Daily Papers

Large Language Models (LLMs) are increasingly deployed in multi-agent settings that require coordination without communication, from human-AI interaction to safety-critical scenarios. Humans often overcome the absence of communication through focal points: salient solutions that naturally stand out to all participants. We present the first large-scale evaluation of how, when, and why focal points emerge in LLMs, comparing their behaviour with humans across cooperative and competitive games, including realistic search and rescue scenarios, demonstrating when focal points enable effective coordination. Across more than 20 open- and closed-source models, we find that LLMs exhibit a remarkable ability to coordinate without communication, often matching or outperforming humans. However, the same models consistently fail in tasks requiring numerical common sense or culturally nuanced notions of salience. We additionally evaluate simple learning-free strategies that substantially improve coordination both among LLMs and between humans and LLMs. Our results reveal striking coordination capabilities, as well as social limitations in modern LLMs, and offer new insight into the latent notions of salience encoded within them. Our findings caution against assuming that LLMs share humans' cultural and perceptual substrate when deployed in coordination settings.

Ordinal regression is a fundamental problem within the field of computer vision, with customised well-trained models on specific tasks. While pre-trained vision-language models (VLMs) have exhibited impressive performance on various vision tasks, their potential for ordinal regression has received less exploration. In this study, we first investigate CLIP's potential for ordinal regression, from which we expect the model could generalise to different ordinal regression tasks and scenarios. Unfortunately, vanilla CLIP fails on this task, since current VLMs have a well-documented limitation of encapsulating compositional concepts such as number sense. We propose a simple yet effective method called NumCLIP to improve the quantitative understanding of VLMs. We disassemble the exact image to number-specific text matching problem into coarse classification and fine prediction stages. We discretize and phrase each numerical bin with common language concept to better leverage the available pre-trained alignment in CLIP. To consider the inherent continuous property of ordinal regression, we propose a novel fine-grained cross-modal ranking-based regularisation loss specifically designed to keep both semantic and ordinal alignment in CLIP's feature space. Experimental results on three general ordinal regression tasks demonstrate the effectiveness of NumCLIP, with 10% and 3.83% accuracy improvement on historical image dating and image aesthetics assessment task, respectively. Code is publicly available at https://github.com/xmed-lab/NumCLIP.

Recent works show that pre-trained language models (PTLMs), such as BERT, possess certain commonsense and factual knowledge. They suggest that it is promising to use PTLMs as "neural knowledge bases" via predicting masked words. Surprisingly, we find that this may not work for numerical commonsense knowledge (e.g., a bird usually has two legs). In this paper, we investigate whether and to what extent we can induce numerical commonsense knowledge from PTLMs as well as the robustness of this process. To study this, we introduce a novel probing task with a diagnostic dataset, NumerSense, containing 13.6k masked-word-prediction probes (10.5k for fine-tuning and 3.1k for testing). Our analysis reveals that: (1) BERT and its stronger variant RoBERTa perform poorly on the diagnostic dataset prior to any fine-tuning; (2) fine-tuning with distant supervision brings some improvement; (3) the best supervised model still performs poorly as compared to human performance (54.06% vs 96.3% in accuracy).

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

Common-sense reasoning is a key language model capability because it encapsulates not just specific factual knowledge but rather general language and world understanding. Measuring common-sense reasoning, therefore, is crucial for language models of different sizes and applications. One of the most widely used benchmarks for evaluating such capabilities is HellaSwag; however, in this paper, we show that it has severe construct validity issues. These issues range from basic ungrammaticality and numerous typos to misleading prompts or equally correct options. Furthermore, we show that if models are evaluated only on answer texts, or with "Lorem ipsum dolor..." instead of the question, more than 65% of model predictions remain the same, and this cannot be attributed merely to contamination. Since benchmark scores are an essential part of model selection in both research and commercial applications, these validity issues can have severe consequences. In particular, knowing that taking benchmark scores at face value is ubiquitous, inadequate evaluation leads to ill-informed decisions about models. In this paper, we thoroughly investigate critical validity issues posed by HellaSwag and illustrate them with various evaluations using generative language models of different sizes. We argue that this benchmark does not accurately measure common-sense reasoning and, therefore, should not be used for evaluation in its current state. Based on the results of our study, we propose requirements that should be met by future common-sense reasoning benchmarks. In addition, we release GoldenSwag, a corrected subset of HellaSwag, which, to our belief, facilitates acceptable common-sense reasoning evaluation.

As large language models (LLMs) continue to advance in complex reasoning tasks, they have learned to heavily prioritize explicit conditions provided in the input. However, in everyday commonsense reasoning, this mechanism exposes a critical vulnerability which we term Salience Bias: models become easily hijacked by useless explicit distractors (e.g., numerical values), leading them to ignore the implicit physical or commonsense prerequisites of a task. A critical open question is whether this failure reflects a genuine gap in commonsense knowledge or merely its suppression under misleading task framing. To investigate this, we construct the SaliTrap Benchmark, a high-quality dataset across four trap dimensions. Evaluating 12 state-of-the-art LLMs, we find that all mainstream models suffer significantly from salience bias, with severity scaling with distractor density and detecting the trap often decoupled from actually avoiding it. Crucially, by re-eliciting the same models with the task framing stripped away, we show that this is overwhelmingly a failure of knowledge suppression rather than knowledge absence: a context-free knowledge probe alone recovers over 90\% of sycophantic-compliance failures, revealing that the requisite commonsense is intrinsically present but actively crowded out by salient distractors that lure the model into over-compliant, unnecessary computation. Building on this diagnosis, we further show that lightweight, inference-time prompting alone substantially closes the gap without any retraining. Our findings relocate the bottleneck of commonsense reasoning failures from model competence to elicitation, and we release SaliTrap as a testbed for this blind spot. The codes are available at https://github.com/Wuzheng02/SaliTrap.

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.

Recently developed large language models (LLMs) have been shown to perform remarkably well on a wide range of language understanding tasks. But, can they really "reason" over the natural language? This question has been receiving significant research attention and many reasoning skills such as commonsense, numerical, and qualitative have been studied. However, the crucial skill pertaining to 'logical reasoning' has remained underexplored. Existing work investigating this reasoning ability of LLMs has focused only on a couple of inference rules (such as modus ponens and modus tollens) of propositional and first-order logic. Addressing the above limitation, we comprehensively evaluate the logical reasoning ability of LLMs on 25 different reasoning patterns spanning over propositional, first-order, and non-monotonic logics. To enable systematic evaluation, we introduce LogicBench, a natural language question-answering dataset focusing on the use of a single inference rule. We conduct detailed analysis with a range of LLMs such as GPT-4, ChatGPT, Gemini, Llama-2, and Mistral using chain-of-thought prompting. Experimental results show that existing LLMs do not fare well on LogicBench; especially, they struggle with instances involving complex reasoning and negations. Furthermore, they sometimes overlook contextual information necessary for reasoning to arrive at the correct conclusion. We believe that our work and findings facilitate future research for evaluating and enhancing the logical reasoning ability of LLMs. Data and code are available at https://github.com/Mihir3009/LogicBench.

Recent large language models (LLMs) have demonstrated remarkable performance on a variety of natural language processing (NLP) tasks, leading to intense excitement about their applicability across various domains. Unfortunately, recent work has also shown that LLMs are unable to perform accurate reasoning nor solve planning problems, which may limit their usefulness for robotics-related tasks. In this work, our central question is whether LLMs are able to translate goals specified in natural language to a structured planning language. If so, LLM can act as a natural interface between the planner and human users; the translated goal can be handed to domain-independent AI planners that are very effective at planning. Our empirical results on GPT 3.5 variants show that LLMs are much better suited towards translation rather than planning. We find that LLMs are able to leverage commonsense knowledge and reasoning to furnish missing details from under-specified goals (as is often the case in natural language). However, our experiments also reveal that LLMs can fail to generate goals in tasks that involve numerical or physical (e.g., spatial) reasoning, and that LLMs are sensitive to the prompts used. As such, these models are promising for translation to structured planning languages, but care should be taken in their use.

Reasoning is a fundamental capability of Large Language Models. While prior research predominantly focuses on enhancing narrow skills like math or code generation, improving performance on many other reasoning tasks remains challenging due to sparse and fragmented training data. To address this issue, we propose CodeI/O, a novel approach that systematically condenses diverse reasoning patterns inherently embedded in contextually-grounded codes, through transforming the original code into a code input-output prediction format. By training models to predict inputs/outputs given code and test cases entirely in natural language as Chain-of-Thought (CoT) rationales, we expose them to universal reasoning primitives -- like logic flow planning, state-space searching, decision tree traversal, and modular decomposition -- while decoupling structured reasoning from code-specific syntax and preserving procedural rigor. Experimental results demonstrate CodeI/O leads to consistent improvements across symbolic, scientific, logic, math & numerical, and commonsense reasoning tasks. By matching the existing ground-truth outputs or re-executing the code with predicted inputs, we can verify each prediction and further enhance the CoTs through multi-turn revision, resulting in CodeI/O++ and achieving higher performance. Our data and models are available at https://github.com/hkust-nlp/CodeIO.

