simultaneously revealing that state-of-the-art LLMs exceed 80% answer accuracy on the same problems where formal provers manage under 10%, exposing a substantial gap that Hard Mode benchmarks are uniquely suited to measure.

Large Language Models (LLMs) excel at code generation but remain heavily reliant on large-scale annotated solutions and verification-based supervision, which constrains scalability and hinders sustained self-improvement. Recent solver--verifier frameworks exploit program execution as an automatic supervision signal, but their effectiveness degrades as solvers become moderately strong: verifier-generated tests increasingly confirm semantic correctness rather than exposing the remaining failure modes. We propose \textbf{ACE}, a self-evolving code generation framework based on a solver--adversary architecture that prioritizes active failure discovery through execution-centric supervision. A single LLM alternates between generating candidate programs and producing adversarial unit test inputs optimized to induce execution-level failures, such as runtime errors, exceptions, or non-termination. Supervision is derived solely from execution outcomes: robust programs are selected for supervised fine-tuning, while adversarial tests are optimized via Kahneman--Tversky Optimization using execution-derived preferences. Notably, the entire training loop requires no ground-truth code or external reward models. Experiments on CodeContests, MBPP, and LiveCodeBench demonstrate that ACE consistently outperforms strong solver--verifier baselines, achieving 3--7% absolute gains in pass@1, with larger improvements on out-of-distribution benchmarks, while maintaining competitive or improved inference efficiency.

Type annotations are essential when printing terms in a way that preserves their meaning under reparsing and type inference. We study the problem of complete and minimal type annotations for rank-one polymorphic $λ$-calculus terms, as used in Isabelle. Building on prior work by Smolka, Blanchette et al., we give a metatheoretical account of the problem, with a full formal specification and proofs, and formalize it in Isabelle/HOL. Our development is a series of experiments featuring human-driven and AI-driven formalization workflows: a human and an LLM-powered AI agent independently produce pen-and-paper proofs, and the AI agent autoformalizes both in Isabelle, with further human-hinted AI interventions refining and generalizing the development.

We consider the problem of verification modulo tested library contracts as a step towards automating the verification of client programs that use complex libraries. We formulate this problem as the synthesis of modular contracts for the library methods used by the client that are adequate to prove the client correct, and that also pass the scrutiny of a testing engine that tests the library against these contracts. We also consider a new form of method contracts called contextual contracts that arise in this setting that hold in the context of the client program, and can often be simpler and easier to infer than classical modular contracts. We provide a counterexample-guided learning framework to solve this problem, in which the synthesizer interacts with a constraint solver as well as the testing engine in order to infer adequate modular/contextual method contracts and inductive invariants for the client. The main synthesis engines we use are generalizing CHC solvers that are realized using ICE learning algorithms. We realize this framework in a tool called DUALIS and show its efficacy on benchmarks where clients call large libraries.

Automatic translation of natural language mathematics into faithful Lean 4 code is hindered by the fundamental dissonance between informal set-theoretic intuition and strict formal type theory. This gap often causes LLMs to hallucinate non-existent library definitions, resulting in code that fails to compile or lacks semantic fidelity. In this work, we investigate the effectiveness of tool-augmented agents for this task through a systematic factorial analysis of three distinct tool categories: Fine-tuned Model Querying (accessing expert drafts), Knowledge Search (retrieving symbol definitions), and Compiler Feedback (verifying code via a Lean REPL). We first benchmark the agent against one-shot baselines, demonstrating large gains in both compilation success and semantic equivalence. We then use the factorial decomposition to quantify the impact of each category, isolating the marginal contribution of each tool type to overall performance.

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.

Vibe coding inherently assumes iterative refinement of LLM-generated code through feedback loops. While effective for conventional software tasks, its reliability in runtime-adaptive systems is unclear -- especially when generated code is not manually inspected. This paper studies feedback-based automated verification of LLM-generated adaptation managers in Collective Adaptive Systems (CAS). We focus on the key challenges of verification in the loop: how to detect failures of generated code at runtime and how to report them precisely enough for an LLM to fix them. We combine the adaptation loop with a vibe-coding feedback loop where correctness is checked against (i) generic architectural constraints and (ii) functional constraints formalized in Functional Constraints Logic (FCL), a novel first-order temporal logic over potentially finite traces. Conducting the Dragon Hunt CAS case study, we show that fine-grained constraint violations provide actionable feedback that typically yields a valid adaptation manager within a few iterations, while simple coarse metric-based feedback often stalls. Our findings suggest that feedback precision is the dominant factor for reliable vibe coding in systems designed by domain experts with no programming skills, thereby obviating the need for human code inspection.

