Daily Papers

In this paper we present a Java-to-Python (J2P) and Python-to-Java (P2J) back-to-back code translation method, and associated tool called CoTran, based on large language models (LLMs). Our method leverages the attention mechanism of LLMs, compilation, and symbolic execution-based test generation for equivalence testing between the input and output programs. More precisely, we modify the typical LLM training loop to incorporate compiler and symbolic execution loss. Via extensive experiments comparing CoTran with 10 other transpilers and LLM-based translation tools over a benchmark of more than 57,000 Java-Python equivalent pairs, we show that CoTran outperforms them on relevant metrics such as compilation and runtime equivalence accuracy. For example, our tool gets 97.43% compilation accuracy and 49.66% runtime equivalence accuracy for J2P translation, whereas the nearest competing tool only gets 96.44% and 6.8% respectively.

Music Emotion Recognition involves the automatic identification of emotional elements within music tracks, and it has garnered significant attention due to its broad applicability in the field of Music Information Retrieval. It can also be used as the upstream task of many other human-related tasks such as emotional music generation and music recommendation. Due to existing psychology research, music emotion is determined by multiple factors such as the Timbre, Velocity, and Structure of the music. Incorporating multiple factors in MER helps achieve more interpretable and finer-grained methods. However, most prior works were uni-domain and showed weak consistency between arousal modeling performance and valence modeling performance. Based on this background, we designed a multi-domain emotion modeling method for instrumental music that combines symbolic analysis and acoustic analysis. At the same time, because of the rarity of music data and the difficulty of labeling, our multi-domain approach can make full use of limited data. Our approach was implemented and assessed using the publicly available piano dataset EMOPIA, resulting in a notable improvement over our baseline model with a 2.4% increase in overall accuracy, establishing its state-of-the-art performance.

The correctness of computations remains a significant challenge in computer science, with traditional approaches relying on automated testing or formal verification. Self-testing/correcting programs introduce an alternative paradigm, allowing a program to verify and correct its own outputs via randomized reductions, a concept that previously required manual derivation. In this paper, we present Bitween, a method and tool for automated learning of randomized (self)-reductions and program properties in numerical programs. Bitween combines symbolic analysis and machine learning, with a surprising finding: polynomial-time linear regression, a basic optimization method, is not only sufficient but also highly effective for deriving complex randomized self-reductions and program invariants, often outperforming sophisticated mixed-integer linear programming solvers. We establish a theoretical framework for learning these reductions and introduce RSR-Bench, a benchmark suite for evaluating Bitween's capabilities on scientific and machine learning functions. Our empirical results show that Bitween surpasses state-of-the-art tools in scalability, stability, and sample efficiency when evaluated on nonlinear invariant benchmarks like NLA-DigBench. Bitween is open-source as a Python package and accessible via a web interface that supports C language programs.

Open-ended learning benefits immensely from the use of symbolic methods for goal representation as they offer ways to structure knowledge for efficient and transferable learning. However, the existing Hierarchical Reinforcement Learning (HRL) approaches relying on symbolic reasoning are often limited as they require a manual goal representation. The challenge in autonomously discovering a symbolic goal representation is that it must preserve critical information, such as the environment dynamics. In this paper, we propose a developmental mechanism for goal discovery via an emergent representation that abstracts (i.e., groups together) sets of environment states that have similar roles in the task. We introduce a Feudal HRL algorithm that concurrently learns both the goal representation and a hierarchical policy. The algorithm uses symbolic reachability analysis for neural networks to approximate the transition relation among sets of states and to refine the goal representation. We evaluate our approach on complex navigation tasks, showing the learned representation is interpretable, transferrable and results in data efficient learning.

Roman Numeral analysis is the important task of identifying chords and their functional context in pieces of tonal music. This paper presents a new approach to automatic Roman Numeral analysis in symbolic music. While existing techniques rely on an intermediate lossy representation of the score, we propose a new method based on Graph Neural Networks (GNNs) that enable the direct description and processing of each individual note in the score. The proposed architecture can leverage notewise features and interdependencies between notes but yield onset-wise representation by virtue of our novel edge contraction algorithm. Our results demonstrate that ChordGNN outperforms existing state-of-the-art models, achieving higher accuracy in Roman Numeral analysis on the reference datasets. In addition, we investigate variants of our model using proposed techniques such as NADE, and post-processing of the chord predictions. The full source code for this work is available at https://github.com/manoskary/chordgnn

In symbolic regression, the goal is to find an analytical expression that accurately fits experimental data with the minimal use of mathematical symbols such as operators, variables, and constants. However, the combinatorial space of possible expressions can make it challenging for traditional evolutionary algorithms to find the correct expression in a reasonable amount of time. To address this issue, Neural Symbolic Regression (NSR) algorithms have been developed that can quickly identify patterns in the data and generate analytical expressions. However, these methods, in their current form, lack the capability to incorporate user-defined prior knowledge, which is often required in natural sciences and engineering fields. To overcome this limitation, we propose a novel neural symbolic regression method, named Neural Symbolic Regression with Hypothesis (NSRwH) that enables the explicit incorporation of assumptions about the expected structure of the ground-truth expression into the prediction process. Our experiments demonstrate that the proposed conditioned deep learning model outperforms its unconditioned counterparts in terms of accuracy while also providing control over the predicted expression structure.

Symbolic regression (SR) aims to discover closed-form mathematical expressions that accurately describe data, offering interpretability and analytical insight beyond standard black-box models. Existing SR methods often rely on population-based search or autoregressive modeling, which struggle with scalability and symbolic consistency. We introduce LIES (Logarithm, Identity, Exponential, Sine), a fixed neural network architecture with interpretable primitive activations that are optimized to model symbolic expressions. We develop a framework to extract compact formulae from LIES networks by training with an appropriate oversampling strategy and a tailored loss function to promote sparsity and to prevent gradient instability. After training, it applies additional pruning strategies to further simplify the learned expressions into compact formulae. Our experiments on SR benchmarks show that the LIES framework consistently produces sparse and accurate symbolic formulae outperforming all baselines. We also demonstrate the importance of each design component through ablation studies.

Hyperdimensional computing (HDC) is a biologically-inspired framework which represents symbols with high-dimensional vectors, and uses vector operations to manipulate them. The ensemble of a particular vector space and a prescribed set of vector operations (including one addition-like for "bundling" and one outer-product-like for "binding") form a *vector symbolic architecture* (VSA). While VSAs have been employed in numerous applications and have been studied empirically, many theoretical questions about VSAs remain open. We analyze the *representation capacities* of four common VSAs: MAP-I, MAP-B, and two VSAs based on sparse binary vectors. "Representation capacity' here refers to bounds on the dimensions of the VSA vectors required to perform certain symbolic tasks, such as testing for set membership i in S and estimating set intersection sizes |X cap Y| for two sets of symbols X and Y, to a given degree of accuracy. We also analyze the ability of a novel variant of a Hopfield network (a simple model of associative memory) to perform some of the same tasks that are typically asked of VSAs. In addition to providing new bounds on VSA capacities, our analyses establish and leverage connections between VSAs, "sketching" (dimensionality reduction) algorithms, and Bloom filters.

Neurosymbolic (NeSy) frameworks combine neural representations and learning with symbolic representations and reasoning. Combining the reasoning capacities, explainability, and interpretability of symbolic processing with the flexibility and power of neural computing allows us to solve complex problems with more reliability while being data-efficient. However, this recently growing topic poses a challenge to developers with its learning curve, lack of user-friendly tools, libraries, and unifying frameworks. In this paper, we characterize the technical facets of existing NeSy frameworks, such as the symbolic representation language, integration with neural models, and the underlying algorithms. A majority of the NeSy research focuses on algorithms instead of providing generic frameworks for declarative problem specification to leverage problem solving. To highlight the key aspects of Neurosymbolic modeling, we showcase three generic NeSy frameworks - DeepProbLog, Scallop, and DomiKnowS. We identify the challenges within each facet that lay the foundation for identifying the expressivity of each framework in solving a variety of problems. Building on this foundation, we aim to spark transformative action and encourage the community to rethink this problem in novel ways.

Transformers demonstrate impressive performance on a range of reasoning benchmarks. To evaluate the degree to which these abilities are a result of actual reasoning, existing work has focused on developing sophisticated benchmarks for behavioral studies. However, these studies do not provide insights into the internal mechanisms driving the observed capabilities. To improve our understanding of the internal mechanisms of transformers, we present a comprehensive mechanistic analysis of a transformer trained on a synthetic reasoning task. We identify a set of interpretable mechanisms the model uses to solve the task, and validate our findings using correlational and causal evidence. Our results suggest that it implements a depth-bounded recurrent mechanisms that operates in parallel and stores intermediate results in selected token positions. We anticipate that the motifs we identified in our synthetic setting can provide valuable insights into the broader operating principles of transformers and thus provide a basis for understanding more complex models.

Symbolic regression plays a crucial role in modern scientific research thanks to its capability of discovering concise and interpretable mathematical expressions from data. A grand challenge lies in the arduous search for parsimonious and generalizable mathematical formulas, in an infinite search space, while intending to fit the training data. Existing algorithms have faced a critical bottleneck of accuracy and efficiency over a decade when handling problems of complexity, which essentially hinders the pace of applying symbolic regression for scientific exploration across interdisciplinary domains. To this end, we introduce a parallelized tree search (PTS) model to efficiently distill generic mathematical expressions from limited data. Through a series of extensive experiments, we demonstrate the superior accuracy and efficiency of PTS for equation discovery, which greatly outperforms the state-of-the-art baseline models on over 80 synthetic and experimental datasets (e.g., lifting its performance by up to 99% accuracy improvement and one-order of magnitude speed up). PTS represents a key advance in accurate and efficient data-driven discovery of symbolic, interpretable models (e.g., underlying physical laws) and marks a pivotal transition towards scalable symbolic learning.

Symbolic equations are at the core of scientific discovery. The task of discovering the underlying equation from a set of input-output pairs is called symbolic regression. Traditionally, symbolic regression methods use hand-designed strategies that do not improve with experience. In this paper, we introduce the first symbolic regression method that leverages large scale pre-training. We procedurally generate an unbounded set of equations, and simultaneously pre-train a Transformer to predict the symbolic equation from a corresponding set of input-output-pairs. At test time, we query the model on a new set of points and use its output to guide the search for the equation. We show empirically that this approach can re-discover a set of well-known physical equations, and that it improves over time with more data and compute.

Symbolic regression is the task of identifying a mathematical expression that best fits a provided dataset of input and output values. Due to the richness of the space of mathematical expressions, symbolic regression is generally a challenging problem. While conventional approaches based on genetic evolution algorithms have been used for decades, deep learning-based methods are relatively new and an active research area. In this work, we present SymbolicGPT, a novel transformer-based language model for symbolic regression. This model exploits the advantages of probabilistic language models like GPT, including strength in performance and flexibility. Through comprehensive experiments, we show that our model performs strongly compared to competing models with respect to the accuracy, running time, and data efficiency.

Symbolic regression (SR) is a powerful technique for discovering the underlying mathematical expressions from observed data. Inspired by the success of deep learning, recent efforts have focused on two categories for SR methods. One is using a neural network or genetic programming to search the expression tree directly. Although this has shown promising results, the large search space poses difficulties in learning constant factors and processing high-dimensional problems. Another approach is leveraging a transformer-based model training on synthetic data and offers advantages in inference speed. However, this method is limited to fixed small numbers of dimensions and may encounter inference problems when given data is out-of-distribution compared to the synthetic data. In this work, we propose DySymNet, a novel neural-guided Dynamic Symbolic Network for SR. Instead of searching for expressions within a large search space, we explore DySymNet with various structures and optimize them to identify expressions that better-fitting the data. With a topology structure like neural networks, DySymNet not only tackles the challenge of high-dimensional problems but also proves effective in optimizing constants. Based on extensive numerical experiments using low-dimensional public standard benchmarks and the well-known SRBench with more variables, our method achieves state-of-the-art performance in terms of fitting accuracy and robustness to noise.

Symbolic regression (SR) is a challenging task in machine learning that involves finding a mathematical expression for a function based on its values. Recent advancements in SR have demonstrated the effectiveness of pretrained transformer-based models in generating equations as sequences, leveraging large-scale pretraining on synthetic datasets and offering notable advantages in terms of inference time over GP-based methods. However, these models primarily rely on supervised pretraining goals borrowed from text generation and overlook equation-specific objectives like accuracy and complexity. To address this, we propose TPSR, a Transformer-based Planning strategy for Symbolic Regression that incorporates Monte Carlo Tree Search into the transformer decoding process. Unlike conventional decoding strategies, TPSR enables the integration of non-differentiable feedback, such as fitting accuracy and complexity, as external sources of knowledge into the transformer-based equation generation process. Extensive experiments on various datasets show that our approach outperforms state-of-the-art methods, enhancing the model's fitting-complexity trade-off, extrapolation abilities, and robustness to noise

We present a method to formulate algorithm discovery as program search, and apply it to discover optimization algorithms for deep neural network training. We leverage efficient search techniques to explore an infinite and sparse program space. To bridge the large generalization gap between proxy and target tasks, we also introduce program selection and simplification strategies. Our method discovers a simple and effective optimization algorithm, Lion (Evo\textbf{Lved Sign Momentum}). It is more memory-efficient than Adam as it only keeps track of the momentum. Different from adaptive optimizers, its update has the same magnitude for each parameter calculated through the sign operation. We compare Lion with widely used optimizers, such as Adam and Adafactor, for training a variety of models on different tasks. On image classification, Lion boosts the accuracy of ViT by up to 2% on ImageNet and saves up to 5x the pre-training compute on JFT. On vision-language contrastive learning, we achieve 88.3% zero-shot and 91.1% fine-tuning accuracy on ImageNet, surpassing the previous best results by 2% and 0.1%, respectively. On diffusion models, Lion outperforms Adam by achieving a better FID score and reducing the training compute by up to 2.3x. For autoregressive, masked language modeling, and fine-tuning, Lion exhibits a similar or better performance compared to Adam. Our analysis of Lion reveals that its performance gain grows with the training batch size. It also requires a smaller learning rate than Adam due to the larger norm of the update produced by the sign function. Additionally, we examine the limitations of Lion and identify scenarios where its improvements are small or not statistically significant. The implementation of Lion is publicly available.

Recent advancements have highlighted that Large Language Models (LLMs) are prone to hallucinations when solving complex reasoning problems, leading to erroneous results. To tackle this issue, researchers incorporate Knowledge Graphs (KGs) to improve the reasoning ability of LLMs. However, existing methods face two limitations: 1) they typically assume that all answers to the questions are contained in KGs, neglecting the incompleteness issue of KGs, and 2) they treat the KG as a static repository and overlook the implicit logical reasoning structures inherent in KGs. In this paper, we introduce SymAgent, an innovative neural-symbolic agent framework that achieves collaborative augmentation between KGs and LLMs. We conceptualize KGs as dynamic environments and transform complex reasoning tasks into a multi-step interactive process, enabling KGs to participate deeply in the reasoning process. SymAgent consists of two modules: Agent-Planner and Agent-Executor. The Agent-Planner leverages LLM's inductive reasoning capability to extract symbolic rules from KGs, guiding efficient question decomposition. The Agent-Executor autonomously invokes predefined action tools to integrate information from KGs and external documents, addressing the issues of KG incompleteness. Furthermore, we design a self-learning framework comprising online exploration and offline iterative policy updating phases, enabling the agent to automatically synthesize reasoning trajectories and improve performance. Experimental results demonstrate that SymAgent with weak LLM backbones (i.e., 7B series) yields better or comparable performance compared to various strong baselines. Further analysis reveals that our agent can identify missing triples, facilitating automatic KG updates.

Large language models (LLMs) are rapidly approaching the level of proficiency in university-level symbolic mathematics required for applications in advanced science and technology. However, existing benchmarks fall short in assessing the core skills of LLMs in symbolic mathematics-such as integration, differential equations, and algebraic simplification. To address this gap, we introduce ASyMOB, a novel assessment framework focused exclusively on symbolic manipulation, featuring 17,092 unique math challenges, organized by similarity and complexity. ASyMOB enables analysis of LLM generalization capabilities by comparing performance in problems that differ by simple numerical or symbolic `perturbations'. Evaluated LLMs exhibit substantial degradation in performance for all perturbation types (up to -70.3%), suggesting reliance on memorized patterns rather than deeper understanding of symbolic math, even among models achieving high baseline accuracy. Comparing LLM performance to computer algebra systems, we identify examples where they fail while LLMs succeed, as well as problems solved only by combining both approaches. Models capable of integrated code execution yielded higher accuracy compared to their performance without code, particularly stabilizing weaker models (up to +33.1% for certain perturbation types). Notably, the most advanced models (o4-mini, Gemini 2.5 Flash) demonstrate not only high symbolic math proficiency (scoring 96.8% and 97.6% on the unperturbed set), but also remarkable robustness against perturbations, (-21.7% and -21.2% vs. average -50.4% for the other models). This may indicate a recent "phase transition" in the generalization capabilities of frontier LLMs. It remains to be seen whether the path forward lies in deeper integration with sophisticated external tools, or in developing models so capable that symbolic math systems like CAS become unnecessary.

Recent advancements in AI safety have led to increased efforts in training and red-teaming large language models (LLMs) to mitigate unsafe content generation. However, these safety mechanisms may not be comprehensive, leaving potential vulnerabilities unexplored. This paper introduces MathPrompt, a novel jailbreaking technique that exploits LLMs' advanced capabilities in symbolic mathematics to bypass their safety mechanisms. By encoding harmful natural language prompts into mathematical problems, we demonstrate a critical vulnerability in current AI safety measures. Our experiments across 13 state-of-the-art LLMs reveal an average attack success rate of 73.6\%, highlighting the inability of existing safety training mechanisms to generalize to mathematically encoded inputs. Analysis of embedding vectors shows a substantial semantic shift between original and encoded prompts, helping explain the attack's success. This work emphasizes the importance of a holistic approach to AI safety, calling for expanded red-teaming efforts to develop robust safeguards across all potential input types and their associated risks.

Software is prone to security vulnerabilities. Program analysis tools to detect them have limited effectiveness in practice due to their reliance on human labeled specifications. Large language models (or LLMs) have shown impressive code generation capabilities but they cannot do complex reasoning over code to detect such vulnerabilities especially since this task requires whole-repository analysis. We propose IRIS, a neuro-symbolic approach that systematically combines LLMs with static analysis to perform whole-repository reasoning for security vulnerability detection. Specifically, IRIS leverages LLMs to infer taint specifications and perform contextual analysis, alleviating needs for human specifications and inspection. For evaluation, we curate a new dataset, CWE-Bench-Java, comprising 120 manually validated security vulnerabilities in real-world Java projects. A state-of-the-art static analysis tool CodeQL detects only 27 of these vulnerabilities whereas IRIS with GPT-4 detects 55 (+28) and improves upon CodeQL's average false discovery rate by 5% points. Furthermore, IRIS identifies 4 previously unknown vulnerabilities which cannot be found by existing tools. IRIS is available publicly at https://github.com/iris-sast/iris.

