Daily Papers

Large language models have achieved near-expert performance in structured reasoning domains like mathematics and programming, yet their ability to perform compositional multi-hop reasoning in specialized scientific fields remains limited. We propose a bottom-up learning paradigm in which models are grounded in axiomatic domain facts and compose them to solve complex, unseen tasks. To this end, we present a post-training pipeline, based on a combination of supervised fine-tuning and reinforcement learning (RL), in which knowledge graphs act as implicit reward models. By deriving novel reward signals from knowledge graph paths, we provide verifiable, scalable, and grounded supervision that encourages models to compose intermediate axioms rather than optimize only final answers during RL. We validate this approach in the medical domain, training a 14B model on short-hop reasoning paths (1-3 hops) and evaluating its zero-shot generalization to complex multi-hop queries (4-5 hops). Our experiments show that path-derived rewards act as a "compositional bridge", enabling our model to significantly outperform much larger models and frontier systems like GPT-5.2 and Gemini 3 Pro, on the most difficult reasoning tasks. Furthermore, we demonstrate the robustness of our approach to adversarial perturbations against option-shuffling stress tests. This work suggests that grounding the reasoning process in structured knowledge is a scalable and efficient path toward intelligent reasoning.

We present AXIOM, a trust-first neuro-symbolic execution architecture for natural-language mathematical reasoning. In AXIOM, the language model functions strictly as a canonicalizer: it rewrites informal problem text into a narrow schema consumed by a deterministic Computer-Algebra-System (CAS) pipeline, which derives and verifies the answer or abstains as a first-class output. Routing follows a 1:1:1 alignment between problem-shape regex, schema-specific prompt, and closed-form CAS handler, with 3,100+ such routes shipped and zero LOST_CORRECT regressions across 250+ consecutive ship commits. We report empirical results on 4 MATH categories with a cumulative correctness of 94.36% (2,592/2,747) at 100.00% trust on parseable (zero confident-wrong answers across the full 2,747-record benchmark), all four domains above the per-domain 70/90/70 floor with per-domain trust at 100.0%, and median latency of 1 ms on rule-only handlers (88% of records on the lm-eval arithmetic 20,000-record benchmark). The architecture has served ~30,000 production queries through a public deployment. The contribution we emphasize is not a final accuracy figure but the forward dynamic the architecture establishes: every logged abstain in production is a candidate correct after one ship cycle, since new tasks compose without regressing the registry. The operational discipline behind this property -- math-template bucketing, LOST_CORRECT scan as regression oracle, parseable-first onboarding, and abstain as first-class output -- constitutes a transferable framework for trustworthy neuro-symbolic systems beyond mathematics.

For text-based AI systems to interact in the real world, causal reasoning is an essential skill. Since interventional data is costly to generate, we study to what extent an agent can learn causal reasoning from passive data. Specifically, we consider an axiomatic training setup where an agent learns from multiple demonstrations of a causal axiom (or rule), rather than incorporating the axiom as an inductive bias or inferring it from data values. A key question is whether the agent would learn to generalize from the axiom demonstrations to new scenarios. For example, if a transformer model is trained on demonstrations of the causal transitivity axiom over small graphs, would it generalize to applying the transitivity axiom over large graphs? Our results, based on a novel axiomatic training scheme, indicate that such generalization is possible. We consider the task of inferring whether a variable causes another variable, given a causal graph structure. We find that a 67 million parameter transformer model, when trained on linear causal chains (along with some noisy variations) can generalize well to new kinds of graphs, including longer causal chains, causal chains with reversed order, and graphs with branching; even when it is not explicitly trained for such settings. Our model performs at par (or even better) than many larger language models such as GPT-4, Gemini Pro, and Phi-3. Overall, our axiomatic training framework provides a new paradigm of learning causal reasoning from passive data that can be used to learn arbitrary axioms, as long as sufficient demonstrations can be generated.

This article establishes a complete approximate axiomatization for the real-closed field R expanded with all differentially-defined functions, including special functions such as sin(x), cos(x), e^x, dots. Every true sentence is provable up to some numerical approximation, and the truth of such approximations converge under mild conditions. Such an axiomatization is a fragment of the axiomatization for differential dynamic logic, and is therefore a finite extension of the axiomatization of real-closed fields. Furthermore, the numerical approximations approximate formulas containing special function symbols by FOL_{R} formulas, improving upon earlier decidability results only concerning closed sentences.

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 common assumption holds that enough observational and interventional data, given to a strong enough predictor, suffices. We report a failure mode that contradicts it. Across hundreds of structural causal models, on identified quantities a strong predictor and a Bayesian baseline both succeed, but on unidentified quantities (the couplings between counterfactual worlds) the predictor collapses to a point, on 28% of models to one no valid model can produce, while the truth is an admissible interval more data never narrows. The gap is structural: prediction cannot represent uncertainty over counterfactual couplings. We cast a world model as a single positive semidefinite coupling kernel K(T,T') over admissible worlds, whose diagonal is the ordinary posterior (what a predictor recovers) and whose off-diagonal is the cross-world coupling it cannot, which every counterfactual reads. The paper is the theory of that off-diagonal. It is real: two states with identical posteriors differ on a cross-world query, and the off-diagonal is the coupling that fixes counterfactuals. It can be bounded: positive semidefiniteness is partial-identifying information the marginals lack, and enforcing it bounds counterfactuals in polynomial time where the exact response-type program is intractable. Logical structure sharpens it: ontology axioms tighten the bound by up to a third, propagating to couplings they never touch. It can be acquired: targeted scars, constraints learned from encountered infeasibilities, close the gap several times faster than untargeted ones. Its full reconstruction is approximate counting of the admissible worlds, tractable below the Sly-Sun threshold and inapproximable above; we do not claim to beat the worst case.

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

Agentic Entity Resolution (ER) systems iteratively plan actions, retrieve evidence, revise hypotheses, and propose graph mutations. An erroneous identity hypothesis may condition later retrieval and produce apparent confirmation. This paper proposes a Digital Identity Optimization (DIO) and Ontology of Digital Identity (ODI)-informed authorization boundary between evidence available for agent reasoning and evidence eligible to authorize persistent identity-changing mutations. A domain-governed constraint-binding function maps selected reconstruction-failure distinctions to explicit identity-preserving constraints. An abstract two-phase supervisor permits a commit only when claim-specific authorization-eligible evidence supports every required proposition, relevant conflicts are resolved, the operation remains within its authorized scope, and relevant state and policy versions remain valid. Admission as UNVERIFIED does not confer authorization eligibility. Separate claim-specific promotion governs evidentiary use; insufficient support or unresolved material conflict leads to deferral. The contribution is a specification connecting diagnostic-to-constraint mapping, evidence promotion, proposition coverage, conflict handling, scope binding, and commit-time version integrity. A proposed evaluation protocol is designed to assess the mapping's incremental value beyond generic authorization and conventional identifier rules. This is a conceptual architecture without implementation or empirical validation, not a theorem of semantic correctness. Its property depends on domain-policy adequacy, evidence promotion, and enforcement. It does not prevent erroneous hypotheses or hypothesis-conditioned retrieval; dependency-aware evidence flow remains future research.

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.

This paper introduces a novel method for merging open-domain terminological knowledge. It takes advantage of the Region Connection Calculus (RCC5), a formalism used to represent regions in a topological space and to reason about their set-theoretic relationships. To this end, we first propose a faithful translation of terminological knowledge provided by several and potentially conflicting sources into region spaces. The merging is then performed on these spaces, and the result is translated back into the underlying language of the input sources. Our approach allows us to benefit from the expressivity and the flexibility of RCC5 while dealing with conflicting knowledge in a principled way.

Current mathematical reasoning benchmarks for large language models (LLMs) are approaching saturation, with some achieving > 90% accuracy, and are increasingly compromised by training-set contamination. We introduce Putnam-AXIOM, a benchmark of 522 university-level competition problems drawn from the prestigious William Lowell Putnam Mathematical Competition, and Putnam-AXIOM Variation, an unseen companion set of 100 functional variants generated by programmatically perturbing variables and constants. The variation protocol produces an unlimited stream of equally difficult, unseen instances -- yielding a contamination-resilient test bed. On the Original set, OpenAI's o1-preview -- the strongest evaluated model -- scores 41.9%, but its accuracy drops by 19.6% (46.8% relative decrease) on the paired Variations. The remaining eighteen models show the same downward trend, ten of them with non-overlapping 95% confidence intervals. These gaps suggest memorization and highlight the necessity of dynamic benchmarks. We complement "boxed" accuracy with Teacher-Forced Accuracy (TFA), a lightweight metric that directly scores reasoning traces and automates natural language proof evaluations. Putnam-AXIOM therefore provides a rigorous, contamination-resilient evaluation framework for assessing advanced mathematical reasoning of LLMs. Data and evaluation code are publicly available at https://github.com/brando90/putnam-axiom.

We study the impact on mechanisms for facility location of moving from one dimension to two (or more) dimensions and Euclidean or Manhattan distances. We consider three fundamental axiomatic properties: anonymity which is a basic fairness property, Pareto optimality which is one of the most important efficiency properties, and strategy proofness which ensures agents do not have an incentive to mis-report. We also consider how well such mechanisms can approximate the optimal welfare. Our results are somewhat negative. Moving from one dimension to two (or more) dimensions often makes these axiomatic properties more difficult to achieve. For example, with two facilities in Euclidean space or with just a single facility in Manhattan space, no mechanism is anonymous, Pareto optimal and strategy proof. By contrast, mechanisms on the line exist with all three properties.We also show that approximation ratios may increase when moving to two (or more) dimensions. All our impossibility results are minimal. If we drop one of the three axioms (anonymity, Pareto optimality or strategy proofness) multiple mechanisms satisfy the other two axioms.

The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.

Large language models (LLMs) have been reported to linearly encode truthfulness, yet recent work questions this finding's generality. We reconcile these views with the truthfulness spectrum hypothesis: the representational space contains directions ranging from broadly domain-general to narrowly domain-specific. To test this hypothesis, we systematically evaluate probe generalization across five truth types (definitional, empirical, logical, fictional, and ethical), sycophantic and expectation-inverted lying, and existing honesty benchmarks. Linear probes generalize well across most domains but fail on sycophantic and expectation-inverted lying. Yet training on all domains jointly recovers strong performance, confirming that domain-general directions exist despite poor pairwise transfer. The geometry of probe directions explains these patterns: Mahalanobis cosine similarity between probes near-perfectly predicts cross-domain generalization (R^2=0.98). Concept-erasure methods further isolate truth directions that are (1) domain-general, (2) domain-specific, or (3) shared only across particular domain subsets. Causal interventions reveal that domain-specific directions steer more effectively than domain-general ones. Finally, post-training reshapes truth geometry, pushing sycophantic lying further from other truth types, suggesting a representational basis for chat models' sycophantic tendencies. Together, our results support the truthfulness spectrum hypothesis: truth directions of varying generality coexist in representational space, with post-training reshaping their geometry. Code for all experiments is provided in https://github.com/zfying/truth_spec.

Recent research-level physics-reasoning benchmarks for LLMs reveal a failure mode that lies not in knowledge or execution, but in modeling the world being reasoned about. Models often silently alter given quantities, fabricate laws, or inject unstated assumptions to force a problem into familiar patterns, and then reason inside this corrupted world. We define this premise-level divergence as modeling drift: the gap between a model's internal axiomatic world model (AWM) and the true physical world specified in the problem. Although RLVR and PRM have yielded substantial gains in mathematics and code, their reward shapes do not directly supervise world modeling in axiomatic domains such as physics. We propose Reinforcement Learning with Axiomatic World Modeling (RLAWM), which optimizes the policy by penalizing divergence between its inferred axiomatic world model and the true physics world. RLAWM probes physics reasoning at two hierarchical levels (modeling and solving), each scored by a physically grounded reward. The modeling reward targets abstract knowledge formulation, evaluating whether the model captures the true axiomatic world before attempting calculation. The solving reward targets instantiated knowledge reasoning, checking whether the model's concrete derivation remains consistent with the axiomatic world. This two-level structure separates what world the model believes it is reasoning in from how it reasons within that world. The framework operates exclusively during training, with no extra cost during inference. RLAWM improves performance by up to +12.2 points on PhysReason and +5.2 on PHYSICS over the baseline. It also generalizes zero-shot across difficulty, source, and domain, with a 2.45x gain and +10 points over the strongest baseline on olympiad-level physics and math.