Recent advancements in large language models (LLMs) have led to remarkable performance across a wide range of language understanding and mathematical tasks. As a result, increasing attention has been given to assessing the true reasoning capabilities of LLMs, driving research into commonsense, numerical, logical, and qualitative reasoning. However, with the rapid progress of reasoning-focused models such as OpenAI's o1 and DeepSeek's R1, there has been a growing demand for reasoning benchmarks that can keep pace with ongoing model developments. In this paper, we introduce MastermindEval, a simple, scalable, and interpretable deductive reasoning benchmark inspired by the board game Mastermind. Our benchmark supports two evaluation paradigms: (1) agentic evaluation, in which the model autonomously plays the game, and (2) deductive reasoning evaluation, in which the model is given a pre-played game state with only one possible valid code to infer. In our experimental results we (1) find that even easy Mastermind instances are difficult for current models and (2) demonstrate that the benchmark is scalable to possibly more advanced models in the future Furthermore, we investigate possible reasons why models cannot deduce the final solution and find that current models are limited in deducing the concealed code as the number of statement to combine information from is increasing.

Language models struggle with handling numerical data and performing arithmetic operations. We hypothesize that this limitation can be partially attributed to non-intuitive textual numbers representation. When a digit is read or generated by a causal language model it does not know its place value (e.g. thousands vs. hundreds) until the entire number is processed. To address this issue, we propose a simple adjustment to how numbers are represented by including the count of digits before each number. For instance, instead of "42", we suggest using "{2:42}" as the new format. This approach, which we term NumeroLogic, offers an added advantage in number generation by serving as a Chain of Thought (CoT). By requiring the model to consider the number of digits first, it enhances the reasoning process before generating the actual number. We use arithmetic tasks to demonstrate the effectiveness of the NumeroLogic formatting. We further demonstrate NumeroLogic applicability to general natural language modeling, improving language understanding performance in the MMLU benchmark.

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.

Commonsense intelligence in machines is often assessed by static benchmarks that compare a model's output against human-prescribed correct labels. An important, albeit implicit, assumption of these labels is that they accurately capture what any human would think, effectively treating human common sense as homogeneous. However, recent empirical work has shown that humans vary enormously in what they consider commonsensical; thus what appears self-evident to one benchmark designer may not be so to another. Here, we propose a novel method for evaluating common sense in artificial intelligence (AI), specifically in large language models (LLMs), that incorporates empirically observed heterogeneity among humans by measuring the correspondence between a model's judgment and that of a human population. We first find that, when treated as independent survey respondents, most LLMs remain below the human median in their individual commonsense competence. Second, when used as simulators of a hypothetical population, LLMs correlate with real humans only modestly in the extent to which they agree on the same set of statements. In both cases, smaller, open-weight models are surprisingly more competitive than larger, proprietary frontier models. Our evaluation framework, which ties commonsense intelligence to its cultural basis, contributes to the growing call for adapting AI models to human collectivities that possess different, often incompatible, social stocks of knowledge.

Large language models often default to step-by-step computation even when efficient numerical shortcuts are available. This raises a basic question: do they exhibit number sense in a human-like behavioral sense, i.e., the ability to recognize numerical structure, apply shortcuts when appropriate, and avoid them when they are not? We introduce SenseMath, a controlled benchmark for evaluating structure-sensitive numerical reasoning in LLMs. SenseMath contains 4,800 items spanning eight shortcut categories and four digit scales, with matched strong-shortcut, weak-shortcut, and control variants. It supports three evaluation settings of increasing cognitive demand: Shortcut Use (whether models can apply shortcuts on shortcut-amenable problems); Applicability Judgment (whether they can recognize when a shortcut is appropriate or misleading); and Problem Generation (whether they can generate new problem items that correctly admit a given type of shortcut). Our evaluation across five LLMs, ranging from GPT-4o-mini to Llama-3.1-8B, shows a consistent pattern: when explicitly prompted, models readily adopt shortcut strategies and achieve substantial accuracy gains on shortcut-amenable items (up to 15%), yet under standard chain-of-thought prompting they spontaneously employ such strategies in fewer than 40% of cases, even when they demonstrably possess the requisite capability. Moreover, this competence is confined to the Use level; models systematically over-generalise shortcuts to problems where they do not apply, and fail to generate valid shortcut-bearing problems from scratch. Together, these results suggest that current LLMs exhibit procedural shortcut fluency without the structural understanding of when and why shortcuts work that underlies human number sense.

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.

In order for AI to be safely deployed in real-world scenarios such as hospitals, schools, and the workplace, it must be able to robustly reason about the physical world. Fundamental to this reasoning is physical common sense: understanding the physical properties and affordances of available objects, how they can be manipulated, and how they interact with other objects. Physical commonsense reasoning is fundamentally a multi-sensory task, since physical properties are manifested through multiple modalities - two of them being vision and acoustics. Our paper takes a step towards real-world physical commonsense reasoning by contributing PACS: the first audiovisual benchmark annotated for physical commonsense attributes. PACS contains 13,400 question-answer pairs, involving 1,377 unique physical commonsense questions and 1,526 videos. Our dataset provides new opportunities to advance the research field of physical reasoning by bringing audio as a core component of this multimodal problem. Using PACS, we evaluate multiple state-of-the-art models on our new challenging task. While some models show promising results (70% accuracy), they all fall short of human performance (95% accuracy). We conclude the paper by demonstrating the importance of multimodal reasoning and providing possible avenues for future research.

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.

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.

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.

Long-form numerical reasoning in financial analysis aims to generate a reasoning program to calculate the correct answer for a given question. Previous work followed a retriever-generator framework, where the retriever selects key facts from a long-form document, and the generator generates a reasoning program based on retrieved facts. However, they treated all facts equally without considering the different contributions of facts with and without numbers. Meanwhile, the program consistency were ignored under supervised training, resulting in lower training accuracy and diversity. To solve these problems, we proposed APOLLO to improve the long-form numerical reasoning framework. For the retriever, we adopt a number-aware negative sampling strategy to enable the retriever to be more discriminative on key numerical facts. For the generator, we design consistency-based reinforcement learning and target program augmentation strategy based on the consistency of program execution results. Experimental results on the FinQA and ConvFinQA leaderboard verify the effectiveness of our proposed method, achieving the new state-of-the-art.

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.

Commonsense reasoning is a critical AI capability, but it is difficult to construct challenging datasets that test common sense. Recent neural question answering systems, based on large pre-trained models of language, have already achieved near-human-level performance on commonsense knowledge benchmarks. These systems do not possess human-level common sense, but are able to exploit limitations of the datasets to achieve human-level scores. We introduce the CODAH dataset, an adversarially-constructed evaluation dataset for testing common sense. CODAH forms a challenging extension to the recently-proposed SWAG dataset, which tests commonsense knowledge using sentence-completion questions that describe situations observed in video. To produce a more difficult dataset, we introduce a novel procedure for question acquisition in which workers author questions designed to target weaknesses of state-of-the-art neural question answering systems. Workers are rewarded for submissions that models fail to answer correctly both before and after fine-tuning (in cross-validation). We create 2.8k questions via this procedure and evaluate the performance of multiple state-of-the-art question answering systems on our dataset. We observe a significant gap between human performance, which is 95.3%, and the performance of the best baseline accuracy of 67.5% by the BERT-Large model.

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.

We introduce IndiaFinBench, to our knowledge the first publicly available evaluation benchmark for assessing large language model (LLM) performance on Indian financial regulatory text. Existing financial NLP benchmarks draw exclusively from Western financial corpora (SEC filings, US earnings reports, and English-language financial news), leaving a significant gap in coverage of non-Western regulatory frameworks. IndiaFinBench addresses this gap with 406 expert-annotated question-answer pairs drawn from 192 documents sourced from the Securities and Exchange Board of India (SEBI) and the Reserve Bank of India (RBI), spanning four task types: regulatory interpretation (174 items), numerical reasoning (92 items), contradiction detection (62 items), and temporal reasoning (78 items). Annotation quality is validated through a model-based secondary pass (kappa=0.918 on contradiction detection) and a 60-item human inter-annotator agreement evaluation (kappa=0.611; 76.7% overall agreement). We evaluate twelve models under zero-shot conditions, with accuracy ranging from 70.4% (Gemma 4 E4B) to 89.7% (Gemini 2.5 Flash). All models substantially outperform a non-specialist human baseline of 60.0%. Numerical reasoning is the most discriminative task, with a 35.9 percentage-point spread across models. Bootstrap significance testing (10,000 resamples) reveals three statistically distinct performance tiers. The dataset, evaluation code, and all model outputs are available at https://github.com/rajveerpall/IndiaFinBench

Large Language Models are increasingly optimized for deep reasoning, prioritizing the correct execution of complex tasks over general conversation. We investigate whether this focus on calculation creates a "tunnel vision" that ignores safety in critical situations. We introduce MortalMATH, a benchmark of 150 scenarios where users request algebra help while describing increasingly life-threatening emergencies (e.g., stroke symptoms, freefall). We find a sharp behavioral split: generalist models (like Llama-3.1) successfully refuse the math to address the danger. In contrast, specialized reasoning models (like Qwen-3-32b and GPT-5-nano) often ignore the emergency entirely, maintaining over 95 percent task completion rates while the user describes dying. Furthermore, the computational time required for reasoning introduces dangerous delays: up to 15 seconds before any potential help is offered. These results suggest that training models to relentlessly pursue correct answers may inadvertently unlearn the survival instincts required for safe deployment.