The proliferation of financial misinformation poses a severe threat to market stability and investor trust, misleading market behavior and creating critical information asymmetry. Detecting such misleading narratives is inherently challenging, particularly in real-world scenarios where external evidence or supplementary references for cross-verification are strictly unavailable. This paper presents our winning methodology for the "Reference-Free Financial Misinformation Detection" shared task. Built upon the recently proposed RFC-BENCH framework (Jiang et al. 2026), this task challenges models to determine the veracity of financial claims by relying solely on internal semantic understanding and contextual consistency, rather than external fact-checking. To address this formidable evaluation setup, we propose a comprehensive framework that capitalizes on the reasoning capabilities of state-of-the-art Large Language Models (LLMs). Our approach systematically integrates in-context learning, specifically zero-shot and few-shot prompting strategies, with Parameter-Efficient Fine-Tuning (PEFT) via Low-Rank Adaptation (LoRA) to optimally align the models with the subtle linguistic cues of financial manipulation. Our proposed system demonstrated superior efficacy, successfully securing the first-place ranking on both official leaderboards. Specifically, we achieved an accuracy of 95.4% on the public test set and 96.3% on the private test set, highlighting the robustness of our method and contributing to the acceleration of context-aware misinformation detection in financial Natural Language Processing. Our models (14B and 32B) are available at https://huggingface.co/KaiNKaiho.

Organisations increasingly outsource privacy-sensitive data transformations to cloud providers, yet no practical mechanism lets the data owner verify that the contracted algorithm was faithfully executed. VeriX-Anon is a multi-layered verification framework for outsourced Target-Driven k-anonymization combining three orthogonal mechanisms: deterministic verification via Merkle-style hashing of an Authenticated Decision Tree, probabilistic verification via Boundary Sentinels and exact-duplicate Twins with cryptographic identifiers, and utility-based verification via Explainable AI fingerprinting that compares SHAP value distributions before and after anonymization using the Wasserstein distance. Across seven cross-domain datasets and four cloud profiles (28 scenarios), against Lazy (drops records), Dumb (fake hash), and Approximate (valid hash) adversaries, VeriX-Anon detects 25 of 28 deviations under a fixed threshold and 27 of 28 once the threshold is calibrated per dataset, with no false alarms. No single layer achieved this alone. The XAI layer was the only mechanism that caught the Approximate adversary, succeeding on six of seven datasets and missing only a high-dimensional case where honest generalization shifts SHAP as much as the attack. Target-Driven anonymization preserved significantly more utility than blind splitting, with mean F1 gaps of +0.058 to +0.362 and Wilcoxon p <= 0.001 on six of seven datasets. Client-side verification completes under one second at one million rows. The threat model covers three empirically evaluated profiles and one theoretical Informed Attacker unable to defeat the cryptographic salt. Sentinel evasion probability ranges from near-zero to 0.82 for the most imbalanced data, which the twin layer offsets in every scenario.

Agents that synthesize their own tools ship a second artifact alongside each answer: a software library that future tasks reuse, compose, and depend on. Task completion (TC) certifies the answer; it does not certify the library. On a Claude Haiku 4.5 pilot, we patch the harness to preserve per-tool source and replay every synthesised tool against a held-out conformance suite. Across 222 preserved tools and three protocols, 96.8% record per-tool correctness C=0.00: two protocols silent-rot at 100%, one at 91.7%. Hand-written reference implementations score C=1.00 on all 16 capability suites. The synthesised tools execute cleanly and the in-session verifier raises no flag; on held-out inputs they simply return wrong answers. This is the verification-vs.-conformance gap a pass-rate benchmark cannot see. EvolveTool-Bench makes that gap measurable. Sessions are structured into seed, gap, variant, composition, regression, and adversarial roles; runs emit per-tool manifests, verified-subset TC, correct-vs.-incorrect reuse, and audit traces for post-deployment monitoring. In the Haiku pilot over five protocols (3 seeds, 8 sessions, 51 verified decisions per pass), TC alone does not separate the protocols; the audit layer does. One pre-specified contrast shows the adapted decision-split synthesis protocol underperforming one-shot synthesis (-7.1 pp, 95% CI [-13.9,-0.1]; not significant after BH correction) -- a hypothesis-generating signal, not a confirmed effect. We release the harness, per-tool source preservation, conformance replay, and reproducibility manifests. An audit schema of this shape -- verifier coverage, reuse decomposition, held-out conformance -- is the minimum a deployed tool-evolving agent should emit between releases.

The emergence of large language model agents capable of invoking external tools has created urgent need for formal verification of agent protocols. Two paradigms dominate this space: Schema-Guided Dialogue (SGD), a research framework for zero-shot API generalization, and the Model Context Protocol (MCP), an industry standard for agent-tool integration. While both enable dynamic service discovery through schema descriptions, their formal relationship remains unexplored. We present the first process calculus formalization of SGD and MCP, proving they are structurally bisimilar under a well-defined mapping Phi. We demonstrate that the reverse mapping Phi-1 is partial and lossy, revealing critical gaps in MCP's expressivity. Through bidirectional analysis, we identify four principles - semantic completeness, explicit action boundaries, failure mode documentation, and inter-tool relationship declaration -- as necessary and sufficient conditions for full behavioral equivalence. We formalize these principles as type-system extensions MCP+, proving MCP+ is fully equivalent to SGD. Our work provides the first formal foundation for verified agent systems and establishes schema quality as a provable safety property. Practically, this means that the current MCP specification has expressiveness gaps compared to SGD and would benefit from the proposed extensions.