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

Reverse derivative categories (RDCs) have recently been shown to be a suitable semantic framework for studying machine learning algorithms. Whereas emphasis has been put on training methodologies, less attention has been devoted to particular model classes: the concrete categories whose morphisms represent machine learning models. In this paper we study presentations by generators and equations of classes of RDCs. In particular, we propose polynomial circuits as a suitable machine learning model. We give an axiomatisation for these circuits and prove a functional completeness result. Finally, we discuss the use of polynomial circuits over specific semirings to perform machine learning with discrete values.

Recent proprietary models (e.g., o3) have begun to demonstrate strong multimodal reasoning capabilities. Yet, most existing open-source research concentrates on training text-only reasoning models, with evaluations limited to mainly mathematical and general-domain tasks. Therefore, it remains unclear how to effectively extend reasoning capabilities beyond text input and general domains. This paper explores a fundamental research question: Is reasoning generalizable across modalities and domains? Our findings support an affirmative answer: General-domain text-based post-training can enable such strong generalizable reasoning. Leveraging this finding, we introduce X-Reasoner, a vision-language model post-trained solely on general-domain text for generalizable reasoning, using a two-stage approach: an initial supervised fine-tuning phase with distilled long chain-of-thoughts, followed by reinforcement learning with verifiable rewards. Experiments show that X-Reasoner successfully transfers reasoning capabilities to both multimodal and out-of-domain settings, outperforming existing state-of-the-art models trained with in-domain and multimodal data across various general and medical benchmarks (Figure 1). Additionally, we find that X-Reasoner's performance in specialized domains can be further enhanced through continued training on domain-specific text-only data. Building upon this, we introduce X-Reasoner-Med, a medical-specialized variant that achieves new state of the art on numerous text-only and multimodal medical benchmarks.

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.

We introduce an axiomatic evaluation framework for latent thought representations in LLMs, comprising metrics that are independent of downstream benchmark scores and reveal representational failures that benchmark accuracy masks. Existing evaluations conflate representation quality with model capacity. Therefore, failures cannot be attributed to the representation rather than to the model that processes it. We formalize four functional axioms (Causality, Minimality, Separability, and Stability) and define a quantitative measure for each, computed directly on the representation independently of downstream accuracy. We audit open-weight LLMs across 23 reasoning tasks (e.g., Spatial Reasoning, Factual QA). We find that no candidate satisfies all four axioms simultaneously, that the representations distinguish task type reliably but cannot distinguish between two questions within the same task, and that the representations encode little information beyond what is already present in the input embedding. The failure is consistent across dense, reasoning-distilled, and RL-trained model families, indicating that the gap is structural rather than a property of model size or training procedure.

The ability to summarize and organize knowledge into abstract concepts is key to learning and reasoning. Many industrial applications rely on the consistent and systematic use of concepts, especially when dealing with decision-critical knowledge. However, we demonstrate that, when methodically questioned, large language models (LLMs) often display and demonstrate significant inconsistencies in their knowledge. Computationally, the basic aspects of the conceptualization of a given domain can be represented as Is-A hierarchies in a knowledge graph (KG) or ontology, together with a few properties or axioms that enable straightforward reasoning. We show that even simple ontologies can be used to reveal conceptual inconsistencies across several LLMs. We also propose strategies that domain experts can use to evaluate and improve the coverage of key domain concepts in LLMs of various sizes. In particular, we have been able to significantly enhance the performance of LLMs of various sizes with openly available weights using simple knowledge-graph (KG) based prompting strategies.

Many critical reasoning tasks, including clinical diagnosis, legal judgment, and industrial fault diagnosis, require step-dependent causal chains in which early errors propagate and correct conclusions can mask invalid reasoning. Although large language models perform well on such tasks, privacy, latency, and controllability motivate distillation into locally deployable models. Standard trajectory imitation does not correct process errors on the student's own rollout distribution. We propose CausalOPD, a curriculum online process distillation framework. A knowledge-augmented teacher first provides trajectories grounded in domain-specific causal rules, entity relations, and structural constraints. The student then generates on-policy trajectories, and the teacher identifies the first wrong step, defined as the earliest transition that verifiably violates available constraints. Starting from the verified prefix, short-horizon reinforcement learning repairs this localized failure. A causal-stage curriculum advances from evidence-level to mechanism-level and conclusion-level errors, following their propagation order. Across three domains, CausalOPD improves average path correctness by 23.4 percentage points over sequence-level online process distillation and reduces the right-label-wrong-reasoning rate from 15.7% to 4.4%. The domain-specific 8B students also surpass both evaluated proprietary references in path correctness across all domains.

We introduce PREM (Private Relative Error Multiplicative weight update), a new framework for generating synthetic data that achieves a relative error guarantee for statistical queries under (varepsilon, δ) differential privacy (DP). Namely, for a domain {cal X}, a family {cal F} of queries f : {cal X} to {0, 1}, and ζ> 0, our framework yields a mechanism that on input dataset D cal X^n outputs a synthetic dataset D cal X^n such that all statistical queries in {cal F} on D, namely sum_{x in D} f(x) for f cal F, are within a 1 pm ζ multiplicative factor of the corresponding value on D up to an additive error that is polynomial in log |{cal F}|, log |{cal X}|, log n, log(1/δ), 1/varepsilon, and 1/ζ. In contrast, any (varepsilon, δ)-DP mechanism is known to require worst-case additive error that is polynomial in at least one of n, |{cal F}|, or |{cal X}|. We complement our algorithm with nearly matching lower bounds.

Answering counterfactual queries has many important applications such as knowledge discovery and explainability, but is challenging when causal variables are unobserved and we only see a projection onto an observation space, for instance, image pixels. One approach is to recover the latent Structural Causal Model (SCM), but this typically needs unrealistic assumptions, such as linearity of the causal mechanisms. Another approach is to use na\"ive ML approximations, such as generative models, to generate counterfactual samples; however, these lack guarantees of accuracy. In this work, we strive to strike a balance between practicality and theoretical guarantees by focusing on a specific type of causal query called domain counterfactuals, which hypothesizes what a sample would have looked like if it had been generated in a different domain (or environment). Concretely, by only assuming invertibility, sparse domain interventions and access to observational data from different domains, we aim to improve domain counterfactual estimation both theoretically and practically with less restrictive assumptions. We define domain counterfactually equivalent models and prove necessary and sufficient properties for equivalent models that provide a tight characterization of the domain counterfactual equivalence classes. Building upon this result, we prove that every equivalence class contains a model where all intervened variables are at the end when topologically sorted by the causal DAG. This surprising result suggests that a model design that only allows intervention in the last k latent variables may improve model estimation for counterfactuals. We then test this model design on extensive simulated and image-based experiments which show the sparse canonical model indeed improves counterfactual estimation over baseline non-sparse models.

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.

Recent advances in large language models (LLMs) have demonstrated impressive capabilities in formal theorem proving, particularly on contest-based mathematical benchmarks like the IMO. However, these contests do not reflect the depth, breadth, and abstraction of modern mathematical research. To bridge this gap, we introduce FATE (Formal Algebra Theorem Evaluation), a new benchmark series in formal algebra designed to chart a course toward advanced mathematical reasoning. We present two new components, FATE-H and FATE-X, each with 100 problems in abstract and commutative algebra. The FATE series spans a difficulty spectrum from undergraduate exercises to problems exceeding PhD qualifying exams. Notably, FATE-X is the first formal benchmark to surpass both PhD-level exam difficulty and the coverage of the Mathlib library. Our evaluations of state-of-the-art LLM provers on this new benchmark reveal a stark performance gap compared to contest math: the best model achieves only 3% (pass@64) accuracy on FATE-H and 0% on FATE-X. Our two-stage evaluation reveals that models' natural-language reasoning is notably more accurate than their ability to formalize this reasoning. We systematically classify the common errors that arise during this formalization process. Furthermore, a comparative study shows that a specialized prover can exhibit less effective reflection than general-purpose models, reducing its accuracy at the natural-language stage. We believe FATE provides a robust and challenging benchmark that establishes essential checkpoints on the path toward research-level formal mathematical reasoning.

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

Inductive reasoning is a core component of human intelligence. In the past research of inductive reasoning within computer science, formal language is used as representations of knowledge (facts and rules, more specifically). However, formal language can cause systematic problems for inductive reasoning such as disability of handling raw input such as natural language, sensitiveness to mislabeled data, and incapacity to handle ambiguous input. To this end, we propose a new paradigm (task) for inductive reasoning, which is to induce natural language rules from natural language facts, and create a dataset termed DEER containing 1.2k rule-fact pairs for the task, where rules and facts are written in natural language. New automatic metrics are also proposed and analysed for the evaluation of this task. With DEER, we investigate a modern approach for inductive reasoning where we use natural language as representation for knowledge instead of formal language and use pretrained language models as ''reasoners''. Moreover, we provide the first and comprehensive analysis of how well pretrained language models can induce natural language rules from natural language facts. We also propose a new framework drawing insights from philosophy literature for this task, which we show in the experiment section that surpasses baselines in both automatic and human evaluations. We discuss about our future perspectives for inductive reasoning in Section 7. Dataset and code are available at https://github.com/ZonglinY/Inductive_Reasoning.

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.

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

We introduce k-quantifier logics -- logics with access to k-tuples of elements and very general quantification patterns for transitions between k-tuples. The framework is very expressive and encompasses e.g. the k-variable fragments of first-order logic, modal logic, and monotone neighbourhood semantics. We introduce a corresponding notion of bisimulation and prove variants of the classical Ehrenfeucht-Fraisse and Hennessy-Milner theorem. Finally, we show a Lindstrom-style characterisation for k-quantifier logics that satisfy Los' theorem by proving that they are the unique maximally expressive logics that satisfy Los' theorem and are invariant under the associated bisimulation relations.

Reference-free faithfulness metrics verify each atomic claim a model makes against ground truth, and are increasingly used to evaluate grounded generation. We show they share a blind spot: they measure only precision -- are the stated claims supported? -- and therefore reward abstention, since a model can score near-perfect faithfulness by saying almost nothing. We make this measurable using Formula 1 telemetry, a domain where strategic ground truth is derived deterministically and, crucially, completely: for each decision we know the full set of facts that mattered. This completeness -- absent in open-domain faithfulness benchmarks -- lets us measure recall (coverage of the relevant facts) exactly, alongside precision. On a multilingual (EN/ES/PT) benchmark of 7,253 decision instances spanning 150 races, the most precise frontier model covers under half of the relevant facts and ranks last by F1, so requiring coverage reorders the systems; the same effect reappears in a second complete-oracle domain (NOAA weather forecasts). A prompt ablation shows the low coverage is not an under-prompting artifact: explicitly asking models to be thorough does not close the gap. We pair faithfulness with coverage into a single score, validate the metric (controlled perturbation; agreement across a model-free regex extractor and a cross-family LLM extractor, system-level Spearman 1.0), and give a verifier-guided generation method that improves precision and recall without references. We release the benchmark, structured annotations, metric, baselines, and an interactive demo.

The recent LLMs like GPT-4 and PaLM-2 have made tremendous progress in solving fundamental math problems like GSM8K by achieving over 90\% accuracy. However, their capabilities to solve more challenging math problems which require domain-specific knowledge (i.e. theorem) have yet to be investigated. In this paper, we introduce TheoremQA, the first theorem-driven question-answering dataset designed to evaluate AI models' capabilities to apply theorems to solve challenging science problems. \dataset is curated by domain experts containing 800 high-quality questions covering 350 theoremse.g. Taylor's theorem, Lagrange's theorem, Huffman coding, Quantum Theorem, Elasticity Theorem, etc from Math, Physics, EE\&CS, and Finance. We evaluate a wide spectrum of 16 large language and code models with different prompting strategies like Chain-of-Thoughts and Program-of-Thoughts. We found that GPT-4's capabilities to solve these problems are unparalleled, achieving an accuracy of 51\% with Program-of-Thoughts Prompting. All the existing open-sourced models are below 15\%, barely surpassing the random-guess baseline. Given the diversity and broad coverage of \dataset, we believe it can be used as a better benchmark to evaluate LLMs' capabilities to solve challenging science problems. The data and code are released in https://github.com/wenhuchen/TheoremQA.

