Reasoning Errors

The reasoning ability of large language models (LLMs) is a critical factor for practical LLM-based applications. To investigate the current reasoning capability of LLMs, we clarify the types of errors that arise in LLMs' reasoning processes on mathematical datasets. We focus on problems where LLMs produce an incorrect answer. We define errors in the reasoning process as reasoning errors and manually analyze the features of reasoning errors. We defined and classified 21 error classes and identified the frequently occurring classes among them. Beyond qualitative evaluation, we leverage the evaluation results to improve the reasoning capability. We designed a prompt that explicitly focuses on eight error classes. The experiments demonstrate that this prompt effectively improves reasoning performance. Furthermore, the results suggest that the frequent reasoning errors identified in this paper are common across LLMs of comparable scale.

As language models are increasingly used for tasks that require verifiable reasoning, reliably distinguishing sound reasoning from flawed reasoning has become an important practical problem. Recent trajectory-based methods seek this signal in layerwise residual-stream displacements, which capture how representations change while attenuating some stable, token-specific information. However, displacement omits the state from which an update originates, whereas restoring the full state risks reintroducing shortcut-prone information. We identify this trade-off and propose a three-stream detector that combines motion with two restricted views of location. A coarse region reader based on vector quantization and a fine direction reader over normalized multi-layer states. This design restores enough state context to interpret the motion without returning to full-state probing. On reasoning benchmarks unseen during training, our method improves selection accuracy by up to 12% over the displacement-only state of the art and 21% over single-layer probing baselines. Although trained only on reasoning benchmarks, it also reads factual completion and fact verification, ahead of every detector we compare against, which places the signal on correctness rather than on a kind of reasoning. Ablations further show that motion, region, and direction provide complementary signals. These results suggest that reasoning validity is better read from state-conditioned motion than from either static states or decontextualized trajectories alone.

We assess 18 chapter-specific reviews of the ten mathematical results announced by OpenAI on 1 August 2026, alongside review standards, Lean formalizations, subsequent research, and mathematical references. The article audits this review record without claiming a complete reconstruction of all ten proofs. No confirmed substantive mathematical error in a principal result remains in the examined assessments, although review depth varies and some dependencies remain partly checked. Chapter 8 presents the strongest reservation: a specialist review requests major revision of compressed analytic arguments. In Chapter 6, an apparent polarity error was withdrawn after an overbar lost during PDF extraction was recovered from the typeset source. Subsequent research independently reuses the Chapter 3 proof mechanism and confirms that Connes's rigidity conjecture is false, without independently reproducing Chapter 4's stronger infinite-family result. Among the cited follow-ups, Chapter 7 receives the strongest direct theorem-level corroboration through a stronger hardness theorem. Related equality results in Chapter 8 do not verify the analytic inequality proof. Some follow-ups disclose material AI assistance. We argue that confidence should combine formal checking, human reconstruction, independent mathematical use, and a public record supporting correction of both proofs and reviews.

Latent reasoning allows language models to carry out intermediate reasoning in continuous latent representations rather than fully externalizing it as discrete chains of thought. However, assigning credit to such latent thoughts from answer-only rewards is difficult: a single final answer mixes thought quality with answer-sampling noise. We propose \textbf{Latent Thought Credit (LTC)}, a hierarchical credit-assignment framework for latent reasoning. For each prompt, LTC samples multiple latent thoughts, fixes the context after each thought, and estimates thought-level expected reward by averaging rewards over multiple answers generated from that fixed context. LTC uses thought-level advantages to optimize the latent-thought phase, answer-level advantages to optimize the answer phase, and an advantage-weighted thought-matching objective that helps the policy reproduce high-credit latent thoughts. We instantiate LTC in a GRPO-style on-policy training framework and evaluate it across mathematical reasoning and STEM multiple-choice tasks. LTC achieves the best average accuracy among the compared methods, while ablations and fixed-context diagnostics show that multi-answer estimation reduces reward-estimation error and mitigates ambiguous or incorrect thought-level credit.

