Daily Papers

The application of Large Language Models (LLMs) in accelerating scientific discovery has garnered increasing attention, with a key focus on constructing research agents endowed with innovative capability, i.e., the ability to autonomously generate novel and significant research ideas. Existing approaches predominantly rely on sophisticated prompt engineering and lack a systematic training paradigm. To address this, we propose DeepInnovator, a training framework designed to trigger the innovative capability of LLMs. Our approach comprises two core components. (1) ``Standing on the shoulders of giants''. We construct an automated data extraction pipeline to extract and organize structured research knowledge from a vast corpus of unlabeled scientific literature. (2) ``Conjectures and refutations''. We introduce a ``Next Idea Prediction'' training paradigm, which models the generation of research ideas as an iterative process of continuously predicting, evaluating, and refining plausible and novel next idea. Both automatic and expert evaluations demonstrate that our DeepInnovator-14B significantly outperforms untrained baselines, achieving win rates of 80.53\%-93.81\%, and attains performance comparable to that of current leading LLMs. This work provides a scalable training pathway toward building research agents with genuine, originative innovative capability, and will open-source the dataset to foster community advancement. Source code and data are available at: https://github.com/HKUDS/DeepInnovator.

Large language models (LLMs) struggle with complex, long-horizon reasoning due to instability caused by their frozen policy assumption. Current test-time scaling methods treat execution feedback merely as an external signal for filtering or rewriting trajectories, without internalizing it to improve the underlying reasoning strategy. Inspired by Popper's epistemology of "conjectures and refutations," we argue that intelligence requires real-time evolution of the model's policy through learning from failed attempts. We introduce Policy of Thoughts (PoT), a framework that recasts reasoning as a within-instance online optimization process. PoT first generates diverse candidate solutions via an efficient exploration mechanism, then uses Group Relative Policy Optimization (GRPO) to update a transient LoRA adapter based on execution feedback. This closed-loop design enables dynamic, instance-specific refinement of the model's reasoning priors. Experiments show that PoT dramatically boosts performance: a 4B model achieves 49.71% accuracy on LiveCodeBench, outperforming GPT-4o and DeepSeek-V3 despite being over 50 smaller.

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

We resolve a $1000 Erdős prize problem, complete with formal verification generated by a large language model. In over a dozen papers, beginning in 1976 and spanning two decades, Paul Erdős repeatedly posed one of his "favourite" conjectures: every finite Sidon set can be extended to a finite perfect difference set. We establish that {1, 2, 4, 8, 13} is a counterexample to this conjecture. During the preparation of this paper, we discovered that although this problem was presumed to be open for half a century, Marshall Hall, Jr. published a different counterexample three decades before Erdős first posed the problem. With a healthy skepticism of this apparent oversight, and out of an abundance of caution, we used ChatGPT to vibe code a Lean proof of both Hall's and our counterexamples.

There is growing excitement about the potential of Language Models (LMs) to accelerate scientific discovery. Falsifying hypotheses is key to scientific progress, as it allows claims to be iteratively refined over time. This process requires significant researcher effort, reasoning, and ingenuity. Yet current benchmarks for LMs predominantly assess their ability to generate solutions rather than challenge them. We advocate for developing benchmarks that evaluate this inverse capability - creating counterexamples for subtly incorrect solutions. To demonstrate this approach, we start with the domain of algorithmic problem solving, where counterexamples can be evaluated automatically using code execution. Specifically, we introduce REFUTE, a dynamically updating benchmark that includes recent problems and incorrect submissions from programming competitions, where human experts successfully identified counterexamples. Our analysis finds that the best reasoning agents, even OpenAI o3-mini (high) with code execution feedback, can create counterexamples for only <9% of incorrect solutions in REFUTE, even though ratings indicate its ability to solve up to 48% of these problems from scratch. We hope our work spurs progress in evaluating and enhancing LMs' ability to falsify incorrect solutions - a capability that is crucial for both accelerating research and making models self-improve through reliable reflective reasoning.

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

We present a case study in semi-autonomous mathematics discovery, using Gemini to systematically evaluate 700 conjectures labeled 'Open' in Bloom's Erdős Problems database. We employ a hybrid methodology: AI-driven natural language verification to narrow the search space, followed by human expert evaluation to gauge correctness and novelty. We address 13 problems that were marked 'Open' in the database: 5 through seemingly novel autonomous solutions, and 8 through identification of previous solutions in the existing literature. Our findings suggest that the 'Open' status of the problems was through obscurity rather than difficulty. We also identify and discuss issues arising in applying AI to math conjectures at scale, highlighting the difficulty of literature identification and the risk of ''subconscious plagiarism'' by AI. We reflect on the takeaways from AI-assisted efforts on the Erdős Problems.

A recent paper (van Rooij et al. 2024) claims to have proved that achieving human-like intelligence using learning from data is intractable in a complexity-theoretic sense. We identify that the proof relies on an unjustified assumption about the distribution of (input, output) pairs to the system. We briefly discuss that assumption in the context of two fundamental barriers to repairing the proof: the need to precisely define ``human-like," and the need to account for the fact that a particular machine learning system will have particular inductive biases that are key to the analysis.

Using a long-standing conjecture from combinatorial group theory, we explore, from multiple perspectives, the challenges of finding rare instances carrying disproportionately high rewards. Based on lessons learned in the context defined by the Andrews-Curtis conjecture, we propose algorithmic enhancements and a topological hardness measure with implications for a broad class of search problems. As part of our study, we also address several open mathematical questions. Notably, we demonstrate the length reducibility of all but two presentations in the Akbulut-Kirby series (1981), and resolve various potential counterexamples in the Miller-Schupp series (1991), including three infinite subfamilies.

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.

Let F_{k,d}(n) be the maximal size of a set {A}subseteq [n] such that the equation \[a_1a_2\dots a_k=x^d, \; a_1<a_2<\ldots<a_k\] has no solution with a_1,a_2,ldots,a_kA and integer x. Erdos, S\'ark\"ozy and T. S\'os studied F_{k,2}, and gave bounds when k=2,3,4,6 and also in the general case. We study the problem for d=3, and provide bounds for k=2,3,4,6 and 9, furthermore, in the general case, as well. In particular, we refute an 18 years old conjecture of Verstra\"ete. We also introduce another function f_{k,d} closely related to F_{k,d}: While the original problem requires a_1, ldots , a_k to all be distinct, we can relax this and only require that the multiset of the a_i's cannot be partitioned into d-tuples where each d-tuple consists of d copies of the same number.

In their recent Nature Human Behaviour paper, "Emergent analogical reasoning in large language models," (Webb, Holyoak, and Lu, 2023) the authors argue that "large language models such as GPT-3 have acquired an emergent ability to find zero-shot solutions to a broad range of analogy problems." In this response, we provide counterexamples of the letter string analogies. In our tests, GPT-3 fails to solve even the easiest variants of the problems presented in the original paper. Zero-shot reasoning is an extraordinary claim that requires extraordinary evidence. We do not see that evidence in our experiments. To strengthen claims of humanlike reasoning such as zero-shot reasoning, it is important that the field develop approaches that rule out data memorization.

It has been shown that Large Reasoning Models (LRMs) may not *say what they think*: they do not always volunteer information about how certain parts of the input influence their reasoning. But it is one thing for a model to *omit* such information and another, worse thing to *lie* about it. Here, we extend the work of Chen et al. (2025) to show that LRMs will do just this: they will flatly deny relying on hints provided in the prompt in answering multiple choice questions -- even when directly asked to reflect on unusual (i.e. hinted) prompt content, even when allowed to use hints, and even though experiments *show* them to be using the hints. Our results thus have discouraging implications for CoT monitoring and interpretability.

Can you find an xy-equation that, when graphed, writes itself on the plane? This idea became internet-famous when a Wikipedia article on Tupper's self-referential formula went viral in 2012. Under scrutiny, the question has two flaws: it is meaningless (it depends on typography) and it is trivial (for reasons we will explain). We fix these flaws by formalizing the problem, and we give a very general solution using techniques from computability theory.

One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed case that one would be wrong. In particular, I argue that computational complexity theory -- the field that studies the resources (such as time, space, and randomness) needed to solve computational problems -- leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume's problem of induction, Goodman's grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing aspects of complexity theory itself that could benefit from philosophical analysis.

Misinformation about climate change is a complex societal issue requiring holistic, interdisciplinary solutions at the intersection between technology and psychology. One proposed solution is a "technocognitive" approach, involving the synthesis of psychological and computer science research. Psychological research has identified that interventions in response to misinformation require both fact-based (e.g., factual explanations) and technique-based (e.g., explanations of misleading techniques) content. However, little progress has been made on documenting and detecting fallacies in climate misinformation. In this study, we apply a previously developed critical thinking methodology for deconstructing climate misinformation, in order to develop a dataset mapping different types of climate misinformation to reasoning fallacies. This dataset is used to train a model to detect fallacies in climate misinformation. Our study shows F1 scores that are 2.5 to 3.5 better than previous works. The fallacies that are easiest to detect include fake experts and anecdotal arguments, while fallacies that require background knowledge, such as oversimplification, misrepresentation, and slothful induction, are relatively more difficult to detect. This research lays the groundwork for development of solutions where automatically detected climate misinformation can be countered with generative technique-based corrections.

We present the surprising finding that a language model's reasoning capabilities can be improved by training on synthetic datasets of chain-of-thought (CoT) traces from more capable models, even when all of those traces lead to an incorrect final answer. Our experiments show this approach can yield better performance on reasoning tasks than training on human-annotated datasets. We hypothesize that two key factors explain this phenomenon: first, the distribution of synthetic data is inherently closer to the language model's own distribution, making it more amenable to learning. Second, these `incorrect' traces are often only partially flawed and contain valid reasoning steps from which the model can learn. To further test the first hypothesis, we use a language model to paraphrase human-annotated traces -- shifting their distribution closer to the model's own distribution -- and show that this improves performance. For the second hypothesis, we introduce increasingly flawed CoT traces and study to what extent models are tolerant to these flaws. We demonstrate our findings across various reasoning domains like math, algorithmic reasoning and code generation using MATH, GSM8K, Countdown and MBPP datasets on various language models ranging from 1.5B to 9B across Qwen, Llama, and Gemma models. Our study shows that curating datasets that are closer to the model's distribution is a critical aspect to consider. We also show that a correct final answer is not always a reliable indicator of a faithful reasoning process.

Recent advancements in machine learning research, i.e., deep learning, introduced methods that excel conventional algorithms as well as humans in several complex tasks, ranging from detection of objects in images and speech recognition to playing difficult strategic games. However, the current methodology of machine learning research and consequently, implementations of the real-world applications of such algorithms, seems to have a recurring HARKing (Hypothesizing After the Results are Known) issue. In this work, we elaborate on the algorithmic, economic and social reasons and consequences of this phenomenon. We present examples from current common practices of conducting machine learning research (e.g. avoidance of reporting negative results) and failure of generalization ability of the proposed algorithms and datasets in actual real-life usage. Furthermore, a potential future trajectory of machine learning research and development from the perspective of accountable, unbiased, ethical and privacy-aware algorithmic decision making is discussed. We would like to emphasize that with this discussion we neither claim to provide an exhaustive argumentation nor blame any specific institution or individual on the raised issues. This is simply a discussion put forth by us, insiders of the machine learning field, reflecting on us.

Counterfactual statements, which describe events that did not or cannot take place, are beneficial to numerous NLP applications. Hence, we consider the problem of counterfactual detection (CFD) and seek to enhance the CFD models. Previous models are reliant on clue phrases to predict counterfactuality, so they suffer from significant performance drop when clue phrase hints do not exist during testing. Moreover, these models tend to predict non-counterfactuals over counterfactuals. To address these issues, we propose to integrate neural topic model into the CFD model to capture the global semantics of the input statement. We continue to causally intervene the hidden representations of the CFD model to balance the effect of the class labels. Extensive experiments show that our approach outperforms previous state-of-the-art CFD and bias-resolving methods in both the CFD and other bias-sensitive tasks.

