Daily Papers

Recent progress has pushed AI frontiers from pattern recognition tasks toward problems that require step by step, System2 style reasoning, especially with large language models. Yet, unlike learning, where generalization and out of distribution (OoD) evaluation concepts are well formalized, there is no clear, consistent definition or metric for reasoning ability. We propose Complexity Out of Distribution (Complexity OoD) generalization as a framework and problem setting to define and measure reasoning. A model exhibits Complexity OoD generalization when it maintains performance on test instances whose minimal required solution complexity, either representational (richer solution structure) or computational (more reasoning steps/program length), exceeds that of all training examples. We formalize complexity via solution description Kolmogorov complexity and operational proxies (e.g., object/relation counts; reasoning step counts), clarifying how Complexity OoD differs from length and compositional OoD. This lens unifies learning and reasoning: many cases solvable with System1 like processing at low complexity become System2 like under complexity pressure, while System2 can be viewed as generalization over solution structures. We translate this perspective into practice with recommendations for operationalizing Complexity OoD across the stack: incorporating complexity into benchmark and evaluation metric design, rethinking supervision to target solution traces, seeking and designing inductive biases for Complexity OoD generalization, addressing learning to reason spillovers such as spurious shortcuts, semantic robustness, catastrophic forgetting, and step wise calibration. Because Complexity OoD cannot be solved by scaling data alone, progress toward robust reasoning will require architectures and training regimes that explicitly model and allocate computation with respect to complexity.

Closed-source Large Language Models (LLMs) have become increasingly popular, with impressive performance across a wide range of natural language tasks. These models can be fine-tuned to further improve performance, but this often results in the models learning from dataset-specific heuristics that reduce their robustness on out-of-distribution (OOD) data. Existing methods to improve robustness either perform poorly, or are non-applicable to closed-source models because they assume access to model internals, or the ability to change the model's training procedure. In this work, we investigate strategies to improve the robustness of closed-source LLMs through data-centric methods that do not require access to model internals. We find that the optimal strategy depends on the complexity of the OOD data. For highly complex OOD datasets, upsampling more challenging training examples can improve robustness by up to 1.5%. For less complex OOD datasets, replacing a portion of the training set with LLM-generated examples can improve robustness by 3.7%. More broadly, we find that large-scale closed-source autoregressive LLMs are substantially more robust than commonly used encoder models, and are a more appropriate choice of baseline going forward.

While large language models (LLMs) have shown exceptional capabilities in understanding complex queries and performing sophisticated tasks, their generalization abilities are often deeply entangled with memorization, necessitating more precise evaluation. To address this challenge, we introduce Scylla, a dynamic evaluation framework that quantitatively measures the generalization abilities of LLMs. Scylla disentangles generalization from memorization via assessing model performance on both in-distribution (ID) and out-of-distribution (OOD) data through 20 tasks across 5 levels of complexity. Through extensive experiments, we uncover a non-monotonic relationship between task complexity and the performance gap between ID and OOD data, which we term the generalization valley. Specifically, this phenomenon reveals a critical threshold - referred to as critical complexity - where reliance on non-generalizable behavior peaks, indicating the upper bound of LLMs' generalization capabilities. As model size increases, the critical complexity shifts toward higher levels of task complexity, suggesting that larger models can handle more complex reasoning tasks before over-relying on memorization. Leveraging Scylla and the concept of critical complexity, we benchmark 28LLMs including both open-sourced models such as LLaMA and Qwen families, and close-sourced models like Claude and GPT, providing a more robust evaluation and establishing a clearer understanding of LLMs' generalization capabilities.

Action-conditioned world models (ACWMs) have shown strong promise for video prediction and decision-making. However, existing benchmarks are largely restricted to egocentric navigation or narrow, task-specific robotics datasets, offering only limited coverage of the rich physical interactions required for generalized world understanding. We introduce ACWM-Phys, a new benchmark for evaluating action-conditioned prediction under diverse physical dynamics in a clean, controllable simulation environment with a carefully designed action space. ACWM-Phys contains training and evaluation data spanning rigid-body dynamics, kinematics, deformable-object interactions, and particle dynamics. To evaluate both interpolation and generalization, we design in-distribution and out-of-distribution protocols with controlled shifts in interaction patterns or scene configurations. By building the benchmark in a fully controllable simulator, ACWM-Phys enables precise data collection, reproducible evaluation, and systematic analysis of model capabilities for physically grounded world modeling. Through systematic experiments on ACWM-DiT, we find that OoD generalization depends not only on the physical regime but also on effective task complexity: models generalize well on visually simple, low-dimensional interactions with clear geometric structure, but suffer larger drops on deformable contacts, high-dimensional control, and complex articulated motion. This suggests that the model still relies heavily on visual appearance patterns instead of fully learning the underlying physics. Ablations show that cross-attention improves high-dimensional action conditioning, causal VAEs outperform frame-wise encoders, and larger action spaces are harder to model but can improve generalization by providing richer control signals. These findings guide the design of physically grounded world models.

In contrast to entropy, which increases monotonically, the "complexity" or "interestingness" of closed systems seems intuitively to increase at first and then decrease as equilibrium is approached. For example, our universe lacked complex structures at the Big Bang and will also lack them after black holes evaporate and particles are dispersed. This paper makes an initial attempt to quantify this pattern. As a model system, we use a simple, two-dimensional cellular automaton that simulates the mixing of two liquids ("coffee" and "cream"). A plausible complexity measure is then the Kolmogorov complexity of a coarse-grained approximation of the automaton's state, which we dub the "apparent complexity." We study this complexity measure, and show analytically that it never becomes large when the liquid particles are non-interacting. By contrast, when the particles do interact, we give numerical evidence that the complexity reaches a maximum comparable to the "coffee cup's" horizontal dimension. We raise the problem of proving this behavior analytically.

The complexity of a graph is the number of its labeled spanning trees. In this work complexity is studied in settings that admit regular graphs. An exact formula is established linking complexity of the complement of a regular graph to numbers of closed walks in the graph by way of an infinite alternating series. Some consequences of this result yield infinite classes of lower and upper bounds on the complexity of such graphs. Applications of these mathematical results to biological problems on neuronal activity are described.

Training large language models (LLMs) with open-domain instruction data has yielded remarkable success in aligning to end tasks and human preferences. Extensive research has highlighted the importance of the quality and diversity of instruction data. However, the impact of data complexity, as a crucial metric, remains relatively unexplored from three aspects: (1)where the sustainability of performance improvements with increasing complexity is uncertain; (2)whether the improvement brought by complexity merely comes from introducing more training tokens; and (3)where the potential benefits of incorporating instructions from easy to difficult are not yet fully understood. In this paper, we propose Tree-Instruct to systematically enhance the instruction complexity in a controllable manner. By adding a specified number of nodes to instructions' semantic trees, this approach not only yields new instruction data from the modified tree but also allows us to control the difficulty level of modified instructions. Our preliminary experiments reveal the following insights: (1)Increasing complexity consistently leads to sustained performance improvements of LLMs. (2)Under the same token budget, a few complex instructions outperform diverse yet simple instructions. (3)Curriculum instruction tuning might not yield the anticipated results; focusing on increasing complexity appears to be the key.

Computational complexity has often been ignored in philosophy of mind, in philosophical artificial intelligence studies. The purpose of this paper is threefold. First and foremost, to show the importance of complexity rather than computability in philosophical and AI problems. Second, to rephrase the notion of computability in terms of solvability, i.e. treating computability as non-sufficient for establishing intelligence. The Church-Turing thesis is therefore revisited and rephrased in order to capture the ontological background of spatial and temporal complexity. Third, to emphasize ontological differences between different time complexities, which seem to provide a solid base towards better understanding of artificial intelligence in general.

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

We present algorithms for diffusion model sampling which obtain δ-error in polylog(1/δ) steps, given access to widetilde O(δ)-accurate score estimates in L^2. This is an exponential improvement over all previous results. Specifically, under minimal data assumptions, the complexity is widetilde O(d_star polylog(1/δ)) where d_star is the intrinsic dimension of the data. Further, under a non-uniform L-Lipschitz condition, the complexity reduces to widetilde O(L polylog(1/δ)). Our approach also yields the first polylog(1/δ) complexity sampler for general log-concave distributions using only gradient evaluations.

We explore the emergence of intelligent behavior in artificial systems by investigating how the complexity of rule-based systems influences the capabilities of models trained to predict these rules. Our study focuses on elementary cellular automata (ECA), simple yet powerful one-dimensional systems that generate behaviors ranging from trivial to highly complex. By training distinct Large Language Models (LLMs) on different ECAs, we evaluated the relationship between the complexity of the rules' behavior and the intelligence exhibited by the LLMs, as reflected in their performance on downstream tasks. Our findings reveal that rules with higher complexity lead to models exhibiting greater intelligence, as demonstrated by their performance on reasoning and chess move prediction tasks. Both uniform and periodic systems, and often also highly chaotic systems, resulted in poorer downstream performance, highlighting a sweet spot of complexity conducive to intelligence. We conjecture that intelligence arises from the ability to predict complexity and that creating intelligence may require only exposure to complexity.

In the realm of embodied artificial intelligence, the reasoning capabilities of Large Language Models (LLMs) play a pivotal role. Although there are effective methods like program-of-thought prompting for LLMs which uses programming language to tackle complex reasoning tasks, the specific impact of code data on the improvement of reasoning capabilities remains under-explored. To address this gap, we propose complexity-impacted reasoning score (CIRS), which combines structural and logical attributes, to measure the correlation between code and reasoning abilities. Specifically, we use the abstract syntax tree to encode the structural information and calculate logical complexity by considering the difficulty and the cyclomatic complexity. Through an empirical analysis, we find not all code data of complexity can be learned or understood by LLMs. Optimal level of complexity is critical to the improvement of reasoning abilities by program-aided prompting. Then we design an auto-synthesizing and stratifying algorithm, and apply it to instruction generation for mathematical reasoning and code data filtering for code generation tasks. Extensive results demonstrates the effectiveness of our proposed approach. Code will be integrated into the EasyInstruct framework at https://github.com/zjunlp/EasyInstruct.

When agents are acting together, they may need a simple mechanism to decide on joint actions. One possibility is to have the agents express their preferences in the form of a ballot and use a voting rule to decide the winning action(s). Unfortunately, agents may try to manipulate such an election by misreporting their preferences. Fortunately, it has been shown that it is NP-hard to compute how to manipulate a number of different voting rules. However, NP-hardness only bounds the worst-case complexity. Recent theoretical results suggest that manipulation may often be easy in practice. To address this issue, I suggest studying empirically if computational complexity is in practice a barrier to manipulation. The basic tool used in my investigations is the identification of computational "phase transitions". Such an approach has been fruitful in identifying hard instances of propositional satisfiability and other NP-hard problems. I show that phase transition behaviour gives insight into the hardness of manipulating voting rules, increasing concern that computational complexity is indeed any sort of barrier. Finally, I look at the problem of computing manipulation of other, related problems like stable marriage and tournament problems.

We find ourselves in the midst of an explosion in artificial intelligence research, particularly with large language models (LLMs). These models have diverse applications spanning finance, commonsense knowledge graphs, medicine, and visual analysis. In the world of Object-Oriented Programming(OOP), a robust body of knowledge and methods has been developed for managing complex tasks through object-oriented thinking. However, the intersection of LLMs with OOP remains an underexplored territory. Empirically, we currently possess limited understanding of how LLMs can enhance the effectiveness of OOP learning and code writing, as well as how we can evaluate such AI-powered tools. Our work aims to address this gap by presenting a vision from the perspectives of key stakeholders involved in an OOP task: programmers, mariners, and experienced programmers. We identify critical junctures within typical coding workflows where the integration of LLMs can offer significant benefits. Furthermore, we propose ways to augment existing logical reasoning and code writing, ultimately enhancing the programming experience.

Large Language Models (LLMs) increasingly exhibit strong reasoning abilities, often attributed to their capacity to generate chain-of-thought-style intermediate reasoning. Recent work suggests that exposure to code can further enhance these skills, but existing studies largely treat code as a generic training signal, leaving open the question of which properties of code actually contribute to improved reasoning. To address this gap, we study the structural complexity of code, which captures control flow and compositional structure that may shape how models internalise multi-step reasoning during fine-tuning. We examine two complementary settings: solution-driven complexity, where complexity varies across multiple solutions to the same problem, and problem-driven complexity, where complexity reflects variation in the underlying tasks. Using cyclomatic complexity and logical lines of code to construct controlled fine-tuning datasets, we evaluate a range of open-weight LLMs on diverse reasoning benchmarks. Our findings show that although code can improve reasoning, structural properties strongly determine its usefulness. In 83% of experiments, restricting fine-tuning data to a specific structural complexity range outperforms training on structurally diverse code, pointing to a data-centric path for improving reasoning beyond scaling.

While machine learning has advanced through massive parallelization, we identify a critical blind spot: some problems are fundamentally sequential. These "inherently serial" problems-from mathematical reasoning to physical simulations to sequential decision-making-require dependent computational steps that cannot be parallelized. Drawing from complexity theory, we formalize this distinction and demonstrate that current parallel-centric architectures face fundamental limitations on such tasks. We argue that recognizing the serial nature of computation holds profound implications on machine learning, model design, hardware development. As AI tackles increasingly complex reasoning, deliberately scaling serial computation-not just parallel computation-is essential for continued progress.