In an era where symbolic mathematical equations are indispensable for modeling complex natural phenomena, scientific inquiry often involves collecting observations and translating them into mathematical expressions. Recently, deep learning has emerged as a powerful tool for extracting insights from data. However, existing models typically specialize in either numeric or symbolic domains, and are usually trained in a supervised manner tailored to specific tasks. This approach neglects the substantial benefits that could arise from a task-agnostic unified understanding between symbolic equations and their numeric counterparts. To bridge the gap, we introduce SNIP, a Symbolic-Numeric Integrated Pre-training, which employs joint contrastive learning between symbolic and numeric domains, enhancing their mutual similarities in the pre-trained embeddings. By performing latent space analysis, we observe that SNIP provides cross-domain insights into the representations, revealing that symbolic supervision enhances the embeddings of numeric data and vice versa. We evaluate SNIP across diverse tasks, including symbolic-to-numeric mathematical property prediction and numeric-to-symbolic equation discovery, commonly known as symbolic regression. Results show that SNIP effectively transfers to various tasks, consistently outperforming fully supervised baselines and competing strongly with established task-specific methods, especially in few-shot learning scenarios where available data is limited.

In this paper, we present MusPy, an open source Python library for symbolic music generation. MusPy provides easy-to-use tools for essential components in a music generation system, including dataset management, data I/O, data preprocessing and model evaluation. In order to showcase its potential, we present statistical analysis of the eleven datasets currently supported by MusPy. Moreover, we conduct a cross-dataset generalizability experiment by training an autoregressive model on each dataset and measuring held-out likelihood on the others---a process which is made easier by MusPy's dataset management system. The results provide a map of domain overlap between various commonly used datasets and show that some datasets contain more representative cross-genre samples than others. Along with the dataset analysis, these results might serve as a guide for choosing datasets in future research. Source code and documentation are available at https://github.com/salu133445/muspy .

Multimodal symbolic logical reasoning, which aims to deduce new facts from multimodal input via formal logic, is critical in high-stakes applications such as autonomous driving and medical diagnosis, as its rigorous, deterministic reasoning helps prevent serious consequences. To evaluate such capabilities of current state-of-the-art vision language models (VLMs), we introduce the first benchmark MuSLR for multimodal symbolic logical reasoning grounded in formal logical rules. MuSLR comprises 1,093 instances across 7 domains, including 35 atomic symbolic logic and 976 logical combinations, with reasoning depths ranging from 2 to 9. We evaluate 7 state-of-the-art VLMs on MuSLR and find that they all struggle with multimodal symbolic reasoning, with the best model, GPT-4.1, achieving only 46.8%. Thus, we propose LogiCAM, a modular framework that applies formal logical rules to multimodal inputs, boosting GPT-4.1's Chain-of-Thought performance by 14.13%, and delivering even larger gains on complex logics such as first-order logic. We also conduct a comprehensive error analysis, showing that around 70% of failures stem from logical misalignment between modalities, offering key insights to guide future improvements. All data and code are publicly available at https://llm-symbol.github.io/MuSLR.

Complex Query Answering (CQA) over incomplete Knowledge Graphs (KGs) is a challenging task. Recently, a line of message-passing-based research has been proposed to solve CQA. However, they perform unsatisfactorily on negative queries and fail to address the noisy messages between variable nodes in the query graph. Moreover, they offer little interpretability and require complex query data and resource-intensive training. In this paper, we propose a Neural-Symbolic Message Passing (NSMP) framework based on pre-trained neural link predictors. By introducing symbolic reasoning and fuzzy logic, NSMP can generalize to arbitrary existential first order logic queries without requiring training while providing interpretable answers. Furthermore, we introduce a dynamic pruning strategy to filter out noisy messages between variable nodes. Experimental results show that NSMP achieves a strong performance. Additionally, through complexity analysis and empirical verification, we demonstrate the superiority of NSMP in inference time over the current state-of-the-art neural-symbolic method. Compared to this approach, NSMP demonstrates faster inference times across all query types on benchmark datasets, with speedup ranging from 2times to over 150times.

Autonomous robots deployed in the real world will need control policies that rapidly adapt to environmental changes. To this end, we propose AutoRobotics-Zero (ARZ), a method based on AutoML-Zero that discovers zero-shot adaptable policies from scratch. In contrast to neural network adaption policies, where only model parameters are optimized, ARZ can build control algorithms with the full expressive power of a linear register machine. We evolve modular policies that tune their model parameters and alter their inference algorithm on-the-fly to adapt to sudden environmental changes. We demonstrate our method on a realistic simulated quadruped robot, for which we evolve safe control policies that avoid falling when individual limbs suddenly break. This is a challenging task in which two popular neural network baselines fail. Finally, we conduct a detailed analysis of our method on a novel and challenging non-stationary control task dubbed Cataclysmic Cartpole. Results confirm our findings that ARZ is significantly more robust to sudden environmental changes and can build simple, interpretable control policies.

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

We introduce an extensive new dataset of MIDI files, created by transcribing audio recordings of piano performances into their constituent notes. The data pipeline we use is multi-stage, employing a language model to autonomously crawl and score audio recordings from the internet based on their metadata, followed by a stage of pruning and segmentation using an audio classifier. The resulting dataset contains over one million distinct MIDI files, comprising roughly 100,000 hours of transcribed audio. We provide an in-depth analysis of our techniques, offering statistical insights, and investigate the content by extracting metadata tags, which we also provide. Dataset available at https://github.com/loubbrad/aria-midi.

This paper presents a concrete and a symbolic rewriting logic semantics for parametric time Petri nets with inhibitor arcs (PITPNs), a flexible model of timed systems where parameters are allowed in firing bounds. We prove that our semantics is bisimilar to the "standard" semantics of PITPNs. This allows us to use the rewriting logic tool Maude, combined with SMT solving, to provide sound and complete formal analyses for PITPNs. We develop and implement a new general folding approach for symbolic reachability, so that Maude-with-SMT reachability analysis terminates whenever the parametric state-class graph of the PITPN is finite. Our work opens up the possibility of using the many formal analysis capabilities of Maude -- including full LTL model checking, analysis with user-defined analysis strategies, and even statistical model checking -- for such nets. We illustrate this by explaining how almost all formal analysis and parameter synthesis methods supported by the state-of-the-art PITPN tool Romeo can be performed using Maude with SMT. In addition, we also support analysis and parameter synthesis from parametric initial markings, as well as full LTL model checking and analysis with user-defined execution strategies. Experiments show that our methods outperform Romeo in many cases.

This report overviews our ongoing work in enriching chain-of-thoughts datasets requiring arithmetical reasoning with the integration of non-parametric components, such as a calculator. We conduct an analysis of prominent relevant datasets such as GSM8K, Ape210K, AQuA-RAT, and MathQA and propose a machine-processable HTML-like format specifically tailored for working with semi-structured chains. By converting the datasets into this unified format, we enable the effective integration of large language models and symbolic systems, empowering them to tackle arithmetical reasoning tasks more efficiently.

Partitura is a lightweight Python package for handling symbolic musical information. It provides easy access to features commonly used in music information retrieval tasks, like note arrays (lists of timed pitched events) and 2D piano roll matrices, as well as other score elements such as time and key signatures, performance directives, and repeat structures. Partitura can load musical scores (in MEI, MusicXML, Kern, and MIDI formats), MIDI performances, and score-to-performance alignments. The package includes some tools for music analysis, such as automatic pitch spelling, key signature identification, and voice separation. Partitura is an open-source project and is available at https://github.com/CPJKU/partitura/.

One of the primary driving forces contributing to the superior performance of Large Language Models (LLMs) is the extensive availability of human-annotated natural language data, which is used for alignment fine-tuning. This inspired researchers to investigate self-training methods to mitigate the extensive reliance on human annotations. However, the current success of self-training has been primarily observed in natural language scenarios, rather than in the increasingly important neural-symbolic scenarios. To this end, we propose an environment-guided neural-symbolic self-training framework named ENVISIONS. It aims to overcome two main challenges: (1) the scarcity of symbolic data, and (2) the limited proficiency of LLMs in processing symbolic language. Extensive evaluations conducted on three distinct domains demonstrate the effectiveness of our approach. Additionally, we have conducted a comprehensive analysis to uncover the factors contributing to ENVISIONS's success, thereby offering valuable insights for future research in this area. Code will be available at https://github.com/xufangzhi/ENVISIONS.

Symbolic execution is a powerful technique for software testing, but suffers from limitations when encountering external functions, such as native methods or third-party libraries. Existing solutions often require additional context, expensive SMT solvers, or manual intervention to approximate these functions through symbolic stubs. In this work, we propose a novel approach to automatically generate symbolic stubs for external functions during symbolic execution that leverages Genetic Programming. When the symbolic executor encounters an external function, AutoStub generates training data by executing the function on randomly generated inputs and collecting the outputs. Genetic Programming then derives expressions that approximate the behavior of the function, serving as symbolic stubs. These automatically generated stubs allow the symbolic executor to continue the analysis without manual intervention, enabling the exploration of program paths that were previously intractable. We demonstrate that AutoStub can automatically approximate external functions with over 90% accuracy for 55% of the functions evaluated, and can infer language-specific behaviors that reveal edge cases crucial for software testing.

Large multimodal models have demonstrated impressive problem-solving abilities in vision and language tasks, and have the potential to encode extensive world knowledge. However, it remains an open challenge for these models to perceive, reason, plan, and act in realistic environments. In this work, we introduce Can-Do, a benchmark dataset designed to evaluate embodied planning abilities through more diverse and complex scenarios than previous datasets. Our dataset includes 400 multimodal samples, each consisting of natural language user instructions, visual images depicting the environment, state changes, and corresponding action plans. The data encompasses diverse aspects of commonsense knowledge, physical understanding, and safety awareness. Our fine-grained analysis reveals that state-of-the-art models, including GPT-4V, face bottlenecks in visual perception, comprehension, and reasoning abilities. To address these challenges, we propose NeuroGround, a neurosymbolic framework that first grounds the plan generation in the perceived environment states and then leverages symbolic planning engines to augment the model-generated plans. Experimental results demonstrate the effectiveness of our framework compared to strong baselines. Our code and dataset are available at https://embodied-planning.github.io.

Recent years have seen a boom in computational approaches to music analysis, yet each one is typically tailored to a specific analytical domain. In this work, we introduce AnalysisGNN, a novel graph neural network framework that leverages a data-shuffling strategy with a custom weighted multi-task loss and logit fusion between task-specific classifiers to integrate heterogeneously annotated symbolic datasets for comprehensive score analysis. We further integrate a Non-Chord-Tone prediction module, which identifies and excludes passing and non-functional notes from all tasks, thereby improving the consistency of label signals. Experimental evaluations demonstrate that AnalysisGNN achieves performance comparable to traditional static-dataset approaches, while showing increased resilience to domain shifts and annotation inconsistencies across multiple heterogeneous corpora.

This paper explores the integration of two AI subdisciplines employed in the development of artificial agents that exhibit intelligent behavior: Large Language Models (LLMs) and Cognitive Architectures (CAs). We present three integration approaches, each grounded in theoretical models and supported by preliminary empirical evidence. The modular approach, which introduces four models with varying degrees of integration, makes use of chain-of-thought prompting, and draws inspiration from augmented LLMs, the Common Model of Cognition, and the simulation theory of cognition. The agency approach, motivated by the Society of Mind theory and the LIDA cognitive architecture, proposes the formation of agent collections that interact at micro and macro cognitive levels, driven by either LLMs or symbolic components. The neuro-symbolic approach, which takes inspiration from the CLARION cognitive architecture, proposes a model where bottom-up learning extracts symbolic representations from an LLM layer and top-down guidance utilizes symbolic representations to direct prompt engineering in the LLM layer. These approaches aim to harness the strengths of both LLMs and CAs, while mitigating their weaknesses, thereby advancing the development of more robust AI systems. We discuss the tradeoffs and challenges associated with each approach.

Large language models (LLMs) have shown remarkable reasoning capabilities given chain-of-thought prompts (examples with intermediate reasoning steps). Existing benchmarks measure reasoning ability indirectly, by evaluating accuracy on downstream tasks such as mathematical reasoning. However, it is unclear how these models obtain the answers and whether they rely on simple heuristics rather than the generated chain-of-thought. To enable systematic exploration of the reasoning ability of LLMs, we present a new synthetic question-answering dataset called PrOntoQA, where each example is generated from a synthetic world model represented in first-order logic. This allows us to parse the generated chain-of-thought into symbolic proofs for formal analysis. Our analysis on InstructGPT and GPT-3 shows that LLMs are quite capable of making correct individual deduction steps, and so are generally capable of reasoning, even in fictional contexts. However, they have difficulty with proof planning: When multiple valid deduction steps are available, they are not able to systematically explore the different options.

Despite the remarkable success achieved by neural networks, particularly those represented by MLP and Transformer, we reveal that they exhibit potential flaws in the modeling and reasoning of periodicity, i.e., they tend to memorize the periodic data rather than genuinely understanding the underlying principles of periodicity. However, periodicity is a crucial trait in various forms of reasoning and generalization, underpinning predictability across natural and engineered systems through recurring patterns in observations. In this paper, we propose FAN, a novel network architecture based on Fourier Analysis, which empowers the ability to efficiently model and reason about periodic phenomena. By introducing Fourier Series, the periodicity is naturally integrated into the structure and computational processes of the neural network, thus achieving a more accurate expression and prediction of periodic patterns. As a promising substitute to multi-layer perceptron (MLP), FAN can seamlessly replace MLP in various models with fewer parameters and FLOPs. Through extensive experiments, we demonstrate the effectiveness of FAN in modeling and reasoning about periodic functions, and the superiority and generalizability of FAN across a range of real-world tasks, including symbolic formula representation, time series forecasting, and language modeling.

Formal proofs are challenging to write even for experienced experts. Recent progress in Neural Theorem Proving (NTP) shows promise in expediting this process. However, the formal corpora available on the Internet are limited compared to the general text, posing a significant data scarcity challenge for NTP. To address this issue, this work proposes Alchemy, a general framework for data synthesis that constructs formal theorems through symbolic mutation. Specifically, for each candidate theorem in Mathlib, we identify all invocable theorems that can be used to rewrite or apply to it. Subsequently, we mutate the candidate theorem by replacing the corresponding term in the statement with its equivalent form or antecedent. As a result, our method increases the number of theorems in Mathlib by an order of magnitude, from 110k to 6M. Furthermore, we perform continual pretraining and supervised finetuning on this augmented corpus for large language models. Experimental results demonstrate the effectiveness of our approach, achieving a 5% absolute performance improvement on Leandojo benchmark. Additionally, our synthetic data achieve a 2.5% absolute performance gain on the out-of-distribution miniF2F benchmark. To provide further insights, we conduct a comprehensive analysis of synthetic data composition and the training paradigm, offering valuable guidance for developing a strong theorem prover.

The emergent few-shot reasoning capabilities of Large Language Models (LLMs) have excited the natural language and machine learning community over recent years. Despite of numerous successful applications, the underlying mechanism of such in-context capabilities still remains unclear. In this work, we hypothesize that the learned semantics of language tokens do the most heavy lifting during the reasoning process. Different from human's symbolic reasoning process, the semantic representations of LLMs could create strong connections among tokens, thus composing a superficial logical chain. To test our hypothesis, we decouple semantics from the language reasoning process and evaluate three kinds of reasoning abilities, i.e., deduction, induction and abduction. Our findings reveal that semantics play a vital role in LLMs' in-context reasoning -- LLMs perform significantly better when semantics are consistent with commonsense but struggle to solve symbolic or counter-commonsense reasoning tasks by leveraging in-context new knowledge. The surprising observations question whether modern LLMs have mastered the inductive, deductive and abductive reasoning abilities as in human intelligence, and motivate research on unveiling the magic existing within the black-box LLMs. On the whole, our analysis provides a novel perspective on the role of semantics in developing and evaluating language models' reasoning abilities. Code is available at {https://github.com/XiaojuanTang/ICSR}.

We propose a novel symbolic music representation and Generative Adversarial Network (GAN) framework specially designed for symbolic multitrack music generation. The main theme of symbolic music generation primarily encompasses the preprocessing of music data and the implementation of a deep learning framework. Current techniques dedicated to symbolic music generation generally encounter two significant challenges: training data's lack of information about chords and scales and the requirement of specially designed model architecture adapted to the unique format of symbolic music representation. In this paper, we solve the above problems by introducing new symbolic music representation with MusicLang chord analysis model. We propose our MMT-BERT architecture adapting to the representation. To build a robust multitrack music generator, we fine-tune a pre-trained MusicBERT model to serve as the discriminator, and incorporate relativistic standard loss. This approach, supported by the in-depth understanding of symbolic music encoded within MusicBERT, fortifies the consonance and humanity of music generated by our method. Experimental results demonstrate the effectiveness of our approach which strictly follows the state-of-the-art methods.

Recent advances in Multimodal Large Language Models (MLLMs) have demonstrated impressive capabilities across various vision-language tasks. However, their reasoning abilities in the multimodal symbolic music domain remain largely unexplored. We introduce WildScore, the first in-the-wild multimodal symbolic music reasoning and analysis benchmark, designed to evaluate MLLMs' capacity to interpret real-world music scores and answer complex musicological queries. Each instance in WildScore is sourced from genuine musical compositions and accompanied by authentic user-generated questions and discussions, capturing the intricacies of practical music analysis. To facilitate systematic evaluation, we propose a systematic taxonomy, comprising both high-level and fine-grained musicological ontologies. Furthermore, we frame complex music reasoning as multiple-choice question answering, enabling controlled and scalable assessment of MLLMs' symbolic music understanding. Empirical benchmarking of state-of-the-art MLLMs on WildScore reveals intriguing patterns in their visual-symbolic reasoning, uncovering both promising directions and persistent challenges for MLLMs in symbolic music reasoning and analysis. We release the dataset and code.