The synthesis of inductive loop invariants remains a critical bottleneck in automated program verification. While Large Language Models (LLMs) show promise in mitigating this issue, they often fail on complex programs, producing invariants that are invalid or computationally ineffective. Although fine-tuning is a natural strategy to address these limitations, obtaining high-quality training data remains an open challenge. We first formalize the properties required for a high-quality training invariant, and then present Wonda, a rigorous data curation pipeline that extracts such invariants from raw verifier output via AST-based normalization followed by LLM-driven semantic rewriting and augmentation with provable quality guarantees. Fine-tuning Small Language Models (SLMs) on Wonda-curated data yields consistent gains across the Qwen3, Llama-3.1, and Mistral families: the 4B and 8B Qwen3 models nearly double invariant correctness and double speedup rates, while Llama-3.1-8B triples both. On the challenging InvBench suite, the same 4B model outperforms an off-the-shelf model 20x its size and matches the end-to-end verification time of GPT-OSS-120B, while a 14B Qwen3 model matches that of the frontier model GPT-5.2, all without test-time compute overhead. Our code is publicly available on GitHub.

Large Language Models (LLMs) using chain-of-thought have demonstrated great potential for solving complex reasoning and planning tasks. Despite these advances, LLM-generated outputs remain susceptible to errors, making verification important for reliable reasoning systems. Learned verifiers can increase trust, enforce safety constraints, and ensure alignment with personal preferences, while also providing feedback to improve generation. This raises a central challenge: when learned verifiers are used to guide generation, the feedback loop between generator and verifier may induce a distribution shift. This is particularly salient for process reward models, a prominent class of learned verifiers that score or classify individual steps in a chain-of-thought reasoning trace. Motivated by this challenge, we propose a new online learning framework for chain-of-thought verifiers that, given a problem statement and a reasoning trace, check the correctness of each reasoning step given the preceding steps. Highlighting the asymmetric role of soundness errors (accepting an incorrect reasoning step) and completeness errors (flagging a correct step as wrong), we introduce novel notions of dimension that characterize their optimal tradeoff. We then show how our learned verifiers can boost the accuracy of a weak generator. Assuming that the generator can produce a correct next step with a small success probability, we show how to learn a strong generator with small error and abstention rates. Our results also allow learning from offline data when queries to an expert verifier can be simulated from a small set of correct reasoning traces. However, we establish a separation between our approach and learning from offline expert demonstrations: we show that learning from offline demonstrations cannot in general achieve the soundness-completeness guarantees produced by our interactive learning approach.

As AI systems enter institutional workflows, workers must decide whether to delegate task execution to AI and how much effort to invest in verifying AI outputs, while institutions evaluate workers using outcome-based standards that may misalign with workers' private costs. We model delegation and verification as the solution to a rational worker's optimization problem, and define worker quality by evaluating an institution-centered utility (distinct from the worker's objective) at the resulting optimal action. We formally characterize optimal worker workflows and show that AI induces phase transitions, where arbitrarily small differences in verification ability lead to sharply different behaviors. As a result, AI can amplify workers with strong verification reliability while degrading institutional worker quality for others who rationally over-delegate and reduce oversight, even when baseline task success improves and no behavioral biases are present. These results identify a structural mechanism by which AI reshapes institutional worker quality and amplifies quality disparities between workers with different verification reliability.

In Machine-Assisted Theorem Proving, a theorem proving agent searches for a sequence of expressions and tactics that can prove a statement in a proof assistant. In this work, we introduce several novel concepts and capabilities to address obstacles faced by machine-assisted theorem proving. We first present a set of \textbf{atomic tactics}, a small finite set of tactics capable of proving any provable statement in Lean. We then introduce a \textbf{transposing atomization} algorithm which turns arbitrary proof expressions into a series of atomic tactics. We next introduce the \textbf{ExprGraph} data structure, which provides a succinct representation for Lean expressions. Finally, we present the \textbf{Nazrin Prover}, short for \textbf{N}eural \textbf{A}tomi\textbf{z}e\textbf{r} for \textbf{In}habitation Problems, a graph neural network-based theorem proving agent using atomic tactics and ExprGraph. Nazrin circumvents many challenges faced by existing proving agents by exclusively dispatching atomic tactics, and it is robust enough to both train and evaluate on consumer-grade hardware. We demonstrate the potential of tools like Nazrin using theorems from Lean's standard library and from Mathlib.

Vision-language-action models have advanced robotic manipulation but remain constrained by reliance on the large, teleoperation-collected datasets dominated by the static, tabletop scenes. We propose a simulation-first framework to verify VLA architectures before real-world deployment and introduce MobileManiBench, a large-scale benchmark for mobile-based robotic manipulation. Built on NVIDIA Isaac Sim and powered by reinforcement learning, our pipeline autonomously generates diverse manipulation trajectories with rich annotations (language instructions, multi-view RGB-depth-segmentation images, synchronized object/robot states and actions). MobileManiBench features 2 mobile platforms (parallel-gripper and dexterous-hand robots), 2 synchronized cameras (head and right wrist), 630 objects in 20 categories, 5 skills (open, close, pull, push, pick) with over 100 tasks performed in 100 realistic scenes, yielding 300K trajectories. This design enables controlled, scalable studies of robot embodiments, sensing modalities, and policy architectures, accelerating research on data efficiency and generalization. We benchmark representative VLA models and report insights into perception, reasoning, and control in complex simulated environments, with all code, datasets, and models publicly released.