Traditional ontologies describe domain structure but cannot generate novel artifacts. Large language models generate fluently but produce outputs lacking structural validity, hallucinating mechanisms without components, goals without end conditions. We introduce Generative Ontology, a framework synthesizing these complementary strengths: ontology provides the grammar; the LLM provides the creativity. Generative Ontology encodes domain knowledge as executable Pydantic schemas constraining LLM generation via DSPy signatures. A multi-agent pipeline assigns specialized roles: a Mechanics Architect designs game systems, a Theme Weaver integrates narrative, a Balance Critic identifies exploits, each carrying a professional "anxiety" that prevents shallow outputs. Retrieval-augmented generation grounds designs in precedents from existing exemplars. We demonstrate the framework through GameGrammar, generating complete tabletop game designs, and present three empirical studies. An ablation study (120 designs, 4 conditions) shows multi-agent specialization produces the largest quality gains (fun d=1.12, depth d=1.59; p<.001), while schema validation eliminates structural errors (d=4.78). A benchmark against 20 published board games reveals structural parity but a bounded creative gap (fun d=1.86): generated designs score 7-8 while published games score 8-9. A test-retest study (50 evaluations) validates the LLM-based evaluator, with 7/9 metrics achieving Good-to-Excellent reliability (ICC 0.836-0.989). The pattern generalizes beyond games. Any domain with expert vocabulary, validity constraints, and accumulated exemplars is a candidate for Generative Ontology.

With recent dramatic increases in AI system capabilities, there has been growing interest in utilizing machine learning for reasoning-heavy, quantitative tasks, particularly mathematics. While there are many resources capturing mathematics at the high-school, undergraduate, and graduate level, there are far fewer resources available that align with the level of difficulty and open endedness encountered by professional mathematicians working on open problems. To address this, we introduce a new collection of datasets, the Algebraic Combinatorics Dataset Repository (ACD Repo), representing either foundational results or open problems in algebraic combinatorics, a subfield of mathematics that studies discrete structures arising from abstract algebra. Further differentiating our dataset collection is the fact that it aims at the conjecturing process. Each dataset includes an open-ended research-level question and a large collection of examples (up to 10M in some cases) from which conjectures should be generated. We describe all nine datasets, the different ways machine learning models can be applied to them (e.g., training with narrow models followed by interpretability analysis or program synthesis with LLMs), and discuss some of the challenges involved in designing datasets like these.

Neural Networks are indispensable to natural sciences and society. Their impact extends from applications in public health to workforce productivity. Here, we introduce Modal Logic Neural Networks (MLNNs) -- an end-to-end differentiable logical neural network realisation of modal logic which evaluates a learnable truth function across possible-world semantics. This neural architecture handles para-consistency and inconsistency via a learnable world accessibility relation and valuation function. Because the modality is fixed by which frame axioms the relation satisfies rather than by the operator, one differentiable engine covers the epistemic, doxastic, deontic and temporal readings, with applications from verification of reactive and distributed systems to legal discourse and microeconomic utility models. In this paper, we introduce a model of differentiable Kripke semantics, and establish their soundness, convergence, and structural guarantees. We show four applications, in which the learned relation reads as a trust matrix, an operating-regime embedding with safety bounds, a temporal precedence order, and a recovered constraint graph.

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

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

We present a set-theoretic, proof-irrelevant model for Calculus of Constructions (CC) with predicative induction and judgmental equality in Zermelo-Fraenkel set theory with an axiom for countably many inaccessible cardinals. We use Aczel's trace encoding which is universally defined for any function type, regardless of being impredicative. Direct and concrete interpretations of simultaneous induction and mutually recursive functions are also provided by extending Dybjer's interpretations on the basis of Aczel's rule sets. Our model can be regarded as a higher-order generalization of the truth-table methods. We provide a relatively simple consistency proof of type theory, which can be used as the basis for a theorem prover.

In the propositional modal (and algebraic) treatment of two-variable first-order logic equality is modelled by a `diagonal' constant, interpreted in square products of universal frames as the identity (also known as the `diagonal') relation. Here we study the decision problem of products of two arbitrary modal logics equipped with such a diagonal. As the presence or absence of equality in two-variable first-order logic does not influence the complexity of its satisfiability problem, one might expect that adding a diagonal to product logics in general is similarly harmless. We show that this is far from being the case, and there can be quite a big jump in complexity, even from decidable to the highly undecidable. Our undecidable logics can also be viewed as new fragments of first- order logic where adding equality changes a decidable fragment to undecidable. We prove our results by a novel application of counter machine problems. While our formalism apparently cannot force reliable counter machine computations directly, the presence of a unique diagonal in the models makes it possible to encode both lossy and insertion-error computations, for the same sequence of instructions. We show that, given such a pair of faulty computations, it is then possible to reconstruct a reliable run from them.

Large language models (LLMs) increasingly underpin scientific AI applications that reason over structured knowledge, from biomedical question answering to materials informatics. However, their logical reasoning often falls short, producing factual inaccuracies unacceptable in these settings. Reliable evaluation remains challenging: manual dataset construction scales poorly, and LLM-based generation risks embedding the very flaws it aims to measure. High-quality benchmarks must ground both correct and incorrect labelled examples in explicit background knowledge, formally verifiable by a standard reasoner. We propose a pipeline that automatically generates ontology-grounded multiple-choice question (MCQ) benchmarks from any sufficiently axiomatised OWL 2 ontology, with correct answers grounded in the ontology by design. Distractors are generated by perturbing the right-hand-side class expressions of class definition axioms, and their incorrectness is formally verified by an OWL reasoner via entailment checks. We evaluate the pipeline on three ontologies: Pizza (small, academic), PMDco (complex, materials science), and DOID (large, biomedical), generating 112, 2,491, and 15,216 MCQs respectively. Distractors span four semantic categories from class unsatisfiability to weakened subsumptions, enabling diagnostic evaluation of specific reasoning failures. Items meet natural language quality standards: mean LLM judge scores of 4.02, 4.36, and 3.36 out of 5 confirm fluency, and correct-answer-to-distractor similarity above 0.8 shows that wrong options cannot be dismissed on surface form alone. Six LLMs evaluated zero-shot achieve 41.1-76.8% accuracy, well above the 25% random-guessing baseline, indicating the benchmarks are challenging and discriminative. This work is a step towards more reliable benchmarks for assessing logical reasoning in scientific AI.

Large language models (LLMs) have recently driven striking performance improvements across a range of natural language processing tasks. The factual knowledge acquired during pretraining and instruction tuning can be useful in various downstream tasks, such as question answering, and language generation. Unlike conventional Knowledge Bases (KBs) that explicitly store factual knowledge, LLMs implicitly store facts in their parameters. Content generated by the LLMs can often exhibit inaccuracies or deviations from the truth, due to facts that can be incorrectly induced or become obsolete over time. To this end, we aim to comprehensively evaluate the extent and scope of factual knowledge within LLMs by designing the benchmark Pinocchio. Pinocchio contains 20K diverse factual questions that span different sources, timelines, domains, regions, and languages. Furthermore, we investigate whether LLMs are able to compose multiple facts, update factual knowledge temporally, reason over multiple pieces of facts, identify subtle factual differences, and resist adversarial examples. Extensive experiments on different sizes and types of LLMs show that existing LLMs still lack factual knowledge and suffer from various spurious correlations. We believe this is a critical bottleneck for realizing trustworthy artificial intelligence. The dataset Pinocchio and our codes will be publicly available.

We present BIS Reasoning 1.0, the first large-scale Japanese dataset of syllogistic reasoning problems explicitly designed to evaluate belief-inconsistent reasoning in large language models (LLMs). Unlike prior datasets such as NeuBAROCO and JFLD, which focus on general or belief-aligned reasoning, BIS Reasoning 1.0 introduces logically valid yet belief-inconsistent syllogisms to uncover reasoning biases in LLMs trained on human-aligned corpora. We benchmark state-of-the-art models - including GPT models, Claude models, and leading Japanese LLMs - revealing significant variance in performance, with GPT-4o achieving 79.54% accuracy. Our analysis identifies critical weaknesses in current LLMs when handling logically valid but belief-conflicting inputs. These findings have important implications for deploying LLMs in high-stakes domains such as law, healthcare, and scientific literature, where truth must override intuitive belief to ensure integrity and safety.

The attribution problem, that is the problem of attributing a model's prediction to its base features, is well-studied. We extend the notion of attribution to also apply to feature interactions. The Shapley value is a commonly used method to attribute a model's prediction to its base features. We propose a generalization of the Shapley value called Shapley-Taylor index that attributes the model's prediction to interactions of subsets of features up to some size k. The method is analogous to how the truncated Taylor Series decomposes the function value at a certain point using its derivatives at a different point. In fact, we show that the Shapley Taylor index is equal to the Taylor Series of the multilinear extension of the set-theoretic behavior of the model. We axiomatize this method using the standard Shapley axioms -- linearity, dummy, symmetry and efficiency -- and an additional axiom that we call the interaction distribution axiom. This new axiom explicitly characterizes how interactions are distributed for a class of functions that model pure interaction. We contrast the Shapley-Taylor index against the previously proposed Shapley Interaction index (cf. [9]) from the cooperative game theory literature. We also apply the Shapley Taylor index to three models and identify interesting qualitative insights.

In order to properly train a machine learning model, data must be properly collected. To guarantee a proper data collection, verifying that the collected data set holds certain properties is a possible solution. For example, guaranteeing that the data set contains samples across the whole input space, or that the data set is balanced w.r.t. different classes. We present a formal approach for verifying a set of arbitrarily stated properties over a data set. The proposed approach relies on the transformation of the data set into a first order logic formula, which can be later verified w.r.t. the different properties also stated in the same logic. A prototype tool, which uses the z3 solver, has been developed; the prototype can take as an input a set of properties stated in a formal language and formally verify a given data set w.r.t. to the given set of properties. Preliminary experimental results show the feasibility and performance of the proposed approach, and furthermore the flexibility for expressing properties of interest.

Actual causality (AC), a fundamental aspect of causal reasoning (CR), is responsible for attribution and responsibility assignment in real-world scenarios. However, existing LLM-based methods lack grounding in formal AC theory, resulting in limited interpretability. Therefore, we propose AC-Reason, a semi-formal reasoning framework that identifies causally relevant events within an AC scenario, infers the values of their formal causal factors (e.g., sufficiency, necessity, and normality), and answers AC queries via a theory-guided algorithm with explanations. While AC-Reason does not explicitly construct a causal graph, it operates over variables in the underlying causal structure to support principled reasoning. To enable comprehensive evaluation, we introduce AC-Bench, a new benchmark built upon and substantially extending Big-Bench Hard Causal Judgment (BBH-CJ). AC-Bench comprises ~1K carefully annotated samples, each with detailed reasoning steps and focuses solely on actual causation. The case study shows that synthesized samples in AC-Bench present greater challenges for LLMs. Extensive experiments on BBH-CJ and AC-Bench show that AC-Reason consistently improves LLM performance over baselines. On BBH-CJ, all tested LLMs surpass the average human rater accuracy of 69.60%, with GPT-4 + AC-Reason achieving 75.04%. On AC-Bench, GPT-4 + AC-Reason again achieves the highest accuracy of 71.82%. AC-Bench further enables fine-grained analysis of reasoning faithfulness, revealing that only Qwen-2.5-72B-Instruct, Claude-3.5-Sonnet, and GPT-4o exhibit faithful reasoning, whereas GPT-4 tends to exploit shortcuts. Finally, our ablation study proves that integrating AC theory into LLMs is highly effective, with the proposed algorithm contributing the most significant performance gains.

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a powerful paradigm for enhancing the reasoning capabilities of LLMs. Existing research has predominantly concentrated on isolated reasoning domains such as mathematical problem-solving, coding tasks, or logical reasoning. However, real world reasoning scenarios inherently demand an integrated application of multiple cognitive skills. Despite this, the interplay among these reasoning skills under reinforcement learning remains poorly understood. To bridge this gap, we present a systematic investigation of multi-domain reasoning within the RLVR framework, explicitly focusing on three primary domains: mathematical reasoning, code generation, and logical puzzle solving. We conduct a comprehensive study comprising four key components: (1) Leveraging the GRPO algorithm and the Qwen-2.5-7B model family, our study thoroughly evaluates the models' in-domain improvements and cross-domain generalization capabilities when trained on single-domain datasets. (2) Additionally, we examine the intricate interactions including mutual enhancements and conflicts that emerge during combined cross-domain training. (3) To further understand the influence of SFT on RL, we also analyze and compare performance differences between base and instruct models under identical RL configurations. (4) Furthermore, we delve into critical RL training details, systematically exploring the impacts of curriculum learning strategies, variations in reward design, and language-specific factors. Through extensive experiments, our results offer significant insights into the dynamics governing domain interactions, revealing key factors influencing both specialized and generalizable reasoning performance. These findings provide valuable guidance for optimizing RL methodologies to foster comprehensive, multi-domain reasoning capabilities in LLMs.