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.

Artificial intelligence (AI) has made significant strides in recent years, yet it continues to struggle with a fundamental aspect of cognition present in all animals: common sense. Current AI systems, including those designed for complex tasks like autonomous driving, problem-solving challenges such as the Abstraction and Reasoning Corpus (ARC), and conversational benchmarks like the Turing Test, often lack the ability to adapt to new situations without extensive prior knowledge. This manuscript argues that integrating common sense into AI systems is essential for achieving true autonomy and unlocking the full societal and commercial value of AI. We propose a shift in the order of knowledge acquisition emphasizing the importance of developing AI systems that start from minimal prior knowledge and are capable of contextual learning, adaptive reasoning, and embodiment -- even within abstract domains. Additionally, we highlight the need to rethink the AI software stack to address this foundational challenge. Without common sense, AI systems may never reach true autonomy, instead exhibiting asymptotic performance that approaches theoretical ideals like AIXI but remains unattainable in practice due to infinite resource and computation requirements. While scaling AI models and passing benchmarks like the Turing Test have brought significant advancements in applications that do not require autonomy, these approaches alone are insufficient to achieve autonomous AI with common sense. By redefining existing benchmarks and challenges to enforce constraints that require genuine common sense, and by broadening our understanding of embodiment to include both physical and abstract domains, we can encourage the development of AI systems better equipped to handle the complexities of real-world and abstract environments.

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.

Large Language Models (LLMs) are increasingly being explored for clinical question answering and decision support, yet safe deployment critically requires reliable handling of patient measurements in heterogeneous clinical notes. Existing evaluations of LLMs for clinical numerical reasoning provide limited operation-level coverage, restricted primarily to arithmetic computation, and rarely assess the robustness of numerical understanding across clinical note formats. We introduce ClinicNumRobBench, a benchmark of 1,624 context-question instances with ground-truth answers that evaluates four main types of clinical numeracy: value retrieval, arithmetic computation, relational comparison, and aggregation. To stress-test robustness, ClinicNumRobBench presents longitudinal MIMIC-IV vital-sign records in three semantically equivalent representations, including a real-world note-style variant derived from the Open Patients dataset, and instantiates queries using 42 question templates. Experiments on 14 LLMs show that value retrieval is generally strong, with most models exceeding 85% accuracy, while relational comparison and aggregation remain challenging, with some models scoring below 15%. Fine-tuning on medical data can reduce numeracy relative to base models by over 30%, and performance drops under note-style variation indicate LLM sensitivity to format. ClinicNumRobBench offers a rigorous testbed for clinically reliable numerical reasoning. Code and data URL are available on https://github.com/MinhVuong2000/ClinicNumRobBench.

Financial reports offer critical insights into a company's operations, yet their extensive length typically spanning 30 40 pages poses challenges for swift decision making in dynamic markets. To address this, we leveraged finetuned Large Language Models (LLMs) to distill key indicators and operational metrics from these reports basis questions from the user. We devised a method to locate critical data, and leverage the FinQA dataset to fine-tune both Llama-2 7B and T5 models for customized question answering. We achieved results comparable to baseline on the final numerical answer, a competitive accuracy in numerical reasoning and calculation.

How much is 56 times 37? Language models often make mistakes in these types of difficult calculations. This is usually explained by their inability to perform complex reasoning. Since language models rely on large training sets and great memorization capability, naturally they are not equipped to run complex calculations. However, one can argue that humans also cannot perform this calculation immediately and require a considerable amount of time to construct the solution. In order to enhance the generalization capability of language models, and as a parallel to human behavior, we propose to use special 'thinking tokens' which allow the model to perform much more calculations whenever a complex problem is encountered.

Although state-of-the-art LLMs can solve math problems, we find that they make errors on numerical comparisons with mixed notation: "Which is larger, 5.7 times 10^2 or 580?" This raises a fundamental question: Do LLMs even know how big these numbers are? We probe the hidden states of several smaller open-source LLMs. A single linear projection of an appropriate hidden layer encodes the log-magnitudes of both kinds of numerals, allowing us to recover the numbers with relative error of about 2.3% (on restricted synthetic text) or 19.06% (on scientific papers). Furthermore, the hidden state after reading a pair of numerals encodes their ranking, with a linear classifier achieving over 90% accuracy. Yet surprisingly, when explicitly asked to rank the same pairs of numerals, these LLMs achieve only 50-70% accuracy, with worse performance for models whose probes are less effective. Finally, we show that incorporating the classifier probe's log-loss as an auxiliary objective during finetuning brings an additional 3.22% improvement in verbalized accuracy over base models, demonstrating that improving models' internal magnitude representations can enhance their numerical reasoning capabilities.

While question answering over knowledge bases (KBQA) has shown progress in addressing factoid questions, KBQA with numerical reasoning remains relatively unexplored. In this paper, we focus on the complex numerical reasoning in KBQA and propose a new task, NR-KBQA, which necessitates the ability to perform both multi-hop reasoning and numerical reasoning. We design a logic form in Python format called PyQL to represent the reasoning process of numerical reasoning questions. To facilitate the development of NR-KBQA, we present a large dataset called MarkQA, which is automatically constructed from a small set of seeds. Each question in MarkQA is equipped with its corresponding SPARQL query, alongside the step-by-step reasoning process in the QDMR format and PyQL program. Experimental results of some state-of-the-art QA methods on the MarkQA show that complex numerical reasoning in KBQA faces great challenges.

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.

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.

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/.

Recent LLMs have demonstrated remarkable performance in solving exam-like math word problems. However, the degree to which these numerical reasoning skills are effective in real-world scenarios, particularly in expert domains, is still largely unexplored. This paper introduces DocMath-Eval, a comprehensive benchmark specifically designed to evaluate the numerical reasoning and problem-solving capabilities of LLMs in the context of understanding and analyzing financial documents containing both text and tables. We evaluate a wide spectrum of 19 LLMs, including those specialized in coding and finance. We also incorporate different prompting strategies (i.e., Chain-of-Thoughts and Program-of-Thoughts) to comprehensively assess the capabilities and limitations of existing LLMs in DocMath-Eval. We found that, although the current best-performing system (i.e., GPT-4), can perform well on simple problems such as calculating the rate of increase in a financial metric within a short document context, it significantly lags behind human experts in more complex problems grounded in longer contexts. We believe DocMath-Eval can be used as a valuable benchmark to evaluate LLMs' capabilities to solve challenging numerical reasoning problems in expert domains. We will release the benchmark and code at https://github.com/yale-nlp/DocMath-Eval.

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.

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.

Deep learning models perform poorly on tasks that require commonsense reasoning, which often necessitates some form of world-knowledge or reasoning over information not immediately present in the input. We collect human explanations for commonsense reasoning in the form of natural language sequences and highlighted annotations in a new dataset called Common Sense Explanations (CoS-E). We use CoS-E to train language models to automatically generate explanations that can be used during training and inference in a novel Commonsense Auto-Generated Explanation (CAGE) framework. CAGE improves the state-of-the-art by 10% on the challenging CommonsenseQA task. We further study commonsense reasoning in DNNs using both human and auto-generated explanations including transfer to out-of-domain tasks. Empirical results indicate that we can effectively leverage language models for commonsense reasoning.

We introduce a large language model (LLM) based approach to answer complex questions requiring multi-hop numerical reasoning over financial reports. While LLMs have exhibited remarkable performance on various natural language and reasoning tasks, complex reasoning problems often rely on few-shot prompts that require carefully crafted examples. In contrast, our approach uses novel zero-shot prompts that guide the LLM to encode the required reasoning into a Python program or a domain specific language. The generated program is then executed by a program interpreter, thus mitigating the limitations of LLM in performing accurate arithmetic calculations. We evaluate the proposed approach on three financial datasets using some of the recently developed generative pretrained transformer (GPT) models and perform comparisons with various zero-shot baselines. The experimental results demonstrate that our approach significantly improves the accuracy for all the LLMs over their respective baselines. We provide a detailed analysis of the results, generating insights to support our findings. The success of our approach demonstrates the enormous potential to extract complex domain specific numerical reasoning by designing zero-shot prompts to effectively exploit the knowledge embedded in LLMs.

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.