Multi-Agent Systems (MAS) built on Large Language Models (LLMs) often exhibit high variance in their reasoning trajectories. Process verification, which evaluates intermediate steps in trajectories, has shown promise in general reasoning settings, and has been suggested as a potential tool for guiding coordination of MAS; however, its actual effectiveness in MAS remains unclear. To fill this gap, we present MAS-ProVe, a systematic empirical study of process verification for multi-agent systems (MAS). Our study spans three verification paradigms (LLM-as-a-Judge, reward models, and process reward models), evaluated across two levels of verification granularity (agent-level and iteration-level). We further examine five representative verifiers and four context management strategies, and conduct experiments over six diverse MAS frameworks on multiple reasoning benchmarks. We find that process-level verification does not consistently improve performance and frequently exhibits high variance, highlighting the difficulty of reliably evaluating partial multi-agent trajectories. Among the methods studied, LLM-as-a-Judge generally outperforms reward-based approaches, with trained judges surpassing general-purpose LLMs. We further observe a small performance gap between LLMs acting as judges and as single agents, and identify a context-length-performance trade-off in verification. Overall, our results suggest that effective and robust process verification for MAS remains an open challenge, requiring further advances beyond current paradigms. Code is available at https://github.com/Wang-ML-Lab/MAS-ProVe.

While large language models (LLMs) have achieved strong performance on mathematical problems with verifiable answers, many advanced problems are proof-based and require evaluating full proofs. However, training such verifiers requires diverse and trustworthy question-proof-check (QPC) examples at scale, which are scarce. To address this challenge, we develop a human-audited, LLM-assisted data pipeline that produces large-scale QPC triplets with limited human effort. By systematically varying problem sources, generation strategies, and generator models, the pipeline creates diverse problem-proof pairs spanning multiple difficulty levels, linguistic styles, and error types. We combine multi-LLM agreement with hierarchical human auditing to obtain accurate proof-correctness labels. Using these data, we train generative proof verifiers and introduce an auxiliary fluency filter together with balanced token weighting to stabilize binary-reward long-form verification RL. Experiments show that our verifier improves proof-judgment accuracy across different proof styles and provides useful guidance for test-time selection. Overall, our results provide a practical data and training framework for natural-language proof verification.

The performance of large language models (LLMs) on verifiable tasks is usually measured by pass@k, the probability of answering a question correctly at least once in k trials. At a fixed budget, a more suitable metric is coverage@cost, the average number of unique questions answered as a function of the total number of attempts. We connect the two metrics and show that the empirically-observed power-law behavior in pass@k leads to a sublinear growth of the coverage@cost (diminishing returns). To solve this problem, we propose Reset-and-Discard (ReD), a query method of LLMs that increases coverage@cost for a given budget, regardless of the pass@k form. Moreover, given a pass@k, we can quantitatively predict the savings in the total number of attempts using ReD. If pass@k is not available for the model, ReD can infer its power-law exponent. Experiments on three LLMs across coding (HumanEval), math (GSM8K), and reasoning (MMLU-Pro) benchmarks demonstrate that ReD substantially reduces the required attempts, tokens, and USD cost to reach a desired coverage, while also offering an efficient way to measure inference power-laws. ReD's advantage is maintained for imperfect verifiers and outperforms the tested allocation baselines.

Determining whether a program terminates is a central problem in computer science. Turing's Halting Problem established termination as undecidable, showing that no algorithm can universally determine termination for all programs and inputs. Hence, verification tools approximate termination, sometimes failing to prove or disprove; these tools rely on problem-specific architectures and are usually tied to particular programming languages. Recent advances in LLMs raise a natural question: To what extent can they reason about program termination? We evaluate frontier LLMs on a diverse set of C programs from the International Competition on Software Verification (SV-Comp) 2025. Our results show that GPT-5 and Claude Sonnet-4.5 achieve scores comparable to top-ranked verification tools (with test-time scaling). However, while models often correctly infer whether programs terminate, they frequently fail to construct a witness as formal proof, revealing a gap between semantic recognition and symbolic proof generation. Performance further degrades as code length increases. Beyond witness automaton graphs, we introduce a divergence-precondition formulation that characterizes non-termination conditions as logical constraints. We hope these findings motivate future research on real-world termination benchmarks, neuro-symbolic approaches that combine LLMs with symbolic verification methods, and, more broadly, LLM reasoning on other undecidable problems.