With the rapid advancement of large language models (LLMs), modern systems not only possess strong foundational capabilities and extensive knowledge, but can also solve complex problems via long, multi-step reasoning. However, as reasoning traces become longer, LLMs may produce a substantial amount of hallucinated content during the reasoning process, which is often difficult to detect. In this work, we conduct a fine-grained analysis of hallucinations arising in LLM reasoning and find that the reasoning traces are particularly prone to Context-Sensitive Factual Hallucinations: cases where the model actually has the relevant knowledge, yet makes factual errors due to contextual interference during reasoning. To address this issue, we propose Step-level Self-Consistency Group Relative Policy Optimization (SSC-GRPO), which assigns step-level rewards to reasoning traces by computing self-consistency scores of individual steps across multiple rollouts. Compared with prior methods, SSC-GRPO achieves state-of-the-art performance on both mathematical reasoning benchmarks and hallucination leaderboards. Our results offer a new perspective for detecting and mitigating hallucinations in the reasoning process of large language models.

The emergence of Chain-of-Thought (CoT) reasoning has significantly enhanced the ability of large language models (LLMs) to tackle complex, multi-step tasks. However, when errors occur, current interaction approaches typically involve re-generating another response that may make mistakes again, or users laboriously flag the faulty step in follow-up turns that may get responses <You are right, I made a mistake here> followed by similar errors recurring. To address this issue, we propose an efficient human intervention mechanism for precisely correcting reasoning errors in LLMs, termed Deep Interaction. Our approach enables direct editing of the original response, allowing erroneous parts to be corrected while preserving accurate reasoning steps. We refine the edited CoT into a distilled prompt, which then steers the LLM along the corrected reasoning path. Experimental results show that our method achieves over a 25% improvement in correction success rate and reduces token usage by approximately 40% on STEM tasks reasoning compared to baseline approaches.

Chain-of-Thought (CoT) prompting has significantly advanced the reasoning capabilities of Large Language Models (LLMs), yet it often incurs substantial computational costs due to over-reasoning: the generation of redundant, verbose, or irrelevant steps. While existing reasoning step evaluators effectively detect logical fallacies and factual errors, our analysis reveals a critical blind spot: they fail to penalize valid but inefficient reasoning steps that inflate token usage without contributing to the solution. To systematically diagnose this limitation, we introduce RIV-GSM8K, a diagnostic benchmark injected with five distinct types of inefficiencies, including circular reasoning and excessive decomposition. Diagnostic experiments reveal that state-of-the-art evaluators struggle to distinguish these inefficiencies from necessary reasoning. To address this gap, we propose CAID (Context-Aware Information Density), a training-free metric grounded in information theory that identifies low-utility steps. To validate the metric's practical utility, we apply it within PACE, a post-hoc compression strategy. Additional control experiments show that the gains of PACE are not explained by trivial pruning: compared with random step removal and PRM-based compression baselines, it preserves accuracy at substantially higher compression rates. Empirical results on GSM8K, StrategyQA, and ARC-Challenge demonstrate that PACE reduces token consumption by 31-53% while maintaining accuracy, confirming that CAID successfully distills informational froth from reasoning chains without compromising deductive validity.

Physics reasoning fails structurally in small language models: an error at any step propagates forward, corrupting every inference that follows. Limited domain knowledge, hallucination under multi-step derivation, and distributional sensitivity compound this failure. We propose a step-level reward framework that identifies the first reasoning error, generates targeted structured feedback, and trains the model to revise its solution via policy gradient with KL regularization, without exposing it to ground truth solutions as generation targets. Unlike annotation-dependent step-level methods, no preference data construction is required and the external verifier operates exclusively at training time. Across five physics benchmarks, our framework delivers accuracy gains of 17-20% over CoT prompting and 10-16% over the strongest baseline, reduces calculation errors from 56.9% to 23.5%, and reduces miscomprehension errors from 22.3% to 12.0% in the best observed cases. Conceptual errors reduce from 89.7% to 68.7%, yet persist as the hardest failure mode across all conditions.