Radiology report generation (RRG) is a challenging task, as it requires a thorough understanding of medical images, integration of multiple temporal inputs, and accurate report generation. Effective interpretation of medical images, such as chest X-rays (CXRs), demands sophisticated visual-language reasoning to map visual findings to structured reports. Recent studies have shown that multimodal large language models (MLLMs) can acquire multimodal capabilities by aligning with pre-trained vision encoders. However, current approaches predominantly focus on single-image analysis or utilise rule-based symbolic processing to handle multiple images, thereby overlooking the essential temporal information derived from comparing current images with prior ones. To overcome this critical limitation, we introduce Libra, a temporal-aware MLLM tailored for CXR report generation using temporal images. Libra integrates a radiology-specific image encoder with a MLLM and utilises a novel Temporal Alignment Connector to capture and synthesise temporal information of images across different time points with unprecedented precision. Extensive experiments show that Libra achieves new state-of-the-art performance among the same parameter scale MLLMs for RRG tasks on the MIMIC-CXR. Specifically, Libra improves the RadCliQ metric by 12.9% and makes substantial gains across all lexical metrics compared to previous models.

Large Language Models (LLMs) have demonstrated remarkable capabilities across various tasks, yet they often struggle with spatial reasoning. This paper presents a novel neural-symbolic framework that enhances LLMs' spatial reasoning abilities through iterative feedback between LLMs and Answer Set Programming (ASP). We evaluate our approach on two benchmark datasets: StepGame and SparQA, implementing three distinct strategies: (1) direct prompting baseline, (2) Facts+Rules prompting, and (3) DSPy-based LLM+ASP pipeline with iterative refinement. Our experimental results demonstrate that the LLM+ASP pipeline significantly outperforms baseline methods, achieving an average 82% accuracy on StepGame and 69% on SparQA, marking improvements of 40-50% and 8-15% respectively over direct prompting. The success stems from three key innovations: (1) effective separation of semantic parsing and logical reasoning through a modular pipeline, (2) iterative feedback mechanism between LLMs and ASP solvers that improves program rate, and (3) robust error handling that addresses parsing, grounding, and solving failures. Additionally, we propose Facts+Rules as a lightweight alternative that achieves comparable performance on complex SparQA dataset, while reducing computational overhead.Our analysis across different LLM architectures (Deepseek, Llama3-70B, GPT-4.0 mini) demonstrates the framework's generalizability and provides insights into the trade-offs between implementation complexity and reasoning capability, contributing to the development of more interpretable and reliable AI systems.

Progress in the task of symbolic music generation may be lagging behind other tasks like audio and text generation, in part because of the scarcity of symbolic training data. In this paper, we leverage the greater scale of audio music data by applying pre-trained MIR models (for transcription, beat tracking, structure analysis, etc.) to extract symbolic events and encode them into token sequences. To the best of our knowledge, this work is the first to demonstrate the feasibility of training symbolic generation models solely from auto-transcribed audio data. Furthermore, to enhance the controllability of the trained model, we introduce SymPAC (Symbolic Music Language Model with Prompting And Constrained Generation), which is distinguished by using (a) prompt bars in encoding and (b) a technique called Constrained Generation via Finite State Machines (FSMs) during inference time. We show the flexibility and controllability of this approach, which may be critical in making music AI useful to creators and users.

Large Language Models (LLMs) excel in complex reasoning tasks but struggle with consistent rule application, exception handling, and explainability, particularly in domains like legal analysis that require both natural language understanding and precise logical inference. This paper introduces a structured prompting framework that decomposes reasoning into three verifiable steps: entity identification, property extraction, and symbolic rule application. By integrating neural and symbolic approaches, our method leverages LLMs' interpretive flexibility while ensuring logical consistency through formal verification. The framework externalizes task definitions, enabling domain experts to refine logical structures without altering the architecture. Evaluated on the LegalBench hearsay determination task, our approach significantly outperformed baselines, with OpenAI o-family models showing substantial improvements - o1 achieving an F1 score of 0.929 and o3-mini reaching 0.867 using structured decomposition with complementary predicates, compared to their few-shot baselines of 0.714 and 0.74 respectively. This hybrid neural-symbolic system offers a promising pathway for transparent and consistent rule-based reasoning, suggesting potential for explainable AI applications in structured legal reasoning tasks.

Data-driven methods such as reinforcement and imitation learning have achieved remarkable success in robot autonomy. However, their data-centric nature still hinders them from generalizing well to ever-changing environments. Moreover, collecting large datasets for robotic tasks is often impractical and expensive. To overcome these challenges, we introduce a new self-supervised neural-symbolic (NeSy) computational framework, imperative learning (IL), for robot autonomy, leveraging the generalization abilities of symbolic reasoning. The framework of IL consists of three primary components: a neural module, a reasoning engine, and a memory system. We formulate IL as a special bilevel optimization (BLO), which enables reciprocal learning over the three modules. This overcomes the label-intensive obstacles associated with data-driven approaches and takes advantage of symbolic reasoning concerning logical reasoning, physical principles, geometric analysis, etc. We discuss several optimization techniques for IL and verify their effectiveness in five distinct robot autonomy tasks including path planning, rule induction, optimal control, visual odometry, and multi-robot routing. Through various experiments, we show that IL can significantly enhance robot autonomy capabilities and we anticipate that it will catalyze further research across diverse domains.

Engineering design operates through hierarchical abstraction from system specifications to component implementations, requiring visual understanding coupled with mathematical reasoning at each level. While Multi-modal Large Language Models (MLLMs) excel at natural image tasks, their ability to extract mathematical models from technical diagrams remains unexplored. We present CircuitSense, a comprehensive benchmark evaluating circuit understanding across this hierarchy through 8,006+ problems spanning component-level schematics to system-level block diagrams. Our benchmark uniquely examines the complete engineering workflow: Perception, Analysis, and Design, with a particular emphasis on the critical but underexplored capability of deriving symbolic equations from visual inputs. We introduce a hierarchical synthetic generation pipeline consisting of a grid-based schematic generator and a block diagram generator with auto-derived symbolic equation labels. Comprehensive evaluation of six state-of-the-art MLLMs, including both closed-source and open-source models, reveals fundamental limitations in visual-to-mathematical reasoning. Closed-source models achieve over 85\% accuracy on perception tasks involving component recognition and topology identification, yet their performance on symbolic derivation and analytical reasoning falls below 19\%, exposing a critical gap between visual parsing and symbolic reasoning. Models with stronger symbolic reasoning capabilities consistently achieve higher design task accuracy, confirming the fundamental role of mathematical understanding in circuit synthesis and establishing symbolic reasoning as the key metric for engineering competence.

Dynamical systems (DS) theory is fundamental for many areas of science and engineering. It can provide deep insights into the behavior of systems evolving in time, as typically described by differential or recursive equations. A common approach to facilitate mathematical tractability and interpretability of DS models involves decomposing nonlinear DS into multiple linear DS separated by switching manifolds, i.e. piecewise linear (PWL) systems. PWL models are popular in engineering and a frequent choice in mathematics for analyzing the topological properties of DS. However, hand-crafting such models is tedious and only possible for very low-dimensional scenarios, while inferring them from data usually gives rise to unnecessarily complex representations with very many linear subregions. Here we introduce Almost-Linear Recurrent Neural Networks (AL-RNNs) which automatically and robustly produce most parsimonious PWL representations of DS from time series data, using as few PWL nonlinearities as possible. AL-RNNs can be efficiently trained with any SOTA algorithm for dynamical systems reconstruction (DSR), and naturally give rise to a symbolic encoding of the underlying DS that provably preserves important topological properties. We show that for the Lorenz and R\"ossler systems, AL-RNNs discover, in a purely data-driven way, the known topologically minimal PWL representations of the corresponding chaotic attractors. We further illustrate on two challenging empirical datasets that interpretable symbolic encodings of the dynamics can be achieved, tremendously facilitating mathematical and computational analysis of the underlying systems.

Chain-of-thought (CoT) via prompting is the de facto method for eliciting reasoning capabilities from large language models (LLMs). But for what kinds of tasks is this extra ``thinking'' really helpful? To analyze this, we conducted a quantitative meta-analysis covering over 100 papers using CoT and ran our own evaluations of 20 datasets across 14 models. Our results show that CoT gives strong performance benefits primarily on tasks involving math or logic, with much smaller gains on other types of tasks. On MMLU, directly generating the answer without CoT leads to almost identical accuracy as CoT unless the question or model's response contains an equals sign, indicating symbolic operations and reasoning. Following this finding, we analyze the behavior of CoT on these problems by separating planning and execution and comparing against tool-augmented LLMs. Much of CoT's gain comes from improving symbolic execution, but it underperforms relative to using a symbolic solver. Our results indicate that CoT can be applied selectively, maintaining performance while saving inference costs. Furthermore, they suggest a need to move beyond prompt-based CoT to new paradigms that better leverage intermediate computation across the whole range of LLM applications.

Neuro-symbolic artificial intelligence (NSAI) represents a transformative approach in artificial intelligence (AI) by combining deep learning's ability to handle large-scale and unstructured data with the structured reasoning of symbolic methods. By leveraging their complementary strengths, NSAI enhances generalization, reasoning, and scalability while addressing key challenges such as transparency and data efficiency. This paper systematically studies diverse NSAI architectures, highlighting their unique approaches to integrating neural and symbolic components. It examines the alignment of contemporary AI techniques such as retrieval-augmented generation, graph neural networks, reinforcement learning, and multi-agent systems with NSAI paradigms. This study then evaluates these architectures against comprehensive set of criteria, including generalization, reasoning capabilities, transferability, and interpretability, therefore providing a comparative analysis of their respective strengths and limitations. Notably, the Neuro > Symbolic < Neuro model consistently outperforms its counterparts across all evaluation metrics. This result aligns with state-of-the-art research that highlight the efficacy of such architectures in harnessing advanced technologies like multi-agent systems.

PySR is an open-source library for practical symbolic regression, a type of machine learning which aims to discover human-interpretable symbolic models. PySR was developed to democratize and popularize symbolic regression for the sciences, and is built on a high-performance distributed back-end, a flexible search algorithm, and interfaces with several deep learning packages. PySR's internal search algorithm is a multi-population evolutionary algorithm, which consists of a unique evolve-simplify-optimize loop, designed for optimization of unknown scalar constants in newly-discovered empirical expressions. PySR's backend is the extremely optimized Julia library SymbolicRegression.jl, which can be used directly from Julia. It is capable of fusing user-defined operators into SIMD kernels at runtime, performing automatic differentiation, and distributing populations of expressions to thousands of cores across a cluster. In describing this software, we also introduce a new benchmark, "EmpiricalBench," to quantify the applicability of symbolic regression algorithms in science. This benchmark measures recovery of historical empirical equations from original and synthetic datasets.

As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. We demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We showcase an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstrate how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation.

In nature, the behaviors of many complex systems can be described by parsimonious math equations. Automatically distilling these equations from limited data is cast as a symbolic regression process which hitherto remains a grand challenge. Keen efforts in recent years have been placed on tackling this issue and demonstrated success in symbolic regression. However, there still exist bottlenecks that current methods struggle to break when the discrete search space tends toward infinity and especially when the underlying math formula is intricate. To this end, we propose a novel Reinforcement Symbolic Regression Machine (RSRM) that masters the capability of uncovering complex math equations from only scarce data. The RSRM model is composed of three key modules: (1) a Monte Carlo tree search (MCTS) agent that explores optimal math expression trees consisting of pre-defined math operators and variables, (2) a Double Q-learning block that helps reduce the feasible search space of MCTS via properly understanding the distribution of reward, and (3) a modulated sub-tree discovery block that heuristically learns and defines new math operators to improve representation ability of math expression trees. Biding of these modules yields the state-of-the-art performance of RSRM in symbolic regression as demonstrated by multiple sets of benchmark examples. The RSRM model shows clear superiority over several representative baseline models.

We develop a general approach to distill symbolic representations of a learned deep model by introducing strong inductive biases. We focus on Graph Neural Networks (GNNs). The technique works as follows: we first encourage sparse latent representations when we train a GNN in a supervised setting, then we apply symbolic regression to components of the learned model to extract explicit physical relations. We find the correct known equations, including force laws and Hamiltonians, can be extracted from the neural network. We then apply our method to a non-trivial cosmology example-a detailed dark matter simulation-and discover a new analytic formula which can predict the concentration of dark matter from the mass distribution of nearby cosmic structures. The symbolic expressions extracted from the GNN using our technique also generalized to out-of-distribution data better than the GNN itself. Our approach offers alternative directions for interpreting neural networks and discovering novel physical principles from the representations they learn.

Neural networks have a reputation for being better at solving statistical or approximate problems than at performing calculations or working with symbolic data. In this paper, we show that they can be surprisingly good at more elaborated tasks in mathematics, such as symbolic integration and solving differential equations. We propose a syntax for representing mathematical problems, and methods for generating large datasets that can be used to train sequence-to-sequence models. We achieve results that outperform commercial Computer Algebra Systems such as Matlab or Mathematica.

Mathematical problem-solving is a key field in artificial intelligence (AI) and a critical benchmark for evaluating the capabilities of large language models (LLMs). While extensive research has focused on mathematical problem-solving, most existing work and datasets concentrate on computational tasks, leaving gaps in areas like mathematical analysis, which demands rigorous proofs and formal reasoning. We developed the DEMI-MathAnalysis dataset, comprising proof-based problems from mathematical analysis topics such as Sequences and Limits, Infinite Series, and Convex Functions. We also designed a guiding framework to rigorously enhance LLMs' ability to solve these problems. Through fine-tuning LLMs on this dataset and employing our framework, we observed significant improvements in their capability to generate logical, complete, and elegant proofs. This work addresses critical gaps in mathematical reasoning and contributes to advancing trustworthy AI capable of handling formalized mathematical language. The code is publicly accessible at LLMs for Mathematical Analysis.

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

State of the art Symbolic Regression (SR) methods currently build specialized models, while the application of Large Language Models (LLMs) remains largely unexplored. In this work, we introduce the first comprehensive framework that utilizes LLMs for the task of SR. We propose In-Context Symbolic Regression (ICSR), an SR method which iteratively refines a functional form with an LLM and determines its coefficients with an external optimizer. ICSR leverages LLMs' strong mathematical prior both to propose an initial set of possible functions given the observations and to refine them based on their errors. Our findings reveal that LLMs are able to successfully find symbolic equations that fit the given data, matching or outperforming the overall performance of the best SR baselines on four popular benchmarks, while yielding simpler equations with better out of distribution generalization.

Symbolic regression (SR) is the problem of learning a symbolic expression from numerical data. Recently, deep neural models trained on procedurally-generated synthetic datasets showed competitive performance compared to more classical Genetic Programming (GP) algorithms. Unlike their GP counterparts, these neural approaches are trained to generate expressions from datasets given as context. This allows them to produce accurate expressions in a single forward pass at test time. However, they usually do not benefit from search abilities, which result in low performance compared to GP on out-of-distribution datasets. In this paper, we propose a novel method which provides the best of both worlds, based on a Monte-Carlo Tree Search procedure using a context-aware neural mutation model, which is initially pre-trained to learn promising mutations, and further refined from successful experiences in an online fashion. The approach demonstrates state-of-the-art performance on the well-known SRBench benchmark.

This paper revisits datasets and evaluation criteria for Symbolic Regression, a task of expressing given data using mathematical equations, specifically focused on its potential for scientific discovery. Focused on a set of formulas used in the existing datasets based on Feynman Lectures on Physics, we recreate 120 datasets to discuss the performance of symbolic regression for scientific discovery (SRSD). For each of the 120 SRSD datasets, we carefully review the properties of the formula and its variables to design reasonably realistic sampling range of values so that our new SRSD datasets can be used for evaluating the potential of SRSD such as whether or not an SR method can (re)discover physical laws from such datasets. As an evaluation metric, we also propose to use normalized edit distances between a predicted equation and the ground-truth equation trees. While existing metrics are either binary or errors between the target values and an SR model's predicted values for a given input, normalized edit distances evaluate a sort of similarity between the ground-truth and predicted equation trees. We have conducted experiments on our new SRSD datasets using five state-of-the-art SR methods in SRBench and a simple baseline based on a recent Transformer architecture. The results show that we provide a more realistic performance evaluation and open up a new machine learning-based approach for scientific discovery. Our datasets and code repository are publicly available.

Assessing the capabilities of large language models (LLMs) is often challenging, in part, because it is hard to find tasks to which they have not been exposed during training. We take one step to address this challenge by turning to a new task: focusing on symbolic graphics programs, which are a popular representation for graphics content that procedurally generates visual data. LLMs have shown exciting promise towards program synthesis, but do they understand symbolic graphics programs? Unlike conventional programs, symbolic graphics programs can be translated to graphics content. Here, we characterize an LLM's understanding of symbolic programs in terms of their ability to answer questions related to the graphics content. This task is challenging as the questions are difficult to answer from the symbolic programs alone -- yet, they would be easy to answer from the corresponding graphics content as we verify through a human experiment. To understand symbolic programs, LLMs may need to possess the ability to imagine how the corresponding graphics content would look without directly accessing the rendered visual content. We use this task to evaluate LLMs by creating a large benchmark for the semantic understanding of symbolic graphics programs. This benchmark is built via program-graphics correspondence, hence requiring minimal human efforts. We evaluate current LLMs on our benchmark to elucidate a preliminary assessment of their ability to reason about visual scenes from programs. We find that this task distinguishes existing LLMs and models considered good at reasoning perform better. Lastly, we introduce Symbolic Instruction Tuning (SIT) to improve this ability. Specifically, we query GPT4-o with questions and images generated by symbolic programs. Such data are then used to finetune an LLM. We also find that SIT data can improve the general instruction following ability of LLMs.

Given ample experimental data from a system governed by differential equations, it is possible to use deep learning techniques to construct the underlying differential operators. In this work we perform symbolic discovery of differential operators in a situation where there is sparse experimental data. This small data regime in machine learning can be made tractable by providing our algorithms with prior information about the underlying dynamics. Physics Informed Neural Networks (PINNs) have been very successful in this regime (reconstructing entire ODE solutions using only a single point or entire PDE solutions with very few measurements of the initial condition). We modify the PINN approach by adding a neural network that learns a representation of unknown hidden terms in the differential equation. The algorithm yields both a surrogate solution to the differential equation and a black-box representation of the hidden terms. These hidden term neural networks can then be converted into symbolic equations using symbolic regression techniques like AI Feynman. In order to achieve convergence of these neural networks, we provide our algorithms with (noisy) measurements of both the initial condition as well as (synthetic) experimental data obtained at later times. We demonstrate strong performance of this approach even when provided with very few measurements of noisy data in both the ODE and PDE regime.