Out-of-distribution (OOD) detection is crucial to modern deep learning applications by identifying and alerting about the OOD samples that should not be tested or used for making predictions. Current OOD detection methods have made significant progress when in-distribution (ID) and OOD samples are drawn from static distributions. However, this can be unrealistic when applied to real-world systems which often undergo continuous variations and shifts in ID and OOD distributions over time. Therefore, for an effective application in real-world systems, the development of OOD detection methods that can adapt to these dynamic and evolving distributions is essential. In this paper, we propose a novel and more realistic setting called continuously adaptive out-of-distribution (CAOOD) detection which targets on developing an OOD detection model that enables dynamic and quick adaptation to a new arriving distribution, with insufficient ID samples during deployment time. To address CAOOD, we develop meta OOD learning (MOL) by designing a learning-to-adapt diagram such that a good initialized OOD detection model is learned during the training process. In the testing process, MOL ensures OOD detection performance over shifting distributions by quickly adapting to new distributions with a few adaptations. Extensive experiments on several OOD benchmarks endorse the effectiveness of our method in preserving both ID classification accuracy and OOD detection performance on continuously shifting distributions.

Out-of-distribution (OOD) detection is vital to safety-critical machine learning applications and has thus been extensively studied, with a plethora of methods developed in the literature. However, the field currently lacks a unified, strictly formulated, and comprehensive benchmark, which often results in unfair comparisons and inconclusive results. From the problem setting perspective, OOD detection is closely related to neighboring fields including anomaly detection (AD), open set recognition (OSR), and model uncertainty, since methods developed for one domain are often applicable to each other. To help the community to improve the evaluation and advance, we build a unified, well-structured codebase called OpenOOD, which implements over 30 methods developed in relevant fields and provides a comprehensive benchmark under the recently proposed generalized OOD detection framework. With a comprehensive comparison of these methods, we are gratified that the field has progressed significantly over the past few years, where both preprocessing methods and the orthogonal post-hoc methods show strong potential.

Human intelligence emerged through the process of natural selection and evolution on Earth. We investigate what it would take to re-create this process in silico. While past work has often focused on low-level processes (such as simulating physics or chemistry), we instead take a more targeted approach, aiming to evolve agents that can accumulate open-ended culture and technologies across generations. Towards this, we present JaxLife: an artificial life simulator in which embodied agents, parameterized by deep neural networks, must learn to survive in an expressive world containing programmable systems. First, we describe the environment and show that it can facilitate meaningful Turing-complete computation. We then analyze the evolved emergent agents' behavior, such as rudimentary communication protocols, agriculture, and tool use. Finally, we investigate how complexity scales with the amount of compute used. We believe JaxLife takes a step towards studying evolved behavior in more open-ended simulations. Our code is available at https://github.com/luchris429/JaxLife

Class incremental learning (CIL) is a challenging setting of continual learning, which learns a series of tasks sequentially. Each task consists of a set of unique classes. The key feature of CIL is that no task identifier (or task-id) is provided at test time. Predicting the task-id for each test sample is a challenging problem. An emerging theory-guided approach (called TIL+OOD) is to train a task-specific model for each task in a shared network for all tasks based on a task-incremental learning (TIL) method to deal with catastrophic forgetting. The model for each task is an out-of-distribution (OOD) detector rather than a conventional classifier. The OOD detector can perform both within-task (in-distribution (IND)) class prediction and OOD detection. The OOD detection capability is the key to task-id prediction during inference. However, this paper argues that using a traditional OOD detector for task-id prediction is sub-optimal because additional information (e.g., the replay data and the learned tasks) available in CIL can be exploited to design a better and principled method for task-id prediction. We call the new method TPL (Task-id Prediction based on Likelihood Ratio). TPL markedly outperforms strong CIL baselines and has negligible catastrophic forgetting. The code of TPL is publicly available at https://github.com/linhaowei1/TPL.

Why should moral philosophers, moral psychologists, and machine ethicists care about computational complexity? Debates on whether artificial intelligence (AI) can or should be used to solve problems in ethical domains have mainly been driven by what AI can or cannot do in terms of human capacities. In this paper, we tackle the problem from the other end by exploring what kind of moral machines are possible based on what computational systems can or cannot do. To do so, we analyze normative ethics through the lens of computational complexity. First, we introduce computational complexity for the uninitiated reader and discuss how the complexity of ethical problems can be framed within Marr's three levels of analysis. We then study a range of ethical problems based on consequentialism, deontology, and virtue ethics, with the aim of elucidating the complexity associated with the problems themselves (e.g., due to combinatorics, uncertainty, strategic dynamics), the computational methods employed (e.g., probability, logic, learning), and the available resources (e.g., time, knowledge, learning). The results indicate that most problems the normative frameworks pose lead to tractability issues in every category analyzed. Our investigation also provides several insights about the computational nature of normative ethics, including the differences between rule- and outcome-based moral strategies, and the implementation-variance with regard to moral resources. We then discuss the consequences complexity results have for the prospect of moral machines in virtue of the trade-off between optimality and efficiency. Finally, we elucidate how computational complexity can be used to inform both philosophical and cognitive-psychological research on human morality by advancing the Moral Tractability Thesis (MTT).

Language models (LMs), like other neural networks, often favor shortcut heuristics based on surface-level patterns. Although LMs behave like n-gram models early in training, they must eventually learn hierarchical syntactic representations to correctly apply grammatical rules out-of-distribution (OOD). In this work, we use case studies of English grammar to explore how complex, diverse training data drives models to generalize OOD. We construct a framework that unifies our understanding of random variation with training dynamics, rule selection with memorization, and data diversity with complexity. We show that these factors are nuanced, and that intermediate levels of diversity and complexity lead to inconsistent behavior across random seeds and to unstable training dynamics. Our findings emphasize the critical role of training data in shaping generalization patterns and illuminate how competing model strategies lead to inconsistent generalization outcomes across random seeds. Code is available at https://github.com/sunnytqin/concept_comp.git.

Among the most important properties of algorithms investigated in computer science are soundness, completeness, and complexity. These properties, however, are rarely analyzed for the vast collection of recently proposed methods for planning with large language models. In this work, we alleviate this gap. We analyse these properties of using LLMs for planning and highlight that recent trends abandon both soundness and completeness for the sake of inefficiency. We propose a significantly more efficient approach that can, at the same time, maintain both soundness and completeness. We exemplify on four representative search problems, comparing to the LLM-based solutions from the literature that attempt to solve these problems. We show that by using LLMs to produce the code for the search components we can solve the entire datasets with 100\% accuracy with only a few calls to the LLM. We argue for a responsible use of compute resources; urging research community to investigate sound and complete LLM-based approaches that uphold efficiency.

LLM-based agents are increasingly expected to handle real-world assistant tasks, yet existing benchmarks typically evaluate them under isolated sources of difficulty, such as a single environment or fully specified instructions. This leaves a substantial gap between current evaluation settings and the compositional challenges that arise in practical deployment. To address this gap, we introduce LiveClawBench, a benchmark to evaluate LLM agents on real-world assistant tasks. Based on an analysis of various real OpenClaw usage cases, we derive a Triple-Axis Complexity Framework that characterizes task difficulty along three dimensions: Environment Complexity, Cognitive Demand, and Runtime Adaptability. Guided by this framework, we construct a pilot benchmark with explicit complexity-factor annotations, covering real-world assistant tasks with compositional difficulty. Together, the framework and benchmark provide a principled foundation for evaluating LLM agents in realistic assistant settings, and establish a basis for future expansion across task domains and complexity axes. We are continuing to enrich our case collections to achieve more comprehensive domain and complexity coverage. The project page is at https://github.com/Mosi-AI/LiveClawBench.

The complexity of learning a concept class under Gaussian marginals in the difficult agnostic model is closely related to its L_1-approximability by low-degree polynomials. For any concept class with Gaussian surface area at most Γ, Klivans et al. (2008) show that degree d = O(Γ^2 / varepsilon^4) suffices to achieve an varepsilon-approximation. This leads to the best-known bounds on the complexity of learning a variety of concept classes. In this note, we improve their analysis by showing that degree d = tilde O (Γ^2 / varepsilon^2) is enough. In light of lower bounds due to Diakonikolas et al. (2021), this yields (near) optimal bounds on the complexity of agnostically learning polynomial threshold functions in the statistical query model. Our proof relies on a direct analogue of a construction of Feldman et al. (2020), who considered L_1-approximation on the Boolean hypercube.

We consider concept generalization at a large scale in the diverse and natural visual spectrum. Established computational modes (i.e., rule-based or similarity-based) are primarily studied isolated and focus on confined and abstract problem spaces. In this work, we study these two modes when the problem space scales up, and the complexity of concepts becomes diverse. Specifically, at the representational level, we seek to answer how the complexity varies when a visual concept is mapped to the representation space. Prior psychology literature has shown that two types of complexities (i.e., subjective complexity and visual complexity) (Griffiths and Tenenbaum, 2003) build an inverted-U relation (Donderi, 2006; Sun and Firestone, 2021). Leveraging Representativeness of Attribute (RoA), we computationally confirm the following observation: Models use attributes with high RoA to describe visual concepts, and the description length falls in an inverted-U relation with the increment in visual complexity. At the computational level, we aim to answer how the complexity of representation affects the shift between the rule- and similarity-based generalization. We hypothesize that category-conditioned visual modeling estimates the co-occurrence frequency between visual and categorical attributes, thus potentially serving as the prior for the natural visual world. Experimental results show that representations with relatively high subjective complexity outperform those with relatively low subjective complexity in the rule-based generalization, while the trend is the opposite in the similarity-based generalization.

Understanding how humans perceive visual complexity is a key area of study in visual cognition. Previous approaches to modeling visual complexity assessments have often resulted in intricate, difficult-to-interpret algorithms that employ numerous features or sophisticated deep learning architectures. While these complex models achieve high performance on specific datasets, they often sacrifice interpretability, making it challenging to understand the factors driving human perception of complexity. Recently (Shen, et al. 2024) proposed an interpretable segmentation-based model that accurately predicted complexity across various datasets, supporting the idea that complexity can be explained simply. In this work, we investigate the failure of their model to capture structural, color and surprisal contributions to complexity. To this end, we propose Multi-Scale Sobel Gradient (MSG) which measures spatial intensity variations, Multi-Scale Unique Color (MUC) which quantifies colorfulness across multiple scales, and surprise scores generated using a Large Language Model. We test our features on existing benchmarks and a novel dataset (Surprising Visual Genome) containing surprising images from Visual Genome. Our experiments demonstrate that modeling complexity accurately is not as simple as previously thought, requiring additional perceptual and semantic factors to address dataset biases. Our model improves predictive performance while maintaining interpretability, offering deeper insights into how visual complexity is perceived and assessed. Our code, analysis and data are available at https://github.com/Complexity-Project/Complexity-in-Complexity.

Training large language models for complex reasoning is bottlenecked by the scarcity of verifiable, high-quality data. In domains like physics, standard text augmentation often introduces hallucinations, while static benchmarks lack the reasoning traces required for fine-tuning. We introduce the Infinite Problem Generator (IPG), an agentic framework that synthesizes physics problems with guaranteed solvability through a Formula-as-Code paradigm. Unlike probabilistic text generation, IPG constructs solutions as executable Python programs, enforcing strict mathematical consistency. As a proof-of-concept, we release ClassicalMechanicsV1, a high-fidelity corpus of 1,335 classical mechanics problems expanded from 165 expert seeds. The corpus demonstrates high structural diversity, spanning 102 unique physical formulas with an average complexity of 3.05 formulas per problem. Furthermore, we identify a Complexity Blueprint, demonstrating a strong linear correlation (R^2 approx 0.95) between formula count and verification code length. This relationship establishes code complexity as a precise, proxy-free metric for problem difficulty, enabling controllable curriculum generation. We release the full IPG pipeline, the ClassicalMechanicsV1 dataset, and our evaluation report to support reproducible research in reasoning-intensive domains.

This paper uses the concept of algorithmic efficiency to present a unified theory of intelligence. Intelligence is defined informally, formally, and computationally. We introduce the concept of Dimensional complexity in algorithmic efficiency and deduce that an optimally efficient algorithm has zero Time complexity, zero Space complexity, and an infinite Dimensional complexity. This algorithm is used to generate the number line.

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

Theory of Mind (ToM) can be used to assess the capabilities of Large Language Models (LLMs) in complex scenarios where social reasoning is required. While the research community has proposed many ToM benchmarks, their hardness varies greatly, and their complexity is not well defined. This work proposes a framework inspired by cognitive load theory to measure the complexity of ToM tasks. We quantify a problem's complexity as the number of states necessary to solve it correctly. Our complexity measure also accounts for spurious states of a ToM problem designed to make it apparently harder. We use our method to assess the complexity of five widely adopted ToM benchmarks. On top of this framework, we design a prompting technique that augments the information available to a model with a description of how the environment changes with the agents' interactions. We name this technique Discrete World Models (DWM) and show how it elicits superior performance on ToM tasks.

