Daily Papers

Tokenization, the division of input text into input tokens, is an often overlooked aspect of the large language model (LLM) pipeline and could be the source of useful or harmful inductive biases. Historically, LLMs have relied on byte pair encoding, without care to specific input domains. With the increased use of LLMs for reasoning, various number-specific tokenization schemes have been adopted, with popular models like LLaMa and PaLM opting for single-digit tokenization while GPT-3.5 and GPT-4 have separate tokens for each 1-, 2-, and 3-digit numbers. In this work, we study the effect this choice has on numerical reasoning through the use of arithmetic tasks. We consider left-to-right and right-to-left tokenization for GPT-3.5 and -4, finding that right-to-left tokenization (enforced by comma separating numbers at inference time) leads to largely improved performance. Furthermore, we find that model errors when using standard left-to-right tokenization follow stereotyped error patterns, suggesting that model computations are systematic rather than approximate. We show that the model is able to convert between tokenizations easily, thus allowing chain-of-thought-inspired approaches to recover performance on left-to-right tokenized inputs. We also find the gap between tokenization directions decreases when models are scaled, possibly indicating that larger models are better able to override this tokenization-dependent inductive bias. In summary, our work performs the first study of how number tokenization choices lead to differences in model performance on arithmetic tasks, accompanied by a thorough analysis of error patterns. We hope this work inspires practitioners to more carefully ablate number tokenization-related choices when working towards general models of numerical reasoning.

Large pre-trained language models (LMs) are known to encode substantial amounts of linguistic information. However, high-level reasoning skills, such as numerical reasoning, are difficult to learn from a language-modeling objective only. Consequently, existing models for numerical reasoning have used specialized architectures with limited flexibility. In this work, we show that numerical reasoning is amenable to automatic data generation, and thus one can inject this skill into pre-trained LMs, by generating large amounts of data, and training in a multi-task setup. We show that pre-training our model, GenBERT, on this data, dramatically improves performance on DROP (49.3 rightarrow 72.3 F1), reaching performance that matches state-of-the-art models of comparable size, while using a simple and general-purpose encoder-decoder architecture. Moreover, GenBERT generalizes well to math word problem datasets, while maintaining high performance on standard RC tasks. Our approach provides a general recipe for injecting skills into large pre-trained LMs, whenever the skill is amenable to automatic data augmentation.

Mathematical reasoning in Large Language Models (LLMs) is often evaluated using benchmarks with limited numerical ranges, failing to reflect real-world problem-solving across diverse scales. Furthermore, most existing evaluation methods only compare model outputs to ground-truth answers, obscuring insights into reasoning processes. To address these limitations, we introduce GSM-Ranges, a dataset generator derived from GSM8K that systematically perturbs numerical values in math problems to assess model robustness across varying numerical scales. Additionally, we propose a novel grading methodology that distinguishes between logical and non-logical errors, offering a more precise evaluation of reasoning processes beyond computational accuracy. Our experiments with various models reveal a significant increase in logical error rates-up to 14 percentage points-as numerical complexity rises, demonstrating a general weakness in reasoning with out-of-distribution numerical values. Moreover, while models demonstrate high accuracy on standalone arithmetic tasks, their performance deteriorates substantially when computations are embedded within word problems. These findings provide a comprehensive evaluation of LLMs' mathematical reasoning capabilities and inform future research directions for improving numerical generalization in language models.

Faithful evaluation of language model capabilities is crucial for deriving actionable insights that can inform model development. However, rigorous causal evaluations in this domain face significant methodological challenges, including complex confounding effects and prohibitive computational costs associated with extensive retraining. To tackle these challenges, we propose a causal representation learning framework wherein observed benchmark performance is modeled as a linear transformation of a few latent capability factors. Crucially, these latent factors are identified as causally interrelated after appropriately controlling for the base model as a common confounder. Applying this approach to a comprehensive dataset encompassing over 1500 models evaluated across six benchmarks from the Open LLM Leaderboard, we identify a concise three-node linear causal structure that reliably explains the observed performance variations. Further interpretation of this causal structure provides substantial scientific insights beyond simple numerical rankings: specifically, we reveal a clear causal direction starting from general problem-solving capabilities, advancing through instruction-following proficiency, and culminating in mathematical reasoning ability. Our results underscore the essential role of carefully controlling base model variations during evaluation, a step critical to accurately uncovering the underlying causal relationships among latent model capabilities.

Large language models (LLMs) are displaying emergent abilities for math reasoning tasks,and there is a growing attention on enhancing the ability of open-source LLMs through supervised fine-tuning (SFT).In this paper, we aim to explore a general data strategy for supervised data to help optimize and expand math reasoning ability.Firstly, we determine the ability boundary of reasoning paths augmentation by identifying these paths' minimal optimal set.Secondly, we validate that different abilities of the model can be cumulatively enhanced by Mix of Minimal Optimal Sets of corresponding types of data, while our models MMOS achieve SOTA performance on series base models under much lower construction costs.Besides, we point out GSM-HARD is not really hard and today's LLMs no longer lack numerical robustness.Also, we provide an Auto Problem Generator for robustness testing and educational applications.Our code and data are publicly available at https://github.com/cyzhh/MMOS.

It remains an open question whether incorporating external knowledge benefits commonsense reasoning while maintaining the flexibility of pretrained sequence models. To investigate this question, we develop generated knowledge prompting, which consists of generating knowledge from a language model, then providing the knowledge as additional input when answering a question. Our method does not require task-specific supervision for knowledge integration, or access to a structured knowledge base, yet it improves performance of large-scale, state-of-the-art models on four commonsense reasoning tasks, achieving state-of-the-art results on numerical commonsense (NumerSense), general commonsense (CommonsenseQA 2.0), and scientific commonsense (QASC) benchmarks. Generated knowledge prompting highlights large-scale language models as flexible sources of external knowledge for improving commonsense reasoning. Our code is available at https://github.com/liujch1998/GKP

Multimodal large language models have various practical applications that demand strong reasoning abilities. Despite recent advancements, these models still struggle to solve complex geometric problems. A key challenge stems from the lack of high-quality image-text pair datasets for understanding geometric images. Furthermore, most template-based data synthesis pipelines typically fail to generalize to questions beyond their predefined templates. In this paper, we bridge this gap by introducing a complementary process of Reinforcement Learning with Verifiable Rewards (RLVR) into the data generation pipeline. By adopting RLVR to refine captions for geometric images synthesized from 50 basic geometric relations and using reward signals derived from mathematical problem-solving tasks, our pipeline successfully captures the key features of geometry problem-solving. This enables better task generalization and yields non-trivial improvements. Furthermore, even in out-of-distribution scenarios, the generated dataset enhances the general reasoning capabilities of multimodal large language models, yielding accuracy improvements of 2.8%-4.8% in statistics, arithmetic, algebraic, and numerical tasks with non-geometric input images of MathVista and MathVerse, along with 2.4%-3.9% improvements in Art, Design, Tech, and Engineering tasks in MMMU.

This paper investigates the mathematical reasoning capabilities of large language models (LLMs) using 50 newly constructed high-school-level word problems. Unlike prior studies that focus solely on answer correctness, we rigorously analyze both final answers and solution steps to identify reasoning failures. Evaluating eight state-of-the-art models - including Mixtral, Llama, Gemini, GPT-4o, and OpenAI's o1 variants - we find that while newer models (e.g., o3-mini, deepseek-r1) achieve higher accuracy, all models exhibit errors in spatial reasoning, strategic planning, and arithmetic, sometimes producing correct answers through flawed logic. Common failure modes include unwarranted assumptions, over-reliance on numerical patterns, and difficulty translating physical intuition into mathematical steps. Manual analysis reveals that models struggle with problems requiring multi-step deduction or real-world knowledge, despite possessing broad mathematical knowledge. Our results underscore the importance of evaluating reasoning processes, not just answers, and caution against overestimating LLMs' problem-solving proficiency. The study highlights persistent gaps in LLMs' generalization abilities, emphasizing the need for targeted improvements in structured reasoning and constraint handling.

Mathematical reasoning---a core ability within human intelligence---presents some unique challenges as a domain: we do not come to understand and solve mathematical problems primarily on the back of experience and evidence, but on the basis of inferring, learning, and exploiting laws, axioms, and symbol manipulation rules. In this paper, we present a new challenge for the evaluation (and eventually the design) of neural architectures and similar system, developing a task suite of mathematics problems involving sequential questions and answers in a free-form textual input/output format. The structured nature of the mathematics domain, covering arithmetic, algebra, probability and calculus, enables the construction of training and test splits designed to clearly illuminate the capabilities and failure-modes of different architectures, as well as evaluate their ability to compose and relate knowledge and learned processes. Having described the data generation process and its potential future expansions, we conduct a comprehensive analysis of models from two broad classes of the most powerful sequence-to-sequence architectures and find notable differences in their ability to resolve mathematical problems and generalize their knowledge.

Mathematical reasoning has long represented one of the most fundamental and challenging frontiers in artificial intelligence research. In recent years, large language models (LLMs) have achieved significant advances in this area. This survey examines the development of mathematical reasoning abilities in LLMs through two high-level cognitive phases: comprehension, where models gain mathematical understanding via diverse pretraining strategies, and answer generation, which has progressed from direct prediction to step-by-step Chain-of-Thought (CoT) reasoning. We review methods for enhancing mathematical reasoning, ranging from training-free prompting to fine-tuning approaches such as supervised fine-tuning and reinforcement learning, and discuss recent work on extended CoT and "test-time scaling". Despite notable progress, fundamental challenges remain in terms of capacity, efficiency, and generalization. To address these issues, we highlight promising research directions, including advanced pretraining and knowledge augmentation techniques, formal reasoning frameworks, and meta-generalization through principled learning paradigms. This survey tries to provide some insights for researchers interested in enhancing reasoning capabilities of LLMs and for those seeking to apply these techniques to other domains.

Conducting contamination-free evaluation of mathematical capabilities can be difficult for two reasons: models may memorize a test set once it is made public, and current mathematical benchmarks are prone to overfitting due to having limited diversity of symbols and rules, coupled with closed-ended answers. This paper proposes a method to leverage these shortcomings as useful features to a construct dynamic, counterfactual benchmark, which can be used to both reveal overfitting and measure true reasoning. We demonstrate this via MatheMagic, which generates math test instances with the interpretations of numbers and operators altered, yet has automatically verifiable answers. Test instances are randomly seeded and constructed at test time to evaluate a model's induction or deduction capability, offering stability, extensibility, comparability, and robustness to overfitting. Our experiments find that models solve deduction more easily than induction, but they revert to standard math. Further analysis reveals that math-adapted models fail to exhibit a general "skill" of reasoning, and fine-tuning on induction tasks generalizes poorly.

Numerical reasoning over natural language has been a long-standing goal for the research community. However, cutting-edge language models have proven difficult to reliably generalize to a broad range of numbers, although they have shown proficiency in reasoning over common and simple numbers. In this paper, we propose a novel method to elicit and exploit the numerical reasoning knowledge hidden in pre-trained language models using simple anchor numbers. Concretely, we first leverage simple numbers as anchors to probe the implicitly inferred arithmetic expressions from language models, and then explicitly apply the expressions on complex numbers to get corresponding answers. To inversely elicit arithmetic expressions, we transform and formulate the task as an analytically solvable linear system. Experimental results on several numerical reasoning benchmarks demonstrate that our approach significantly improves numerical reasoning capabilities of existing LMs. More importantly, our approach is training-free and simply works in the inference phase, making it highly portable and achieving consistent performance benefits across a variety of language models (GPT-3, T5, BART, etc) in all zero-shot, few-shot, and fine-tuning scenarios.

Mathematical reasoning, a core aspect of human cognition, is vital across many domains, from educational problem-solving to scientific advancements. As artificial general intelligence (AGI) progresses, integrating large language models (LLMs) with mathematical reasoning tasks is becoming increasingly significant. This survey provides the first comprehensive analysis of mathematical reasoning in the era of multimodal large language models (MLLMs). We review over 200 studies published since 2021, and examine the state-of-the-art developments in Math-LLMs, with a focus on multimodal settings. We categorize the field into three dimensions: benchmarks, methodologies, and challenges. In particular, we explore multimodal mathematical reasoning pipeline, as well as the role of (M)LLMs and the associated methodologies. Finally, we identify five major challenges hindering the realization of AGI in this domain, offering insights into the future direction for enhancing multimodal reasoning capabilities. This survey serves as a critical resource for the research community in advancing the capabilities of LLMs to tackle complex multimodal reasoning tasks.

While Large Reasoning Models (LRMs) generate extensive chain-of-thought reasoning, we lack a principled framework for understanding how these thoughts are structured. In this paper, we introduce a novel approach by applying Schoenfeld's Episode Theory, a classic cognitive framework for human mathematical problem-solving, to analyze the reasoning traces of LRMs. We annotated thousands of sentences and paragraphs from model-generated solutions to math problems using seven cognitive labels (e.g., Plan, Implement, Verify). The result is the first publicly available benchmark for the fine-grained analysis of machine reasoning, including a large annotated corpus and detailed annotation guidebooks. Our preliminary analysis reveals distinct patterns in LRM reasoning, such as the transition dynamics between cognitive states. This framework provides a theoretically grounded methodology for interpreting LRM cognition and enables future work on more controllable and transparent reasoning systems.

Math reasoning has become the poster child of progress in large language models (LLMs), with new models rapidly surpassing human-level performance on benchmarks like MATH and AIME. But as math leaderboards improve week by week, it is worth asking: do these gains reflect broader problem-solving ability or just narrow overfitting? To answer this question, we evaluate over 20 open-weight reasoning-tuned models across a broad suite of tasks, including math, scientific QA, agent planning, coding, and standard instruction-following. We surprisingly find that most models that succeed in math fail to transfer their gains to other domains. To rigorously study this phenomenon, we conduct controlled experiments on Qwen3-14B models using math-only data but different tuning methods. We find that reinforcement learning (RL)-tuned models generalize well across domains, while supervised fine-tuning (SFT)-tuned models often forget general capabilities. Latent-space representation and token-space distribution shift analyses reveal that SFT induces substantial representation and output drift, while RL preserves general-domain structure. Our results suggest a need to rethink standard post-training recipes, particularly the reliance on SFT-distilled data for advancing reasoning models.

Visual mathematical reasoning, as a fundamental visual reasoning ability, has received widespread attention from the Large Multimodal Models (LMMs) community. Existing benchmarks, such as MathVista and MathVerse, focus more on the result-oriented performance but neglect the underlying principles in knowledge acquisition and generalization. Inspired by human-like mathematical reasoning, we introduce WE-MATH, the first benchmark specifically designed to explore the problem-solving principles beyond end-to-end performance. We meticulously collect and categorize 6.5K visual math problems, spanning 67 hierarchical knowledge concepts and five layers of knowledge granularity. We decompose composite problems into sub-problems according to the required knowledge concepts and introduce a novel four-dimensional metric, namely Insufficient Knowledge (IK), Inadequate Generalization (IG), Complete Mastery (CM), and Rote Memorization (RM), to hierarchically assess inherent issues in LMMs' reasoning process. With WE-MATH, we conduct a thorough evaluation of existing LMMs in visual mathematical reasoning and reveal a negative correlation between solving steps and problem-specific performance. We confirm the IK issue of LMMs can be effectively improved via knowledge augmentation strategies. More notably, the primary challenge of GPT-4o has significantly transitioned from IK to IG, establishing it as the first LMM advancing towards the knowledge generalization stage. In contrast, other LMMs exhibit a marked inclination towards Rote Memorization - they correctly solve composite problems involving multiple knowledge concepts yet fail to answer sub-problems. We anticipate that WE-MATH will open new pathways for advancements in visual mathematical reasoning for LMMs. The WE-MATH data and evaluation code are available at https://github.com/We-Math/We-Math.

Although large language models demonstrate emergent abilities in solving math word problems, there is a challenging task in complex multi-step mathematical reasoning tasks. To improve model performance on mathematical reasoning tasks, previous work has conducted supervised fine-tuning on open-source models by improving the quality and quantity of data. In this paper, we propose a novel approach, named Brain, to imitate human thought processes to enhance mathematical reasoning abilities, using the Frontal Lobe Model to generate plans, and then employing the Parietal Lobe Model to generate code and execute to obtain answers. First, we achieve SOTA performance in comparison with Code LLaMA 7B based models through this method. Secondly, we find that plans can be explicitly extracted from natural language, code, or formal language. Our code and data are publicly available at https://github.com/cyzhh/Brain.

Recent large-scale language models (LLMs) with long Chain-of-Thought reasoning-such as DeepSeek-R1-have achieved impressive results on Olympiad-level mathematics benchmarks. However, they often rely on a narrow set of strategies and struggle with problems that require a novel way of thinking. To systematically investigate these limitations, we introduce OMEGA-Out-of-distribution Math Problems Evaluation with 3 Generalization Axes-a controlled yet diverse benchmark designed to evaluate three axes of out-of-distribution generalization, inspired by Boden's typology of creativity: (1) Exploratory-applying known problem solving skills to more complex instances within the same problem domain; (2) Compositional-combining distinct reasoning skills, previously learned in isolation, to solve novel problems that require integrating these skills in new and coherent ways; and (3) Transformative-adopting novel, often unconventional strategies by moving beyond familiar approaches to solve problems more effectively. OMEGA consists of programmatically generated training-test pairs derived from templated problem generators across geometry, number theory, algebra, combinatorics, logic, and puzzles, with solutions verified using symbolic, numerical, or graphical methods. We evaluate frontier (or top-tier) LLMs and observe sharp performance degradation as problem complexity increases. Moreover, we fine-tune the Qwen-series models across all generalization settings and observe notable improvements in exploratory generalization, while compositional generalization remains limited and transformative reasoning shows little to no improvement. By isolating and quantifying these fine-grained failures, OMEGA lays the groundwork for advancing LLMs toward genuine mathematical creativity beyond mechanical proficiency.

Do large language models (LLMs) solve reasoning tasks by learning robust generalizable algorithms, or do they memorize training data? To investigate this question, we use arithmetic reasoning as a representative task. Using causal analysis, we identify a subset of the model (a circuit) that explains most of the model's behavior for basic arithmetic logic and examine its functionality. By zooming in on the level of individual circuit neurons, we discover a sparse set of important neurons that implement simple heuristics. Each heuristic identifies a numerical input pattern and outputs corresponding answers. We hypothesize that the combination of these heuristic neurons is the mechanism used to produce correct arithmetic answers. To test this, we categorize each neuron into several heuristic types-such as neurons that activate when an operand falls within a certain range-and find that the unordered combination of these heuristic types is the mechanism that explains most of the model's accuracy on arithmetic prompts. Finally, we demonstrate that this mechanism appears as the main source of arithmetic accuracy early in training. Overall, our experimental results across several LLMs show that LLMs perform arithmetic using neither robust algorithms nor memorization; rather, they rely on a "bag of heuristics".

We construct evaluation tasks where extending the reasoning length of Large Reasoning Models (LRMs) deteriorates performance, exhibiting an inverse scaling relationship between test-time compute and accuracy. Our evaluation tasks span four categories: simple counting tasks with distractors, regression tasks with spurious features, deduction tasks with constraint tracking, and advanced AI risks. We identify five distinct failure modes when models reason for longer: 1) Claude models become increasingly distracted by irrelevant information; 2) OpenAI o-series models resist distractors but overfit to problem framings; 3) models shift from reasonable priors to spurious correlations; 4) all models show difficulties in maintaining focus on complex deductive tasks; and 5) extended reasoning may amplify concerning behaviors, with Claude Sonnet 4 showing increased expressions of self-preservation. These findings suggest that while test-time compute scaling remains promising for improving model capabilities, it may inadvertently reinforce problematic reasoning patterns. Our results demonstrate the importance of evaluating models across diverse reasoning lengths to identify and address these failure modes in LRMs.

Mathematical reasoning remains one of the most challenging domains for large language models (LLMs), requiring not only linguistic understanding but also structured logical deduction and numerical precision. While recent LLMs demonstrate strong general-purpose reasoning abilities, their mathematical competence across diverse languages remains underexplored. Existing benchmarks primarily focus on English or a narrow subset of high-resource languages, leaving significant gaps in assessing multilingual and cross-lingual mathematical reasoning. To address this, we introduce MathMist, a parallel multilingual benchmark for mathematical problem solving and reasoning. MathMist encompasses over 21K aligned question-answer pairs across seven languages, representing a balanced coverage of high-, medium-, and low-resource linguistic settings. The dataset captures linguistic variety, multiple types of problem settings, and solution synthesizing capabilities. We systematically evaluate a diverse suite of models, including open-source small and medium LLMs, proprietary systems, and multilingual-reasoning-focused models, under zero-shot, chain-of-thought (CoT), and code-switched reasoning paradigms. Our results reveal persistent deficiencies in LLMs' ability to perform consistent and interpretable mathematical reasoning across languages, with pronounced degradation in low-resource settings. All the codes and data are available at GitHub: https://github.com/mahbubhimel/MathMist

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 a fundamental discovery that challenges our understanding of how complex reasoning emerges in large language models. While conventional wisdom suggests that sophisticated reasoning tasks demand extensive training data (>100,000 examples), we demonstrate that complex mathematical reasoning abilities can be effectively elicited with surprisingly few examples. Through comprehensive experiments, our proposed model LIMO demonstrates unprecedented performance in mathematical reasoning. With merely 817 curated training samples, LIMO achieves 57.1% accuracy on AIME and 94.8% on MATH, improving from previous SFT-based models' 6.5% and 59.2% respectively, while only using 1% of the training data required by previous approaches. LIMO demonstrates exceptional out-of-distribution generalization, achieving 40.5% absolute improvement across 10 diverse benchmarks, outperforming models trained on 100x more data, challenging the notion that SFT leads to memorization rather than generalization. Based on these results, we propose the Less-Is-More Reasoning Hypothesis (LIMO Hypothesis): In foundation models where domain knowledge has been comprehensively encoded during pre-training, sophisticated reasoning capabilities can emerge through minimal but precisely orchestrated demonstrations of cognitive processes. This hypothesis posits that the elicitation threshold for complex reasoning is determined by two key factors: (1) the completeness of the model's encoded knowledge foundation during pre-training, and (2) the effectiveness of post-training examples as "cognitive templates" that show the model how to utilize its knowledge base to solve complex reasoning tasks. To facilitate reproducibility and future research in data-efficient reasoning, we release LIMO as a comprehensive open-source suite at https://github.com/GAIR-NLP/LIMO.