In medicine, claims remain valid when supported by empirical evidence grounded in stable biological reality. In law, by contrast, truth is contingent, defined by jurisdiction, temporal validity, and the hierarchy of authoritative sources. The recent success of large language models (LLMs) on medical licensing examinations has encouraged an expectation of comparable legal competence. This analogy, however, obscures a critical distinction between domains. Unlike in medicine, legal performance often depends less on inference than on determining when external authority is applicable, valid, and non-contradictory. We introduce a comparative diagnostic framework evaluating legal reasoning against medical baselines along four axes (knowledge recall, grounding, confidence, and robustness), uncovering a sharp domain asymmetry when applied to a new benchmark that encodes temporal validity and normative relationships. While medical LLMs reliably benefit from verified sources, legal LLMs struggle to assess when retrieved citations are useful or misleading, exhibiting overconfidence in perturbed contexts and sensitivity to superficial formatting cues. Increased model scale amplifies this tendency, revealing that stronger instruction following can coincide with weaker resistance to authoritative perturbations. These findings show that LLMs treat law as unstructured text rather than binding precedent, while revealing a tendency to over-trust authoritative but false information when external references conflict with a model's internal knowledge.

In this paper we introduce the olog, or ontology log, a category-theoretic model for knowledge representation (KR). Grounded in formal mathematics, ologs can be rigorously formulated and cross-compared in ways that other KR models (such as semantic networks) cannot. An olog is similar to a relational database schema; in fact an olog can serve as a data repository if desired. Unlike database schemas, which are generally difficult to create or modify, ologs are designed to be user-friendly enough that authoring or reconfiguring an olog is a matter of course rather than a difficult chore. It is hoped that learning to author ologs is much simpler than learning a database definition language, despite their similarity. We describe ologs carefully and illustrate with many examples. As an application we show that any primitive recursive function can be described by an olog. We also show that ologs can be aligned or connected together into a larger network using functors. The various methods of information flow and institutions can then be used to integrate local and global world-views. We finish by providing several different avenues for future research.

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

The remarkable abilities of large language models (LLMs) like GPT-4 partially stem from post-training processes like Reinforcement Learning from Human Feedback (RLHF) involving human preferences encoded in a reward model. However, these reward models (RMs) often lack direct knowledge of why, or under what principles, the preferences annotations were made. In this study, we identify principles that guide RMs to better align with human preferences, and then develop an axiomatic framework to generate a rich variety of preference signals to uphold them. We use these axiomatic signals to train a model for scoring answers to longform questions. Our approach yields a Preference Model with only about 220M parameters that agrees with gold human-annotated preference labels more often than GPT-4. The contributions of this work include: training a standalone preference model that can score human- and LLM-generated answers on the same scale; developing an axiomatic framework for generating training data pairs tailored to certain principles; and showing that a small amount of axiomatic signals can help small models outperform GPT-4 in preference scoring. We release our model on huggingface: https://huggingface.co/corbyrosset/axiomatic_preference_model

Large Language Models (LLMs) are widely applied to downstream domains. However, current LLMs for high-stakes domain tasks, such as financial investment and legal QA, typically generate brief answers without reasoning processes and explanations. This limits users' confidence in making decisions based on their responses. While original CoT shows promise, it lacks self-correction mechanisms during reasoning. This work introduces Domaino1s, which enhances LLMs' reasoning capabilities on domain tasks through supervised fine-tuning and tree search. We construct CoT-stock-2k and CoT-legal-2k datasets for fine-tuning models that activate domain-specific reasoning steps based on their judgment. Additionally, we propose Selective Tree Exploration to spontaneously explore solution spaces and sample optimal reasoning paths to improve performance. We also introduce PROOF-Score, a new metric for evaluating domain models' explainability, complementing traditional accuracy metrics with richer assessment dimensions. Extensive experiments on stock investment recommendation and legal reasoning QA tasks demonstrate Domaino1s's leading performance and explainability. Our code is available at https://anonymous.4open.science/r/Domaino1s-006F/.

When an agent can articulate why something works, we typically take this as evidence of genuine understanding. This presupposes that effective action and correct explanation covary, and that coherent explanation reliably signals both. I argue that this assumption fails for contemporary Large Language Models (LLMs). I introduce what I call the Bidirectional Coherence Paradox: competence and grounding not only dissociate but invert across epistemic conditions. In low-observability domains, LLMs often act successfully while misidentifying the mechanisms that produce their success. In high-observability domains, they frequently generate explanations that accurately track observable causal structure yet fail to translate those diagnoses into effective intervention. In both cases, explanatory coherence remains intact, obscuring the underlying dissociation. Drawing on experiments in compiler optimization and hyperparameter tuning, I develop the Epistemic Triangle, a model of how priors, signals, and domain knowledge interact under varying observability. The results suggest that neither behavioral success nor explanatory accuracy alone suffices for attributing understanding. I argue that evaluating artificial epistemic agents requires a tripartite framework -- coherence, grounding, and a proper basing relation linking explanation to action. The systematic separation of knowing-that and knowing-how in LLMs thus challenges assumptions inherited from both epistemology and current AI evaluation practice.

The growing popularity of neuro symbolic reasoning has led to the adoption of various forms of differentiable (i.e., fuzzy) first order logic. We introduce PyReason, a software framework based on generalized annotated logic that both captures the current cohort of differentiable logics and temporal extensions to support inference over finite periods of time with capabilities for open world reasoning. Further, PyReason is implemented to directly support reasoning over graphical structures (e.g., knowledge graphs, social networks, biological networks, etc.), produces fully explainable traces of inference, and includes various practical features such as type checking and a memory-efficient implementation. This paper reviews various extensions of generalized annotated logic integrated into our implementation, our modern, efficient Python-based implementation that conducts exact yet scalable deductive inference, and a suite of experiments. PyReason is available at: github.com/lab-v2/pyreason.

Automated Planning algorithms require a model of the domain that specifies the preconditions and effects of each action. Obtaining such a domain model is notoriously hard. Algorithms for learning domain models exist, yet it remains unclear whether learning a domain model and planning is an effective approach for numeric planning environments, i.e., where states include discrete and numeric state variables. In this work, we explore the benefits of learning a numeric domain model and compare it with alternative model-free solutions. As a case study, we use two tasks in Minecraft, a popular sandbox game that has been used as an AI challenge. First, we consider an offline learning setting, where a set of expert trajectories are available to learn from. This is the standard setting for learning domain models. We used the Numeric Safe Action Model Learning (NSAM) algorithm to learn a numeric domain model and solve new problems with the learned domain model and a numeric planner. We call this model-based solution NSAM_(+p), and compare it to several model-free Imitation Learning (IL) and Offline Reinforcement Learning (RL) algorithms. Empirical results show that some IL algorithms can learn faster to solve simple tasks, while NSAM_(+p) allows solving tasks that require long-term planning and enables generalizing to solve problems in larger environments. Then, we consider an online learning setting, where learning is done by moving an agent in the environment. For this setting, we introduce RAMP. In RAMP, observations collected during the agent's execution are used to simultaneously train an RL policy and learn a planning domain action model. This forms a positive feedback loop between the RL policy and the learned domain model. We demonstrate experimentally the benefits of using RAMP, showing that it finds more efficient plans and solves more problems than several RL baselines.

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

Studies have underscored how, regardless of the recent breakthrough and swift advances in AI research, even state-of-the-art Large Language models (LLMs) continue to struggle when performing logical and mathematical reasoning. The results seem to suggest that LLMs still work as (highly advanced) data pattern identifiers, scoring poorly when attempting to generalise and solve reasoning problems the models have never previously seen or that are not close to samples presented in their training data. To address this compelling concern, this paper makes use of the notion of critical questions from the literature on argumentation theory, focusing in particular on Toulmin's model of argumentation. We show that employing these critical questions can improve the reasoning capabilities of LLMs. By probing the rationale behind the models' reasoning process, the LLM can assess whether some logical mistake is occurring and correct it before providing the final reply to the user prompt. The underlying idea is drawn from the gold standard of any valid argumentative procedure: the conclusion is valid if it is entailed by accepted premises. Or, to paraphrase such Aristotelian principle in a real-world approximation, characterised by incomplete information and presumptive logic, the conclusion is valid if not proved otherwise. This approach successfully steers the models' output through a reasoning pipeline, resulting in better performance against the baseline and its Chain-of-Thought (CoT) implementation. To this end, an extensive evaluation of the proposed approach on the MT-Bench Reasoning and Math tasks across a range of LLMs is provided.

The aim of Reverse Mathematics(RM for short)is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak system of computable mathematics. The Big Five phenomenon of RM is the observation that a large number of theorems from ordinary mathematics are either provable in the base theory or equivalent to one of only four systems; these five systems together are called the 'Big Five'. The aim of this paper is to greatly extend the Big Five phenomenon as follows: there are two supposedly fundamentally different approaches to RM where the main difference is whether the language is restricted to second-order objects or if one allows third-order objects. In this paper, we unite these two strands of RM by establishing numerous equivalences involving the second-order Big Five systems on one hand, and well-known third-order theorems from analysis about (possibly) discontinuous functions on the other hand. We both study relatively tame notions, like cadlag or Baire 1, and potentially wild ones, like quasi-continuity. We also show that slight generalisations and variations of the aforementioned third-order theorems fall far outside of the Big Five.

Most scientific materials compress reasoning, presenting conclusions while omitting the derivational chains that justify them. This compression hinders verification by lacking explicit, step-wise justifications and inhibits cross-domain links by collapsing the very pathways that establish the logical and causal connections between concepts. We introduce a scalable framework that decompresses scientific reasoning, constructing a verifiable Long Chain-of-Thought (LCoT) knowledge base and projecting it into an emergent encyclopedia, SciencePedia. Our pipeline operationalizes an endpoint-driven, reductionist strategy: a Socratic agent, guided by a curriculum of around 200 courses, generates approximately 3 million first-principles questions. To ensure high fidelity, multiple independent solver models generate LCoTs, which are then rigorously filtered by prompt sanitization and cross-model answer consensus, retaining only those with verifiable endpoints. This verified corpus powers the Brainstorm Search Engine, which performs inverse knowledge search -- retrieving diverse, first-principles derivations that culminate in a target concept. This engine, in turn, feeds the Plato synthesizer, which narrates these verified chains into coherent articles. The initial SciencePedia comprises approximately 200,000 fine-grained entries spanning mathematics, physics, chemistry, biology, engineering, and computation. In evaluations across six disciplines, Plato-synthesized articles (conditioned on retrieved LCoTs) exhibit substantially higher knowledge-point density and significantly lower factual error rates than an equally-prompted baseline without retrieval (as judged by an external LLM). Built on this verifiable LCoT knowledge base, this reasoning-centric approach enables trustworthy, cross-domain scientific synthesis at scale and establishes the foundation for an ever-expanding encyclopedia.

Open Information Extraction (OIE) methods extract facts from natural language text in the form of ("subject"; "relation"; "object") triples. These facts are, however, merely surface forms, the ambiguity of which impedes their downstream usage; e.g., the surface phrase "Michael Jordan" may refer to either the former basketball player or the university professor. Knowledge Graphs (KGs), on the other hand, contain facts in a canonical (i.e., unambiguous) form, but their coverage is limited by a static schema (i.e., a fixed set of entities and predicates). To bridge this gap, we need the best of both worlds: (i) high coverage of free-text OIEs, and (ii) semantic precision (i.e., monosemy) of KGs. In order to achieve this goal, we propose a new benchmark with novel evaluation protocols that can, for example, measure fact linking performance on a granular triple slot level, while also measuring if a system has the ability to recognize that a surface form has no match in the existing KG. Our extensive evaluation of several baselines show that detection of out-of-KG entities and predicates is more difficult than accurate linking to existing ones, thus calling for more research efforts on this difficult task. We publicly release all resources (data, benchmark and code) on https://github.com/nec-research/fact-linking.