Despite the superior performance of Large Reasoning Models (LRMs), their reasoning behaviors are often counterintuitive, leading to suboptimal reasoning capabilities. To theoretically formalize the desired reasoning behaviors, this paper presents the Laws of Reasoning (LoRe), a unified framework that characterizes intrinsic reasoning patterns in LRMs. We first propose compute law with the hypothesis that the reasoning compute should scale linearly with question complexity. Beyond compute, we extend LoRe with a supplementary accuracy law. Since the question complexity is difficult to quantify in practice, we examine these hypotheses by two properties of the laws, monotonicity and compositionality. We therefore introduce LoRe-Bench, a benchmark that systematically measures these two tractable properties for large reasoning models. Evaluation shows that most reasoning models exhibit reasonable monotonicity but lack compositionality. In response, we develop an effective finetuning approach that enforces compute-law compositionality. Extensive empirical studies demonstrate that better compliance with compute laws yields consistently improved reasoning performance on multiple benchmarks, and uncovers synergistic effects across properties and laws. Project page: https://lore-project.github.io/

Prior work typically describes out-of-domain (OOD) or out-of-distribution (OODist) samples as those that originate from dataset(s) or source(s) different from the training set but for the same task. When compared to in-domain (ID) samples, the models have been known to usually perform poorer on OOD samples, although this observation is not consistent. Another thread of research has focused on OOD detection, albeit mostly using supervised approaches. In this work, we first consolidate and present a systematic analysis of multiple definitions of OOD and OODist as discussed in prior literature. Then, we analyze the performance of a model under ID and OOD/OODist settings in a principled way. Finally, we seek to identify an unsupervised method for reliably identifying OOD/OODist samples without using a trained model. The results of our extensive evaluation using 12 datasets from 4 different tasks suggest the promising potential of unsupervised metrics in this task.

Complex hierarchical structures composed of simple nanoscale building blocks form the basis of most biological materials. Here we demonstrate how analogies between seemingly different fields enable the understanding of general principles by which functional properties in hierarchical systems emerge, similar to an analogy learning process. Specifically, natural hierarchical materials like spider silk exhibit properties comparable to classical music in terms of their hierarchical structure and function. As a comparative tool here we apply hierarchical ontology logs (olog) that follow a rigorous mathematical formulation based on category theory to provide an insightful system representation by expressing knowledge in a conceptual map. We explain the process of analogy creation, draw connections at several levels of hierarchy and identify similar patterns that govern the structure of the hierarchical systems silk and music and discuss the impact of the derived analogy for nanotechnology.

There often is a dilemma between ease of optimization and robust out-of-distribution (OoD) generalization. For instance, many OoD methods rely on penalty terms whose optimization is challenging. They are either too strong to optimize reliably or too weak to achieve their goals. We propose to initialize the networks with a rich representation containing a palette of potentially useful features, ready to be used by even simple models. On the one hand, a rich representation provides a good initialization for the optimizer. On the other hand, it also provides an inductive bias that helps OoD generalization. Such a representation is constructed with the Rich Feature Construction (RFC) algorithm, also called the Bonsai algorithm, which consists of a succession of training episodes. During discovery episodes, we craft a multi-objective optimization criterion and its associated datasets in a manner that prevents the network from using the features constructed in the previous iterations. During synthesis episodes, we use knowledge distillation to force the network to simultaneously represent all the previously discovered features. Initializing the networks with Bonsai representations consistently helps six OoD methods achieve top performance on ColoredMNIST benchmark. The same technique substantially outperforms comparable results on the Wilds Camelyon17 task, eliminates the high result variance that plagues other methods, and makes hyperparameter tuning and model selection more reliable.

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

Deep learning has achieved tremendous success with independent and identically distributed (i.i.d.) data. However, the performance of neural networks often degenerates drastically when encountering out-of-distribution (OoD) data, i.e., when training and test data are sampled from different distributions. While a plethora of algorithms have been proposed for OoD generalization, our understanding of the data used to train and evaluate these algorithms remains stagnant. In this work, we first identify and measure two distinct kinds of distribution shifts that are ubiquitous in various datasets. Next, through extensive experiments, we compare OoD generalization algorithms across two groups of benchmarks, each dominated by one of the distribution shifts, revealing their strengths on one shift as well as limitations on the other shift. Overall, we position existing datasets and algorithms from different research areas seemingly unconnected into the same coherent picture. It may serve as a foothold that can be resorted to by future OoD generalization research. Our code is available at https://github.com/ynysjtu/ood_bench.

This comprehensive study evaluates the performance of OpenAI's o1-preview large language model across a diverse array of complex reasoning tasks, spanning multiple domains, including computer science, mathematics, natural sciences, medicine, linguistics, and social sciences. Through rigorous testing, o1-preview demonstrated remarkable capabilities, often achieving human-level or superior performance in areas ranging from coding challenges to scientific reasoning and from language processing to creative problem-solving. Key findings include: -83.3% success rate in solving complex competitive programming problems, surpassing many human experts. -Superior ability in generating coherent and accurate radiology reports, outperforming other evaluated models. -100% accuracy in high school-level mathematical reasoning tasks, providing detailed step-by-step solutions. -Advanced natural language inference capabilities across general and specialized domains like medicine. -Impressive performance in chip design tasks, outperforming specialized models in areas such as EDA script generation and bug analysis. -Remarkable proficiency in anthropology and geology, demonstrating deep understanding and reasoning in these specialized fields. -Strong capabilities in quantitative investing. O1 has comprehensive financial knowledge and statistical modeling skills. -Effective performance in social media analysis, including sentiment analysis and emotion recognition. The model excelled particularly in tasks requiring intricate reasoning and knowledge integration across various fields. While some limitations were observed, including occasional errors on simpler problems and challenges with certain highly specialized concepts, the overall results indicate significant progress towards artificial general intelligence.

We study the representation complexity of model-based and model-free reinforcement learning (RL) in the context of circuit complexity. We prove theoretically that there exists a broad class of MDPs such that their underlying transition and reward functions can be represented by constant depth circuits with polynomial size, while the optimal Q-function suffers an exponential circuit complexity in constant-depth circuits. By drawing attention to the approximation errors and building connections to complexity theory, our theory provides unique insights into why model-based algorithms usually enjoy better sample complexity than model-free algorithms from a novel representation complexity perspective: in some cases, the ground-truth rule (model) of the environment is simple to represent, while other quantities, such as Q-function, appear complex. We empirically corroborate our theory by comparing the approximation error of the transition kernel, reward function, and optimal Q-function in various Mujoco environments, which demonstrates that the approximation errors of the transition kernel and reward function are consistently lower than those of the optimal Q-function. To the best of our knowledge, this work is the first to study the circuit complexity of RL, which also provides a rigorous framework for future research.

In the context of distributed synchronous computing, processors perform in rounds, and the time-complexity of a distributed algorithm is classically defined as the number of rounds before all computing nodes have output. Hence, this complexity measure captures the running time of the slowest node(s). In this paper, we are interested in the running time of the ordinary nodes, to be compared with the running time of the slowest nodes. The node-averaged time-complexity of a distributed algorithm on a given instance is defined as the average, taken over every node of the instance, of the number of rounds before that node output. We compare the node-averaged time-complexity with the classical one in the standard LOCAL model for distributed network computing. We show that there can be an exponential gap between the node-averaged time-complexity and the classical time-complexity, as witnessed by, e.g., leader election. Our first main result is a positive one, stating that, in fact, the two time-complexities behave the same for a large class of problems on very sparse graphs. In particular, we show that, for LCL problems on cycles, the node-averaged time complexity is of the same order of magnitude as the slowest node time-complexity. In addition, in the LOCAL model, the time-complexity is computed as a worst case over all possible identity assignments to the nodes of the network. In this paper, we also investigate the ID-averaged time-complexity, when the number of rounds is averaged over all possible identity assignments. Our second main result is that the ID-averaged time-complexity is essentially the same as the expected time-complexity of randomized algorithms (where the expectation is taken over all possible random bits used by the nodes, and the number of rounds is measured for the worst-case identity assignment). Finally, we study the node-averaged ID-averaged time-complexity.

Large Reasoning Models (LRMs) have demonstrated impressive performance on complex tasks, including logical puzzle games that require deriving solutions satisfying all constraints. However, whether they can flexibly apply appropriate rules to varying conditions, particularly when faced with non-canonical game variants, remains an open question. Existing corpora focus on popular puzzles like 9x9 Sudoku, risking overfitting to canonical formats and memorization of solution patterns, which can mask deficiencies in understanding novel rules or adapting strategies to new variants. To address this, we introduce HardcoreLogic, a challenging benchmark of over 5,000 puzzles across 10 games, designed to test the robustness of LRMs on the "long-tail" of logical games. HardcoreLogic systematically transforms canonical puzzles through three dimensions: Increased Complexity (IC), Uncommon Elements (UE), and Unsolvable Puzzles (UP), reducing reliance on shortcut memorization. Evaluations on a diverse set of LRMs reveal significant performance drops, even for models achieving top scores on existing benchmarks, indicating heavy reliance on memorized stereotypes. While increased complexity is the dominant source of difficulty, models also struggle with subtle rule variations that do not necessarily increase puzzle difficulty. Our systematic error analysis on solvable and unsolvable puzzles further highlights gaps in genuine reasoning. Overall, HardcoreLogic exposes the limitations of current LRMs and establishes a benchmark for advancing high-level logical reasoning.

Rapidly learning abstract concepts from limited examples is a hallmark of human intelligence. This work investigates whether gradient-based meta-learning can equip neural networks with inductive biases for efficient few-shot acquisition of discrete concepts. I compare meta-learning methods against a supervised learning baseline on Boolean concepts (logical statements) generated by a probabilistic context-free grammar (PCFG). By systematically varying concept dimensionality (number of features) and recursive compositionality (depth of grammar recursion), I delineate between complexity regimes in which meta-learning robustly improves few-shot concept learning and regimes in which it does not. Meta-learners are much better able to handle compositional complexity than featural complexity. I highlight some reasons for this with a representational analysis of the weights of meta-learners and a loss landscape analysis demonstrating how featural complexity increases the roughness of loss trajectories, allowing curvature-aware optimization to be more effective than first-order methods. I find improvements in out-of-distribution generalization on complex concepts by increasing the number of adaptation steps in meta-SGD, where adaptation acts as a way of encouraging exploration of rougher loss basins. Overall, this work highlights the intricacies of learning compositional versus featural complexity in high dimensional concept spaces and provides a road to understanding the role of 2nd order methods and extended gradient adaptation in few-shot concept learning.

Wordle is a single-player word-guessing game where the goal is to discover a secret word w that has been chosen from a dictionary D. In order to discover w, the player can make at most ell guesses, which must also be words from D, all words in D having the same length k. After each guess, the player is notified of the positions in which their guess matches the secret word, as well as letters in the guess that appear in the secret word in a different position. We study the game of Wordle from a complexity perspective, proving NP-hardness of its natural formalization: to decide given a dictionary D and an integer ell if the player can guarantee to discover the secret word within ell guesses. Moreover, we prove that hardness holds even over instances where words have length k = 5, and that even in this case it is NP-hard to approximate the minimum number of guesses required to guarantee discovering the secret word. We also present results regarding its parameterized complexity and offer some related open problems.

Evolutionary program synthesis systems such as AlphaEvolve, OpenEvolve, and ShinkaEvolve offer a new approach to AI-assisted mathematical discovery. These systems utilize teams of large language models (LLMs) to generate candidate solutions to a problem as human readable code. These candidate solutions are then 'evolved' with the goal of improving them beyond what an LLM can produce in a single shot. While existing mathematical applications have mostly focused on problems of establishing bounds (e.g., sphere packing), the program synthesis approach is well suited to any problem where the solution takes the form of an explicit construction. With this in mind, in this paper we explore the use of OpenEvolve for combinatorial bijection discovery. We describe the results of applying OpenEvolve to three bijection construction problems involving Dyck paths, two of which are known and one of which is open. We find that while systems like OpenEvolve show promise as a valuable tool for combinatorialists, the problem of finding novel, research-level bijections remains a challenging task for current frontier systems, reinforcing the need for human mathematicians in the loop. We describe some lessons learned for others in the field interested in exploring the use of these systems.

We show that a variety of modern deep learning tasks exhibit a "double-descent" phenomenon where, as we increase model size, performance first gets worse and then gets better. Moreover, we show that double descent occurs not just as a function of model size, but also as a function of the number of training epochs. We unify the above phenomena by defining a new complexity measure we call the effective model complexity and conjecture a generalized double descent with respect to this measure. Furthermore, our notion of model complexity allows us to identify certain regimes where increasing (even quadrupling) the number of train samples actually hurts test performance.

Several studies have compared the in-distribution (ID) and out-of-distribution (OOD) performance of models in computer vision and NLP. They report a frequent positive correlation and some surprisingly never even observe an inverse correlation indicative of a necessary trade-off. The possibility of inverse patterns is important to determine whether ID performance can serve as a proxy for OOD generalization capabilities. This paper shows with multiple datasets that inverse correlations between ID and OOD performance do happen in real-world data - not only in theoretical worst-case settings. We also explain theoretically how these cases can arise even in a minimal linear setting, and why past studies could miss such cases due to a biased selection of models. Our observations lead to recommendations that contradict those found in much of the current literature. - High OOD performance sometimes requires trading off ID performance. - Focusing on ID performance alone may not lead to optimal OOD performance. It may produce diminishing (eventually negative) returns in OOD performance. - In these cases, studies on OOD generalization that use ID performance for model selection (a common recommended practice) will necessarily miss the best-performing models, making these studies blind to a whole range of phenomena.