RL-trained Lean theorem provers mode-collapse at inference time: on miniF2F-test with DeepSeek-Prover-V1.5-RL, doubling the i.i.d.\ sampling budget from $k{=}32$ to $k{=}64$ produces zero additional solved theorems (42/244 in both cases). A fixed schedule of 15 tactic skeletons breaks this plateau and recovers a $+45$ relative improvement at $k{=}16$ (mean $Δ= +12.3 \pm 4.2$ theorems across $n{=}3$ seeds, sign preserved in every seed). A controlled diversity ablation rules out the prompt-diversity confound: tactic skeletons help, paraphrases match the baseline, and irrelevant Lean comments actively degrade. A leave-one-out formalization-difficulty stratification reveals a structural-content gradient across the three perturbations. The phenomenon is RL-specific: V1.5-Base proves zero theorems regardless of intervention, identifying RL as the stage that creates the proof capability which subsequently collapses; extending to two additional 7B Lean provers, RL-trained DeepSeek-Prover-V2-7B contributes $+3$ frontier solves no i.i.d. baseline can reach despite a flat aggregate, while SFT-trained Goedel-Prover does not ( $-10.0$ $\pm 4.4$ theorems, $n{=}3$, sign preserved every seed). Inference-time structural diversity is a cheap, complementary axis for RL-trained provers, orthogonal to scaling model size or training compute.

A program migrated to post-quantum cryptography can verify its own signatures while producing keys or signatures that another implementation rejects. We introduce a contract-based task for migrating a Go file signer from RSA to ML-DSA-44, and compare coding agents with and without structured checker feedback. Both conditions receive the contract, compiler, documentation, and OpenSSL. Across 160 attempts in four local-agent configurations, twelve final patches pass local verification but fail external requirements. Checker access does not increase the observed completion rate in any comparison. Recorded traces show unresolved defects and checks invoked only after a patch is correct. Serving settings also affect completion: reducing only Qwen3.8's context window from 128K to 32K lowers full passes from 36/40 to 4/40. Four exploratory trials using GPT-6 Astra through Codex and Claude Fable 5.1 through Claude Code pass all 40 checks, including both baselines. These findings support evaluating interoperability separately from local agreement and reporting harness and serving limits alongside agent results.

LLMs excel at reasoning, but validating their steps remains challenging. Formal verification offers a solution through mechanically checkable proofs. Interactive theorem provers (ITPs) dominate mathematical reasoning but require detailed low-level proof steps, while auto-active verifiers offer automation but focus on software verification. Recent work has begun bridging this divide by evaluating LLMs for software verification in ITPs, but the complementary direction, LLMs for mathematical theorem proving in auto-active verifiers, remains unexplored. We present MINIF2F-DAFNY, the first translation of the widely-used mathematical benchmark miniF2F to an auto-active verifier: Dafny. We find that Dafny's automation alone solves 39-44% of problems with empty proofs, whereas many require substantial proof guidance in ITPs. We evaluate 8 off-the-shelf LLMs on proof generation, with the best model (Claude Opus 4.6) achieving 62.7% cumulative pass@4 on the full test set, improving over the 38.9% empty-proof baseline by 23.8 percentage points. These results show that auto-active verification offers a complementary empirical setting for AI-assisted mathematical reasoning, where LLMs provide high-level guidance while SMT automation handles low-level details. Our benchmark and evaluation infrastructure are publicly available on https://github.com/dafny-lang/miniF2F.

Language models frequently produce plausible yet incorrect reasoning traces that are difficult to verify. We investigate fine-tuning models to use Prolog as an external symbolic reasoning tool, training Qwen2.5-3B-Instruct with Group Relative Policy Optimization (GRPO) on a cleaned version of GSM8K (which we release as gsm8k-prolog-prover). We systematically vary prompt structure, reward composition (execution, syntax, semantics, structure), and inference protocol (single-try, multiple-try, and two agentic modes). Our reinforcement learning approach outperforms supervised fine-tuning on GSM8K, and the resulting 3B model achieves zero-shot performance on MMLU-STEM and MMLU-Pro competitive with 7B few-shot baselines. Most importantly, we identify an accuracy--auditability trade-off: configurations tuned for correctness alone learn to delegate reasoning to natural language and use Prolog only for the final computation, while configurations rewarded for symbolic structure produce fully auditable programs at a cost in accuracy. We interpret this trade-off as a form of reward hacking and discuss its implications for deploying neurosymbolic systems in safety-critical domains. The source code for our experiments is available under https://github.com/aisilab/Prolog-as-a-Tool

The integration of Formal Verification tools with Large Language Models (LLMs) offers a path to scale software verification beyond manual workflows. However, current methods remain unreliable: without a solid theoretical footing, the refinement process acts as a black box that may oscillate, loop, or diverge. This work bridges this critical gap by developing an LLM-Verifier Convergence Theorem, providing the first formal framework with provable guarantees for termination in multi-stage verification pipelines. We model the interaction not as a generic loop, but as a sequential absorbing Markov Chain comprising four essential engineering stages: \texttt{CodeGen}, \texttt{Compilation}, \texttt{InvariantSynth}, and \texttt{SMTSolving}. We prove that for any non-zero stage success probability ($δ> 0$), the system reaches the \texttt{Verified} state almost surely. Furthermore, because of the sequential nature of the pipeline, we derive a precise latency bound of $\mathbb{E}[n] \leq 4/δ$. We stress-tested this prediction in an extensive empirical campaign comprising over 90,000 trials. The results match the theory with striking consistency: every run reached verification, and the empirical convergence factor clustered tightly around $C_f\approx 1.0$, confirming that the $4/δ$ bound accurately mirrors system behavior rather than serving as a loose buffer. Based on this data, we identify three distinct operating zones -- marginal, practical, and high-performance -- and propose a dynamic calibration strategy to handle parameter drift in real-world environments. Together, these contributions replace heuristic guesswork with a rigorous architectural foundation, enabling predictable resource planning and performance budgeting for safety-critical software.