Recent advances in language models have demonstrated their capability to solve mathematical reasoning problems, achieving near-perfect accuracy on grade-school level math benchmarks like GSM8K. In this paper, we formally study how language models solve these problems. We design a series of controlled experiments to address several fundamental questions: (1) Can language models truly develop reasoning skills, or do they simply memorize templates? (2) What is the model's hidden (mental) reasoning process? (3) Do models solve math questions using skills similar to or different from humans? (4) Do models trained on GSM8K-like datasets develop reasoning skills beyond those necessary for solving GSM8K problems? (5) What mental process causes models to make reasoning mistakes? (6) How large or deep must a model be to effectively solve GSM8K-level math questions? Our study uncovers many hidden mechanisms by which language models solve mathematical questions, providing insights that extend beyond current understandings of LLMs.

Large Language Models (LLMs) have been applied to Math Word Problems (MWPs) with transformative impacts, revolutionizing how these complex problems are approached and solved in various domains including educational settings. However, the evaluation of these models often prioritizes final accuracy, overlooking the crucial aspect of reasoning capabilities. This work addresses this gap by focusing on the ability of LLMs to detect and correct reasoning mistakes. We introduce a novel dataset MWP-MISTAKE, incorporating MWPs with both correct and incorrect reasoning steps generated through rule-based methods and smaller language models. Our comprehensive benchmarking reveals significant insights into the strengths and weaknesses of state-of-the-art models, such as GPT-4o, GPT-4, GPT-3.5Turbo, and others. We highlight GPT-$o's superior performance in mistake detection and rectification and the persistent challenges faced by smaller models. Additionally, we identify issues related to data contamination and memorization, impacting the reliability of LLMs in real-world applications. Our findings emphasize the importance of rigorous evaluation of reasoning processes and propose future directions to enhance the generalization and robustness of LLMs in mathematical problem-solving.

The surprising ability of Large Language Models (LLMs) to perform well on complex reasoning with only few-shot chain-of-thought prompts is believed to emerge only in very large-scale models (100+ billion parameters). We show that such abilities can, in fact, be distilled down from GPT-3.5 (ge 175B) to T5 variants (le 11B). We propose model specialization, to specialize the model's ability towards a target task. The hypothesis is that large models (commonly viewed as larger than 100B) have strong modeling power, but are spread on a large spectrum of tasks. Small models (commonly viewed as smaller than 10B) have limited model capacity, but if we concentrate their capacity on a specific target task, the model can achieve a decent improved performance. We use multi-step math reasoning as our testbed because it is a very typical emergent ability. We show two important aspects of model abilities: (1). there exists a very complex balance/ tradeoff between language models' multi-dimensional abilities; (2). by paying the price of decreased generic ability, we can clearly lift up the scaling curve of models smaller than 10B towards a specialized multi-step math reasoning ability. We further give comprehensive discussions about important design choices for better generalization, including the tuning data format, the start model checkpoint, and a new model selection method. We hope our practice and discoveries can serve as an important attempt towards specialized smaller models in the new research paradigm set by LLMs.

Whether neural networks can learn abstract reasoning or whether they merely rely on superficial statistics is a topic of recent debate. Here, we propose a dataset and challenge designed to probe abstract reasoning, inspired by a well-known human IQ test. To succeed at this challenge, models must cope with various generalisation `regimes' in which the training and test data differ in clearly-defined ways. We show that popular models such as ResNets perform poorly, even when the training and test sets differ only minimally, and we present a novel architecture, with a structure designed to encourage reasoning, that does significantly better. When we vary the way in which the test questions and training data differ, we find that our model is notably proficient at certain forms of generalisation, but notably weak at others. We further show that the model's ability to generalise improves markedly if it is trained to predict symbolic explanations for its answers. Altogether, we introduce and explore ways to both measure and induce stronger abstract reasoning in neural networks. Our freely-available dataset should motivate further progress in this direction.

Recent advancements in Large Language Models (LLMs) have sparked interest in their formal reasoning capabilities, particularly in mathematics. The GSM8K benchmark is widely used to assess the mathematical reasoning of models on grade-school-level questions. While the performance of LLMs on GSM8K has significantly improved in recent years, it remains unclear whether their mathematical reasoning capabilities have genuinely advanced, raising questions about the reliability of the reported metrics. To address these concerns, we conduct a large-scale study on several SOTA open and closed models. To overcome the limitations of existing evaluations, we introduce GSM-Symbolic, an improved benchmark created from symbolic templates that allow for the generation of a diverse set of questions. GSM-Symbolic enables more controllable evaluations, providing key insights and more reliable metrics for measuring the reasoning capabilities of models.Our findings reveal that LLMs exhibit noticeable variance when responding to different instantiations of the same question. Specifically, the performance of all models declines when only the numerical values in the question are altered in the GSM-Symbolic benchmark. Furthermore, we investigate the fragility of mathematical reasoning in these models and show that their performance significantly deteriorates as the number of clauses in a question increases. We hypothesize that this decline is because current LLMs cannot perform genuine logical reasoning; they replicate reasoning steps from their training data. Adding a single clause that seems relevant to the question causes significant performance drops (up to 65%) across all state-of-the-art models, even though the clause doesn't contribute to the reasoning chain needed for the final answer. Overall, our work offers a more nuanced understanding of LLMs' capabilities and limitations in mathematical reasoning.

Large language models have shown impressive results for multi-hop mathematical reasoning when the input question is only textual. Many mathematical reasoning problems, however, contain both text and image. With the ever-increasing adoption of vision language models (VLMs), understanding their reasoning abilities for such problems is crucial. In this paper, we evaluate the reasoning capabilities of VLMs along various axes through the lens of geometry problems. We procedurally create a synthetic dataset of geometry questions with controllable difficulty levels along multiple axes, thus enabling a systematic evaluation. The empirical results obtained using our benchmark for state-of-the-art VLMs indicate that these models are not as capable in subjects like geometry (and, by generalization, other topics requiring similar reasoning) as suggested by previous benchmarks. This is made especially clear by the construction of our benchmark at various depth levels, since solving higher-depth problems requires long chains of reasoning rather than additional memorized knowledge. We release the dataset for further research in this area.

Mathematical reasoning represents a critical frontier in advancing large language models (LLMs). While step-by-step approaches have emerged as the dominant paradigm for mathematical problem-solving in LLMs, the quality of reasoning steps in training data fundamentally constrains the performance of the models. Recent studies has demonstrated that more detailed intermediate steps can enhance model performance, yet existing methods for step expansion either require more powerful external models or incur substantial computational costs. In this paper, we introduce MathFimer, a novel framework for mathematical reasoning step expansion inspired by the "Fill-in-the-middle" task from code completion. By decomposing solution chains into prefix-suffix pairs and training models to reconstruct missing intermediate steps, we develop a specialized model, MathFimer-7B, on our carefully curated NuminaMath-FIM dataset. We then apply these models to enhance existing mathematical reasoning datasets by inserting detailed intermediate steps into their solution chains, creating MathFimer-expanded versions. Through comprehensive experiments on multiple mathematical reasoning datasets, including MathInstruct, MetaMathQA and etc., we demonstrate that models trained on MathFimer-expanded data consistently outperform their counterparts trained on original data across various benchmarks such as GSM8K and MATH. Our approach offers a practical, scalable solution for enhancing mathematical reasoning capabilities in LLMs without relying on powerful external models or expensive inference procedures.

There is a substantial body of literature examining the mathematical reasoning capabilities of large language models (LLMs), particularly their performance on precise arithmetic operations in autoregressive architectures. However, their ability to perform approximate reasoning in informal, fast-paced mathematical operations has received far less attention, especially among non-autoregressive decoder models. Our work addresses this gap by introducing StreetMath, a benchmark designed to evaluate models' approximation abilities under real-world approximation scenarios. We conduct extensive evaluations across different LLM architectures: Qwen3-4B-Instruct-2507, Qwen3-4B-Thinking-2507, Dream-v0-Instruct-7B, Falcon-Mamba-7B-Instruct, and Mamba-GPT-3B. Furthermore, we apply mechanistic interpretability techniques to probe their internal computational states. Our analysis reveals that LLMs generally attempt to compute exact values or invoke external tools even in tasks that call for approximation. Moreover, while models sometimes reach the correct answer in early layers or steps, they still consume more tokens when solving approximation tasks. Additional experiments indicate that exact and approximate arithmetic operations rely on largely separate neural components. Drawing upon research on cognitive psychology, we argue that LLMs do not exhibit cognitive miserliness in the same way humans do in street math settings. We open source our work https://github.com/ctseng777/StreetMath

Learning from mistakes is a fundamental feature of human intelligence. Previous work has shown that Large Language Models (LLMs) can also learn from incorrect answers when provided with a comprehensive rationale detailing why an answer is wrong or how to correct it. In this work, we examine whether LLMs can learn from mistakes in mathematical reasoning tasks when these explanations are not provided. We investigate if LLMs are able to implicitly infer such rationales simply from observing both incorrect and correct answers. Surprisingly, we find that LLMs perform better, on average, when rationales are eliminated from the context and incorrect answers are simply shown alongside correct ones. This approach also substantially outperforms chain-of-thought prompting in our evaluations. We show that these results are consistent across LLMs of different sizes and varying reasoning abilities. Further, we carry out an in-depth analysis, and show that prompting with both wrong and correct answers leads to greater performance and better generalisation than introducing additional, more diverse question-answer pairs into the context. Finally, we show that new rationales generated by models that have only observed incorrect and correct answers are scored equally as highly by humans as those produced with the aid of exemplar rationales. Our results demonstrate that LLMs are indeed capable of in-context implicit learning.

In this paper we look at the ability of recent large language models (LLMs) at solving mathematical problems in combinatorics. We compare models LLaMA-2, LLaMA-3.1, GPT-4, and Mixtral against each other and against human pupils and undergraduates with prior experience in mathematical olympiads. To facilitate these comparisons we introduce the Combi-Puzzles dataset, which contains 125 problem variants based on 25 combinatorial reasoning problems. Each problem is presented in one of five distinct forms, created by systematically manipulating the problem statements through adversarial additions, numeric parameter changes, and linguistic obfuscation. Our variations preserve the mathematical core and are designed to measure the generalisability of LLM problem-solving abilities, while also increasing confidence that problems are submitted to LLMs in forms that have not been seen as training instances. We found that a model based on GPT-4 outperformed all other models in producing correct responses, and performed significantly better in the mathematical variation of the problems than humans. We also found that modifications to problem statements significantly impact the LLM's performance, while human performance remains unaffected.

Mathematical reasoning is a cornerstone of artificial general intelligence and a primary benchmark for evaluating the capabilities of Large Language Models (LLMs). While state-of-the-art models show promise, they often falter when faced with complex problems that demand deep conceptual understanding and intricate, multi-step deliberation. To address this challenge, we introduce JT-Math-8B, a series of open-source models comprising base, instruct, and thinking versions, built upon a systematic, multi-stage optimization framework. Our pre-training corpus is a high-quality, 210B-token dataset curated through a dedicated data pipeline that uses model-based validation to ensure quality and diversity. The Instruct Model is optimized for direct, concise answers through Supervised Fine-Tuning (SFT) and a GRPO-based reinforcement learning (RL) method. The Thinking Model is trained for complex problem-solving using a Long Chain-of-Thought (Long CoT) approach, combining SFT with a novel, multi-stage RL curriculum that progressively increases task difficulty and context length up to 32K tokens. JT-Math-8B achieves state-of-the-art results among open-source models of similar size, surpassing prominent models like OpenAI's O1-mini and GPT-4o , and demonstrating superior performance on competition-level mathematics.

Mathematical reasoning skills are essential for general-purpose intelligent systems to perform tasks from grocery shopping to climate modeling. Towards evaluating and improving AI systems in this domain, we propose LILA, a unified mathematical reasoning benchmark consisting of 23 diverse tasks along four dimensions: (i) mathematical abilities e.g., arithmetic, calculus (ii) language format e.g., question-answering, fill-in-the-blanks (iii) language diversity e.g., no language, simple language (iv) external knowledge e.g., commonsense, physics. We construct our benchmark by extending 20 datasets benchmark by collecting task instructions and solutions in the form of Python programs, thereby obtaining explainable solutions in addition to the correct answer. We additionally introduce two evaluation datasets to measure out-of-distribution performance and robustness to language perturbation. Finally, we introduce BHASKARA, a general-purpose mathematical reasoning model trained on LILA. Importantly, we find that multi-tasking leads to significant improvements (average relative improvement of 21.83% F1 score vs. single-task models), while the best performing model only obtains 60.40%, indicating the room for improvement in general mathematical reasoning and understanding.

The standard methodology of evaluating large language models (LLMs) based on static pairs of inputs and outputs is insufficient for developing assistants: this kind of assessments fails to take into account the essential interactive element in their deployment, and therefore limits how we understand language model capabilities. We introduce CheckMate, an adaptable prototype platform for humans to interact with and evaluate LLMs. We conduct a study with CheckMate to evaluate three language models~(InstructGPT, ChatGPT, and GPT-4) as assistants in proving undergraduate-level mathematics, with a mixed cohort of participants from undergraduate students to professors of mathematics. We release the resulting interaction and rating dataset, MathConverse. By analysing MathConverse, we derive a preliminary taxonomy of human behaviours and uncover that despite a generally positive correlation, there are notable instances of divergence between correctness and perceived helpfulness in LLM generations, amongst other findings. Further, we identify useful scenarios and existing issues of GPT-4 in mathematical reasoning through a series of case studies contributed by expert mathematicians. We conclude with actionable takeaways for ML practitioners and mathematicians: models which communicate uncertainty, respond well to user corrections, are more interpretable and concise may constitute better assistants; interactive evaluation is a promising way to continually navigate the capability of these models; humans should be aware of language models' algebraic fallibility, and for that reason discern where they should be used.

Recent supervised fine-tuning (SFT) approaches have significantly improved language models' performance on mathematical reasoning tasks, even when models are trained at a small scale. However, the specific capabilities enhanced through such fine-tuning remain poorly understood. In this paper, we conduct a detailed analysis of model performance on the AIME24 dataset to understand how reasoning capabilities evolve. We discover a ladder-like structure in problem difficulty, categorize questions into four tiers (Easy, Medium, Hard, and Extremely Hard (Exh)), and identify the specific requirements for advancing between tiers. We find that progression from Easy to Medium tier requires adopting an R1 reasoning style with minimal SFT (500-1K instances), while Hard-level questions suffer from frequent model's errors at each step of the reasoning chain, with accuracy plateauing at around 65% despite logarithmic scaling. Exh-level questions present a fundamentally different challenge; they require unconventional problem-solving skills that current models uniformly struggle with. Additional findings reveal that carefully curated small-scale datasets offer limited advantage-scaling dataset size proves far more effective. Our analysis provides a clearer roadmap for advancing language model capabilities in mathematical reasoning.

Mathematical capabilities were previously believed to emerge in common language models only at a very large scale or require extensive math-related pre-training. This paper shows that the LLaMA-2 7B model with common pre-training already exhibits strong mathematical abilities, as evidenced by its impressive accuracy of 97.7% and 72.0% on the GSM8K and MATH benchmarks, respectively, when selecting the best response from 256 random generations. The primary issue with the current base model is the difficulty in consistently eliciting its inherent mathematical capabilities. Notably, the accuracy for the first answer drops to 49.5% and 7.9% on the GSM8K and MATH benchmarks, respectively. We find that simply scaling up the SFT data can significantly enhance the reliability of generating correct answers. However, the potential for extensive scaling is constrained by the scarcity of publicly available math questions. To overcome this limitation, we employ synthetic data, which proves to be nearly as effective as real data and shows no clear saturation when scaled up to approximately one million samples. This straightforward approach achieves an accuracy of 82.6% on GSM8K and 40.6% on MATH using LLaMA-2 7B models, surpassing previous models by 14.2% and 20.8%, respectively. We also provide insights into scaling behaviors across different reasoning complexities and error types.

Metacognitive knowledge refers to humans' intuitive knowledge of their own thinking and reasoning processes. Today's best LLMs clearly possess some reasoning processes. The paper gives evidence that they also have metacognitive knowledge, including ability to name skills and procedures to apply given a task. We explore this primarily in context of math reasoning, developing a prompt-guided interaction procedure to get a powerful LLM to assign sensible skill labels to math questions, followed by having it perform semantic clustering to obtain coarser families of skill labels. These coarse skill labels look interpretable to humans. To validate that these skill labels are meaningful and relevant to the LLM's reasoning processes we perform the following experiments. (a) We ask GPT-4 to assign skill labels to training questions in math datasets GSM8K and MATH. (b) When using an LLM to solve the test questions, we present it with the full list of skill labels and ask it to identify the skill needed. Then it is presented with randomly selected exemplar solved questions associated with that skill label. This improves accuracy on GSM8k and MATH for several strong LLMs, including code-assisted models. The methodology presented is domain-agnostic, even though this article applies it to math problems.

Large language models (LLMs) have achieved impressive performance across various mathematical reasoning benchmarks. However, there are increasing debates regarding whether these models truly understand and apply mathematical knowledge or merely rely on shortcuts for mathematical reasoning. One essential and frequently occurring evidence is that when the math questions are slightly changed, LLMs can behave incorrectly. This motivates us to evaluate the robustness of LLMs' math reasoning capability by testing a wide range of question variations. We introduce the adversarial grade school math (\datasetname) dataset, an extension of GSM8K augmented with various mathematical perturbations. Our experiments on 25 LLMs and 4 prompting techniques show that while LLMs exhibit different levels of math reasoning abilities, their performances are far from robust. In particular, even for problems that have been solved in GSM8K, LLMs can make mistakes when new statements are added or the question targets are altered. We also explore whether more robust performance can be achieved by composing existing prompting methods, in which we try an iterative method that generates and verifies each intermediate thought based on its reasoning goal and calculation result. Code and data are available at https://github.com/qtli/GSM-Plus.

Mathematical reasoning in large language models (LMs) has garnered significant attention in recent work, but there is a limited understanding of how these models process and store information related to arithmetic tasks within their architecture. In order to improve our understanding of this aspect of language models, we present a mechanistic interpretation of Transformer-based LMs on arithmetic questions using a causal mediation analysis framework. By intervening on the activations of specific model components and measuring the resulting changes in predicted probabilities, we identify the subset of parameters responsible for specific predictions. This provides insights into how information related to arithmetic is processed by LMs. Our experimental results indicate that LMs process the input by transmitting the information relevant to the query from mid-sequence early layers to the final token using the attention mechanism. Then, this information is processed by a set of MLP modules, which generate result-related information that is incorporated into the residual stream. To assess the specificity of the observed activation dynamics, we compare the effects of different model components on arithmetic queries with other tasks, including number retrieval from prompts and factual knowledge questions.

Large Language Models (LLMs) have demonstrated impressive capabilities in natural language processing tasks, such as text generation and semantic understanding. However, their performance on numerical reasoning tasks, such as basic arithmetic, numerical retrieval, and magnitude comparison, remains surprisingly poor. This gap arises from their reliance on surface-level statistical patterns rather than understanding numbers as continuous magnitudes. Existing benchmarks primarily focus on either linguistic competence or structured mathematical problem-solving, neglecting fundamental numerical reasoning required in real-world scenarios. To bridge this gap, we propose NumericBench, a comprehensive benchmark to evaluate six fundamental numerical capabilities: number recognition, arithmetic operations, contextual retrieval, comparison, summary, and logical reasoning. NumericBench includes datasets ranging from synthetic number lists to the crawled real-world data, addressing challenges like long contexts, noise, and multi-step reasoning. Extensive experiments on state-of-the-art LLMs, including GPT-4 and DeepSeek, reveal persistent weaknesses in numerical reasoning, highlighting the urgent need to improve numerically-aware language modeling. The benchmark is released in: https://github.com/TreeAI-Lab/NumericBench.

We present Hermes 4, a family of hybrid reasoning models that combine structured, multi-turn reasoning with broad instruction-following ability. We describe the challenges encountered during data curation, synthesis, training, and evaluation, and outline the solutions employed to address these challenges at scale. We comprehensively evaluate across mathematical reasoning, coding, knowledge, comprehension, and alignment benchmarks, and we report both quantitative performance and qualitative behavioral analysis. To support open research, all model weights are published publicly at https://huggingface.co/collections/NousResearch/hermes-4-collection-68a731bfd452e20816725728

Neural networks can learn to represent and manipulate numerical information, but they seldom generalize well outside of the range of numerical values encountered during training. To encourage more systematic numerical extrapolation, we propose an architecture that represents numerical quantities as linear activations which are manipulated using primitive arithmetic operators, controlled by learned gates. We call this module a neural arithmetic logic unit (NALU), by analogy to the arithmetic logic unit in traditional processors. Experiments show that NALU-enhanced neural networks can learn to track time, perform arithmetic over images of numbers, translate numerical language into real-valued scalars, execute computer code, and count objects in images. In contrast to conventional architectures, we obtain substantially better generalization both inside and outside of the range of numerical values encountered during training, often extrapolating orders of magnitude beyond trained numerical ranges.