Among the hardest tasks for humans are those found in competitive programming where problems require sophisticated algorithmic thinking, puzzle solving, and the creation of effective code. As a domain to assess language models (LMs), it has not received enough attention, though. This study presents the ICPC benchmark, which consists of 254 international collegiate programming contest (ICPC) tasks. Each problem includes official analysis, reference code, and sample, high-quality unit, and hidden tests. We are able to develop and evaluate a variety of LM inference techniques for competitive programming with these resources. With zero-shot chain-of-thought prompting, we find that o1 only achieves a 19.1\% pass@1 solve rate. With our best inference technique, which combines multi-turn self-judge with reflection and retrieval over episodic information, raises this to 42.2\%. Furthermore, we conduct a new human-in-the-loop investigation to gain a deeper understanding of the remaining difficulties. Surprisingly, we discover that o1 can solve 17 out of 18 problems that were previously unsolvable by any model or technique with just a few specific instructions. A footstep toward LMs with grounded, imaginative, and algorithmic thinking is provided by our quantitative findings and qualitative research. We open-source our code and data at https://github.com/kraritt/zolve.

We resolve the open question regarding the sample complexity of policy learning for maximizing the long-run average reward associated with a uniformly ergodic Markov decision process (MDP), assuming a generative model. In this context, the existing literature provides a sample complexity upper bound of widetilde O(|S||A|t_{mix}^2 epsilon^{-2}) and a lower bound of Omega(|S||A|t_{mix} epsilon^{-2}). In these expressions, |S| and |A| denote the cardinalities of the state and action spaces respectively, t_{mix} serves as a uniform upper limit for the total variation mixing times, and epsilon signifies the error tolerance. Therefore, a notable gap of t_{mix} still remains to be bridged. Our primary contribution is the development of an estimator for the optimal policy of average reward MDPs with a sample complexity of widetilde O(|S||A|t_{mix}epsilon^{-2}). This marks the first algorithm and analysis to reach the literature's lower bound. Our new algorithm draws inspiration from ideas in Li et al. (2020), Jin and Sidford (2021), and Wang et al. (2023). Additionally, we conduct numerical experiments to validate our theoretical findings.

Generalizing to out-of-distribution (OOD) data or unseen domain, termed OOD generalization, still lacks appropriate theoretical guarantees. Canonical OOD bounds focus on different distance measurements between source and target domains but fail to consider the optimization property of the learned model. As empirically shown in recent work, the sharpness of learned minima influences OOD generalization. To bridge this gap between optimization and OOD generalization, we study the effect of sharpness on how a model tolerates data change in domain shift which is usually captured by "robustness" in generalization. In this paper, we give a rigorous connection between sharpness and robustness, which gives better OOD guarantees for robust algorithms. It also provides a theoretical backing for "flat minima leads to better OOD generalization". Overall, we propose a sharpness-based OOD generalization bound by taking robustness into consideration, resulting in a tighter bound than non-robust guarantees. Our findings are supported by the experiments on a ridge regression model, as well as the experiments on deep learning classification tasks.

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

Systematic, compositional generalization beyond the training distribution remains a core challenge in machine learning -- and a critical bottleneck for the emergent reasoning abilities of modern language models. This work investigates out-of-distribution (OOD) generalization in Transformer networks using a GSM8K-style modular arithmetic on computational graphs task as a testbed. We introduce and explore a set of four architectural mechanisms aimed at enhancing OOD generalization: (i) input-adaptive recurrence; (ii) algorithmic supervision; (iii) anchored latent representations via a discrete bottleneck; and (iv) an explicit error-correction mechanism. Collectively, these mechanisms yield an architectural approach for native and scalable latent space reasoning in Transformer networks with robust algorithmic generalization capabilities. We complement these empirical results with a detailed mechanistic interpretability analysis that reveals how these mechanisms give rise to robust OOD generalization abilities.

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

We introduce a new predator-prey model by replacing the growth and predation constant by a square matrix, and the population density as a population vector. The classical Lotka-Volterra model describes a population that either modulates or converges. Stability analysis of such models have been extensively studied by the works of Merdan (https://doi.org/10.1016/j.chaos.2007.06.062). The new model adds complexity by introducing an age group structure where the population of each age group evolves as prescribed by the Leslie matrix. The added complexity changes the behavior of the model such that the population either displays roughly an exponential growth or decay. We first provide an exact equation that describes a time evolution and use analytic techniques to obtain an approximate growth factor. We also discuss the variants of the Leslie model, i.e., the complex value predator-prey model and the competitive model. We then prove the Last Species Standing theorem that determines the dominant population in the large time limit. The recursive structure of the model denies the application of simple regression. We discuss a machine learning scheme that allows an admissible fit for the population evolution of Paramecium Aurelia and Paramecium Caudatum. Another potential avenue to simplify the computation is to use the machinery of quantum operators. We demonstrate the potential of this approach by computing the Hamiltonian of a simple Leslie system.

We develop a discipline-agnostic emergence calculus that treats theories as fixed points of idempotent operators acting on descriptions. We show that, once processes are composable but access to the underlying system is mediated by a bounded observational interface, a canonical toolkit of six closure-changing primitives (P1--P6) is unavoidable. The framework unifies order-theoretic closure operators with dynamics-induced endomaps E_{τ,f} built from a Markov kernel, a coarse-graining lens, and a time scale τ. We introduce a computable total-variation idempotence defect for E_{τ,f}; small retention error implies approximate idempotence and yields stable "objects" packaged at the chosen τ within a fixed lens. For directionality, we define an arrow-of-time functional as the path-space KL divergence between forward and time-reversed trajectories and prove it is monotone under coarse-graining (data processing); we also formalize a protocol-trap audit showing that protocol holonomy alone cannot sustain asymmetry without a genuine affinity in the lifted dynamics. Finally, we prove a finite forcing-style counting lemma: relative to a partition-based theory, definable predicate extensions are exponentially rare, giving a clean anti-saturation mechanism for strict ladder climbing.

What are the root causes of hallucinations in large language models (LLMs)? We use Communication Complexity to prove that the Transformer layer is incapable of composing functions (e.g., identify a grandparent of a person in a genealogy) if the domains of the functions are large enough; we show through examples that this inability is already empirically present when the domains are quite small. We also point out that several mathematical tasks that are at the core of the so-called compositional tasks thought to be hard for LLMs are unlikely to be solvable by Transformers, for large enough instances and assuming that certain well accepted conjectures in the field of Computational Complexity are true.

Large Language Models (LLMs) suffer from reliability issues on complex tasks, as existing decomposition methods are heuristic and rely on agent or manual decomposition. This work introduces a novel, systematic decomposition framework that we call Analysis of CONstraint-Induced Complexity (ACONIC), which models the task as a constraint problem and leveraging formal complexity measures to guide decomposition. On combinatorial (SATBench) and LLM database querying tasks (Spider), we find that by decomposing the tasks following the measure of complexity, agent can perform considerably better (10-40 percentage point).

The International Mathematical Olympiad (IMO) is widely regarded as the world championship of high-school mathematics. IMO problems are renowned for their difficulty and novelty, demanding deep insight, creativity, and rigor. Although large language models perform well on many mathematical benchmarks, they often struggle with Olympiad-level problems. Using carefully designed prompts, we construct a model-agnostic, verification-and-refinement pipeline. We demonstrate its effectiveness on the recent IMO 2025, avoiding data contamination for models released before the competition. Equipped with any of the three leading models -- Gemini 2.5 Pro, Grok-4, or GPT-5 -- our pipeline correctly solved 5 out of the 6 problems (approx85.7% accuracy). This is in sharp contrast to their baseline accuracies: 31.6% (Gemini 2.5 Pro), 21.4% (Grok-4), and 38.1% (GPT-5), obtained by selecting the best of 32 candidate solutions. The substantial improvement underscores that the path to advanced AI reasoning requires not only developing more powerful base models but also designing effective methodologies to harness their full potential for complex tasks.

Building upon our previous investigations of O1 replication (Part 1: Journey Learning [Qin et al., 2024] and Part 2: Distillation [Huang et al., 2024]), this work explores the potential of inference-time scaling in large language models (LLMs) for medical reasoning tasks, ranging from diagnostic decision-making to treatment planning. Through extensive experiments on medical benchmarks of varying complexity (MedQA, Medbullets, and JAMA Clinical Challenges), our investigation reveals several key insights: (1) Increasing inference time does lead to improved performance. With a modest training set of 500 samples, our model yields substantial performance improvements of 6%-11%. (2) Task complexity directly correlates with the required length of reasoning chains, confirming the necessity of extended thought processes for challenging problems. (3) The differential diagnoses generated by our model adhere to the principles of the hypothetico-deductive method, producing a list of potential conditions that may explain a patient's symptoms and systematically narrowing these possibilities by evaluating the evidence. These findings demonstrate the promising synergy between inference-time scaling and journey learning in advancing LLMs' real-world clinical reasoning capabilities.

In this work, we investigate how Large Language Models (LLMs) adapt their internal representations when encountering inputs of increasing difficulty, quantified as the degree of out-of-distribution (OOD) shift. We reveal a consistent and quantifiable phenomenon: as task difficulty increases, whether through harder reasoning questions, longer contexts, or adding answer choices, the last hidden states of LLMs become substantially sparser. In short, \textit{the farther the shift, the sparser the representations}. This sparsity--difficulty relation is observable across diverse models and domains, suggesting that language models respond to unfamiliar or complex inputs by concentrating computation into specialized subspaces in the last hidden state. Through a series of controlled analyses with a learning dynamic explanation, we demonstrate that this sparsity is not incidental but an adaptive mechanism for stabilizing reasoning under OOD. Leveraging this insight, we design Sparsity-Guided Curriculum In-Context Learning (SG-ICL), a strategy that explicitly uses representation sparsity to schedule few-shot demonstrations, leading to considerable performance enhancements. Our study provides new mechanistic insights into how LLMs internalize OOD challenges. The source code is available at the URL: https://github.com/MingyuJ666/sparsityLLM.

We prove that any Turing machine running on inputs of arbitrary length can be simulated by a constant bit-size transformer, as long as the context window is sufficiently long. This improves previous works, which require scaling up either the model's precision or the number of parameters on longer inputs. Furthermore, we prove that the complexity class SPACE[s(n)] exactly characterizes the expressive power of a constant bit-size transformer with a context window of length s(n). Our approach relies on simulating Post machines, a Turing-complete computational model. Post machines can be modeled as automata equipped with a queue, exhibiting computational behaviors naturally aligned with those of transformers. The behavioral similarity between transformers and Post machines may offer new insights into the mechanisms underlying the reasoning abilities of transformers.

Large language models (LLMs) have shown remarkable improvements in reasoning and many existing benchmarks have been addressed by models such as o1 and o3 either fully or partially. However, a majority of these benchmarks emphasize deductive reasoning, including mathematical and coding tasks in which rules such as mathematical axioms or programming syntax are clearly defined, based on which LLMs can plan and apply these rules to arrive at a solution. In contrast, inductive reasoning, where one infers the underlying rules from observed data, remains less explored. Such inductive processes lie at the heart of scientific discovery, as they enable researchers to extract general principles from empirical observations. To assess whether LLMs possess this capacity, we introduce InductionBench, a new benchmark designed to evaluate the inductive reasoning ability of LLMs. Our experimental findings reveal that even the most advanced models available struggle to master the simplest complexity classes within the subregular hierarchy of functions, highlighting a notable deficiency in current LLMs' inductive reasoning capabilities. Coda and data are available https://github.com/Wenyueh/inductive_reasoning_benchmark.

We design and implement AXLearn, a production deep learning system that facilitates scalable and high-performance training of large deep learning models. Compared to other state-of-the-art deep learning systems, AXLearn has a unique focus on modularity and support for heterogeneous hardware infrastructure. AXLearn's internal interfaces between software components follow strict encapsulation, allowing different components to be assembled to facilitate rapid model development and experimentation on heterogeneous compute infrastructure. We introduce a novel method of quantifying modularity via Lines-of-Code (LoC)-complexity, which demonstrates how our system maintains constant complexity as we scale the components in the system, compared to linear or quadratic complexity in other systems. This allows integrating features such as Rotary Position Embeddings (RoPE) into AXLearn across hundred of modules with just 10 lines of code, compared to hundreds as required in other systems. At the same time, AXLearn maintains equivalent performance compared to state-of-the-art training systems. Finally, we share our experience in the development and operation of AXLearn.

The superficial alignment hypothesis (SAH) posits that large language models learn most of their knowledge during pre-training, and that post-training merely surfaces this knowledge. The SAH, however, lacks a precise definition, which has led to (i) different and seemingly orthogonal arguments supporting it, and (ii) important critiques to it. We propose a new metric called task complexity: the length of the shortest program that achieves a target performance on a task. In this framework, the SAH simply claims that pre-trained models drastically reduce the complexity of achieving high performance on many tasks. Our definition unifies prior arguments supporting the SAH, interpreting them as different strategies to find such short programs. Experimentally, we estimate the task complexity of mathematical reasoning, machine translation, and instruction following; we then show that these complexities can be remarkably low when conditioned on a pre-trained model. Further, we find that pre-training enables access to strong performances on our tasks, but it can require programs of gigabytes of length to access them. Post-training, on the other hand, collapses the complexity of reaching this same performance by several orders of magnitude. Overall, our results highlight that task adaptation often requires surprisingly little information -- often just a few kilobytes.