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

Large Language Models (LLMs) have emerged as transformative tools in artificial intelligence, capable of processing and understanding extensive human knowledge to enhance problem-solving across various domains. This paper explores the potential of LLMs to drive the discovery of symbolic solutions within scientific and engineering disciplines, where such solutions are crucial for advancing theoretical and practical applications. We propose a novel framework that utilizes LLMs in an evolutionary search methodology, augmented by a dynamic knowledge library that integrates and refines insights in an open-ended manner. This approach aims to tackle the dual challenges of efficiently navigating complex symbolic representation spaces and leveraging both existing and newly generated knowledge to foster open-ended innovation. By enabling LLMs to interact with and expand upon a knowledge library, we facilitate the continuous generation of novel solutions in diverse forms such as language, code, and mathematical expressions. Our experimental results demonstrate that this method not only enhances the efficiency of searching for symbolic solutions but also supports the ongoing discovery process, akin to human scientific endeavors. This study represents a first effort in conceptualizing the search for symbolic solutions as a lifelong, iterative process, marking a significant step towards harnessing AI in the perpetual pursuit of scientific and engineering breakthroughs. We have open-sourced our code and data, please visit https://github.com/pgg3/CoEvo for more information.

This paper approaches the problem of separating the notes from a quantized symbolic music piece (e.g., a MIDI file) into multiple voices and staves. This is a fundamental part of the larger task of music score engraving (or score typesetting), which aims to produce readable musical scores for human performers. We focus on piano music and support homophonic voices, i.e., voices that can contain chords, and cross-staff voices, which are notably difficult tasks that have often been overlooked in previous research. We propose an end-to-end system based on graph neural networks that clusters notes that belong to the same chord and connects them with edges if they are part of a voice. Our results show clear and consistent improvements over a previous approach on two datasets of different styles. To aid the qualitative analysis of our results, we support the export in symbolic music formats and provide a direct visualization of our outputs graph over the musical score. All code and pre-trained models are available at https://github.com/CPJKU/piano_svsep

Large Language Models (LLMs) have shown to be capable of various tasks, yet their capability in interpreting and reasoning over tabular data remains an underexplored area. In this context, this study investigates from three core perspectives: the robustness of LLMs to structural perturbations in tables, the comparative analysis of textual and symbolic reasoning on tables, and the potential of boosting model performance through the aggregation of multiple reasoning pathways. We discover that structural variance of tables presenting the same content reveals a notable performance decline, particularly in symbolic reasoning tasks. This prompts the proposal of a method for table structure normalization. Moreover, textual reasoning slightly edges out symbolic reasoning, and a detailed error analysis reveals that each exhibits different strengths depending on the specific tasks. Notably, the aggregation of textual and symbolic reasoning pathways, bolstered by a mix self-consistency mechanism, resulted in achieving SOTA performance, with an accuracy of 73.6% on WIKITABLEQUESTIONS, representing a substantial advancement over previous existing table processing paradigms of LLMs.

Visual reasoning is critical for a wide range of computer vision tasks that go beyond surface-level object detection and classification. Despite notable advances in relational, symbolic, temporal, causal, and commonsense reasoning, existing surveys often address these directions in isolation, lacking a unified analysis and comparison across reasoning types, methodologies, and evaluation protocols. This survey aims to address this gap by categorizing visual reasoning into five major types (relational, symbolic, temporal, causal, and commonsense) and systematically examining their implementation through architectures such as graph-based models, memory networks, attention mechanisms, and neuro-symbolic systems. We review evaluation protocols designed to assess functional correctness, structural consistency, and causal validity, and critically analyze their limitations in terms of generalizability, reproducibility, and explanatory power. Beyond evaluation, we identify key open challenges in visual reasoning, including scalability to complex scenes, deeper integration of symbolic and neural paradigms, the lack of comprehensive benchmark datasets, and reasoning under weak supervision. Finally, we outline a forward-looking research agenda for next-generation vision systems, emphasizing that bridging perception and reasoning is essential for building transparent, trustworthy, and cross-domain adaptive AI systems, particularly in critical domains such as autonomous driving and medical diagnostics.

Large Language Models (LLMs) have shown state-of-the-art performance in a variety of tasks, including arithmetic and reasoning; however, to gauge the intellectual capabilities of LLMs, causal reasoning has become a reliable proxy for validating a general understanding of the mechanics and intricacies of the world similar to humans. Previous works in natural language processing (NLP) have either focused on open-ended causal reasoning via causal commonsense reasoning (CCR) or framed a symbolic representation-based question answering for theoretically backed-up analysis via a causal inference engine. The former adds an advantage of real-world grounding but lacks theoretically backed-up analysis/validation, whereas the latter is far from real-world grounding. In this work, we bridge this gap by proposing the COLD (Causal reasOning in cLosed Daily activities) framework, which is built upon human understanding of daily real-world activities to reason about the causal nature of events. We show that the proposed framework facilitates the creation of enormous causal queries (~ 9 million) and comes close to the mini-turing test, simulating causal reasoning to evaluate the understanding of a daily real-world task. We evaluate multiple LLMs on the created causal queries and find that causal reasoning is challenging even for activities trivial to humans. We further explore (the causal reasoning abilities of LLMs) using the backdoor criterion to determine the causal strength between events.

Large language models routinely hallucinate APIs and mislocalize edits, while language servers compute verified, IDE-grade facts about real code. We present Lanser-CLI, a CLI-first orchestration layer that pins and mediates a Language Server Protocol (LSP) server for coding agents and CI, exposing deterministic, replayable workflows. Our position is that language servers provide not only structural information (definitions, references, types, diagnostics) but also an actionable process reward: machine-checked, step-wise signals that align an agent's planning loop with program reality. In this work, Lanser-CLI contributes: (i) a robust addressing scheme beyond brittle "file:line:col" via a Selector DSL (symbolic, AST-path, and content-anchored selectors) with a principled relocation algorithm; (ii) deterministic Analysis Bundles that normalize Language Server responses and capture environment/capability metadata with stable content hashes; (iii) a safety envelope for mutating operations (rename, code actions) with preview, workspace jails, and Git-aware, transactional apply; and (iv) a process-reward functional derived from Language Server facts (diagnostic deltas, disambiguation confidence, and safe-apply checks) that is computable online and replayable offline. We formalize determinism under frozen snapshots and establish a monotonicity property for the process reward, making it suitable for process supervision and counterfactual analysis. Project Page: https://github.com/yifanzhang-pro/lanser-cli

Symbolic regression (SR) is an area of interpretable machine learning that aims to identify mathematical expressions, often composed of simple functions, that best fit in a given set of covariates X and response y. In recent years, deep symbolic regression (DSR) has emerged as a popular method in the field by leveraging deep reinforcement learning to solve the complicated combinatorial search problem. In this work, we propose an alternative framework (GFN-SR) to approach SR with deep learning. We model the construction of an expression tree as traversing through a directed acyclic graph (DAG) so that GFlowNet can learn a stochastic policy to generate such trees sequentially. Enhanced with an adaptive reward baseline, our method is capable of generating a diverse set of best-fitting expressions. Notably, we observe that GFN-SR outperforms other SR algorithms in noisy data regimes, owing to its ability to learn a distribution of rewards over a space of candidate solutions.

How can we perform computations over natural language representations to solve tasks that require symbolic and numeric reasoning? We propose natural language embedded programs (NLEP) as a unifying framework for addressing math/symbolic reasoning, natural language understanding, and instruction following tasks. Our approach prompts a language model to generate full Python programs that define functions over data structures which contain natural language representations of structured knowledge. A Python interpreter then executes the generated code and prints the output. Despite using a task-general prompt, we find that this approach can improve upon strong baselines across a range of different tasks including math and symbolic reasoning, text classification, question answering, and instruction following. We further find the generated programs are often interpretable and enable post-hoc verification of the intermediate reasoning steps.

Representation learning has become an invaluable approach for learning from symbolic data such as text and graphs. However, while complex symbolic datasets often exhibit a latent hierarchical structure, state-of-the-art methods typically learn embeddings in Euclidean vector spaces, which do not account for this property. For this purpose, we introduce a new approach for learning hierarchical representations of symbolic data by embedding them into hyperbolic space -- or more precisely into an n-dimensional Poincar\'e ball. Due to the underlying hyperbolic geometry, this allows us to learn parsimonious representations of symbolic data by simultaneously capturing hierarchy and similarity. We introduce an efficient algorithm to learn the embeddings based on Riemannian optimization and show experimentally that Poincar\'e embeddings outperform Euclidean embeddings significantly on data with latent hierarchies, both in terms of representation capacity and in terms of generalization ability.

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

With recent dramatic increases in AI system capabilities, there has been growing interest in utilizing machine learning for reasoning-heavy, quantitative tasks, particularly mathematics. While there are many resources capturing mathematics at the high-school, undergraduate, and graduate level, there are far fewer resources available that align with the level of difficulty and open endedness encountered by professional mathematicians working on open problems. To address this, we introduce a new collection of datasets, the Algebraic Combinatorics Dataset Repository (ACD Repo), representing either foundational results or open problems in algebraic combinatorics, a subfield of mathematics that studies discrete structures arising from abstract algebra. Further differentiating our dataset collection is the fact that it aims at the conjecturing process. Each dataset includes an open-ended research-level question and a large collection of examples (up to 10M in some cases) from which conjectures should be generated. We describe all nine datasets, the different ways machine learning models can be applied to them (e.g., training with narrow models followed by interpretability analysis or program synthesis with LLMs), and discuss some of the challenges involved in designing datasets like these.

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

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

Mathematical symbols and descriptions appear in various forms across document section boundaries without explicit markup. In this paper, we present a new large-scale dataset that emphasizes extracting symbols and descriptions in scientific documents. Symlink annotates scientific papers of 5 different domains (i.e., computer science, biology, physics, mathematics, and economics). Our experiments on Symlink demonstrate the challenges of the symbol-description linking task for existing models and call for further research effort in this area. We will publicly release Symlink to facilitate future research.

Ongoing efforts that span over decades show a rise of AI methods for accelerating scientific discovery, yet accelerating discovery in mathematics remains a persistent challenge for AI. Specifically, AI methods were not effective in creation of formulas for mathematical constants because each such formula must be correct for infinite digits of precision, with "near-true" formulas providing no insight toward the correct ones. Consequently, formula discovery lacks a clear distance metric needed to guide automated discovery in this realm. In this work, we propose a systematic methodology for categorization, characterization, and pattern identification of such formulas. The key to our methodology is introducing metrics based on the convergence dynamics of the formulas, rather than on the numerical value of the formula. These metrics enable the first automated clustering of mathematical formulas. We demonstrate this methodology on Polynomial Continued Fraction formulas, which are ubiquitous in their intrinsic connections to mathematical constants, and generalize many mathematical functions and structures. We test our methodology on a set of 1,768,900 such formulas, identifying many known formulas for mathematical constants, and discover previously unknown formulas for pi, ln(2), Gauss', and Lemniscate's constants. The uncovered patterns enable a direct generalization of individual formulas to infinite families, unveiling rich mathematical structures. This success paves the way towards a generative model that creates formulas fulfilling specified mathematical properties, accelerating the rate of discovery of useful formulas.

We present semantic correctness proofs of Automatic Differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming, based on diffeological spaces. We show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is, in essence, a logical relations argument. Finally, we sketch how the analysis extends to other AD methods by considering a continuation-based method.

The synergy between symbolic knowledge, often represented by Knowledge Graphs (KGs), and the generative capabilities of neural networks is central to advancing neurosymbolic AI. A primary bottleneck in realizing this potential is the difficulty of automating KG construction, which faces challenges related to output reliability, consistency, and verifiability. These issues can manifest as structural inconsistencies within the generated graphs, such as the formation of disconnected isolated islands of data or the inaccurate conflation of abstract classes with specific instances. To address these challenges, we propose HyDRA, a Hybrid-Driven Reasoning Architecture designed for verifiable KG automation. Given a domain or an initial set of documents, HyDRA first constructs an ontology via a panel of collaborative neurosymbolic agents. These agents collaboratively agree on a set of competency questions (CQs) that define the scope and requirements the ontology must be able to answer. Given these CQs, we build an ontology graph that subsequently guides the automated extraction of triplets for KG generation from arbitrary documents. Inspired by design-by-contracts (DbC) principles, our method leverages verifiable contracts as the primary control mechanism to steer the generative process of Large Language Models (LLMs). To verify the output of our approach, we extend beyond standard benchmarks and propose an evaluation framework that assesses the functional correctness of the resulting KG by leveraging symbolic verifications as described by the neurosymbolic AI framework, SymbolicAI. This work contributes a hybrid-driven architecture for improving the reliability of automated KG construction and the exploration of evaluation methods for measuring the functional integrity of its output. The code is publicly available.

The advancement of machine learning and symbolic approaches have underscored their strengths and weaknesses in Natural Language Processing (NLP). While machine learning approaches are powerful in identifying patterns in data, they often fall short in learning commonsense and the factual knowledge required for the NLP tasks. Meanwhile, the symbolic methods excel in representing knowledge-rich data. However, they struggle to adapt dynamic data and generalize the knowledge. Bridging these two paradigms through hybrid approaches enables the alleviation of weaknesses in both while preserving their strengths. Recent studies extol the virtues of this union, showcasing promising results in a wide range of NLP tasks. In this paper, we present an overview of hybrid approaches used for NLP. Specifically, we delve into the state-of-the-art hybrid approaches used for a broad spectrum of NLP tasks requiring natural language understanding, generation, and reasoning. Furthermore, we discuss the existing resources available for hybrid approaches for NLP along with the challenges and future directions, offering a roadmap for future research avenues.

Neuro-symbolic learning was proposed to address challenges with training neural networks for complex reasoning tasks with the added benefits of interpretability, reliability, and efficiency. Neuro-symbolic learning methods traditionally train neural models in conjunction with symbolic programs, but they face significant challenges that limit them to simplistic problems. On the other hand, purely-neural foundation models now reach state-of-the-art performance through prompting rather than training, but they are often unreliable and lack interpretability. Supplementing foundation models with symbolic programs, which we call neuro-symbolic prompting, provides a way to use these models for complex reasoning tasks. Doing so raises the question: What role does specialized model training as part of neuro-symbolic learning have in the age of foundation models? To explore this question, we highlight three pitfalls of traditional neuro-symbolic learning with respect to the compute, data, and programs leading to generalization problems. This position paper argues that foundation models enable generalizable neuro-symbolic solutions, offering a path towards achieving the original goals of neuro-symbolic learning without the downsides of training from scratch.

The advancement in generative AI could be boosted with more accessible mathematics. Beyond human-AI chat, large language models (LLMs) are emerging in programming, algorithm discovery, and theorem proving, yet their genomics application is limited. This project introduces Math Agents and mathematical embedding as fresh entries to the "Moore's Law of Mathematics", using a GPT-based workflow to convert equations from literature into LaTeX and Python formats. While many digital equation representations exist, there's a lack of automated large-scale evaluation tools. LLMs are pivotal as linguistic user interfaces, providing natural language access for human-AI chat and formal languages for large-scale AI-assisted computational infrastructure. Given the infinite formal possibility spaces, Math Agents, which interact with math, could potentially shift us from "big data" to "big math". Math, unlike the more flexible natural language, has properties subject to proof, enabling its use beyond traditional applications like high-validation math-certified icons for AI alignment aims. This project aims to use Math Agents and mathematical embeddings to address the ageing issue in information systems biology by applying multiscalar physics mathematics to disease models and genomic data. Generative AI with episodic memory could help analyse causal relations in longitudinal health records, using SIR Precision Health models. Genomic data is suggested for addressing the unsolved Alzheimer's disease problem.

Recently, large language models have presented promising results in aiding formal mathematical reasoning. However, their performance is restricted due to the scarcity of formal theorem-proving data, which requires additional effort to be extracted from raw formal language corpora. Meanwhile, a significant amount of human-written formal language corpora remains underutilized. To address this issue, we propose LEAN-GitHub, a dataset consisting of large-scale formal data extracted from almost all Lean 4 repositories on GitHub. After fine-tuning InternLM-math-plus on this dataset, our model achieved accuracies of 48.8% with a single pass and 54.5% with 64 passes on the Lean 4 miniF2F test, surpassing state-of-the-art method at 52%. And it also achieves state-of-the-art on two other Lean 4 benchmarks (ProofNet and Putnam) targeting different fields/levels of math. These results demonstrate that our proposed dataset is beneficial for formal reasoning on a wide range of math topics. We open-source our model at https://GitHub. com/InternLM/InternLM-Math and our data at https://huggingface.co/ datasets/InternLM/Lean-GitHub

We introduce SymbolicAI, a versatile and modular framework employing a logic-based approach to concept learning and flow management in generative processes. SymbolicAI enables the seamless integration of generative models with a diverse range of solvers by treating large language models (LLMs) as semantic parsers that execute tasks based on both natural and formal language instructions, thus bridging the gap between symbolic reasoning and generative AI. We leverage probabilistic programming principles to tackle complex tasks, and utilize differentiable and classical programming paradigms with their respective strengths. The framework introduces a set of polymorphic, compositional, and self-referential operations for data stream manipulation, aligning LLM outputs with user objectives. As a result, we can transition between the capabilities of various foundation models endowed with zero- and few-shot learning capabilities and specialized, fine-tuned models or solvers proficient in addressing specific problems. In turn, the framework facilitates the creation and evaluation of explainable computational graphs. We conclude by introducing a quality measure and its empirical score for evaluating these computational graphs, and propose a benchmark that compares various state-of-the-art LLMs across a set of complex workflows. We refer to the empirical score as the "Vector Embedding for Relational Trajectory Evaluation through Cross-similarity", or VERTEX score for short. The framework codebase and benchmark are linked below.

The latest Deep Learning (DL) models for detection and classification have achieved an unprecedented performance over classical machine learning algorithms. However, DL models are black-box methods hard to debug, interpret, and certify. DL alone cannot provide explanations that can be validated by a non technical audience. In contrast, symbolic AI systems that convert concepts into rules or symbols -- such as knowledge graphs -- are easier to explain. However, they present lower generalisation and scaling capabilities. A very important challenge is to fuse DL representations with expert knowledge. One way to address this challenge, as well as the performance-explainability trade-off is by leveraging the best of both streams without obviating domain expert knowledge. We tackle such problem by considering the symbolic knowledge is expressed in form of a domain expert knowledge graph. We present the eXplainable Neural-symbolic learning (X-NeSyL) methodology, designed to learn both symbolic and deep representations, together with an explainability metric to assess the level of alignment of machine and human expert explanations. The ultimate objective is to fuse DL representations with expert domain knowledge during the learning process to serve as a sound basis for explainability. X-NeSyL methodology involves the concrete use of two notions of explanation at inference and training time respectively: 1) EXPLANet: Expert-aligned eXplainable Part-based cLAssifier NETwork Architecture, a compositional CNN that makes use of symbolic representations, and 2) SHAP-Backprop, an explainable AI-informed training procedure that guides the DL process to align with such symbolic representations in form of knowledge graphs. We showcase X-NeSyL methodology using MonuMAI dataset for monument facade image classification, and demonstrate that our approach improves explainability and performance.