Science progresses by iteratively advancing and correcting humanity's understanding of the world. In machine learning (ML) research, rapid advancements have led to an explosion of publications, but have also led to misleading, incorrect, flawed or perhaps even fraudulent studies being accepted and sometimes highlighted at ML conferences due to the fallibility of peer review. While such mistakes are understandable, ML conferences do not offer robust processes to help the field systematically correct when such errors are made. This position paper argues that ML conferences should establish a dedicated "Refutations and Critiques" (R&C) Track. This R&C Track would provide a high-profile, reputable platform to support vital research that critically challenges prior research, thereby fostering a dynamic self-correcting research ecosystem. We discuss key considerations including track design, review principles, potential pitfalls, and provide an illustrative example submission concerning a recent ICLR 2025 Oral. We conclude that ML conferences should create official, reputable mechanisms to help ML research self-correct.

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.

There has been considerable divergence of opinion on the reasoning abilities of Large Language Models (LLMs). While the initial optimism that reasoning might emerge automatically with scale has been tempered thanks to a slew of counterexamples, a wide spread belief in their iterative self-critique capabilities persists. In this paper, we set out to systematically investigate the effectiveness of iterative prompting of LLMs in the context of Graph Coloring, a canonical NP-complete reasoning problem that is related to propositional satisfiability as well as practical problems like scheduling and allocation. We present a principled empirical study of the performance of GPT4 in solving graph coloring instances or verifying the correctness of candidate colorings. In iterative modes, we experiment with the model critiquing its own answers and an external correct reasoner verifying proposed solutions. In both cases, we analyze whether the content of the criticisms actually affects bottom line performance. The study seems to indicate that (i) LLMs are bad at solving graph coloring instances (ii) they are no better at verifying a solution--and thus are not effective in iterative modes with LLMs critiquing LLM-generated solutions (iii) the correctness and content of the criticisms--whether by LLMs or external solvers--seems largely irrelevant to the performance of iterative prompting. We show that the observed increase in effectiveness is largely due to the correct solution being fortuitously present in the top-k completions of the prompt (and being recognized as such by an external verifier). Our results thus call into question claims about the self-critiquing capabilities of state of the art LLMs.

As Large Language Models become more ubiquitous across domains, it becomes important to examine their inherent limitations critically. This work argues that hallucinations in language models are not just occasional errors but an inevitable feature of these systems. We demonstrate that hallucinations stem from the fundamental mathematical and logical structure of LLMs. It is, therefore, impossible to eliminate them through architectural improvements, dataset enhancements, or fact-checking mechanisms. Our analysis draws on computational theory and Godel's First Incompleteness Theorem, which references the undecidability of problems like the Halting, Emptiness, and Acceptance Problems. We demonstrate that every stage of the LLM process-from training data compilation to fact retrieval, intent classification, and text generation-will have a non-zero probability of producing hallucinations. This work introduces the concept of Structural Hallucination as an intrinsic nature of these systems. By establishing the mathematical certainty of hallucinations, we challenge the prevailing notion that they can be fully mitigated.

Naturally occurring information-seeking questions often contain questionable assumptions -- assumptions that are false or unverifiable. Questions containing questionable assumptions are challenging because they require a distinct answer strategy that deviates from typical answers for information-seeking questions. For instance, the question "When did Marie Curie discover Uranium?" cannot be answered as a typical "when" question without addressing the false assumption "Marie Curie discovered Uranium". In this work, we propose (QA)^2 (Question Answering with Questionable Assumptions), an open-domain evaluation dataset consisting of naturally occurring search engine queries that may or may not contain questionable assumptions. To be successful on (QA)^2, systems must be able to detect questionable assumptions and also be able to produce adequate responses for both typical information-seeking questions and ones with questionable assumptions. Through human rater acceptability on end-to-end QA with (QA)^2, we find that current models do struggle with handling questionable assumptions, leaving substantial headroom for progress.

With the growing importance of detecting misinformation, many studies have focused on verifying factual claims by retrieving evidence. However, canonical fact verification tasks do not apply to catching subtle differences in factually consistent claims, which might still bias the readers, especially on contentious political or economic issues. Our underlying assumption is that among the trusted sources, one's argument is not necessarily more true than the other, requiring comparison rather than verification. In this study, we propose ClaimDiff, a novel dataset that primarily focuses on comparing the nuance between claim pairs. In ClaimDiff, we provide 2,941 annotated claim pairs from 268 news articles. We observe that while humans are capable of detecting the nuances between claims, strong baselines struggle to detect them, showing over a 19% absolute gap with the humans. We hope this initial study could help readers to gain an unbiased grasp of contentious issues through machine-aided comparison.

Research is a fundamental process driving the advancement of human civilization, yet it demands substantial time and effort from researchers. In recent years, the rapid development of artificial intelligence (AI) technologies has inspired researchers to explore how AI can accelerate and enhance research. To monitor relevant advancements, this paper presents a systematic review of the progress in this domain. Specifically, we organize the relevant studies into three main categories: hypothesis formulation, hypothesis validation, and manuscript publication. Hypothesis formulation involves knowledge synthesis and hypothesis generation. Hypothesis validation includes the verification of scientific claims, theorem proving, and experiment validation. Manuscript publication encompasses manuscript writing and the peer review process. Furthermore, we identify and discuss the current challenges faced in these areas, as well as potential future directions for research. Finally, we also offer a comprehensive overview of existing benchmarks and tools across various domains that support the integration of AI into the research process. We hope this paper serves as an introduction for beginners and fosters future research. Resources have been made publicly available at https://github.com/zkzhou126/AI-for-Research.

Interestingly, LLMs yet struggle with some basic tasks that humans find trivial to handle, e.g., counting the number of character r's in the word "strawberry". There are several popular conjectures (e.g., tokenization, architecture and training data) regarding the reason for deficiency of LLMs in simple word-based counting problems, sharing the similar belief that such failure stems from model pretraining hence probably inevitable during deployment. In this paper, we carefully design multiple evaluation settings to investigate validity of prevalent conjectures. Meanwhile, we measure transferability of advanced mathematical and coding reasoning capabilities from specialized LLMs to simple counting tasks. Although specialized LLMs suffer from counting problems as well, we find conjectures about inherent deficiency of LLMs invalid and further seek opportunities to elicit knowledge and capabilities from LLMs that are beneficial to counting tasks. Compared with strategies such as finetuning and in-context learning that are commonly adopted to enhance performance on new or challenging tasks, we show that engaging reasoning is the most robust and efficient way to help LLMs better perceive tasks with more accurate responses. We hope our conjecture validation design could provide insights into the study of future critical failure modes of LLMs. Based on challenges in transferring advanced capabilities to much simpler tasks, we call for more attention to model capability acquisition and evaluation. We also highlight the importance of cultivating consciousness of "reasoning before responding" during model pretraining.

This paper provides theoretical insights into why and how deep learning can generalize well, despite its large capacity, complexity, possible algorithmic instability, nonrobustness, and sharp minima, responding to an open question in the literature. We also discuss approaches to provide non-vacuous generalization guarantees for deep learning. Based on theoretical observations, we propose new open problems and discuss the limitations of our results.

Retrieval-augmented language models are being increasingly tasked with subjective, contentious, and conflicting queries such as "is aspartame linked to cancer". To resolve these ambiguous queries, one must search through a large range of websites and consider "which, if any, of this evidence do I find convincing?". In this work, we study how LLMs answer this question. In particular, we construct ConflictingQA, a dataset that pairs controversial queries with a series of real-world evidence documents that contain different facts (e.g., quantitative results), argument styles (e.g., appeals to authority), and answers (Yes or No). We use this dataset to perform sensitivity and counterfactual analyses to explore which text features most affect LLM predictions. Overall, we find that current models rely heavily on the relevance of a website to the query, while largely ignoring stylistic features that humans find important such as whether a text contains scientific references or is written with a neutral tone. Taken together, these results highlight the importance of RAG corpus quality (e.g., the need to filter misinformation), and possibly even a shift in how LLMs are trained to better align with human judgements.

This is the third paper of the CayleyPy project applying artificial intelligence to problems in group theory. We announce the first public release of CayleyPy, an open source Python library for computations with Cayley and Schreier graphs. Compared with systems such as GAP and Sage, CayleyPy handles much larger graphs and performs several orders of magnitude faster. Using CayleyPy we obtained about 200 new conjectures on Cayley and Schreier graphs, focused on diameters and growth. For many Cayley graphs of symmetric groups Sn we observe quasi polynomial diameter formulas: a small set of quadratic or linear polynomials indexed by n mod s. We conjecture that this is a general phenomenon, giving efficient diameter computation despite the problem being NP hard. We propose a refinement of the Babai type conjecture on diameters of Sn: n^2/2 + 4n upper bounds in the undirected case, compared to previous O(n^2) bounds. We also provide explicit generator families, related to involutions in a square with whiskers pattern, conjectured to maximize the diameter; search confirms this for all n up to 15. We further conjecture an answer to a question posed by V M Glushkov in 1968 on directed Cayley graphs generated by a cyclic shift and a transposition. For nilpotent groups we conjecture an improvement of J S Ellenberg's results on upper unitriangular matrices over Z/pZ, showing linear dependence of diameter on p. Moreover. Some conjectures are LLM friendly, naturally stated as sorting problems verifiable by algorithms or Python code. To benchmark path finding we created more than 10 Kaggle datasets. CayleyPy works with arbitrary permutation or matrix groups and includes over 100 predefined generators. Our growth computation code outperforms GAP and Sage up to 1000 times in speed and size.

Research on causality extraction from text has so far almost entirely neglected counterclaims. Existing causality extraction datasets focus solely on "procausal" claims, i.e., statements that support a relationship. "Concausal" claims, i.e., statements that refute a relationship, are entirely ignored or even accidentally annotated as procausal. We address this shortcoming by developing a new dataset that integrates concausality. Based on an extensive literature review, we first show that concausality is an integral part of causal reasoning on incomplete knowledge. We operationalize this theory in the form of a rigorous guideline for annotation and then augment the Causal News Corpus with concausal statements, obtaining a substantial inter-annotator agreement of Cohen's κ=0.74. To demonstrate the importance of integrating concausal statements, we show that models trained without concausal relationships tend to misclassify these as procausal instead. Based on our new dataset, this mistake can be mitigated, enabling transformers to effectively distinguish pro- and concausality.

We argue that progress toward AGI is theory limited rather than data or scale limited. Building on the critical rationalism of Popper and Deutsch, we challenge the Platonic Representation Hypothesis. Observationally equivalent worlds can diverge under interventions, so observational adequacy alone cannot guarantee interventional competence. We begin by laying foundations, definitions of knowledge, learning, intelligence, counterfactual competence and AGI, and then analyze the limits of observational learning that motivate an error centric shift. We recast the problem as three questions about how explicit and implicit errors evolve under an agent's actions, which errors are unreachable within a fixed hypothesis space, and how conjecture and criticism expand that space. From these questions we propose Causal Mechanics, a mechanisms first program in which hypothesis space change is a first class operation and probabilistic structure is used when useful rather than presumed. We advance structural principles that make error discovery and correction tractable, including a differential Locality and Autonomy Principle for modular interventions, a gauge invariant form of Independent Causal Mechanisms for separability, and the Compositional Autonomy Principle for analogy preservation, together with actionable diagnostics. The aim is a scaffold for systems that can convert unreachable errors into reachable ones and correct them.

Counterfactual explanations are attracting significant attention due to the flourishing applications of machine learning models in consequential domains. A counterfactual plan consists of multiple possibilities to modify a given instance so that the model's prediction will be altered. As the predictive model can be updated subject to the future arrival of new data, a counterfactual plan may become ineffective or infeasible with respect to the future values of the model parameters. In this work, we study the counterfactual plans under model uncertainty, in which the distribution of the model parameters is partially prescribed using only the first- and second-moment information. First, we propose an uncertainty quantification tool to compute the lower and upper bounds of the probability of validity for any given counterfactual plan. We then provide corrective methods to adjust the counterfactual plan to improve the validity measure. The numerical experiments validate our bounds and demonstrate that our correction increases the robustness of the counterfactual plans in different real-world datasets.