The k-Coloring problem on hereditary graph classes has been a deeply researched problem over the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs. We say that a graph is (H_1,H_2,ldots)-free if it does not contain any of H_1,H_2,ldots as induced subgraphs. The complexity landscape of the problem remains unclear even when restricting to the case k=4 and classes defined by a few forbidden induced subgraphs. While the case of only one forbidden induced subgraph has been completely resolved lately, the complexity when considering two forbidden induced subgraphs still has a couple of unknown cases. In particular, 4-Coloring on (P_6,C_3)-free graphs is polynomial while it is NP-hard on (P_{22},C_3)-free graphs. We provide a reduction showing NP-completeness of 4-Coloring on (P_t,C_3)-free graphs for 19leq tleq 21, thus constricting the gap of cases whose complexity remains unknown. Our proof includes a computer search ensuring that the graph family obtained through the reduction is indeed P_{19}-free.

Gradient-free/zeroth-order methods for black-box convex optimization have been extensively studied in the last decade with the main focus on oracle calls complexity. In this paper, besides the oracle complexity, we focus also on iteration complexity, and propose a generic approach that, based on optimal first-order methods, allows to obtain in a black-box fashion new zeroth-order algorithms for non-smooth convex optimization problems. Our approach not only leads to optimal oracle complexity, but also allows to obtain iteration complexity similar to first-order methods, which, in turn, allows to exploit parallel computations to accelerate the convergence of our algorithms. We also elaborate on extensions for stochastic optimization problems, saddle-point problems, and distributed optimization.

Intelligence appears under different names in different fields: as data compression in statistics and machine learning, as universal computation in dynamical systems, and as adaptive behavior in agents. Each field carries its own objective, and the two most influential drives often fail in mirror image: novelty search, which seeks surprise, is transfixed by a noisy television screen, while the free-energy principle, which avoids surprise, is most content in a dark room. Both failures have a single cause: each objective treats as one quantity the surprise a learner can convert into knowledge and the surprise it never can. Here we show that the learnable part of that information, which we call learnable novelty, yields the seemingly disparate projections of intelligence, and we give a closed-form estimator of it built on a cheap and differentiable reservoir computer. Used as a measure, with no supervision of any kind, the estimator recovers decades of complexity classification, ranking the Turing-complete rule~110 highest among the elementary cellular automata. Used as an objective, its gradient carries a neural cellular automaton from simple dynamics into a regime of solitons, the traveling, colliding structures by which rule~110 computes, as well as organizes the representation of an image encoder around the ten digit classes of MNIST, fully unsupervised: no label ever enters training. Handed to a reinforcement-learning agent as an intrinsic reward, it supplies the exploration that task rewards lack, improving on the task baseline in nine of ten environments and collapsing in none. Complexity generation, abstraction, and exploration, ordinarily pursued with unrelated objectives in separate fields, thus emerge from ascent on one differentiable quantity, and the projections of intelligence gain a common quantitative footing.

Instruction tuning has been widely used to unleash the complete potential of large language models. Notably, complex and diverse instructions are of significant importance as they can effectively align models with various downstream tasks. However, current approaches to constructing large-scale instructions predominantly favour powerful models such as GPT-4 or those with over 70 billion parameters, under the empirical presumption that such larger language models (LLMs) inherently possess enhanced capabilities. In this study, we question this prevalent assumption and conduct an in-depth exploration into the potential of smaller language models (SLMs) in the context of instruction evolution. Extensive experiments across three scenarios of instruction evolution reveal that smaller language models (SLMs) can synthesize more effective instructions than LLMs. Further analysis demonstrates that SLMs possess a broader output space during instruction evolution, resulting in more complex and diverse variants. We also observe that the existing metrics fail to focus on the impact of the instructions. Thus, we propose Instruction Complex-Aware IFD (IC-IFD), which introduces instruction complexity in the original IFD score to evaluate the effectiveness of instruction data more accurately. Our source code is available at: https://github.com/HypherX/Evolution-Analysis{https://github.com/HypherX/Evolution-Analysis}

The Right to Explanation and the Right to be Forgotten are two important principles outlined to regulate algorithmic decision making and data usage in real-world applications. While the right to explanation allows individuals to request an actionable explanation for an algorithmic decision, the right to be forgotten grants them the right to ask for their data to be deleted from all the databases and models of an organization. Intuitively, enforcing the right to be forgotten may trigger model updates which in turn invalidate previously provided explanations, thus violating the right to explanation. In this work, we investigate the technical implications arising due to the interference between the two aforementioned regulatory principles, and propose the first algorithmic framework to resolve the tension between them. To this end, we formulate a novel optimization problem to generate explanations that are robust to model updates due to the removal of training data instances by data deletion requests. We then derive an efficient approximation algorithm to handle the combinatorial complexity of this optimization problem. We theoretically demonstrate that our method generates explanations that are provably robust to worst-case data deletion requests with bounded costs in case of linear models and certain classes of non-linear models. Extensive experimentation with real-world datasets demonstrates the efficacy of the proposed framework.

When a text is translated, does the translation retain the complexity of the original? We introduce ComplexityMT, a new challenge for assessing how text complexity and machine translation interact with and influence each other, using the Common European Framework of Reference for Languages (CEFR) levels as the measure of text complexity. Across six languages, including Arabic, Dutch, English, French, Hindi, and Russian, we evaluate three open-weight models, one closed model, and a commercial machine translation system on two tasks: i) correlation of CEFR with translation difficulty, and ii) shifts in CEFR levels of the source texts. Our experiments show that higher CEFR levels make texts more difficult to translate, and that machine translation shifts the CEFR level of the target text compared to the original source, for most languages. These findings provide new insights for researchers and practitioners working on multilingual pedagogical content generation and machine translation difficulty estimation.

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

To make AI systems broadly useful for challenging real-world tasks, we need them to learn complex human goals and preferences. One approach to specifying complex goals asks humans to judge during training which agent behaviors are safe and useful, but this approach can fail if the task is too complicated for a human to directly judge. To help address this concern, we propose training agents via self play on a zero sum debate game. Given a question or proposed action, two agents take turns making short statements up to a limit, then a human judges which of the agents gave the most true, useful information. In an analogy to complexity theory, debate with optimal play can answer any question in PSPACE given polynomial time judges (direct judging answers only NP questions). In practice, whether debate works involves empirical questions about humans and the tasks we want AIs to perform, plus theoretical questions about the meaning of AI alignment. We report results on an initial MNIST experiment where agents compete to convince a sparse classifier, boosting the classifier's accuracy from 59.4% to 88.9% given 6 pixels and from 48.2% to 85.2% given 4 pixels. Finally, we discuss theoretical and practical aspects of the debate model, focusing on potential weaknesses as the model scales up, and we propose future human and computer experiments to test these properties.

Out-of-Distribution (OOD) detection is critical for the reliable operation of open-world intelligent systems. Despite the emergence of an increasing number of OOD detection methods, the evaluation inconsistencies present challenges for tracking the progress in this field. OpenOOD v1 initiated the unification of the OOD detection evaluation but faced limitations in scalability and usability. In response, this paper presents OpenOOD v1.5, a significant improvement from its predecessor that ensures accurate, standardized, and user-friendly evaluation of OOD detection methodologies. Notably, OpenOOD v1.5 extends its evaluation capabilities to large-scale datasets such as ImageNet, investigates full-spectrum OOD detection which is important yet underexplored, and introduces new features including an online leaderboard and an easy-to-use evaluator. This work also contributes in-depth analysis and insights derived from comprehensive experimental results, thereby enriching the knowledge pool of OOD detection methodologies. With these enhancements, OpenOOD v1.5 aims to drive advancements and offer a more robust and comprehensive evaluation benchmark for OOD detection research.

Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over more traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground state properties of gapped Hamiltonians in finite spatial dimensions, after learning from data obtained by measuring other Hamiltonians in the same quantum phase of matter. In contrast, under widely accepted complexity theory assumptions, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases of matter. Our arguments are based on the concept of a classical shadow, a succinct classical description of a many-body quantum state that can be constructed in feasible quantum experiments and be used to predict many properties of the state. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, 2D random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.

We introduce WiCkeD, a simple method to increase the complexity of existing multiple-choice benchmarks by randomly replacing a choice with "None of the above", a method often used in educational tests. We show that WiCkeD can be automatically applied to any existing benchmark, making it more challenging. We apply WiCkeD to 6 popular benchmarks and use it to evaluate 18 open-weight LLMs. The performance of the models drops 12.1 points on average with respect to the original versions of the datasets. When using chain-of-thought on 3 MMLU datasets, the performance drop for the WiCkeD variant is similar to the one observed when using the LLMs directly, showing that WiCkeD is also challenging for models with enhanced reasoning abilities. WiCkeD also uncovers that some models are more sensitive to the extra reasoning required, providing additional information with respect to the original benchmarks. We relase our code and data at https://github.com/ahmedselhady/wicked-benchmarks.

Large language models (LLMs) fine-tuned with reinforcement learning from human feedback (RLHF) have been used in some of the most widely deployed AI models to date, such as OpenAI's ChatGPT, Anthropic's Claude, or Meta's LLaMA-2. While there has been significant work developing these methods, our understanding of the benefits and downsides of each stage in RLHF is still limited. To fill this gap, we present an extensive analysis of how each stage of the process (i.e. supervised fine-tuning (SFT), reward modelling, and RLHF) affects two key properties: out-of-distribution (OOD) generalisation and output diversity. OOD generalisation is crucial given the wide range of real-world scenarios in which these models are being used, while output diversity refers to the model's ability to generate varied outputs and is important for a variety of use cases. We perform our analysis across two base models on both summarisation and instruction following tasks, the latter being highly relevant for current LLM use cases. We find that RLHF generalises better than SFT to new inputs, particularly as the distribution shift between train and test becomes larger. However, RLHF significantly reduces output diversity compared to SFT across a variety of measures, implying a tradeoff in current LLM fine-tuning methods between generalisation and diversity. Our results provide guidance on which fine-tuning method should be used depending on the application, and show that more research is needed to improve the trade-off between generalisation and diversity.

Understanding the origin(s) of life (OoL) is a fundamental challenge for science in the 21st century. Research on OoL spans many disciplines, including chemistry, physics, biology, planetary sciences, computer science, mathematics and philosophy. The sheer number of different scientific perspectives relevant to the problem has resulted in the coexistence of diverse tools, techniques, data, and software in OoL studies. This has made communication between the disciplines relevant to the OoL extremely difficult because the interpretation of data, analyses, or standards of evidence can vary dramatically. Here, we hope to bridge this wide field of study by providing common ground via the consolidation of tools and techniques rather than positing a unifying view on how life emerges. We review the common tools and techniques that have been used significantly in OoL studies in recent years. In particular, we aim to identify which information is most relevant for comparing and integrating the results of experimental analyses into mathematical and computational models. This review aims to provide a baseline expectation and understanding of technical aspects of origins research, rather than being a primer on any particular topic. As such, it spans broadly -- from analytical chemistry to mathematical models -- and highlights areas of future work that will benefit from a multidisciplinary approach to tackling the mystery of life's origin. Ultimately, we hope to empower a new generation of OoL scientists by reviewing how they can investigate life's origin, rather than dictating how to think about the problem.

We study multi-agent reinforcement learning in the setting of episodic Markov decision processes, where multiple agents cooperate via communication through a central server. We propose a provably efficient algorithm based on value iteration that enable asynchronous communication while ensuring the advantage of cooperation with low communication overhead. With linear function approximation, we prove that our algorithm enjoys an mathcal{O}(d^{3/2}H^2K) regret with mathcal{O}(dHM^2) communication complexity, where d is the feature dimension, H is the horizon length, M is the total number of agents, and K is the total number of episodes. We also provide a lower bound showing that a minimal Omega(dM) communication complexity is required to improve the performance through collaboration.

Out-of-distribution (OOD) detection is critical to ensuring the reliability and safety of machine learning systems. For instance, in autonomous driving, we would like the driving system to issue an alert and hand over the control to humans when it detects unusual scenes or objects that it has never seen during training time and cannot make a safe decision. The term, OOD detection, first emerged in 2017 and since then has received increasing attention from the research community, leading to a plethora of methods developed, ranging from classification-based to density-based to distance-based ones. Meanwhile, several other problems, including anomaly detection (AD), novelty detection (ND), open set recognition (OSR), and outlier detection (OD), are closely related to OOD detection in terms of motivation and methodology. Despite common goals, these topics develop in isolation, and their subtle differences in definition and problem setting often confuse readers and practitioners. In this survey, we first present a unified framework called generalized OOD detection, which encompasses the five aforementioned problems, i.e., AD, ND, OSR, OOD detection, and OD. Under our framework, these five problems can be seen as special cases or sub-tasks, and are easier to distinguish. We then review each of these five areas by summarizing their recent technical developments, with a special focus on OOD detection methodologies. We conclude this survey with open challenges and potential research directions.

The generation of sustained, open-ended complexity from local interactions remains a fundamental challenge in artificial life. Differentiable multi-agent systems, such as Petri Dish Neural Cellular Automata (PD-NCA), exhibit rich self-organization driven purely by spatial competition; however, they are highly sensitive to hyperparameters and frequently collapse into uninteresting patterns and dynamics, such as frozen equilibria or structureless noise. In this paper, we introduce PBT-NCA, a meta-evolutionary algorithm that evolves a population of PD-NCAs subject to a composite objective that rewards both historical behavioral novelty and contemporary visual diversity. Driven by this continuous evolutionary pressure, PBT-NCA spontaneously generates a plethora of emergent lifelike phenomena over extended horizons-a hallmark of true open-endedness. Strikingly, the substrate autonomously discovers diverse morphological survival and self-organization strategies. We observe highly regular, coordinated periodic waves; spore-like scattering where homogeneous groups eject cell-like clusters to colonize distant territories; and fluid, shape-shifting macro-structures that migrate across the substrate, maintaining stable outer boundaries that enclose highly active interiors. By actively penalizing monocultures and dead states, PBT-NCA sustains a state of effective complexity that is neither globally ordered nor globally random, operating persistently at the "edge of chaos".