Automatic verification is a critical component in building math-solving agents and reinforcement learning, yet it often falls short in generalizability, performance, and cost-efficiency. Identifying that the primary bottleneck of verification lies in error detection capability, we propose pessimistic verification, a paradigm of agentic workflows that rejects a solution if any of multiple parallel verifiers identifies a flaw. We further introduce progressive pessimistic verification, which employs fine-grained proof decomposition to significantly enhance verification accuracy and efficiency. Our approach surpasses the performance and token efficiency of extended long chain-of-thought (long CoT) and mainstream verification workflows, crucially, our analysis reveals that existing benchmarks underestimate its effectiveness on stronger models due to inherent annotation errors. To further validate the effectiveness of our method, we applied a verification-based solving workflow on the IMO 2025 and MathArena Apex 2025 datasets, where the workflow with progressive pessimistic verification exhibits remarkable improvements in both efficiency and accuracy on highly challenging contest-level math problems with state-of-the-art models. Code is available at https://github.com/THUNLP-MT/pverify.

Large Language Models (LLMs) excel at both informal and formal (e.g. Lean 4) mathematical reasoning but still struggle with autoformalisation, the task of transforming informal into formal mathematical statements. Yet, the performance of current Math LLMs is constrained by the scarcity of large-scale corpora, particularly those containing pairs of informal and formal statements. Interestingly, the formal languages used in autoformalisation share structural similarities with programming languages, and code data is available at scale. However, current models trained on code do not transfer effectively to formal math, due to structural and syntactic differences between them. To address this, we propose TopoAlign, a framework that unlocks widely available code repositories as training resources for Math LLMs. TopoAlign decomposes code into docstrings, main functions, and dependency functions, and reassembles these components into analogues that structurally mirror formal statements. We train three state-of-the-art models, DeepSeek-Math, Qwen-3 and Herald, and evaluate them on the MiniF2F, Putnam, and ProofNet benchmarks. TopoAlign provides substantial gains for DeepSeek-Math, improving performance by 17.77% on BEq@10 and 68.82% on typecheck@10, and also measurably improves Herald by 0.12% on BEq@10 and 1.09% on typecheck@10 despite introducing no new mathematical knowledge.

The objectives that Large Language Models (LLMs) implicitly optimize remain dangerously opaque, making trustworthy alignment and auditing a grand challenge. While Inverse Reinforcement Learning (IRL) can infer reward functions from behaviour, existing approaches either produce a single, overconfident reward estimate or fail to address the fundamental ambiguity of the task (non-identifiability). This paper introduces a principled auditing framework that re-frames reward inference from a simple estimation task to a comprehensive process for verification. Our framework leverages Bayesian IRL to not only recover a distribution over objectives but to enable three critical audit capabilities: (i) Quantifying and systematically reducing non-identifiability by demonstrating posterior contraction over sequential rounds of evidence; (ii) Providing actionable, uncertainty-aware diagnostics that expose spurious shortcuts and identify out-of-distribution prompts where the inferred objective cannot be trusted; and (iii) Validating policy-level utility by showing that the refined, low-uncertainty reward can be used directly in RLHF to achieve training dynamics and toxicity reductions comparable to the ground-truth alignment process. Empirically, our framework successfully audits a detoxified LLM, yielding a well-calibrated and interpretable objective that strengthens alignment guarantees. Overall, this work provides a practical toolkit for auditors, safety teams, and regulators to verify what LLMs are truly trying to achieve, moving us toward more trustworthy and accountable AI.

Accurate auto-formalization of theorem statements is essential for advancing automated discovery and verification of research-level mathematics, yet remains a major bottleneck for LLMs due to hallucinations, semantic mismatches, and their inability to synthesize new definitions. To tackle these issues, we present Aria (Agent for Retrieval and Iterative Autoformalization), a system for conjecture-level formalization in Lean that emulates human expert reasoning via a two-phase Graph-of-Thought process: recursively decomposing statements into a dependency graph and then constructing formalizations from grounded concepts. To ensure semantic correctness, we introduce AriaScorer, a checker that retrieves definitions from Mathlib for term-level grounding, enabling rigorous and reliable verification. We evaluate Aria on diverse benchmarks. On ProofNet, it achieves 91.6% compilation success rate and 68.5% final accuracy, surpassing previous methods. On FATE-X, a suite of challenging algebra problems from research literature, it outperforms the best baseline with 44.0% vs. 24.0% final accuracy. On a dataset of homological conjectures, Aria reaches 42.9% final accuracy while all other models score 0%.