Large reasoning models (e.g., R1, o3) have demonstrated remarkable mathematical problem-solving abilities. However, the high reported accuracy of these advanced models on popular datasets, reliance on purely numerical evaluation and potential benchmark leakage, often masks their true reasoning shortcomings. To address this, we propose leveraging the inherent rigor and methodological complexity of mathematical proofs as a diagnostic tool to expose these hidden failures. Specifically, we introduce the RFMDataset (Reveal Failure Modes), a collection of 200 diverse mathematical proof problems, and thoroughly evaluate advanced models' performance on it. Our in-depth analysis of their failures uncovers 10 fine-grained error types, which shows fundamental limitations in current large reasoning models: 1) large reasoning models grapple profoundly with mathematical proofs, with some generating entirely correct proofs for less than 20% of problems and failing even on basic ones; 2) models exhibit a diverse spectrum of reasoning failures, prominently demonstrating the lack of guarantees for the correctness and rigor of single-step reasoning; and 3) models show hallucination and incompleteness during the reasoning process. Our findings reveal that models' self-reflection is insufficient to resolve the current logical dilemmas, necessitating formalized and fine-grained logical training.

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

With the widespread adoption of large language models (LLMs), hallucinations, which are non-factual fabrications in model outputs, have become serious concerns. Reasoning capabilities have received attention as a self-verification process to improve output reliability. However, the effect of reasoning within a closed system where LLMs cannot rely on external tools or knowledge has yet to be clarified. We therefore conduct experiments under strict constraints (recommending peer-reviewed journal articles in computer science) to examine the effect of reasoning across multiple models (GPT-5.2 and Gemini 3 Flash). Our results reveal a problematic trade-off between constraint compliance and factual accuracy. Non-reasoning models exhibit high constraint violation rates (66-75%) but maintain factual accuracy, while reasoning models reduce violations (13-26%) but systematically distort known facts to satisfy constraints and increase complete fabrication. This trade-off pattern is consistent across both models despite different architectures, indicating a fundamental limitation of reasoning. Furthermore, reasoning does not uniformly improve output authenticity: effects diverge by model, reflecting different allocations of the compliance-truthfulness trade-off. These findings challenge the assumption that reasoning universally improves reliability: reasoning models trade honest constraint violations for detection-resistant distortions.

Chain-of-Thought (CoT) reasoning has significantly advanced state-of-the-art AI capabilities. However, recent studies have shown that CoT reasoning is not always faithful when models face an explicit bias in their prompts, i.e., the CoT can give an incorrect picture of how models arrive at conclusions. We go further and show that unfaithful CoT can also occur on realistic prompts with no artificial bias. We find that when separately presented with the questions "Is X bigger than Y?" and "Is Y bigger than X?", models sometimes produce superficially coherent arguments to justify systematically answering Yes to both questions or No to both questions, despite such responses being logically contradictory. We show preliminary evidence that this is due to models' implicit biases towards Yes or No, thus labeling this unfaithfulness as Implicit Post-Hoc Rationalization. Our results reveal that several production models exhibit surprisingly high rates of post-hoc rationalization in our settings: GPT-4o-mini (13%) and Haiku 3.5 (7%). While frontier models are more faithful, especially thinking ones, none are entirely faithful: Gemini 2.5 Flash (2.17%), ChatGPT-4o (0.49%), DeepSeek R1 (0.37%), Gemini 2.5 Pro (0.14%), and Sonnet 3.7 with thinking (0.04%). We also investigate Unfaithful Illogical Shortcuts, where models use subtly illogical reasoning to try to make a speculative answer to hard maths problems seem rigorously proven. Our findings raise challenges for strategies for detecting undesired behavior in LLMs via the chain of thought.

In this work, we use large language models (LLMs) to augment and accelerate research on the P versus NP problem, one of the most important open problems in theoretical computer science and mathematics. Specifically, we propose Socratic reasoning, a general framework that promotes in-depth thinking with LLMs for complex problem-solving. Socratic reasoning encourages LLMs to recursively discover, solve, and integrate problems while facilitating self-evaluation and refinement. Our pilot study on the P vs. NP problem shows that GPT-4 successfully produces a proof schema and engages in rigorous reasoning throughout 97 dialogue turns, concluding "P neq NP", which is in alignment with (Xu and Zhou, 2023). The investigation uncovers novel insights within the extensive solution space of LLMs, shedding light on LLM for Science.

We introduce a benchmark to evaluate the capability of AI to solve problems in theoretical physics, focusing on high-energy theory and cosmology. The first iteration of our benchmark consists of 57 problems of varying difficulty, from undergraduate to research level. These problems are novel in the sense that they do not come from public problem collections. We evaluate our data set on various open and closed language models, including o3-mini, o1, DeepSeek-R1, GPT-4o and versions of Llama and Qwen. While we find impressive progress in model performance with the most recent models, our research-level difficulty problems are mostly unsolved. We address challenges of auto-verifiability and grading, and discuss common failure modes. While currently state-of-the art models are still of limited use for researchers, our results show that AI assisted theoretical physics research may become possible in the near future. We discuss the main obstacles towards this goal and possible strategies to overcome them. The public problems and solutions, results for various models, and updates to the data set and score distribution, are available on the website of the dataset tpbench.org.

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.

Large language models (LLMs) have recently shown strong performance on mathematical benchmarks. At the same time, they are prone to hallucination and sycophancy, often providing convincing but flawed proofs for incorrect mathematical statements provided by users. This significantly limits the applicability of LLMs in theorem proving, as verification of these flawed proofs must be done manually by expert mathematicians. However, existing benchmarks that measure sycophancy in mathematics are limited: they focus solely on final-answer problems, rely on very simple and often contaminated datasets, and construct benchmark samples using synthetic modifications that create ill-posed questions rather than well-posed questions that are demonstrably false. To address these issues, we introduce BrokenMath, the first benchmark for evaluating sycophantic behavior in LLMs within the context of natural language theorem proving. BrokenMath is built from advanced 2025 competition problems, which are perturbed with an LLM to produce false statements and subsequently refined through expert review. Using an LLM-as-a-judge framework, we evaluate state-of-the-art LLMs and agentic systems and find that sycophancy is widespread, with the best model, GPT-5, producing sycophantic answers 29% of the time. We further investigate several mitigation strategies, including test-time interventions and supervised fine-tuning on curated sycophantic examples. These approaches substantially reduce, but do not eliminate, sycophantic behavior.

Large language models (LLMs) have witnessed rapid advancements, demonstrating remarkable capabilities. However, a notable vulnerability persists: LLMs often uncritically accept flawed or contradictory premises, leading to inefficient reasoning and unreliable outputs. This emphasizes the significance of possessing the Premise Critique Ability for LLMs, defined as the capacity to proactively identify and articulate errors in input premises. Most existing studies assess LLMs' reasoning ability in ideal settings, largely ignoring their vulnerabilities when faced with flawed premises. Thus, we introduce the Premise Critique Bench (PCBench), designed by incorporating four error types across three difficulty levels, paired with multi-faceted evaluation metrics. We conducted systematic evaluations of 15 representative LLMs. Our findings reveal: (1) Most models rely heavily on explicit prompts to detect errors, with limited autonomous critique; (2) Premise critique ability depends on question difficulty and error type, with direct contradictions being easier to detect than complex or procedural errors; (3) Reasoning ability does not consistently correlate with the premise critique ability; (4) Flawed premises trigger overthinking in reasoning models, markedly lengthening responses due to repeated attempts at resolving conflicts. These insights underscore the urgent need to enhance LLMs' proactive evaluation of input validity, positioning premise critique as a foundational capability for developing reliable, human-centric systems. The code is available at https://github.com/MLGroupJLU/Premise_Critique.

Recent advances in large language models (LLMs) have opened new avenues for accelerating scientific research. While models are increasingly capable of assisting with routine tasks, their ability to contribute to novel, expert-level mathematical discovery is less understood. We present a collection of case studies demonstrating how researchers have successfully collaborated with advanced AI models, specifically Google's Gemini-based models (in particular Gemini Deep Think and its advanced variants), to solve open problems, refute conjectures, and generate new proofs across diverse areas in theoretical computer science, as well as other areas such as economics, optimization, and physics. Based on these experiences, we extract common techniques for effective human-AI collaboration in theoretical research, such as iterative refinement, problem decomposition, and cross-disciplinary knowledge transfer. While the majority of our results stem from this interactive, conversational methodology, we also highlight specific instances that push beyond standard chat interfaces. These include deploying the model as a rigorous adversarial reviewer to detect subtle flaws in existing proofs, and embedding it within a "neuro-symbolic" loop that autonomously writes and executes code to verify complex derivations. Together, these examples highlight the potential of AI not just as a tool for automation, but as a versatile, genuine partner in the creative process of scientific discovery.

Recent advancements in language models have led to significant improvements in mathematical reasoning across various benchmarks. However, most of these benchmarks rely on automatic evaluation methods that only compare final answers using heuristics, without verifying the underlying reasoning steps. This limitation results in false positive solutions, where models may produce correct final answers but with flawed deduction paths. In this paper, we systematically examine the prevalence of false positive solutions in mathematical problem solving for language models. We analyze the characteristics and extent of this issue across different open-source models, datasets of varying difficulty levels, and decoding strategies. Specifically, we explore how false positives influence the inference time scaling behavior of language models. Our experimental results reveal that: (1) false positive solutions persist across different models, datasets, and decoding methods, (2) sampling-based inference time scaling methods do not alleviate the problem, and (3) the pass@N evaluation metric is more susceptible to false positives, suggesting a significantly lower scaling ceiling than what automatic evaluations indicate. Additionally, we analyze specific instances of false positives and discuss potential limitations in self-improvement techniques and synthetic data generation under such conditions. Our data and code are publicly available at https://github.com/Wloner0809/False-Positives-in-Math.

When AI systems explain their reasoning step-by-step, practitioners often assume these explanations reveal what actually influenced the AI's answer. We tested this assumption by embedding hints into questions and measuring whether models mentioned them. In a study of over 9,000 test cases across 11 leading AI models, we found a troubling pattern: models almost never mention hints spontaneously, yet when asked directly, they admit noticing them. This suggests models see influential information but choose not to report it. Telling models they are being watched does not help. Forcing models to report hints works, but causes them to report hints even when none exist and reduces their accuracy. We also found that hints appealing to user preferences are especially dangerous-models follow them most often while reporting them least. These findings suggest that simply watching AI reasoning is not enough to catch hidden influences.

Large Language Models have demonstrated remarkable capabilities across diverse tasks, yet they frequently generate hallucinations outputs that are fluent but factually incorrect or unsupported. We propose Counterfactual Probing, a novel approach for detecting and mitigating hallucinations in LLM outputs. Our method dynamically generates counterfactual statements that appear plausible but contain subtle factual errors, then evaluates the model's sensitivity to these perturbations. We hypothesize that genuine knowledge exhibits robustness to counterfactual variations, while hallucinated content shows inconsistent confidence patterns when confronted with plausible alternatives. Our comprehensive evaluation on TruthfulQA, factual statement datasets, and curated hallucination examples demonstrates that counterfactual probing achieves superior detection performance compared to baseline methods, while our adaptive mitigation strategies reduce hallucination scores by an average of 24.5%. The approach requires no model retraining and can be integrated into existing LLM pipelines as a realtime verification mechanism.