Real-world knowledge graphs (KGs) contain not only standard triple-based facts, but also more complex, heterogeneous types of facts, such as hyper-relational facts with auxiliary key-value pairs, temporal facts with additional timestamps, and nested facts that imply relationships between facts. These richer forms of representation have attracted significant attention due to their enhanced expressiveness and capacity to model complex semantics in real-world scenarios. However, most existing studies suffer from two main limitations: (1) they typically focus on modeling only specific types of facts, thus making it difficult to generalize to real-world scenarios with multiple fact types; and (2) they struggle to achieve generalizable hierarchical (inter-fact and intra-fact) modeling due to the complexity of these representations. To overcome these limitations, we propose UniHR, a Unified Hierarchical Representation learning framework, which consists of a learning-optimized Hierarchical Data Representation (HiDR) module and a unified Hierarchical Structure Learning (HiSL) module. The HiDR module unifies hyper-relational KGs, temporal KGs, and nested factual KGs into triple-based representations. Then HiSL incorporates intra-fact and inter-fact message passing, focusing on enhancing both semantic information within individual facts and enriching the structural information between facts. To go beyond the unified method itself, we further explore the potential of unified representation in complex real-world scenarios. Extensive experiments on 9 datasets across 5 types of KGs demonstrate the effectiveness of UniHR and highlight the strong potential of unified representations. Code and data are available at https://github.com/zjukg/UniHR.

OWL (Web Ontology Language) ontologies which are able to formally represent complex knowledge and support semantic reasoning have been widely adopted across various domains such as healthcare and bioinformatics. Recently, ontology embeddings have gained wide attention due to its potential to infer plausible new knowledge and approximate complex reasoning. However, existing methods face notable limitations: geometric model-based embeddings typically overlook valuable textual information, resulting in suboptimal performance, while the approaches that incorporate text, which are often based on language models, fail to preserve the logical structure. In this work, we propose a new ontology embedding method OnT, which tunes a Pretrained Language Model (PLM) via geometric modeling in a hyperbolic space for effectively incorporating textual labels and simultaneously preserving class hierarchies and other logical relationships of Description Logic EL. Extensive experiments on four real-world ontologies show that OnT consistently outperforms the baselines including the state-of-the-art across both tasks of prediction and inference of axioms. OnT also demonstrates strong potential in real-world applications, indicated by its robust transfer learning abilities and effectiveness in real cases of constructing a new ontology from SNOMED CT. Data and code are available at https://github.com/HuiYang1997/OnT.

In real world applications, knowledge graphs (KG) are widely used in various domains (e.g. medical applications and dialogue agents). However, for fact verification, KGs have not been adequately utilized as a knowledge source. KGs can be a valuable knowledge source in fact verification due to their reliability and broad applicability. A KG consists of nodes and edges which makes it clear how concepts are linked together, allowing machines to reason over chains of topics. However, there are many challenges in understanding how these machine-readable concepts map to information in text. To enable the community to better use KGs, we introduce a new dataset, FactKG: Fact Verification via Reasoning on Knowledge Graphs. It consists of 108k natural language claims with five types of reasoning: One-hop, Conjunction, Existence, Multi-hop, and Negation. Furthermore, FactKG contains various linguistic patterns, including colloquial style claims as well as written style claims to increase practicality. Lastly, we develop a baseline approach and analyze FactKG over these reasoning types. We believe FactKG can advance both reliability and practicality in KG-based fact verification.

We consider three monads on Top, the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them, and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads from V to H. In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. Third, we show that V can be restricted to a submonad of tau-smooth probability measures on Top. By composing these two morphisms of monads, we obtain that taking the support of a tau-smooth probability measure is also a morphism of monads.

Explaining an inferred fact under rule evaluation can require identifying the inclusion-minimal input sets that suffice for the inference and the deletions that make the fact disappear. For a fixed union of conjunctive queries, every minimal support is bounded by the query body. For recursive rules, the same answer may depend on large supports, and the number of minimal supports may be exponential in the input. We study the gap through deletion-based explanation, using inclusion-minimal endogenous input facts that entail the atom together with fixed background facts. We organize these supports as a hypergraph and prove that it determines actual causes, counterfactual causes, responsibility, and deletion robustness. The resulting view separates nonrecursive queries from recursive Datalog at the level of minimal input explanations. For positive-length reachability, minimal supports are exactly simple directed paths, and deletion robustness is the minimum directed edge cut. We also prove invariance under fixed-goal equivalent positive Datalog programs and an NP-hardness calibration for the robustness threshold problem.

This position paper argues that AI-assisted software engineering requires explicit mechanisms for tracking the epistemic status and temporal validity of architectural decisions. LLM coding assistants generate decisions faster than teams can validate them, yet no widely-adopted framework distinguishes conjecture from verified knowledge, prevents trust inflation through conservative aggregation, or detects when evidence expires. We propose three requirements for responsible AI-assisted engineering: (1) epistemic layers that separate unverified hypotheses from empirically validated claims, (2) conservative assurance aggregation grounded in the Gödel t-norm that prevents weak evidence from inflating confidence, and (3) automated evidence decay tracking that surfaces stale assumptions before they cause failures. We formalize these requirements as the First Principles Framework (FPF), ground its aggregation semantics in fuzzy logic, and define a quintet of invariants that any valid aggregation operator must satisfy. Our retrospective audit applying FPF criteria to two internal projects found that 20-25% of architectural decisions had stale evidence within two months, validating the need for temporal accountability. We outline research directions including learnable aggregation operators, federated evidence sharing, and SMT-based claim validation.

AI for Mathematics (AI4Math) has emerged as a distinct field that leverages machine learning to navigate mathematical landscapes historically intractable for early symbolic systems. While mid-20th-century symbolic approaches successfully automated formal logic, they faced severe scalability limitations due to the combinatorial explosion of the search space. The recent integration of data-driven approaches has revitalized this pursuit. In this review, we provide a systematic overview of AI4Math, highlighting its primary focus on developing AI models to support mathematical research. Crucially, we emphasize that this is not merely the application of AI to mathematical activities; it also encompasses the development of stronger AI systems where the rigorous nature of mathematics serves as a premier testbed for advancing general reasoning capabilities. We categorize existing research into two complementary directions: problem-specific modeling, involving the design of specialized architectures for distinct mathematical tasks, and general-purpose modeling, focusing on foundation models capable of broader reasoning, retrieval, and exploratory workflows. We conclude by discussing key challenges and prospects, advocating for AI systems that go beyond facilitating formal correctness to enabling the discovery of meaningful results and unified theories, recognizing that the true value of a proof lies in the insights and tools it offers to the broader mathematical landscape.

We propose a many-sorted modal logic for reasoning about knowledge in multi-agent systems. Our logic introduces a clear distinction between participating agents and the environment. This allows to express local properties of agents and global properties of worlds in a uniform way, as well as to talk about the presence or absence of agents in a world. The logic subsumes the standard epistemic logic and is a conservative extension of it. The semantics is given in chromatic hypergraphs, a generalization of chromatic simplicial complexes, which were recently used to model knowledge in distributed systems. We show that the logic is sound and complete with respect to the intended semantics. We also show a further connection of chromatic hypergraphs with neighborhood frames.

Single domain generalization aims to learn a model from a single training domain (source domain) and apply it to multiple unseen test domains (target domains). Existing methods focus on expanding the distribution of the training domain to cover the target domains, but without estimating the domain shift between the source and target domains. In this paper, we propose a new learning paradigm, namely simulate-analyze-reduce, which first simulates the domain shift by building an auxiliary domain as the target domain, then learns to analyze the causes of domain shift, and finally learns to reduce the domain shift for model adaptation. Under this paradigm, we propose a meta-causal learning method to learn meta-knowledge, that is, how to infer the causes of domain shift between the auxiliary and source domains during training. We use the meta-knowledge to analyze the shift between the target and source domains during testing. Specifically, we perform multiple transformations on source data to generate the auxiliary domain, perform counterfactual inference to learn to discover the causal factors of the shift between the auxiliary and source domains, and incorporate the inferred causality into factor-aware domain alignments. Extensive experiments on several benchmarks of image classification show the effectiveness of our method.

Domain ontologies are important information sources for knowledge-based systems. Yet, building domain ontologies from scratch is known to be a very labor-intensive process. In this study, we present our semi-automatic approach to building an ontology for the domain of wind energy which is an important type of renewable energy with a growing share in electricity generation all over the world. Related Wikipedia articles are first processed in an automated manner to determine the basic concepts of the domain together with their properties and next the concepts, properties, and relationships are organized to arrive at the ultimate ontology. We also provide pointers to other engineering ontologies which could be utilized together with the proposed wind energy ontology in addition to its prospective application areas. The current study is significant as, to the best of our knowledge, it proposes the first considerably wide-coverage ontology for the wind energy domain and the ontology is built through a semi-automatic process which makes use of the related Web resources, thereby reducing the overall cost of the ontology building process.

Bold claims about AI's role in science-from "AGI will cure all diseases" to promises of radically accelerated discovery-raise a central epistemic question: do large language models (LLMs) truly generate new knowledge, or do they merely remix memorized fragments? We propose unlearning-as-ablation as a falsifiable probe of constructive scientific discovery. The idea is to systematically remove a target result together with its forget-closure (supporting lemmas, paraphrases, and multi-hop entailments) and then evaluate whether the model can re-derive the result from only permitted axioms and tools. Success would indicate generative capability beyond recall; failure would expose current limits. Unlike prevailing motivations for unlearning-privacy, copyright, or safety-our framing repositions it as an epistemic probe for AI-for-Science. We outline a minimal pilot in mathematics and algorithms to illustrate feasibility, and sketch how the same approach could later be extended to domains such as physics or chemistry. This is a position paper: our contribution is conceptual and methodological, not empirical. We aim to stimulate discussion on how principled ablation tests could help distinguish models that reconstruct knowledge from those that merely retrieve it, and how such probes might guide the next generation of AI-for-Science benchmarks.

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.

Formal theorem proving, a field at the intersection of mathematics and computer science, has seen renewed interest with advancements in large language models (LLMs). This paper introduces SubgoalXL, a novel approach that synergizes subgoal-based proofs with expert learning to enhance LLMs' capabilities in formal theorem proving within the Isabelle environment. SubgoalXL addresses two critical challenges: the scarcity of specialized mathematics and theorem-proving data, and the need for improved multi-step reasoning abilities in LLMs. By optimizing data efficiency and employing subgoal-level supervision, SubgoalXL extracts richer information from limited human-generated proofs. The framework integrates subgoal-oriented proof strategies with an expert learning system, iteratively refining formal statement, proof, and subgoal generators. Leveraging the Isabelle environment's advantages in subgoal-based proofs, SubgoalXL achieves a new state-of-the-art performance of 56.1\% in Isabelle on the standard miniF2F dataset, marking an absolute improvement of 4.9\%. Notably, SubgoalXL successfully solves 41 AMC12, 9 AIME, and 3 IMO problems from miniF2F. These results underscore the effectiveness of maximizing limited data utility and employing targeted guidance for complex reasoning in formal theorem proving, contributing to the ongoing advancement of AI reasoning capabilities. The implementation is available at https://github.com/zhaoxlpku/SubgoalXL.

Happiness computing based on large-scale online web data and machine learning methods is an emerging research topic that underpins a range of issues, from personal growth to social stability. Many advanced Machine Learning (ML) models with explanations are used to compute the happiness online assessment while maintaining high accuracy of results. However, domain knowledge constraints, such as the primary and secondary relations of happiness factors, are absent from these models, which limits the association between computing results and the right reasons for why they occurred. This article attempts to provide new insights into the explanation consistency from an empirical study perspective. Then we study how to represent and introduce domain knowledge constraints to make ML models more trustworthy. We achieve this through: (1) proving that multiple prediction models with additive factor attributions will have the desirable property of primary and secondary relations consistency, and (2) showing that factor relations with quantity can be represented as an importance distribution for encoding domain knowledge. Factor explanation difference is penalized by the Kullback-Leibler divergence-based loss among computing models. Experimental results using two online web datasets show that domain knowledge of stable factor relations exists. Using this knowledge not only improves happiness computing accuracy but also reveals more significative happiness factors for assisting decisions well.