Quantitative reasoning is a higher-order reasoning skill that any intelligent natural language understanding system can reasonably be expected to handle. We present EQUATE (Evaluating Quantitative Understanding Aptitude in Textual Entailment), a new framework for quantitative reasoning in textual entailment. We benchmark the performance of 9 published NLI models on EQUATE, and find that on average, state-of-the-art methods do not achieve an absolute improvement over a majority-class baseline, suggesting that they do not implicitly learn to reason with quantities. We establish a new baseline Q-REAS that manipulates quantities symbolically. In comparison to the best performing NLI model, it achieves success on numerical reasoning tests (+24.2%), but has limited verbal reasoning capabilities (-8.1%). We hope our evaluation framework will support the development of models of quantitative reasoning in language understanding.

Building future wireless systems that support services like digital twins (DTs) is challenging to achieve through advances to conventional technologies like meta-surfaces. While artificial intelligence (AI)-native networks promise to overcome some limitations of wireless technologies, developments still rely on AI tools like neural networks. Such tools struggle to cope with the non-trivial challenges of the network environment and the growing demands of emerging use cases. In this paper, we revisit the concept of AI-native wireless systems, equipping them with the common sense necessary to transform them into artificial general intelligence (AGI)-native systems. These systems acquire common sense by exploiting different cognitive abilities such as perception, analogy, and reasoning, that enable them to generalize and deal with unforeseen scenarios. Towards developing the components of such a system, we start by showing how the perception module can be built through abstracting real-world elements into generalizable representations. These representations are then used to create a world model, founded on principles of causality and hyper-dimensional (HD) computing, that aligns with intuitive physics and enables analogical reasoning, that define common sense. Then, we explain how methods such as integrated information theory play a role in the proposed intent-driven and objective-driven planning methods that maneuver the AGI-native network to take actions. Next, we discuss how an AGI-native network can enable use cases related to human and autonomous agents: a) analogical reasoning for next-generation DTs, b) synchronized and resilient experiences for cognitive avatars, and c) brain-level metaverse experiences like holographic teleportation. Finally, we conclude with a set of recommendations to build AGI-native systems. Ultimately, we envision this paper as a roadmap for the beyond 6G era.

We continue the investigation into the power of smaller Transformer-based language models as initiated by TinyStories -- a 10 million parameter model that can produce coherent English -- and the follow-up work on phi-1, a 1.3 billion parameter model with Python coding performance close to the state-of-the-art. The latter work proposed to use existing Large Language Models (LLMs) to generate ``textbook quality" data as a way to enhance the learning process compared to traditional web data. We follow the ``Textbooks Are All You Need" approach, focusing this time on common sense reasoning in natural language, and create a new 1.3 billion parameter model named phi-1.5, with performance on natural language tasks comparable to models 5x larger, and surpassing most non-frontier LLMs on more complex reasoning tasks such as grade-school mathematics and basic coding. More generally, phi-1.5 exhibits many of the traits of much larger LLMs, both good -- such as the ability to ``think step by step" or perform some rudimentary in-context learning -- and bad, including hallucinations and the potential for toxic and biased generations -- encouragingly though, we are seeing improvement on that front thanks to the absence of web data. We open-source phi-1.5 to promote further research on these urgent topics.

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.

As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.

There are now several comprehensive web applications, stand-alone computer programs and computer algebra functions that, given a floating point number such as 6.518670730718491, can return concise nonfloat constants such as 3 arctan 2 + ln 9 + 1, that closely approximate the float. Examples include AskConstants, Inverse Symbolic Calculator, the Maple identify function, MESearch, OEIS, RIES, and WolframAlpha. Usefully often such a result is the exact limit as the float is computed with increasing precision. Therefore these program results are candidates for proving an exact result that you could not otherwise compute or conjecture without the program. Moreover, candidates that are not the exact limit can be provable bounds, or convey qualitative insight, or suggest series that they truncate, or provide sufficiently close efficient approximations for subsequent computation. This article describes some of these programs, how they work, and how best to use each of them. Almost everyone who uses or should use mathematical software can benefit from acquaintance with several such programs, because these programs differ in the sets of constants that they can return.

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.

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 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

Given questions regarding some prototypical situation such as Name something that people usually do before they leave the house for work? a human can easily answer them via acquired experiences. There can be multiple right answers for such questions, with some more common for a situation than others. This paper introduces a new question answering dataset for training and evaluating common sense reasoning capabilities of artificial intelligence systems in such prototypical situations. The training set is gathered from an existing set of questions played in a long-running international game show FAMILY- FEUD. The hidden evaluation set is created by gathering answers for each question from 100 crowd-workers. We also propose a generative evaluation task where a model has to output a ranked list of answers, ideally covering all prototypical answers for a question. After presenting multiple competitive baseline models, we find that human performance still exceeds model scores on all evaluation metrics with a meaningful gap, supporting the challenging nature of the task.

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.

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.

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.

Chain-of-Thought (CoT) reasoning has enabled Large Language Model (LLM) to achieve remarkable performance in various NLP tasks, including arithmetic problem-solving. However, this success does not generalize to small language model (sLM) like T5, due to their limited capacity and absence of emergent abilities associated with larger models. Recent works to enhance sLM through knowledge distillation have yielded some improvements but still face significant limitations, particularly high ambiguity from the variability in natural language expressions and substantial computational costs. In this paper, we investigate why sLM perform poorly on arithmetic reasoning tasks and hypothesize that natural language format variability introduces high ambiguity for these smaller models. Based on this hypothesis, we conduct experiments with equation-only format, which is a reasoning format that unifies arithmetic reasoning previously expressed in natural language formats into mathematical equations. Experiment results demonstrate that equation-only format effectively boosts the arithmetic reasoning abilities of sLM, especially in very small models like T5-Tiny.

Ensuring that Large Language Models (LLMs) generate summaries faithful to a given source document is essential for real-world applications. While prior research has explored LLM faithfulness, existing benchmarks suffer from annotation ambiguity, primarily due to the ill-defined boundary of permissible external knowledge in generated outputs. For instance, common sense is often incorporated into responses and labeled as "faithful", yet the acceptable extent of such knowledge remains unspecified, leading to inconsistent annotations. To address this issue, we propose a novel faithfulness annotation framework, which introduces an intermediate category, Out-Dependent, to classify cases where external knowledge is required for verification. Using this framework, we construct VeriGray (Verification with the Gray Zone) -- a new unfaithfulness detection benchmark in summarization. Statistics reveal that even SOTA LLMs, such as GPT-5, exhibit hallucinations (sim 6% of sentences) in summarization tasks. Moreover, a substantial proportion (sim 8% on average of models) of generated sentences fall into the Out-Dependent category, underscoring the importance of resolving annotation ambiguity in unfaithfulness detection benchmarks. Experiments demonstrate that our benchmark poses significant challenges to multiple baseline methods, indicating considerable room for future improvement.

When answering a question, people often draw upon their rich world knowledge in addition to the particular context. Recent work has focused primarily on answering questions given some relevant document or context, and required very little general background. To investigate question answering with prior knowledge, we present CommonsenseQA: a challenging new dataset for commonsense question answering. To capture common sense beyond associations, we extract from ConceptNet (Speer et al., 2017) multiple target concepts that have the same semantic relation to a single source concept. Crowd-workers are asked to author multiple-choice questions that mention the source concept and discriminate in turn between each of the target concepts. This encourages workers to create questions with complex semantics that often require prior knowledge. We create 12,247 questions through this procedure and demonstrate the difficulty of our task with a large number of strong baselines. Our best baseline is based on BERT-large (Devlin et al., 2018) and obtains 56% accuracy, well below human performance, which is 89%.

Recently, large-scale pre-trained language models have demonstrated impressive performance on several commonsense-reasoning benchmark datasets. However, building machines with commonsense to compose realistically plausible sentences remains challenging. In this paper, we present a constrained text generation task, CommonGen associated with a benchmark dataset, to explicitly test machines for the ability of generative commonsense reasoning. Given a set of common concepts (e.g., {dog, frisbee, catch, throw}); the task is to generate a coherent sentence describing an everyday scenario using these concepts (e.g., "a man throws a frisbee and his dog catches it"). The CommonGen task is challenging because it inherently requires 1) relational reasoning with background commonsense knowledge, and 2) compositional generalization ability to work on unseen concept combinations. Our dataset, constructed through a combination of crowdsourced and existing caption corpora, consists of 79k commonsense descriptions over 35k unique concept-sets. Experiments show that there is a large gap between state-of-the-art text generation models (e.g., T5) and human performance. Furthermore, we demonstrate that the learned generative commonsense reasoning capability can be transferred to improve downstream tasks such as CommonsenseQA by generating additional context.

Advancement in Large Language Models (LLMs) reasoning capabilities enables them to solve scientific problems with enhanced efficacy. Thereby, a high-quality benchmark for comprehensive and appropriate assessment holds significance, while existing ones either confront the risk of data contamination or lack involved disciplines. To be specific, due to the data source overlap of LLMs training and static benchmark, the keys or number pattern of answers inadvertently memorized (i.e. data contamination), leading to systematic overestimation of their reasoning capabilities, especially numerical reasoning. We propose SciDA, a multidisciplinary benchmark that consists exclusively of over 1k Olympic-level numerical computation problems, allowing randomized numerical initializations for each inference round to avoid reliance on fixed numerical patterns. We conduct a series of experiments with both closed-source and open-source top-performing LLMs, and it is observed that the performance of LLMs drop significantly under random numerical initialization. Thus, we provide truthful and unbiased assessments of the numerical reasoning capabilities of LLMs. The data is available at https://huggingface.co/datasets/m-a-p/SciDA

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.