Given the ubiquitous nature of numbers in text, reasoning with numbers to perform simple calculations is an important skill of AI systems. While many datasets and models have been developed to this end, state-of-the-art AI systems are brittle; failing to perform the underlying mathematical reasoning when they appear in a slightly different scenario. Drawing inspiration from GLUE that was proposed in the context of natural language understanding, we propose NumGLUE, a multi-task benchmark that evaluates the performance of AI systems on eight different tasks, that at their core require simple arithmetic understanding. We show that this benchmark is far from being solved with neural models including state-of-the-art large-scale language models performing significantly worse than humans (lower by 46.4%). Further, NumGLUE promotes sharing knowledge across tasks, especially those with limited training data as evidenced by the superior performance (average gain of 3.4% on each task) when a model is jointly trained on all the tasks as opposed to task-specific modeling. Finally, we hope that NumGLUE will encourage systems that perform robust and general arithmetic reasoning within language, a first step towards being able to perform more complex mathematical reasoning.

Large language models (LLMs) can solve an increasing number of complex reasoning tasks while making surprising mistakes in basic numerical understanding and processing (such as 9.11 > 9.9). The latter ability is essential for tackling complex arithmetic and mathematical problems and serves as a foundation for most reasoning tasks, but previous work paid little attention to it or only discussed several restricted tasks (like integer addition). In this paper, we comprehensively investigate the numerical understanding and processing ability (NUPA) of LLMs. Firstly, we introduce a benchmark covering four common numerical representations and 17 distinct numerical tasks in four major categories, resulting in 41 meaningful combinations in total. These tasks are derived from primary and secondary education curricula, encompassing nearly all everyday numerical understanding and processing scenarios, and the rules of these tasks are very simple and clear. Through the benchmark, we find that current LLMs fail frequently in many of the tasks. To study the problem, we train small models with existing and potential techniques for enhancing NUPA (such as tokenizers, PEs, and number formats), comprehensively evaluating their effectiveness using our testbed. We also finetune practical-scale LLMs on our proposed NUPA tasks and find that 1) naive finetuning can improve NUPA a lot on many but not all tasks, and 2) surprisingly, techniques designed to enhance NUPA prove ineffective for finetuning pretrained models. We further explore the impact of chain-of-thought techniques on NUPA. Our work provides a more detailed and comprehensive understanding of NUPA in LLMs. Our benchmark and code are released at https://github.com/GraphPKU/number_cookbook.

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.

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.

Numerical reasoning over text (NRoT) presents unique challenges that are not well addressed by existing pre-training objectives. We explore five sequential training schedules that adapt a pre-trained T5 model for NRoT. Our final model is adapted from T5, but further pre-trained on three datasets designed to strengthen skills necessary for NRoT and general reading comprehension before being fine-tuned on the Discrete Reasoning over Text (DROP) dataset. The training improves DROP's adjusted F1 performance (a numeracy-focused score) from 45.90 to 70.83. Our model closes in on GenBERT (72.4), a custom BERT-Base model using the same datasets with significantly more parameters. We show that training the T5 multitasking framework with multiple numerical reasoning datasets of increasing difficulty, good performance on DROP can be achieved without manually engineering partitioned functionality between distributed and symbol modules.

Human cognition exhibits systematic compositionality, the algebraic ability to generate infinite novel combinations from finite learned components, which is the key to understanding and reasoning about complex logic. In this work, we investigate the compositionality of large language models (LLMs) in mathematical reasoning. Specifically, we construct a new dataset MathTrap by introducing carefully designed logical traps into the problem descriptions of MATH and GSM8K. Since problems with logical flaws are quite rare in the real world, these represent "unseen" cases to LLMs. Solving these requires the models to systematically compose (1) the mathematical knowledge involved in the original problems with (2) knowledge related to the introduced traps. Our experiments show that while LLMs possess both components of requisite knowledge, they do not spontaneously combine them to handle these novel cases. We explore several methods to mitigate this deficiency, such as natural language prompts, few-shot demonstrations, and fine-tuning. Additionally, we test the recently released OpenAI o1 model and find that human-like `slow thinking' helps improve the compositionality of LLMs. Overall, systematic compositionality remains an open challenge for large language models.

Reasoning in mathematical domains remains a significant challenge for relatively small language models (LMs). Many current methods focus on specializing LMs in mathematical reasoning and rely heavily on knowledge distillation from powerful but inefficient large LMs (LLMs). In this work, we explore a new direction that avoids over-reliance on LLM teachers, introducing a multi-view fine-tuning method that efficiently exploits existing mathematical problem datasets with diverse annotation styles. Our approach uniquely considers the various annotation formats as different "views" and leverages them in training the model. By postpending distinct instructions to input questions, models can learn to generate solutions in diverse formats in a flexible manner. Experimental results show that our strategy enables a LLaMA-7B model to outperform prior approaches that utilize knowledge distillation, as well as carefully established baselines. Additionally, the proposed method grants the models promising generalization ability across various views and datasets, and the capability to learn from inaccurate or incomplete noisy data. We hope our multi-view training paradigm could inspire future studies in other machine reasoning domains.

Recent research has shown that smaller language models can acquire substantial reasoning abilities when fine-tuned with reasoning exemplars crafted by a significantly larger teacher model. We explore this paradigm for the financial domain, focusing on the challenge of answering questions that require multi-hop numerical reasoning over financial texts. We assess the performance of several smaller models that have been fine-tuned to generate programs that encode the required financial reasoning and calculations. Our findings demonstrate that these fine-tuned smaller models approach the performance of the teacher model. To provide a granular analysis of model performance, we propose an approach to investigate the specific student model capabilities that are enhanced by fine-tuning. Our empirical analysis indicates that fine-tuning refines the student models ability to express and apply the required financial concepts along with adapting the entity extraction for the specific data format. In addition, we hypothesize and demonstrate that comparable financial reasoning capability can be induced using relatively smaller datasets.

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.

The capabilities and limitations of Large Language Models have been sketched out in great detail in recent years, providing an intriguing yet conflicting picture. On the one hand, LLMs demonstrate a general ability to solve problems. On the other hand, they show surprising reasoning gaps when compared to humans, casting doubt on the robustness of their generalisation strategies. The sheer volume of data used in the design of LLMs has precluded us from applying the method traditionally used to measure generalisation: train-test set separation. To overcome this, we study what kind of generalisation strategies LLMs employ when performing reasoning tasks by investigating the pretraining data they rely on. For two models of different sizes (7B and 35B) and 2.5B of their pretraining tokens, we identify what documents influence the model outputs for three simple mathematical reasoning tasks and contrast this to the data that are influential for answering factual questions. We find that, while the models rely on mostly distinct sets of data for each factual question, a document often has a similar influence across different reasoning questions within the same task, indicating the presence of procedural knowledge. We further find that the answers to factual questions often show up in the most influential data. However, for reasoning questions the answers usually do not show up as highly influential, nor do the answers to the intermediate reasoning steps. When we characterise the top ranked documents for the reasoning questions qualitatively, we confirm that the influential documents often contain procedural knowledge, like demonstrating how to obtain a solution using formulae or code. Our findings indicate that the approach to reasoning the models use is unlike retrieval, and more like a generalisable strategy that synthesises procedural knowledge from documents doing a similar form of reasoning.

Many challenging reasoning tasks require not just rapid, intuitive responses, but a more deliberate, multi-step approach. Recent progress in large language models (LLMs) highlights an important shift from the "System 1" way of quick reactions to the "System 2" style of reflection-and-correction problem solving. However, current benchmarks heavily rely on the final-answer accuracy, leaving much of a model's intermediate reasoning steps unexamined. This fails to assess the model's ability to reflect and rectify mistakes within the reasoning process. To bridge this gap, we introduce FINEREASON, a logic-puzzle benchmark for fine-grained evaluation of LLMs' reasoning capabilities. Each puzzle can be decomposed into atomic steps, making it ideal for rigorous validation of intermediate correctness. Building on this, we introduce two tasks: state checking, and state transition, for a comprehensive evaluation of how models assess the current situation and plan the next move. To support broader research, we also provide a puzzle training set aimed at enhancing performance on general mathematical tasks. We show that models trained on our state checking and transition data demonstrate gains in math reasoning by up to 5.1% on GSM8K.

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.

Mathematical reasoning, a core ability of human intelligence, presents unique challenges for machines in abstract thinking and logical reasoning. Recent large pre-trained language models such as GPT-3 have achieved remarkable progress on mathematical reasoning tasks written in text form, such as math word problems (MWP). However, it is unknown if the models can handle more complex problems that involve math reasoning over heterogeneous information, such as tabular data. To fill the gap, we present Tabular Math Word Problems (TabMWP), a new dataset containing 38,431 open-domain grade-level problems that require mathematical reasoning on both textual and tabular data. Each question in TabMWP is aligned with a tabular context, which is presented as an image, semi-structured text, and a structured table. There are two types of questions: free-text and multi-choice, and each problem is annotated with gold solutions to reveal the multi-step reasoning process. We evaluate different pre-trained models on TabMWP, including the GPT-3 model in a few-shot setting. As earlier studies suggest, since few-shot GPT-3 relies on the selection of in-context examples, its performance is unstable and can degrade to near chance. The unstable issue is more severe when handling complex problems like TabMWP. To mitigate this, we further propose a novel approach, PromptPG, which utilizes policy gradient to learn to select in-context examples from a small amount of training data and then constructs the corresponding prompt for the test example. Experimental results show that our method outperforms the best baseline by 5.31% on the accuracy metric and reduces the prediction variance significantly compared to random selection, which verifies its effectiveness in selecting in-context examples.

We propose a framework for robust evaluation of reasoning capabilities of language models, using functional variants of benchmarks. Models that solve a reasoning test should exhibit no difference in performance over the static version of a problem compared to a snapshot of the functional variant. We have rewritten the relevant fragment of the MATH benchmark into its functional variant MATH(), with functionalization of other benchmarks to follow. When evaluating current state-of-the-art models over snapshots of MATH(), we find a reasoning gap -- the percentage difference between the static and functional accuracies. We find reasoning gaps from 58.35% to 80.31% among the state-of-the-art closed and open weights models that perform well on static benchmarks, with the caveat that the gaps are likely to be smaller with more sophisticated prompting strategies. Here we show that models which anecdotally have good reasoning performance over real-world tasks, have quantifiable lower gaps, motivating the open problem of building "gap 0" models. Code for evaluation and new evaluation datasets, three MATH() snapshots, are publicly available at https://github.com/consequentai/fneval/.

The nature of abstract reasoning is a matter of debate. Modern artificial neural network (ANN) models, like large language models, demonstrate impressive success when tested on abstract reasoning problems. However, it has been argued that their success reflects some form of memorization of similar problems (data contamination) rather than a general-purpose abstract reasoning capability. This concern is supported by evidence of brittleness, and the requirement of extensive training. In our study, we explored whether abstract reasoning can be achieved using the toolbox of ANNs, without prior training. Specifically, we studied an ANN model in which the weights of a naive network are optimized during the solution of the problem, using the problem data itself, rather than any prior knowledge. We tested this modeling approach on visual reasoning problems and found that it performs relatively well. Crucially, this success does not rely on memorization of similar problems. We further suggest an explanation of how it works. Finally, as problem solving is performed by changing the ANN weights, we explored the connection between problem solving and the accumulation of knowledge in the ANNs.

Large language models (LLMs) have seen considerable advancements in natural language understanding tasks, yet there remains a gap to bridge before attaining true artificial general intelligence, especially concerning shortcomings in mathematical reasoning capabilities. We postulate that the inherent nature of LLM training, which focuses on predicting probabilities of next token, presents challenges in effectively modeling mathematical reasoning that demands exact calculations, both from data-driven and theoretical standpoints. In this paper, we address this challenge by enriching the data landscape and introducing a novel math dataset, enhanced with a capability to utilize a Python code interpreter. This dataset is derived from GSM8K and MATH and has been further refined through a combination of GPT-4 annotations, human review, and self-training processes, where the errors in the original GSM8K training set have been fixed. Additionally, we propose a tentative, easily replicable protocol for the fine-tuning of math-specific LLMs, which has led to a significant improvement in the performance of a 7B-parameter LLM on the GSM8K and MATH datasets. We are committed to advancing the field of mathematical reasoning in LLMs and, to that end, we have made the model checkpoints and will make the dataset publicly available. We hope this will facilitate further research and development within the community.

The ability to understand and work with numbers (numeracy) is critical for many complex reasoning tasks. Currently, most NLP models treat numbers in text in the same way as other tokens---they embed them as distributed vectors. Is this enough to capture numeracy? We begin by investigating the numerical reasoning capabilities of a state-of-the-art question answering model on the DROP dataset. We find this model excels on questions that require numerical reasoning, i.e., it already captures numeracy. To understand how this capability emerges, we probe token embedding methods (e.g., BERT, GloVe) on synthetic list maximum, number decoding, and addition tasks. A surprising degree of numeracy is naturally present in standard embeddings. For example, GloVe and word2vec accurately encode magnitude for numbers up to 1,000. Furthermore, character-level embeddings are even more precise---ELMo captures numeracy the best for all pre-trained methods---but BERT, which uses sub-word units, is less exact.

Large Language Models (LLMs) have shown remarkable performance in various natural language processing tasks but face challenges in mathematical reasoning, where complex problem-solving requires both linguistic understanding and mathematical reasoning skills. Existing approaches to address this challenge often rely on ensemble methods and suffer from the problem of data scarcity in target domains. In this work, we present a novel method to enhance LLMs' capabilities in mathematical reasoning tasks. Motivated by the need to bridge this gap, our approach incorporates a question paraphrase strategy, which aims at diversifying the linguistic forms of mathematical questions to improve generalization. Additionally, specialized training objectives are employed to guide the model's learning process, focusing on enhancing its understanding of mathematical concepts and reasoning processes. We conduct experiments on four datasets using different LLMs, and demonstrate the effectiveness of our approach in improving LLMs' performance on mathematical reasoning tasks. Our findings underscore the significance of our methodology in the advancement of large language models and its potential implications for real-world applications that require mathematical reasoning abilities.

The ability to extrapolate from short problem instances to longer ones is an important form of out-of-distribution generalization in reasoning tasks, and is crucial when learning from datasets where longer problem instances are rare. These include theorem proving, solving quantitative mathematics problems, and reading/summarizing novels. In this paper, we run careful empirical studies exploring the length generalization capabilities of transformer-based language models. We first establish that naively finetuning transformers on length generalization tasks shows significant generalization deficiencies independent of model scale. We then show that combining pretrained large language models' in-context learning abilities with scratchpad prompting (asking the model to output solution steps before producing an answer) results in a dramatic improvement in length generalization. We run careful failure analyses on each of the learning modalities and identify common sources of mistakes that highlight opportunities in equipping language models with the ability to generalize to longer problems.

Pretrained language models have shown superior performance on many natural language processing tasks, yet they still struggle at multi-step formal reasoning tasks like grade school math problems. One key challenge of finetuning them to solve such math reasoning problems is that many existing datasets only contain one reference solution for each problem, despite the fact that there are often alternative solutions resembling different reasoning paths to the final answer. This way, the finetuned models are biased towards the limited reference solutions, which limits their generalization to unseen examples. To mitigate this issue, we propose to let the model perform sampling during training and learn from both self-sampled fully-correct solutions, which yield the correct answer upon execution, and partially-correct solutions, whose intermediate state matches an intermediate state of a known correct solution. We show that our use of self-sampled correct and partially-correct solutions can benefit learning and help guide the sampling process, leading to more efficient exploration of the solution space. Additionally, we explore various training objectives to support learning from multiple solutions per example and find they greatly affect the performance. Experiments on two math reasoning datasets show the effectiveness of our method compared to learning from a single reference solution with MLE, where we improve PASS@100 from 35.5% to 44.5% for GSM8K, and 27.6% to 36.2% PASS@80 for MathQA. Such improvements are also consistent across different model sizes. Our code is available at https://github.com/microsoft/TraceCodegen.

Test-time compute is emerging as a new paradigm for enhancing language models' complex multi-step reasoning capabilities, as demonstrated by the success of OpenAI's o1 and o3, as well as DeepSeek's R1. Compared to explicit reasoning in test-time compute, implicit reasoning is more inference-efficient, requiring fewer generated tokens. However, why does the advanced reasoning capability fail to emerge in the implicit reasoning style? In this work, we train GPT-2 from scratch on a curated multi-step mathematical reasoning dataset and conduct analytical experiments to investigate how language models perform implicit reasoning in multi-step tasks. Our findings reveal: 1) Language models can perform step-by-step reasoning and achieve high accuracy in both in-domain and out-of-domain tests via implicit reasoning. However, this capability only emerges when trained on fixed-pattern data. 2) Conversely, implicit reasoning abilities emerging from training on unfixed-pattern data tend to overfit a specific pattern and fail to generalize further. Notably, this limitation is also observed in state-of-the-art large language models. These findings suggest that language models acquire implicit reasoning through shortcut learning, enabling strong performance on tasks with similar patterns while lacking generalization.

Do large language models (LLMs) anticipate when they will answer correctly? To study this, we extract activations after a question is read but before any tokens are generated, and train linear probes to predict whether the model's forthcoming answer will be correct. Across three open-source model families ranging from 7 to 70 billion parameters, projections on this "in-advance correctness direction" trained on generic trivia questions predict success in distribution and on diverse out-of-distribution knowledge datasets, outperforming black-box baselines and verbalised predicted confidence. Predictive power saturates in intermediate layers, suggesting that self-assessment emerges mid-computation. Notably, generalisation falters on questions requiring mathematical reasoning. Moreover, for models responding "I don't know", doing so strongly correlates with the probe score, indicating that the same direction also captures confidence. By complementing previous results on truthfulness and other behaviours obtained with probes and sparse auto-encoders, our work contributes essential findings to elucidate LLM internals.

Large language models (LLMs) have demonstrated remarkable capabilities in language generation, understanding, and few-shot learning in recent years. An extensive body of work has explored how their performance may be further improved through the tools of prompting, ranging from verification, self-consistency, or intermediate scratchpads. In this paper, we present a complementary approach to improve language responses where multiple language model instances propose and debate their individual responses and reasoning processes over multiple rounds to arrive at a common final answer. Our findings indicate that this approach significantly enhances mathematical and strategic reasoning across a number of tasks. We also demonstrate that our approach improves the factual validity of generated content, reducing fallacious answers and hallucinations that contemporary models are prone to. Our approach may be directly applied to existing black-box models and uses identical procedure and prompts for all tasks we investigate. Overall, our findings suggest that such "society of minds" approach has the potential to significantly advance the capabilities of LLMs and pave the way for further breakthroughs in language generation and understanding.

The derivation of mathematical results in specialised fields using Large Language Models (LLMs) is an emerging research direction that can help identify models' limitations, and potentially support mathematical discovery. In this paper, we leverage a symbolic engine to generate derivations of equations at scale, and investigate the capabilities of LLMs when deriving goal equations from premises. Specifically, we employ in-context learning for GPT and fine-tune a range of T5 models to compare the robustness and generalisation of pre-training strategies to specialised models. Empirical results show that fine-tuned FLAN-T5-large (MathT5) outperforms GPT models on all static and out-of-distribution test sets in terms of absolute performance. However, an in-depth analysis reveals that the fine-tuned models are more sensitive to perturbations involving unseen symbols and (to a lesser extent) changes to equation structure. In addition, we analyse 1.7K equations and over 200 derivations to highlight common reasoning errors such as the inclusion of incorrect, irrelevant, and redundant equations, along with the tendency to skip derivation steps. Finally, we explore the suitability of existing metrics for evaluating mathematical derivations finding evidence that, while they capture general properties such as sensitivity to perturbations, they fail to highlight fine-grained reasoning errors and essential differences between models. Overall, this work demonstrates that training models on synthetic data can improve their mathematical capabilities beyond larger architectures.

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.

We evaluate the reasoning abilities of large language models in multilingual settings. We introduce the Multilingual Grade School Math (MGSM) benchmark, by manually translating 250 grade-school math problems from the GSM8K dataset (Cobbe et al., 2021) into ten typologically diverse languages. We find that the ability to solve MGSM problems via chain-of-thought prompting emerges with increasing model scale, and that models have strikingly strong multilingual reasoning abilities, even in underrepresented languages such as Bengali and Swahili. Finally, we show that the multilingual reasoning abilities of language models extend to other tasks such as commonsense reasoning and word-in-context semantic judgment. The MGSM benchmark is publicly available at https://github.com/google-research/url-nlp.

Large language models have demonstrated impressive performance on challenging mathematical reasoning tasks, which has triggered the discussion of whether the performance is achieved by true reasoning capability or memorization. To investigate this question, prior work has constructed mathematical benchmarks when questions undergo simple perturbations -- modifications that still preserve the underlying reasoning patterns of the solutions. However, no work has explored hard perturbations, which fundamentally change the nature of the problem so that the original solution steps do not apply. To bridge the gap, we construct MATH-P-Simple and MATH-P-Hard via simple perturbation and hard perturbation, respectively. Each consists of 279 perturbed math problems derived from level-5 (hardest) problems in the MATH dataset (Hendrycksmath et. al., 2021). We observe significant performance drops on MATH-P-Hard across various models, including o1-mini (-16.49%) and gemini-2.0-flash-thinking (-12.9%). We also raise concerns about a novel form of memorization where models blindly apply learned problem-solving skills without assessing their applicability to modified contexts. This issue is amplified when using original problems for in-context learning. We call for research efforts to address this challenge, which is critical for developing more robust and reliable reasoning models.