We investigate the mathematical capabilities of ChatGPT by testing it on publicly available datasets, as well as hand-crafted ones, and measuring its performance against other models trained on a mathematical corpus, such as Minerva. We also test whether ChatGPT can be a useful assistant to professional mathematicians by emulating various use cases that come up in the daily professional activities of mathematicians (question answering, theorem searching). In contrast to formal mathematics, where large databases of formal proofs are available (e.g., the Lean Mathematical Library), current datasets of natural-language mathematics, used to benchmark language models, only cover elementary mathematics. We address this issue by introducing a new dataset: GHOSTS. It is the first natural-language dataset made and curated by working researchers in mathematics that (1) aims to cover graduate-level mathematics and (2) provides a holistic overview of the mathematical capabilities of language models. We benchmark ChatGPT on GHOSTS and evaluate performance against fine-grained criteria. We make this new dataset publicly available to assist a community-driven comparison of ChatGPT with (future) large language models in terms of advanced mathematical comprehension. We conclude that contrary to many positive reports in the media (a potential case of selection bias), ChatGPT's mathematical abilities are significantly below those of an average mathematics graduate student. Our results show that ChatGPT often understands the question but fails to provide correct solutions. Hence, if your goal is to use it to pass a university exam, you would be better off copying from your average peer!

Optimizing Register Transfer Level (RTL) code is crucial for improving the power, performance, and area (PPA) of digital circuits in the early stages of synthesis. Manual rewriting, guided by synthesis feedback, can yield high-quality results but is time-consuming and error-prone. Most existing compiler-based approaches have difficulty handling complex design constraints. Large Language Model (LLM)-based methods have emerged as a promising alternative to address these challenges. However, LLM-based approaches often face difficulties in ensuring alignment between the generated code and the provided prompts. This paper presents SymRTLO, a novel neuron-symbolic RTL optimization framework that seamlessly integrates LLM-based code rewriting with symbolic reasoning techniques. Our method incorporates a retrieval-augmented generation (RAG) system of optimization rules and Abstract Syntax Tree (AST)-based templates, enabling LLM-based rewriting that maintains syntactic correctness while minimizing undesired circuit behaviors. A symbolic module is proposed for analyzing and optimizing finite state machine (FSM) logic, allowing fine-grained state merging and partial specification handling beyond the scope of pattern-based compilers. Furthermore, a fast verification pipeline, combining formal equivalence checks with test-driven validation, further reduces the complexity of verification. Experiments on the RTL-Rewriter benchmark with Synopsys Design Compiler and Yosys show that SymRTLO improves power, performance, and area (PPA) by up to 43.9%, 62.5%, and 51.1%, respectively, compared to the state-of-the-art methods.

We enhance the calculus of string diagrams for monoidal categories with hierarchical features in order to capture closed monoidal (and cartesian closed) structure. Using this new syntax we formulate an automatic differentiation algorithm for (applied) simply typed lambda calculus in the style of [Pearlmutter and Siskind 2008] and we prove for the first time its soundness. To give an efficient yet principled implementation of the AD algorithm we define a sound and complete representation of hierarchical string diagrams as a class of hierarchical hypergraphs we call hypernets.

The research in AI-based formal mathematical reasoning has shown an unstoppable growth trend. These studies have excelled in mathematical competitions like IMO, showing significant progress. However, these studies intertwined multiple skills simultaneously, i.e., problem-solving, reasoning, and writing formal specifications, making it hard to precisely identify the LLMs' strengths and weaknesses in each task. This paper focuses on formal verification, an immediate application scenario of formal reasoning, and decomposes it into six sub-tasks. We constructed 18k high-quality instruction-response pairs across five mainstream formal specification languages (Coq, Lean4, Dafny, ACSL, and TLA+) in six formal-verification-related tasks by distilling GPT-4o. They are split into a 14k+ fine-tuning dataset FM-alpaca and a 4k benchmark FM-Bench. We found that LLMs are good at writing proof segments when given either the code, or the detailed description of proof steps. Also, the fine-tuning brought about a nearly threefold improvement at most. Interestingly, we observed that fine-tuning with formal data also enhances mathematics, reasoning, and coding abilities. We hope our findings inspire further research. Fine-tuned models are released to facilitate subsequent studies

CONTEXT The R programming language has a huge and active community, especially in the area of statistical computing. Its interpreted nature allows for several interesting constructs, like the manipulation of functions at run-time, that hinder the static analysis of R programs. At the same time, there is a lack of existing research regarding how these features, or even the R language as a whole are used in practice. OBJECTIVE In this paper, we conduct a large-scale, static analysis of more than 50 million lines of real-world R programs and packages to identify their characteristics and the features that are actually used. Moreover, we compare the similarities and differences between the scripts of R users and the implementations of package authors. We provide insights for static analysis tools like the lintr package as well as potential interpreter optimizations and uncover areas for future research. METHOD We analyze 4230 R scripts submitted alongside publications and the sources of 19450 CRAN packages for over 350000 R files, collecting and summarizing quantitative information for features of interest. RESULTS We find a high frequency of name-based indexing operations, assignments, and loops, but a low frequency for most of R's reflective functions. Furthermore, we find neither testing functions nor many calls to R's foreign function interface (FFI) in the publication submissions. CONCLUSION R scripts and package sources differ, for example, in their size, the way they include other packages, and their usage of R's reflective capabilities. We provide features that are used frequently and should be prioritized by static analysis tools, like operator assignments, function calls, and certain reflective functions like load.

In recent years, there has been remarkable progress in leveraging Language Models (LMs), encompassing Pre-trained Language Models (PLMs) and Large-scale Language Models (LLMs), within the domain of mathematics. This paper conducts a comprehensive survey of mathematical LMs, systematically categorizing pivotal research endeavors from two distinct perspectives: tasks and methodologies. The landscape reveals a large number of proposed mathematical LLMs, which are further delineated into instruction learning, tool-based methods, fundamental CoT techniques, and advanced CoT methodologies. In addition, our survey entails the compilation of over 60 mathematical datasets, including training datasets, benchmark datasets, and augmented datasets. Addressing the primary challenges and delineating future trajectories within the field of mathematical LMs, this survey is positioned as a valuable resource, poised to facilitate and inspire future innovation among researchers invested in advancing this domain.

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

This paper does not introduce a novel method. Instead, it offers a fairer and more comprehensive comparison of KAN and MLP models across various tasks, including machine learning, computer vision, audio processing, natural language processing, and symbolic formula representation. Specifically, we control the number of parameters and FLOPs to compare the performance of KAN and MLP. Our main observation is that, except for symbolic formula representation tasks, MLP generally outperforms KAN. We also conduct ablation studies on KAN and find that its advantage in symbolic formula representation mainly stems from its B-spline activation function. When B-spline is applied to MLP, performance in symbolic formula representation significantly improves, surpassing or matching that of KAN. However, in other tasks where MLP already excels over KAN, B-spline does not substantially enhance MLP's performance. Furthermore, we find that KAN's forgetting issue is more severe than that of MLP in a standard class-incremental continual learning setting, which differs from the findings reported in the KAN paper. We hope these results provide insights for future research on KAN and other MLP alternatives. Project link: https://github.com/yu-rp/KANbeFair

We introduce ProofNet, a benchmark for autoformalization and formal proving of undergraduate-level mathematics. The ProofNet benchmarks consists of 371 examples, each consisting of a formal theorem statement in Lean 3, a natural language theorem statement, and a natural language proof. The problems are primarily drawn from popular undergraduate pure mathematics textbooks and cover topics such as real and complex analysis, linear algebra, abstract algebra, and topology. We intend for ProofNet to be a challenging benchmark that will drive progress in autoformalization and automatic theorem proving. We report baseline results on statement autoformalization via in-context learning. Moreover, we introduce two novel statement autoformalization methods: prompt retrieval and distilled backtranslation.

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

Automata learning is a popular technique used to automatically construct an automaton model from queries. Much research went into devising ad hoc adaptations of algorithms for different types of automata. The CALF project seeks to unify these using category theory in order to ease correctness proofs and guide the design of new algorithms. In this paper, we extend CALF to cover learning of algebraic structures that may not have a coalgebraic presentation. Furthermore, we provide a detailed algorithmic account of an abstract version of the popular L* algorithm, which was missing from CALF. We instantiate the abstract theory to a large class of Set functors, by which we recover for the first time practical tree automata learning algorithms from an abstract framework and at the same time obtain new algorithms to learn algebras of quotiented polynomial functors.

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

Large Language Models (LLMs) have demonstrated exceptional capabilities in various natural language tasks, often achieving performances that surpass those of humans. Despite these advancements, the domain of mathematics presents a distinctive challenge, primarily due to its specialized structure and the precision it demands. In this study, we adopted a two-step approach for investigating the proficiency of LLMs in answering mathematical questions. First, we employ the most effective LLMs, as identified by their performance on math question-answer benchmarks, to generate answers to 78 questions from the Math Stack Exchange (MSE). Second, a case analysis is conducted on the LLM that showed the highest performance, focusing on the quality and accuracy of its answers through manual evaluation. We found that GPT-4 performs best (nDCG of 0.48 and P@10 of 0.37) amongst existing LLMs fine-tuned for answering mathematics questions and outperforms the current best approach on ArqMATH3 Task1, considering P@10. Our Case analysis indicates that while the GPT-4 can generate relevant responses in certain instances, it does not consistently answer all questions accurately. This paper explores the current limitations of LLMs in navigating complex mathematical problem-solving. Through case analysis, we shed light on the gaps in LLM capabilities within mathematics, thereby setting the stage for future research and advancements in AI-driven mathematical reasoning. We make our code and findings publicly available for research: https://github.com/gipplab/LLM-Investig-MathStackExchange

The application of machine learning algorithms to source code has grown in the past years. Since these algorithms are quite sensitive to input data, it is not surprising that researchers experiment with input representations. Nowadays, a popular starting point to represent code is abstract syntax trees (ASTs). Abstract syntax trees have been used for a long time in various software engineering domains, and in particular in IDEs. The API of modern IDEs allows to manipulate and traverse ASTs, resolve references between code elements, etc. Such algorithms can enrich ASTs with new data and therefore may be useful in ML-based code analysis. In this work, we present PSIMiner - a tool for processing PSI trees from the IntelliJ Platform. PSI trees contain code syntax trees as well as functions to work with them, and therefore can be used to enrich code representation using static analysis algorithms of modern IDEs. To showcase this idea, we use our tool to infer types of identifiers in Java ASTs and extend the code2seq model for the method name prediction problem.

In this work, we introduce Boolformer, the first Transformer architecture trained to perform end-to-end symbolic regression of Boolean functions. First, we show that it can predict compact formulas for complex functions which were not seen during training, when provided a clean truth table. Then, we demonstrate its ability to find approximate expressions when provided incomplete and noisy observations. We evaluate the Boolformer on a broad set of real-world binary classification datasets, demonstrating its potential as an interpretable alternative to classic machine learning methods. Finally, we apply it to the widespread task of modelling the dynamics of gene regulatory networks. Using a recent benchmark, we show that Boolformer is competitive with state-of-the art genetic algorithms with a speedup of several orders of magnitude. Our code and models are available publicly.

Large Language Models (LLMs) have greatly propelled the progress in natural language(NL)-centric tasks based on NL interface. However, the NL form is not enough for world knowledge. Current works focus on this question by injecting specific symbolic knowledge into LLM, which ignore two critical challenges: the interrelations between various symbols and the balance between symbolic-centric and NL-centric capabilities. In this work, we tackle these challenges from both a data and framework perspective and introduce Symbol-LLM series models. First, we collect 34 symbolic tasks, covering ~20 different forms, which are unified to capture symbol interrelations. Then, a two-stage tuning framework succeeds in injecting symbolic knowledge without loss of the generality ability. Extensive experiments on both symbol- and NL-centric tasks demonstrate the balanced and superior performances of Symbol-LLM series models.

We present a neuro-symbolic (NeSy) workflow combining a symbolic-based learning technique with a large language model (LLM) agent to generate synthetic data for code comment classification in the C programming language. We also show how generating controlled synthetic data using this workflow fixes some of the notable weaknesses of LLM-based generation and increases the performance of classical machine learning models on the code comment classification task. Our best model, a Neural Network, achieves a Macro-F1 score of 91.412% with an increase of 1.033% after data augmentation.

Chain-of-Thought (CoT) prompting has become the de facto method to elicit reasoning capabilities from large language models (LLMs). However, to mitigate hallucinations in CoT that are notoriously difficult to detect, current methods such as process reward models (PRMs) or self-consistency operate as opaque boxes and do not provide checkable evidence for their judgments, possibly limiting their effectiveness. To address this issue, we draw inspiration from the idea that "the gold standard for supporting a mathematical claim is to provide a proof". We propose a retrospective, step-aware formal verification framework Safe. Rather than assigning arbitrary scores, we strive to articulate mathematical claims in formal mathematical language Lean 4 at each reasoning step and provide formal proofs to identify hallucinations. We evaluate our framework Safe across multiple language models and various mathematical datasets, demonstrating a significant performance improvement while offering interpretable and verifiable evidence. We also propose FormalStep as a benchmark for step correctness theorem proving with 30,809 formal statements. To the best of our knowledge, our work represents the first endeavor to utilize formal mathematical language Lean 4 for verifying natural language content generated by LLMs, aligning with the reason why formal mathematical languages were created in the first place: to provide a robust foundation for hallucination-prone human-written proofs.

Mathematical reasoning lies at the heart of artificial intelligence, underpinning applications in education, program verification, and research-level mathematical discovery. Mathematical competitions, in particular, present two challenging problem types: theorem proving, which requires rigorous proofs of stated conclusions, and answer construction, which involves hypothesizing and formally verifying mathematical objects. Large Language Models (LLMs) effectively generate creative candidate answers but struggle with formal verification, while symbolic provers ensure rigor but cannot efficiently handle creative conjecture generation. We introduce the Enumerate-Conjecture-Prove (ECP) framework, a modular neuro-symbolic method integrating LLM-based enumeration and pattern-driven conjecturing with formal theorem proving. We present ConstructiveBench, a dataset of 3,431 answer-construction problems in various math competitions with verified Lean formalizations. On the ConstructiveBench dataset, ECP improves the accuracy of answer construction from a Chain-of-Thought (CoT) baseline of 14.54% to 45.06% with the gpt-4.1-mini model. Moreover, combined with ECP's constructed answers, the state-of-the-art DeepSeek-Prover-V2-7B model generates correct proofs for 858 of the 3,431 constructive problems in Lean, achieving 25.01% accuracy compared to 9.86% for symbolic-only baselines. Our code and dataset are publicly available at https://github.com/JackSun200312/ECP.

The AI community has been exploring a pathway to artificial general intelligence (AGI) by developing "language agents", which are complex large language models (LLMs) pipelines involving both prompting techniques and tool usage methods. While language agents have demonstrated impressive capabilities for many real-world tasks, a fundamental limitation of current language agents research is that they are model-centric, or engineering-centric. That's to say, the progress on prompts, tools, and pipelines of language agents requires substantial manual engineering efforts from human experts rather than automatically learning from data. We believe the transition from model-centric, or engineering-centric, to data-centric, i.e., the ability of language agents to autonomously learn and evolve in environments, is the key for them to possibly achieve AGI. In this work, we introduce agent symbolic learning, a systematic framework that enables language agents to optimize themselves on their own in a data-centric way using symbolic optimizers. Specifically, we consider agents as symbolic networks where learnable weights are defined by prompts, tools, and the way they are stacked together. Agent symbolic learning is designed to optimize the symbolic network within language agents by mimicking two fundamental algorithms in connectionist learning: back-propagation and gradient descent. Instead of dealing with numeric weights, agent symbolic learning works with natural language simulacrums of weights, loss, and gradients. We conduct proof-of-concept experiments on both standard benchmarks and complex real-world tasks and show that agent symbolic learning enables language agents to update themselves after being created and deployed in the wild, resulting in "self-evolving agents".

We present semantic correctness proofs of automatic differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming, based on diffeological spaces. We show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is, in essence, a logical relations argument. Throughout, we show how the analysis extends to AD methods for computing higher order derivatives using a Taylor approximation.

Zero-shot learning (ZSL) which aims at predicting classes that have never appeared during the training using external knowledge (a.k.a. side information) has been widely investigated. In this paper we present a literature review towards ZSL in the perspective of external knowledge, where we categorize the external knowledge, review their methods and compare different external knowledge. With the literature review, we further discuss and outlook the role of symbolic knowledge in addressing ZSL and other machine learning sample shortage issues.

Verifiable formal languages like Lean have profoundly impacted mathematical reasoning, particularly through the use of large language models (LLMs) for automated reasoning. A significant challenge in training LLMs for these formal languages is the lack of parallel datasets that align natural language with formal language proofs. To address this challenge, this paper introduces a novel framework for translating the Mathlib4 corpus (a unified library of mathematics in formal language Lean 4) into natural language. Building upon this, we employ a dual augmentation strategy that combines tactic-based and informal-based approaches, leveraging the Lean-jixia system, a Lean 4 analyzer. We present the results of this pipeline on Mathlib4 as Herald (Hierarchy and Retrieval-based Translated Lean Dataset). We also propose the Herald Translator, which is fine-tuned on Herald. Herald translator achieves a 93.2% accuracy (Pass@128) on formalizing statements in the miniF2F-test and a 22.5% accuracy on our internal graduate-level textbook dataset, outperforming InternLM2-Math-Plus-7B (74.0% and 7.5%) and TheoremLlama (50.1% and 4.0%). Furthermore, we propose a section-level translation framework for real-world applications. As a direct application of Herald translator, we have successfully translated a template section in the Stack project, marking a notable progress in the automatic formalization of graduate-level mathematical literature. Our model, along with the datasets, will be open-sourced to the public soon.