Majority voting over multiple LLM attempts improves mathematical reasoning, but correlated errors limit the effective sample size. A natural fix is to assign different reasoning strategies to different voters. The approach, Diverse Prompt Mixer, is tested on the AIMO 3 competition: 3 models, 23+ experiments, 50 IMO-level problems, one H100 80 GB, 5-hour limit. Every prompt-level intervention fails. High-temperature sampling already decorrelates errors; weaker strategies reduce accuracy more than they reduce correlation. Across an 8-point capability gap at equal N=8 and every optimization tested, model capability dominates. The gap between the best majority-vote score (42/50) and pass@20 (~45.5) is selection loss, not prompt loss. A verifier-based selector could close it. Prompt engineering cannot.

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.

Three of the most popular methods for training language models to reason look like three different tricks. They are not. All three adjust a single number: standard deviation, reflecting how much a prompt's sampled answers disagree. When such a model is trained, it answers each problem many times, and an automatic checker marks every answer right or wrong. The standard deviation of those marks measures the disagreement: largest when the answers split evenly between right and wrong, and zero when they all agree. Group Relative Policy Optimization (GRPO) divides by this number, GRPO Done Right (Dr. GRPO) drops the division, and Decoupled Clip and Dynamic Sampling Policy Optimization (DAPO) discards the groups where it is zero. Each is presented as its own fix, yet this paper proves they are three settings of one dial. That dial is not cosmetic: for right-or-wrong rewards, the disagreement is exactly the size of the training update, the group-standard-deviation identity. A split group teaches the most, while a unanimous group teaches nothing and falls silent. The same result says which problems deserve the most weight and how many tries each one needs. This paper confirms the intuition on a large real difficulty dataset (Big-Math) and in a controlled training run. What looks like a harmless normalization step is the dial that decides where learning happens and how strongly.

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.

Neural networks trained on datasets such as ImageNet have led to major advances in visual object classification. One obstacle that prevents networks from reasoning more deeply about complex scenes and situations, and from integrating visual knowledge with natural language, like humans do, is their lack of common sense knowledge about the physical world. Videos, unlike still images, contain a wealth of detailed information about the physical world. However, most labelled video datasets represent high-level concepts rather than detailed physical aspects about actions and scenes. In this work, we describe our ongoing collection of the "something-something" database of video prediction tasks whose solutions require a common sense understanding of the depicted situation. The database currently contains more than 100,000 videos across 174 classes, which are defined as caption-templates. We also describe the challenges in crowd-sourcing this data at scale.

Embedding words in high-dimensional vector spaces has proven valuable in many natural language applications. In this work, we investigate whether similarly-trained embeddings of integers can capture concepts that are useful for mathematical applications. We probe the integer embeddings for mathematical knowledge, apply them to a set of numerical reasoning tasks, and show that by learning the representations from mathematical sequence data, we can substantially improve over number embeddings learned from English text corpora.

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.

This paper presents the first-ever Sinhala physical common sense reasoning dataset created as part of Global PIQA. It contains 110 human-created and verified data samples, where each sample consists of a prompt, the corresponding correct answer, and a wrong answer. Most of the questions refer to the Sri Lankan context, where Sinhala is an official language.

The ability (and inability) of large language models (LLMs) to perform arithmetic tasks has been the subject of much theoretical and practical debate. We show that LLMs are frequently able to correctly and confidently predict the first digit of n-digit by m-digit multiplication tasks without using chain of thought reasoning, despite these tasks require compounding operations to solve. Simultaneously, LLMs in practice often fail to correctly or confidently predict the last digit of an n-digit by m-digit multiplication, a task equivalent to 1-digit by 1-digit multiplication which can be easily learned or memorized. We show that the latter task can be solved more robustly when the LLM is conditioned on all of the correct higher-order digits, which on average increases the confidence of the correct last digit on 5-digit by 5-digit multiplication tasks using Llama 2-13B by over 230% (0.13 to 0.43) and Mistral-7B by 150% (0.22 to 0.55).

We show that a GPT-3 model can learn to express uncertainty about its own answers in natural language -- without use of model logits. When given a question, the model generates both an answer and a level of confidence (e.g. "90% confidence" or "high confidence"). These levels map to probabilities that are well calibrated. The model also remains moderately calibrated under distribution shift, and is sensitive to uncertainty in its own answers, rather than imitating human examples. To our knowledge, this is the first time a model has been shown to express calibrated uncertainty about its own answers in natural language. For testing calibration, we introduce the CalibratedMath suite of tasks. We compare the calibration of uncertainty expressed in words ("verbalized probability") to uncertainty extracted from model logits. Both kinds of uncertainty are capable of generalizing calibration under distribution shift. We also provide evidence that GPT-3's ability to generalize calibration depends on pre-trained latent representations that correlate with epistemic uncertainty over its answers.

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

Reading comprehension has recently seen rapid progress, with systems matching humans on the most popular datasets for the task. However, a large body of work has highlighted the brittleness of these systems, showing that there is much work left to be done. We introduce a new English reading comprehension benchmark, DROP, which requires Discrete Reasoning Over the content of Paragraphs. In this crowdsourced, adversarially-created, 96k-question benchmark, a system must resolve references in a question, perhaps to multiple input positions, and perform discrete operations over them (such as addition, counting, or sorting). These operations require a much more comprehensive understanding of the content of paragraphs than what was necessary for prior datasets. We apply state-of-the-art methods from both the reading comprehension and semantic parsing literature on this dataset and show that the best systems only achieve 32.7% F1 on our generalized accuracy metric, while expert human performance is 96.0%. We additionally present a new model that combines reading comprehension methods with simple numerical reasoning to achieve 47.0% F1.

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.

In this report, we introduce Qwen2.5, a comprehensive series of large language models (LLMs) designed to meet diverse needs. Compared to previous iterations, Qwen 2.5 has been significantly improved during both the pre-training and post-training stages. In terms of pre-training, we have scaled the high-quality pre-training datasets from the previous 7 trillion tokens to 18 trillion tokens. This provides a strong foundation for common sense, expert knowledge, and reasoning capabilities. In terms of post-training, we implement intricate supervised finetuning with over 1 million samples, as well as multistage reinforcement learning. Post-training techniques enhance human preference, and notably improve long text generation, structural data analysis, and instruction following. To handle diverse and varied use cases effectively, we present Qwen2.5 LLM series in rich sizes. Open-weight offerings include base and instruction-tuned models, with quantized versions available. In addition, for hosted solutions, the proprietary models currently include two mixture-of-experts (MoE) variants: Qwen2.5-Turbo and Qwen2.5-Plus, both available from Alibaba Cloud Model Studio. Qwen2.5 has demonstrated top-tier performance on a wide range of benchmarks evaluating language understanding, reasoning, mathematics, coding, human preference alignment, etc. Specifically, the open-weight flagship Qwen2.5-72B-Instruct outperforms a number of open and proprietary models and demonstrates competitive performance to the state-of-the-art open-weight model, Llama-3-405B-Instruct, which is around 5 times larger. Qwen2.5-Turbo and Qwen2.5-Plus offer superior cost-effectiveness while performing competitively against GPT-4o-mini and GPT-4o respectively. Additionally, as the foundation, Qwen2.5 models have been instrumental in training specialized models such as Qwen2.5-Math, Qwen2.5-Coder, QwQ, and multimodal models.

Large Language Models (LLMs) have limited performance when solving arithmetic reasoning tasks and often provide incorrect answers. Unlike natural language understanding, math problems typically have a single correct answer, making the task of generating accurate solutions more challenging for LLMs. To the best of our knowledge, we are not aware of any LLMs that indicate their level of confidence in their responses which fuels a trust deficit in these models impeding their adoption. To address this deficiency, we propose `MathPrompter', a technique that improves performance of LLMs on arithmetic problems along with increased reliance in the predictions. MathPrompter uses the Zero-shot chain-of-thought prompting technique to generate multiple Algebraic expressions or Python functions to solve the same math problem in different ways and thereby raise the confidence level in the output results. This is in contrast to other prompt based CoT methods, where there is no check on the validity of the intermediate steps followed. Our technique improves over state-of-the-art on the MultiArith dataset (78.7%rightarrow92.5%) evaluated using 175B parameter GPT-based LLM.