Humans flexibly adapt their reasoning strategies to the requirements of a given problem. Large language models (LLMs) have performed well on many cognitive tasks, however, it is unclear whether this accuracy is a result of pattern matching from training data or flexible reasoning. Here, we introduce a novel paradigm to test this question: the riddle riddle paradigm. Riddle riddles are word problems written to mimic popular riddles, but altered so their answers only require literal interpretations. Identifying correct answers requires looking past the structure of each question and flexibly apply different reasoning strategies based on the content. If LLMs respond to surface features, such as form, a riddle-like structure should cause models to use an inventive reasoning strategy even when a literal interpretation suffices. Alternatively, if LLMs reason based on content, they should flexibly switch strategies when appropriate. Across two experiments with nine state-of-the-art LLMs and 100 human participants, we show humans and LLMs fail on this paradigm in opposite directions. LLMs were far more accurate on genuine riddles than on riddle riddles (84.9% vs. 50.7%); whereas humans showed the reverse effect (50.5% vs. 80.5%). Error analysis shows that 90.8% of LLM errors on riddle riddles (the condition where they show diminished performance) were due to inappropriate use of inventive reasoning while only 57.6% of human errors on genuine riddles were due to overextending literal reasoning. Thus, while both groups make mistakes, reasoning mistakes are made more often by LLMs than by humans. Overall, LLMs' strong performance on genuine riddles may reflect memory retrieval rather than flexible strategy selection, and without stimuli designed to elicit this contrast, it becomes easy to conflate LLM-generated outputs that look like reasoning with genuine reasoning.

Large language models produce fluent but often incorrect multi-step reasoning, and naive correction methods risk degrading already-correct answers. We introduce Denoising Iterative Self-Correction (DISC), a test-time procedure that treats verification question outputs as noisy measurements of where a solution may be corrupted. Using these signals, DISC progressively reduces errors across multiple verify-judge-correct passes, analogous to traditional iterative denoising. A binary judgment gate controls correction precision by blocking rewrites that would damage already-correct answers while the verifier and corrector together repair errors. We evaluate this trade-off using two paired diagnostics: an improvement-to-degradation ratio (precision) and a repair rate (recall). Across three benchmarks (BIG-Bench Mistake, HotpotQA, GPQA Diamond) and four models, DISC dominates Chain-of-Verification and Self-Refine on the precision-recall trade-off, reaching 81.6% accuracy with 13x more improvements per degradation than Chain-of-Verification and 5x more than Self-Refine on BIG-Bench Mistake (Sonnet~4.5). On GPQA Diamond, we identify a capability floor below which judges acknowledge contradictions in evidence but cannot translate that recognition into a correction. We further show that cross-model role allocation -- assigning verification and judgment to a model different from the generator -- mitigates self-confirmation bias.

Understanding student errors in the programming is a cornerstone of programming education, yet obtaining a representative set of student errors for any newly designed task remains slow and costly, since authentic submissions only accumulate after extensive classroom deployment. This paper explores whether large language models (LLMs) can serve as scalable proxies for students by simulating realistic logical errors in code submissions. Using the CodeWorkout dataset of 74,000+ unique student Java submissions across 37 problems, we evaluate five LLMs under three mainstream prompting strategies: Input-Output (IO), Chain-of-Thought (CoT), and iterative Self-Refine. We assess performance along two key dimensions: diversity (the range of distinct error patterns) and alignment (alignment with authentic student mistakes), and examine how these vary by struggling level of programming tasks. Our quantitative findings reveal that while all models generate diverse errors, their alignment to human submissions diverges: Claude Sonnet 4 achieves the most balanced performance. In addition, we conducted a blinded expert annotation study (N = 401) comparing synthetic and authentic errors. This qualitative analysis confirms that the generated errors are functionally indistinguishable from authentic student errors. Moreover, higher-struggling-level problems elicit more diverse but less student-like errors. These results highlight trade-offs in using LLMs to simulate human learners and suggest design considerations for integrating synthetic errors into teachable agents, intelligent tutoring systems, and large-scale learning analytics.