To achieve faithful reasoning that aligns with human expectations, large language models (LLMs) need to ground their reasoning to real-world knowledge (e.g., web facts, math and physical rules). Tools help LLMs access this external knowledge, but there remains challenges for fine-tuning LLM agents (e.g., Toolformer) to invoke tools in multi-step reasoning problems, where inter-connected tool calls require holistic and efficient tool usage planning. In this work, we propose a new method for LLMs to better leverage tools in multi-step reasoning. Our method, Chain-of-Abstraction (CoA), trains LLMs to first decode reasoning chains with abstract placeholders, and then call domain tools to reify each reasoning chain by filling in specific knowledge. This planning with abstract chains enables LLMs to learn more general reasoning strategies, which are robust to shifts of domain knowledge (e.g., math results) relevant to different reasoning questions. It also allows LLMs to perform decoding and calling of external tools in parallel, which avoids the inference delay caused by waiting for tool responses. In mathematical reasoning and Wiki QA domains, we show that our method consistently outperforms previous chain-of-thought and tool-augmented baselines on both in-distribution and out-of-distribution test sets, with an average ~6% absolute QA accuracy improvement. LLM agents trained with our method also show more efficient tool use, with inference speed being on average ~1.4x faster than baseline tool-augmented LLMs.

In this paper, a short survey about the concepts underlying general logics is given. In particular, a novel rigorous definition of a fuzzy negation as an operation acting on a lattice to render it into a fuzzy logic is presented. According to this definition, a fuzzy negation satisfies the weak double negation condition, requiring double negation to be expansive, the antitony condition, being equivalent to the disjunctive De Morgan law and thus warranting compatibility of negation with the lattice operations, and the Boolean boundary condition stating that the universal bounds of the lattice are the negation of each other. From this perspective, the most general logics are fuzzy logics, containing as special cases paraconsistent (quantum) logics, quantum logics, intuitionistic logics, and Boolean logics, each of which given by its own algebraic restrictions. New examples of a non-contradictory logic violating the conjunctive De Morgan law, and of a typical non-orthomodular fuzzy logic along with its explicit lattice representation are given.

Mathematical proofs are both paradigms of certainty and some of the most explicitly-justified arguments that we have in the cultural record. Their very explicitness, however, leads to a paradox, because the probability of error grows exponentially as the argument expands. When a mathematician encounters a proof, how does she come to believe it? Here we show that, under a cognitively-plausible belief formation mechanism combining deductive and abductive reasoning, belief in mathematical arguments can undergo what we call an epistemic phase transition: a dramatic and rapidly-propagating jump from uncertainty to near-complete confidence at reasonable levels of claim-to-claim error rates. To show this, we analyze an unusual dataset of forty-eight machine-aided proofs from the formalized reasoning system Coq, including major theorems ranging from ancient to 21st Century mathematics, along with five hand-constructed cases including Euclid, Apollonius, Hernstein's Topics in Algebra, and Andrew Wiles's proof of Fermat's Last Theorem. Our results bear both on recent work in the history and philosophy of mathematics on how we understand proofs, and on a question, basic to cognitive science, of how we justify complex beliefs.

In this paper, we present a new theoretical approach for enabling domain knowledge acquisition by intelligent systems. We introduce a hybrid model that starts with minimal input knowledge in the form of an upper ontology of concepts, stores and reasons over this knowledge through a knowledge graph database and learns new information through a Logic Neural Network. We study the behavior of this architecture when handling new data and show that the final system is capable of enriching its current knowledge as well as extending it to new domains.

Exam-style accuracy does not establish whether large language models (LLMs) reason well over clinical records. We define clinical reasoning as integrating and updating evidence across time and sources to form, revise and justify a patient's problem representation and a defensible plan. This structured narrative review maps three literatures: medical education assessment instruments, clinical LLM benchmarks published from 2023 onwards, and general-domain methods for evaluating long-form generation. We examine six dimensions: problem representation, temporal synthesis, differential and management reasoning, counterfactual reasoning, calibrated uncertainty, and reasoning faithfulness. Preprints are included and flagged. No single instrument covers all six dimensions. Problem representation and differential or management reasoning are reasonably covered, although reliability varies by instrument and setting. TIMER-Eval targets temporal synthesis, and ER-Reason assesses sequential diagnostic belief updating. Dedicated uncertainty and counterfactual evaluations are emerging, but their applicability to longitudinal free-text reasoning remains limited. Factual completeness is well theorised in general-domain evaluation, with early clinical evidence of important omissions. Faithfulness remains the weakest dimension, with one identified clinical causal-ablation study on multiple-choice questions. Existing tools should be combined through binary rubric items, separate completeness and correctness scores, case-specific importance weighting with non-compensable safety caps, temporal order-consistency checks, and chance-corrected reliability reporting. Further design work is needed for calibrated uncertainty, counterfactual reasoning and faithfulness over longitudinal free-text records. This review provides a design rationale, not a validated instrument.

We focus on a simple, one-dimensional collective decision problem (often referred to as the facility location problem) and explore issues of strategyproofness and proportionality-based fairness. We introduce and analyze a hierarchy of proportionality-based fairness axioms of varying strength: Individual Fair Share (IFS), Unanimous Fair Share (UFS), Proportionality (as in Freeman et al, 2021), and Proportional Fairness (PF). For each axiom, we characterize the family of mechanisms that satisfy the axiom and strategyproofness. We show that imposing strategyproofness renders many of the axioms to be equivalent: the family of mechanisms that satisfy proportionality, unanimity, and strategyproofness is equivalent to the family of mechanisms that satisfy UFS and strategyproofness, which, in turn, is equivalent to the family of mechanisms that satisfy PF and strategyproofness. Furthermore, there is a unique such mechanism: the Uniform Phantom mechanism, which is studied in Freeman et al. (2021). We also characterize the outcomes of the Uniform Phantom mechanism as the unique (pure) equilibrium outcome for any mechanism that satisfies continuity, strict monotonicity, and UFS. Finally, we analyze the approximation guarantees, in terms of optimal social welfare and minimum total cost, obtained by mechanisms that are strategyproof and satisfy each proportionality-based fairness axiom. We show that the Uniform Phantom mechanism provides the best approximation of the optimal social welfare (and also minimum total cost) among all mechanisms that satisfy UFS.

Leveraging mathematical Large Language Models (LLMs) for proof generation is a fundamental topic in LLMs research. We argue that the ability of current LLMs to prove statements largely depends on whether they have encountered the relevant proof process during training. This reliance limits their deeper understanding of mathematical theorems and related concepts. Inspired by the pedagogical method of "proof by counterexamples" commonly used in human mathematics education, our work aims to enhance LLMs' ability to conduct mathematical reasoning and proof through counterexamples. Specifically, we manually create a high-quality, university-level mathematical benchmark, CounterMATH, which requires LLMs to prove mathematical statements by providing counterexamples, thereby assessing their grasp of mathematical concepts. Additionally, we develop a data engineering framework to automatically obtain training data for further model improvement. Extensive experiments and detailed analyses demonstrate that CounterMATH is challenging, indicating that LLMs, such as OpenAI o1, have insufficient counterexample-driven proof capabilities. Moreover, our exploration into model training reveals that strengthening LLMs' counterexample-driven conceptual reasoning abilities is crucial for improving their overall mathematical capabilities. We believe that our work offers new perspectives on the community of mathematical LLMs.

Integer sequences are of central importance to the modeling of concepts admitting complete finitary descriptions. We introduce a novel view on the learning of such concepts and lay down a set of benchmarking tasks aimed at conceptual understanding by machine learning models. These tasks indirectly assess model ability to abstract, and challenge them to reason both interpolatively and extrapolatively from the knowledge gained by observing representative examples. To further aid research in knowledge representation and reasoning, we present FACT, the Finitary Abstraction Comprehension Toolkit. The toolkit surrounds a large dataset of integer sequences comprising both organic and synthetic entries, a library for data pre-processing and generation, a set of model performance evaluation tools, and a collection of baseline model implementations, enabling the making of the future advancements with ease.

What makes a short story gripping; a news article newsworthy; or a math proof elegant? These constructs resist articulation or verification; their meaning is at least partially tacit. However, modern AI models are improved primarily via articulated constitutions, rubrics and verifiers (i.e. in RLAIF and RLVR); tacit components of preferences are typically understudied. We introduce a large, labeled preference dataset CreativePreferences, containing 2.8M texts labeled by 317M human preference judgments across 7 creative domains, with 42 benchmark tasks. We model these labels with executable programs, rubric banks and densely trained models (V, A and VAT, respectively). We observe robust articulability gaps, VAT-VA; and verifiability gaps, VAT-V; we estimate upper and lower bounds for each gap with a novel measurement approach that discovers articulable and verifiable metrics, identifies spurious variables and estimates the value of undiscovered metrics using capture-recapture. These gaps occur across all domains, even in domains traditionally treated as fully verifiable: correctness-centered domains (i.e. mathematics and software engineering) and claim- and novelty-centric domains (i.e. news, patents, peer review). The size of the gap varies based on domain (e.g. peer review and creative writing have the largest articulability gaps) and widens as more people take part in the judgment, consistent with Collins' collective tacit knowledge. We show two consequences: (1) on human generations, the full model more closely matches human preferences, often in disagreement with articulated criteria, and (2) in an analogy to Goodhart's law, articulating preference shifts it away from the tacit dimension. Articulability and verifiability gaps are consequential; we give recommendations on when tasks can be prompted; how learning mechanisms might improve; and when to leave judgments with humans.

Multi-hop logical reasoning over knowledge graph (KG) plays a fundamental role in many artificial intelligence tasks. Recent complex query embedding (CQE) methods for reasoning focus on static KGs, while temporal knowledge graphs (TKGs) have not been fully explored. Reasoning over TKGs has two challenges: 1. The query should answer entities or timestamps; 2. The operators should consider both set logic on entity set and temporal logic on timestamp set. To bridge this gap, we define the multi-hop logical reasoning problem on TKGs. With generated three datasets, we propose the first temporal CQE named Temporal Feature-Logic Embedding framework (TFLEX) to answer the temporal complex queries. We utilize vector logic to compute the logic part of Temporal Feature-Logic embeddings, thus naturally modeling all First-Order Logic (FOL) operations on entity set. In addition, our framework extends vector logic on timestamp set to cope with three extra temporal operators (After, Before and Between). Experiments on numerous query patterns demonstrate the effectiveness of our method.

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.

In the context of reinforcement learning from human feedback (RLHF), the reward function is generally derived from maximum likelihood estimation of a random utility model based on pairwise comparisons made by humans. The problem of learning a reward function is one of preference aggregation that, we argue, largely falls within the scope of social choice theory. From this perspective, we can evaluate different aggregation methods via established axioms, examining whether these methods meet or fail well-known standards. We demonstrate that both the Bradley-Terry-Luce Model and its broad generalizations fail to meet basic axioms. In response, we develop novel rules for learning reward functions with strong axiomatic guarantees. A key innovation from the standpoint of social choice is that our problem has a linear structure, which greatly restricts the space of feasible rules and leads to a new paradigm that we call linear social choice.

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

Knights and knaves problems represent a classic genre of logical puzzles where characters either tell the truth or lie. The objective is to logically deduce each character's identity based on their statements. The challenge arises from the truth-telling or lying behavior, which influences the logical implications of each statement. Solving these puzzles requires not only direct deductions from individual statements, but the ability to assess the truthfulness of statements by reasoning through various hypothetical scenarios. As such, knights and knaves puzzles serve as compelling examples of suppositional reasoning. In this paper, we introduce TruthQuest, a benchmark for suppositional reasoning based on the principles of knights and knaves puzzles. Our benchmark presents problems of varying complexity, considering both the number of characters and the types of logical statements involved. Evaluations on TruthQuest show that large language models like Llama 3 and Mixtral-8x7B exhibit significant difficulties solving these tasks. A detailed error analysis of the models' output reveals that lower-performing models exhibit a diverse range of reasoning errors, frequently failing to grasp the concept of truth and lies. In comparison, more proficient models primarily struggle with accurately inferring the logical implications of potentially false statements.