Recent advances in Deep Reinforcement Learning (DRL) have largely focused on improving the performance of agents with the aim of replacing humans in known and well-defined environments. The use of these techniques as a game design tool for video game production, where the aim is instead to create Non-Player Character (NPC) behaviors, has received relatively little attention until recently. Turn-based strategy games like Roguelikes, for example, present unique challenges to DRL. In particular, the categorical nature of their complex game state, composed of many entities with different attributes, requires agents able to learn how to compare and prioritize these entities. Moreover, this complexity often leads to agents that overfit to states seen during training and that are unable to generalize in the face of design changes made during development. In this paper we propose two network architectures which, when combined with a procedural loot generation system, are able to better handle complex categorical state spaces and to mitigate the need for retraining forced by design decisions. The first is based on a dense embedding of the categorical input space that abstracts the discrete observation model and renders trained agents more able to generalize. The second proposed architecture is more general and is based on a Transformer network able to reason relationally about input and input attributes. Our experimental evaluation demonstrates that new agents have better adaptation capacity with respect to a baseline architecture, making this framework more robust to dynamic gameplay changes during development. Based on the results shown in this paper, we believe that these solutions represent a step forward towards making DRL more accessible to the gaming industry.

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

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

Enabling Large Language Models (LLMs) to handle a wider range of complex tasks (e.g., coding, math) has drawn great attention from many researchers. As LLMs continue to evolve, merely increasing the number of model parameters yields diminishing performance improvements and heavy computational costs. Recently, OpenAI's o1 model has shown that inference strategies (i.e., Test-time Compute methods) can also significantly enhance the reasoning capabilities of LLMs. However, the mechanisms behind these methods are still unexplored. In our work, to investigate the reasoning patterns of o1, we compare o1 with existing Test-time Compute methods (BoN, Step-wise BoN, Agent Workflow, and Self-Refine) by using OpenAI's GPT-4o as a backbone on general reasoning benchmarks in three domains (i.e., math, coding, commonsense reasoning). Specifically, first, our experiments show that the o1 model has achieved the best performance on most datasets. Second, as for the methods of searching diverse responses (e.g., BoN), we find the reward models' capability and the search space both limit the upper boundary of these methods. Third, as for the methods that break the problem into many sub-problems, the Agent Workflow has achieved better performance than Step-wise BoN due to the domain-specific system prompt for planning better reasoning processes. Fourth, it is worth mentioning that we have summarized six reasoning patterns of o1, and provided a detailed analysis on several reasoning benchmarks.

Open World Object Detection (OWOD) is a new and challenging computer vision task that bridges the gap between classic object detection (OD) benchmarks and object detection in the real world. In addition to detecting and classifying seen/labeled objects, OWOD algorithms are expected to detect novel/unknown objects - which can be classified and incrementally learned. In standard OD, object proposals not overlapping with a labeled object are automatically classified as background. Therefore, simply applying OD methods to OWOD fails as unknown objects would be predicted as background. The challenge of detecting unknown objects stems from the lack of supervision in distinguishing unknown objects and background object proposals. Previous OWOD methods have attempted to overcome this issue by generating supervision using pseudo-labeling - however, unknown object detection has remained low. Probabilistic/generative models may provide a solution for this challenge. Herein, we introduce a novel probabilistic framework for objectness estimation, where we alternate between probability distribution estimation and objectness likelihood maximization of known objects in the embedded feature space - ultimately allowing us to estimate the objectness probability of different proposals. The resulting Probabilistic Objectness transformer-based open-world detector, PROB, integrates our framework into traditional object detection models, adapting them for the open-world setting. Comprehensive experiments on OWOD benchmarks show that PROB outperforms all existing OWOD methods in both unknown object detection (sim 2times unknown recall) and known object detection (sim 10% mAP). Our code will be made available upon publication at https://github.com/orrzohar/PROB.

Solving complex reasoning tasks may involve visual understanding, domain knowledge retrieval, numerical calculation, and multi-step reasoning. Existing methods augment large language models (LLMs) with external tools but are restricted to specialized domains, limited tool types, or require additional training data. In this paper, we introduce OctoTools, a training-free, user-friendly, and easily extensible open-source agentic framework designed to tackle complex reasoning across diverse domains. OctoTools introduces standardized tool cards to encapsulate tool functionality, a planner for both high-level and low-level planning, and an executor to carry out tool usage. We validate OctoTools' generality across 16 diverse tasks (including MathVista, MMLU-Pro, MedQA, and GAIA-Text), achieving substantial average accuracy gains of 9.3% over GPT-4o. Furthermore, OctoTools outperforms AutoGen, GPT-Functions and LangChain by up to 10.6% when given the same set of tools. Through comprehensive analysis and ablations, OctoTools demonstrates advantages in task planning, effective tool usage, and multi-step problem solving.

It is almost always easier to find an accurate-but-complex model than an accurate-yet-simple model. Finding optimal, sparse, accurate models of various forms (linear models with integer coefficients, decision sets, rule lists, decision trees) is generally NP-hard. We often do not know whether the search for a simpler model will be worthwhile, and thus we do not go to the trouble of searching for one. In this work, we ask an important practical question: can accurate-yet-simple models be proven to exist, or shown likely to exist, before explicitly searching for them? We hypothesize that there is an important reason that simple-yet-accurate models often do exist. This hypothesis is that the size of the Rashomon set is often large, where the Rashomon set is the set of almost-equally-accurate models from a function class. If the Rashomon set is large, it contains numerous accurate models, and perhaps at least one of them is the simple model we desire. In this work, we formally present the Rashomon ratio as a new gauge of simplicity for a learning problem, depending on a function class and a data set. The Rashomon ratio is the ratio of the volume of the set of accurate models to the volume of the hypothesis space, and it is different from standard complexity measures from statistical learning theory. Insight from studying the Rashomon ratio provides an easy way to check whether a simpler model might exist for a problem before finding it, namely whether several different machine learning methods achieve similar performance on the data. In that sense, the Rashomon ratio is a powerful tool for understanding why and when an accurate-yet-simple model might exist. If, as we hypothesize in this work, many real-world data sets admit large Rashomon sets, the implications are vast: it means that simple or interpretable models may often be used for high-stakes decisions without losing accuracy.

We study the task of prompting large-scale language models to perform multi-step reasoning. Existing work shows that when prompted with a chain of thoughts (CoT), sequences of short sentences describing intermediate reasoning steps towards a final answer, large language models can generate new reasoning chains and predict answers for new inputs. A central question is which reasoning examples make the most effective prompts. In this work, we propose complexity-based prompting, a simple and effective example selection scheme for multi-step reasoning. We show that prompts with higher reasoning complexity, i.e., chains with more reasoning steps, achieve substantially better performance on multi-step reasoning tasks over strong baselines. We further extend our complexity-based criteria from prompting (selecting inputs) to decoding (selecting outputs), where we sample multiple reasoning chains from the model, then choose the majority of generated answers from complex reasoning chains (over simple chains). When used to prompt GPT-3 and Codex, our approach substantially improves multi-step reasoning accuracy and achieves new state-of-the-art (SOTA) performance on three math benchmarks (GSM8K, MultiArith, and MathQA) and two BigBenchHard tasks (Date Understanding and Penguins), with an average +5.3 and up to +18 accuracy improvements. Compared with existing example selection schemes like manual tuning or retrieval-based selection, selection based on reasoning complexity is intuitive, easy to implement, and annotation-efficient. Further results demonstrate the robustness of performance gains from complex prompts under format perturbation and distribution shift.

The promise of Large Language Models in automated software engineering is often measured by functional correctness, overlooking the critical issue of long term maintainability. This paper presents a systematic audit of technical debt in AI-generated software, revealing that AI does not eliminate flaws but rather introduces a distinct machine signature of defects. Our multi-scale analysis, spanning single-file algorithmic tasks and complex, agent generated systems, identifies a fundamental Reasoning-Complexity Trade-off: as models become more capable, they generate increasingly bloated and coupled code. This architectural decay is so pronounced that we establish a Volume-Quality Inverse Law, where code volume is a near perfect predictor of structural degradation. Crucially, we demonstrate that neither functional correctness nor detailed prompting mitigates this decay. These findings challenge the current paradigm of prompt-driven generation, reframing the central problem of AI-based software engineering from one of code generation to one of architectural complexity management. We conclude that future progress depends on equipping agents with explicit architectural foresight to ensure the software they build is not just functional, but also maintainable.

Large language models (LLMs) such as GPT-4 sometimes appear to be creative, solving novel tasks often with a few demonstrations in the prompt. These tasks require the models to generalize on distributions different from those from training data -- which is known as out-of-distribution (OOD) generalization. Despite the tremendous success of LLMs, how they approach OOD generalization remains an open and underexplored question. We examine OOD generalization in settings where instances are generated according to hidden rules, including in-context learning with symbolic reasoning. Models are required to infer the hidden rules behind input prompts without any fine-tuning. We empirically examined the training dynamics of Transformers on a synthetic example and conducted extensive experiments on a variety of pretrained LLMs, focusing on a type of components known as induction heads. We found that OOD generalization and composition are tied together -- models can learn rules by composing two self-attention layers, thereby achieving OOD generalization. Furthermore, a shared latent subspace in the embedding (or feature) space acts as a bridge for composition by aligning early layers and later layers, which we refer to as the common bridge representation hypothesis.

True self-evolution requires agents to act as lifelong learners that internalize novel experiences to solve future problems. However, rigorously measuring this foundational capability is hindered by two obstacles: the entanglement of prior knowledge, where ``new'' knowledge may appear in pre-training data, and the entanglement of reasoning complexity, where failures may stem from problem difficulty rather than an inability to recall learned knowledge. We introduce SE-Bench, a diagnostic environment that obfuscates the NumPy library and its API doc into a pseudo-novel package with randomized identifiers. Agents are trained to internalize this package and evaluated on simple coding tasks without access to documentation, yielding a clean setting where tasks are trivial with the new API doc but impossible for base models without it. Our investigation reveals three insights: (1) the Open-Book Paradox, where training with reference documentation inhibits retention, requiring "Closed-Book Training" to force knowledge compression into weights; (2) the RL Gap, where standard RL fails to internalize new knowledge completely due to PPO clipping and negative gradients; and (3) the viability of Self-Play for internalization, proving models can learn from self-generated, noisy tasks when coupled with SFT, but not RL. Overall, SE-Bench establishes a rigorous diagnostic platform for self-evolution with knowledge internalization. Our code and dataset can be found at https://github.com/thunlp/SE-Bench.

We introduce a new type of programming challenge called programming puzzles, as an objective and comprehensive evaluation of program synthesis, and release an open-source dataset of Python Programming Puzzles (P3). Each puzzle is defined by a short Python program f, and the goal is to find an input which makes f return True. The puzzles are objective in that each one is specified entirely by the source code of its verifier f, so evaluating f is all that is needed to test a candidate solution. They do not require an answer key or input/output examples, nor do they depend on natural language understanding. The dataset is comprehensive in that it spans problems of a range of difficulties and domains, ranging from trivial string manipulation problems, to classic programming puzzles (e.g., Tower of Hanoi), to interview/competitive-programming problems (e.g., dynamic programming), to longstanding open problems in algorithms and mathematics (e.g., factoring). We develop baseline enumerative program synthesis, GPT-3 and Codex solvers that are capable of solving puzzles -- even without access to any reference solutions -- by learning from their own past solutions. Codex performs best, solving up to 18% of 397 test problems with a single try and 80% of the problems with 1,000 tries per problem. In a small user study, we find a positive correlation between puzzle-solving performance and coding experience, and between the puzzle difficulty for humans and AI solvers. Therefore, further improvements on P3 could have a significant impact on many program synthesis areas.

Procedures are inherently hierarchical. To "make videos", one may need to "purchase a camera", which in turn may require one to "set a budget". While such hierarchical knowledge is critical for reasoning about complex procedures, most existing work has treated procedures as shallow structures without modeling the parent-child relation. In this work, we attempt to construct an open-domain hierarchical knowledge-base (KB) of procedures based on wikiHow, a website containing more than 110k instructional articles, each documenting the steps to carry out a complex procedure. To this end, we develop a simple and efficient method that links steps (e.g., "purchase a camera") in an article to other articles with similar goals (e.g., "how to choose a camera"), recursively constructing the KB. Our method significantly outperforms several strong baselines according to automatic evaluation, human judgment, and application to downstream tasks such as instructional video retrieval. A demo with partial data can be found at https://wikihow-hierarchy.github.io. The code and the data are at https://github.com/shuyanzhou/wikihow_hierarchy.