In order to properly train a machine learning model, data must be properly collected. To guarantee a proper data collection, verifying that the collected data set holds certain properties is a possible solution. For example, guaranteeing that the data set contains samples across the whole input space, or that the data set is balanced w.r.t. different classes. We present a formal approach for verifying a set of arbitrarily stated properties over a data set. The proposed approach relies on the transformation of the data set into a first order logic formula, which can be later verified w.r.t. the different properties also stated in the same logic. A prototype tool, which uses the z3 solver, has been developed; the prototype can take as an input a set of properties stated in a formal language and formally verify a given data set w.r.t. to the given set of properties. Preliminary experimental results show the feasibility and performance of the proposed approach, and furthermore the flexibility for expressing properties of interest.

Mathematical reasoning is a fundamental aspect of human intelligence and is applicable in various fields, including science, engineering, finance, and everyday life. The development of artificial intelligence (AI) systems capable of solving math problems and proving theorems has garnered significant interest in the fields of machine learning and natural language processing. For example, mathematics serves as a testbed for aspects of reasoning that are challenging for powerful deep learning models, driving new algorithmic and modeling advances. On the other hand, recent advances in large-scale neural language models have opened up new benchmarks and opportunities to use deep learning for mathematical reasoning. In this survey paper, we review the key tasks, datasets, and methods at the intersection of mathematical reasoning and deep learning over the past decade. We also evaluate existing benchmarks and methods, and discuss future research directions in this domain.

This paper presents MathBode, a dynamic diagnostic for mathematical reasoning in large language models (LLMs). Instead of one-shot accuracy, MathBode treats each parametric problem as a system: we drive a single parameter sinusoidally and fit first-harmonic responses of model outputs and exact solutions. This yields interpretable, frequency-resolved metrics -- gain (amplitude tracking) and phase (lag) -- that form Bode-style fingerprints. Across five closed-form families (linear solve, ratio/saturation, compound interest, 2x2 linear systems, similar triangles), the diagnostic surfaces systematic low-pass behavior and growing phase lag that accuracy alone obscures. We compare several models against a symbolic baseline that calibrates the instrument (G approx 1, phi approx 0). Results separate frontier from mid-tier models on dynamics, providing a compact, reproducible protocol that complements standard benchmarks with actionable measurements of reasoning fidelity and consistency. We open-source the dataset and code to enable further research and adoption.

The integration of knowledge extracted from diverse models, whether described by domain experts or generated by machine learning algorithms, has historically been challenged by the absence of a suitable framework for specifying and integrating structures, learning processes, data transformations, and data models or rules. In this work, we extend algebraic specification methods to address these challenges within such a framework. In our work, we tackle the challenging task of developing a comprehensive framework for defining and analyzing deep learning architectures. We believe that previous efforts have fallen short by failing to establish a clear connection between the constraints a model must adhere to and its actual implementation. Our methodology employs graphical structures that resemble Ehresmann's sketches, interpreted within a universe of fuzzy sets. This approach offers a unified theory that elegantly encompasses both deterministic and non-deterministic neural network designs. Furthermore, we highlight how this theory naturally incorporates fundamental concepts from computer science and automata theory. Our extended algebraic specification framework, grounded in graphical structures akin to Ehresmann's sketches, offers a promising solution for integrating knowledge across disparate models and domains. By bridging the gap between domain-specific expertise and machine-generated insights, we pave the way for more comprehensive, collaborative, and effective approaches to knowledge integration and modeling.

Neural networks are sensitive to hyper-parameter and architecture choices. Automated Machine Learning (AutoML) is a promising paradigm for automating these choices. Current ML software libraries, however, are quite limited in handling the dynamic interactions among the components of AutoML. For example, efficientNAS algorithms, such as ENAS and DARTS, typically require an implementation coupling between the search space and search algorithm, the two key components in AutoML. Furthermore, implementing a complex search flow, such as searching architectures within a loop of searching hardware configurations, is difficult. To summarize, changing the search space, search algorithm, or search flow in current ML libraries usually requires a significant change in the program logic. In this paper, we introduce a new way of programming AutoML based on symbolic programming. Under this paradigm, ML programs are mutable, thus can be manipulated easily by another program. As a result, AutoML can be reformulated as an automated process of symbolic manipulation. With this formulation, we decouple the triangle of the search algorithm, the search space and the child program. This decoupling makes it easy to change the search space and search algorithm (without and with weight sharing), as well as to add search capabilities to existing code and implement complex search flows. We then introduce PyGlove, a new Python library that implements this paradigm. Through case studies on ImageNet and NAS-Bench-101, we show that with PyGlove users can easily convert a static program into a search space, quickly iterate on the search spaces and search algorithms, and craft complex search flows to achieve better results.

We present Llemma, a large language model for mathematics. We continue pretraining Code Llama on the Proof-Pile-2, a mixture of scientific papers, web data containing mathematics, and mathematical code, yielding Llemma. On the MATH benchmark Llemma outperforms all known open base models, as well as the unreleased Minerva model suite on an equi-parameter basis. Moreover, Llemma is capable of tool use and formal theorem proving without any further finetuning. We openly release all artifacts, including 7 billion and 34 billion parameter models, the Proof-Pile-2, and code to replicate our experiments.

We present Azimuth, an open-source and easy-to-use tool to perform error analysis for text classification. Compared to other stages of the ML development cycle, such as model training and hyper-parameter tuning, the process and tooling for the error analysis stage are less mature. However, this stage is critical for the development of reliable and trustworthy AI systems. To make error analysis more systematic, we propose an approach comprising dataset analysis and model quality assessment, which Azimuth facilitates. We aim to help AI practitioners discover and address areas where the model does not generalize by leveraging and integrating a range of ML techniques, such as saliency maps, similarity, uncertainty, and behavioral analyses, all in one tool. Our code and documentation are available at github.com/servicenow/azimuth.

Code has been shown to be effective in enhancing the mathematical reasoning abilities of large language models due to its precision and accuracy. Previous works involving continued mathematical pretraining often include code that utilizes math-related packages, which are primarily designed for fields such as engineering, machine learning, signal processing, or module testing, rather than being directly focused on mathematical reasoning. In this paper, we introduce a novel method for generating mathematical code accompanied with corresponding reasoning steps for continued pretraining. Our approach begins with the construction of a high-quality mathematical continued pretraining dataset by incorporating math-related web data, code using mathematical packages, math textbooks, and synthetic data. Next, we construct reasoning steps by extracting LaTeX expressions, the conditions needed for the expressions, and the results of the expressions from the previously collected dataset. Based on this extracted information, we generate corresponding code to accurately capture the mathematical reasoning process. Appending the generated code to each reasoning step results in data consisting of paired natural language reasoning steps and their corresponding code. Combining this data with the original dataset results in a 19.2B-token high-performing mathematical pretraining corpus, which we name MathCode-Pile. Training several popular base models with this corpus significantly improves their mathematical abilities, leading to the creation of the MathCoder2 family of models. All of our data processing and training code is open-sourced, ensuring full transparency and easy reproducibility of the entire data collection and training pipeline. The code is released at https://github.com/mathllm/MathCoder2 .

Mathematical reasoning serves as a cornerstone for assessing the fundamental cognitive capabilities of human intelligence. In recent times, there has been a notable surge in the development of Large Language Models (LLMs) geared towards the automated resolution of mathematical problems. However, the landscape of mathematical problem types is vast and varied, with LLM-oriented techniques undergoing evaluation across diverse datasets and settings. This diversity makes it challenging to discern the true advancements and obstacles within this burgeoning field. This survey endeavors to address four pivotal dimensions: i) a comprehensive exploration of the various mathematical problems and their corresponding datasets that have been investigated; ii) an examination of the spectrum of LLM-oriented techniques that have been proposed for mathematical problem-solving; iii) an overview of factors and concerns affecting LLMs in solving math; and iv) an elucidation of the persisting challenges within this domain. To the best of our knowledge, this survey stands as one of the first extensive examinations of the landscape of LLMs in the realm of mathematics, providing a holistic perspective on the current state, accomplishments, and future challenges in this rapidly evolving field.

Theorem proving in natural mathematical language - the mixture of symbolic and natural language used by humans - plays a central role in mathematical advances and education, and tests aspects of reasoning that are core to intelligence. Yet it has remained underexplored with modern generative models. We study large-scale language models on two new generation tasks: suggesting the next step in a mathematical proof, and full proof generation. We develop NaturalProver, a language model that generates proofs by conditioning on background references (e.g. theorems and definitions that are either retrieved or human-provided), and optionally enforces their presence with constrained decoding. On theorems from the NaturalProofs benchmark, NaturalProver improves the quality of next-step suggestions and generated proofs over fine-tuned GPT-3, according to human evaluations from university-level mathematics students. NaturalProver is capable of proving some theorems that require short (2-6 step) proofs, and providing next-step suggestions that are rated as correct and useful over 40% of the time, which is to our knowledge the first demonstration of these capabilities using neural language models.

Tool-augmented Large Language Models (TALM) are known to enhance the skillset of large language models (LLM), thereby, leading to their improved reasoning abilities across many tasks. While, TALMs have been successfully employed in different question-answering benchmarks, their efficacy on complex mathematical reasoning benchmarks, and the potential complimentary benefits offered by tools for knowledge retrieval and mathematical equation solving, are open research questions. In this work, we present MATHSENSEI, a tool-augmented large language model for mathematical reasoning. Augmented with tools for knowledge retrieval (Bing Web Search), program execution (Python), and symbolic equation solving (Wolfram-Alpha), we study the complimentary benefits of these tools through evaluations on mathematical reasoning datasets. We perform exhaustive ablations on MATH,a popular dataset for evaluating mathematical reasoning on diverse mathematical disciplines. We also conduct experiments involving well-known tool planners to study the impact of tool sequencing on the model performance. MATHSENSEI achieves 13.5% better accuracy over gpt-3.5-turbo with chain-of-thought on the MATH dataset. We further observe that TALMs are not as effective for simpler math word problems (in GSM-8k), and the benefit increases as the complexity and required knowledge increases (progressively over AQuA, MMLU-Math, and higher level complex questions in MATH). The code and data are available at https://github.com/Debrup-61/MathSensei.

We develop tools for explicitly constructing categories enriched over generating data and that compose via ordinary scalar and matrix arithmetic arithmetic operations. We characterize meaningful size maps, weightings, and magnitude that reveal features analogous to outliers that these same notions have previously been shown to reveal in the context of metric spaces. Throughout, we provide examples of such "outlier detection" relevant to the analysis of computer programs, neural networks, cyber-physical systems, and networks of communications channels.

We introduce Explanatory Learning (EL), a framework to let machines use existing knowledge buried in symbolic sequences -- e.g. explanations written in hieroglyphic -- by autonomously learning to interpret them. In EL, the burden of interpreting symbols is not left to humans or rigid human-coded compilers, as done in Program Synthesis. Rather, EL calls for a learned interpreter, built upon a limited collection of symbolic sequences paired with observations of several phenomena. This interpreter can be used to make predictions on a novel phenomenon given its explanation, and even to find that explanation using only a handful of observations, like human scientists do. We formulate the EL problem as a simple binary classification task, so that common end-to-end approaches aligned with the dominant empiricist view of machine learning could, in principle, solve it. To these models, we oppose Critical Rationalist Networks (CRNs), which instead embrace a rationalist view on the acquisition of knowledge. CRNs express several desired properties by construction, they are truly explainable, can adjust their processing at test-time for harder inferences, and can offer strong confidence guarantees on their predictions. As a final contribution, we introduce Odeen, a basic EL environment that simulates a small flatland-style universe full of phenomena to explain. Using Odeen as a testbed, we show how CRNs outperform empiricist end-to-end approaches of similar size and architecture (Transformers) in discovering explanations for novel phenomena.

Quantifying the similarity between two mathematical structures or datasets constitutes a particularly interesting and useful operation in several theoretical and applied problems. Aimed at this specific objective, the Jaccard index has been extensively used in the most diverse types of problems, also motivating some respective generalizations. The present work addresses further generalizations of this index, including its modification into a coincidence index capable of accounting also for the level of relative interiority between the two compared entities, as well as respective extensions for sets in continuous vector spaces, the generalization to multiset addition, densities and generic scalar fields, as well as a means to quantify the joint interdependence between two random variables. The also interesting possibility to take into account more than two sets has also been addressed, including the description of an index capable of quantifying the level of chaining between three structures. Several of the described and suggested eneralizations have been illustrated with respect to numeric case examples. It is also posited that these indices can play an important role while analyzing and integrating datasets in modeling approaches and pattern recognition activities, including as a measurement of clusters similarity or separation and as a resource for representing and analyzing complex networks.

The scarcity of high-quality, logically sound data is a critical bottleneck for advancing the mathematical reasoning of Large Language Models (LLMs). Our work confronts this challenge by turning decades of automated theorem proving research into a scalable data engine. Rather than relying on error-prone LLMs or complex proof-assistant syntax like Lean and Isabelle, our framework leverages E-prover's saturation capabilities on the vast TPTP axiom library to derive a massive, guaranteed-valid corpus of theorems. Our pipeline is principled and simple: saturate axioms, filter for "interesting" theorems, and generate tasks. With no LLMs in the loop, we eliminate factual errors by construction. This purely symbolic data is then transformed into three difficulty-controlled challenges: entailment verification, premise selection, and proof reconstruction. Our zero-shot experiments on frontier models reveal a clear weakness: performance collapses on tasks requiring deep, structural reasoning. Our framework provides both the diagnostic tool to measure this gap and a scalable source of symbolic training data to address it. We make the code and data publicly available. https://github.com/sileod/reasoning_core https://hf.co/datasets/reasoning-core/rc1

Leveraging mathematical Large Language Models (LLMs) for proof generation is a fundamental topic in LLMs research. We argue that the ability of current LLMs to prove statements largely depends on whether they have encountered the relevant proof process during training. This reliance limits their deeper understanding of mathematical theorems and related concepts. Inspired by the pedagogical method of "proof by counterexamples" commonly used in human mathematics education, our work aims to enhance LLMs' ability to conduct mathematical reasoning and proof through counterexamples. Specifically, we manually create a high-quality, university-level mathematical benchmark, CounterMATH, which requires LLMs to prove mathematical statements by providing counterexamples, thereby assessing their grasp of mathematical concepts. Additionally, we develop a data engineering framework to automatically obtain training data for further model improvement. Extensive experiments and detailed analyses demonstrate that CounterMATH is challenging, indicating that LLMs, such as OpenAI o1, have insufficient counterexample-driven proof capabilities. Moreover, our exploration into model training reveals that strengthening LLMs' counterexample-driven conceptual reasoning abilities is crucial for improving their overall mathematical capabilities. We believe that our work offers new perspectives on the community of mathematical LLMs.

Large Language Models (LLMs) have shown remarkable abilities in structured reasoning and symbolic tasks, with coding emerging as a particular area of strength. This success has sparked growing interest in applying LLMs to mathematics, both in informal problem-solving and formal theorem proving. However, progress in formal mathematics has proven to be significantly more difficult, despite surface-level similarities between programming and proof construction. This discrepancy raises important questions about how LLMs ``reason'', how they are supervised, and whether they internally track a notion of computational or deductive state. In this article, we address the state-of-the-art of the discipline, focusing on recent models and benchmarks, and explore three central issues at the intersection of machine learning and mathematical cognition: (i) the trade-offs between formal and informal mathematics as training domains; (ii) the deeper reasons why proof generation remains more brittle than code synthesis; (iii) and the question of whether LLMs represent, or merely mimic, a notion of evolving logical state. Our goal is not to draw hard boundaries, but to identify where the current limits lie, and how they might be extended.

Maintaining legacy software requires many software and systems engineering hours. Assembly code programs, which demand low-level control over the computer machine state and have no variable names, are particularly difficult for humans to analyze. Existing conventional program translators guarantee correctness, but are hand-engineered for the source and target programming languages in question. Learned transpilation, i.e. automatic translation of code, offers an alternative to manual re-writing and engineering efforts. Automated symbolic program translation approaches guarantee correctness but struggle to scale to longer programs due to the exponentially large search space. Their rigid rule-based systems also limit their expressivity, so they can only reason about a reduced space of programs. Probabilistic neural language models (LMs) produce plausible outputs for every input, but do so at the cost of guaranteed correctness. In this work, we leverage the strengths of LMs and symbolic solvers in a neurosymbolic approach to learned transpilation for assembly code. Assembly code is an appropriate setting for a neurosymbolic approach, since assembly code can be divided into shorter non-branching basic blocks amenable to the use of symbolic methods. Guess & Sketch extracts alignment and confidence information from features of the LM then passes it to a symbolic solver to resolve semantic equivalence of the transpilation input and output. We test Guess & Sketch on three different test sets of assembly transpilation tasks, varying in difficulty, and show that it successfully transpiles 57.6% more examples than GPT-4 and 39.6% more examples than an engineered transpiler. We also share a training and evaluation dataset for this task.

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

Vector Symbolic Architectures (VSAs) give a way to represent a complex object as a single fixed-length vector, so that similar objects have similar vector representations. These vector representations then become easy to use for machine learning or nearest-neighbor search. We review a previously proposed VSA method, MBAT (Matrix Binding of Additive Terms), which uses multiplication by random matrices for binding related terms. However, multiplying by such matrices introduces instabilities which can harm performance. Making the random matrices be orthogonal matrices provably fixes this problem. With respect to larger scale applications, we see how to apply MBAT vector representations for any data expressed in JSON. JSON is used in numerous programming languages to express complex data, but its native format appears highly unsuited for machine learning. Expressing JSON as a fixed-length vector makes it readily usable for machine learning and nearest-neighbor search. Creating such JSON vectors also shows that a VSA needs to employ binding operations that are non-commutative. VSAs are now ready to try with full-scale practical applications, including healthcare, pharmaceuticals, and genomics. Keywords: MBAT (Matrix Binding of Additive Terms), VSA (Vector Symbolic Architecture), HDC (Hyperdimensional Computing), Distributed Representations, Binding, Orthogonal Matrices, Recurrent Connections, Machine Learning, Search, JSON, VSA Applications

Graph representations of programs are commonly a central element of machine learning for code research. We introduce an open source Python library python_graphs that applies static analysis to construct graph representations of Python programs suitable for training machine learning models. Our library admits the construction of control-flow graphs, data-flow graphs, and composite ``program graphs'' that combine control-flow, data-flow, syntactic, and lexical information about a program. We present the capabilities and limitations of the library, perform a case study applying the library to millions of competitive programming submissions, and showcase the library's utility for machine learning research.