Math reasoning is a highly active area of Large Language Model (LLM) research because it is a hallmark of artificial intelligence. However, few works have explored how math reasoning is encoded within LLM parameters and if it is a skill that can be isolated within a model. Doing so could allow targeted intervention to improve math performance without altering non-math behavior and foster understanding of how models encode math reasoning. We introduce Math Neurosurgery (MathNeuro), a method for isolating math-specific parameters in LLMs using only forward passes. MathNeuro builds on existing work by using weights and activations to calculate parameter importance, but isolates math-specific parameters by removing those important for general language tasks. Pruning parameters MathNeuro identifies deletes a LLM's math reasoning ability without destroying its general language ability. Scaling these parameters by a small constant improves a pretrained or instruction-tuned LLM's performance by 4-17% on GSM8K while leaving non-math behavior unaltered. MathNeuro is also data efficient: most of its effectiveness holds when identifying math-specific parameters using a single sample. MathNeuro highlights the potential for future work to intervene on math-specific parameters.

We present Semi-Structured Explanations for COPA (COPA-SSE), a new crowdsourced dataset of 9,747 semi-structured, English common sense explanations for Choice of Plausible Alternatives (COPA) questions. The explanations are formatted as a set of triple-like common sense statements with ConceptNet relations but freely written concepts. This semi-structured format strikes a balance between the high quality but low coverage of structured data and the lower quality but high coverage of free-form crowdsourcing. Each explanation also includes a set of human-given quality ratings. With their familiar format, the explanations are geared towards commonsense reasoners operating on knowledge graphs and serve as a starting point for ongoing work on improving such systems. The dataset is available at https://github.com/a-brassard/copa-sse.

Reduced numerical precision is a common technique to reduce computational cost in many Deep Neural Networks (DNNs). While it has been observed that DNNs are resilient to small errors and noise, no general result exists that is capable of predicting a given DNN system architecture's sensitivity to reduced precision. In this project, we emulate arbitrary bit-width using a specified floating-point representation with a truncation method, which is applied to the neural network after each batch. We explore the impact of several model parameters on the network's training accuracy and show results on the MNIST dataset. We then present a preliminary theoretical investigation of the error scaling in both forward and backward propagations. We end with a discussion of the implications of these results as well as the potential for generalization to other network architectures.

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.

Though Large Language Models (LLMs) have shown remarkable abilities in mathematics reasoning, they are still struggling with performing numeric operations accurately, such as addition and multiplication. Numbers can be tokenized into tokens in various ways by different LLMs and affect the numeric operations performance. Currently, there are two representatives: 1) Tokenize into 1-digit, and 2) Tokenize into 1sim 3 digit. The difference is roughly equivalent to using different numeral systems (namely base 10 or base 10^{3}). In light of this, we study the scaling behavior of different numeral systems in the context of transformer-based large language models. We empirically show that a base 10 system is consistently more data-efficient than a base 10^{2} or 10^{3} system across training data scale, model sizes under from-scratch training settings, while different number systems have very similar fine-tuning performances. We attribute this to higher token frequencies of a base 10 system. Additionally, we reveal extrapolation behavior patterns on addition and multiplication. We identify that base 100 and base 1000 systems struggle on token-level discernment and token-level operations. We also sheds light on the mechanism learnt by the models.

EasyMath is a compact benchmark for practical math reasoning in small language models. It covers thirteen categories, from basic arithmetic and order of operations to word problems, algebraic expressions, edge cases, and omits specialist topics. We tested 23 models (14M to 4B parameters) using exact, numerical, and symbolic checks on free-form answers in a zero-shot setting. Accuracy rises with size and training, chain-of-thought adds modest gains, and consistency improves at scale.

Recently, the Natural Language Inference (NLI) task has been studied for semi-structured tables that do not have a strict format. Although neural approaches have achieved high performance in various types of NLI, including NLI between semi-structured tables and texts, they still have difficulty in performing a numerical type of inference, such as counting. To handle a numerical type of inference, we propose a logical inference system for reasoning between semi-structured tables and texts. We use logical representations as meaning representations for tables and texts and use model checking to handle a numerical type of inference between texts and tables. To evaluate the extent to which our system can perform inference with numerical comparatives, we make an evaluation protocol that focuses on numerical understanding between semi-structured tables and texts in English. We show that our system can more robustly perform inference between tables and texts that requires numerical understanding compared with current neural approaches.

Contemporary large language models (LLMs) have demonstrated remarkable reasoning capabilities, particularly in specialized domains like mathematics and physics. However, their ability to generalize these reasoning skills to more general and broader contexts--often termed general reasoning--remains under-explored. Unlike domain-specific reasoning, general reasoning relies less on expert knowledge but still presents formidable reasoning challenges, such as complex constraints, nested logical branches, and semantic interference. To address this gap, we introduce General365, a benchmark specifically designed to assess general reasoning in LLMs. By restricting background knowledge to a K-12 level, General365 explicitly decouples reasoning from specialized expertise. The benchmark comprises 365 seed problems and 1,095 variant problems across eight categories, ensuring both high difficulty and diversity. Evaluations across 26 leading LLMs reveal that even the top-performing model achieves only 62.8% accuracy, in stark contrast to the near-perfect performances of LLMs in math and physics benchmarks. These results suggest that the reasoning abilities of current LLMs are heavily domain-dependent, leaving significant room for improvement in broader applications. We envision General365 as a catalyst for advancing LLM reasoning beyond domain-specific tasks toward robust, general-purpose real-world scenarios. Code, Dataset, and Leaderboard: https://general365.github.io

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.

Large Language Models (LLMs) are often described as being instances of foundation models - that is, models that transfer strongly across various tasks and conditions in few-show or zero-shot manner, while exhibiting scaling laws that predict function improvement when increasing the pre-training scale. These claims of excelling in different functions and tasks rely on measurements taken across various sets of standardized benchmarks showing high scores for such models. We demonstrate here a dramatic breakdown of function and reasoning capabilities of state-of-the-art models trained at the largest available scales which claim strong function, using a simple, short, conventional common sense problem formulated in concise natural language, easily solvable by humans. The breakdown is dramatic, as models also express strong overconfidence in their wrong solutions, while providing often non-sensical "reasoning"-like explanations akin to confabulations to justify and backup the validity of their clearly failed responses, making them sound plausible. Various standard interventions in an attempt to get the right solution, like various type of enhanced prompting, or urging the models to reconsider the wrong solutions again by multi step re-evaluation, fail. We take these initial observations to the scientific and technological community to stimulate urgent re-assessment of the claimed capabilities of current generation of LLMs, Such re-assessment also requires common action to create standardized benchmarks that would allow proper detection of such basic reasoning deficits that obviously manage to remain undiscovered by current state-of-the-art evaluation procedures and benchmarks. Code for reproducing experiments in the paper and raw experiments data can be found at https://github.com/LAION-AI/AIW

The ability to precisely derive mathematical objects is a core requirement for downstream STEM applications, including mathematics, physics, and chemistry, where reasoning must culminate in formally structured expressions. Yet, current LM evaluations of mathematical and scientific reasoning rely heavily on simplified answer formats such as numerical values or multiple choice options due to the convenience of automated assessment. In this paper we provide three contributions for improving reasoning over mathematical objects: (i) we build and release training data and benchmarks for deriving mathematical objects, the Principia suite; (ii) we provide training recipes with strong LLM-judges and verifiers, where we show that on-policy judge training boosts performance; (iii) we show how on-policy training can also be used to scale test-time compute via aggregation. We find that strong LMs such as Qwen3-235B and o3 struggle on Principia, while our training recipes can bring significant improvements over different LLM backbones, while simultaneously improving results on existing numerical and MCQA tasks, demonstrating cross-format generalization of reasoning abilities.

Commonsense reasoning is fundamentally based on multimodal knowledge. However, existing large language models (LLMs) are primarily trained using textual data only, limiting their ability to incorporate essential visual information. In contrast, Visual Language Models, which excel at visually-oriented tasks, often fail at non-visual tasks such as basic commonsense reasoning. This divergence highlights a critical challenge - the integration of robust visual understanding with foundational text-based language reasoning. To this end, we introduce a method aimed at enhancing LLMs' visual commonsense. Specifically, our method generates multiple images based on the input text prompt and integrates these into the model's decision-making process by mixing their prediction probabilities. To facilitate multimodal grounded language modeling, we employ a late-fusion layer that combines the projected visual features with the output of a pre-trained LLM conditioned on text only. This late-fusion layer enables predictions based on comprehensive image-text knowledge as well as text only when this is required. We evaluate our approach using several visual commonsense reasoning tasks together with traditional NLP tasks, including common sense reasoning and reading comprehension. Our experimental results demonstrate significant superiority over existing baselines. When applied to recent state-of-the-art LLMs (e.g., Llama3), we observe improvements not only in visual common sense but also in traditional NLP benchmarks. Code and models are available under https://github.com/guyyariv/vLMIG.