Verifiability is a core content policy of Wikipedia: claims that are likely to be challenged need to be backed by citations. There are millions of articles available online and thousands of new articles are released each month. For this reason, finding relevant sources is a difficult task: many claims do not have any references that support them. Furthermore, even existing citations might not support a given claim or become obsolete once the original source is updated or deleted. Hence, maintaining and improving the quality of Wikipedia references is an important challenge and there is a pressing need for better tools to assist humans in this effort. Here, we show that the process of improving references can be tackled with the help of artificial intelligence (AI). We develop a neural network based system, called Side, to identify Wikipedia citations that are unlikely to support their claims, and subsequently recommend better ones from the web. We train this model on existing Wikipedia references, therefore learning from the contributions and combined wisdom of thousands of Wikipedia editors. Using crowd-sourcing, we observe that for the top 10% most likely citations to be tagged as unverifiable by our system, humans prefer our system's suggested alternatives compared to the originally cited reference 70% of the time. To validate the applicability of our system, we built a demo to engage with the English-speaking Wikipedia community and find that Side's first citation recommendation collects over 60% more preferences than existing Wikipedia citations for the same top 10% most likely unverifiable claims according to Side. Our results indicate that an AI-based system could be used, in tandem with humans, to improve the verifiability of Wikipedia. More generally, we hope that our work can be used to assist fact checking efforts and increase the general trustworthiness of information online.

In this paper we introduce a new publicly available dataset for verification against textual sources, FEVER: Fact Extraction and VERification. It consists of 185,445 claims generated by altering sentences extracted from Wikipedia and subsequently verified without knowledge of the sentence they were derived from. The claims are classified as Supported, Refuted or NotEnoughInfo by annotators achieving 0.6841 in Fleiss kappa. For the first two classes, the annotators also recorded the sentence(s) forming the necessary evidence for their judgment. To characterize the challenge of the dataset presented, we develop a pipeline approach and compare it to suitably designed oracles. The best accuracy we achieve on labeling a claim accompanied by the correct evidence is 31.87%, while if we ignore the evidence we achieve 50.91%. Thus we believe that FEVER is a challenging testbed that will help stimulate progress on claim verification against textual sources.

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.

Large language models (LLMs) have shown significant potential in scientific disciplines such as biomedicine, particularly in hypothesis generation, where they can analyze vast literature, identify patterns, and suggest research directions. However, a key challenge lies in evaluating the truthfulness of generated hypotheses, as verifying their accuracy often requires substantial time and resources. Additionally, the hallucination problem in LLMs can lead to the generation of hypotheses that appear plausible but are ultimately incorrect, undermining their reliability. To facilitate the systematic study of these challenges, we introduce TruthHypo, a benchmark for assessing the capabilities of LLMs in generating truthful biomedical hypotheses, and KnowHD, a knowledge-based hallucination detector to evaluate how well hypotheses are grounded in existing knowledge. Our results show that LLMs struggle to generate truthful hypotheses. By analyzing hallucinations in reasoning steps, we demonstrate that the groundedness scores provided by KnowHD serve as an effective metric for filtering truthful hypotheses from the diverse outputs of LLMs. Human evaluations further validate the utility of KnowHD in identifying truthful hypotheses and accelerating scientific discovery. Our data and source code are available at https://github.com/Teddy-XiongGZ/TruthHypo.

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

Verifying complex political claims is a challenging task, especially when politicians use various tactics to subtly misrepresent the facts. Automatic fact-checking systems fall short here, and their predictions like "half-true" are not very useful in isolation, since we have no idea which parts of the claim are true and which are not. In this work, we focus on decomposing a complex claim into a comprehensive set of yes-no subquestions whose answers influence the veracity of the claim. We present ClaimDecomp, a dataset of decompositions for over 1000 claims. Given a claim and its verification paragraph written by fact-checkers, our trained annotators write subquestions covering both explicit propositions of the original claim and its implicit facets, such as asking about additional political context that changes our view of the claim's veracity. We study whether state-of-the-art models can generate such subquestions, showing that these models generate reasonable questions to ask, but predicting the comprehensive set of subquestions from the original claim without evidence remains challenging. We further show that these subquestions can help identify relevant evidence to fact-check the full claim and derive the veracity through their answers, suggesting that they can be useful pieces of a fact-checking pipeline.

With the exponential growth of research facilitated by modern technology and improved accessibility, scientific discoveries have become increasingly fragmented within and across fields. This makes it challenging to assess the significance, novelty, incremental findings, and equivalent ideas between related works, particularly those from different research communities. Large language models (LLMs) have recently demonstrated strong quantitative and qualitative reasoning abilities, and multi-agent LLM debates have shown promise in handling complex reasoning tasks by exploring diverse perspectives and reasoning paths. Inspired by this, we introduce Tree-of-Debate (ToD), a framework which converts scientific papers into LLM personas that debate their respective novelties. To emphasize structured, critical reasoning rather than focusing solely on outcomes, ToD dynamically constructs a debate tree, enabling fine-grained analysis of independent novelty arguments within scholarly articles. Through experiments on scientific literature across various domains, evaluated by expert researchers, we demonstrate that ToD generates informative arguments, effectively contrasts papers, and supports researchers in their literature review.

Although counterfactual reasoning is a fundamental aspect of intelligence, the lack of large-scale counterfactual open-domain question-answering (QA) benchmarks makes it difficult to evaluate and improve models on this ability. To address this void, we introduce the first such dataset, named IfQA, where each question is based on a counterfactual presupposition via an "if" clause. For example, if Los Angeles was on the east coast of the U.S., what would be the time difference between Los Angeles and Paris? Such questions require models to go beyond retrieving direct factual knowledge from the Web: they must identify the right information to retrieve and reason about an imagined situation that may even go against the facts built into their parameters. The IfQA dataset contains over 3,800 questions that were annotated annotated by crowdworkers on relevant Wikipedia passages. Empirical analysis reveals that the IfQA dataset is highly challenging for existing open-domain QA methods, including supervised retrieve-then-read pipeline methods (EM score 36.2), as well as recent few-shot approaches such as chain-of-thought prompting with GPT-3 (EM score 27.4). The unique challenges posed by the IfQA benchmark will push open-domain QA research on both retrieval and counterfactual reasoning fronts.

This position paper provides a critical but constructive discussion of current practices in benchmarking and evaluative practices in the field of formal reasoning and automated theorem proving. We take the position that open code, open data, and benchmarks that are complete and error-free will accelerate progress in this field. We identify practices that create barriers to contributing to this field and suggest ways to remove them. We also discuss some of the practices that might produce misleading evaluative information. We aim to create discussions that bring together people from various groups contributing to automated theorem proving, autoformalization, and informal reasoning.

Despite alarm over the reliance of machine learning systems on so-called spurious patterns, the term lacks coherent meaning in standard statistical frameworks. However, the language of causality offers clarity: spurious associations are due to confounding (e.g., a common cause), but not direct or indirect causal effects. In this paper, we focus on natural language processing, introducing methods and resources for training models less sensitive to spurious patterns. Given documents and their initial labels, we task humans with revising each document so that it (i) accords with a counterfactual target label; (ii) retains internal coherence; and (iii) avoids unnecessary changes. Interestingly, on sentiment analysis and natural language inference tasks, classifiers trained on original data fail on their counterfactually-revised counterparts and vice versa. Classifiers trained on combined datasets perform remarkably well, just shy of those specialized to either domain. While classifiers trained on either original or manipulated data alone are sensitive to spurious features (e.g., mentions of genre), models trained on the combined data are less sensitive to this signal. Both datasets are publicly available.

When building artificial intelligence systems that can reason and answer questions about visual data, we need diagnostic tests to analyze our progress and discover shortcomings. Existing benchmarks for visual question answering can help, but have strong biases that models can exploit to correctly answer questions without reasoning. They also conflate multiple sources of error, making it hard to pinpoint model weaknesses. We present a diagnostic dataset that tests a range of visual reasoning abilities. It contains minimal biases and has detailed annotations describing the kind of reasoning each question requires. We use this dataset to analyze a variety of modern visual reasoning systems, providing novel insights into their abilities and limitations.

Shojaee et al. (2025) report that Large Reasoning Models (LRMs) exhibit "accuracy collapse" on planning puzzles beyond certain complexity thresholds. We demonstrate that their findings primarily reflect experimental design limitations rather than fundamental reasoning failures. Our analysis reveals three critical issues: (1) Tower of Hanoi experiments systematically exceed model output token limits at reported failure points, with models explicitly acknowledging these constraints in their outputs; (2) The authors' automated evaluation framework fails to distinguish between reasoning failures and practical constraints, leading to misclassification of model capabilities; (3) Most concerningly, their River Crossing benchmarks include mathematically impossible instances for N > 5 due to insufficient boat capacity, yet models are scored as failures for not solving these unsolvable problems. When we control for these experimental artifacts, by requesting generating functions instead of exhaustive move lists, preliminary experiments across multiple models indicate high accuracy on Tower of Hanoi instances previously reported as complete failures. These findings highlight the importance of careful experimental design when evaluating AI reasoning capabilities.

Large Language Models (LLMs) often exhibit sycophancy, distorting responses to align with user beliefs, notably by readily agreeing with user counterarguments. Paradoxically, LLMs are increasingly adopted as successful evaluative agents for tasks such as grading and adjudicating claims. This research investigates that tension: why do LLMs show sycophancy when challenged in subsequent conversational turns, yet perform well when evaluating conflicting arguments presented simultaneously? We empirically tested these contrasting scenarios by varying key interaction patterns. We find that state-of-the-art models: (1) are more likely to endorse a user's counterargument when framed as a follow-up from a user, rather than when both responses are presented simultaneously for evaluation; (2) show increased susceptibility to persuasion when the user's rebuttal includes detailed reasoning, even when the conclusion of the reasoning is incorrect; and (3) are more readily swayed by casually phrased feedback than by formal critiques, even when the casual input lacks justification. Our results highlight the risk of relying on LLMs for judgment tasks without accounting for conversational framing.

In this paper, we sharpen results obtained by the author in 2023. The new results reduce the Mathieu Conjecture on SU(N) (formulated for all compact connected Lie groups by O. Mathieu in 1997) to a conjecture involving only functions on R^ntimes (S^1)^m with n,m non-negative integers instead of involving functions on R^ntimes (S^1setminus{1})^m. The proofs rely on a more recent work of the author (2024) and a specific KAK decomposition. Finally, with these results we can also improve the results on the groups Sp(N) and G_2 in the latter paper, since they relied on the construction introduced in the 2023 paper.

Fair decisions require ignoring irrelevant, potentially biasing, information. To achieve this, decision-makers need to approximate what decision they would have made had they not known certain facts, such as the gender or race of a job candidate. This counterfactual self-simulation is notoriously hard for humans, leading to biased judgments even by well-meaning actors. Here we show that large language models (LLMs) suffer from similar limitations in their ability to approximate what decisions they would make under counterfactual knowledge in offsetting gender and race biases and overcoming sycophancy. We show that prompting models to ignore or pretend not to know biasing information fails to offset these biases and occasionally backfires. However, unlike humans, LLMs can be given access to a ground-truth model of their own counterfactual cognition -- their own API. We show that this access to the responses of a blinded replica enables fairer decisions, while providing greater transparency to distinguish implicit from intentionally biased behavior.

As neural theorem provers become increasingly agentic, the ability to interpret and act on compiler feedback is critical. However, existing Lean datasets consist almost exclusively of correct proofs, offering little supervision for understanding and repairing failures. We study Lean proof repair as a supervised learning problem: given an erroneous proof and compiler feedback, predict both a corrected proof and a natural-language diagnosis grounded in the same feedback. We introduce APRIL (Automated Proof Repair in Lean), a dataset of 260,000 supervised tuples pairing systematically generated proof failures with compiler diagnostics and aligned repair and explanation targets. Training language models on APRIL substantially improves repair accuracy and feedback-conditioned reasoning; in our single-shot repair evaluation setting, a finetuned 4B-parameter model outperforms the strongest open-source baseline. We view diagnostic-conditioned supervision as a complementary training signal for feedback-using provers. Our dataset is available at https://huggingface.co/datasets/uw-math-ai/APRIL{this link}.

