We present a fundamental discovery that challenges our understanding of how complex reasoning emerges in large language models. While conventional wisdom suggests that sophisticated reasoning tasks demand extensive training data (often > 100, 000 examples), we demonstrate a striking phenomenon: complex mathematical reasoning abilities can be effectively elicited with surprisingly few examples. This finding challenges not only the assumption of massive data requirements but also the common belief that supervised fine-tuning primarily leads to memorization rather than generalization. Through comprehensive experiments, our proposed model LIMO demonstrates unprecedented performance and efficiency in mathematical reasoning. With merely 817 curated training samples, LIMO achieves 57.1% accuracy on the highly challenging AIME benchmark and 94.8% on MATH, improving the performance of previous strong SFT-based models from 6.5% to 57.1% on AIME and from 59.2% to 94.8% on MATH, while only using 1% of the training data required by previous approaches. Most remarkably, LIMO demonstrates exceptional out-of-distribution generalization, achieving 40.5% absolute improvement across 10 diverse benchmarks, outperforming models trained on 100x more data, directly challenging the prevailing notion that SFT inherently leads to memorization rather than generalization. Synthesizing these pioneering results, we propose the Less-Is-More Reasoning Hypothesis (LIMO Hypothesis): In foundation models where domain knowledge has been comprehensively encoded during pre-training, sophisticated reasoning capabilities can emerge through minimal but precisely orchestrated demonstrations of cognitive processes. This hypothesis posits that the elicitation threshold for complex reasoning is not inherently bounded by the complexity of the target reasoning task, but fundamentally determined by two key factors: (1) the completeness of the model's encoded knowledge foundation during pre-training, and (2) the effectiveness of post-training examples, which serve as "cognitive templates" that show the model how to effectively utilize its existing knowledge base to solve complex reasoning tasks. To facilitate reproducibility and future research in data-efficient reasoning, we release LIMO as a comprehensive open-source suite at https://github.com/GAIR-NLP/LIMO.

Graph neural networks (GNNs) have been widely used in graph-related contexts. It is known that the separation power of GNNs is equivalent to that of the Weisfeiler-Lehman (WL) test; hence, GNNs are imperfect at identifying all non-isomorphic graphs, which severely limits their expressive power. This work investigates $k$-hop subgraph GNNs that aggregate information from neighbors with distances up to $k$ and incorporate the subgraph structure. We prove that under appropriate assumptions, the $k$-hop subgraph GNNs can approximate any permutation-invariant/equivariant continuous function over graphs without cycles of length greater than $2k+1$ within any error tolerance. We also provide an extension to $k$-hop GNNs without incorporating the subgraph structure. Our numerical experiments on established benchmarks and novel architectures validate our theory on the relationship between the information aggregation distance and the cycle size.

The rapid evolution of Large Language Models (LLMs) has transformed natural language processing but raises critical concerns about biases inherent in their deployment and use across diverse linguistic and sociocultural contexts. This paper presents a framework named ASCenD BDS (Adaptable, Stochastic and Context-aware framework for Detection of Bias, Discrimination and Stereotyping). The framework presents approach to detecting bias, discrimination, stereotyping across various categories such as gender, caste, age, disability, socioeconomic status, linguistic variations, etc., using an approach which is Adaptive, Stochastic and Context-Aware. The existing frameworks rely heavily on usage of datasets to generate scenarios for detection of Bias, Discrimination and Stereotyping. Examples include datasets such as Civil Comments, Wino Gender, WinoBias, BOLD, CrowS Pairs and BBQ. However, such an approach provides point solutions. As a result, these datasets provide a finite number of scenarios for assessment. The current framework overcomes this limitation by having features which enable Adaptability, Stochasticity, Context Awareness. Context awareness can be customized for any nation or culture or sub-culture (for example an organization's unique culture). In this paper, context awareness in the Indian context has been established. Content has been leveraged from Indian Census 2011 to have a commonality of categorization. A framework has been developed using Category, Sub-Category, STEM, X-Factor, Synonym to enable the features for Adaptability, Stochasticity and Context awareness. The framework has been described in detail in Section 3. Overall 800 plus STEMs, 10 Categories, 31 unique SubCategories were developed by a team of consultants at Saint Fox Consultancy Private Ltd. The concept has been tested out in SFCLabs as part of product development.