Exceptional mathematical reasoning ability is one of the key features that demonstrate the power of large language models (LLMs). How to comprehensively define and evaluate the mathematical abilities of LLMs, and even reflect the user experience in real-world scenarios, has emerged as a critical issue. Current benchmarks predominantly concentrate on problem-solving capabilities, which presents a substantial risk of model overfitting and fails to accurately represent genuine mathematical reasoning abilities. In this paper, we argue that if a model really understands a problem, it should be robustly and readily applied across a diverse array of tasks. Motivated by this, we introduce MATHCHECK, a well-designed checklist for testing task generalization and reasoning robustness, as well as an automatic tool to generate checklists efficiently. MATHCHECK includes multiple mathematical reasoning tasks and robustness test types to facilitate a comprehensive evaluation of both mathematical reasoning ability and behavior testing. Utilizing MATHCHECK, we develop MATHCHECK-GSM and MATHCHECK-GEO to assess mathematical textual reasoning and multi-modal reasoning capabilities, respectively, serving as upgraded versions of benchmarks including GSM8k, GeoQA, UniGeo, and Geometry3K. We adopt MATHCHECK-GSM and MATHCHECK-GEO to evaluate over 20 LLMs and 11 MLLMs, assessing their comprehensive mathematical reasoning abilities. Our results demonstrate that while frontier LLMs like GPT-4o continue to excel in various abilities on the checklist, many other model families exhibit a significant decline. Further experiments indicate that, compared to traditional math benchmarks, MATHCHECK better reflects true mathematical abilities and represents mathematical intelligence more linearly, thereby supporting our design. On our MATHCHECK, we can easily conduct detailed behavior analysis to deeply investigate models.

Multimodal large language models (MLLMs) have demonstrated remarkable capabilities in vision-language answering tasks. Despite their strengths, these models often encounter challenges in achieving complex reasoning tasks such as mathematical problem-solving. Previous works have focused on fine-tuning on specialized mathematical datasets. However, these datasets are typically distilled directly from teacher models, which capture only static reasoning patterns and leaving substantial gaps compared to student models. This reliance on fixed teacher-derived datasets not only restricts the model's ability to adapt to novel or more intricate questions that extend beyond the confines of the training data, but also lacks the iterative depth needed for robust generalization. To overcome these limitations, we propose \method, a Mathematical Self-Evolving framework for MLLMs. In contrast to traditional one-shot fine-tuning paradigms, \method iteratively refines the model through cycles of inference, reflection, and reward-based feedback. Specifically, we leverage iterative fine-tuning by incorporating correct reasoning paths derived from previous-stage inference and integrating reflections from a specialized Outcome Reward Model (ORM). To verify the effectiveness of \method, we evaluate it on a suite of challenging benchmarks, demonstrating significant performance gains over backbone models. Notably, our experimental results on MathVL-test surpass the leading open-source multimodal mathematical reasoning model QVQ. Our code and models are available at https://zheny2751\allowbreak-dotcom.github.io/\allowbreak MathSE.github.io/.

Large language models have achieved significant advancements in complex mathematical reasoning benchmarks, such as MATH. However, their substantial computational requirements present challenges for practical deployment. Model quantization has emerged as an effective strategy to reduce memory usage and computational costs by employing lower precision and bit-width representations. In this study, we systematically evaluate the impact of quantization on mathematical reasoning tasks. We introduce a multidimensional evaluation framework that qualitatively assesses specific capability dimensions and conduct quantitative analyses on the step-by-step outputs of various quantization methods. Our results demonstrate that quantization differentially affects numerical computation and reasoning planning abilities, identifying key areas where quantized models experience performance degradation.

Mathematical reasoning has been challenging for large language models (LLMs). However, the introduction of step-by-step Chain-of-Thought (CoT) inference has significantly advanced the mathematical capabilities of LLMs. Despite this progress, current approaches either necessitate extensive inference datasets for training or depend on few-shot methods that frequently compromise computational accuracy. To address these bottlenecks in mathematical reasoning, we propose a novel method called Step Guidied Reasoning, which is more stable and generalizable than few-shot methods and does not involve further fine-tuning of the model. In this approach, LLMs reflect on small reasoning steps, similar to how humans deliberate and focus attention on what to do next. By incorporating this reflective process into the inference stage, LLMs can effectively guide their reasoning from one step to the next. Through extensive experiments, we demonstrate the significant effect of Step Guidied Reasoning in augmenting mathematical performance in state-of-the-art language models. Qwen2-72B-Instruct outperforms its math-specific counterpart, Qwen2.5-72B-Math-Instruct, on MMLU- STEM with a score of 90.9%, compared to 87.3%. The average scores of Qwen2-7B-Instruct and Qwen2-72B-Instruct increase from 27.1% to 36.3% and from 36.5% to 47.4% on the mathematics domain, respectively.

We developed a benchmark set to assess the generalization of state-of-the-art large language models on problems beyond linguistic tasks and evaluate it on a systematic progression of GPT models (GPT-3.5, GPT-4, GPT-4o, GPT-4o-mini). Using simple games like Tic-Tac-Toe, Connect Four, Battleship, and a Shape Recognition Game, all encoded in ASCII, we test strategic capabilities and spatial reasoning, core abilities any artificial intelligence would need to master for solving problems in chemistry. To probe generalization, we introduce two new games for spatial logic: LEGO Connect Language (LCL) and Guess-the-SMILES (GtS), a operationally simple chemistry benchmark. Our results show that GPT models provide meaningful responses for several tasks but, generally, perform poorly. A systematic performance progression with increased model capabilities (GPT-3.5, GPT-4, GPT-4o) is only observed for 4 out of the 7 benchmark tasks. All models consistently struggle with Battleship, LCL, and GtS. This suggests that while GPT models can emulate conversational proficiency and basic rule comprehension, they have limited generalization with respect to strategy and spatial reasoning. Particularly poor performance is observed for interpreting molecular graphs when encoded in ASCII. The results provided by our open-source benchmark suite (https://github.com/BlueVelvetSackOfGoldPotatoes/child-play{ChildPlay GitHub Repository}) caution against claims of emergent intelligence in GPT models, which appear more specialized than general.

Recent math benchmarks for large language models (LLMs) such as MathArena indicate that state-of-the-art reasoning models achieve impressive performance on mathematical competitions like AIME, with the leading model, o3-mini, achieving scores comparable to top human competitors. However, these benchmarks evaluate models solely based on final numerical answers, neglecting rigorous reasoning and proof generation which are essential for real-world mathematical tasks. To address this, we introduce the first comprehensive evaluation of full-solution reasoning for challenging mathematical problems. Using expert human annotators, we evaluated several state-of-the-art reasoning models on the six problems from the 2025 USAMO within hours of their release. Our results reveal that all tested models struggled significantly, achieving less than 5% on average. Through detailed analysis of reasoning traces, we identify the most common failure modes and find several unwanted artifacts arising from the optimization strategies employed during model training. Overall, our results suggest that current LLMs are inadequate for rigorous mathematical reasoning tasks, highlighting the need for substantial improvements in reasoning and proof generation capabilities.

Despite the rapid progress in the capabilities of Large Language Models (LLMs), they continue to have difficulty following relatively simple, unambiguous instructions, especially when compositions are involved. In this paper, we take inspiration from recent work that suggests that models may follow instructions better when they are expressed in pseudo-code. However, writing pseudo-code programs can be tedious and using few-shot demonstrations to craft code representations for use in inference can be unnatural for non-expert users of LLMs. To overcome these limitations, we propose fine-tuning LLMs with instruction-tuning data that additionally includes instructions re-expressed in pseudo-code along with the final response. We evaluate models trained using our method on 11 publicly available benchmarks comprising of tasks related to instruction-following, mathematics, and common-sense reasoning. We conduct rigorous experiments with 5 different models and find that not only do models follow instructions better when trained with pseudo-code, they also retain their capabilities on the other tasks related to mathematical and common sense reasoning. Specifically, we observe a relative gain of 3--19% on instruction-following benchmark, and an average gain of upto 14% across all tasks.

This paper considers the learning of logical (Boolean) functions with focus on the generalization on the unseen (GOTU) setting, a strong case of out-of-distribution generalization. This is motivated by the fact that the rich combinatorial nature of data in certain reasoning tasks (e.g., arithmetic/logic) makes representative data sampling challenging, and learning successfully under GOTU gives a first vignette of an 'extrapolating' or 'reasoning' learner. We then study how different network architectures trained by (S)GD perform under GOTU and provide both theoretical and experimental evidence that for a class of network models including instances of Transformers, random features models, and diagonal linear networks, a min-degree-interpolator (MDI) is learned on the unseen. We also provide evidence that other instances with larger learning rates or mean-field networks reach leaky MDIs. These findings lead to two implications: (1) we provide an explanation to the length generalization problem (e.g., Anil et al. 2022); (2) we introduce a curriculum learning algorithm called Degree-Curriculum that learns monomials more efficiently by incrementing supports.

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

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.

The recent advent of large language models has reinvigorated debate over whether human cognitive capacities might emerge in such generic models given sufficient training data. Of particular interest is the ability of these models to reason about novel problems zero-shot, without any direct training. In human cognition, this capacity is closely tied to an ability to reason by analogy. Here, we performed a direct comparison between human reasoners and a large language model (the text-davinci-003 variant of GPT-3) on a range of analogical tasks, including a non-visual matrix reasoning task based on the rule structure of Raven's Standard Progressive Matrices. We found that GPT-3 displayed a surprisingly strong capacity for abstract pattern induction, matching or even surpassing human capabilities in most settings; preliminary tests of GPT-4 indicated even better performance. Our results indicate 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.

Reasoning capabilities of large language models are primarily studied for English, even when pretrained models are multilingual. In this work, we investigate to what extent English reasoning finetuning with long chain-of-thoughts (CoTs) can generalize across languages. First, we find that scaling up inference compute for English-centric reasoning language models (RLMs) improves multilingual mathematical reasoning across many languages including low-resource languages, to an extent where they outperform models twice their size. Second, we reveal that while English-centric RLM's CoTs are naturally predominantly English, they consistently follow a quote-and-think pattern to reason about quoted non-English inputs. Third, we discover an effective strategy to control the language of long CoT reasoning, and we observe that models reason better and more efficiently in high-resource languages. Finally, we observe poor out-of-domain reasoning generalization, in particular from STEM to cultural commonsense knowledge, even for English. Overall, we demonstrate the potentials, study the mechanisms and outline the limitations of crosslingual generalization of English reasoning test-time scaling. We conclude that practitioners should let English-centric RLMs reason in high-resource languages, while further work is needed to improve reasoning in low-resource languages and out-of-domain contexts.

Instruction-following is essential for aligning large language models (LLMs) with user intent. While recent reasoning-oriented models exhibit impressive performance on complex mathematical problems, their ability to adhere to natural language instructions remains underexplored. In this work, we introduce MathIF, a dedicated benchmark for evaluating instruction-following in mathematical reasoning tasks. Our empirical analysis reveals a consistent tension between scaling up reasoning capacity and maintaining controllability, as models that reason more effectively often struggle to comply with user directives. We find that models tuned on distilled long chains-of-thought or trained with reasoning-oriented reinforcement learning often degrade in instruction adherence, especially when generation length increases. Furthermore, we show that even simple interventions can partially recover obedience, though at the cost of reasoning performance. These findings highlight a fundamental tension in current LLM training paradigms and motivate the need for more instruction-aware reasoning models. We release the code and data at https://github.com/TingchenFu/MathIF.

As language models regularly make mistakes when solving math problems, automated identification of errors in the reasoning process becomes increasingly significant for their scalable oversight. In this paper, we introduce ProcessBench for measuring the ability to identify erroneous steps in mathematical reasoning. It consists of 3,400 test cases, primarily focused on competition- and Olympiad-level math problems. Each test case contains a step-by-step solution with error location annotated by human experts. Models are required to identify the earliest step that contains an error, or conclude that all steps are correct. We conduct extensive evaluation on ProcessBench, involving two types of models: process reward models (PRMs) and critic models, where for the latter we prompt general language models to critique each solution step by step. We draw two main observations: (1) Existing PRMs typically fail to generalize to more challenging math problems beyond GSM8K and MATH. They underperform both critic models (i.e., prompted general language models) and our own trained PRM that is straightforwardly fine-tuned on the PRM800K dataset. (2) The best open-source model, QwQ-32B-Preview, has demonstrated the critique capability competitive with the proprietary model GPT-4o, despite that it still lags behind the reasoning-specialized o1-mini. We hope ProcessBench can foster future research in reasoning process assessment, paving the way toward scalable oversight of language models.

The mathematical problem-solving capabilities of large language models have become a focal point of research, with growing interests in leveraging self-generated reasoning paths as a promising way to refine and enhance these models. These paths capture step-by-step logical processes while requiring only the correct answer for supervision. The self-training method has been shown to be effective in reasoning tasks while eliminating the need for external models and manual annotations. However, optimizing the use of self-generated data for model training remains an open challenge. In this work, we propose Entropy-Based Adaptive Weighting for Self-Training (EAST), an adaptive weighting strategy designed to prioritize uncertain data during self-training. Specifically, EAST employs a mapping function with a tunable parameter that controls the sharpness of the weighting, assigning higher weights to data where the model exhibits greater uncertainty. This approach guides the model to focus on more informative and challenging examples, thereby enhancing its reasoning ability. We evaluate our approach on GSM8K and MATH benchmarks. Empirical results show that, while the vanilla method yields virtually no improvement (0%) on MATH, EAST achieves around a 1% gain over backbone model. On GSM8K, EAST attains a further 1-2% performance boost compared to the vanilla method.

Solving math word problems depends on how to articulate the problems, the lens through which models view human linguistic expressions. Real-world settings count on such a method even more due to the diverse practices of the same mathematical operations. Earlier works constrain available thinking processes by limited prediction strategies without considering their significance in acquiring mathematical knowledge. We introduce Attention-based THought Expansion Network Architecture (ATHENA) to tackle the challenges of real-world practices by mimicking human thought expansion mechanisms in the form of neural network propagation. A thought expansion recurrently generates the candidates carrying the thoughts of possible math expressions driven from the previous step and yields reasonable thoughts by selecting the valid pathways to the goal. Our experiments show that ATHENA achieves a new state-of-the-art stage toward the ideal model that is compelling in variant questions even when the informativeness in training examples is restricted.

Large Language Models (LLMs) exhibit impressive reasoning abilities, yet their reliance on structured step-by-step processing reveals a critical limitation. While human cognition fluidly adapts between intuitive, heuristic (System 1) and analytical, deliberative (System 2) reasoning depending on the context, LLMs lack this dynamic flexibility. This rigidity can lead to brittle and unreliable performance when faced with tasks that deviate from their trained patterns. To address this, we create a dataset of 2,000 samples with valid System 1 and System 2 answers, explicitly align LLMs with these reasoning styles, and evaluate their performance across reasoning benchmarks. Our results reveal an accuracy-efficiency trade-off: System 2-aligned models excel in arithmetic and symbolic reasoning, while System 1-aligned models perform better in commonsense tasks. A mechanistic analysis of model responses shows that System 1 models employ more definitive answers, whereas System 2 models demonstrate greater uncertainty. Interpolating between these extremes produces a monotonic transition in reasoning accuracy, preserving coherence. This work challenges the assumption that step-by-step reasoning is always optimal and highlights the need for adapting reasoning strategies based on task demands.

To augment language models with the ability to reason, researchers usually prompt or finetune them to produce chain of thought reasoning steps before producing the final answer. However, although people use natural language to reason effectively, it may be that LMs could reason more effectively with some intermediate computation that is not in natural language. In this work, we explore an alternative reasoning approach: instead of explicitly producing the chain of thought reasoning steps, we use the language model's internal hidden states to perform implicit reasoning. The implicit reasoning steps are distilled from a teacher model trained on explicit chain-of-thought reasoning, and instead of doing reasoning "horizontally" by producing intermediate words one-by-one, we distill it such that the reasoning happens "vertically" among the hidden states in different layers. We conduct experiments on a multi-digit multiplication task and a grade school math problem dataset and find that this approach enables solving tasks previously not solvable without explicit chain-of-thought, at a speed comparable to no chain-of-thought.

Large Reasoning Models (LRMs) have demonstrated remarkable problem-solving abilities in mathematics, as evaluated by existing benchmarks exclusively on well-defined problems. However, such evaluation setup constitutes a critical gap, since a genuine intelligent agent should not only solve problems (as a math quiz solver), but also be able~to ask for information when the problems lack sufficient information, enabling proactivity in responding users' requests. To bridge such gap, we proposes a new dataset consisting of two types of incomplete problems with diverse contexts. Based on the dataset, our systematical evaluation of LRMs reveals their inability in proactively asking for information. In addition, we uncover the behaviors related to overthinking and hallucination of LRMs, and highlight the potential and challenges of supervised fine-tuning in learning such ability. We hope to provide new insights in developing LRMs with genuine intelligence, rather than just solving problems.

Large Language Models (LLMs) have demonstrated impressive real-world utility, exemplifying artificial useful intelligence (AUI). However, their ability to reason adaptively and robustly -- the hallmarks of artificial general intelligence (AGI) -- remains fragile. While LLMs seemingly succeed in commonsense reasoning, programming, and mathematics, they struggle to generalize algorithmic understanding across novel contexts. Our experiments with algorithmic tasks in esoteric programming languages reveal that LLM's reasoning overfits to the training data and is limited in its transferability. We hypothesize that the core issue underlying such limited transferability is the coupling of reasoning and knowledge in LLMs. To transition from AUI to AGI, we propose disentangling knowledge and reasoning through three key directions: (1) pretaining to reason using RL from scratch as an alternative to the widely used next-token prediction pretraining, (2) using a curriculum of synthetic tasks to ease the learning of a reasoning prior for RL that can then be transferred to natural language tasks, and (3) learning more generalizable reasoning functions using a small context window to reduce exploiting spurious correlations between tokens. Such a reasoning system coupled with a trained retrieval system and a large external memory bank as a knowledge store can overcome several limitations of existing architectures at learning to reason in novel scenarios.

We study whether transformers can learn to implicitly reason over parametric knowledge, a skill that even the most capable language models struggle with. Focusing on two representative reasoning types, composition and comparison, we consistently find that transformers can learn implicit reasoning, but only through grokking, i.e., extended training far beyond overfitting. The levels of generalization also vary across reasoning types: when faced with out-of-distribution examples, transformers fail to systematically generalize for composition but succeed for comparison. We delve into the model's internals throughout training, conducting analytical experiments that reveal: 1) the mechanism behind grokking, such as the formation of the generalizing circuit and its relation to the relative efficiency of generalizing and memorizing circuits, and 2) the connection between systematicity and the configuration of the generalizing circuit. Our findings guide data and training setup to better induce implicit reasoning and suggest potential improvements to the transformer architecture, such as encouraging cross-layer knowledge sharing. Furthermore, we demonstrate that for a challenging reasoning task with a large search space, GPT-4-Turbo and Gemini-1.5-Pro based on non-parametric memory fail badly regardless of prompting styles or retrieval augmentation, while a fully grokked transformer can achieve near-perfect accuracy, showcasing the power of parametric memory for complex reasoning.

The evaluation of mathematical reasoning capabilities is essential for advancing Artificial General Intelligence (AGI). While Large Language Models (LLMs) have shown impressive performance in solving mathematical problems, existing benchmarks such as GSM8K and MATH present limitations, including narrow problem definitions with specific numbers and reliance on predetermined rules that hinder accurate assessments of reasoning and adaptability. This paper introduces the UTMath Benchmark, which robustly evaluates the models through extensive unit tests. It consists of 1,053 problems across 9 mathematical domains, with over 68 test cases per problem. We propose an innovative evaluation framework inspired by unit testing in software development, focusing on both accuracy and reliability of results. Furthermore, we introduce the Reasoning-to-Coding of Thoughts (RCoT) approach, which encourages LLMs to perform explicit reasoning before generating code, leading to generating more advanced solution and improved performance. Furthermore, we are releasing not only the UTMath benchmark but also the UTMath-Train training dataset (more than 70k samples), to support the community in further exploring mathematical reasoning.

Although Large Language Models (LLMs) achieve remarkable performance across various tasks, they often struggle with complex reasoning tasks, such as answering mathematical questions. Recent efforts to address this issue have primarily focused on leveraging mathematical datasets through supervised fine-tuning or self-improvement techniques. However, these methods often depend on high-quality datasets that are difficult to prepare, or they require substantial computational resources for fine-tuning. Inspired by findings that LLMs know how to produce the right answer but struggle to select the correct reasoning path, we propose a purely inference-based searching method -- MindStar (M*). This method formulates reasoning tasks as searching problems and proposes two search ideas to identify the optimal reasoning paths. We evaluate the M* framework on both the GSM8K and MATH datasets, comparing its performance with existing open and closed-source LLMs. Our results demonstrate that M* significantly enhances the reasoning abilities of open-source models, such as Llama-2-13B and Mistral-7B, and achieves comparable performance to GPT-3.5 and Grok-1, but with substantially reduced model size and computational costs.