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

Proof-oriented programs mix computational content with proofs of program correctness. However, the human effort involved in programming and proving is still substantial, despite the use of Satisfiability Modulo Theories (SMT) solvers to automate proofs in languages such as F*. Seeking to spur research on using AI to automate the construction of proof-oriented programs, we curate a dataset of 600K lines of open-source F* programs and proofs, including software used in production systems ranging from Windows and Linux, to Python and Firefox. Our dataset includes around 32K top-level F* definitions, each representing a type-directed program and proof synthesis problem -- producing a definition given a formal specification expressed as an F* type. We provide a program-fragment checker that queries F* to check the correctness of candidate solutions. We believe this is the largest corpus of SMT-assisted program proofs coupled with a reproducible program-fragment checker. Grounded in this dataset, we investigate the use of AI to synthesize programs and their proofs in F*, with promising results. Our main finding in that the performance of fine-tuned smaller language models (such as Phi-2 or StarCoder) compare favorably with large language models (such as GPT-4), at a much lower computational cost. We also identify various type-based retrieval augmentation techniques and find that they boost performance significantly. With detailed error analysis and case studies, we identify potential strengths and weaknesses of models and techniques and suggest directions for future improvements.

Large Language Models (LLMs) have shown remarkable abilities in structured reasoning and symbolic tasks, with coding emerging as a particular area of strength. This success has sparked growing interest in applying LLMs to mathematics, both in informal problem-solving and formal theorem proving. However, progress in formal mathematics has proven to be significantly more difficult, despite surface-level similarities between programming and proof construction. This discrepancy raises important questions about how LLMs ``reason'', how they are supervised, and whether they internally track a notion of computational or deductive state. In this article, we address the state-of-the-art of the discipline, focusing on recent models and benchmarks, and explore three central issues at the intersection of machine learning and mathematical cognition: (i) the trade-offs between formal and informal mathematics as training domains; (ii) the deeper reasons why proof generation remains more brittle than code synthesis; (iii) and the question of whether LLMs represent, or merely mimic, a notion of evolving logical state. Our goal is not to draw hard boundaries, but to identify where the current limits lie, and how they might be extended.

Although much research has focused on AI explanations to support decisions in complex information-seeking tasks such as fact-checking, the role of evidence is surprisingly under-researched. In our study, we systematically varied explanation type, AI prediction certainty, and correctness of AI system advice for non-expert participants, who evaluated the veracity of claims and AI system predictions. Participants were provided the option of easily inspecting the underlying evidence. We found that participants consistently relied on evidence to validate AI claims across all experimental conditions. When participants were presented with natural language explanations, evidence was used less frequently although they relied on it when these explanations seemed insufficient or flawed. Qualitative data suggests that participants attempted to infer evidence source reliability, despite source identities being deliberately omitted. Our results demonstrate that evidence is a key ingredient in how people evaluate the reliability of information presented by an AI system and, in combination with natural language explanations, offers valuable support for decision-making. Further research is urgently needed to understand how evidence ought to be presented and how people engage with it in practice.

Critiques are important for enhancing the performance of Large Language Models (LLMs), enabling both self-improvement and constructive feedback for others by identifying flaws and suggesting improvements. However, evaluating the critique capabilities of LLMs presents a significant challenge due to the open-ended nature of the task. In this work, we introduce a new benchmark designed to assess the critique capabilities of LLMs. Unlike existing benchmarks, which typically function in an open-loop fashion, our approach employs a closed-loop methodology that evaluates the quality of corrections generated from critiques. Moreover, the benchmark incorporates features such as self-critique, cross-critique, and iterative critique, which are crucial for distinguishing the abilities of advanced reasoning models from more classical ones. We implement this benchmark using eight challenging reasoning tasks. We have several interesting findings. First, despite demonstrating comparable performance in direct chain-of-thought generation, classical LLMs significantly lag behind the advanced reasoning-based model o1-mini across all critique scenarios. Second, in self-critique and iterative critique settings, classical LLMs may even underperform relative to their baseline capabilities. We hope that this benchmark will serve as a valuable resource to guide future advancements. The code and data are available at https://github.com/tangzhy/RealCritic.

Large language models (LLMs) are trained to imitate humans to explain human decisions. However, do LLMs explain themselves? Can they help humans build mental models of how LLMs process different inputs? To answer these questions, we propose to evaluate counterfactual simulatability of natural language explanations: whether an explanation can enable humans to precisely infer the model's outputs on diverse counterfactuals of the explained input. For example, if a model answers "yes" to the input question "Can eagles fly?" with the explanation "all birds can fly", then humans would infer from the explanation that it would also answer "yes" to the counterfactual input "Can penguins fly?". If the explanation is precise, then the model's answer should match humans' expectations. We implemented two metrics based on counterfactual simulatability: precision and generality. We generated diverse counterfactuals automatically using LLMs. We then used these metrics to evaluate state-of-the-art LLMs (e.g., GPT-4) on two tasks: multi-hop factual reasoning and reward modeling. We found that LLM's explanations have low precision and that precision does not correlate with plausibility. Therefore, naively optimizing human approvals (e.g., RLHF) may not be a sufficient solution.

Can AI make progress on important, unsolved mathematical problems? Large language models are now capable of sophisticated mathematical and scientific reasoning, but whether they can perform novel research is still widely debated and underexplored. We introduce HorizonMath, a benchmark of over 100 predominantly unsolved problems spanning 8 domains in computational and applied mathematics, paired with an open-source evaluation framework for automated verification. Our benchmark targets a class of problems where discovery is hard, requiring meaningful mathematical insight, but verification is computationally efficient and simple. Because these solutions are unknown, HorizonMath is immune to data contamination, and most state-of-the-art models score near 0%. Existing research-level benchmarks instead rely on formal proof verification or manual review, both of which are expensive to scale. Using this platform, we find two problems for which GPT 5.4 Pro proposes solutions that improve on the best-known published results, representing potential novel contributions (pending expert review). We release HorizonMath as an open challenge and a growing community resource, where correct solutions to problems in the unsolved problem classes could constitute novel results in the mathematical literature.

One well motivated explanation method for classifiers leverages counterfactuals which are hypothetical events identical to real observations in all aspects except for one categorical feature. Constructing such counterfactual poses specific challenges for texts, however, as some attribute values may not necessarily align with plausible real-world events. In this paper we propose a simple method for generating counterfactuals by intervening in the space of text representations which bypasses this limitation. We argue that our interventions are minimally disruptive and that they are theoretically sound as they align with counterfactuals as defined in Pearl's causal inference framework. To validate our method, we first conduct experiments on a synthetic dataset of counterfactuals, allowing for a direct comparison between classifier predictions based on ground truth counterfactuals (obtained through explicit text interventions) and our counterfactuals, derived through interventions in the representation space. Second, we study a real world scenario where our counterfactuals can be leveraged both for explaining a classifier and for bias mitigation.

Large language models have demonstrated remarkable proficiency in long and complex reasoning tasks. However, they frequently exhibit a problematic reliance on familiar reasoning patterns, a phenomenon we term reasoning rigidity. Despite explicit instructions from users, these models often override clearly stated conditions and default to habitual reasoning trajectories, leading to incorrect conclusions. This behavior presents significant challenges, particularly in domains such as mathematics and logic puzzle, where precise adherence to specified constraints is critical. To systematically investigate reasoning rigidity, a behavior largely unexplored in prior work, we introduce a expert-curated diagnostic set, . Our dataset includes specially modified variants of existing mathematical benchmarks, namely AIME and MATH500, as well as well-known puzzles deliberately redesigned to require deviation from familiar reasoning strategies. Using this dataset, we identify recurring contamination patterns that occur when models default to ingrained reasoning. Specifically, we categorize this contamination into three distinctive modes: (i) Interpretation Overload, (ii) Input Distrust, and (iii) Partial Instruction Attention, each causing models to ignore or distort provided instructions. We publicly release our diagnostic set to facilitate future research on mitigating reasoning rigidity in language models.

Machine learning systems are known to be sensitive to spurious correlations between biased features of the inputs (e.g., background, texture, and secondary objects) and the corresponding labels. These features and their correlations with the labels are known as "spurious" because they tend to change with shifts in real-world data distributions, which can negatively impact the model's generalization and robustness. In this survey, we provide a comprehensive review of this issue, along with a taxonomy of current state-of-the-art methods for addressing spurious correlations in machine learning models. Additionally, we summarize existing datasets, benchmarks, and metrics to aid future research. The paper concludes with a discussion of the recent advancements and future research challenges in this field, aiming to provide valuable insights for researchers in the related domains.

The formalization of existing mathematical proofs is a notoriously difficult process. Despite decades of research on automation and proof assistants, writing formal proofs remains arduous and only accessible to a few experts. While previous studies to automate formalization focused on powerful search algorithms, no attempts were made to take advantage of available informal proofs. In this work, we introduce Draft, Sketch, and Prove (DSP), a method that maps informal proofs to formal proof sketches, and uses the sketches to guide an automated prover by directing its search to easier sub-problems. We investigate two relevant setups where informal proofs are either written by humans or generated by a language model. Our experiments and ablation studies show that large language models are able to produce well-structured formal sketches that follow the same reasoning steps as the informal proofs. Guiding an automated prover with these sketches enhances its performance from 20.9% to 39.3% on a collection of mathematical competition problems.

Recent large language models (LLMs) have shown indications of mathematical reasoning ability. However it has not been clear how they would fare on more challenging competition-level problems. And while self-generated verbalizations of intermediate reasoning steps (i.e., chain-of-thought prompting) have been shown to be helpful, whether LLMs can make use of helpful side information such as problem-specific hints has not been investigated before. In this paper, we propose a challenging benchmark dataset for enabling such analyses. The Concept and Hint-Annotated Math Problems (CHAMP) consists of high school math competition problems, annotated with concepts, or general math facts, and hints, or problem-specific tricks. These annotations allow us to explore the effects of additional information, such as relevant hints, misleading concepts, or related problems. This benchmark is difficult, with the best model only scoring 58.1% in standard settings. With concepts and hints, performance sometimes improves, indicating that some models can make use of such side information. We further annotate model-generated solutions for their correctness. Using this corpus, we find that models often arrive at the correct final answer through wrong reasoning steps. In addition, we test whether models are able to verify these solutions, and find that most models struggle. The dataset and code are available on the project website.

Deductive reasoning plays a pivotal role in the formulation of sound and cohesive arguments. It allows individuals to draw conclusions that logically follow, given the truth value of the information provided. Recent progress in the domain of large language models (LLMs) has showcased their capability in executing deductive reasoning tasks. Nonetheless, a significant portion of research primarily assesses the accuracy of LLMs in solving such tasks, often overlooking a deeper analysis of their reasoning behavior. In this study, we draw upon principles from cognitive psychology to examine inferential strategies employed by LLMs, through a detailed evaluation of their responses to propositional logic problems. Our findings indicate that LLMs display reasoning patterns akin to those observed in humans, including strategies like supposition following or chain construction. Moreover, our research demonstrates that the architecture and scale of the model significantly affect its preferred method of reasoning, with more advanced models tending to adopt strategies more frequently than less sophisticated ones. Importantly, we assert that a model's accuracy, that is the correctness of its final conclusion, does not necessarily reflect the validity of its reasoning process. This distinction underscores the necessity for more nuanced evaluation procedures in the field.

A patent must be deemed novel and non-obvious in order to be granted by the US Patent Office (USPTO). If it is not, a US patent examiner will cite the prior work, or prior art, that invalidates the novelty and issue a non-final rejection. Predicting what claims of the invention should change given the prior art is an essential and crucial step in securing invention rights, yet has not been studied before as a learnable task. In this work we introduce the PatentEdits dataset, which contains 105K examples of successful revisions that overcome objections to novelty. We design algorithms to label edits sentence by sentence, then establish how well these edits can be predicted with large language models (LLMs). We demonstrate that evaluating textual entailment between cited references and draft sentences is especially effective in predicting which inventive claims remained unchanged or are novel in relation to prior art.