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

Recognizing the prevalence of domain shift as a common challenge in machine learning, various domain generalization (DG) techniques have been developed to enhance the performance of machine learning systems when dealing with out-of-distribution (OOD) data. Furthermore, in real-world scenarios, data distributions can gradually change across a sequence of sequential domains. While current methodologies primarily focus on improving model effectiveness within these new domains, they often overlook fairness issues throughout the learning process. In response, we introduce an innovative framework called Counterfactual Fairness-Aware Domain Generalization with Sequential Autoencoder (CDSAE). This approach effectively separates environmental information and sensitive attributes from the embedded representation of classification features. This concurrent separation not only greatly improves model generalization across diverse and unfamiliar domains but also effectively addresses challenges related to unfair classification. Our strategy is rooted in the principles of causal inference to tackle these dual issues. To examine the intricate relationship between semantic information, sensitive attributes, and environmental cues, we systematically categorize exogenous uncertainty factors into four latent variables: 1) semantic information influenced by sensitive attributes, 2) semantic information unaffected by sensitive attributes, 3) environmental cues influenced by sensitive attributes, and 4) environmental cues unaffected by sensitive attributes. By incorporating fairness regularization, we exclusively employ semantic information for classification purposes. Empirical validation on synthetic and real-world datasets substantiates the effectiveness of our approach, demonstrating improved accuracy levels while ensuring the preservation of fairness in the evolving landscape of continuous domains.

Understanding domain-specific theorems often requires more than just text-based reasoning; effective communication through structured visual explanations is crucial for deeper comprehension. While large language models (LLMs) demonstrate strong performance in text-based theorem reasoning, their ability to generate coherent and pedagogically meaningful visual explanations remains an open challenge. In this work, we introduce TheoremExplainAgent, an agentic approach for generating long-form theorem explanation videos (over 5 minutes) using Manim animations. To systematically evaluate multimodal theorem explanations, we propose TheoremExplainBench, a benchmark covering 240 theorems across multiple STEM disciplines, along with 5 automated evaluation metrics. Our results reveal that agentic planning is essential for generating detailed long-form videos, and the o3-mini agent achieves a success rate of 93.8% and an overall score of 0.77. However, our quantitative and qualitative studies show that most of the videos produced exhibit minor issues with visual element layout. Furthermore, multimodal explanations expose deeper reasoning flaws that text-based explanations fail to reveal, highlighting the importance of multimodal explanations.

Building general-purpose reasoning models using reinforcement learning with verifiable rewards (RLVR) across diverse domains has been widely adopted by frontier open-weight models. However, their training recipes and domain mixtures are often not disclosed. Joint optimization across domains poses significant challenges: domains vary widely in rollout length, problem difficulty and sample efficiency. Further, models with long chain-of-thought traces increase inference cost and latency, making efficiency critical for practical deployment. We present Apriel-Reasoner, trained with a fully reproducible multi-domain RL post-training recipe on Apriel-Base, a 15B-parameter open-weight LLM, across five domains using public datasets: mathematics, code generation, instruction following, logical puzzles and function calling. We introduce an adaptive domain sampling mechanism that preserves target domain ratios despite heterogeneous rollout dynamics, and a difficulty-aware extension of the standard length penalty that, with no additional training overhead, encourages longer reasoning for difficult problems and shorter traces for easy ones. Trained with a strict 16K-token output budget, Apriel-Reasoner generalizes to 32K tokens at inference and improves over Apriel-Base on AIME 2025, GPQA, MMLU-Pro, and LiveCodeBench while producing 30-50% shorter reasoning traces. It matches strong open-weight models of similar size at lower token cost, thereby pushing the Pareto frontier of accuracy versus token budget.

Reasoning in knowledge-intensive domains remains challenging as intermediate steps are often not locally verifiable: unlike math or code, evaluating step correctness may require synthesizing clues across large external knowledge sources. As a result, subtle errors can propagate through reasoning traces, potentially never to be detected. Prior work has proposed process reward models (PRMs), including retrieval-augmented variants, but these methods operate post hoc, scoring completed trajectories, which prevents their integration into dynamic inference procedures. Here, we introduce Process Reward Agents (PRA), a test-time method for providing domain-grounded, online, step-wise rewards to a frozen policy. In contrast to prior retrieval-augmented PRMs, PRA enables search-based decoding to rank and prune candidate trajectories at every generation step. Experiments on multiple medical reasoning benchmarks demonstrate that PRA consistently outperforms strong baselines, achieving 80.8% accuracy on MedQA with Qwen3-4B, a new state of the art at the 4B scale. Importantly, PRA generalizes to unseen frozen policy models ranging from 0.5B to 8B parameters, improving their accuracy by up to 25.7% without any policy model updates. More broadly, PRA suggests a paradigm in which frozen reasoners are decoupled from domain-specific reward modules, allowing the deployment of new backbones in complex domains without retraining.

The integration of knowledge extracted from diverse models, whether described by domain experts or generated by machine learning algorithms, has historically been challenged by the absence of a suitable framework for specifying and integrating structures, learning processes, data transformations, and data models or rules. In this work, we extend algebraic specification methods to address these challenges within such a framework. In our work, we tackle the challenging task of developing a comprehensive framework for defining and analyzing deep learning architectures. We believe that previous efforts have fallen short by failing to establish a clear connection between the constraints a model must adhere to and its actual implementation. Our methodology employs graphical structures that resemble Ehresmann's sketches, interpreted within a universe of fuzzy sets. This approach offers a unified theory that elegantly encompasses both deterministic and non-deterministic neural network designs. Furthermore, we highlight how this theory naturally incorporates fundamental concepts from computer science and automata theory. Our extended algebraic specification framework, grounded in graphical structures akin to Ehresmann's sketches, offers a promising solution for integrating knowledge across disparate models and domains. By bridging the gap between domain-specific expertise and machine-generated insights, we pave the way for more comprehensive, collaborative, and effective approaches to knowledge integration and modeling.

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

AGI has become the Holly Grail of AI with the promise of level intelligence and the major Tech companies around the world are investing unprecedented amounts of resources in its pursuit. Yet, there does not exist a single formal definition and only some empirical AGI benchmarking frameworks currently exist. The main purpose of this paper is to develop a general, algebraic and category theoretic framework for describing, comparing and analysing different possible AGI architectures. Thus, this Category theoretic formalization would also allow to compare different possible candidate AGI architectures, such as, RL, Universal AI, Active Inference, CRL, Schema based Learning, etc. It will allow to unambiguously expose their commonalities and differences, and what is even more important, expose areas for future research. From the applied Category theoretic point of view, we take as inspiration Machines in a Category to provide a modern view of AGI Architectures in a Category. More specifically, this first position paper provides, on one hand, a first exercise on RL, Causal RL and SBL Architectures in a Category, and on the other hand, it is a first step on a broader research program that seeks to provide a unified formal foundation for AGI systems, integrating architectural structure, informational organization, agent realization, agent and environment interaction, behavioural development over time, and the empirical evaluation of properties. This framework is also intended to support the definition of architectural properties, both syntactic and informational, as well as semantic properties of agents and their assessment in environments with explicitly characterized features. We claim that Category Theory and AGI will have a very symbiotic relation.

Rule-based reasoning, a fundamental type of legal reasoning, enables us to draw conclusions by accurately applying a rule to a set of facts. We explore causal language models as rule-based reasoners, specifically with respect to compositional rules - rules consisting of multiple elements which form a complex logical expression. Reasoning about compositional rules is challenging because it requires multiple reasoning steps, and attending to the logical relationships between elements. We introduce a new prompting method, Chain of Logic, which elicits rule-based reasoning through decomposition (solving elements as independent threads of logic), and recomposition (recombining these sub-answers to resolve the underlying logical expression). This method was inspired by the IRAC (Issue, Rule, Application, Conclusion) framework, a sequential reasoning approach used by lawyers. We evaluate chain of logic across eight rule-based reasoning tasks involving three distinct compositional rules from the LegalBench benchmark and demonstrate it consistently outperforms other prompting methods, including chain of thought and self-ask, using open-source and commercial language models.

Reinforcement learning (RL) has emerged as a promising approach to improve large language model (LLM) reasoning, yet most open efforts focus narrowly on math and code, limiting our understanding of its broader applicability to general reasoning. A key challenge lies in the lack of reliable, scalable RL reward signals across diverse reasoning domains. We introduce Guru, a curated RL reasoning corpus of 92K verifiable examples spanning six reasoning domains--Math, Code, Science, Logic, Simulation, and Tabular--each built through domain-specific reward design, deduplication, and filtering to ensure reliability and effectiveness for RL training. Based on Guru, we systematically revisit established findings in RL for LLM reasoning and observe significant variation across domains. For example, while prior work suggests that RL primarily elicits existing knowledge from pretrained models, our results reveal a more nuanced pattern: domains frequently seen during pretraining (Math, Code, Science) easily benefit from cross-domain RL training, while domains with limited pretraining exposure (Logic, Simulation, and Tabular) require in-domain training to achieve meaningful performance gains, suggesting that RL is likely to facilitate genuine skill acquisition. Finally, we present Guru-7B and Guru-32B, two models that achieve state-of-the-art performance among open models RL-trained with publicly available data, outperforming best baselines by 7.9% and 6.7% on our 17-task evaluation suite across six reasoning domains. We also show that our models effectively improve the Pass@k performance of their base models, particularly on complex tasks less likely to appear in pretraining data. We release data, models, training and evaluation code to facilitate general-purpose reasoning at: https://github.com/LLM360/Reasoning360

As Large Language Model (LLM) agents are increasingly tasked with high-stakes autonomous decision-making, the transparency of their reasoning processes has become a critical safety concern. While Chain-of-Thought (CoT) prompting allows agents to generate human-readable reasoning traces, it remains unclear whether these traces are faithful generative drivers of the model's output or merely post-hoc rationalizations. We introduce Project Ariadne, a novel XAI framework that utilizes Structural Causal Models (SCMs) and counterfactual logic to audit the causal integrity of agentic reasoning. Unlike existing interpretability methods that rely on surface-level textual similarity, Project Ariadne performs hard interventions (do-calculus) on intermediate reasoning nodes -- systematically inverting logic, negating premises, and reversing factual claims -- to measure the Causal Sensitivity (φ) of the terminal answer. Our empirical evaluation of state-of-the-art models reveals a persistent Faithfulness Gap. We define and detect a widespread failure mode termed Causal Decoupling, where agents exhibit a violation density (ρ) of up to 0.77 in factual and scientific domains. In these instances, agents arrive at identical conclusions despite contradictory internal logic, proving that their reasoning traces function as "Reasoning Theater" while decision-making is governed by latent parametric priors. Our findings suggest that current agentic architectures are inherently prone to unfaithful explanation, and we propose the Ariadne Score as a new benchmark for aligning stated logic with model action.

There is no doubt that safety alignment is an essential step in LLM training. However, conceptually it does not distinguish between various domains and the level of potential harm of a query, which creates significant complications in the fields like cyber security, where a model should not be constrained by its safety circuits to accomplish the goals of legitimate, authorized operations. In this work, we share our findings from a large scale abliteration experiment on 24 open-source LLMs and show that domain-specific abliteration is achievable with standard methodology on the example of a 1T-parameter Kimi K2. Building on recent work showing that refusal in LLMs occupies a multi-dimensional subspace within layers, we find that it is also distributed widely across layers, especially in trillion-parameter MoE architectures, and so we aim to capture the part of it that represents harmful concepts in the cybersecurity domain exclusively. We also investigate the correlation between models' features and the effect of domain-specific abliteration, identifying that the type of safety training and architecture are the most reliable predictors. Finally, we classify the models into 3 abliteration susceptibility tiers and put forward a set of conjectures as to why a particular effect from this intervention might be observed in a given model.

Large Language Models (LLMs) have demonstrated impressive reasoning capabilities, leading to their adoption in high-stakes domains such as healthcare, law, and scientific research. However, their reasoning often contains subtle logical errors masked by fluent language, posing significant risks for critical applications. While existing approaches like fact-checking, self-consistency methods, and rule-based validation provide partial solutions, they fail to detect complex logical flaws in multi-step reasoning. To overcome these challenges, we present MATP, an evaluation framework for systematically verifying LLM reasoning via Multi-step Automatic Theorem Proving. MATP translates natural language reasoning into First-Order Logic (FOL) and applies automated theorem provers to assess step-by-step logical validity. This approach identifies hidden logical errors and provides fine-grained classifications of reasoning correctness. Evaluations on a benchmark comprising 10,830 reasoning instances generated by 10 LLMs across tasks from PrOntoQA-OOD, ProofWriter, and FOLIO show that MATP surpasses prompting-based baselines by over 42 percentage points in reasoning step verification. It further reveals model-level disparities, with reasoning models generating more logically coherent outputs than general models. These results demonstrate MATP's potential to enhance the trustworthiness of LLM-generated reasoning.