Neural networks, as currently designed, fall short of achieving true logical intelligence. Modern AI models rely on standard neural computation-inner-product-based transformations and nonlinear activations-to approximate patterns from data. While effective for inductive learning, this architecture lacks the structural guarantees necessary for deductive inference and logical consistency. As a result, deep networks struggle with rule-based reasoning, structured generalization, and interpretability without extensive post-hoc modifications. This position paper argues that standard neural layers must be fundamentally rethought to integrate logical reasoning. We advocate for Logical Neural Units (LNUs)-modular components that embed differentiable approximations of logical operations (e.g., AND, OR, NOT) directly within neural architectures. We critique existing neurosymbolic approaches, highlight the limitations of standard neural computation for logical inference, and present LNUs as a necessary paradigm shift in AI. Finally, we outline a roadmap for implementation, discussing theoretical foundations, architectural integration, and key challenges for future research.

Large language models (LLMs) have demonstrated remarkable Large language models (LLMs) have demonstrated performance across a wide range of reasoning remarkable reasoning capabilities across tasks, including mathematical problems (Cobbe et al., 2021; diverse domains. Recent studies have shown that Hendrycks et al., 2021a), programming (Chen et al., 2021; increasing test-time computation enhances LLMs' Zhuo et al., 2024) and logical reasoning (Han et al., 2024; reasoning capabilities. This typically involves extensive Liu et al., 2020). One of the key techniques enabling these sampling at inference time guided by an strong reasoning capabilities is Chain-of-Thought (CoT) external LLM verifier, resulting in a two-player prompting (Wei et al., 2022), which allows LLMs to address system. Despite external guidance, the effectiveness complex tasks by generating a series of intermediate of this system demonstrates the potential of reasoning steps. As a result, many early efforts focus on finetuning a single LLM to tackle complex tasks. Thus, we LLMs using large-scale, high-quality CoT reasoning pose a new research problem: Can we internalize chains, either through human annotation (Hendrycks et al., the searching capabilities to fundamentally 2021a; Yue et al., 2024) or by distilling synthetic data from enhance the reasoning abilities of a single LLM? more advanced models (Yu et al., 2024; Toshniwal et al., This work explores an orthogonal direction focusing 2024a; Ding et al., 2024). However, human annotation is on post-training LLMs for autoregressive extremely labor intensive, and distillation often limits the searching (i.e., an extended reasoning process model's reasoning capabilities to certain level.

Ensuring the safety and harmlessness of Large Language Models (LLMs) has become equally critical as their performance in applications. However, existing safety alignment methods typically suffer from safety-performance trade-offs and the susceptibility to jailbreak attacks, primarily due to their reliance on direct refusals for malicious queries. In this paper, we propose STAIR, a novel framework that integrates SafeTy Alignment with Itrospective Reasoning. We enable LLMs to identify safety risks through step-by-step analysis by self-improving chain-of-thought (CoT) reasoning with safety awareness. STAIR first equips the model with a structured reasoning capability and then advances safety alignment via iterative preference optimization on step-level reasoning data generated using our newly proposed Safety-Informed Monte Carlo Tree Search (SI-MCTS). We further train a process reward model on this data to guide test-time searches for improved responses. Extensive experiments show that STAIR effectively mitigates harmful outputs while better preserving helpfulness, compared to instinctive alignment strategies. With test-time scaling, STAIR achieves a safety performance comparable to Claude-3.5 against popular jailbreak attacks. Relevant resources in this work are available at https://github.com/thu-ml/STAIR.