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

What is reasoning? This question has driven centuries of philosophical inquiry, from Aristotle's syllogisms to modern computational complexity theory. In the age of large language models achieving superhuman performance on benchmarks like GSM8K (95\% accuracy) and HumanEval (90\% pass@1), we must ask: have these systems learned to reason, or have they learned to pattern-match over reasoning traces? This paper argues for a specific answer: reasoning is iterative operator application in state spaces, converging to fixed points. This definition is not merely philosophical -- it has concrete architectural implications that explain both the failures of current systems and the path to genuine reasoning capabilities. Our investigation begins with a puzzle (OpenXOR), progresses through theory (OpenOperator), and culminates in a working solution (OpenLM) that achieves 76\% accuracy where state-of-the-art LLMs achieve 0\%. This is not about criticizing existing systems, but about understanding what reasoning requires and building architectures that provide it.

Retrieval-enhanced methods have become a primary approach in fact verification (FV); it requires reasoning over multiple retrieved pieces of evidence to verify the integrity of a claim. To retrieve evidence, existing work often employs off-the-shelf retrieval models whose design is based on the probability ranking principle. We argue that, rather than relevance, for FV we need to focus on the utility that a claim verifier derives from the retrieved evidence. We introduce the feedback-based evidence retriever(FER) that optimizes the evidence retrieval process by incorporating feedback from the claim verifier. As a feedback signal we use the divergence in utility between how effectively the verifier utilizes the retrieved evidence and the ground-truth evidence to produce the final claim label. Empirical studies demonstrate the superiority of FER over prevailing baselines.

Large Reasoning Models (LRMs) have significantly enhanced their capabilities in complex problem-solving by introducing a thinking draft that enables multi-path Chain-of-Thought explorations before producing final answers. Ensuring the faithfulness of these intermediate reasoning processes is crucial for reliable monitoring, interpretation, and effective control. In this paper, we propose a systematic counterfactual intervention framework to rigorously evaluate thinking draft faithfulness. Our approach focuses on two complementary dimensions: (1) Intra-Draft Faithfulness, which assesses whether individual reasoning steps causally influence subsequent steps and the final draft conclusion through counterfactual step insertions; and (2) Draft-to-Answer Faithfulness, which evaluates whether final answers are logically consistent with and dependent on the thinking draft, by perturbing the draft's concluding logic. We conduct extensive experiments across six state-of-the-art LRMs. Our findings show that current LRMs demonstrate selective faithfulness to intermediate reasoning steps and frequently fail to faithfully align with the draft conclusions. These results underscore the need for more faithful and interpretable reasoning in advanced LRMs.

Fact-checking is the task of verifying the veracity of claims by assessing their assertions against credible evidence. The vast majority of fact-checking studies focus exclusively on political claims. Very little research explores fact-checking for other topics, specifically subject matters for which expertise is required. We present the first study of explainable fact-checking for claims which require specific expertise. For our case study we choose the setting of public health. To support this case study we construct a new dataset PUBHEALTH of 11.8K claims accompanied by journalist crafted, gold standard explanations (i.e., judgments) to support the fact-check labels for claims. We explore two tasks: veracity prediction and explanation generation. We also define and evaluate, with humans and computationally, three coherence properties of explanation quality. Our results indicate that, by training on in-domain data, gains can be made in explainable, automated fact-checking for claims which require specific expertise.

Prior work has shown that presupposition in generated questions can introduce unverified assumptions, leading to inconsistencies in claim verification. Additionally, prompt sensitivity remains a significant challenge for large language models (LLMs), resulting in performance variance as high as 3-6%. While recent advancements have reduced this gap, our study demonstrates that prompt sensitivity remains a persistent issue. To address this, we propose a structured and robust claim verification framework that reasons through presupposition-free, decomposed questions. Extensive experiments across multiple prompts, datasets, and LLMs reveal that even state-of-the-art models remain susceptible to prompt variance and presupposition. Our method consistently mitigates these issues, achieving up to a 2-5% 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.

While there are numerous benchmarks comparing the performance of modern language models (LMs), end-task evaluations often conflate notions of *factual accuracy* ("truth") and *reasoning ability* ("rationality", or "honesty" in the sense of correctly reporting implications of beliefs). Our goal is a dataset that clearly distinguishes these two notions. Our approach is to leverage and extend a collection of human-annotated *entailment trees*, engineered to express both good and bad chains of reasoning, and using a mixture of true and false facts, in particular including counterfactual examples, to avoid belief bias (also known as the "content effect"). The resulting dataset, called BaRDa, contains 3000 entailments (1787 valid, 1213 invalid), using 6681 true and 2319 false statements. Testing on four GPT-series models, GPT3(curie)/GPT3(davinici)/3.5/4, we find factual accuracy (truth) scores of 74.1/80.6/82.6/87.1 and reasoning accuracy scores of 63.1/78.0/71.8/79.2. This shows the clear progression of models towards improved factual accuracy and entailment reasoning, and the dataset provides a new benchmark that more cleanly separates and quantifies these two notions.

State-of-the-art language models (LMs) are notoriously susceptible to generating hallucinated information. Such inaccurate outputs not only undermine the reliability of these models but also limit their use and raise serious concerns about misinformation and propaganda. In this work, we focus on hallucinated book and article references and present them as the "model organism" of language model hallucination research, due to their frequent and easy-to-discern nature. We posit that if a language model cites a particular reference in its output, then it should ideally possess sufficient information about its authors and content, among other relevant details. Using this basic insight, we illustrate that one can identify hallucinated references without ever consulting any external resources, by asking a set of direct or indirect queries to the language model about the references. These queries can be considered as "consistency checks." Our findings highlight that while LMs, including GPT-4, often produce inconsistent author lists for hallucinated references, they also often accurately recall the authors of real references. In this sense, the LM can be said to "know" when it is hallucinating references. Furthermore, these findings show how hallucinated references can be dissected to shed light on their nature. Replication code and results can be found at https://github.com/microsoft/hallucinated-references.

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.

Machine learning plays a role in many deployed decision systems, often in ways that are difficult or impossible to understand by human stakeholders. Explaining, in a human-understandable way, the relationship between the input and output of machine learning models is essential to the development of trustworthy machine learning based systems. A burgeoning body of research seeks to define the goals and methods of explainability in machine learning. In this paper, we seek to review and categorize research on counterfactual explanations, a specific class of explanation that provides a link between what could have happened had input to a model been changed in a particular way. Modern approaches to counterfactual explainability in machine learning draw connections to the established legal doctrine in many countries, making them appealing to fielded systems in high-impact areas such as finance and healthcare. Thus, we design a rubric with desirable properties of counterfactual explanation algorithms and comprehensively evaluate all currently proposed algorithms against that rubric. Our rubric provides easy comparison and comprehension of the advantages and disadvantages of different approaches and serves as an introduction to major research themes in this field. We also identify gaps and discuss promising research directions in the space of counterfactual explainability.

Mathematical reasoning remains a significant challenge for Large Language Models (LLMs) due to hallucinations. When combined with formal proof assistants like Lean, these hallucinations can be eliminated through rigorous verification, making theorem proving reliable. However, even with formal verification, LLMs still struggle with long proofs and complex mathematical formalizations. While Lean with LLMs offers valuable assistance with retrieving lemmas, generating tactics, or even complete proofs, it lacks a crucial capability: providing a sense of proof progress. This limitation particularly impacts the overall development efficiency in large formalization projects. We introduce LeanProgress, a method that predicts the progress in the proof. Training and evaluating our models made on a large corpus of Lean proofs from Lean Workbook Plus and Mathlib4 and how many steps remain to complete it, we employ data preprocessing and balancing techniques to handle the skewed distribution of proof lengths. Our experiments show that LeanProgress achieves an overall prediction accuracy of 75.1\% in predicting the amount of progress and, hence, the remaining number of steps. When integrated into a best-first search framework using Reprover, our method shows a 3.8\% improvement on Mathlib4 compared to baseline performances of 41.2\%, particularly for longer proofs. These results demonstrate how proof progress prediction can enhance both automated and interactive theorem proving, enabling users to make more informed decisions about proof strategies.

Neural language models (LMs) can be used to evaluate the truth of factual statements in two ways: they can be either queried for statement probabilities, or probed for internal representations of truthfulness. Past work has found that these two procedures sometimes disagree, and that probes tend to be more accurate than LM outputs. This has led some researchers to conclude that LMs "lie" or otherwise encode non-cooperative communicative intents. Is this an accurate description of today's LMs, or can query-probe disagreement arise in other ways? We identify three different classes of disagreement, which we term confabulation, deception, and heterogeneity. In many cases, the superiority of probes is simply attributable to better calibration on uncertain answers rather than a greater fraction of correct, high-confidence answers. In some cases, queries and probes perform better on different subsets of inputs, and accuracy can further be improved by ensembling the two. Code is available at github.com/lingo-mit/lm-truthfulness.

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 survey a current, heated debate in the AI research community on whether large pre-trained language models can be said to "understand" language -- and the physical and social situations language encodes -- in any important sense. We describe arguments that have been made for and against such understanding, and key questions for the broader sciences of intelligence that have arisen in light of these arguments. We contend that a new science of intelligence can be developed that will provide insight into distinct modes of understanding, their strengths and limitations, and the challenge of integrating diverse forms of cognition.

Hypothetical induction is recognized as the main reasoning type when scientists make observations about the world and try to propose hypotheses to explain those observations. Past research on hypothetical induction is under a constrained setting: (1) the observation annotations in the dataset are carefully manually handpicked sentences (resulting in a close-domain setting); and (2) the ground truth hypotheses are mostly commonsense knowledge, making the task less challenging. In this work, we tackle these problems by proposing the first dataset for social science academic hypotheses discovery, with the final goal to create systems that automatically generate valid, novel, and helpful scientific hypotheses, given only a pile of raw web corpus. Unlike previous settings, the new dataset requires (1) using open-domain data (raw web corpus) as observations; and (2) proposing hypotheses even new to humanity. A multi-module framework is developed for the task, including three different feedback mechanisms to boost performance, which exhibits superior performance in terms of both GPT-4 based and expert-based evaluation. To the best of our knowledge, this is the first work showing that LLMs are able to generate novel (''not existing in literature'') and valid (''reflecting reality'') scientific hypotheses.

In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and algorithms become more powerful, an intriguing possibility arises - the interplay between human intuition and computer algorithms can lead to discoveries of novel mathematical concepts that would otherwise remain elusive. To realize this perspective, we have developed a massively parallel computer algorithm that discovers an unprecedented number of continued fraction formulas for fundamental mathematical constants. The sheer number of formulas discovered by the algorithm unveils a novel mathematical structure that we call the conservative matrix field. Such matrix fields (1) unify thousands of existing formulas, (2) generate infinitely many new formulas, and most importantly, (3) lead to unexpected relations between different mathematical constants, including multiple integer values of the Riemann zeta function. Conservative matrix fields also enable new mathematical proofs of irrationality. In particular, we can use them to generalize the celebrated proof by Ap\'ery for the irrationality of zeta(3). Utilizing thousands of personal computers worldwide, our computer-supported research strategy demonstrates the power of experimental mathematics, highlighting the prospects of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science.

The pervasiveness of large language models and generative AI in online media has amplified the need for effective automated fact-checking to assist fact-checkers in tackling the increasing volume and sophistication of misinformation. The complex nature of fact-checking demands that automated fact-checking systems provide explanations that enable fact-checkers to scrutinise their outputs. However, it is unclear how these explanations should align with the decision-making and reasoning processes of fact-checkers to be effectively integrated into their workflows. Through semi-structured interviews with fact-checking professionals, we bridge this gap by: (i) providing an account of how fact-checkers assess evidence, make decisions, and explain their processes; (ii) examining how fact-checkers use automated tools in practice; and (iii) identifying fact-checker explanation requirements for automated fact-checking tools. The findings show unmet explanation needs and identify important criteria for replicable fact-checking explanations that trace the model's reasoning path, reference specific evidence, and highlight uncertainty and information gaps.