Recent advancements in large reasoning models (LRMs) have significantly enhanced language models' capabilities in complex problem-solving by emulating human-like deliberative thinking. However, these models often exhibit overthinking (i.e., the generation of unnecessarily verbose and redundant content), which hinders efficiency and inflates inference cost. In this work, we explore the representational and behavioral origins of this inefficiency, revealing that LRMs inherently possess the capacity for more concise reasoning. Empirical analyses show that correct reasoning paths vary significantly in length, and the shortest correct responses often suffice, indicating untapped efficiency potential. Exploiting these findings, we propose two lightweight methods to enhance LRM efficiency. First, we introduce Efficiency Steering, a training-free activation steering technique that modulates reasoning behavior via a single direction in the model's representation space. Second, we develop Self-Rewarded Efficiency RL, a reinforcement learning framework that dynamically balances task accuracy and brevity by rewarding concise correct solutions. Extensive experiments on seven LRM backbones across multiple mathematical reasoning benchmarks demonstrate that our methods significantly reduce reasoning length while preserving or improving task performance. Our results highlight that reasoning efficiency can be improved by leveraging and guiding the intrinsic capabilities of existing models in a self-guided manner.

Large Language Models (LLMs) have demonstrated impressive reasoning abilities through test-time computation (TTC) techniques such as chain-of-thought prompting and tree-based reasoning. However, we argue that current reasoning LLMs (RLLMs) lack the ability to systematically explore the solution space. This paper formalizes what constitutes systematic problem solving and identifies common failure modes that reveal reasoning LLMs to be wanderers rather than systematic explorers. Through qualitative and quantitative analysis across multiple state-of-the-art LLMs, we uncover persistent issues: invalid reasoning steps, redundant explorations, hallucinated or unfaithful conclusions, and so on. Our findings suggest that current models' performance can appear to be competent on simple tasks yet degrade sharply as complexity increases. Based on the findings, we advocate for new metrics and tools that evaluate not just final outputs but the structure of the reasoning process itself.

Reasoning models are large language models that emit a long chain-of-thought before answering, providing both higher accuracy and explicit reasoning for their response. A major question has been whether language model reasoning generalizes beyond mathematics, programming, and logic, where most previous work has focused. We demonstrate that reasoning models can be post-trained for chemistry without additional domain pretraining, and require substantially less data compared to contemporary domain-specific models. We report ether0, a 24B parameter LLM (based on Mistral-Small-24B) that can reason in natural language and respond with chemical structures. This reasoning model was trained with reinforcement learning on 640,730 experimentally-grounded chemistry problems across 375 tasks ranging from synthesizability, to blood-brain barrier permeability, to human receptor activity, to scent. Our model exceeds general-purpose chemistry models, frontier models, and human experts on molecular design tasks. It is also more data efficient relative to specialized models. We anticipate that this method can be applied to train data-efficient language models specialized for tasks across a wide variety of scientific domains.

Large reasoning models (LRMs) have demonstrated impressive reasoning capabilities across a broad range of tasks including Olympiad-level mathematical problems, indicating evidence of their complex reasoning abilities. While many reasoning benchmarks focus on the STEM domain, the ability of LRMs to reason correctly in broader task domains remains underexplored. In this work, we introduce TTT-Bench, a new benchmark that is designed to evaluate basic strategic, spatial, and logical reasoning abilities in LRMs through a suite of four two-player Tic-Tac-Toe-style games that humans can effortlessly solve from a young age. We propose a simple yet scalable programmatic approach for generating verifiable two-player game problems for TTT-Bench. Although these games are trivial for humans, they require reasoning about the intentions of the opponent, as well as the game board's spatial configurations, to ensure a win. We evaluate a diverse set of state-of-the-art LRMs, and discover that the models that excel at hard math problems frequently fail at these simple reasoning games. Further testing reveals that our evaluated reasoning models score on average downarrow 41\% \& downarrow 5\% lower on TTT-Bench compared to MATH 500 \& AIME 2024 respectively, with larger models achieving higher performance using shorter reasoning traces, where most of the models struggle on long-term strategic reasoning situations on simple and new TTT-Bench tasks.

Large language models (LLMs) have shown increasing in-context learning capabilities through scaling up model and data size. Despite this progress, LLMs are still unable to solve algorithmic reasoning problems. While providing a rationale with the final answer has led to further improvements in multi-step reasoning problems, Anil et al. 2022 showed that even simple algorithmic reasoning tasks such as parity are far from solved. In this work, we identify and study four key stages for successfully teaching algorithmic reasoning to LLMs: (1) formulating algorithms as skills, (2) teaching multiple skills simultaneously (skill accumulation), (3) teaching how to combine skills (skill composition) and (4) teaching how to use skills as tools. We show that it is possible to teach algorithmic reasoning to LLMs via in-context learning, which we refer to as algorithmic prompting. We evaluate our approach on a variety of arithmetic and quantitative reasoning tasks, and demonstrate significant boosts in performance over existing prompting techniques. In particular, for long parity, addition, multiplication and subtraction, we achieve an error reduction of approximately 10x, 9x, 5x and 2x respectively compared to the best available baselines.

The impressive performance of recent language models across a wide range of tasks suggests that they possess a degree of abstract reasoning skills. Are these skills general and transferable, or specialized to specific tasks seen during pretraining? To disentangle these effects, we propose an evaluation framework based on "counterfactual" task variants that deviate from the default assumptions underlying standard tasks. Across a suite of 11 tasks, we observe nontrivial performance on the counterfactual variants, but nevertheless find that performance substantially and consistently degrades compared to the default conditions. This suggests that while current LMs may possess abstract task-solving skills to a degree, they often also rely on narrow, non-transferable procedures for task-solving. These results motivate a more careful interpretation of language model performance that teases apart these aspects of behavior.

We have recently witnessed a number of impressive results on hard mathematical reasoning problems with language models. At the same time, the robustness of these models has also been called into question; recent works have shown that models can rely on shallow patterns in the problem description when generating a solution. Building on the idea of behavioral testing, we propose a novel framework, which pins down the causal effect of various factors in the input, e.g., the surface form of the problem text, the operands, and math operators on the output solution. By grounding the behavioral analysis in a causal graph describing an intuitive reasoning process, we study the behavior of language models in terms of robustness and sensitivity to direct interventions in the input space. We apply our framework on a test bed of math word problems. Our analysis shows that robustness does not appear to continuously improve as a function of size, but the GPT-3 Davinci models (175B) achieve a dramatic improvement in both robustness and sensitivity compared to all other GPT variants.

Why do thinking language models like DeepSeek R1 outperform their base counterparts? Despite consistent performance gains, it remains unclear to what extent thinking models learn entirely new reasoning capabilities or repurpose pre-existing base model ones. In this work, we propose a hybrid model where we activate reasoning mechanisms in base models at the right time to elicit thinking-model-level reasoning chains, implying that thinking models exploit already existing capabilities. To ground our analysis, we introduce an unsupervised, bottom-up approach for uncovering human-interpretable reasoning behaviors in thinking models. This approach provides an unbiased method to discover reasoning behaviors without imposing manual or LLM-derived assumptions. Across three base and four thinking models, using GSM8K and MATH500, our hybrid model recovers up to 91% of the performance gap to thinking models without any weight updates while steering only 12% of tokens. Concretely, our empirical setup provides a simple, causal way to test the effectiveness of existing reasoning mechanisms in base models by invoking them directly and measuring the resulting task performance. More broadly, these results reframe our understanding of how thinking models are trained: pre-training is when models acquire most of their reasoning mechanisms, and post-training teaches efficient deployment of these mechanisms at the right time, enabling efficient use of their inference-time compute.

Large Language Models (LLMs) exhibit impressive performance across various domains but still struggle with arithmetic reasoning tasks. Recent work shows the effectiveness of prompt design methods in enhancing reasoning capabilities. However, these approaches overlook crucial requirements for prior knowledge of specific concepts, theorems, and tricks to tackle most arithmetic reasoning problems successfully. To address this issue, we propose a novel and effective Teaching-Inspired Integrated Framework, which emulates the instructional process of a teacher guiding students. This method equips LLMs with essential concepts, relevant theorems, and similar problems with analogous solution approaches, facilitating the enhancement of reasoning abilities. Additionally, we introduce two new Chinese datasets, MathMC and MathToF, both with detailed explanations and answers. Experiments are conducted on nine benchmarks which demonstrates that our approach improves the reasoning accuracy of LLMs. With GPT-4 and our framework, we achieve new state-of-the-art performance on four math benchmarks (AddSub, SVAMP, Math23K and AQuA) with accuracies of 98.2% (+3.3%), 93.9% (+0.2%), 94.3% (+7.2%) and 81.1% (+1.2%). Our data and code are available at https://github.com/SallyTan13/Teaching-Inspired-Prompting.

Math Word Problems (MWPs) play a vital role in assessing the capabilities of Large Language Models (LLMs), yet current research primarily focuses on questions with concise contexts. The impact of longer contexts on mathematical reasoning remains under-explored. This study pioneers the investigation of Context Length Generalizability (CoLeG), which refers to the ability of LLMs to solve MWPs with extended narratives. We introduce Extended Grade-School Math (E-GSM), a collection of MWPs featuring lengthy narratives, and propose two novel metrics to evaluate the efficacy and resilience of LLMs in tackling these problems. Our analysis of existing zero-shot prompting techniques with proprietary LLMs along with open-source LLMs reveals a general deficiency in CoLeG. To alleviate these issues, we propose tailored approaches for different categories of LLMs. For proprietary LLMs, we introduce a new instructional prompt designed to mitigate the impact of long contexts. For open-source LLMs, we develop a novel auxiliary task for fine-tuning to enhance CoLeG. Our comprehensive results demonstrate the effectiveness of our proposed methods, showing improved performance on E-GSM. Additionally, we conduct an in-depth analysis to differentiate the effects of semantic understanding and reasoning efficacy, showing that our methods improves the latter. We also establish the generalizability of our methods across several other MWP benchmarks. Our findings highlight the limitations of current LLMs and offer practical solutions correspondingly, paving the way for further exploration of model generalizability and training methodologies.

We study the depth of grade-school math (GSM) problem-solving capabilities of LLMs. To this end, we evaluate their performance on pairs of existing math word problems together so that the answer to the second problem depends on correctly answering the first problem. Our findings reveal a significant reasoning gap in most LLMs, that is performance difference between solving the compositional pairs and solving each question independently. This gap is more pronounced in smaller, more cost-efficient, and math-specialized models. Moreover, instruction-tuning recipes and code generation have varying effects across LLM sizes, while finetuning on GSM can lead to task overfitting. Our analysis indicates that large reasoning gaps are not because of test-set leakage, but due to distraction from additional context and poor second-hop reasoning. Overall, LLMs exhibit systematic differences in their reasoning abilities, despite what their performance on standard benchmarks indicates.

Chain-of-thought (CoT) prompting enhances reasoning in large language models (LLMs) but often leads to verbose and redundant outputs, thus increasing inference cost. We hypothesize that many reasoning steps are unnecessary for producing correct answers. To investigate this, we start with a systematic study to examine what is the minimum reasoning required for a model to reach a stable decision. We find that on math reasoning tasks like math, models typically converge to their final answers after 60\% of the reasoning steps, suggesting substantial redundancy in the remaining content. Based on these insights, we propose three inference-time strategies to improve efficiency: (1) early stopping via answer consistency, (2) boosting the probability of generating end-of-reasoning signals, and (3) a supervised method that learns when to stop based on internal activations. Experiments across five benchmarks and five open-weights LLMs show that our methods significantly reduce token usage with little or no accuracy drop. In particular, on NaturalQuestions, Answer Consistency reduces tokens by over 40\% while further improving accuracy. Our work underscores the importance of cost-effective reasoning methods that operate at inference time, offering practical benefits for real-world applications.

In this work, we introduce a novel evaluation paradigm for Large Language Models, one that challenges them to engage in meta-reasoning. This approach addresses critical shortcomings in existing math problem-solving benchmarks, traditionally used to evaluate the cognitive capabilities of agents. Our paradigm shifts the focus from result-oriented assessments, which often overlook the reasoning process, to a more holistic evaluation that effectively differentiates the cognitive capabilities among models. For example, in our benchmark, GPT-4 demonstrates a performance ten times more accurate than GPT3-5. The significance of this new paradigm lies in its ability to reveal potential cognitive deficiencies in LLMs that current benchmarks, such as GSM8K, fail to uncover due to their saturation and lack of effective differentiation among varying reasoning abilities. Our comprehensive analysis includes several state-of-the-art math models from both open-source and closed-source communities, uncovering fundamental deficiencies in their training and evaluation approaches. This paper not only advocates for a paradigm shift in the assessment of LLMs but also contributes to the ongoing discourse on the trajectory towards Artificial General Intelligence (AGI). By promoting the adoption of meta-reasoning evaluation methods similar to ours, we aim to facilitate a more accurate assessment of the true cognitive abilities of LLMs.

Reasoning models enhance performance by tackling problems in a step-by-step manner, decomposing them into sub-problems and exploring long chains of thought before producing an answer. However, applying extended reasoning to every step introduces substantial redundancy, as sub-problems vary widely in difficulty and complexity: a small number of pivotal steps are genuinely challenging and decisive for the final answer, while many others only involve straightforward revisions or simple computations. Therefore, a natural idea is to endow reasoning models with the ability to adaptively respond to this variation, rather than treating all steps with the same level of elaboration. To this end, we propose MixReasoning, a framework that dynamically adjusts the depth of reasoning within a single response. The resulting chain of thought then becomes a mixture of detailed reasoning on difficult steps and concise inference on simpler ones. Experiments on GSM8K, MATH-500, and AIME show that MixReasoning shortens reasoning length and substantially improves efficiency without compromising accuracy.

Large Language Models (LLMs) have shown excellent performance in language understanding, text generation, code synthesis, and many other tasks, while they still struggle in complex multi-step reasoning problems, such as mathematical reasoning. In this paper, through a newly proposed arithmetical puzzle problem, we show that the model can perform well on multi-step reasoning tasks via fine-tuning on high-quality synthetic data. Experimental results with the open-llama-3B model on three different test datasets show that not only the model can reach a zero-shot pass@1 at 0.44 on the in-domain dataset, it also demonstrates certain generalization capabilities on the out-of-domain datasets. Specifically, this paper has designed two out-of-domain datasets in the form of extending the numerical range and the composing components of the arithmetical puzzle problem separately. The fine-tuned models have shown encouraging performance on these two far more difficult tasks with the zero-shot pass@1 at 0.33 and 0.35, respectively.

Large Language Models (LLMs) have demonstrated promising capabilities in solving mathematical reasoning tasks, leveraging Chain-of-Thought (CoT) data as a vital component in guiding answer generation. Current paradigms typically generate CoT and answers directly for a given problem, diverging from human problem-solving strategies to some extent. Humans often solve problems by recalling analogous cases and leveraging their solutions to reason about the current task. Inspired by this cognitive process, we propose MetaLadder, a novel framework that explicitly prompts LLMs to recall and reflect on meta-problems, those structurally or semantically analogous problems, alongside their CoT solutions before addressing the target problem. Additionally, we introduce a problem-restating mechanism to enhance the model's comprehension of the target problem by regenerating the original question, which further improves reasoning accuracy. Therefore, the model can achieve reasoning transfer from analogical problems, mimicking human-like "learning from examples" and generalization abilities. Extensive experiments on mathematical benchmarks demonstrate that our MetaLadder significantly boosts LLMs' problem-solving accuracy, largely outperforming standard CoT-based methods (10.3\% accuracy gain) and other methods. Our code and data has been released at https://github.com/LHL3341/MetaLadder.

Recent generations of language models have introduced Large Reasoning Models (LRMs) that generate detailed thinking processes before providing answers. While these models demonstrate improved performance on reasoning benchmarks, their fundamental capabilities, scaling properties, and limitations remain insufficiently understood. Current evaluations primarily focus on established math and coding benchmarks, emphasizing final answer accuracy. However, this evaluation paradigm often suffers from contamination and does not provide insights into the reasoning traces. In this work, we systematically investigate these gaps with the help of controllable puzzle environments that allow precise manipulation of complexity while maintaining consistent logical structures. This setup enables the analysis of not only final answers but also the internal reasoning traces, offering insights into how LRMs think. Through extensive experiments, we show that LRMs face a complete accuracy collapse beyond certain complexities. Moreover, they exhibit a counterintuitive scaling limit: their reasoning effort increases with problem complexity up to a point, then declines despite having remaining token budget. By comparing LRMs with their standard LLM counterparts under same inference compute, we identify three performance regimes: (1) low-complexity tasks where standard models outperform LRMs, (2) medium-complexity tasks where LRMs demonstrates advantage, and (3) high-complexity tasks where both models face complete collapse. We found that LRMs have limitations in exact computation: they fail to use explicit algorithms and reason inconsistently across scales. We also investigate the reasoning traces in more depth, studying the patterns of explored solutions and analyzing the models' computational behavior, shedding light on their strengths, limitations, and raising questions about their reasoning capabilities.

Large Language Models (LLMs) have made notable progress in mathematical reasoning, yet they often rely on single-paradigm reasoning that limits their effectiveness across diverse tasks. In this paper, we introduce Chain-of-Reasoning (CoR), a novel unified framework that integrates multiple reasoning paradigms--Natural Language Reasoning (NLR), Algorithmic Reasoning (AR), and Symbolic Reasoning (SR)--to enable synergistic collaboration. CoR generates multiple potential answers using different reasoning paradigms and synthesizes them into a coherent final solution. We propose a Progressive Paradigm Training (PPT) strategy that allows models to progressively master these paradigms, culminating in the development of CoR-Math-7B. Experimental results demonstrate that CoR-Math-7B significantly outperforms current SOTA models, achieving up to a 41.0% absolute improvement over GPT-4 in theorem proving tasks and a 7.9% improvement over RL-based methods in arithmetic tasks. These results showcase the enhanced mathematical comprehensive ability of our model, achieving significant performance gains on specific tasks and enabling zero-shot generalization across tasks.

Our work presents a novel angle for evaluating language models' (LMs) mathematical abilities, by investigating whether they can discern skills and concepts enabled by math content. We contribute two datasets: one consisting of 385 fine-grained descriptions of K-12 math skills and concepts, or standards, from Achieve the Core (ATC), and another of 9.9K problems labeled with these standards (MathFish). Working with experienced teachers, we find that LMs struggle to tag and verify standards linked to problems, and instead predict labels that are close to ground truth, but differ in subtle ways. We also show that LMs often generate problems that do not fully align with standards described in prompts. Finally, we categorize problems in GSM8k using math standards, allowing us to better understand why some problems are more difficult to solve for models than others.

Mathematical understanding and reasoning are crucial tasks for assessing the capabilities of artificial intelligence (AI). However, existing benchmarks either require just a few steps of reasoning, or only contain a small amount of data in one specific topic, making it hard to analyse AI's behaviour with reference to different problems within a specific topic in detail. In this work, we propose Conic10K, a challenging math problem dataset on conic sections in Chinese senior high school education. Our dataset contains various problems with different reasoning depths, while only the knowledge from conic sections is required. Since the dataset only involves a narrow range of knowledge, it is easy to separately analyse the knowledge a model possesses and the reasoning ability it has. For each problem, we provide a high-quality formal representation, the reasoning steps, and the final solution. Experiments show that existing large language models, including GPT-4, exhibit weak performance on complex reasoning. We hope that our findings could inspire more advanced techniques for precise natural language understanding and reasoning. Our dataset and codes are available at https://github.com/whyNLP/Conic10K.

Multimodal Large Language Models (MLLMs) have shown promising capabilities in mathematical reasoning within visual contexts across various datasets. However, most existing multimodal math benchmarks are limited to single-visual contexts, which diverges from the multi-visual scenarios commonly encountered in real-world mathematical applications. To address this gap, we introduce MV-MATH: a meticulously curated dataset of 2,009 high-quality mathematical problems. Each problem integrates multiple images interleaved with text, derived from authentic K-12 scenarios, and enriched with detailed annotations. MV-MATH includes multiple-choice, free-form, and multi-step questions, covering 11 subject areas across 3 difficulty levels, and serves as a comprehensive and rigorous benchmark for assessing MLLMs' mathematical reasoning in multi-visual contexts. Through extensive experimentation, we observe that MLLMs encounter substantial challenges in multi-visual math tasks, with a considerable performance gap relative to human capabilities on MV-MATH. Furthermore, we analyze the performance and error patterns of various models, providing insights into MLLMs' mathematical reasoning capabilities within multi-visual settings.

Large language models (LLMs) have demonstrated impressive reasoning capabilities, particularly in textual mathematical problem-solving. However, existing open-source image instruction fine-tuning datasets, containing limited question-answer pairs per image, do not fully exploit visual information to enhance the multimodal mathematical reasoning capabilities of Multimodal LLMs (MLLMs). To bridge this gap, we address the lack of high-quality, diverse multimodal mathematical datasets by collecting 40K high-quality images with question-answer pairs from 24 existing datasets and synthesizing 320K new pairs, creating the MathV360K dataset, which enhances both the breadth and depth of multimodal mathematical questions. We introduce Math-LLaVA, a LLaVA-1.5-based model fine-tuned with MathV360K. This novel approach significantly improves the multimodal mathematical reasoning capabilities of LLaVA-1.5, achieving a 19-point increase and comparable performance to GPT-4V on MathVista's minitest split. Furthermore, Math-LLaVA demonstrates enhanced generalizability, showing substantial improvements on the MMMU benchmark. Our research highlights the importance of dataset diversity and synthesis in advancing MLLMs' mathematical reasoning abilities. The code and data are available at: https://github.com/HZQ950419/Math-LLaVA.