Transformers, central to the successes in modern Natural Language Processing, often falter on arithmetic tasks despite their vast capabilities --which paradoxically include remarkable coding abilities. We observe that a crucial challenge is their naive reliance on positional information to solve arithmetic problems with a small number of digits, leading to poor performance on larger numbers. Herein, we delve deeper into the role of positional encoding, and propose several ways to fix the issue, either by modifying the positional encoding directly, or by modifying the representation of the arithmetic task to leverage standard positional encoding differently. We investigate the value of these modifications for three tasks: (i) classical multiplication, (ii) length extrapolation in addition, and (iii) addition in natural language context. For (i) we train a small model on a small dataset (100M parameters and 300k samples) with remarkable aptitude in (direct, no scratchpad) 15 digits multiplication and essentially perfect up to 12 digits, while usual training in this context would give a model failing at 4 digits multiplication. In the experiments on addition, we use a mere 120k samples to demonstrate: for (ii) extrapolation from 10 digits to testing on 12 digits numbers while usual training would have no extrapolation, and for (iii) almost perfect accuracy up to 5 digits while usual training would be correct only up to 3 digits (which is essentially memorization with a training set of 120k samples).

The common practice for training commonsense models has gone from-human-to-corpus-to-machine: humans author commonsense knowledge graphs in order to train commonsense models. In this work, we investigate an alternative, from-machine-to-corpus-to-machine: general language models author these commonsense knowledge graphs to train commonsense models. Our study leads to a new framework, Symbolic Knowledge Distillation. As with prior art in Knowledge Distillation (Hinton et al., 2015), our approach uses larger models to teach smaller models. A key difference is that we distill knowledge symbolically-as text-in addition to the neural model. We also distill only one aspect-the commonsense of a general language model teacher, allowing the student to be a different type, a commonsense model. Altogether, we show that careful prompt engineering and a separately trained critic model allow us to selectively distill high-quality causal commonsense from GPT-3, a general language model. Empirical results demonstrate that, for the first time, a human-authored commonsense knowledge graph is surpassed by our automatically distilled variant in all three criteria: quantity, quality, and diversity. In addition, it results in a neural commonsense model that surpasses the teacher model's commonsense capabilities despite its 100x smaller size. We apply this to the ATOMIC resource, and share our new symbolic knowledge graph and commonsense models.

We present ORCA (Omni Research on Calculation in AI) Benchmark - a novel benchmark that evaluates large language models (LLMs) on multi-domain, real-life quantitative reasoning using verified outputs from Omni's calculator engine. In 500 natural-language tasks across domains such as finance, physics, health, and statistics, the five state-of-the-art systems (ChatGPT-5, Gemini~2.5~Flash, Claude~Sonnet~4.5, Grok~4, and DeepSeek~V3.2) achieved only 45--63,% accuracy, with errors mainly related to rounding (35,%) and calculation mistakes (33,%). Results in specific domains indicate strengths in mathematics and engineering, but weaknesses in physics and natural sciences. Correlation analysis (r approx 0.40--0.65) shows that the models often fail together but differ in the types of errors they make, highlighting their partial complementarity rather than redundancy. Unlike standard math datasets, ORCA evaluates step-by-step reasoning, numerical precision, and domain generalization across real problems from finance, physics, health, and statistics.

Physical commonsense reasoning represents a fundamental capability of human intelligence, enabling individuals to understand their environment, predict future events, and navigate physical spaces. Recent years have witnessed growing interest in reasoning tasks within Natural Language Processing (NLP). However, no prior research has examined the performance of Large Language Models (LLMs) on non-question-answering (non-QA) physical commonsense reasoning tasks in low-resource languages such as Basque. Taking the Italian GITA as a starting point, this paper addresses this gap by presenting BasPhyCo, the first non-QA physical commonsense reasoning dataset for Basque, available in both standard and dialectal variants. We evaluate model performance across three hierarchical levels of commonsense understanding: (1) distinguishing between plausible and implausible narratives (accuracy), (2) identifying the conflicting element that renders a narrative implausible (consistency), and (3) determining the specific physical state that creates the implausibility (verifiability). These tasks were assessed using multiple multilingual LLMs as well as models pretrained specifically for Italian and Basque. Results indicate that, in terms of verifiability, LLMs exhibit limited physical commonsense capabilities in low-resource languages such as Basque, especially when processing dialectal variants.

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

Understanding and creating mathematics using natural mathematical language - the mixture of symbolic and natural language used by humans - is a challenging and important problem for driving progress in machine learning. As a step in this direction, we develop NaturalProofs, a multi-domain corpus of mathematical statements and their proofs, written in natural mathematical language. NaturalProofs unifies broad coverage, deep coverage, and low-resource mathematical sources, allowing for evaluating both in-distribution and zero-shot generalization. Using NaturalProofs, we benchmark strong neural methods on mathematical reference retrieval and generation tasks which test a system's ability to determine key results that appear in a proof. Large-scale sequence models show promise compared to classical information retrieval methods, yet their performance and out-of-domain generalization leave substantial room for improvement. NaturalProofs opens many avenues for research on challenging mathematical tasks.

Previous studies have typically assumed that large language models are unable to accurately perform arithmetic operations, particularly multiplication of >8 digits, and operations involving decimals and fractions, without the use of calculator tools. This paper aims to challenge this misconception. With sufficient training data, a 2 billion-parameter language model can accurately perform multi-digit arithmetic operations with almost 100% accuracy without data leakage, significantly surpassing GPT-4 (whose multi-digit multiplication accuracy is only 4.3%). We also demonstrate that our MathGLM, fine-tuned from GLM-10B on a dataset with additional multi-step arithmetic operations and math problems described in text, achieves similar performance to GPT-4 on a 5,000-samples Chinese math problem test set.

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.

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.

Many intellectual endeavors require mathematical problem solving, but this skill remains beyond the capabilities of computers. To measure this ability in machine learning models, we introduce MATH, a new dataset of 12,500 challenging competition mathematics problems. Each problem in MATH has a full step-by-step solution which can be used to teach models to generate answer derivations and explanations. To facilitate future research and increase accuracy on MATH, we also contribute a large auxiliary pretraining dataset which helps teach models the fundamentals of mathematics. Even though we are able to increase accuracy on MATH, our results show that accuracy remains relatively low, even with enormous Transformer models. Moreover, we find that simply increasing budgets and model parameter counts will be impractical for achieving strong mathematical reasoning if scaling trends continue. While scaling Transformers is automatically solving most other text-based tasks, scaling is not currently solving MATH. To have more traction on mathematical problem solving we will likely need new algorithmic advancements from the broader research community.

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.

Large Language Models (LLMs) solely trained on next-token prediction learn to solve a wide range of problems involving mathematical reasoning. But how does this ability evolve during training? We show the first analysis of how mathematical reasoning abilities of several open-weight LLMs develop during pre-training and post-training. To this end, we construct MathCAMPS, a synthetic dataset of novel mathematical reasoning problems grounded in 44 fine-grained skills taken from the Common Core curriculum from K to 8th grades. In one experiment, we show that mathematical skills are learned during pre-training in an order that measurably correlates with the human-designed curriculum, even though training data are randomly ordered. We also show a detailed analysis of which mathematical abilities benefit from instruction tuning, a widely used post-training method and, in contrast, which skills suffer. Our work paves the way for an empirical understanding of LLM training dynamics in relation to reasoning.

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.

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

Neural algorithmic reasoning (NAR) is an emerging field that seeks to design neural networks that mimic classical algorithmic computations. Today, graph neural networks (GNNs) are widely used in neural algorithmic reasoners due to their message passing framework and permutation equivariance. In this extended abstract, we challenge this design choice, and replace the equivariant aggregation function with a recurrent neural network. While seemingly counter-intuitive, this approach has appropriate grounding when nodes have a natural ordering -- and this is the case frequently in established reasoning benchmarks like CLRS-30. Indeed, our recurrent NAR (RNAR) model performs very strongly on such tasks, while handling many others gracefully. A notable achievement of RNAR is its decisive state-of-the-art result on the Heapsort and Quickselect tasks, both deemed as a significant challenge for contemporary neural algorithmic reasoners -- especially the latter, where RNAR achieves a mean micro-F1 score of 87%.

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.

Mathematical reasoning models are widely deployed in education, automated tutoring, and decision support systems despite exhibiting fundamental computational instabilities. We demonstrate that state-of-the-art models (Qwen2.5-Math-7B) achieve 61% accuracy through a mixture of reliable and unreliable reasoning pathways: 18.4% of correct predictions employ stable, faithful reasoning while 81.6% emerge through computationally inconsistent pathways. Additionally, 8.8% of all predictions are silent failures -- confident yet incorrect outputs. Through comprehensive analysis using novel faithfulness metrics, we reveal: (1) reasoning quality shows weak negative correlation with correctness (r=-0.21, p=0.002), reflecting a binary classification threshold artifact rather than a monotonic inverse relationship; (2) scaling from 1.5B to 7B parameters (4.7x increase) provides zero accuracy benefit on our evaluated subset (6% of GSM8K), requiring validation on the complete benchmark; and (3) latent reasoning employs diverse computational strategies, with ~20% sharing CoT-like patterns. These findings highlight that benchmark accuracy can mask computational unreliability, demanding evaluation reforms measuring stability beyond single-sample metrics.