While test-time scaling has revolutionized reasoning in large language models, generative video reasoning remains bottlenecked by a single-shot paradigm. We demonstrate that searching over denoising steps cannot rescue logically flawed rollouts because spatial trajectories commit early in the diffusion process. Root-level Best-of-N (BoN) sampling is similarly inefficient: reasoning errors cluster early in the temporal axis, and resampling blindly discards verified upstream progress. To unlock effective test-time scaling for video models, we introduce Temporal Backtracking Search (TBS), which shifts the search space to the temporal axis. TBS transforms video generation into an iterative generate-verify-restart loop via three core mechanisms: (1) variable-K conditioning to resume generation from arbitrary clean prefixes; (2) temporal process verification to localize failures and extract valid restart anchors; and (3) prefix-based search to reallocate compute toward extending correct trajectories rather than root resampling. Across algorithmic, navigation, and robotics domains, TBS Pareto-dominates matched-budget BoN. In a strict out-of-distribution setting where one-shot generation collapses (0.7% for BoN), TBS achieves 22.7%, with every solved episode stemming from a restarted branch. Ultimately, TBS reveals that the local reasoning competence of video models far exceeds what single-shot rollouts indicate, providing a scalable test-time framework to unlock it.

When large language models (LLMs) fail to generalize or make haphazard errors in reasoning, it is often taken as evidence that LLMs are not truly reasoning, but rather performing a kind of pattern matching. The implication is that people's behavior does not exhibit the same types of failures because human reasoning uses principled and abstract world models. We evaluate human participants and 25 LLMs on their ability to engage in common-sense reasoning about a variety of everyday situations and observe similar patterns of errors in both people and models. We then identify the set of attention heads driving LLM responses and find that these heads implement a form of pattern-matching. These attention heads allow us to predict seemingly inexplicable reasoning errors in people caused by ostensibly irrelevant prompt details. Taken together, our results suggest that everyday causal reasoning in people and LLMs is more consistent with a form of pattern-matching than with abstract world models.

Large Language Models (LLMs) struggle to rigorously verify complex mathematical proofs. Standard global evaluation approaches suffer from "context poisoning," in which superficially plausible statements mask subtle logical flaws, leading to hallucination or over-skepticism. To address this, we shift from global evaluation to strict step-level verification: our framework maintains detailed context for each deduction step and strictly constrains the sources of applied theorems. We evaluate on a carefully curated adversarial diagnostic suite of research-level proofs drawn from the FirstProof challenge. A systematic ablation study demonstrates that these deductive constraints are indispensable, as unconstrained global prompting consistently fails to localize subtle logical errors. Beyond outperforming global evaluation, our approach fundamentally alters the failure taxonomy. Error analysis reveals that, rather than exhibiting severe logical hallucinations, remaining rejections are primarily instances of "pedantic hyper-rigor" stemming from unstated domain conventions, effectively exposing implicit ambiguities within the expert benchmark itself. Our findings suggest that prompting agents to organize their verification notes in a cautious, human-mathematician-like manner can substantially improve their ability to distinguish rigorous proofs from flawed ones, with the potential to strengthen agentic reasoning on frontier mathematical concepts that the base model does not already know well, and to lay a theoretical foundation for future automated proof-review systems. Code and prompts are available at GitHub.

Recent Large Language Models (LLMs) have shown impressive reasoning abilities; but they are still susceptible to hallucinations, intermediate reasoning mistakes, and unreliable reasoning results in complex mathematical reasoning problems. In this study, we introduce a critic-based heterogeneous multi-agent approach to improve the dependability of mathematical reasoning. This framework incorporates several LLM agents of different specialties and employs a critic-driven adaptive learning system to assess and guide the reasoning process based on intermediate feedback. The system adopts a generator-validator framework, with the validator not only determining correctness but also offering critiques to guide regeneration of solutions. This allows for adaptive error correction and prevents error cascading. Our experiments on the GSM8K benchmark show that the proposed method achieves up to 13% accuracy improvement over single-shot and non-critic models. Additionally, findings suggest that heterogeneity and critique reduce the need for large models, allowing smaller models to perform on par. Ablation studies reveal the main performance gains are due to the critic-based feedback loop and not model size. In summary, the proposed approach showcases the benefits of combining heterogeneous multi-agent collaboration and critique to obtain reliable and interpretable reasoning systems.