Large language models (LLMs) have demonstrated remarkable capabilities in problem-solving. However, their proficiency in solving mathematical problems remains inadequate. We propose MathScale, a simple and scalable method to create high-quality mathematical reasoning data using frontier LLMs (e.g., {\tt GPT-3.5}). Inspired by the cognitive mechanism in human mathematical learning, it first extracts topics and knowledge points from seed math questions and then build a concept graph, which is subsequently used to generate new math questions. MathScale exhibits effective scalability along the size axis of the math dataset that we generate. As a result, we create a mathematical reasoning dataset (MathScaleQA) containing two million math question-answer pairs. To evaluate mathematical reasoning abilities of LLMs comprehensively, we construct {\sc MwpBench}, a benchmark of Math Word Problems, which is a collection of ten datasets (including GSM8K and MATH) covering K-12, college, and competition level math problems. We apply MathScaleQA to fine-tune open-source LLMs (e.g., LLaMA-2 and Mistral), resulting in significantly improved capabilities in mathematical reasoning. Evaluated on {\sc MwpBench}, MathScale-7B achieves state-of-the-art performance across all datasets, surpassing its best peers of equivalent size by 42.9\% in micro average accuracy and 43.7\% in macro average accuracy, respectively.

Recently, there have been notable advancements in large language models (LLMs), demonstrating their growing abilities in complex reasoning. However, existing research largely overlooks a thorough and systematic comparison of these models' reasoning processes and outputs, particularly regarding their self-reflection pattern (also termed "Aha moment") and the interconnections across diverse domains. This paper proposes a novel framework for analyzing the reasoning characteristics of four cutting-edge large reasoning models (GPT-o1, DeepSeek-R1, Kimi-k1.5, and Grok-3) using keywords statistic and LLM-as-a-judge paradigm. Our approach connects their internal thinking processes with their final outputs. A diverse dataset consists of real-world scenario-based questions covering logical deduction, causal inference, and multi-step problem-solving. Additionally, a set of metrics is put forward to assess both the coherence of reasoning and the accuracy of the outputs. The research results uncover various patterns of how these models balance exploration and exploitation, deal with problems, and reach conclusions during the reasoning process. Through quantitative and qualitative comparisons, disparities among these models are identified in aspects such as the depth of reasoning, the reliance on intermediate steps, and the degree of similarity between their thinking processes and output patterns and those of GPT-o1. This work offers valuable insights into the trade-off between computational efficiency and reasoning robustness and provides practical recommendations for enhancing model design and evaluation in practical applications. We publicly release our project at: https://github.com/ChangWenhan/FromThinking2Output

Recent Large Language Models (LLMs) have reported high accuracy on reasoning benchmarks. However, it is still unclear whether the observed results arise from true reasoning or from statistical recall of the training set. Inspired by the ladder of causation (Pearl, 2009) and its three levels (associations, interventions and counterfactuals), this paper introduces RE-IMAGINE, a framework to characterize a hierarchy of reasoning ability in LLMs, alongside an automated pipeline to generate problem variations at different levels of the hierarchy. By altering problems in an intermediate symbolic representation, RE-IMAGINE generates arbitrarily many problems that are not solvable using memorization alone. Moreover, the framework is general and can work across reasoning domains, including math, code, and logic. We demonstrate our framework on four widely-used benchmarks to evaluate several families of LLMs, and observe reductions in performance when the models are queried with problem variations. These assessments indicate a degree of reliance on statistical recall for past performance, and open the door to further research targeting skills across the reasoning hierarchy.

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 reasoning models (LRMs) have recently demonstrated impressive capabilities in complex reasoning tasks by leveraging increased test-time computation and exhibiting behaviors reminiscent of human-like self-reflection. While LRMs show a clear capacity for valuable self-reflection, how this ability interacts with other model behaviors remains underexplored. We investigate this connection by analyzing verbalized confidence, how models articulate their certainty, as a lens into the nature of self-reflection in LRMs. We find that supervised fine-tuning on reasoning traces (i.e., distillation) and reinforcement learning can improve verbalized calibration in reasoning-intensive settings in a progressive, laddered fashion. However, our results also indicate that reasoning models may possess a diminished awareness of their own knowledge boundaries, as evidenced by significantly lower "I don't know" response rates on factuality benchmarks. Moreover, we examine the relationship between verbalized confidence and reasoning chains, finding that models tend to express higher confidence when providing shorter or less elaborate reasoning. Our findings highlight how reasoning-oriented training can enhance performance in reasoning-centric tasks while potentially incurring a "reasoning tax," a cost reflected in the model's reduced ability to accurately recognize the limits of its own knowledge in small-scale models. More broadly, our work showcases how this erosion of knowledge boundaries can compromise model faithfulness, as models grow more confident without a commensurate understanding of when they should abstain.

Recent scholarship on reasoning in LLMs has supplied evidence of impressive performance and flexible adaptation to machine generated or human feedback. Nonmonotonic reasoning, crucial to human cognition for navigating the real world, remains a challenging, yet understudied task. In this work, we study nonmonotonic reasoning capabilities of seven state-of-the-art LLMs in one abstract and one commonsense reasoning task featuring generics, such as 'Birds fly', and exceptions, 'Penguins don't fly' (see Fig. 1). While LLMs exhibit reasoning patterns in accordance with human nonmonotonic reasoning abilities, they fail to maintain stable beliefs on truth conditions of generics at the addition of supporting examples ('Owls fly') or unrelated information ('Lions have manes'). Our findings highlight pitfalls in attributing human reasoning behaviours to LLMs, as well as assessing general capabilities, while consistent reasoning remains elusive.

Recent advances in reasoning models have demonstrated significant improvements in accuracy, particularly for complex tasks such as mathematical reasoning, by employing detailed and comprehensive reasoning processes. However, generating these lengthy reasoning sequences is computationally expensive and time-consuming. To address this inefficiency, we leverage the inherent parallelizability of certain tasks to accelerate the reasoning process. Specifically, when multiple parallel reasoning branches exist, we decode multiple tokens per step using a specialized attention mask, processing them within a single sequence, avoiding additional memory usage. Experimental results show that our method achieves over 100% speedup in decoding time while maintaining the answer quality.

Large language models have demonstrated remarkable progress in mathematical reasoning, leveraging chain-of-thought and test-time compute scaling. However, many open questions remain regarding the interplay between reasoning token usage and accuracy gains. In particular, when comparing models across generations, it is unclear whether improved performance results from longer reasoning chains or more efficient reasoning. We systematically analyze chain-of-thought length across o1-mini and o3-mini variants on the Omni-MATH benchmark, finding that o3-mini (m) achieves superior accuracy without requiring longer reasoning chains than o1-mini. Moreover, we show that accuracy generally declines as reasoning chains grow across all models and compute settings, even when controlling for difficulty of the questions. This accuracy drop is significantly smaller in more proficient models, suggesting that new generations of reasoning models use test-time compute more effectively. Finally, we highlight that while o3-mini (h) achieves a marginal accuracy gain over o3-mini (m), it does so by allocating substantially more reasoning tokens across all problems, even the ones that o3-mini (m) can already solve. These findings provide new insights into the relationship between model capability and reasoning length, with implications for efficiency, scaling, and evaluation methodologies.

Despite high benchmark scores, Large Language Models (LLMs) often fail simple problem, raising a critical question: Do LLMs learn mathematical principles or merely memorize patterns? Rather than designing increasingly complex benchmarks like recent works, we investigate this using elementary two-integer addition (0 to 2^{64}), probing two core properties: commutativity (A+B=B+A) and compositional generalization (via isomorphic symbolic mappings, e.g., 7 rightarrow y). While state-of-the-art LLMs achieve 73.8-99.8\% accuracy on numerical addition, performance collapses to leq7.5\% under symbolic mapping, indicating failure to generalize learned rules. Non-monotonic performance scaling with digit count and frequent commutativity violations (over 1,700 cases of A+B neq B+A) further support this. Explicitly providing addition rules degrades performance by 81.2\% on average, while self-explanation maintains baseline accuracy, suggesting LLM arithmetic processing is misaligned with human-defined principles. Our findings indicate current LLMs rely on memory pattern over genuine rule learning, highlighting architectural limitations and the need for new approaches to achieve true mathematical reasoning.

Neural-symbolic learning, an intersection of neural networks and symbolic reasoning, aims to blend neural networks' learning capabilities with symbolic AI's interpretability and reasoning. This paper introduces an approach designed to improve the performance of neural models in learning reasoning tasks. It achieves this by integrating Answer Set Programming (ASP) solvers and domain-specific expertise, which is an approach that diverges from traditional complex neural-symbolic models. In this paper, a shallow artificial neural network (ANN) is specifically trained to solve Sudoku puzzles with minimal training data. The model has a unique loss function that integrates losses calculated using the ASP solver outputs, effectively enhancing its training efficiency. Most notably, the model shows a significant improvement in solving Sudoku puzzles using only 12 puzzles for training and testing without hyperparameter tuning. This advancement indicates that the model's enhanced reasoning capabilities have practical applications, extending well beyond Sudoku puzzles to potentially include a variety of other domains. The code can be found on GitHub: https://github.com/Fadi2200/ASPEN.

Large language models (LLMs) have scaled up to unlock a wide range of complex reasoning tasks with the aid of various prompting methods. However, current prompting methods generate natural language intermediate steps to help reasoning, which can cause imperfect task reduction and confusion. To mitigate such limitations, we explore code prompting, a neural symbolic prompting method with both zero-shot and few-shot versions which triggers code as intermediate steps. We conduct experiments on 7 widely-used benchmarks involving symbolic reasoning and arithmetic reasoning. Code prompting generally outperforms chain-of-thought (CoT) prompting. To further understand the performance and limitations of code prompting, we perform extensive ablation studies and error analyses, and identify several exclusive advantages of using symbolic promptings compared to natural language. We also consider the ensemble of code prompting and CoT prompting to combine the strengths of both. Finally, we show through experiments how code annotations and their locations affect code prompting.

We introduce MAmmoTH, a series of open-source large language models (LLMs) specifically tailored for general math problem-solving. The MAmmoTH models are trained on MathInstruct, our meticulously curated instruction tuning dataset. MathInstruct is compiled from 13 math datasets with intermediate rationales, six of which have rationales newly curated by us. It presents a unique hybrid of chain-of-thought (CoT) and program-of-thought (PoT) rationales, and also ensures extensive coverage of diverse fields in math. The hybrid of CoT and PoT not only unleashes the potential of tool use but also allows different thought processes for different math problems. As a result, the MAmmoTH series substantially outperform existing open-source models on nine mathematical reasoning datasets across all scales with an average accuracy gain between 13% and 29%. Remarkably, our MAmmoTH-7B model reaches 35% on MATH (a competition-level dataset), which exceeds the best open-source 7B model (WizardMath) by 25%, and the MAmmoTH-34B model achieves 46% accuracy on MATH, even surpassing GPT-4's CoT result. Our work underscores the importance of diverse problem coverage and the use of hybrid rationales in developing superior math generalist models.

The rapid advancements in Vision-Language Models (VLMs) have shown great potential in tackling mathematical reasoning tasks that involve visual context. Unlike humans who can reliably apply solution steps to similar problems with minor modifications, we found that SOTA VLMs like GPT-4o can consistently fail in these scenarios, revealing limitations in their mathematical reasoning capabilities. In this paper, we investigate the mathematical reasoning robustness in VLMs and evaluate how well these models perform under different variants of the same question, such as changes in visual numerical values or function graphs. While several vision-based math benchmarks have been developed to assess VLMs' problem-solving capabilities, these benchmarks contain only static sets of problems and cannot easily evaluate mathematical reasoning robustness. To fill this gap, we introduce DynaMath, a dynamic visual math benchmark designed for in-depth assessment of VLMs. DynaMath includes 501 high-quality, multi-topic seed questions, each represented as a Python program. Those programs are carefully designed and annotated to enable the automatic generation of a much larger set of concrete questions, including many different types of visual and textual variations. DynaMath allows us to evaluate the generalization ability of VLMs, by assessing their performance under varying input conditions of a seed question. We evaluated 14 SOTA VLMs with 5,010 generated concrete questions. Our results show that the worst-case model accuracy, defined as the percentage of correctly answered seed questions in all 10 variants, is significantly lower than the average-case accuracy. Our analysis emphasizes the need to study the robustness of VLMs' reasoning abilities, and DynaMath provides valuable insights to guide the development of more reliable models for mathematical reasoning.

Vision language models (VLMs) achieve unified modeling of images and text, enabling them to accomplish complex real-world tasks through perception, planning, and reasoning. Among these tasks, reasoning is particularly representative, with mathematical reasoning serving as a prominent example. It highlights the high-level capability of VLMs to comprehend mathematical information in images and to perform sophisticated reasoning. Recently, numerous visual mathematical reasoning benchmarks have been proposed, but they are often restricted to geometry, lack coverage of math word problems, and rarely assess reasoning across multiple images. To address these gaps, we introduce GSM8K-V, a purely visual multi-image mathematical reasoning benchmark. GSM8K-V is built by systematically mapping each sample from the widely used text-based GSM8K into visual form. Through a carefully designed automated image-generation pipeline combined with meticulous human annotation, we curate 1,319 high-quality samples. We evaluate a wide range of open-source and closed-source models on GSM8K-V. Results show that although existing VLMs have nearly saturated performance on text-based GSM8K, there remains substantial room for improvement on GSM8K-V. For example, the best-performing model, Gemini-2.5-Pro, achieves 95.22% accuracy on GSM8K but only 46.93% on GSM8K-V. We conduct a comprehensive analysis of GSM8K-V, examining the limitations of current models as well as potential directions for improvement. GSM8K-V offers a new perspective on visual mathematical reasoning and establishes a benchmark to guide the development of more robust and generalizable VLMs.

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.

Reasoning has emerged as the next major frontier for language models (LMs), with rapid advances from both academic and industrial labs. However, this progress often outpaces methodological rigor, with many evaluations relying on benchmarking practices that lack transparency, robustness, or statistical grounding. In this work, we conduct a comprehensive empirical study and find that current mathematical reasoning benchmarks are highly sensitive to subtle implementation choices - including decoding parameters, random seeds, prompt formatting, and even hardware and software-framework configurations. Performance gains reported in recent studies frequently hinge on unclear comparisons or unreported sources of variance. To address these issues, we propose a standardized evaluation framework with clearly defined best practices and reporting standards. Using this framework, we reassess recent methods and find that reinforcement learning (RL) approaches yield only modest improvements - far below prior claims - and are prone to overfitting, especially on small-scale benchmarks like AIME24. In contrast, supervised finetuning (SFT) methods show consistently stronger generalization. To foster reproducibility, we release all code, prompts, and model outputs, for reasoning benchmarks, establishing more rigorous foundations for future work.

While large language models demonstrate impressive performance on static benchmarks, the true potential of large language models as self-learning and reasoning agents in dynamic environments remains unclear. This study systematically evaluates the efficacy of self-reflection, heuristic mutation, and planning as prompting techniques to test the adaptive capabilities of agents. We conduct experiments with various open-source language models in dynamic environments and find that larger models generally outperform smaller ones, but that strategic prompting can close this performance gap. Second, a too-long prompt can negatively impact smaller models on basic reactive tasks, while larger models show more robust behaviour. Third, advanced prompting techniques primarily benefit smaller models on complex games, but offer less improvement for already high-performing large language models. Yet, we find that advanced reasoning methods yield highly variable outcomes: while capable of significantly improving performance when reasoning and decision-making align, they also introduce instability and can lead to big performance drops. Compared to human performance, our findings reveal little evidence of true emergent reasoning. Instead, large language model performance exhibits persistent limitations in crucial areas such as planning, reasoning, and spatial coordination, suggesting that current-generation large language models still suffer fundamental shortcomings that may not be fully overcome through self-reflective prompting alone. Reasoning is a multi-faceted task, and while reasoning methods like Chain of thought improves multi-step reasoning on math word problems, our findings using dynamic benchmarks highlight important shortcomings in general reasoning capabilities, indicating a need to move beyond static benchmarks to capture the complexity of reasoning.

Large Language Models (LLMs) have been shown to achieve breakthrough performance on complex logical reasoning tasks. Nevertheless, most existing research focuses on employing formal language to guide LLMs to derive reliable reasoning paths, while systematic evaluations of these capabilities are still limited. In this paper, we aim to conduct a comprehensive evaluation of LLMs across various logical reasoning problems utilizing formal languages. From the perspective of three dimensions, i.e., spectrum of LLMs, taxonomy of tasks, and format of trajectories, our key findings are: 1) Thinking models significantly outperform Instruct models, especially when formal language is employed; 2) All LLMs exhibit limitations in inductive reasoning capability, irrespective of whether they use a formal language; 3) Data with PoT format achieves the best generalization performance across other languages. Additionally, we also curate the formal-relative training data to further enhance the small language models, and the experimental results indicate that a simple rejected fine-tuning method can better enable LLMs to generalize across formal languages and achieve the best overall performance. Our codes and reports are available at https://github.com/jiangjin1999/FormalEval.

Probing and enhancing large language models' reasoning capacity remains a crucial open question. Here we re-purpose the reverse dictionary task as a case study to probe LLMs' capacity for conceptual inference. We use in-context learning to guide the models to generate the term for an object concept implied in a linguistic description. Models robustly achieve high accuracy in this task, and their representation space encodes information about object categories and fine-grained features. Further experiments suggest that the conceptual inference ability as probed by the reverse-dictionary task predicts model's general reasoning performance across multiple benchmarks, despite similar syntactic generalization behaviors across models. Explorative analyses suggest that prompting LLMs with descriptionRightarrowword examples may induce generalization beyond surface-level differences in task construals and facilitate models on broader commonsense reasoning problems.

Mathematical reasoning is an increasingly important indicator of large language model (LLM) capabilities, yet we lack understanding of how LLMs process even simple mathematical tasks. To address this, we reverse engineer how three mid-sized LLMs compute addition. We first discover that numbers are represented in these LLMs as a generalized helix, which is strongly causally implicated for the tasks of addition and subtraction, and is also causally relevant for integer division, multiplication, and modular arithmetic. We then propose that LLMs compute addition by manipulating this generalized helix using the "Clock" algorithm: to solve a+b, the helices for a and b are manipulated to produce the a+b answer helix which is then read out to model logits. We model influential MLP outputs, attention head outputs, and even individual neuron preactivations with these helices and verify our understanding with causal interventions. By demonstrating that LLMs represent numbers on a helix and manipulate this helix to perform addition, we present the first representation-level explanation of an LLM's mathematical capability.

Large language models (LLMs) have significantly improved their reasoning capabilities; however, they still struggle with complex multi-step mathematical problem-solving due to error propagation, lack of self-correction, and limited adaptability to diverse reasoning styles. Existing methods rely on static fine-tuning or prompt engineering, which fail to generalize across problem complexities, while the scarcity of high-quality preference data further hinders reliable reasoning. We introduce SPHERE, a self-evolving data generation pipeline that enhances reasoning in small language models (SLMs) by iteratively generating, correcting, and diversifying reasoning chains. SPHERE operates in three stages: (i) Self-Generation, where the model autonomously constructs problem-solving steps; (ii) Self-Correction, enabling it to identify and rectify errors; and (iii) Diversity Induction, improving robustness through multiple valid reasoning trajectories. This self-evolution mechanism strengthens mathematical reasoning and enhances model reliability. Evaluations on MATH 500, GSM8K, AIME, AMC, and Olympiad show that SPHERE-trained models achieve significant gains over their base versions and match/surpass GPT-4o on certain benchmarks. Our findings demonstrate that self-evolving models can close the reasoning gap between SLMs and state-of-the-art LLMs, making mathematical AI more reliable, scalable, and efficient.

Mathematical reasoning is regarded as a necessary ability for Language Models (LMs). Recent works demonstrate large LMs' impressive performance in solving math problems. The success is attributed to their Chain-of-Thought (CoT) reasoning abilities, i.e., the ability to decompose complex questions into step-by-step reasoning chains, but such ability seems only to emerge from models with abundant parameters. This work investigates how to incorporate relatively small LMs with the capabilities of multi-step reasoning. We propose to inject such abilities by continually pre-training LMs on a synthetic dataset MsAT which is composed of Multi-step Arithmetic Tasks. Our experiments on four math word problem datasets show the effectiveness of the proposed method in enhancing LMs' math reasoning abilities.

In this paper, we aim to establish a simple, effective, and theoretically grounded benchmark for rigorously probing abstract reasoning in Large Language Models (LLMs). To achieve this, we first develop a mathematic framework that defines abstract reasoning as the ability to: (i) extract essential patterns independent of surface representations, and (ii) apply consistent rules to these abstract patterns. Based on this framework, we introduce two novel complementary metrics: \(\scoreGamma\) measures basic reasoning accuracy, while \(\scoreDelta\) quantifies a model's reliance on specific symbols rather than underlying patterns - a key indicator of true abstraction versus mere memorization. To implement this measurement, we design a benchmark: systematic symbol remapping in rule-based tasks, which forces models to demonstrate genuine pattern recognition beyond superficial token matching. Extensive LLM evaluations using this benchmark (commercial API models, 7B-70B, multi-agent) reveal:1) critical limitations in non-decimal arithmetic and symbolic reasoning; 2) persistent abstraction gaps despite chain-of-thought prompting; and 3) \(\scoreDelta\)'s effectiveness in robustly measuring memory dependence by quantifying performance degradation under symbol remapping, particularly highlighting operand-specific memorization. These findings underscore that current LLMs, despite domain-specific strengths, still lack robust abstract reasoning, highlighting key areas for future improvement.