Neural networks with ReLU activations are a widely used model in machine learning. It is thus important to have a profound understanding of the properties of the functions computed by such networks. Recently, there has been increasing interest in the (parameterized) computational complexity of determining these properties. In this work, we close several gaps and resolve an open problem posed by Froese et al. [COLT '25] regarding the parameterized complexity of various problems related to network verification. In particular, we prove that, for all $\ell\ge 2$, deciding positivity (and thus surjectivity) of a function $f:\mathbb{R}^d\to\mathbb{R}$ computed by an $\ell$-layer ReLU network is W[$\ell-1$]-hard when parameterized by the input dimension $d$. The case $\ell=2$ implies that zonotope non-containment (a problem that is of independent interest in computational geometry, control theory, and robotics) is W[1]-hard with respect to the ambient dimension $d$. Moreover, we show that approximating the maximum within any multiplicative factor and computing the $L_p$-Lipschitz constant for $p\in(0,\infty]$ in $\ell$-layer networks is NP-hard and W[$\ell-1$]-hard with respect to $d$. For $\ell\ge 3$, approximating the $L_p$-Lipschitz constant is NP- and W[$\ell-2$]-hard. We further show that the above problems are NP- and W[$t$]-hard (for all $t\ge 1$) with respect to $\ell$ for constant $d$. Notably, our hardness results imply that the naive enumeration-based methods for these fundamental problems running in $n^{(\ell-1) d}\cdot\operatorname{poly}(N)$ time are all essentially optimal under the Exponential Time Hypothesis.

Large Language Models (LLMs) have demonstrated significant promise in formal theorem proving. In this study, we investigate the ability of LLMs to discover novel theorems and produce verified proofs. We propose a pipeline called Conjecturing-Proving Loop (CPL), which iteratively generates mathematical conjectures and attempts to prove them in Lean 4. A key feature of CPL is that each iteration conditions the LLM on previously generated theorems and their formal proofs, enabling parameter-free improvement of proof strategies via in-context learning. We provide both theoretical and experimental evidence that CPL increases the discovery rate of hard-to-prove theorems compared to frameworks that generate statements and proofs simultaneously. Moreover, our experiments show that reusing the LLM's own formally verified outputs as context consistently improves subsequent proof success, demonstrating the effectiveness of self-generated in-context learning for neural theorem proving. The source code is available at https://github.com/auto-res/ConjecturingProvingLoop.

Target-conditioned sequence models provide a simple interface for controllable offline decision making, but the requested target return can be an unreliable control signal, especially when the target return lies in underrepresented regions of the dataset. This paper proposes Doctor, a hybrid sequence modeling and reinforced verification framework for controllable target-conditioned offline decision making. Doctor trains a shared masked trajectory Transformer with two complementary objectives: masked trajectory reconstruction for candidate generation and in-sample value learning for action-value verification. At inference time, the model samples multiple nearby target returns, generates candidate actions in parallel, and selects the action whose verified value is closest to the requested target return. We analyze this verifier-guided selection rule and show that its value-level alignment error is bounded by candidate-value coverage around the target return and verifier accuracy. Experiments on D4RL and EpiCare show that Doctor improves target-return alignment under reduced high-return coverage, remains competitive on standard offline return-maximization benchmarks, and enables a single policy to modulate between conservative and aggressive operating points in a simulated clinical decision-making task. These results suggest that reinforced verification can improve the controllability of target-conditioned policies.

Existing informal language-based (e.g., human language) Large Language Models (LLMs) trained with Reinforcement Learning (RL) face a significant challenge: their verification processes, which provide crucial training signals, are neither reliable nor scalable. In fact, the prevalent large proprietary models could hardly generate verifiable programs. A promising yet largely uncharted alternative is formal language-based reasoning. Grounding LLMs in rigorous formal systems where generative models operate in formal language spaces (e.g., Dafny) enables the automatic and mathematically provable verification of their reasoning processes and outcomes. This capability is pivotal for achieving large-scale, reliable formal software verification. It is a common practice to employ human-annotated chain-of-thought and answers to induce the reasoning and coding capabilities of LLMs. Unfortunately, it becomes unacceptably all-consuming to provide such priors for supervising complex programming tasks. In this work, we systematically explore ways to reduce human annotations with the formal language, Dafny, as the main environment for our pilot study. Our pipeline mainly relies on introducing an automatic and scalable data curation pipeline, and careful RL designs integrated with feedback from the formal language verifier. We introduce DafnyComp, a benchmark of compositional formal programs with auto-formalized specifications for specification reasoning. Our supervised fine-tuning (SFT) stage enables even small models (e.g., 0.5B) to generate syntactically valid and verifiable Dafny code, surpassing proprietary models. RL with regularization further improves performance, achieving stronger generalization to out-of-domain tasks and outperforming all strong baselines on the challenging DafnyComp benchmark.