Research on LLM technologies is rapidly emerging, with most of them employing a 'fast thinking' approach to inference. Most LLMs generate the final result based solely on a single query and LLM's reasoning capabilities. However, with the advent of OpenAI-o1, 'slow thinking' techniques have garnered increasing attention because its process is closer to the human thought process. Inspired by the human ability to constantly associate and replenish knowledge during thinking, we developed the novel Chain-of-Associated-Thoughts (CoAT) framework, which introduces an innovative synergy between the Monte Carlo Tree Search (MCTS) algorithm and a dynamic mechanism for integrating new key information, termed 'associative memory'. By combining the structured exploration capabilities of MCTS with the adaptive learning capacity of associative memory, CoAT significantly expands the LLM search space, enabling our framework to explore diverse reasoning pathways and dynamically update its knowledge base in real-time. This allows the framework to not only revisit and refine earlier inferences but also adaptively incorporate evolving information, ensuring that the final output is both accurate and comprehensive. To validate the effectiveness of our framework, we conducted extensive experiments across a range of generative and reasoning tasks. These experiments demonstrated that our framework outperforms conventional inference processes on accuracy, coherence, and diversity. The framework's ability to iteratively expand its search space while retaining contextually relevant information results.

Chain-of-thought reasoning and scratchpads have emerged as critical tools for enhancing the computational capabilities of transformers. While theoretical results show that polynomial-length scratchpads can extend transformers' expressivity from $TC^0$ to $PTIME$, their required length remains poorly understood. Empirical evidence even suggests that transformers need scratchpads even for many problems in $TC^0$, such as Parity or Multiplication, challenging optimistic bounds derived from circuit complexity. In this work, we initiate the study of systematic lower bounds for the number of CoT steps across different algorithmic problems, in the hard-attention regime. We study a variety of algorithmic problems, and provide bounds that are tight up to logarithmic factors. Overall, these results contribute to emerging understanding of the power and limitations of chain-of-thought reasoning.

Tables serve as a fundamental format for representing structured relational data. While current language models (LMs) excel at many text-based tasks, they still face challenges in table understanding due to the complex characteristics of tabular data, such as their structured nature. In this paper, we aim to enhance LMs for improved table understanding. We identify four key challenges: 1) difficulty in locating target data, 2) deficiency in table semantics, 3) numerical inaccuracies in textual reasoning, and 4) semantic inflexibility in symbolic reasoning. To address these issues, we propose TableMaster, a recipe and comprehensive framework that integrates multiple solutions to overcome these obstacles. TableMaster first extracts relevant table content and verbalizes it with enriched semantic context. Additionally, we introduce adaptive reasoning, a flexible approach that dynamically adjusts between textual and symbolic reasoning, tailoring the reasoning process to each query. Extensive analyses and experiments demonstrate our findings and the effectiveness of TableMaster. On the WikiTQ dataset, TableMaster achieves an accuracy of 78.13% using GPT-4o-mini, surpassing existing baselines.

We investigate the logical reasoning capabilities of large language models (LLMs) and their scalability in complex non-monotonic reasoning. To this end, we introduce ZebraLogic, a comprehensive evaluation framework for assessing LLM reasoning performance on logic grid puzzles derived from constraint satisfaction problems (CSPs). ZebraLogic enables the generation of puzzles with controllable and quantifiable complexity, facilitating a systematic study of the scaling limits of models such as Llama, o1 models, and DeepSeek-R1. By encompassing a broad range of search space complexities and diverse logical constraints, ZebraLogic provides a structured environment to evaluate reasoning under increasing difficulty. Our results reveal a significant decline in accuracy as problem complexity grows -- a phenomenon we term the curse of complexity. This limitation persists even with larger models and increased inference-time computation, suggesting inherent constraints in current LLM reasoning capabilities. Additionally, we explore strategies to enhance logical reasoning, including Best-of-N sampling, backtracking mechanisms, and self-verification prompts. Our findings offer critical insights into the scalability of LLM reasoning, highlight fundamental limitations, and outline potential directions for improvement.