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

Condorcet domains are subsets of permutations that ensure pairwise majority voting yields acyclic outcomes, and they form an active area of research at the intersection of social choice theory and combinatorics. In this paper, we extend the theory of Condorcet domains to the broader setting of arbitrary finite Coxeter groups. The core contribution of our approach is the introduction of Condorcet root posets, defined on the chosen root systems. Notably, we establish a natural bijection between closed Condorcet domains and Condorcet root posets, which facilitates the study of Condorcet domains. Using this correspondence, we extend the median graph representation of closed Condorcet domains to arbitrary finite Coxeter groups, demonstrating that these domains can be characterized by the skeletons of their associated Condorcet root posets. These results are novel even in type A. Furthermore, these posets give a unified language that efficiently captures a wide range of desirable properties of Condorcet domains, such as being maximal, connected, peak-pit, and of tiling type. Using this framework, we strengthen and generalize several classical results: we establish that a maximal Condorcet domain is connected if and only if it is peak-pit; we prove that the tiling-type property is equivalent to the combination of being maximal and connected, and having maximal width; and we show that strictly positive voting profiles on connected Condorcet domains yield outcomes with only simple ties.

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.

Recent advances in Large Reasoning Models (LRMs) trained with Long Chain-of-Thought (Long CoT) reasoning have demonstrated remarkable cross-domain generalization capabilities. However, the underlying mechanisms supporting such transfer remain poorly understood. We hypothesize that cross-domain generalization arises from shared abstract reasoning prototypes -- fundamental reasoning patterns that capture the essence of problems across domains. These prototypes minimize the nuances of the representation, revealing that seemingly diverse tasks are grounded in shared reasoning structures.Based on this hypothesis, we propose ProtoReasoning, a framework that enhances the reasoning ability of LLMs by leveraging scalable and verifiable prototypical representations (Prolog for logical reasoning, PDDL for planning).ProtoReasoning features: (1) an automated prototype construction pipeline that transforms problems into corresponding prototype representations; (2) a comprehensive verification system providing reliable feedback through Prolog/PDDL interpreters; (3) the scalability to synthesize problems arbitrarily within prototype space while ensuring correctness. Extensive experiments show that ProtoReasoning achieves 4.7% improvement over baseline models on logical reasoning (Enigmata-Eval), 6.3% improvement on planning tasks, 4.0% improvement on general reasoning (MMLU) and 1.0% on mathematics (AIME24). Significantly, our ablation studies confirm that learning in prototype space also demonstrates enhanced generalization to structurally similar problems compared to training solely on natural language representations, validating our hypothesis that reasoning prototypes serve as the foundation for generalizable reasoning in large language models.

We extend the simply-typed guarded lambda-calculus with discrete probabilities and endow it with a program logic for reasoning about relational properties of guarded probabilistic computations. This provides a framework for programming and reasoning about infinite stochastic processes like Markov chains. We demonstrate the logic sound by interpreting its judgements in the topos of trees and by using probabilistic couplings for the semantics of relational assertions over distributions on discrete types. The program logic is designed to support syntax-directed proofs in the style of relational refinement types, but retains the expressiveness of higher-order logic extended with discrete distributions, and the ability to reason relationally about expressions that have different types or syntactic structure. In addition, our proof system leverages a well-known theorem from the coupling literature to justify better proof rules for relational reasoning about probabilistic expressions. We illustrate these benefits with a broad range of examples that were beyond the scope of previous systems, including shift couplings and lump couplings between random walks.

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/ .

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.

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

Large Language Models (LLMs) are predominantly governed by probabilistic frameworks in which the sum of outcome probabilities is constrained to unity. This architectural limitation, often imposed by Softmax layers, leads to a collapse of uncertainty that makes it difficult to differentiate between epistemic uncertainty, paradox, and vagueness. We present an empirical investigation of the application of Neutrosophic Logic, a framework that treats Truth (T), Indeterminacy (I), and Falsity (F) as three independent dimensions, to model epistemic states in LLMs. We conducted experiments on a family of four OpenAI GPT models across five linguistic phenomena: logical paradoxes, epistemic ignorance, vagueness, ethical contradictions, and future contingencies, under three prompting strategies: neutrosophic, probabilistic, and entropy-derived. Our findings reveal that the neutrosophic approach, by allowing T+I+F > 1, a state we term hyper-truth, provides a richer representation of a model's internal state. In 35% of evaluations, hyper-truth emerged spontaneously, predominantly under ethical contradiction and logical paradox. We demonstrate that this approach preserves truth values in fuzzy contexts and offers a robust method for identifying and quantifying internal model conflict. We conclude that the integration of neutrosophic evaluation layers is a critical step toward more transparent, reliable, and ethically aware AI systems.

The solution methods used to realize artificial general intelligence (AGI) may not contain the formalism needed to adequately model and characterize AGI. In particular, current approaches to learning hold notions of problem domain and problem task as fundamental precepts, but it is hardly apparent that an AGI encountered in the wild will be discernable into a set of domain-task pairings. Nor is it apparent that the outcomes of AGI in a system can be well expressed in terms of domain and task, or as consequences thereof. Thus, there is both a practical and theoretical use for meta-theories of learning which do not express themselves explicitly in terms of solution methods. General systems theory offers such a meta-theory. Herein, Mesarovician abstract systems theory is used as a super-structure for learning. Abstract learning systems are formulated. Subsequent elaboration stratifies the assumptions of learning systems into a hierarchy and considers the hierarchy such stratification projects onto learning theory. The presented Mesarovician abstract learning systems theory calls back to the founding motivations of artificial intelligence research by focusing on the thinking participants directly, in this case, learning systems, in contrast to the contemporary focus on the problems thinking participants solve.

Group polarization, the phenomenon where individuals become more extreme after interacting, has been gaining attention, especially with the rise of social media shaping people's opinions. Recent interest has emerged in formal reasoning about group polarization using logical systems. In this work we consider the modal logic PNL that captures the notion of agents agreeing or disagreeing on a given topic. Our contribution involves enhancing PNL with advanced formal reasoning techniques, instead of relying on axiomatic systems for analyzing group polarization. To achieve this, we introduce a semantic game tailored for (hybrid) extensions of PNL. This game fosters dynamic reasoning about concrete network models, aligning with our goal of strengthening PNL's effectiveness in studying group polarization. We show how this semantic game leads to a provability game by systemically exploring the truth in all models. This leads to the first cut-free sequent systems for some variants of PNL. Using polarization of formulas, the proposed calculi can be modularly adapted to consider different frame properties of the underlying model.

Large Language Models (LLMs) have been successful in mathematical reasoning tasks such as formal theorem proving when integrated with interactive proof assistants like Lean. Existing approaches involve training or fine-tuning an LLM on a specific dataset to perform well on particular domains, such as undergraduate-level mathematics. These methods struggle with generalizability to advanced mathematics. A fundamental limitation is that these approaches operate on static domains, failing to capture how mathematicians often work across multiple domains and projects simultaneously or cyclically. We present LeanAgent, a novel lifelong learning framework for theorem proving that continuously generalizes to and improves on ever-expanding mathematical knowledge without forgetting previously learned knowledge. LeanAgent introduces several key innovations, including a curriculum learning strategy that optimizes the learning trajectory in terms of mathematical difficulty, a dynamic database for efficient management of evolving mathematical knowledge, and progressive training to balance stability and plasticity. LeanAgent successfully proves 162 theorems previously unproved by humans across 23 diverse Lean repositories, many from advanced mathematics. It performs up to 11times better than the static LLM baseline, proving challenging theorems in domains like abstract algebra and algebraic topology while showcasing a clear progression of learning from basic concepts to advanced topics. In addition, we analyze LeanAgent's superior performance on key lifelong learning metrics. LeanAgent achieves exceptional scores in stability and backward transfer, where learning new tasks improves performance on previously learned tasks. This emphasizes LeanAgent's continuous generalizability and improvement, explaining its superior theorem proving performance.

In the field of representation learning on knowledge graphs (KGs), a hyper-relational fact consists of a main triple and several auxiliary attribute-value descriptions, which is considered more comprehensive and specific than a triple-based fact. However, currently available hyper-relational KG embedding methods in a single view are limited in application because they weaken the hierarchical structure that represents the affiliation between entities. To overcome this limitation, we propose a dual-view hyper-relational KG structure (DH-KG) that contains a hyper-relational instance view for entities and a hyper-relational ontology view for concepts that are abstracted hierarchically from the entities. This paper defines link prediction and entity typing tasks on DH-KG for the first time and constructs two DH-KG datasets, JW44K-6K, extracted from Wikidata, and HTDM based on medical data. Furthermore, we propose DHGE, a DH-KG embedding model based on GRAN encoders, HGNNs, and joint learning. DHGE outperforms baseline models on DH-KG, according to experimental results. Finally, we provide an example of how this technology can be used to treat hypertension. Our model and new datasets are publicly available.

Scientific reasoning relies not only on logical inference but also on activating prior knowledge and experiential structures. Memory can efficiently reuse knowledge and enhance reasoning consistency and stability. However, existing benchmarks mainly evaluate final answers or step-by-step coherence, overlooking the memory-driven mechanisms that underlie human reasoning, which involves activating anchors and attractors, then integrating them into multi-step inference. To address this gap, we propose A^3-Bench~ https://a3-bench.github.io, a benchmark designed to evaluate scientific reasoning through dual-scale memory-driven activation, grounded in Anchor and Attractor Activation. First, we annotate 2,198 science reasoning problems across domains using the SAPM process(subject, anchor & attractor, problem, and memory developing). Second, we introduce a dual-scale memory evaluation framework utilizing anchors and attractors, along with the AAUI(Anchor--Attractor Utilization Index) metric to measure memory activation rates. Finally, through experiments with various base models and paradigms, we validate A^3-Bench and analyze how memory activation impacts reasoning performance, providing insights into memory-driven scientific reasoning.

We study the problem of attributing the prediction of a deep network to its input features, a problem previously studied by several other works. We identify two fundamental axioms---Sensitivity and Implementation Invariance that attribution methods ought to satisfy. We show that they are not satisfied by most known attribution methods, which we consider to be a fundamental weakness of those methods. We use the axioms to guide the design of a new attribution method called Integrated Gradients. Our method requires no modification to the original network and is extremely simple to implement; it just needs a few calls to the standard gradient operator. We apply this method to a couple of image models, a couple of text models and a chemistry model, demonstrating its ability to debug networks, to extract rules from a network, and to enable users to engage with models better.

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.

One of the fundamental problems in Artificial Intelligence is to perform complex multi-hop logical reasoning over the facts captured by a knowledge graph (KG). This problem is challenging, because KGs can be massive and incomplete. Recent approaches embed KG entities in a low dimensional space and then use these embeddings to find the answer entities. However, it has been an outstanding challenge of how to handle arbitrary first-order logic (FOL) queries as present methods are limited to only a subset of FOL operators. In particular, the negation operator is not supported. An additional limitation of present methods is also that they cannot naturally model uncertainty. Here, we present BetaE, a probabilistic embedding framework for answering arbitrary FOL queries over KGs. BetaE is the first method that can handle a complete set of first-order logical operations: conjunction (wedge), disjunction (vee), and negation (neg). A key insight of BetaE is to use probabilistic distributions with bounded support, specifically the Beta distribution, and embed queries/entities as distributions, which as a consequence allows us to also faithfully model uncertainty. Logical operations are performed in the embedding space by neural operators over the probabilistic embeddings. We demonstrate the performance of BetaE on answering arbitrary FOL queries on three large, incomplete KGs. While being more general, BetaE also increases relative performance by up to 25.4% over the current state-of-the-art KG reasoning methods that can only handle conjunctive queries without negation.

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

Large language models (LLMs) have demonstrated impressive capabilities in natural language understanding and generation, but they exhibit problems with logical consistency in the output they generate. How can we harness LLMs' broad-coverage parametric knowledge in formal reasoning despite their inconsistency? We present a method for directly integrating an LLM into the interpretation function of the formal semantics for a paraconsistent logic. We provide experimental evidence for the feasibility of the method by evaluating the function using datasets created from several short-form factuality benchmarks. Unlike prior work, our method offers a theoretical framework for neuro-symbolic reasoning that leverages an LLM's knowledge while preserving the underlying logic's soundness and completeness properties.