Recent advances in large language models (LLMs) and multimodal LLMs (MLLMs) have led to strong reasoning ability across a wide range of tasks. However, their ability to perform mathematical reasoning from spoken input remains underexplored. Prior studies on speech modality have mostly focused on factual speech understanding or simple audio reasoning tasks, providing limited insight into logical step-by-step reasoning, such as that required for mathematical problem solving. To address this gap, we introduce Spoken Math Question Answering (Spoken-MQA), a new benchmark designed to evaluate the mathematical reasoning capabilities of speech-based models, including both cascade models (ASR + LLMs) and end-to-end speech LLMs. Spoken-MQA covers a diverse set of math problems, including pure arithmetic, single-step and multi-step contextual reasoning, and knowledge-oriented reasoning problems, all presented in unambiguous natural spoken language. Through extensive experiments, we find that: (1) while some speech LLMs perform competitively on contextual reasoning tasks involving basic arithmetic, they still struggle with direct arithmetic problems; (2) current LLMs exhibit a strong bias toward symbolic mathematical expressions written in LaTex and have difficulty interpreting verbalized mathematical expressions; and (3) mathematical knowledge reasoning abilities are significantly degraded in current speech LLMs.

When faced with novel situations, people are able to marshal relevant considerations from a wide range of background knowledge and put these to use in inferences and predictions. What permits us to draw in globally relevant information and reason over it coherently? Here, we explore the hypothesis that people use a combination of distributed and symbolic representations to construct bespoke mental models tailored to novel situations. We propose a computational implementation of this idea -- a ``Model Synthesis Architecture'' (MSA) -- using language models to implement global relevance-based retrieval and model synthesis and probabilistic programs to implement bespoke, coherent world models. We evaluate our MSA as a model of human judgments on a novel reasoning dataset. The dataset -- built around a `Model Olympics` domain of sports vignettes -- tests models' capacity for human-like, open-ended reasoning by requiring (i) judgments about novel causal structures described in language; (ii) drawing on large bodies of background knowledge; and (iii) doing both in light of observations that introduce arbitrary novel variables. Our MSA approach captures human judgments better than language model-only baselines, under both direct and chain-of-thought generations from the LM that supports model synthesis. These results suggest that MSAs can be implemented in a way that mirrors people's ability to deliver locally coherent reasoning over globally relevant variables, offering a path to understanding and replicating human reasoning in open-ended domains.

Cognitive textual and visual reasoning tasks, such as puzzles, series, and analogies, demand the ability to quickly reason, decipher, and evaluate patterns both textually and spatially. While LLMs and VLMs, through extensive training on large amounts of human-curated data, have attained a high level of pseudo-human intelligence in some common sense reasoning tasks, they still struggle with more complex reasoning tasks that require cognitive understanding. In this work, we introduce a new dataset, NTSEBench, designed to evaluate the cognitive multi-modal reasoning and problem-solving skills of large models. The dataset comprises 2,728 multiple-choice questions comprising of a total of 4,642 images across 26 categories sampled from the NTSE examination conducted nationwide in India, featuring both visual and textual general aptitude questions that do not rely on rote learning. We establish baselines on the dataset using state-of-the-art LLMs and VLMs. To facilitate a comparison between open source and propriety models, we propose four distinct modeling strategies to handle different modalities (text and images) in the dataset instances.

Existing math datasets evaluate the reasoning abilities of large language models (LLMs) by either using the final answer or the intermediate reasoning steps derived from static examples. However, the former approach fails to surface model's uses of shortcuts and wrong reasoning while the later poses challenges in accommodating alternative solutions. In this work, we seek to use symbolic programs as a means for automated evaluation if a model can consistently produce correct final answers across various inputs to the program. We begin by extracting programs for popular math datasets (GSM8K and MATH) using GPT4-o. For those executable programs verified using the original input-output pairs, they are found to encapsulate the proper reasoning required to solve the original text questions. We then prompt GPT4-o to generate new questions using alternative input-output pairs based the extracted program. We apply the resulting datasets to evaluate a collection of LLMs. In our experiments, we observe significant accuracy drops using our proposed evaluation compared with original static examples, suggesting the fragility of math reasoning in state-of-the-art LLMs.

Procedural planning, which entails decomposing a high-level goal into a sequence of temporally ordered steps, is an important yet intricate task for machines. It involves integrating common-sense knowledge to reason about complex contextualized situations that are often counterfactual, e.g. "scheduling a doctor's appointment without a phone". While current approaches show encouraging results using large language models (LLMs), they are hindered by drawbacks such as costly API calls and reproducibility issues. In this paper, we advocate planning using smaller language models. We present PlaSma, a novel two-pronged approach to endow small language models with procedural knowledge and (counterfactual) planning capabilities. More concretely, we develop symbolic procedural knowledge distillation to enhance the implicit knowledge in small language models and an inference-time algorithm to facilitate more structured and accurate reasoning. In addition, we introduce a novel task, Counterfactual Planning, that requires a revision of a plan to cope with a counterfactual situation. In both the original and counterfactual setting, we show that orders-of-magnitude smaller models (770M-11B parameters) can compete and often surpass their larger teacher models' capabilities.

The observed similarities in the behavior of humans and Large Language Models (LLMs) have prompted researchers to consider the potential of using LLMs as models of human cognition. However, several significant challenges must be addressed before LLMs can be legitimately regarded as cognitive models. For instance, LLMs are trained on far more data than humans typically encounter, and may have been directly trained on human data in specific cognitive tasks or aligned with human preferences. Consequently, the origins of these behavioral similarities are not well understood. In this paper, we propose a novel way to enhance the utility of LLMs as cognitive models. This approach involves (i) leveraging computationally equivalent tasks that both an LLM and a rational agent need to master for solving a cognitive problem and (ii) examining the specific task distributions required for an LLM to exhibit human-like behaviors. We apply this approach to decision-making -- specifically risky and intertemporal choice -- where the key computationally equivalent task is the arithmetic of expected value calculations. We show that an LLM pretrained on an ecologically valid arithmetic dataset, which we call Arithmetic-GPT, predicts human behavior better than many traditional cognitive models. Pretraining LLMs on ecologically valid arithmetic datasets is sufficient to produce a strong correspondence between these models and human decision-making. Our results also suggest that LLMs used as cognitive models should be carefully investigated via ablation studies of the pretraining data.

In this paper, we investigate the underlying factors that potentially enhance the mathematical reasoning capabilities of large language models (LLMs). We argue that the data scaling law for math reasoning capabilities in modern LLMs is far from being saturated, highlighting how the model's quality improves with increases in data quantity. To support this claim, we introduce the Skywork-Math model series, supervised fine-tuned (SFT) on common 7B LLMs using our proposed 2.5M-instance Skywork-MathQA dataset. Skywork-Math 7B has achieved impressive accuracies of 51.2% on the competition-level MATH benchmark and 83.9% on the GSM8K benchmark using only SFT data, outperforming an early version of GPT-4 on MATH. The superior performance of Skywork-Math models contributes to our novel two-stage data synthesis and model SFT pipelines, which include three different augmentation methods and a diverse seed problem set, ensuring both the quantity and quality of Skywork-MathQA dataset across varying difficulty levels. Most importantly, we provide several practical takeaways to enhance math reasoning abilities in LLMs for both research and industry applications.

Chain-of-thought reasoning, while powerful, can produce unnecessarily verbose output for simpler problems. We present a framework for difficulty-aware reasoning that teaches models to dynamically adjust reasoning depth based on problem complexity. Remarkably, we show that models can be endowed with such dynamic inference pathways without any architectural modifications; we simply post-train on data that is carefully curated to include chain-of-thought traces that are proportional in length to problem difficulty. Our analysis reveals that post-training via supervised fine-tuning (SFT) primarily captures patterns like reasoning length and format, while direct preference optimization (DPO) preserves reasoning accuracy, with their combination reducing length and maintaining or improving performance. Both quantitative metrics and qualitative assessments confirm that models can learn to "think proportionally", reasoning minimally on simple problems while maintaining depth for complex ones.

Can Multimodal Large Language Models (MLLMs) develop an intuitive number sense similar to humans? Targeting this problem, we introduce Visual Number Benchmark (VisNumBench) to evaluate the number sense abilities of MLLMs across a wide range of visual numerical tasks. VisNumBench consists of about 1,900 multiple-choice question-answer pairs derived from both synthetic and real-world visual data, covering seven visual numerical attributes and four types of visual numerical estimation tasks. Our experiments on VisNumBench led to the following key findings: (i) The 17 MLLMs we tested, including open-source models such as Qwen2.5-VL and InternVL2.5, as well as proprietary models like GPT-4o and Gemini 2.0 Flash, perform significantly below human levels in number sense-related tasks. (ii) Multimodal mathematical models and multimodal chain-of-thought (CoT) models did not exhibit significant improvements in number sense abilities. (iii) Stronger MLLMs with larger parameter sizes and broader general abilities demonstrate modest gains in number sense abilities. We believe VisNumBench will serve as a valuable resource for the research community, encouraging further advancements in enhancing MLLMs' number sense abilities. All benchmark resources, including code and datasets, will be publicly available at https://wwwtttjjj.github.io/VisNumBench/.

Supervised fine-tuning (SFT) and reinforcement learning (RL) are widely used post-training techniques for foundation models. However, their roles in enhancing model generalization capabilities remain unclear. This paper studies the difference between SFT and RL on generalization and memorization, focusing on text-based rule variants and visual variants. We introduce GeneralPoints, an arithmetic reasoning card game, and adopt V-IRL, a real-world navigation environment, to assess how models trained with SFT and RL generalize to unseen variants in both textual and visual domains. We show that RL, especially when trained with an outcome-based reward, generalizes across both rule-based textual and visual variants. SFT, in contrast, tends to memorize training data and struggles to generalize out-of-distribution scenarios. Further analysis reveals that RL improves the model's underlying visual recognition capabilities, contributing to its enhanced generalization in the visual domain. Despite RL's superior generalization, we show that SFT remains essential for effective RL training; SFT stabilizes the model's output format, enabling subsequent RL to achieve its performance gains. These findings demonstrates the capability of RL for acquiring generalizable knowledge in complex, multi-modal tasks.

Effective training of language models (LMs) for mathematical reasoning tasks demands high-quality supervised fine-tuning data. Besides obtaining annotations from human experts, a common alternative is sampling from larger and more powerful LMs. However, this knowledge distillation approach can be costly and unstable, particularly when relying on closed-source, proprietary LMs like GPT-4, whose behaviors are often unpredictable. In this work, we demonstrate that the reasoning abilities of small-scale LMs can be enhanced through self-training, a process where models learn from their own outputs. We also show that the conventional self-training can be further augmented by a preference learning algorithm called Direct Preference Optimization (DPO). By integrating DPO into self-training, we leverage preference data to guide LMs towards more accurate and diverse chain-of-thought reasoning. We evaluate our method across various mathematical reasoning tasks using different base models. Our experiments show that this approach not only improves LMs' reasoning performance but also offers a more cost-effective and scalable solution compared to relying on large proprietary LMs.

Large Language Models (LLMs) have significantly advanced various fields, particularly coding, mathematical reasoning, and logical problem solving. However, a critical question remains: Do these mathematical reasoning abilities persist when LLMs are presented with culturally adapted math problems? Specifically, how do LLMs perform when faced with math problems embedded in cultural contexts that have no significant representation in main stream web-scale AI training data? To explore this, we generated six synthetic cultural datasets from GSM8K, a widely used benchmark for assessing LLMs' mathematical reasoning skills. While preserving the mathematical logic and numerical values of the original GSM8K test set, we modify cultural elements such as personal names, food items, place names, etc. These culturally adapted datasets provide a more reliable framework for evaluating LLMs' mathematical reasoning under shifting cultural contexts. Our findings reveal that LLMs struggle with math problems when cultural references change, even though the underlying mathematical structure remains constant. Smaller models exhibit greater performance drops compared to larger models. Interestingly, our results also suggest that cultural familiarity can enhance mathematical reasoning. Even models with no explicit mathematical training but exposure to relevant cultural contexts sometimes outperform larger, mathematically proficient models on culturally embedded math problems. This study highlights the impact of cultural context on the mathematical reasoning abilities of LLMs, underscoring the need for more diverse and representative training data to improve robustness in real-world applications. The benchmark data sets and script for reproducing the results are available at https://github.com/akarim23131/Lost_in_Cultural_Translation

Recent large language models (LLMs) have demonstrated remarkable generalization abilities in mathematics and logical reasoning tasks. Prior research indicates that LLMs pre-trained with programming language data exhibit high mathematical and reasoning abilities; however, this causal relationship has not been rigorously tested. Our research aims to verify which programming languages and features during pre-training affect logical inference performance. Specifically, we pre-trained decoder-based language models from scratch using datasets from ten programming languages (e.g., Python, C, Java) and three natural language datasets (Wikipedia, Fineweb, C4) under identical conditions. Thereafter, we evaluated the trained models in a few-shot in-context learning setting on logical reasoning tasks: FLD and bAbi, which do not require commonsense or world knowledge. The results demonstrate that nearly all models trained with programming languages consistently outperform those trained with natural languages, indicating that programming languages contain factors that elicit logic inference performance. In addition, we found that models trained with programming languages exhibit a better ability to follow instructions compared to those trained with natural languages. Further analysis reveals that the depth of Abstract Syntax Trees representing parsed results of programs also affects logical reasoning performance. These findings will offer insights into the essential elements of pre-training for acquiring the foundational abilities of LLMs.

Large Language Models (LLMs) have demonstrated remarkable capabilities across various domains. Math Word Problems (MWPs) serve as a crucial benchmark for evaluating LLMs' reasoning abilities. While most research primarily focuses on improving accuracy, it often neglects understanding and addressing the underlying patterns of errors. Current error classification methods rely on static and predefined categories, which limit their ability to capture the full spectrum of error patterns in mathematical reasoning. To enable systematic error analysis, we collect error samples from 15 different LLMs of varying sizes across four distinct MWP datasets using multiple sampling strategies. Based on this extensive collection, we introduce MWPES-300K, a comprehensive dataset containing 304,865 error samples that cover diverse error patterns and reasoning paths. To reduce human bias and enable fine-grained analysis of error patterns, we propose a novel framework for automated dynamic error classification in mathematical reasoning. Experimental results demonstrate that dataset characteristics significantly shape error patterns, which evolve from basic to complex manifestations as model capabilities increase. With deeper insights into error patterns, we propose error-aware prompting that incorporates common error patterns as explicit guidance, leading to significant improvements in mathematical reasoning performance.

This study investigates the reasoning robustness of large language models (LLMs) on mathematical problem-solving tasks under systematically introduced input perturbations. Using the GSM8K dataset as a controlled testbed, we evaluate how well state-of-the-art models maintain logical consistency and correctness when confronted with four categories of prompt perturbations: irrelevant context, pathological instructions, factually relevant but non-essential context, and a combination of the latter two. Our experiments, conducted on thirteen open-source and closed-source LLMs, reveal that introducing irrelevant context within the model's context window significantly degrades performance, suggesting that distinguishing essential from extraneous details remains a pressing challenge. Surprisingly, performance regressions are relatively insensitive to the complexity of the reasoning task, as measured by the number of steps required, and are not strictly correlated with model size. Moreover, we observe that certain perturbations inadvertently trigger chain-of-thought-like reasoning behaviors, even without explicit prompting. Our findings highlight critical vulnerabilities in current LLMs and underscore the need for improved robustness against noisy, misleading, and contextually dense inputs, paving the way for more resilient and reliable reasoning in real-world applications.

Large language models (LLMs) have pushed the limits of natural language understanding and exhibited excellent problem-solving ability. Despite the great success, most existing open-source LLMs (\eg, LLaMA-2) are still far away from satisfactory for solving mathematical problem due to the complex reasoning procedures. To bridge this gap, we propose MetaMath, a fine-tuned language model that specializes in mathematical reasoning. Specifically, we start by bootstrapping mathematical questions by rewriting the question from multiple perspectives without extra knowledge, which results in a new dataset called {MetaMathQA}. Then we fine-tune the LLaMA-2 models on MetaMathQA. Experimental results on two popular benchmarks (\ie, GSM8K and MATH) for mathematical reasoning demonstrate that MetaMath outperforms a suite of open-source LLMs by a significant margin. Our MetaMath-7B model achieves 66.4% on GSM8K and 19.4% on MATH, exceeding the state-of-the-art models of the same size by 11.5% and 8.7%. Particularly, {MetaMath-70B} achieves an accuracy of 82.3% on {GSM8K}, slightly better than {GPT-3.5-Turbo}. We release the {MetaMathQA} dataset, the {MetaMath} models with different model sizes and the training code for public use.

The mathematical capabilities of AI systems are complex and multifaceted. Most existing research has predominantly focused on the correctness of AI-generated solutions to mathematical problems. In this work, we argue that beyond producing correct answers, AI systems should also be capable of, or assist humans in, developing novel solutions to mathematical challenges. This study explores the creative potential of Large Language Models (LLMs) in mathematical reasoning, an aspect that has received limited attention in prior research. We introduce a novel framework and benchmark, CreativeMath, which encompasses problems ranging from middle school curricula to Olympic-level competitions, designed to assess LLMs' ability to propose innovative solutions after some known solutions have been provided. Our experiments demonstrate that, while LLMs perform well on standard mathematical tasks, their capacity for creative problem-solving varies considerably. Notably, the Gemini-1.5-Pro model outperformed other LLMs in generating novel solutions. This research opens a new frontier in evaluating AI creativity, shedding light on both the strengths and limitations of LLMs in fostering mathematical innovation, and setting the stage for future developments in AI-assisted mathematical discovery.

Recent studies have shown that Large Language Models (LLMs) augmented with chain-of-thought (CoT) reasoning demonstrate impressive problem-solving abilities. However, in this work, we identify a recurring issue where these models occasionally generate overly short reasoning, leading to degraded performance on even simple mathematical problems. Specifically, we investigate how reasoning length is embedded in the hidden representations of reasoning models and its impact on accuracy. Our analysis reveals that reasoning length is governed by a linear direction in the representation space, allowing us to induce overly short reasoning by steering the model along this direction. Building on this insight, we introduce ThinkEdit, a simple yet effective weight-editing approach to mitigate the issue of overly short reasoning. We first identify a small subset of attention heads (approximately 2%) that predominantly drive short reasoning behavior. We then edit the output projection weights of these heads to suppress the short reasoning direction. With changes to only 0.1% of the model's parameters, ThinkEdit effectively reduces overly short reasoning and yields notable accuracy gains for short reasoning outputs (+5.44%), along with an overall improvement across multiple math benchmarks (+2.43%). Our findings provide new mechanistic insights into how reasoning length is controlled within LLMs and highlight the potential of fine-grained model interventions to improve reasoning quality. Our code is available at https://github.com/Trustworthy-ML-Lab/ThinkEdit

Large language models (LLMs) have demonstrated remarkable reasoning capabilities in math and coding, often bolstered by post-training on the chain-of-thoughts (CoTs) generated by stronger models. However, existing strategies for curating such training data predominantly rely on heuristics, limiting generalizability and failing to capture subtleties underlying in data. To address these limitations, we leverage influence functions to systematically attribute LLMs' reasoning ability on math and coding to individual training examples, sequences, and tokens, enabling deeper insights into effective data characteristics. Our Influence-based Reasoning Attribution (Infra) uncovers nontrivial cross-domain effects across math and coding tasks: high-difficulty math examples improve both math and code reasoning, while low-difficulty code tasks most effectively benefit code reasoning. Based on these findings, we introduce a simple yet effective dataset reweighting strategy by flipping task difficulty, which doubles AIME24 accuracy from 10\% to 20\% and boosts LiveCodeBench accuracy from 33.8\% to 35.3\% for Qwen2.5-7B-Instruct. Moreover, our fine-grained attribution reveals that the sequence-level exploratory behaviors enhance reasoning performance in both math and code, and the token-level influence patterns are distinct for math and code reasoning: the former prefers natural language logic connectors and the latter emphasizes structural syntax.

Despite strong performance on vision-language tasks, Multimodal Large Language Models (MLLMs) struggle with mathematical problem-solving, with both open-source and state-of-the-art models falling short of human performance on visual-math benchmarks. To systematically examine visual-mathematical reasoning in MLLMs, we (1) evaluate their understanding of geometric primitives, (2) test multi-step reasoning, and (3) explore a potential solution to improve visual reasoning capabilities. Our findings reveal fundamental shortcomings in shape recognition, with top models achieving under 50% accuracy in identifying regular polygons. We analyze these failures through the lens of dual-process theory and show that MLLMs rely on System 1 (intuitive, memorized associations) rather than System 2 (deliberate reasoning). Consequently, MLLMs fail to count the sides of both familiar and novel shapes, suggesting they have neither learned the concept of sides nor effectively process visual inputs. Finally, we propose Visually Cued Chain-of-Thought (VC-CoT) prompting, which enhances multi-step mathematical reasoning by explicitly referencing visual annotations in diagrams, boosting GPT-4o's accuracy on an irregular polygon side-counting task from 7% to 93%. Our findings suggest that System 2 reasoning in MLLMs remains an open problem, and visually-guided prompting is essential for successfully engaging visual reasoning. Code available at: https://github.com/rsinghlab/Shape-Blind.

The recent progress in large language models (LLMs), especially the invention of chain-of-thoughts (CoT) prompting, makes it possible to solve reasoning problems. However, even the strongest LLMs are still struggling with more complicated problems that require non-linear thinking and multi-step reasoning. In this work, we explore whether LLMs have the ability to recognize their own errors, without resorting to external resources. In particular, we investigate whether they can be used to identify individual errors within a step-by-step reasoning. To this end, we propose a zero-shot verification scheme to recognize such errors. We then use this verification scheme to improve question-answering performance, by using it to perform weighted voting on different generated answers. We test the method on three math datasets-GSM8K, MathQA, and MATH-and find that it successfully recognizes errors and, in turn, increases final predictive performance.