Single-score translation metrics can conflate legitimate variation with error, a problem especially acute for classical languages where multiple defensible English renderings of the same passage coexist. We audit Pali-to-English output from four flagship large language models (LLMs): GPT-5.5, Claude Sonnet 4.6, Gemini 3.1 Pro, and Grok 4.3, on 1,700 passages from the Pali Canon, using three established human translations by Bhikkhu Sujato, Thanissaro Bhikkhu, and Bhikkhu Bodhi as a local reference envelope rather than a single gold standard. Each candidate's normalized embedding drift from the reference centroid serves as a triage signal, not an error label; the 1,203 candidates above a 1.5 drift threshold are then adjudicated by a blinded three-model LLM judge panel, calibrated against a 300-instance author-adjudicated validation set. Two results stand out. First, drift predicts severity rather than error per se: the major-error rate among adjudicated high-drift candidates rose monotonically from 7.9% in the 1.5-2.0 band to 51.6% above 3.0, while approximately 80% of 1.5-2.0 outliers were judged valid translation variations. Second, model differences were clearest in the high-drift tail: GPT-5.5 had the lowest adjudicated high-drift major-error rate, with confidence intervals overlapping those of Claude Sonnet 4.6 and Gemini 3.1 Pro; Grok 4.3 had both the largest outlier volume and the highest tail major-error rate (27.6% overall, 74.4% above drift 3.0). The dominant major-error categories (e.g. omission or truncation, doctrinal term errors) are precisely the failures most likely to mislead readers of doctrinal text. The contribution is a reusable audit design for classical-to-modern translation: define a local reference envelope from multiple human translators, use embedding drift to prioritize review, and adjudicate the flagged tail rather than treating outlier status as error.

Large language models (LLMs) generate not only reasoning text, but also token-level confidence trajectories that record how uncertainty evolves during inference. Whether these trajectories are relevant to reasoning correctness remains unclear. Here we show that confidence trajectories encode a content-agnostic confidence geometry associated with trace-level final-answer correctness. Using only token-level confidence values, without access to the input question, reasoning text, hidden states, or external verifiers, we find that low-dimensional representations of confidence trajectories separate correct from incorrect reasoning traces. Across GSM8K, MATH, and MMLU, this geometric separation is quantitatively linked to downstream predictability: stronger clustering of correct and incorrect traces, measured by the Davies--Bouldin index, consistently corresponds to higher correctness-discrimination AUC. We further show that correctness-related information is enriched in the tail of reasoning, suggesting that late-stage confidence dynamics carry key correctness signals. We propose NeuralConf, a lightweight estimator that learns from confidence trajectories for correctness evaluation. Under a fixed trace budget, NeuralConf-derived scores improve confidence-weighted answer aggregation over majority voting, tail confidence, and other static baselines. These results reveal that LLMs expose trace-intrinsic statistical signals of correctness through their own confidence dynamics, offering a route to improve inference using information already present within generation.

Most models of machine teaching and learning assume the learner makes no errors in its internal deductive inference. However, humans and large language models in few-shot learning regimes are two important examples of learners where this does not hold. They fail on some consistency checks, and they can fail stochastically. In this paper we introduce a teaching and learning framework that takes these deductive errors into account. We specifically study the case of machine teaching, as different characterizations of the teacher can account for both machine teaching and learning. In an overhauled Probably Approximately Correct (PAC) setting, we study theoretically that, for some estimated error level, the teacher must find a PAC teaching set that with high probability will lead the learner to guess a hypothesis that is approximately correct. We study the computational complexity of six different problems related to computing optimal PAC teaching sets. We give XP algorithms parametrized by size of teaching set, with tight runtime bounds under standard complexity assumptions like ETH. These results are complemented with a small experimental study of which teaching and learning protocols can best represent the observed behavior in some LLM teaching sessions.