Fairness for machine learning predictions is widely required in practice for legal, ethical, and societal reasons. Existing work typically focuses on settings without unobserved confounding, even though unobserved confounding can lead to severe violations of causal fairness and, thus, unfair predictions. In this work, we analyze the sensitivity of causal fairness to unobserved confounding. Our contributions are three-fold. First, we derive bounds for causal fairness metrics under different sources of unobserved confounding. This enables practitioners to examine the sensitivity of their machine learning models to unobserved confounding in fairness-critical applications. Second, we propose a novel neural framework for learning fair predictions, which allows us to offer worst-case guarantees of the extent to which causal fairness can be violated due to unobserved confounding. Third, we demonstrate the effectiveness of our framework in a series of experiments, including a real-world case study about predicting prison sentences. To the best of our knowledge, ours is the first work to study causal fairness under unobserved confounding. To this end, our work is of direct practical value as a refutation strategy to ensure the fairness of predictions in high-stakes applications.

Recent failures such as Google Gemini generating people of color in Nazi-era uniforms illustrate how AI outputs can be factually plausible yet socially harmful. AI models are increasingly evaluated for "fairness," yet existing benchmarks often conflate two fundamentally different dimensions: factual correctness and normative fairness. A model may generate responses that are factually accurate but socially unfair, or conversely, appear fair while distorting factual reality. We argue that identifying the boundary between fact and fair is essential for meaningful fairness evaluation. We introduce Fact-or-Fair, a benchmark with (i) objective queries aligned with descriptive, fact-based judgments, and (ii) subjective queries aligned with normative, fairness-based judgments. Our queries are constructed from 19 statistics and are grounded in cognitive psychology, drawing on representativeness bias, attribution bias, and ingroup-outgroup bias to explain why models often misalign fact and fairness. Experiments across ten frontier models reveal different levels of fact-fair trade-offs. By reframing fairness evaluation, we provide both a new theoretical lens and a practical benchmark to advance the responsible model assessments. Our test suite is publicly available at https://github.com/uclanlp/Fact-or-Fair.

This study evaluates the effect of counterfactual explanations on the interpretation of chest X-rays. We conduct a reader study with two radiologists assessing 240 chest X-ray predictions to rate their confidence that the model's prediction is correct using a 5 point scale. Half of the predictions are false positives. Each prediction is explained twice, once using traditional attribution methods and once with a counterfactual explanation. The overall results indicate that counterfactual explanations allow a radiologist to have more confidence in true positive predictions compared to traditional approaches (0.15pm0.95 with p=0.01) with only a small increase in false positive predictions (0.04pm1.06 with p=0.57). We observe the specific prediction tasks of Mass and Atelectasis appear to benefit the most compared to other tasks.

Large Language Models are trained on vast amounts of text from the internet, which contains both factual and misleading information about the world. Can language models discern truth from falsehood in this contradicting data? Expanding on the view that LLMs can model different agents producing the corpora, we hypothesize that they can cluster truthful text by modeling a truthful persona: a group of agents that are likely to produce truthful text and share similar features. For example, trustworthy sources like Wikipedia and Science usually use formal writing styles and make consistent claims. By modeling this persona, LLMs can generalize truthfulness beyond the specific contexts in which each agent generated the training text. For example, the model can infer that the agent "Wikipedia" will behave truthfully on topics that were only generated by "Science" because they share a persona. We first show evidence for the persona hypothesis via two observations: (1) we can probe whether a model's answer will be truthful before it is generated; (2) finetuning a model on a set of facts improves its truthfulness on unseen topics. Next, using arithmetics as a synthetic environment, we show that language models can separate true and false statements, and generalize truthfulness across agents; but only if agents in the training data share a truthful generative process that enables the creation of a truthful persona. Overall, our findings suggest that models can exploit hierarchical structures in the data to learn abstract concepts like truthfulness.

This technical report analyzes the challenge of "hallucinations" (false information) in LLMs applied to law. It examines their causes, manifestations, and the effectiveness of the RAG mitigation strategy, highlighting its limitations and proposing holistic optimizations. The paper explores the ethical and regulatory implications, emphasizing human oversight as an irreplaceable role. It concludes that the solution lies not in incrementally improving generative models, but in adopting a "consultative" AI paradigm that prioritizes veracity and traceability, acting as a tool to amplify, not replace, professional judgment. -- Este informe t\'ecnico analiza el desaf\'io de las "alucinaciones" (informaci\'on falsa) en los LLMs aplicados al derecho. Se examinan sus causas, manifestaciones y la efectividad de la estrategia de mitigaci\'on RAG, exponiendo sus limitaciones y proponiendo optimizaciones hol\'isticas. Se exploran las implicaciones \'eticas y regulatorias, enfatizando la supervisi\'on humana como un rol insustituible. El documento concluye que la soluci\'on no reside en mejorar incrementalmente los modelos generativos, sino en adoptar un paradigma de IA "consultiva" que priorice la veracidad y la trazabilidad, actuando como una herramienta para amplificar, y no sustituir, el juicio profesional.

As AI capabilities increasingly surpass human proficiency in complex tasks, current alignment techniques including SFT and RLHF face fundamental challenges in ensuring reliable oversight. These methods rely on direct human assessment and become untenable when AI outputs exceed human cognitive thresholds. In response to this challenge, we explore two hypotheses: (1) critique of critique can be easier than critique itself, extending the widely-accepted observation that verification is easier than generation to the critique domain, as critique itself is a specialized form of generation; (2) this difficulty relationship is recursively held, suggesting that when direct evaluation is infeasible, performing high-order critiques (e.g., critique of critique of critique) offers a more tractable supervision pathway. To examine these hypotheses, we perform Human-Human, Human-AI, and AI-AI experiments across multiple tasks. Our results demonstrate encouraging evidence supporting these hypotheses and suggest that recursive self-critiquing is a promising direction for scalable oversight.

Probabilistic text generators have been used to produce fake scientific papers for more than a decade. Such nonsensical papers are easily detected by both human and machine. Now more complex AI-powered generation techniques produce texts indistinguishable from that of humans and the generation of scientific texts from a few keywords has been documented. Our study introduces the concept of tortured phrases: unexpected weird phrases in lieu of established ones, such as 'counterfeit consciousness' instead of 'artificial intelligence.' We combed the literature for tortured phrases and study one reputable journal where these concentrated en masse. Hypothesising the use of advanced language models we ran a detector on the abstracts of recent articles of this journal and on several control sets. The pairwise comparisons reveal a concentration of abstracts flagged as 'synthetic' in the journal. We also highlight irregularities in its operation, such as abrupt changes in editorial timelines. We substantiate our call for investigation by analysing several individual dubious articles, stressing questionable features: tortured writing style, citation of non-existent literature, and unacknowledged image reuse. Surprisingly, some websites offer to rewrite texts for free, generating gobbledegook full of tortured phrases. We believe some authors used rewritten texts to pad their manuscripts. We wish to raise the awareness on publications containing such questionable AI-generated or rewritten texts that passed (poor) peer review. Deception with synthetic texts threatens the integrity of the scientific literature.

Logical reading comprehension is a challenging task that entails grasping the underlying semantics of text and applying reasoning to deduce the correct answer. Prior researches have primarily focused on enhancing logical reasoning capabilities through Chain-of-Thought (CoT) or data augmentation. However, previous work constructing chain-of-thought rationales concentrates solely on analyzing correct options, neglecting the incorrect alternatives. Addtionally, earlier efforts on data augmentation by altering contexts rely on rule-based methods, which result in generated contexts that lack diversity and coherence. To address these issues, we propose a Premise-Oriented Data Augmentation (PODA) framework. This framework can generate CoT rationales including analyses for both correct and incorrect options, while constructing diverse and high-quality counterfactual contexts from incorrect candidate options. We integrate summarizing premises and identifying premises for each option into rationales. Subsequently, we employ multi-step prompts with identified premises to construct counterfactual context. To facilitate the model's capabilities to better differentiate the reasoning process associated with each option, we introduce a novel thought-path contrastive learning method that compares reasoning paths between the original and counterfactual samples. Experimental results on three representative LLMs demonstrate that our method can improve the baselines substantially across two challenging logical reasoning benchmarks (ReClor and LogiQA 2.0). The data and code are released at https://github.com/lalalamdbf/TPReasoner.

Counterfactual statements describe events that did not or cannot take place. We consider the problem of counterfactual detection (CFD) in product reviews. For this purpose, we annotate a multilingual CFD dataset from Amazon product reviews covering counterfactual statements written in English, German, and Japanese languages. The dataset is unique as it contains counterfactuals in multiple languages, covers a new application area of e-commerce reviews, and provides high quality professional annotations. We train CFD models using different text representation methods and classifiers. We find that these models are robust against the selectional biases introduced due to cue phrase-based sentence selection. Moreover, our CFD dataset is compatible with prior datasets and can be merged to learn accurate CFD models. Applying machine translation on English counterfactual examples to create multilingual data performs poorly, demonstrating the language-specificity of this problem, which has been ignored so far.

Thanks to rapid progress in artificial intelligence, we have entered an era when technology and philosophy intersect in interesting ways. Sitting squarely at the centre of this intersection are large language models (LLMs). The more adept LLMs become at mimicking human language, the more vulnerable we become to anthropomorphism, to seeing the systems in which they are embedded as more human-like than they really are. This trend is amplified by the natural tendency to use philosophically loaded terms, such as "knows", "believes", and "thinks", when describing these systems. To mitigate this trend, this paper advocates the practice of repeatedly stepping back to remind ourselves of how LLMs, and the systems of which they form a part, actually work. The hope is that increased scientific precision will encourage more philosophical nuance in the discourse around artificial intelligence, both within the field and in the public sphere.

Large language models (LLMs) are becoming deeply embedded in human communication and decision-making, yet they inherit the ambiguity, bias, and lack of direct access to truth inherent in language itself. While their outputs are fluent, emotionally resonant, and coherent, they are generated through statistical prediction rather than grounded reasoning. This creates the risk of hallucination, responses that sound convincing but lack factual validity. Building on Geoffrey Hinton's observation that AI mirrors human intuition rather than reasoning, this paper argues that LLMs operationalize System 1 cognition at scale: fast, associative, and persuasive, but without reflection or falsification. To address this, we introduce the Rose-Frame, a three-dimensional framework for diagnosing cognitive and epistemic drift in human-AI interaction. The three axes are: (i) Map vs. Territory, which distinguishes representations of reality (epistemology) from reality itself (ontology); (ii) Intuition vs. Reason, drawing on dual-process theory to separate fast, emotional judgments from slow, reflective thinking; and (iii) Conflict vs. Confirmation, which examines whether ideas are critically tested through disagreement or simply reinforced through mutual validation. Each dimension captures a distinct failure mode, and their combination amplifies misalignment. Rose-Frame does not attempt to fix LLMs with more data or rules. Instead, it offers a reflective tool that makes both the model's limitations and the user's assumptions visible, enabling more transparent and critically aware AI deployment. It reframes alignment as cognitive governance: intuition, whether human or artificial, must remain governed by human reason. Only by embedding reflective, falsifiable oversight can we align machine fluency with human understanding.

Large Language Models (LLMs) have impressive capabilities, but are prone to outputting falsehoods. Recent work has developed techniques for inferring whether a LLM is telling the truth by training probes on the LLM's internal activations. However, this line of work is controversial, with some authors pointing out failures of these probes to generalize in basic ways, among other conceptual issues. In this work, we use high-quality datasets of simple true/false statements to study in detail the structure of LLM representations of truth, drawing on three lines of evidence: 1. Visualizations of LLM true/false statement representations, which reveal clear linear structure. 2. Transfer experiments in which probes trained on one dataset generalize to different datasets. 3. Causal evidence obtained by surgically intervening in a LLM's forward pass, causing it to treat false statements as true and vice versa. Overall, we present evidence that at sufficient scale, LLMs linearly represent the truth or falsehood of factual statements. We also show that simple difference-in-mean probes generalize as well as other probing techniques while identifying directions which are more causally implicated in model outputs.