Large Language Models (LLMs) have succeeded remarkably in various natural language processing (NLP) tasks, yet their reasoning capabilities remain a fundamental challenge. While LLMs exhibit impressive fluency and factual recall, their ability to perform complex reasoning-spanning logical deduction, mathematical problem-solving, commonsense inference, and multi-step reasoning-often falls short of human expectations. This survey provides a comprehensive review of emerging techniques enhancing reasoning in LLMs. We categorize existing methods into key approaches, including prompting strategies (e.g., Chain-of-Thought reasoning, Self-Consistency, and Tree-of-Thought reasoning), architectural innovations (e.g., retrieval-augmented models, modular reasoning networks, and neuro-symbolic integration), and learning paradigms (e.g., fine-tuning with reasoning-specific datasets, reinforcement learning, and self-supervised reasoning objectives). Additionally, we explore evaluation frameworks used to assess reasoning in LLMs and highlight open challenges, such as hallucinations, robustness, and reasoning generalization across diverse tasks. By synthesizing recent advancements, this survey aims to provide insights into promising directions for future research and practical applications of reasoning-augmented LLMs.

Large language models (LLMs) like GPT-4, PaLM, and LLaMA have shown significant improvements in various reasoning tasks. However, smaller models such as Llama-3-8B and DeepSeekMath-Base still struggle with complex mathematical reasoning because they fail to effectively identify and correct reasoning errors. Recent reflection-based methods aim to address these issues by enabling self-reflection and self-correction, but they still face challenges in independently detecting errors in their reasoning steps. To overcome these limitations, we propose SuperCorrect, a novel two-stage framework that uses a large teacher model to supervise and correct both the reasoning and reflection processes of a smaller student model. In the first stage, we extract hierarchical high-level and detailed thought templates from the teacher model to guide the student model in eliciting more fine-grained reasoning thoughts. In the second stage, we introduce cross-model collaborative direct preference optimization (DPO) to enhance the self-correction abilities of the student model by following the teacher's correction traces during training. This cross-model DPO approach teaches the student model to effectively locate and resolve erroneous thoughts with error-driven insights from the teacher model, breaking the bottleneck of its thoughts and acquiring new skills and knowledge to tackle challenging problems. Extensive experiments consistently demonstrate our superiority over previous methods. Notably, our SuperCorrect-7B model significantly surpasses powerful DeepSeekMath-7B by 7.8%/5.3% and Qwen2.5-Math-7B by 15.1%/6.3% on MATH/GSM8K benchmarks, achieving new SOTA performance among all 7B models. Code: https://github.com/YangLing0818/SuperCorrect-llm

Large language models (LLMs) can perform reasoning computations both internally within their latent space and externally by generating explicit token sequences like chains of thought. Significant progress in enhancing reasoning abilities has been made by scaling test-time compute. However, understanding and quantifying model-internal reasoning abilities - the inferential "leaps" models make between individual token predictions - remains crucial. This study introduces a benchmark (n = 4,000 items) designed to quantify model-internal reasoning in different domains. We achieve this by having LLMs indicate the correct solution to reasoning problems not through descriptive text, but by selecting a specific language of their initial response token that is different from English, the benchmark language. This not only requires models to reason beyond their context window, but also to overrise their default tendency to respond in the same language as the prompt, thereby posing an additional cognitive strain. We evaluate a set of 18 LLMs, showing significant performance variations, with GPT-4.5 achieving the highest accuracy (74.7%), outperforming models like Grok-2 (67.2%), and Llama 3.1 405B (65.6%). Control experiments and difficulty scaling analyses suggest that while LLMs engage in internal reasoning, we cannot rule out heuristic exploitations under certain conditions, marking an area for future investigation. Our experiments demonstrate that LLMs can "think" via latent-space computations, revealing model-internal inference strategies that need further understanding, especially regarding safety-related concerns such as covert planning, goal-seeking, or deception emerging without explicit token traces.

With the recent advance in large pre-trained language models, researchers have achieved record performances in NLP tasks that mostly focus on language pattern matching. The community is experiencing the shift of the challenge from how to model language to the imitation of complex reasoning abilities like human beings. In this work, we investigate the application domain of finance that involves real-world, complex numerical reasoning. We propose a new large-scale dataset, ConvFinQA, aiming to study the chain of numerical reasoning in conversational question answering. Our dataset poses great challenge in modeling long-range, complex numerical reasoning paths in real-world conversations. We conduct comprehensive experiments and analyses with both the neural symbolic methods and the prompting-based methods, to provide insights into the reasoning mechanisms of these two divisions. We believe our new dataset should serve as a valuable resource to push forward the exploration of real-world, complex reasoning tasks as the next research focus. Our dataset and code is publicly available at https://github.com/czyssrs/ConvFinQA.

Reasoning is central to human intelligence, enabling structured problem-solving across diverse tasks. Recent advances in large language models (LLMs) have greatly enhanced their reasoning abilities in arithmetic, commonsense, and symbolic domains. However, effectively extending these capabilities into multimodal contexts-where models must integrate both visual and textual inputs-continues to be a significant challenge. Multimodal reasoning introduces complexities, such as handling conflicting information across modalities, which require models to adopt advanced interpretative strategies. Addressing these challenges involves not only sophisticated algorithms but also robust methodologies for evaluating reasoning accuracy and coherence. This paper offers a concise yet insightful overview of reasoning techniques in both textual and multimodal LLMs. Through a thorough and up-to-date comparison, we clearly formulate core reasoning challenges and opportunities, highlighting practical methods for post-training optimization and test-time inference. Our work provides valuable insights and guidance, bridging theoretical frameworks and practical implementations, and sets clear directions for future research.

Large Language Models (LLMs) have shown strong performance in solving mathematical problems, with code-based solutions proving particularly effective. However, the best practice to leverage coding instruction data to enhance mathematical reasoning remains underexplored. This study investigates three key questions: (1) How do different coding styles of mathematical code-based rationales impact LLMs' learning performance? (2) Can general-domain coding instructions improve performance? (3) How does integrating textual rationales with code-based ones during training enhance mathematical reasoning abilities? Our findings reveal that code-based rationales with concise comments, descriptive naming, and hardcoded solutions are beneficial, while improvements from general-domain coding instructions and textual rationales are relatively minor. Based on these insights, we propose CoinMath, a learning strategy designed to enhance mathematical reasoning by diversifying the coding styles of code-based rationales. CoinMath generates a variety of code-based rationales incorporating concise comments, descriptive naming conventions, and hardcoded solutions. Experimental results demonstrate that CoinMath significantly outperforms its baseline model, MAmmoTH, one of the SOTA math LLMs.

Length generalization, the ability to solve problems of longer sequences than those observed during training, poses a core challenge of Transformer-based large language models (LLM). Although existing studies have predominantly focused on data-driven approaches for arithmetic operations and symbolic manipulation tasks, these approaches tend to be task-specific with limited overall performance. To pursue a more general solution, this paper focuses on a broader case of reasoning problems that are computable, i.e., problems that algorithms can solve, thus can be solved by the Turing Machine. From this perspective, this paper proposes Turing MAchine Imitation Learning (TAIL) to improve the length generalization ability of LLMs. TAIL synthesizes chain-of-thoughts (CoT) data that imitate the execution process of a Turing Machine by computer programs, which linearly expands the reasoning steps into atomic states to alleviate shortcut learning and explicit memory fetch mechanism to reduce the difficulties of dynamic and long-range data access in elementary operations. To validate the reliability and universality of TAIL, we construct a challenging synthetic dataset covering 8 classes of algorithms and 18 tasks. Without bells and whistles, TAIL significantly improves the length generalization ability as well as the performance of Qwen2.5-7B on various tasks using only synthetic data, surpassing previous methods and DeepSeek-R1. The experimental results reveal that the key concepts in the Turing Machine, instead of the thinking styles, are indispensable for TAIL for length generalization, through which the model exhibits read-and-write behaviors consistent with the properties of the Turing Machine in their attention layers. This work provides a promising direction for future research in the learning of LLM reasoning from synthetic data.

Large language models have achieved substantial progress in mathematical reasoning, yet their advancement is limited by the scarcity of high-quality, high-difficulty training data. Existing synthesis methods largely rely on transforming human-written templates, limiting both diversity and scalability. We propose MathSmith, a novel framework for synthesizing challenging mathematical problems to enhance LLM reasoning. Rather than modifying existing problems, MathSmith constructs new ones from scratch by randomly sampling concept-explanation pairs from PlanetMath, ensuring data independence and avoiding contamination. To increase difficulty, we design nine predefined strategies as soft constraints during rationales. We further adopts reinforcement learning to jointly optimize structural validity, reasoning complexity, and answer consistency. The length of the reasoning trace generated under autoregressive prompting is used to reflect cognitive complexity, encouraging the creation of more demanding problems aligned with long-chain-of-thought reasoning. Experiments across five benchmarks, categorized as easy & medium (GSM8K, MATH-500) and hard (AIME2024, AIME2025, OlympiadBench), show that MathSmith consistently outperforms existing baselines under both short and long CoT settings. Additionally, a weakness-focused variant generation module enables targeted improvement on specific concepts. Overall, MathSmith exhibits strong scalability, generalization, and transferability, highlighting the promise of high-difficulty synthetic data in advancing LLM reasoning capabilities.

Large Language Models (LLMs) are smart but forgetful. Recent studies, (e.g., (Bubeck et al., 2023)) on modern LLMs have shown that they are capable of performing amazing tasks typically necessitating human-level intelligence. However, unlike humans, frozen LLMs do not improve over time; they neither acquire new knowledge nor learn from their successes or failures. Some approaches to improving the intelligence of LLMs include fine-tuning models based on problem-solving performance (Zelikman et al., 2022), and building bigger and more sophisticated models (Bubeck et al., 2023). However, these methods have the drawback of requiring substantial data and computational resources to retrain existing models. In this paper, we explore the use of Retrieval Augmented Generation, also known as RAG (Lewis et al., 2021) to improve problem-solving performance. We propose ARM-RAG (Auxiliary Rationale Memory for Retrieval Augmented Generation), a system that learns from its successes without incurring high training costs. We demonstrate that the storage and subsequent retrieval of reasoning chains have a positive influence on performance in grade-school math problems.

Modern large language models (LLMs) employ various forms of logical inference, both implicitly and explicitly, when addressing reasoning tasks. Understanding how to optimally leverage these inference paradigms is critical for advancing LLMs' reasoning capabilities. This paper adopts an exploratory approach by introducing a controlled evaluation environment for analogical reasoning -- a fundamental cognitive task -- that is systematically parameterized across three dimensions: modality (textual, visual, symbolic), difficulty (easy, medium, hard), and task format (multiple-choice or free-text generation). We analyze the comparative dynamics of inductive, abductive, and deductive inference pipelines across these dimensions, and demonstrate that our findings generalize to broader in-context learning tasks. Additionally, we investigate advanced paradigms such as hypothesis selection, verification, and refinement, revealing their potential to scale up logical inference in LLM reasoning. This exploratory study provides a foundation for future research in enhancing LLM reasoning through systematic logical inference strategies.

Large language models (LLMs) are known to struggle with complicated reasoning tasks such as math word problems (MWPs). In this paper, we present how analogy from similarly structured questions can improve LLMs' problem-solving capabilities for MWPs. Specifically, we rely on the retrieval of problems with similar computational graphs to the given question to serve as exemplars in the prompt, providing the correct reasoning path for the generation model to refer to. Empirical results across six math word problem datasets demonstrate the effectiveness of our proposed method, which achieves a significant improvement of up to 6.7 percent on average in absolute value, compared to baseline methods. These results highlight our method's potential in addressing the reasoning challenges in current LLMs.

We introduce KiloGram, a resource for studying abstract visual reasoning in humans and machines. Drawing on the history of tangram puzzles as stimuli in cognitive science, we build a richly annotated dataset that, with >1k distinct stimuli, is orders of magnitude larger and more diverse than prior resources. It is both visually and linguistically richer, moving beyond whole shape descriptions to include segmentation maps and part labels. We use this resource to evaluate the abstract visual reasoning capacities of recent multi-modal models. We observe that pre-trained weights demonstrate limited abstract reasoning, which dramatically improves with fine-tuning. We also observe that explicitly describing parts aids abstract reasoning for both humans and models, especially when jointly encoding the linguistic and visual inputs. KiloGram is available at https://lil.nlp.cornell.edu/kilogram .

Current multimodal benchmarks often conflate reasoning with domain-specific knowledge, making it difficult to isolate and evaluate general reasoning abilities in non-expert settings. To address this, we introduce VisualPuzzles, a benchmark that targets visual reasoning while deliberately minimizing reliance on specialized knowledge. VisualPuzzles consists of diverse questions spanning five categories: algorithmic, analogical, deductive, inductive, and spatial reasoning. One major source of our questions is manually translated logical reasoning questions from the Chinese Civil Service Examination. Experiments show that VisualPuzzles requires significantly less intensive domain-specific knowledge and more complex reasoning compared to benchmarks like MMMU, enabling us to better evaluate genuine multimodal reasoning. Evaluations show that state-of-the-art multimodal large language models consistently lag behind human performance on VisualPuzzles, and that strong performance on knowledge-intensive benchmarks does not necessarily translate to success on reasoning-focused, knowledge-light tasks. Additionally, reasoning enhancements such as scaling up inference compute (with "thinking" modes) yield inconsistent gains across models and task types, and we observe no clear correlation between model size and performance. We also found that models exhibit different reasoning and answering patterns on VisualPuzzles compared to benchmarks with heavier emphasis on knowledge. VisualPuzzles offers a clearer lens through which to evaluate reasoning capabilities beyond factual recall and domain knowledge.

Recent advancements in reasoning-reinforced Large Language Models (LLMs) have shown remarkable capabilities in complex reasoning tasks. However, the mechanism underlying their utilization of different human reasoning skills remains poorly investigated, especially for multilingual commonsense reasoning that involves everyday knowledge across different languages and cultures. To address this gap, we propose a Multilingual and Scalable Benchmark for Skill-based Commonsense Reasoning (mSCoRe). Our benchmark incorporates three key components that are designed to systematically evaluate LLM's reasoning capabilities, including: (1) a novel taxonomy of reasoning skills that enables fine-grained analysis of models' reasoning processes, (2) a robust data synthesis pipeline tailored specifically for commonsense reasoning evaluation, and (3) a complexity scaling framework allowing task difficulty to scale dynamically alongside future improvements in LLM abilities. Extensive experiments on eights state-of-the-art LLMs of varying sizes and training approaches demonstrate that mSCoRe remains significantly challenging for current models, particularly at higher complexity levels. Our results reveal the limitations of such reasoning-reinforced models when confronted with nuanced multilingual general and cultural commonsense. We further provide detailed analysis on the models' reasoning processes, suggesting future directions for improving multilingual commonsense reasoning capabilities.

Human reasoning relies on constructing and manipulating mental models-simplified internal representations of situations that we use to understand and solve problems. Conceptual diagrams (for example, sketches drawn by humans to aid reasoning) externalize these mental models, abstracting irrelevant details to efficiently capture relational and spatial information. In contrast, Large Language Models (LLMs) and Large Multimodal Models (LMMs) predominantly reason through textual representations, limiting their effectiveness in complex multi-step combinatorial and planning tasks. In this paper, we propose a zero-shot fully automatic framework that enables LMMs to reason through multiple chains of self-generated intermediate conceptual diagrams, significantly enhancing their combinatorial planning capabilities. Our approach does not require any human initialization beyond a natural language description of the task. It integrates both textual and diagrammatic reasoning within an optimized graph-of-thought inference framework, enhanced by beam search and depth-wise backtracking. Evaluated on multiple challenging PDDL planning domains, our method substantially improves GPT-4o's performance (for example, from 35.5% to 90.2% in Blocksworld). On more difficult planning domains with solution depths up to 40, our approach outperforms even the o1-preview reasoning model (for example, over 13% improvement in Parking). These results highlight the value of conceptual diagrams as a complementary reasoning medium in LMMs.

Grokking, i.e., test performance keeps improving long after training loss converged, has been recently witnessed in neural network training, making the mechanism of generalization and other emerging capabilities such as reasoning mysterious. While prior studies usually train small models on a few toy or highly-specific tasks for thousands of epochs, we conduct the first study of grokking on checkpoints during one-pass pretraining of a 7B large language model (LLM), i.e., OLMoE. We compute the training loss and evaluate generalization on diverse benchmark tasks, including math reasoning, code generation, and commonsense/domain-specific knowledge retrieval tasks. Our study, for the first time, verifies that grokking still happens in the pretraining of large-scale foundation models, though different data may enter grokking stages asynchronously. We further demystify grokking's "emergence of generalization" by investigating LLM internal dynamics. Specifically, we find that training samples' pathways (i.e., expert choices across layers) evolve from random, instance-specific to more structured and shareable between samples during grokking. Also, the complexity of a sample's pathway reduces despite the converged loss. These indicate a memorization-to-generalization conversion, providing a mechanistic explanation of delayed generalization. In the study, we develop two novel metrics to quantify pathway distance and the complexity of a single pathway. We show their ability to predict the generalization improvement on diverse downstream tasks. They are efficient, simple to compute and solely dependent on training data. Hence, they have practical value for pretraining, enabling us to monitor the generalization performance without finetuning and test. Theoretically, we show that more structured pathways reduce model complexity and improve the generalization bound.

Standard Neural Networks can learn mathematical operations, but they do not extrapolate. Extrapolation means that the model can apply to larger numbers, well beyond those observed during training. Recent architectures tackle arithmetic operations and can extrapolate; however, the equally important problem of quantitative reasoning remains unaddressed. In this work, we propose a novel architectural element, the Neural Status Register (NSR), for quantitative reasoning over numbers. Our NSR relaxes the discrete bit logic of physical status registers to continuous numbers and allows end-to-end learning with gradient descent. Experiments show that the NSR achieves solutions that extrapolate to numbers many orders of magnitude larger than those in the training set. We successfully train the NSR on number comparisons, piecewise discontinuous functions, counting in sequences, recurrently finding minimums, finding shortest paths in graphs, and comparing digits in images.

Large multimodal models extend the impressive capabilities of large language models by integrating multimodal understanding abilities. However, it is not clear how they can emulate the general intelligence and reasoning ability of humans. As recognizing patterns and abstracting concepts are key to general intelligence, we introduce PuzzleVQA, a collection of puzzles based on abstract patterns. With this dataset, we evaluate large multimodal models with abstract patterns based on fundamental concepts, including colors, numbers, sizes, and shapes. Through our experiments on state-of-the-art large multimodal models, we find that they are not able to generalize well to simple abstract patterns. Notably, even GPT-4V cannot solve more than half of the puzzles. To diagnose the reasoning challenges in large multimodal models, we progressively guide the models with our ground truth reasoning explanations for visual perception, inductive reasoning, and deductive reasoning. Our systematic analysis finds that the main bottlenecks of GPT-4V are weaker visual perception and inductive reasoning abilities. Through this work, we hope to shed light on the limitations of large multimodal models and how they can better emulate human cognitive processes in the future (Our data and code will be released publicly at https://github.com/declare-lab/LLM-PuzzleTest).

Large language models are powerful systems that excel at many tasks, ranging from translation to mathematical reasoning. Yet, at the same time, these models often show unhuman-like characteristics. In the present paper, we address this gap and ask whether large language models can be turned into cognitive models. We find that -- after finetuning them on data from psychological experiments -- these models offer accurate representations of human behavior, even outperforming traditional cognitive models in two decision-making domains. In addition, we show that their representations contain the information necessary to model behavior on the level of individual subjects. Finally, we demonstrate that finetuning on multiple tasks enables large language models to predict human behavior in a previously unseen task. Taken together, these results suggest that large, pre-trained models can be adapted to become generalist cognitive models, thereby opening up new research directions that could transform cognitive psychology and the behavioral sciences as a whole.

Large Language Models (LLMs) display striking surface fluency yet systematically fail at tasks requiring symbolic reasoning, arithmetic accuracy, and logical consistency. This paper offers a structural diagnosis of such failures, revealing a persistent gap between comprehension and competence. Through controlled experiments and architectural analysis, we demonstrate that LLMs often articulate correct principles without reliably applying them--a failure rooted not in knowledge access, but in computational execution. We term this phenomenon the computational split-brain syndrome, where instruction and action pathways are geometrically and functionally dissociated. This core limitation recurs across domains, from mathematical operations to relational inferences, and explains why model behavior remains brittle even under idealized prompting. We argue that LLMs function as powerful pattern completion engines, but lack the architectural scaffolding for principled, compositional reasoning. Our findings delineate the boundary of current LLM capabilities and motivate future models with metacognitive control, principle lifting, and structurally grounded execution. This diagnosis also clarifies why mechanistic interpretability findings may reflect training-specific pattern coordination rather than universal computational principles, and why the geometric separation between instruction and execution pathways suggests limitations in neural introspection and mechanistic analysis.

This study presents a novel learning approach designed to enhance both mathematical reasoning and problem-solving abilities of Large Language Models (LLMs). We focus on integrating the Chain-of-Thought (CoT) and the Program-of-Thought (PoT) learning, hypothesizing that prioritizing the learning of mathematical reasoning ability is helpful for the amplification of problem-solving ability. Thus, the initial learning with CoT is essential for solving challenging mathematical problems. To this end, we propose a sequential learning approach, named SAAS (Solving Ability Amplification Strategy), which strategically transitions from CoT learning to PoT learning. Our empirical study, involving an extensive performance comparison using several benchmarks, demonstrates that our SAAS achieves state-of-the-art (SOTA) performance. The results underscore the effectiveness of our sequential learning approach, marking a significant advancement in the field of mathematical reasoning in LLMs.