Chain-of-thought (CoT) prompting assumes that generated reasoning reflects a model's internal computation. We show this assumption is wrong in a specific, measurable way: models internally detect their own reasoning errors but outwardly express confidence in them. A linear probe on hidden states predicts trace correctness with 0.95 AUROC -- from the very first reasoning step (0.79) -- while verbalized confidence for wrong traces is 4.55/5, nearly identical to correct ones (4.87/5). A text-surface classifier achieves only 0.59 on the same data, confirming a 0.20-point gap invisible in the generated text. This hidden error awareness holds across three model families (Qwen, Llama, Phi), 1.5B-72B parameters, and RL-trained reasoning models (DeepSeek-R1, 0.852 AUROC). The natural question is whether this signal can fix the errors it detects. It cannot. Four interventions -- activation steering, probe-guided best-of-N, self-correction, and activation patching -- all fail; patching destroys output coherence entirely. The signal is diagnostic, not causal: a readout of computation quality, not a lever to redirect it. This delineates a boundary for mechanistic interpretability: error representations during reasoning are fundamentally different from the factual knowledge representations that prior work has successfully edited.

Large language models have achieved remarkable progress on complex reasoning tasks. However, they often implicitly fabricate information when inputs are incomplete, producing confident but unreliable conclusions -- a failure mode we term ungrounded reasoning. We argue that this issue arises not from insufficient reasoning capability, but from the lack of inferential boundary awareness -- the ability to recognize when the necessary premises for valid inference are missing. To address this issue, we propose Grounded Reasoning via Interactive Reinforcement Learning (GRIL), a multi-turn reinforcement learning framework for grounded reasoning under incomplete information. GRIL decomposes the reasoning process into two stages: clarify and pause, which identifies whether the available information is sufficient, and grounded reasoning, which performs task solving once the necessary premises are established. We design stage-specific rewards to penalize hallucinations, enabling models to detect gaps, stop proactively, and resume reasoning after clarification. Experiments on GSM8K-Insufficient and MetaMATH-Insufficient show that GRIL significantly improves premise detection (up to 45%), leading to a 30% increase in task success while reducing average response length by over 20%. Additional analyses confirm robustness to noisy user responses and generalization to out-of-distribution tasks.

Large language models frequently commit unrecoverable reasoning errors mid-generation: once a wrong step is taken, subsequent tokens compound the mistake rather than correct it. We introduce $\textbf{Latent Phase-Shift Rollback}$ (LPSR): at each generation step, we monitor the residual stream at a critical layer lcrit, detect abrupt directional reversals (phase shifts) via a cosine-similarity $+$ entropy dual gate, and respond by rolling back the KV-cache and injecting a pre-computed steering vector. No fine-tuning, gradient computation, or additional forward passes are required. LPSR achieves $\mathbf{44.0\%}$ on MATH-500 with an 8B model versus $28.8\%$ for standard AR ($+15.2$ pp; McNemar $χ^2 = 66.96$, $p < 10^{-15}$). Critically, prompted self-correction, the most natural inference-time baseline, scores only $19.8\%$, below standard AR; LPSR exceeds it by $+24.2$ pp ($χ^2 = 89.4$, $p \approx 0$). LPSR also outperforms Best-of-16 ($+7.8$ pp) at $5.4\times$ lower token cost, and surpasses a standard 70B model ($35.2\%$) with $8.75\times$ fewer parameters at ${\sim}3\times$ the token budget. A 32-layer sweep reveals a novel \textbf{detection-correction dissociation}: error-detection AUC peaks at layer14 ($0.718$) but task accuracy peaks at layer16 ($44.0\%$ vs.\ $29.2\%$), demonstrating that optimal monitoring depth differs for detection and correction.