This paper presents our winning submission to the AI Mathematical Olympiad - Progress Prize 2 (AIMO-2) competition. Our recipe for building state-of-the-art mathematical reasoning models relies on three key pillars. First, we create a large-scale dataset comprising 540K unique high-quality math problems, including olympiad-level problems, and their 3.2M long-reasoning solutions. Second, we develop a novel method to integrate code execution with long reasoning models through iterative training, generation, and quality filtering, resulting in 1.7M high-quality Tool-Integrated Reasoning solutions. Third, we create a pipeline to train models to select the most promising solution from many candidates. We show that such generative solution selection (GenSelect) can significantly improve upon majority voting baseline. Combining these ideas, we train a series of models that achieve state-of-the-art results on mathematical reasoning benchmarks. To facilitate further research, we release our code, models, and the complete OpenMathReasoning dataset under a commercially permissive license.

Large language models have demonstrated impressive capabilities across various natural language processing tasks, especially in solving mathematical problems. However, large language models are not good at math theorem proving using formal languages like Lean. A significant challenge in this area is the scarcity of training data available in these formal languages. To address this issue, we propose a novel pipeline that iteratively generates and filters synthetic data to translate natural language mathematical problems into Lean 4 statements, and vice versa. Our results indicate that the synthetic data pipeline can provide useful training data and improve the performance of LLMs in translating and understanding complex mathematical problems and proofs. Our final dataset contains about 57K formal-informal question pairs along with searched proof from the math contest forum and 21 new IMO questions. We open-source our code at https://github.com/InternLM/InternLM-Math and our data at https://huggingface.co/datasets/InternLM/Lean-Workbook.

Abstract symbolic writing systems are semiotic codes that are ubiquitous in modern society but are otherwise absent in the animal kingdom. Anthropological evidence suggests that the earliest forms of some writing systems originally consisted of iconic pictographs, which signify their referent via visual resemblance. While previous studies have examined the emergence and, separately, the evolution of pictographic writing systems through a computational lens, most employ non-naturalistic methodologies that make it difficult to draw clear analogies to human and animal cognition. We develop a multi-agent reinforcement learning testbed for emergent communication called a Signification Game, and formulate a model of inferential communication that enables agents to leverage visual theory of mind to communicate actions using pictographs. Our model, which is situated within a broader formalism for animal communication, sheds light on the cognitive and cultural processes that led to the development of early writing systems.

Mathematical formulas are a fundamental and widely used component in various scientific fields, serving as a universal language for expressing complex concepts and relationships. While state-of-the-art transformer models excel in processing and understanding natural language, they encounter challenges with mathematical notation, which involves a complex structure and diverse representations. This study focuses on the development of specialized training datasets to enhance the encoding of mathematical content. We introduce Math Mutator (MAMUT), a framework capable of generating equivalent and falsified versions of a given mathematical formula in LaTeX notation, effectively capturing the mathematical variety in notation of the same concept. Based on MAMUT, we have generated four large mathematical datasets containing diverse notation, which can be used to train language models with enhanced mathematical embeddings.

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

Large Language Models (LLMs) have shown human-like reasoning abilities but still struggle with complex logical problems. This paper introduces a novel framework, Logic-LM, which integrates LLMs with symbolic solvers to improve logical problem-solving. Our method first utilizes LLMs to translate a natural language problem into a symbolic formulation. Afterward, a deterministic symbolic solver performs inference on the formulated problem. We also introduce a self-refinement module, which utilizes the symbolic solver's error messages to revise symbolic formalizations. We demonstrate Logic-LM's effectiveness on five logical reasoning datasets: ProofWriter, PrOntoQA, FOLIO, LogicalDeduction, and AR-LSAT. On average, Logic-LM achieves a significant performance boost of 39.2% over using LLM alone with standard prompting and 18.4% over LLM with chain-of-thought prompting. Our findings suggest that Logic-LM, by combining LLMs with symbolic logic, offers a promising avenue for faithful logical reasoning. Code and data are publicly available at https://github.com/teacherpeterpan/Logic-LLM.

Logical reasoning, i.e., deductively inferring the truth value of a conclusion from a set of premises, is an important task for artificial intelligence with wide potential impacts on science, mathematics, and society. While many prompting-based strategies have been proposed to enable Large Language Models (LLMs) to do such reasoning more effectively, they still appear unsatisfactory, often failing in subtle and unpredictable ways. In this work, we investigate the validity of instead reformulating such tasks as modular neurosymbolic programming, which we call LINC: Logical Inference via Neurosymbolic Computation. In LINC, the LLM acts as a semantic parser, translating premises and conclusions from natural language to expressions in first-order logic. These expressions are then offloaded to an external theorem prover, which symbolically performs deductive inference. Leveraging this approach, we observe significant performance gains on FOLIO and a balanced subset of ProofWriter for three different models in nearly all experimental conditions we evaluate. On ProofWriter, augmenting the comparatively small open-source StarCoder+ (15.5B parameters) with LINC even outperforms GPT-3.5 and GPT-4 with Chain-of-Thought (CoT) prompting by an absolute 38% and 10%, respectively. When used with GPT-4, LINC scores 26% higher than CoT on ProofWriter while performing comparatively on FOLIO. Further analysis reveals that although both methods on average succeed roughly equally often on this dataset, they exhibit distinct and complementary failure modes. We thus provide promising evidence for how logical reasoning over natural language can be tackled through jointly leveraging LLMs alongside symbolic provers. All corresponding code is publicly available at https://github.com/benlipkin/linc

We present PutnamBench, a new multilingual benchmark for evaluating the ability of neural theorem-provers to solve competition mathematics problems. PutnamBench consists of 1697 hand-constructed formalizations of 640 theorems sourced from the William Lowell Putnam Mathematical Competition, the premier undergraduate-level mathematics competition in North America. All the theorems have formalizations in Lean 4 and Isabelle; a substantial subset also has Coq formalizations. Proving the theorems requires significant problem-solving ability and proficiency in a broad range of topics taught in undergraduate mathematics courses. We use PutnamBench to evaluate several established neural and symbolic theorem-provers. These approaches can only solve a handful of the PutnamBench problems, establishing the benchmark as a difficult open challenge for research on neural theorem-proving. PutnamBench is available at https://github.com/trishullab/PutnamBench.

Formulas involving fundamental mathematical constants had a great impact on various fields of science and mathematics, for example aiding in proofs of irrationality of constants. However, the discovery of such formulas has historically remained scarce, often perceived as an act of mathematical genius by great mathematicians such as Ramanujan, Euler, and Gauss. Recent efforts to automate the discovery of formulas for mathematical constants, such as the Ramanujan Machine project, relied on exhaustive search. Despite several successful discoveries, exhaustive search remains limited by the space of options that can be covered and by the need for vast amounts of computational resources. Here we propose a fundamentally different method to search for conjectures on mathematical constants: through analysis of integer sequences. We introduce the Enumerated Signed-continued-fraction Massey Approve (ESMA) algorithm, which builds on the Berlekamp-Massey algorithm to identify patterns in integer sequences that represent mathematical constants. The ESMA algorithm found various known formulas for e, e^2, tan(1), and ratios of values of Bessel functions. The algorithm further discovered a large number of new conjectures for these constants, some providing simpler representations and some providing faster numerical convergence than the corresponding simple continued fractions. Along with the algorithm, we present mathematical tools for manipulating continued fractions. These connections enable us to characterize what space of constants can be found by ESMA and quantify its algorithmic advantage in certain scenarios. Altogether, this work continues in the development of augmenting mathematical intuition by computer algorithms, to help reveal mathematical structures and accelerate mathematical research.

Premise selection is a crucial yet challenging step in mathematical formalization, especially for users with limited experience. Due to the lack of available formalization projects, existing approaches that leverage language models often suffer from data scarcity. In this work, we introduce an innovative method for training a premise retriever to support the formalization of mathematics. Our approach employs a BERT model to embed proof states and premises into a shared latent space. The retrieval model is trained within a contrastive learning framework and incorporates a domain-specific tokenizer along with a fine-grained similarity computation method. Experimental results show that our model is highly competitive compared to existing baselines, achieving strong performance while requiring fewer computational resources. Performance is further enhanced through the integration of a re-ranking module. To streamline the formalization process, we will release a search engine that enables users to query Mathlib theorems directly using proof states, significantly improving accessibility and efficiency. Codes are available at https://github.com/ruc-ai4math/Premise-Retrieval.

Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effect of such an intervention as an endofunctor which performs `string diagram surgery' within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on a well-known toy example, where we predict the causal effect of smoking on cancer in the presence of a confounding common cause. After developing this specific example, we show this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature.

Understanding sentences that contain mathematical expressions in text form poses significant challenges. To address this, the importance of converting these expressions into formula images has been highlighted. For instance, the expression ``x equals minus b plus or minus the square root of b squared minus four a c, all over two a'' is more readily comprehensible when displayed as an image x = -b pm sqrt{b^2 - 4ac}{2a}. To develop a text-to-image conversion system, we can break down the process into text-to-LaTeX and LaTeX-to-image conversions, with the latter being managed with by existing various LaTeX engines. However, the former approach has been notably hindered by the severe scarcity of text-to-LaTeX paired data, presenting a significant challenge in the field.In this context, we introduce MathBridge, the first extensive dataset for translating mathematical spoken English into LaTeX, which aims to establish a robust baseline for future research in text-to-LaTeX translation. MathBridge comprises approximately 23 million LaTeX formulas paired with corresponding spoken English expressions. Through comprehensive evaluations, including fine-tuning and testing with data, we discovered that MathBridge significantly enhances pre-trained language models' capabilities for text-to-LaTeX translation. Specifically, for the T5-large model, the sacreBLEU score increased from 4.77 to 46.8, demonstrating substantial enhancement. Our findings indicate the necessity for a new metric specifically for text-to-LaTeX conversion evaluation.

We propose a new definition of higher-order DisCoCat (categorical compositional distributional) models where the meaning of a word is not a diagram, but a diagram-valued higher-order function. Our models can be seen as a variant of Montague semantics based on a lambda calculus where the primitives act on string diagrams rather than logical formulae. As a special case, we show how to translate from the Lambek calculus into Peirce's system beta for first-order logic. This allows us to give a purely diagrammatic treatment of higher-order and non-linear processes in natural language semantics: adverbs, prepositions, negation and quantifiers. The theoretical definition presented in this article comes with a proof-of-concept implementation in DisCoPy, the Python library for string diagrams.

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.

We present an elementary, self-contained proof of Grothendieck's inequality that unifies the real and complex cases and yields both the Krivine and Haagerup bounds, the current best-known explicit bounds for the real and complex Grothendieck constants respectively. This article is intended to be pedagogical, combining and streamlining known ideas of Lindenstrauss--Pe{\l}czy\'nski, Krivine, and Haagerup into a proof that need only univariate calculus, basic complex variables, and a modicum of linear algebra as prerequisites.

Instructing large language models (LLMs) to solve elementary school math problems has shown great success using Chain of Thought (CoT). However, the CoT approach relies on an LLM to generate a sequence of arithmetic calculations which can be prone to cascaded calculation errors. We hypothesize that an LLM should focus on extracting predicates and generating symbolic formulas from the math problem description so that the underlying calculation can be done via an external code interpreter. We investigate using LLM to generate Prolog programs to solve mathematical questions. Experimental results show that our Prolog-based arithmetic problem-solving outperforms CoT generation in the GSM8K benchmark across three distinct LLMs. In addition, given the insensitive ordering of predicates and symbolic formulas in Prolog, we propose to permute the ground truth predicates for more robust LLM training via data augmentation.

We introduce a large-scale dataset of math word problems and an interpretable neural math problem solver that learns to map problems to operation programs. Due to annotation challenges, current datasets in this domain have been either relatively small in scale or did not offer precise operational annotations over diverse problem types. We introduce a new representation language to model precise operation programs corresponding to each math problem that aim to improve both the performance and the interpretability of the learned models. Using this representation language, our new dataset, MathQA, significantly enhances the AQuA dataset with fully-specified operational programs. We additionally introduce a neural sequence-to-program model enhanced with automatic problem categorization. Our experiments show improvements over competitive baselines in our MathQA as well as the AQuA dataset. The results are still significantly lower than human performance indicating that the dataset poses new challenges for future research. Our dataset is available at: https://math-qa.github.io/math-QA/

In the domain of semantic parsing, significant progress has been achieved in Text-to-SQL and question-answering tasks, both of which focus on extracting information from data sources in their native formats. However, the inherent constraints of their formal meaning representations, such as SQL programming language or basic logical forms, hinder their ability to analyze data from various perspectives, such as conducting statistical analyses. To address this limitation and inspire research in this field, we design SIGMA, a new dataset for Text-to-Code semantic parsing with statistical analysis. SIGMA comprises 6000 questions with corresponding Python code labels, spanning across 160 databases. Half of the questions involve query types, which return information in its original format, while the remaining 50% are statistical analysis questions, which perform statistical operations on the data. The Python code labels in our dataset cover 4 types of query types and 40 types of statistical analysis patterns. We evaluated the SIGMA dataset using three different baseline models: LGESQL, SmBoP, and SLSQL. The experimental results show that the LGESQL model with ELECTRA outperforms all other models, achieving 83.37% structure accuracy. In terms of execution accuracy, the SmBoP model, when combined with GraPPa and T5, reaches 76.38%.

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

Automatic program repair seeks to generate correct code from buggy programs, with most approaches searching the correct program in a discrete, symbolic space of source code tokens. This symbolic search is fundamentally limited by its inability to directly reason about program behavior. We introduce Gradient-Based Program Repair (GBPR), a new paradigm that reframes program repair as continuous optimization in a differentiable numerical program space. Our core insight is to compile symbolic programs into differentiable numerical representations, enabling search in the numerical program space directly guided by program behavior. To evaluate GBPR, we present RaspBugs, a new benchmark of 1,466 buggy symbolic RASP programs and their respective numerical representations. Our experiments demonstrate that GBPR can effectively repair buggy symbolic programs by gradient-based optimization in the numerical program space, with convincing repair trajectories. To our knowledge, we are the first to state program repair as continuous optimization in a numerical program space. Our work establishes a new direction for program repair research, bridging two rich worlds: continuous optimization and program behavior.

The remarkable progress of Multi-modal Large Language Models (MLLMs) has garnered unparalleled attention, due to their superior performance in visual contexts. However, their capabilities in visual math problem-solving remain insufficiently evaluated and understood. We investigate current benchmarks to incorporate excessive visual content within textual questions, which potentially assist MLLMs in deducing answers without truly interpreting the input diagrams. To this end, we introduce MathVerse, an all-around visual math benchmark designed for an equitable and in-depth evaluation of MLLMs. We meticulously collect 2,612 high-quality, multi-subject math problems with diagrams from publicly available sources. Each problem is then transformed by human annotators into six distinct versions, each offering varying degrees of information content in multi-modality, contributing to 15K test samples in total. This approach allows MathVerse to comprehensively assess whether and how much MLLMs can truly understand the visual diagrams for mathematical reasoning. In addition, we propose a Chain-of-Thought (CoT) evaluation strategy for a fine-grained assessment of the output answers. Rather than naively judging True or False, we employ GPT-4(V) to adaptively extract crucial reasoning steps, and then score each step with detailed error analysis, which can reveal the intermediate CoT reasoning quality by MLLMs. We hope the MathVerse benchmark may provide unique insights to guide the future development of MLLMs. Project page: https://mathverse-cuhk.github.io

Proving mathematical theorems using computer-verifiable formal languages like Lean significantly impacts mathematical reasoning. One approach to formal theorem proving involves generating complete proofs using Large Language Models (LLMs) based on Natural Language (NL) proofs. Similar methods have shown promising results in code generation. However, most modern LLMs exhibit suboptimal performance due to the scarcity of aligned NL and Formal Language (FL) theorem-proving data. This scarcity results in a paucity of methodologies for training LLMs and techniques to fully utilize their capabilities in composing formal proofs. To address the challenges, this paper proposes **TheoremLlama**, an end-to-end framework to train a general-purpose LLM to become a Lean4 expert. This framework encompasses NL-FL aligned dataset generation methods, training approaches for the LLM formal theorem prover, and techniques for LLM Lean4 proof writing. Using the dataset generation method, we provide *Open Bootstrapped Theorems* (OBT), an NL-FL aligned and bootstrapped dataset. A key innovation in this framework is the NL-FL bootstrapping method, where NL proofs are integrated into Lean4 code for training datasets, leveraging the NL reasoning ability of LLMs for formal reasoning. The **TheoremLlama** framework achieves cumulative accuracies of 36.48% and 33.61% on MiniF2F-Valid and Test datasets respectively, surpassing the GPT-4 baseline of 22.95% and 25.41%. We have also open-sourced our model checkpoints and generated dataset, and will soon make all the code publicly available.

Principal component analysis (PCA) is a mainstay of modern data analysis - a black box that is widely used but (sometimes) poorly understood. The goal of this paper is to dispel the magic behind this black box. This manuscript focuses on building a solid intuition for how and why principal component analysis works. This manuscript crystallizes this knowledge by deriving from simple intuitions, the mathematics behind PCA. This tutorial does not shy away from explaining the ideas informally, nor does it shy away from the mathematics. The hope is that by addressing both aspects, readers of all levels will be able to gain a better understanding of PCA as well as the when, the how and the why of applying this technique.

JSON Schema is an important, evolving standard schema language for families of JSON documents. It is based on a complex combination of structural and Boolean assertions, and features negation and recursion. The static analysis of JSON Schema documents comprises practically relevant problems, including schema satisfiability, inclusion, and equivalence. These three problems can be reduced to witness generation: given a schema, generate an element of the schema, if it exists, and report failure otherwise. Schema satisfiability, inclusion, and equivalence have been shown to be decidable, by reduction to reachability in alternating tree automata. However, no witness generation algorithm has yet been formally described. We contribute a first, direct algorithm for JSON Schema witness generation. We study its effectiveness and efficiency, in experiments over several schema collections, including thousands of real-world schemas. Our focus is on the completeness of the language, where we only exclude the uniqueItems operator, and on the ability of the algorithm to run in a reasonable time on a large set of real-world examples, despite the exponential complexity of the underlying problem.

We extract mathematical concepts from mathematical text using generative large language models (LLMs) like ChatGPT, contributing to the field of automatic term extraction (ATE) and mathematical text processing, and also to the study of LLMs themselves. Our work builds on that of others in that we aim for automatic extraction of terms (keywords) in one mathematical field, category theory, using as a corpus the 755 abstracts from a snapshot of the online journal "Theory and Applications of Categories", circa 2020. Where our study diverges from previous work is in (1) providing a more thorough analysis of what makes mathematical term extraction a difficult problem to begin with; (2) paying close attention to inter-annotator disagreements; (3) providing a set of guidelines which both human and machine annotators could use to standardize the extraction process; (4) introducing a new annotation tool to help humans with ATE, applicable to any mathematical field and even beyond mathematics; (5) using prompts to ChatGPT as part of the extraction process, and proposing best practices for such prompts; and (6) raising the question of whether ChatGPT could be used as an annotator on the same level as human experts. Our overall findings are that the matter of mathematical ATE is an interesting field which can benefit from participation by LLMs, but LLMs themselves cannot at this time surpass human performance on it.