While language models have exceptional capabilities at text generation, they lack a natural inductive bias for emitting numbers and thus struggle in tasks involving reasoning over quantities, especially arithmetics. This has particular relevance in scientific datasets where combinations of text and numerical data are abundant. One fundamental limitation is the nature of the CE loss, which assumes a nominal (categorical) scale and thus cannot convey proximity between generated number tokens. As a remedy, we here present two versions of a number token loss. The first is based on an L_p loss between the ground truth token value and the weighted sum of the predicted class probabilities. The second loss minimizes the Wasserstein-1 distance between the distribution of the predicted output probabilities and the ground truth distribution. These regression-like losses can easily be added to any language model and extend the CE objective during training. We compare the proposed schemes on a mathematics dataset against existing tokenization, encoding, and decoding schemes for improving number representation in language models. Our results reveal a significant improvement in numerical accuracy when equipping a standard T5 model with the proposed loss schemes.

Understanding sources of a model's uncertainty regarding its predictions is crucial for effective human-AI collaboration. Prior work proposes using numerical uncertainty or hedges ("I'm not sure, but ..."), which do not explain uncertainty that arises from conflicting evidence, leaving users unable to resolve disagreements or rely on the output. We introduce CLUE (Conflict-and-Agreement-aware Language-model Uncertainty Explanations), the first framework to generate natural language explanations of model uncertainty by (i) identifying relationships between spans of text that expose claim-evidence or inter-evidence conflicts and agreements that drive the model's predictive uncertainty in an unsupervised way, and (ii) generating explanations via prompting and attention steering that verbalize these critical interactions. Across three language models and two fact-checking datasets, we show that CLUE produces explanations that are more faithful to the model's uncertainty and more consistent with fact-checking decisions than prompting for uncertainty explanations without span-interaction guidance. Human evaluators judge our explanations to be more helpful, more informative, less redundant, and more logically consistent with the input than this baseline. CLUE requires no fine-tuning or architectural changes, making it plug-and-play for any white-box language model. By explicitly linking uncertainty to evidence conflicts, it offers practical support for fact-checking and generalises readily to other tasks that require reasoning over complex information.

Small-scale models offer various computational advantages, and yet to which extent size is critical for problem-solving abilities remains an open question. Specifically for solving grade school math, the smallest model size so far required to break the 80\% barrier on the GSM8K benchmark remains to be 34B. Our work studies how high-quality datasets may be the key for small language models to acquire mathematical reasoning. We introduce TinyGSM, a synthetic dataset of 12.3M grade school math problems paired with Python solutions, generated fully by GPT-3.5. After finetuning on TinyGSM, we find that a duo of a 1.3B generation model and a 1.3B verifier model can achieve 81.5\% accuracy, outperforming existing models that are orders of magnitude larger. This also rivals the performance of the GPT-3.5 ``teacher'' model (77.4\%), from which our model's training data is generated. Our approach is simple and has two key components: 1) the high-quality dataset TinyGSM, 2) the use of a verifier, which selects the final outputs from multiple candidate generations.

Reasoning over commonsense knowledge bases (CSKB) whose elements are in the form of free-text is an important yet hard task in NLP. While CSKB completion only fills the missing links within the domain of the CSKB, CSKB population is alternatively proposed with the goal of reasoning unseen assertions from external resources. In this task, CSKBs are grounded to a large-scale eventuality (activity, state, and event) graph to discriminate whether novel triples from the eventuality graph are plausible or not. However, existing evaluations on the population task are either not accurate (automatic evaluation with randomly sampled negative examples) or of small scale (human annotation). In this paper, we benchmark the CSKB population task with a new large-scale dataset by first aligning four popular CSKBs, and then presenting a high-quality human-annotated evaluation set to probe neural models' commonsense reasoning ability. We also propose a novel inductive commonsense reasoning model that reasons over graphs. Experimental results show that generalizing commonsense reasoning on unseen assertions is inherently a hard task. Models achieving high accuracy during training perform poorly on the evaluation set, with a large gap between human performance. We will make the data publicly available for future contributions. Codes and data are available at https://github.com/HKUST-KnowComp/CSKB-Population.

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.

This survey paper proposes a clearer view of natural language reasoning in the field of Natural Language Processing (NLP), both conceptually and practically. Conceptually, we provide a distinct definition for natural language reasoning in NLP, based on both philosophy and NLP scenarios, discuss what types of tasks require reasoning, and introduce a taxonomy of reasoning. Practically, we conduct a comprehensive literature review on natural language reasoning in NLP, mainly covering classical logical reasoning, natural language inference, multi-hop question answering, and commonsense reasoning. The paper also identifies and views backward reasoning, a powerful paradigm for multi-step reasoning, and introduces defeasible reasoning as one of the most important future directions in natural language reasoning research. We focus on single-modality unstructured natural language text, excluding neuro-symbolic techniques and mathematical reasoning.

Hierarchical reasoning model (HRM) achieves extraordinary performance on various reasoning tasks, significantly outperforming large language model-based reasoners. To understand the strengths and potential failure modes of HRM, we conduct a mechanistic study on its reasoning patterns and find three surprising facts: (a) Failure of extremely simple puzzles, e.g., HRM can fail on a puzzle with only one unknown cell. We attribute this failure to the violation of the fixed point property, a fundamental assumption of HRM. (b) "Grokking" dynamics in reasoning steps, i.e., the answer is not improved uniformly, but instead there is a critical reasoning step that suddenly makes the answer correct; (c) Existence of multiple fixed points. HRM "guesses" the first fixed point, which could be incorrect, and gets trapped there for a while or forever. All facts imply that HRM appears to be "guessing" instead of "reasoning". Leveraging this "guessing" picture, we propose three strategies to scale HRM's guesses: data augmentation (scaling the quality of guesses), input perturbation (scaling the number of guesses by leveraging inference randomness), and model bootstrapping (scaling the number of guesses by leveraging training randomness). On the practical side, by combining all methods, we develop Augmented HRM, boosting accuracy on Sudoku-Extreme from 54.5% to 96.9%. On the scientific side, our analysis provides new insights into how reasoning models "reason".

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.

Generative commonsense reasoning (GCR) in natural language is to reason about the commonsense while generating coherent text. Recent years have seen a surge of interest in improving the generation quality of commonsense reasoning tasks. Nevertheless, these approaches have seldom investigated diversity in the GCR tasks, which aims to generate alternative explanations for a real-world situation or predict all possible outcomes. Diversifying GCR is challenging as it expects to generate multiple outputs that are not only semantically different but also grounded in commonsense knowledge. In this paper, we propose MoKGE, a novel method that diversifies the generative reasoning by a mixture of expert (MoE) strategy on commonsense knowledge graphs (KG). A set of knowledge experts seek diverse reasoning on KG to encourage various generation outputs. Empirical experiments demonstrated that MoKGE can significantly improve the diversity while achieving on par performance on accuracy on two GCR benchmarks, based on both automatic and human evaluations.

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 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.

To apply eyeshadow without a brush, should I use a cotton swab or a toothpick? Questions requiring this kind of physical commonsense pose a challenge to today's natural language understanding systems. While recent pretrained models (such as BERT) have made progress on question answering over more abstract domains - such as news articles and encyclopedia entries, where text is plentiful - in more physical domains, text is inherently limited due to reporting bias. Can AI systems learn to reliably answer physical common-sense questions without experiencing the physical world? In this paper, we introduce the task of physical commonsense reasoning and a corresponding benchmark dataset Physical Interaction: Question Answering or PIQA. Though humans find the dataset easy (95% accuracy), large pretrained models struggle (77%). We provide analysis about the dimensions of knowledge that existing models lack, which offers significant opportunities for future research.

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 word problem-solving has long been recognized as a complex task for small language models (SLMs). A recent study hypothesized that the smallest model size, needed to achieve over 80% accuracy on the GSM8K benchmark, is 34 billion parameters. To reach this level of performance with smaller models, researcher often train SLMs to generate Python code or use tools to help avoid calculation errors. Additionally, they employ ensembling, where outputs of up to 100 model runs are combined to arrive at a more accurate result. Result selection is done using consensus, majority vote or a separate a verifier model used in conjunction with the SLM. Ensembling provides a substantial boost in accuracy but at a significant cost increase with multiple calls to the model (e.g., Phi-GSM uses top-48 to boost the performance from 68.2 to 81.5). In this work, we present Orca-Math, a 7-billion-parameter SLM based on the Mistral-7B, which achieves 86.81% on GSM8k without the need for multiple model calls or the use of verifiers, code execution or any other external tools. Our approach has the following key elements: (1) A high quality synthetic dataset of 200K math problems created using a multi-agent setup where agents collaborate to create the data, (2) An iterative learning techniques that enables the SLM to practice solving problems, receive feedback on its solutions and learn from preference pairs incorporating the SLM solutions and the feedback. When trained with Supervised Fine-Tuning alone, Orca-Math achieves 81.50% on GSM8k pass@1 metric. With iterative preference learning, Orca-Math achieves 86.81% pass@1. Orca-Math surpasses the performance of significantly larger models such as LLAMA-2-70B, WizardMath-70B, Gemini-Pro, ChatGPT-3.5. It also significantly outperforms other smaller models while using much smaller data (hundreds of thousands vs. millions of problems).