As large language models (LLMs) evolve from conversational assistants into agents capable of handling complex tasks, they are increasingly deployed in high-risk domains. However, existing benchmarks largely rely on mixed queries and posterior evaluation, output-level scoring, which quantifies hallucination severity but offers limited insight into where and why hallucinations arise in the generation pipeline. We therefore reformulate hallucination evaluation as a diagnostic problem and propose PRISM, a controlled benchmark that disentangles hallucinations into four dimensions: knowledge missing, knowledge errors, reasoning errors, and instruction-following errors, grounded in three stages of generation (memory, instruction, and reasoning). PRISM contains 9,448 instances across 65 tasks and supports fine-grained, stage-aware diagnostic evaluation. Evaluating 24 mainstream open-source and proprietary LLMs, we uncover consistent trade-offs across instruction following, memory retrieval, and logical reasoning, showing that mitigation strategies often improve specific dimensions at the expense of others. We hope PRISM provides a framework for understanding the specific mechanisms behind LLMs hallucinations, ultimately accelerating the development of trustworthy large language models.

The complexity of Vietnam's legal texts presents a significant barrier to public access to justice. While Large Language Models offer a promising solution for legal text simplification, evaluating their true capabilities requires a multifaceted approach that goes beyond surface-level metrics. This paper introduces a comprehensive dual-aspect evaluation framework to address this need. First, we establish a performance benchmark for four state-of-the-art large language models (GPT-4o, Claude 3 Opus, Gemini 1.5 Pro, and Grok-1) across three key dimensions: Accuracy, Readability, and Consistency. Second, to understand the "why" behind these performance scores, we conduct a large-scale error analysis on a curated dataset of 60 complex Vietnamese legal articles, using a novel, expert-validated error typology. Our results reveal a crucial trade-off: models like Grok-1 excel in Readability and Consistency but compromise on fine-grained legal Accuracy, while models like Claude 3 Opus achieve high Accuracy scores that mask a significant number of subtle but critical reasoning errors. The error analysis pinpoints \textit{Incorrect Example} and \textit{Misinterpretation} as the most prevalent failures, confirming that the primary challenge for current LLMs is not summarization but controlled, accurate legal reasoning. By integrating a quantitative benchmark with a qualitative deep dive, our work provides a holistic and actionable assessment of LLMs for legal applications.

Modeling student misconceptions in a realistic manner is critical for AI in education. In this work, we examine how large language models (LLMs) reason about misconceptions when generating distractor answers for multiple-choice questions (MCQs), a task that requires producing answers that are incorrect, yet plausible. We introduce a taxonomy over reasoning strategies for distractor generation that is grounded in learning-science literature and empirical observation, which we apply to LLM-generated reasoning traces across math and science MCQs. On the math dataset, we find that models follow a misconception-based process with potentially high diagnostic value: they recover the correct solution, articulate student errors, simulate them, and select plausible candidates. On the science dataset, on the other hand, they tend to follow a less robust approach based on semantic similarity to the correct answer. We find the most frequent failure modes to be that the model is unable to generate a correct solution or that it discards plausible distractor candidates when performing selection. Providing the correct solution in the prompt yields a relative improvement of 6.4% in alignment with human-authored distractors, highlighting the critical role of anchoring distractor generation to the correct solution. Together, our findings offer an interpretable view of how LLMs model incorrect student reasoning.

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

Recent advancements in Large Language Models (LLMs) have significantly enhanced their ability to perform complex reasoning tasks, transitioning from fast and intuitive thinking (System 1) to slow and deep reasoning (System 2). While System 2 reasoning improves task accuracy, it often incurs substantial computational costs due to its slow thinking nature and inefficient or unnecessary reasoning behaviors. In contrast, System 1 reasoning is computationally efficient but leads to suboptimal performance. Consequently, it is critical to balance the trade-off between performance (benefits) and computational costs (budgets), giving rise to the concept of reasoning economy. In this survey, we provide a comprehensive analysis of reasoning economy in both the post-training and test-time inference stages of LLMs, encompassing i) the cause of reasoning inefficiency, ii) behavior analysis of different reasoning patterns, and iii) potential solutions to achieve reasoning economy. By offering actionable insights and highlighting open challenges, we aim to shed light on strategies for improving the reasoning economy of LLMs, thereby serving as a valuable resource for advancing research in this evolving area. We also provide a public repository to continually track developments in this fast-evolving field.