Large language models have achieved remarkable success on final-answer mathematical problems, largely due to the ease of applying reinforcement learning with verifiable rewards. However, the reasoning underlying these solutions is often flawed. Advancing to rigorous proof-based mathematics requires reliable proof verification capabilities. We begin by analyzing multiple evaluation setups and show that focusing on a single benchmark can lead to brittle or misleading conclusions. To address this, we evaluate both proof-based and final-answer reasoning to obtain a more reliable measure of model performance. We then scale two major generative verification methods (GenSelect and LLM-as-a-Judge) to millions of tokens and identify their combination as the most effective framework for solution verification and selection. We further show that the choice of prompt for LLM-as-a-Judge significantly affects the model's performance, but reinforcement learning can reduce this sensitivity. However, despite improving proof-level metrics, reinforcement learning does not enhance final-answer precision, indicating that current models often reward stylistic or procedural correctness rather than mathematical validity. Our results establish practical guidelines for designing and evaluating scalable proof-verification and selection systems.

Mathematical reasoning serves as a cornerstone for assessing the fundamental cognitive capabilities of human intelligence. In recent times, there has been a notable surge in the development of Large Language Models (LLMs) geared towards the automated resolution of mathematical problems. However, the landscape of mathematical problem types is vast and varied, with LLM-oriented techniques undergoing evaluation across diverse datasets and settings. This diversity makes it challenging to discern the true advancements and obstacles within this burgeoning field. This survey endeavors to address four pivotal dimensions: i) a comprehensive exploration of the various mathematical problems and their corresponding datasets that have been investigated; ii) an examination of the spectrum of LLM-oriented techniques that have been proposed for mathematical problem-solving; iii) an overview of factors and concerns affecting LLMs in solving math; and iv) an elucidation of the persisting challenges within this domain. To the best of our knowledge, this survey stands as one of the first extensive examinations of the landscape of LLMs in the realm of mathematics, providing a holistic perspective on the current state, accomplishments, and future challenges in this rapidly evolving field.

We present miniF2F, a dataset of formal Olympiad-level mathematics problems statements intended to provide a unified cross-system benchmark for neural theorem proving. The miniF2F benchmark currently targets Metamath, Lean, Isabelle (partially) and HOL Light (partially) and consists of 488 problem statements drawn from the AIME, AMC, and the International Mathematical Olympiad (IMO), as well as material from high-school and undergraduate mathematics courses. We report baseline results using GPT-f, a neural theorem prover based on GPT-3 and provide an analysis of its performance. We intend for miniF2F to be a community-driven effort and hope that our benchmark will help spur advances in neural theorem proving.

In science and medicine, model interpretations may be reported as discoveries of natural phenomena or used to guide patient treatments. In such high-stakes tasks, false discoveries may lead investigators astray. These applications would therefore benefit from control over the finite-sample error rate of interpretations. We reframe black box model interpretability as a multiple hypothesis testing problem. The task is to discover "important" features by testing whether the model prediction is significantly different from what would be expected if the features were replaced with uninformative counterfactuals. We propose two testing methods: one that provably controls the false discovery rate but which is not yet feasible for large-scale applications, and an approximate testing method which can be applied to real-world data sets. In simulation, both tests have high power relative to existing interpretability methods. When applied to state-of-the-art vision and language models, the framework selects features that intuitively explain model predictions. The resulting explanations have the additional advantage that they are themselves easy to interpret.

In this paper, we revisit one of the simplest problems in data structures: the task of inserting elements into an open-addressed hash table so that elements can later be retrieved with as few probes as possible. We show that, even without reordering elements over time, it is possible to construct a hash table that achieves far better expected search complexities (both amortized and worst-case) than were previously thought possible. Along the way, we disprove the central conjecture left by Yao in his seminal paper ``Uniform Hashing is Optimal''. All of our results come with matching lower bounds.

Interpretability research on large language models (LLMs) has yielded important insights into model behaviour, yet recurring pitfalls persist: findings that do not generalise, and causal interpretations that outrun the evidence. Our position is that causal inference specifies what constitutes a valid mapping from model activations to invariant high-level structures, the data or assumptions needed to achieve it, and the inferences it can support. Specifically, Pearl's causal hierarchy clarifies what an interpretability study can justify. Observations establish associations between model behaviour and internal components. Interventions (e.g., ablations or activation patching) support claims how these edits affect a behavioural metric (e.g., average change in token probabilities) over a set of prompts. However, counterfactual claims -- i.e., asking what the model output would have been for the same prompt under an unobserved intervention -- remain largely unverifiable without controlled supervision. We show how causal representation learning (CRL) operationalises this hierarchy, specifying which variables are recoverable from activations and under what assumptions. Together, these motivate a diagnostic framework that helps practitioners select methods and evaluations matching claims to evidence such that findings generalise.

LLM-based formal proof assistants (e.g., in Lean) hold great promise for automating mathematical discovery. But beyond syntactic correctness, do these systems truly understand mathematical structure as humans do? We investigate this question through the lens of mathematical inequalities -- a fundamental tool across many domains. While modern provers can solve basic inequalities, we probe their ability to handle human-intuitive compositionality. We introduce Ineq-Comp, a benchmark built from elementary inequalities through systematic transformations, including variable duplication, algebraic rewriting, and multi-step composition. Although these problems remain easy for humans, we find that most provers -- including Goedel, STP, and Kimina-7B -- struggle significantly. DeepSeek-Prover-V2-7B shows relative robustness -- possibly because it is trained to decompose the problems into sub-problems -- but still suffers a 20\% performance drop (pass@32). Strikingly, performance remains poor for all models even when formal proofs of the constituent parts are provided in context, revealing that the source of weakness is indeed in compositional reasoning. Our results expose a persisting gap between the generalization behavior of current AI provers and human mathematical intuition.

Self-correction has emerged as a promising solution to boost the reasoning performance of large language models (LLMs), where LLMs refine their solutions using self-generated critiques that pinpoint the errors. This work explores whether smaller-size (<= 13B) language models (LMs) have the ability of self-correction on reasoning tasks with minimal inputs from stronger LMs. We propose a novel pipeline that prompts smaller LMs to collect self-correction data that supports the training of self-refinement abilities. First, we leverage correct solutions to guide the model in critiquing their incorrect responses. Second, the generated critiques, after filtering, are used for supervised fine-tuning of the self-correcting reasoner through solution refinement. Our experimental results show improved self-correction abilities of two models on five datasets spanning math and commonsense reasoning, with notable performance gains when paired with a strong GPT-4-based verifier, though limitations are identified when using a weak self-verifier for determining when to correct.

Large language models (LLMs) perform better when they produce step-by-step, "Chain-of-Thought" (CoT) reasoning before answering a question, but it is unclear if the stated reasoning is a faithful explanation of the model's actual reasoning (i.e., its process for answering the question). We investigate hypotheses for how CoT reasoning may be unfaithful, by examining how the model predictions change when we intervene on the CoT (e.g., by adding mistakes or paraphrasing it). Models show large variation across tasks in how strongly they condition on the CoT when predicting their answer, sometimes relying heavily on the CoT and other times primarily ignoring it. CoT's performance boost does not seem to come from CoT's added test-time compute alone or from information encoded via the particular phrasing of the CoT. As models become larger and more capable, they produce less faithful reasoning on most tasks we study. Overall, our results suggest that CoT can be faithful if the circumstances such as the model size and task are carefully chosen.

Mathematical reasoning is a fundamental aspect of human intelligence and is applicable in various fields, including science, engineering, finance, and everyday life. The development of artificial intelligence (AI) systems capable of solving math problems and proving theorems has garnered significant interest in the fields of machine learning and natural language processing. For example, mathematics serves as a testbed for aspects of reasoning that are challenging for powerful deep learning models, driving new algorithmic and modeling advances. On the other hand, recent advances in large-scale neural language models have opened up new benchmarks and opportunities to use deep learning for mathematical reasoning. In this survey paper, we review the key tasks, datasets, and methods at the intersection of mathematical reasoning and deep learning over the past decade. We also evaluate existing benchmarks and methods, and discuss future research directions in this domain.

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

Large Language Models (LLMs) demonstrate impressive reasoning capabilities in familiar contexts, but struggle when the context conflicts with their parametric knowledge. To investigate this phenomenon, we introduce CounterLogic, a dataset containing 1,800 examples across 9 logical schemas, explicitly designed to evaluate logical reasoning through counterfactual (hypothetical knowledge-conflicting) scenarios. Our systematic evaluation of 11 LLMs across 6 different datasets reveals a consistent performance degradation, with accuracies dropping by 27% on average when reasoning through counterfactual information. We propose Self-Segregate, a prompting method enabling metacognitive awareness (explicitly identifying knowledge conflicts) before reasoning. Our method dramatically narrows the average performance gaps from 27% to just 11%, while significantly increasing the overall accuracy (+7.5%). We discuss the implications of these findings and draw parallels to human cognitive processes, particularly on how humans disambiguate conflicting information during reasoning tasks. Our findings offer practical insights for understanding and enhancing LLMs reasoning capabilities in real-world applications, especially where models must logically reason independently of their factual knowledge.

Online misinformation poses a global risk with significant real-world consequences. To combat misinformation, current research relies on professionals like journalists and fact-checkers for annotating and debunking misinformation, and develops automated machine learning methods for detecting misinformation. Complementary to these approaches, recent research has increasingly concentrated on utilizing the power of ordinary social media users, a.k.a. "crowd", who act as eyes-on-the-ground proactively questioning and countering misinformation. Notably, recent studies show that 96% of counter-misinformation responses originate from them. Acknowledging their prominent role, we present the first systematic and comprehensive survey of research papers that actively leverage the crowds to combat misinformation. We first identify 88 papers related to crowd-based efforts, following a meticulous annotation process adhering to the PRISMA framework. We then present key statistics related to misinformation, counter-misinformation, and crowd input in different formats and topics. Upon holistic analysis of the papers, we introduce a novel taxonomy of the roles played by the crowds: (i)annotators who actively identify misinformation; (ii)evaluators who assess counter-misinformation effectiveness; (iii)creators who create counter-misinformation. This taxonomy explores the crowd's capabilities in misinformation detection, identifies prerequisites for effective counter-misinformation, and analyzes crowd-generated counter-misinformation. Then, we delve into (i)distinguishing individual, collaborative, and machine-assisted labeling for annotators; (ii)analyzing the effectiveness of counter-misinformation through surveys, interviews, and in-lab experiments for evaluators; and (iii)characterizing creation patterns and creator profiles for creators. Finally, we outline potential future research in this field.

Current Chain-of-Thought (CoT) verification methods predict reasoning correctness based on outputs (black-box) or activations (gray-box), but offer limited insight into why a computation fails. We introduce a white-box method: Circuit-based Reasoning Verification (CRV). We hypothesize that attribution graphs of correct CoT steps, viewed as execution traces of the model's latent reasoning circuits, possess distinct structural fingerprints from those of incorrect steps. By training a classifier on structural features of these graphs, we show that these traces contain a powerful signal of reasoning errors. Our white-box approach yields novel scientific insights unattainable by other methods. (1) We demonstrate that structural signatures of error are highly predictive, establishing the viability of verifying reasoning directly via its computational graph. (2) We find these signatures to be highly domain-specific, revealing that failures in different reasoning tasks manifest as distinct computational patterns. (3) We provide evidence that these signatures are not merely correlational; by using our analysis to guide targeted interventions on individual transcoder features, we successfully correct the model's faulty reasoning. Our work shows that, by scrutinizing a model's computational process, we can move from simple error detection to a deeper, causal understanding of LLM reasoning.