We present FIMO, an innovative dataset comprising formal mathematical problem statements sourced from the International Mathematical Olympiad (IMO) Shortlisted Problems. Designed to facilitate advanced automated theorem proving at the IMO level, FIMO is currently tailored for the Lean formal language. It comprises 149 formal problem statements, accompanied by both informal problem descriptions and their corresponding LaTeX-based informal proofs. Through initial experiments involving GPT-4, our findings underscore the existing limitations in current methodologies, indicating a substantial journey ahead before achieving satisfactory IMO-level automated theorem proving outcomes.

Large language models display remarkable capabilities in logical and mathematical reasoning, allowing them to solve complex tasks. Interestingly, these abilities emerge in networks trained on the simple task of next-token prediction. In this work, we present a theoretical framework for studying auto-regressive next-token predictors. We demonstrate that even simple models such as linear next-token predictors, trained on Chain-of-Thought (CoT) data, can approximate any function efficiently computed by a Turing machine. We introduce a new complexity measure -- length complexity -- which measures the number of intermediate tokens in a CoT sequence required to approximate some target function, and analyze the interplay between length complexity and other notions of complexity. Finally, we show experimentally that simple next-token predictors, such as linear networks and shallow Multi-Layer Perceptrons (MLPs), display non-trivial performance on text generation and arithmetic tasks. Our results demonstrate that the power of language models can be attributed, to a great extent, to the auto-regressive next-token training scheme, and not necessarily to a particular choice of architecture.

Out-of-distribution (OOD) generalization is critical for machine learning models deployed in the real world. However, achieving this can be fundamentally challenging, as it requires the ability to learn invariant features across different domains or environments. In this paper, we propose a novel framework HYPO (HYPerspherical OOD generalization) that provably learns domain-invariant representations in a hyperspherical space. In particular, our hyperspherical learning algorithm is guided by intra-class variation and inter-class separation principles -- ensuring that features from the same class (across different training domains) are closely aligned with their class prototypes, while different class prototypes are maximally separated. We further provide theoretical justifications on how our prototypical learning objective improves the OOD generalization bound. Through extensive experiments on challenging OOD benchmarks, we demonstrate that our approach outperforms competitive baselines and achieves superior performance. Code is available at https://github.com/deeplearning-wisc/hypo.

Relational reasoning is the ability to infer relations that jointly bind multiple entities, attributes, or variables. This ability is central to scientific reasoning, but existing evaluations of relational reasoning in large language models often focus on structured inputs such as tables, graphs, or synthetic tasks, and do not isolate the difficulty introduced by higher-arity relational binding. We study this problem through the lens of Relational Complexity (RC), which we define as the minimum number of independent entities or operands that must be simultaneously bound to apply a relation. RC provides a principled way to vary reasoning difficulty while controlling for confounders such as input size, vocabulary, and representational choices. Building on RC, we introduce REL, a generative benchmark framework spanning algebra, chemistry, and biology that varies RC within each domain. Across frontier LLMs, performance degrades consistently and monotonically as RC increases, even when the total number of entities is held fixed. This failure mode persists with increased test-time compute and in-context learning, suggesting a limitation tied to the arity of the required relational binding rather than to insufficient inference steps or lack of exposure to examples. Our results identify a regime of higher-arity reasoning in which current models struggle, and motivate re-examining benchmarks through the lens of relational complexity.

We present a smoothly broken power law functional form (that we refer to as a Broken Neural Scaling Law (BNSL)) that accurately models & extrapolates the scaling behaviors of deep neural networks (i.e. how the evaluation metric of interest varies as amount of compute used for training (or inference), number of model parameters, training dataset size, model input size, number of training steps, or upstream performance varies) for various architectures & for each of various tasks within a large & diverse set of upstream & downstream tasks, in zero-shot, prompted, & finetuned settings. This set includes large-scale vision, language, audio, video, diffusion, generative modeling, multimodal learning, contrastive learning, AI alignment, AI capabilities, robotics, out-of-distribution (OOD) generalization, continual learning, transfer learning, uncertainty estimation / calibration, OOD detection, adversarial robustness, distillation, sparsity, retrieval, quantization, pruning, fairness, molecules, computer programming/coding, math word problems, "emergent phase transitions", arithmetic, supervised learning, unsupervised/self-supervised learning, & reinforcement learning (single agent & multi-agent). When compared to other functional forms for neural scaling, this functional form yields extrapolations of scaling behavior that are considerably more accurate on this set. Moreover, this functional form accurately models & extrapolates scaling behavior that other functional forms are incapable of expressing such as the nonmonotonic transitions present in the scaling behavior of phenomena such as double descent & the delayed, sharp inflection points present in the scaling behavior of tasks such as arithmetic. Lastly, we use this functional form to glean insights about the limit of the predictability of scaling behavior. Code is available at https://github.com/ethancaballero/broken_neural_scaling_laws

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

Solving mathematical problems requires advanced reasoning abilities and presents notable challenges for large language models. Previous works usually synthesize data from proprietary models to augment existing datasets, followed by instruction tuning to achieve top-tier results. However, our analysis of these datasets reveals severe biases towards easy queries, with frequent failures to generate any correct response for the most challenging queries. Hypothesizing that difficult queries are crucial to learn complex reasoning, we propose Difficulty-Aware Rejection Tuning (DART), a method that allocates difficult queries more trials during the synthesis phase, enabling more extensive training on difficult samples. Utilizing DART, we have created new datasets for mathematical problem-solving that focus more on difficult queries and are substantially smaller than previous ones. Remarkably, our synthesis process solely relies on a 7B-sized open-weight model, without reliance on the commonly used proprietary GPT-4. We fine-tune various base models on our datasets ranging from 7B to 70B in size, resulting in a series of strong models called DART-MATH. In comprehensive in-domain and out-of-domain evaluation on 6 mathematical benchmarks, DART-MATH outperforms vanilla rejection tuning significantly, being superior or comparable to previous arts, despite using much smaller datasets and no proprietary models. Furthermore, our results position our synthetic datasets as the most effective and cost-efficient publicly available resources for advancing mathematical problem-solving.

Operations research (OR) is widely deployed to solve critical decision-making problems with complex objectives and constraints, impacting manufacturing, logistics, finance, and healthcare outcomes. While Large Language Models (LLMs) have shown promising results in various domains, their practical application in industry-relevant operations research (OR) problems presents significant challenges and opportunities. Preliminary industrial applications of LLMs for operations research face two critical deployment challenges: 1) Self-correction focuses on code syntax rather than mathematical accuracy, causing costly errors; 2) Complex expert selection creates unpredictable workflows that reduce transparency and increase maintenance costs, making them impractical for time-sensitive business applications. To address these business limitations, we introduce ORMind, a cognitive-inspired framework that enhances optimization through counterfactual reasoning. Our approach emulates human cognition, implementing an end-to-end workflow that systematically transforms requirements into mathematical models and executable solver code. It is currently being tested internally in Lenovo's AI Assistant, with plans to enhance optimization capabilities for both business and consumer customers. Experiments demonstrate that ORMind outperforms existing methods, achieving a 9.5\% improvement on the NL4Opt dataset and a 14.6\% improvement on the ComplexOR dataset.

Recent work synthesizes agentic tasks for post-training tool-using LLMs, yet robust generalization under shifts in tasks and toolsets remains an open challenge. We trace this brittleness to insufficient diversity in synthesized tasks. Scaling diversity is difficult because training requires tasks to remain executable and verifiable, while generalization demands coverage of diverse tool types, toolset combinations, and heterogeneous tool-use patterns. We propose DIVE, an evidence-driven recipe that inverts synthesis order, executing diverse, real-world tools first and reverse-deriving tasks strictly entailed by the resulting traces, thereby providing grounding by construction. DIVE scales structural diversity along two controllable axes, tool-pool coverage and per-task toolset variety, and an Evidence Collection--Task Derivation loop further induces rich multi-step tool-use patterns across 373 tools in five domains. Training Qwen3-8B on DIVE data (48k SFT + 3.2k RL) improves by +22 average points across 9 OOD benchmarks and outperforms the strongest 8B baseline by +68. Remarkably, controlled scaling analysis reveals that diversity scaling consistently outperforms quantity scaling for OOD generalization, even with 4x less data.

Composer 2 is a specialized model designed for agentic software engineering. The model demonstrates strong long-term planning and coding intelligence while maintaining the ability to efficiently solve problems for interactive use. The model is trained in two phases: first, continued pretraining to improve the model's knowledge and latent coding ability, followed by large-scale reinforcement learning to improve end-to-end coding performance through stronger reasoning, accurate multi-step execution, and coherence on long-horizon realistic coding problems. We develop infrastructure to support training in the same Cursor harness that is used by the deployed model, with equivalent tools and structure, and use environments that match real problems closely. To measure the ability of the model on increasingly difficult tasks, we introduce a benchmark derived from real software engineering problems in large codebases including our own. Composer 2 is a frontier-level coding model and demonstrates a process for training strong domain-specialized models. On our CursorBench evaluations the model achieves a major improvement in accuracy compared to previous Composer models (61.3). On public benchmarks the model scores 61.7 on Terminal-Bench and 73.7 on SWE-bench Multilingual in our harness, comparable to state-of-the-art systems.

Modern machine learning algorithms, especially deep learning based techniques, typically involve careful hyperparameter tuning to achieve the best performance. Despite the surge of intense interest in practical techniques like Bayesian optimization and random search based approaches to automating this laborious and compute intensive task, the fundamental learning theoretic complexity of tuning hyperparameters for deep neural networks is poorly understood. Inspired by this glaring gap, we initiate the formal study of hyperparameter tuning complexity in deep learning through a recently introduced data driven setting. We assume that we have a series of deep learning tasks, and we have to tune hyperparameters to do well on average over the distribution of tasks. A major difficulty is that the utility function as a function of the hyperparameter is very volatile and furthermore, it is given implicitly by an optimization problem over the model parameters. To tackle this challenge, we introduce a new technique to characterize the discontinuities and oscillations of the utility function on any fixed problem instance as we vary the hyperparameter; our analysis relies on subtle concepts including tools from differential/algebraic geometry and constrained optimization. This can be used to show that the learning theoretic complexity of the corresponding family of utility functions is bounded. We instantiate our results and provide sample complexity bounds for concrete applications tuning a hyperparameter that interpolates neural activation functions and setting the kernel parameter in graph neural networks.

Feature shaping refers to a family of methods that exhibit state-of-the-art performance for out-of-distribution (OOD) detection. These approaches manipulate the feature representation, typically from the penultimate layer of a pre-trained deep learning model, so as to better differentiate between in-distribution (ID) and OOD samples. However, existing feature-shaping methods usually employ rules manually designed for specific model architectures and OOD datasets, which consequently limit their generalization ability. To address this gap, we first formulate an abstract optimization framework for studying feature-shaping methods. We then propose a concrete reduction of the framework with a simple piecewise constant shaping function and show that existing feature-shaping methods approximate the optimal solution to the concrete optimization problem. Further, assuming that OOD data is inaccessible, we propose a formulation that yields a closed-form solution for the piecewise constant shaping function, utilizing solely the ID data. Through extensive experiments, we show that the feature-shaping function optimized by our method improves the generalization ability of OOD detection across a large variety of datasets and model architectures.

Heuristics are widely used for dealing with complex search and optimization problems. However, manual design of heuristics can be often very labour extensive and requires rich working experience and knowledge. This paper proposes Evolution of Heuristic (EoH), a novel evolutionary paradigm that leverages both Large Language Models (LLMs) and Evolutionary Computation (EC) methods for Automatic Heuristic Design (AHD). EoH represents the ideas of heuristics in natural language, termed thoughts. They are then translated into executable codes by LLMs. The evolution of both thoughts and codes in an evolutionary search framework makes it very effective and efficient for generating high-performance heuristics. Experiments on three widely studied combinatorial optimization benchmark problems demonstrate that EoH outperforms commonly used handcrafted heuristics and other recent AHD methods including FunSearch. Particularly, the heuristic produced by EoH with a low computational budget (in terms of the number of queries to LLMs) significantly outperforms widely-used human hand-crafted baseline algorithms for the online bin packing problem.

We introduce a unified theoretical framework for the rigorous analysis and systematic construction of deep neural networks (DNNs). This framework addresses a gap in existing theory by explicitly modeling the structure of tensor operations -- lower level information that is often abstracted. Our framework enables two novel objectives: (1) analysis of the evolution of architectural complexity over deep learning history, and (2) automatic construction of novel architectures based on new types of tensor operations. Our study of DNNs introduced over the past 40 years reveals a connection between groundbreaking architectures and increases in different types of architectural complexity. Moreover, we identify several large classes of higher complexity architectures that have not yet been explored. We then collect a dataset of 3,000+ higher complexity architectures, which we publicly release at: https://github.com/combinatoriallabs/ArchitecturalComplexity.

Advancing complex reasoning in large language models relies on high-quality, verifiable datasets, yet human annotation remains cost-prohibitive and difficult to scale. Current synthesis paradigms often face a recurring trade-off: maintaining structural validity typically restricts problem complexity, while relaxing constraints to increase difficulty frequently leads to inconsistent or unsolvable instances. To address this, we propose Agentic Proposing, a framework that models problem synthesis as a goal-driven sequential decision process where a specialized agent dynamically selects and composes modular reasoning skills. Through an iterative workflow of internal reflection and tool-use, we develop the Agentic-Proposer-4B using Multi-Granularity Policy Optimization (MGPO) to generate high-precision, verifiable training trajectories across mathematics, coding, and science. Empirical results demonstrate that downstream solvers trained on agent-synthesized data significantly outperform leading baselines and exhibit robust cross-domain generalization. Notably, a 30B solver trained on only 11,000 synthesized trajectories achieves a state-of-the-art 91.6% accuracy on AIME25, rivaling frontier-scale proprietary models such as GPT-5 and proving that a small volume of high-quality synthetic signals can effectively substitute for massive human-curated datasets.