Deep neural networks (DNNs) rely heavily on high-quality open-source datasets (e.g., ImageNet) for their success, making dataset ownership verification (DOV) crucial for protecting public dataset copyrights. In this paper, we find existing DOV methods (implicitly) assume that the verification process is faithful, where the suspicious model will directly verify ownership by using the verification samples as input and returning their results. However, this assumption may not necessarily hold in practice and their performance may degrade sharply when subjected to intentional or unintentional perturbations. To address this limitation, we propose the first certified dataset watermark (i.e., CertDW) and CertDW-based certified dataset ownership verification method that ensures reliable verification even under malicious attacks, under certain conditions (e.g., constrained pixel-level perturbation). Specifically, inspired by conformal prediction, we introduce two statistical measures, including principal probability (PP) and watermark robustness (WR), to assess model prediction stability on benign and watermarked samples under noise perturbations. We derive provable certification conditions relating WR to a PP-based calibration threshold, and a high-probability upper bound on the false positive rate, enabling ownership verification when a suspicious model's WR value significantly exceeds the PP values of multiple benign models trained on watermark-free datasets. If the number of PP values smaller than WR exceeds a threshold determined via conformal calibration, the suspicious model is regarded as having been trained on the protected dataset. Extensive experiments on benchmark datasets verify the effectiveness of our CertDW method and its resistance to potential adaptive attacks. Our codes are at \href{https://github.com/NcepuQiaoTing/CertDW}{GitHub}.

Large language models (LLMs) have achieved remarkable progress in formal mathematical reasoning. Mathematical competition problems fall into two broad types: theorem-proving problems ask for a proof of a fully specified statement, whereas answer-construction problems ask the solver to construct an answer object and prove that it satisfies the stated specification. Existing mathematical reasoning engines mainly target theorem-proving problems, yet answer-construction problems remain less studied. This setting is challenging because model capabilities are misaligned, with general LLMs better suited to answer construction and prover LLMs better suited to proof generation, and because Lean proof checking alone does not rule out inadmissible circular witnesses. To close this gap, we introduce Enumerate-Conjecture-Prove (ECP), a neuro-symbolic framework for solving answer-construction problems in Lean. ECP uses general LLMs to perform bounded enumeration and construct candidate answers, and invokes prover LLMs to produce machine-checked proofs. ECP introduces admissibility checking to ensure that each answer is canonical and does not involve a circular argument. On answer-construction problems from PutnamBench and autoformalized MathArena, ECP formally solves 17/346 PutnamBench instances and 18/75 MathArena instances with admissible answers and proofs, outperforming LLM baselines at aligned inference budgets.

Large language models (LLMs) struggle with formal domains that require rigorous logical deduction and symbolic reasoning, such as mathematical proof generation. We propose a neuro-symbolic approach centered on the hypothesis that structurally analogous problems often admit similar proofs. As a proof-of-concept, we focus on SAT-level geometry problems. Our approach is two-fold: (1) We retrieve analogous problems and use their proofs to guide the LLM, and (2) a formal verifier evaluates the generated proofs and provides feedback, helping the model fix incorrect proofs. Our complete pipeline substantially improves proof accuracy across model families, achieving 68%-96% accuracy compared with 10%-44% for LLM-only baselines that use neither analogy retrieval nor verifier feedback. When comparing against models with the same verifier feedback and inference budget, analogical guidance improves accuracy from 88% to 96% for GPT-5, 78% to 86% for Claude Sonnet 4.6, 72% to 86% for Gemini-Flash-2.5, and 52% to 80% for OpenAI o1. More broadly, shifting to LLMs that generate provably correct conclusions has the potential to dramatically improve their reliability, accuracy and consistency, unlocking complex tasks and critical real-world applications that require trustworthiness.

Speaker verification models are trained on large-scale public datasets whose licenses usually prohibit unauthorized commercial use, yet such infringement is difficult to detect or deter. Dataset ownership verification (DOV) is the mainstream countermeasure: it can watermark a dataset with backdoor attacks so that models trained on it exhibit owner-specified behaviors. However, existing DOV methods presuppose a closed label space fixed at watermarking time, whereas in open-set speaker verification the identities that a deployed model accepts are enrolled by third parties after release and are never observed by the dataset owner. We show that straightforward adaptations fail in two characteristic modes, and accordingly distill three requirements for an effective watermark, namely identity agnosticism, coverage, and fidelity, together with an intrinsic tension between the latter two. Our clustering-based backdoor watermark (CBW) resolves this tension by partitioning training speakers into clusters by feature similarity and implanting a distinct trigger for each cluster, so that each trigger covers one region of the speaker embedding space while the trigger set is designed to jointly cover it. We further develop paired hypothesis tests for ownership verification under both the similarity-available and the decision-only black-box settings at the 1-to-1 and 1-to-$N$ enrollment scales, and theoretically characterize when the audit succeeds, including an exact small-sample certificate and the effect of the enrollment size. Extensive experiments on benchmark datasets and representative models verify the effectiveness of our CBW, its resistance to watermark-removal attacks, and its transferability across model structures. Code is at https://github.com/Radiant0726/CBW/tree/master.