This paper presents a critical examination of current approaches to replicating OpenAI's O1 model capabilities, with particular focus on the widespread but often undisclosed use of knowledge distillation techniques. While our previous work explored the fundamental technical path to O1 replication, this study reveals how simple distillation from O1's API, combined with supervised fine-tuning, can achieve superior performance on complex mathematical reasoning tasks. Through extensive experiments, we show that a base model fine-tuned on simply tens of thousands of samples O1-distilled long-thought chains outperforms O1-preview on the American Invitational Mathematics Examination (AIME) with minimal technical complexity. Moreover, our investigation extends beyond mathematical reasoning to explore the generalization capabilities of O1-distilled models across diverse tasks: hallucination, safety and open-domain QA. Notably, despite training only on mathematical problem-solving data, our models demonstrated strong generalization to open-ended QA tasks and became significantly less susceptible to sycophancy after fine-tuning. We deliberately make this finding public to promote transparency in AI research and to challenge the current trend of obscured technical claims in the field. Our work includes: (1) A detailed technical exposition of the distillation process and its effectiveness, (2) A comprehensive benchmark framework for evaluating and categorizing O1 replication attempts based on their technical transparency and reproducibility, (3) A critical discussion of the limitations and potential risks of over-relying on distillation approaches, our analysis culminates in a crucial bitter lesson: while the pursuit of more capable AI systems is important, the development of researchers grounded in first-principles thinking is paramount.

Advancing automated programming necessitates robust and comprehensive code generation benchmarks, yet current evaluation frameworks largely neglect object-oriented programming (OOP) in favor of functional programming (FP), e.g., HumanEval and MBPP. To address this, our study introduces a pioneering OOP-focused benchmark, featuring 431 Python programs that encompass essential OOP concepts and features like classes and encapsulation methods. We propose a novel evaluation metric, pass@o, tailored for OOP, enhancing traditional pass@k measures. Our evaluation of 23 leading large language models (LLMs), including both general and code-specialized models, reveals three key insights: 1) pass@o offers a more relevant and comprehensive assessment for OOP code generation; 2) Despite excelling in FP, code-specialized LLMs like WizardCoder lag in OOP compared to models like ChatGPT; 3) The poor performance of all advanced LLMs on our OOP benchmark highlights a critical need for improvements in this field. Our benchmark and scripts are publicly released at: https://github.com/alphadl/OOP-eval.

While large language models (LLMs) show promise in scientific discovery, existing research focuses on inference or feedback-driven training, leaving the direct modeling of the generative reasoning process, P(hypothesis|background) (P(h|b)), unexplored. We demonstrate that directly training P(h|b) is mathematically intractable due to the combinatorial complexity (O(N^k)) inherent in retrieving and composing inspirations from a vast knowledge base. To break this barrier, we introduce MOOSE-Star, a unified framework enabling tractable training and scalable inference. In the best case, MOOSE-Star reduces complexity from exponential to logarithmic (O(log N)) by (1) training on decomposed subtasks derived from the probabilistic equation of discovery, (2) employing motivation-guided hierarchical search to enable logarithmic retrieval and prune irrelevant subspaces, and (3) utilizing bounded composition for robustness against retrieval noise. To facilitate this, we release TOMATO-Star, a dataset of 108,717 decomposed papers (38,400 GPU hours) for training. Furthermore, we show that while brute-force sampling hits a ''complexity wall,'' MOOSE-Star exhibits continuous test-time scaling.

Chain-of-thought (CoT), tree-of-thought (ToT), and related techniques work surprisingly well in practice for some complex reasoning tasks with Large Language Models (LLMs), but why? This work seeks the underlying reasons by conducting experimental case studies and linking the performance benefits to well-established sample and computational complexity principles in machine learning. We experimented with 6 reasoning tasks, ranging from grade school math, air travel planning, ..., to Blocksworld. The results suggest that (i) both CoT and ToT benefit significantly from task decomposition, which breaks a complex reasoning task into a sequence of steps with low sample complexity and explicitly outlines the reasoning structure, and (ii) for computationally hard reasoning tasks, the more sophisticated tree structure of ToT outperforms the linear structure of CoT. These findings provide useful guidelines for the use of LLM in solving reasoning tasks in practice.

Advances in Large Language Models (LLMs) have sparked interest in their ability to solve Olympiad-level math problems. However, the training and evaluation of these models are constrained by the limited size and quality of available datasets, as creating large-scale data for such advanced problems requires extensive effort from human experts. In addition, current benchmarks are prone to contamination, leading to unreliable evaluations. In this paper, we present an automated pipeline that leverages the rich resources of the Art of Problem Solving (AoPS) forum, which predominantly features Olympiad-level problems and community-driven solutions. Using open-source LLMs, we develop a method to extract question-answer pairs from the forum, resulting in AoPS-Instruct, a dataset of more than 600,000 high-quality QA pairs. Our experiments demonstrate that fine-tuning LLMs on AoPS-Instruct improves their reasoning abilities across various benchmarks. Moreover, we build an automatic pipeline that introduces LiveAoPSBench, an evolving evaluation set with timestamps, derived from the latest forum data, providing a contamination-resistant benchmark for assessing LLM performance. Notably, we observe a significant decline in LLM performance over time, suggesting their success on older examples may stem from pre-training exposure rather than true reasoning ability. Our work presents a scalable approach to creating and maintaining large-scale, high-quality datasets for advanced math reasoning, offering valuable insights into the capabilities and limitations of LLMs in this domain. Our benchmark and code is available at https://github.com/DSL-Lab/aops

Recent advances in reinforcement learning for large language models have converged on increasing complexity: multi-stage training pipelines, dynamic hyperparameter schedules, and curriculum learning strategies. This raises a fundamental question: Is this complexity necessary? We present JustRL, a minimal approach using single-stage training with fixed hyperparameters that achieves state-of-the-art performance on two 1.5B reasoning models (54.9\% and 64.3\% average accuracy across nine mathematical benchmarks) while using 2times less compute than sophisticated approaches. The same hyperparameters transfer across both models without tuning, and training exhibits smooth, monotonic improvement over 4,000+ steps without the collapses or plateaus that typically motivate interventions. Critically, ablations reveal that adding ``standard tricks'' like explicit length penalties and robust verifiers may degrade performance by collapsing exploration. These results suggest that the field may be adding complexity to solve problems that disappear with a stable, scaled-up baseline. We release our models and code to establish a simple, validated baseline for the community.

We introduce BigO(Bench), a novel coding benchmark designed to evaluate the capabilities of generative language models in understanding and generating code with specified time and space complexities. This benchmark addresses the gap in current evaluations that often overlook the ability of models to comprehend and produce code constrained by computational complexity. BigO(Bench) includes tooling to infer the algorithmic complexity of any Python function from profiling measurements, including human- or LLM-generated solutions. BigO(Bench) also includes of set of 3,105 coding problems and 1,190,250 solutions from Code Contests annotated with inferred (synthetic) time and space complexity labels from the complexity framework, as well as corresponding runtime and memory footprint values for a large set of input sizes. We present results from evaluating multiple state-of-the-art language models on this benchmark, highlighting their strengths and weaknesses in handling complexity requirements. In particular, token-space reasoning models are unrivaled in code generation but not in complexity understanding, hinting that they may not generalize well to tasks for which no reward was given at training time.

In this paper, we investigate which questions are challenging for retrieval-based Question Answering (QA). We (i) propose retrieval complexity (RC), a novel metric conditioned on the completeness of retrieved documents, which measures the difficulty of answering questions, and (ii) propose an unsupervised pipeline to measure RC given an arbitrary retrieval system. Our proposed pipeline measures RC more accurately than alternative estimators, including LLMs, on six challenging QA benchmarks. Further investigation reveals that RC scores strongly correlate with both QA performance and expert judgment across five of the six studied benchmarks, indicating that RC is an effective measure of question difficulty. Subsequent categorization of high-RC questions shows that they span a broad set of question shapes, including multi-hop, compositional, and temporal QA, indicating that RC scores can categorize a new subset of complex questions. Our system can also have a major impact on retrieval-based systems by helping to identify more challenging questions on existing datasets.

Large-language-model code generation has shifted from single-shot prompting to multi-agent orchestrations - analyst, coder, tester, and debugger pipelines - and is evaluated almost exclusively on functional correctness. Whether these architectures also affect the structural complexity of the code they produce, and which orchestration layers carry the cost, remains largely unexamined: prior work has documented prompt-level effects on code complexity, but the architecture-level question is open. We compare six widely-used multi-agent configurations (Basic, AC, ACT, Debugger, AC+Debugger, ACT+Debugger) under two models from the GPT-4o family across all 164 HumanEval tasks - 1,968 paired observations - using the five RADON complexity metrics (SLOC, cyclomatic complexity, and Halstead Volume, Difficulty, and Effort). We apply a paired non-parametric statistical pipeline (Friedman omnibus, Wilcoxon signed-rank post-hoc with Holm correction, Kendall's W and matched-pairs rank-biserial effect sizes) in both all-completions and passing-only conditions. The six architectures collapse into two indistinguishable complexity clusters separated by a 50-130% gap, the same partition in both models and under both conditions; among the architectural layers, the analyst-coder split inflates complexity, the runtime debugger does not - and on the analyst-coder background actively deflates it - and the tester re-inflates it. The heavy cluster's additional complexity buys no pass@1 advantage: the leanest architectures match or beat the heaviest on accuracy. Architectural elaboration in LLM code generation should therefore be justified by measured benefit on the dimensions that matter, not assumed.

This manuscript portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.

We introduce Complex-Edit, a comprehensive benchmark designed to systematically evaluate instruction-based image editing models across instructions of varying complexity. To develop this benchmark, we harness GPT-4o to automatically collect a diverse set of editing instructions at scale. Our approach follows a well-structured ``Chain-of-Edit'' pipeline: we first generate individual atomic editing tasks independently and then integrate them to form cohesive, complex instructions. Additionally, we introduce a suite of metrics to assess various aspects of editing performance, along with a VLM-based auto-evaluation pipeline that supports large-scale assessments. Our benchmark yields several notable insights: 1) Open-source models significantly underperform relative to proprietary, closed-source models, with the performance gap widening as instruction complexity increases; 2) Increased instructional complexity primarily impairs the models' ability to retain key elements from the input images and to preserve the overall aesthetic quality; 3) Decomposing a complex instruction into a sequence of atomic steps, executed in a step-by-step manner, substantially degrades performance across multiple metrics; 4) A straightforward Best-of-N selection strategy improves results for both direct editing and the step-by-step sequential approach; and 5) We observe a ``curse of synthetic data'': when synthetic data is involved in model training, the edited images from such models tend to appear increasingly synthetic as the complexity of the editing instructions rises -- a phenomenon that intriguingly also manifests in the latest GPT-4o outputs.

Identifying Out-of-distribution (OOD) data is becoming increasingly critical as the real-world applications of deep learning methods expand. Post-hoc methods modify softmax scores fine-tuned on outlier data or leverage intermediate feature layers to identify distinctive patterns between In-Distribution (ID) and OOD samples. Other methods focus on employing diverse OOD samples to learn discrepancies between ID and OOD. These techniques, however, are typically dependent on the quality of the outlier samples assumed. Density-based methods explicitly model class-conditioned distributions but this requires long training time or retraining the classifier. To tackle these issues, we introduce FlowCon, a new density-based OOD detection technique. Our main innovation lies in efficiently combining the properties of normalizing flow with supervised contrastive learning, ensuring robust representation learning with tractable density estimation. Empirical evaluation shows the enhanced performance of our method across common vision datasets such as CIFAR-10 and CIFAR-100 pretrained on ResNet18 and WideResNet classifiers. We also perform quantitative analysis using likelihood plots and qualitative visualization using UMAP embeddings and demonstrate the robustness of the proposed method under various OOD contexts. Code will be open-sourced post decision.

We study what actually works and what doesn't for training large language models as agents via multi-turn reinforcement learning. Despite rapid progress, existing frameworks and definitions are fragmented, and there is no systematic formulation or analysis of which design choices matter across tasks. We address this gap by first breaking down the design space into three inter-related pillars -- environment, reward, and policy -- and empirically derive a recipe for training LLM agents in situated textual domains. In particular, we test TextWorld and ALFWorld, popular domains for testing situated embodied reasoning, as well as SWE-Gym for more software engineering style tasks. (i) For the environment, we analyze the impacts of task complexity in terms of sizes of the state and action spaces as well as optimal solution length, finding that even simple environments within a domain can provide signal on how well an agent can generalize to more complex tasks. (ii) For the reward, we ablate relative reward sparsity, observing that while dense turn-level rewards accelerate training, performance and stability is highly dependent on the choice of RL algorithm. (iii) And for the agent's policy, we explore the interplay between reward sparsity and biased (PPO, GRPO) and unbiased (RLOO) policy gradient methods in addition to showing how to find the optimal Supervised Fine-tuning (SFT) to RL training ratio given a fixed budget. We distill these findings into a training recipe that guides co-design across the three pillars, facilitating research and practical efforts in multi-turn agentic RL. Code: https://github.com/pearls-lab/meow-tea-taro