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

Formal verification can provably guarantee the correctness of critical system software, but the high proof burden has long hindered its wide adoption. Recently, Large Language Models (LLMs) have shown success in code analysis and synthesis. In this paper, we present a combination of LLMs and static analysis to synthesize invariants, assertions, and other proof structures for a Rust-based formal verification framework called Verus. In a few-shot setting, LLMs demonstrate impressive logical ability in generating postconditions and loop invariants, especially when analyzing short code snippets. However, LLMs lack the ability to retain and propagate context information, a strength of traditional static analysis. Based on these observations, we developed a prototype based on OpenAI's GPT-4 model. Our prototype decomposes the verification task into multiple smaller ones, iteratively queries GPT-4, and combines its output with lightweight static analysis. We evaluated the prototype with a developer in the automation loop on 20 vector-manipulating programs. The results demonstrate that it significantly reduces human effort in writing entry-level proof code.

Advanced applied mathematics problems are underrepresented in existing Large Language Model (LLM) benchmark datasets. To address this, we introduce HARDMath, a dataset inspired by a graduate course on asymptotic methods, featuring challenging applied mathematics problems that require analytical approximation techniques. These problems demand a combination of mathematical reasoning, computational tools, and subjective judgment, making them difficult for LLMs. Our framework auto-generates a large number of problems with solutions validated against numerical ground truths. We evaluate both open- and closed-source LLMs on HARDMath-mini, a sub-sampled test set of 366 problems, as well as on 40 word problems formulated in applied science contexts. Even leading closed-source models like GPT-4 achieve only 43.8% overall accuracy with few-shot Chain-of-Thought prompting, and all models demonstrate significantly lower performance compared to results on existing mathematics benchmark datasets. We additionally conduct a detailed error analysis to gain insights into the failure cases of LLMs. These results demonstrate limitations of current LLM performance on advanced graduate-level applied math problems and underscore the importance of datasets like HARDMath to advance mathematical abilities of LLMs.

Large Language Models (LLMs) have exhibited exceptional performance across a broad range of tasks and domains. However, they still encounter difficulties in solving mathematical problems due to the rigorous and logical nature of mathematics. Previous studies have employed techniques such as supervised fine-tuning (SFT), prompt engineering, and search-based methods to improve the mathematical problem-solving abilities of LLMs. Despite these efforts, their performance remains suboptimal and demands substantial computational resources. To address this issue, we propose a novel approach, BEATS, to enhance mathematical problem-solving abilities. Our method leverages newly designed prompts that guide the model to iteratively rewrite, advance by one step, and generate answers based on previous steps. Additionally, we introduce a new back-verification technique that uses LLMs to validate the correctness of the generated answers. Furthermore, we employ a pruning tree search to optimize search time while achieving strong performance. Notably, our method improves Qwen2-7b-Instruct's score from 36.94 to 61.52, outperforming GPT4's 42.5 on the MATH benchmark.

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

This is the compendium of the cluster algebra and quiver package for sage. The purpose of this package is to provide a platform to work with cluster algebras in graduate courses and to further develop the theory by working on examples, by gathering data, and by exhibiting and testing conjectures. In this compendium, we include the relevant theory to introduce the reader to cluster algebras assuming no prior background; this exposition has been written to be accessible to an interested undergraduate. Throughout this compendium, we include examples that the user can run in the sage notebook or command line, and then close with a detailed description of the data structures and methods in this package.

LLM-based formal proof assistants (e.g., in Lean) hold great promise for automating mathematical discovery. But beyond syntactic correctness, do these systems truly understand mathematical structure as humans do? We investigate this question through the lens of mathematical inequalities -- a fundamental tool across many domains. While modern provers can solve basic inequalities, we probe their ability to handle human-intuitive compositionality. We introduce Ineq-Comp, a benchmark built from elementary inequalities through systematic transformations, including variable duplication, algebraic rewriting, and multi-step composition. Although these problems remain easy for humans, we find that most provers -- including Goedel, STP, and Kimina-7B -- struggle significantly. DeepSeek-Prover-V2-7B shows relative robustness -- possibly because it is trained to decompose the problems into sub-problems -- but still suffers a 20\% performance drop (pass@32). Strikingly, performance remains poor for all models even when formal proofs of the constituent parts are provided in context, revealing that the source of weakness is indeed in compositional reasoning. Our results expose a persisting gap between the generalization behavior of current AI provers and human mathematical intuition.

Progress in AI is hindered by the lack of a programming language with all the requisite features. Libraries like PyTorch and TensorFlow provide automatic differentiation and efficient GPU implementation, but are additions to Python, which was never intended for AI. Their lack of support for automated reasoning and knowledge acquisition has led to a long and costly series of hacky attempts to tack them on. On the other hand, AI languages like LISP an Prolog lack scalability and support for learning. This paper proposes tensor logic, a language that solves these problems by unifying neural and symbolic AI at a fundamental level. The sole construct in tensor logic is the tensor equation, based on the observation that logical rules and Einstein summation are essentially the same operation, and all else can be reduced to them. I show how to elegantly implement key forms of neural, symbolic and statistical AI in tensor logic, including transformers, formal reasoning, kernel machines and graphical models. Most importantly, tensor logic makes new directions possible, such as sound reasoning in embedding space. This combines the scalability and learnability of neural networks with the reliability and transparency of symbolic reasoning, and is potentially a basis for the wider adoption of AI.

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

We present a complete formalization in Isabelle/HOL of the object part of an equivalence between L-mosaics and bounded join-semilattices, employing an AI-assisted methodology that integrates large language models as reasoning assistants throughout the proof development process. The equivalence was originally established by Cangiotti, Linzi, and Talotti in their study of hypercompositional structures related to orthomodular lattices and quantum logic. Our formalization rigorously verifies the main theoretical result and demonstrates the mutual inverse property of the transformations establishing this equivalence. The development showcases both the mathematical depth of multivalued algebraic operations and the potential for AI-enhanced interactive theorem proving in tackling complex formalization projects.

High-quality, large-scale corpora are the cornerstone of building foundation models. In this work, we introduce MathPile, a diverse and high-quality math-centric corpus comprising about 9.5 billion tokens. Throughout its creation, we adhered to the principle of ``less is more'', firmly believing in the supremacy of data quality over quantity, even in the pre-training phase. Our meticulous data collection and processing efforts included a complex suite of preprocessing, prefiltering, language identification, cleaning, filtering, and deduplication, ensuring the high quality of our corpus. Furthermore, we performed data contamination detection on downstream benchmark test sets to eliminate duplicates. We hope our MathPile can help to enhance the mathematical reasoning abilities of language models. We plan to open-source different versions of \mathpile with the scripts used for processing, to facilitate future developments in this field.

The math abilities of large language models can represent their abstract reasoning ability. In this paper, we introduce and open-source our math reasoning LLMs InternLM-Math which is continue pre-trained from InternLM2. We unify chain-of-thought reasoning, reward modeling, formal reasoning, data augmentation, and code interpreter in a unified seq2seq format and supervise our model to be a versatile math reasoner, verifier, prover, and augmenter. These abilities can be used to develop the next math LLMs or self-iteration. InternLM-Math obtains open-sourced state-of-the-art performance under the setting of in-context learning, supervised fine-tuning, and code-assisted reasoning in various informal and formal benchmarks including GSM8K, MATH, Hungary math exam, MathBench-ZH, and MiniF2F. Our pre-trained model achieves 30.3 on the MiniF2F test set without fine-tuning. We further explore how to use LEAN to solve math problems and study its performance under the setting of multi-task learning which shows the possibility of using LEAN as a unified platform for solving and proving in math. Our models, codes, and data are released at https://github.com/InternLM/InternLM-Math.

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.

In 2011, einsum was introduced to NumPy as a practical and convenient notation for tensor expressions in machine learning, quantum circuit simulation, and other fields. It has since been implemented in additional Python frameworks such as PyTorch and TensorFlow, as well as in other programming languages such as Julia. Despite its practical success, the einsum notation still lacks a solid theoretical basis, and is not unified across the different frameworks, limiting opportunities for formal reasoning and systematic optimization. In this work, we discuss the terminology of tensor expressions and provide a formal definition of the einsum language. Based on this definition, we formalize and prove important equivalence rules for tensor expressions and highlight their relevance in practical applications.

The field of Automatic Machine Learning (AutoML) has recently attained impressive results, including the discovery of state-of-the-art machine learning solutions, such as neural image classifiers. This is often done by applying an evolutionary search method, which samples multiple candidate solutions from a large space and evaluates the quality of each candidate through a long training process. As a result, the search tends to be slow. In this paper, we show that large efficiency gains can be obtained by employing a fast unified functional hash, especially through the functional equivalence caching technique, which we also present. The central idea is to detect by hashing when the search method produces equivalent candidates, which occurs very frequently, and this way avoid their costly re-evaluation. Our hash is "functional" in that it identifies equivalent candidates even if they were represented or coded differently, and it is "unified" in that the same algorithm can hash arbitrary representations; e.g. compute graphs, imperative code, or lambda functions. As evidence, we show dramatic improvements on multiple AutoML domains, including neural architecture search and algorithm discovery. Finally, we consider the effect of hash collisions, evaluation noise, and search distribution through empirical analysis. Altogether, we hope this paper may serve as a guide to hashing techniques in AutoML.

Combining existing pre-trained expert LLMs is a promising avenue for scalably tackling large-scale and diverse tasks. However, selecting experts at the task level is often too coarse-grained, as heterogeneous tasks may require different expertise for each instance. To enable adaptive instance-level mixing of pre-trained LLM experts, we propose Symbolic-MoE, a symbolic, text-based, and gradient-free Mixture-of-Experts framework. Symbolic-MoE takes a fine-grained approach to selection by emphasizing skills, e.g., algebra in math or molecular biology in biomedical reasoning. We propose a skill-based recruiting strategy that dynamically selects the most relevant set of expert LLMs for diverse reasoning tasks based on their strengths. Each selected expert then generates its own reasoning, resulting in k outputs from k experts, which are then synthesized into a final high-quality response by an aggregator chosen based on its ability to integrate diverse reasoning outputs. We show that Symbolic-MoE's instance-level expert selection improves performance by a large margin but -- when implemented naively -- can introduce a high computational overhead due to the need for constant model loading and offloading. To address this, we implement a batch inference strategy that groups instances based on their assigned experts, loading each model only once. This allows us to integrate 16 expert models on 1 GPU with a time cost comparable to or better than prior multi-agent baselines using 4 GPUs. Through extensive evaluations on diverse benchmarks (MMLU-Pro, GPQA, AIME, and MedMCQA), we demonstrate that Symbolic-MoE outperforms strong LLMs like GPT4o-mini, as well as multi-agent approaches, with an absolute average improvement of 8.15% over the best multi-agent baseline. Moreover, Symbolic-MoE removes the need for expensive multi-round discussions, outperforming discussion baselines with less computation.

These handouts are designed for people who is just starting involved with the topic artificial neural networks. We show how it works a single artificial neuron (McCulloch & Pitt model), mathematically and graphically. We do explain the delta rule, a learning algorithm to find the neuron weights. We also present some examples in MATLAB/Octave. There are examples for classification task for lineal and non-lineal problems. At the end, we present an artificial neural network, a feed-forward neural network along its learning algorithm backpropagation. ----- Estos apuntes est\'an dise\~nados para personas que por primera vez se introducen en el tema de las redes neuronales artificiales. Se muestra el funcionamiento b\'asico de una neurona, matem\'aticamente y gr\'aficamente. Se explica la Regla Delta, algoritmo deaprendizaje para encontrar los pesos de una neurona. Tambi\'en se muestran ejemplos en MATLAB/Octave. Hay ejemplos para problemas de clasificaci\'on, para problemas lineales y no-lineales. En la parte final se muestra la arquitectura de red neuronal artificial conocida como backpropagation.

Large Language Models (LLMs) are thought to struggle with arithmetic learning due to the inherent differences between language modeling and numerical computation, but concrete evidence has been lacking. This work responds to this claim through a two-side experiment. We first investigate whether LLMs leverage partial products during arithmetic learning. We find that although LLMs can identify some partial products after learning, they fail to leverage them for arithmetic tasks, conversely. We then explore how LLMs approach arithmetic symbolically by breaking tasks into subgroups, hypothesizing that difficulties arise from subgroup complexity and selection. Our results show that when subgroup complexity is fixed, LLMs treat a collection of different arithmetic operations similarly. By analyzing position-level accuracy across different training sizes, we further observe that it follows a U-shaped pattern: LLMs quickly learn the easiest patterns at the first and last positions, while progressively learning the more difficult patterns in the middle positions. This suggests that LLMs select subgroup following an easy-to-hard paradigm during learning. Our work confirms that LLMs are pure symbolic learners in arithmetic tasks and underscores the importance of understanding them deeply through subgroup-level quantification.

We present GenesisGeo, an automated theorem prover in Euclidean geometry. We have open-sourced a large-scale geometry dataset of 21.8 million geometric problems, over 3 million of which contain auxiliary constructions. Specially, we significantly accelerate the symbolic deduction engine DDARN by 120x through theorem matching, combined with a C++ implementation of its core components. Furthermore, we build our neuro-symbolic prover, GenesisGeo, upon Qwen3-0.6B-Base, which solves 24 of 30 problems (IMO silver medal level) in the IMO-AG-30 benchmark using a single model, and achieves 26 problems (IMO gold medal level) with a dual-model ensemble.

We introduce MathWriting, the largest online handwritten mathematical expression dataset to date. It consists of 230k human-written samples and an additional 400k synthetic ones. MathWriting can also be used for offline HME recognition and is larger than all existing offline HME datasets like IM2LATEX-100K. We introduce a benchmark based on MathWriting data in order to advance research on both online and offline HME recognition.

This study introduces an evaluation benchmark for middle school algebra to be used in artificial intelligence(AI) based educational platforms. The goal is to support the design of AI systems that can enhance learner conceptual understanding of algebra by taking into account their current level of algebra comprehension. The data set comprises 55 misconceptions about algebra, common errors, and 220 diagnostic examples identified in previous peer-reviewed studies. We provide an example application using a large language model, observing a range of precision and recall scores depending on the topic and experimental setup that reaches 83.9% when including educator feedback and restricting it by topic. We found that topics such as ratios and proportions prove as difficult for LLMs as they are for students. We included a human assessment of LLMs results and feedback from five middle school math educators on the clarity and occurrence of misconceptions in the dataset and the potential use of AI in conjunction with the dataset. Most educators (80% or more) indicated that they encounter these misconceptions among their students, suggesting the relevance of the data set to teaching middle school algebra. Despite varying familiarity with AI tools, four out of five educators expressed interest in using the data set with AI to diagnose student misconceptions or train teachers. The results emphasize the importance of topic-constrained testing, the need for multimodal approaches, and the relevance of human expertise to gain practical insights when using AI for human learning.

We introduce Goedel-Prover, an open-source large language model (LLM) that achieves the state-of-the-art (SOTA) performance in automated formal proof generation for mathematical problems. The key challenge in this field is the scarcity of formalized math statements and proofs, which we tackle in the following ways. We train statement formalizers to translate the natural language math problems from Numina into formal language (Lean 4), creating a dataset of 1.64 million formal statements. LLMs are used to check that the formal statements accurately preserve the content of the original natural language problems. We then iteratively build a large dataset of formal proofs by training a series of provers. Each prover succeeds in proving many statements that the previous ones could not, and these new proofs are added to the training set for the next prover. The final prover outperforms all existing open-source models in whole-proof generation. On the miniF2F benchmark, it achieves a 57.6% success rate (Pass@32), exceeding the previous best open-source model by 7.6%. On PutnamBench, Goedel-Prover successfully solves 7 problems (Pass@512), ranking first on the leaderboard. Furthermore, it generates 29.7K formal proofs for Lean Workbook problems, nearly doubling the 15.7K produced by earlier works.

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

Large language models (LLMs) have significantly advanced natural language understanding and demonstrated strong problem-solving abilities. Despite these successes, most LLMs still struggle with solving mathematical problems due to the intricate reasoning required. This paper investigates the mathematical problem-solving capabilities of LLMs using the newly developed "MathOdyssey" dataset. The dataset includes diverse mathematical problems at high school and university levels, created by experts from notable institutions to rigorously test LLMs in advanced problem-solving scenarios and cover a wider range of subject areas. By providing the MathOdyssey dataset as a resource to the AI community, we aim to contribute to the understanding and improvement of AI capabilities in complex mathematical problem-solving. We conduct benchmarking on open-source models, such as Llama-3 and DBRX-Instruct, and closed-source models from the GPT series and Gemini models. Our results indicate that while LLMs perform well on routine and moderately difficult tasks, they face significant challenges with Olympiad-level problems and complex university-level questions. Our analysis shows a narrowing performance gap between open-source and closed-source models, yet substantial challenges remain, particularly with the most demanding problems. This study highlights the ongoing need for research to enhance the mathematical reasoning of LLMs. The dataset, results, and code are publicly available.

Machine learning is becoming an increasingly valuable tool in mathematics, enabling one to identify subtle patterns across collections of examples so vast that they would be impossible for a single researcher to feasibly review and analyze. In this work, we use graph neural networks to investigate quiver mutation -- an operation that transforms one quiver (or directed multigraph) into another -- which is central to the theory of cluster algebras with deep connections to geometry, topology, and physics. In the study of cluster algebras, the question of mutation equivalence is of fundamental concern: given two quivers, can one efficiently determine if one quiver can be transformed into the other through a sequence of mutations? In this paper, we use graph neural networks and AI explainability techniques to independently discover mutation equivalence criteria for quivers of type D. Along the way, we also show that even without explicit training to do so, our model captures structure within its hidden representation that allows us to reconstruct known criteria from type D, adding to the growing evidence that modern machine learning models are capable of learning abstract and parsimonious rules from mathematical data.

In this paper, we present a novel framework for the analysis of Riemann Hypothesis [27], which is composed of three key components: a) probabilistic modeling with cross entropy optimization and reasoning; b) the application of the law of large numbers; c) the application of mathematical inductions. The analysis is mainly conducted by virtue of probabilistic modeling of cross entropy optimization and reasoning with rare event simulation techniques. The application of the law of large numbers [2, 3, 6] and the application of mathematical inductions make the analysis of Riemann Hypothesis self-contained and complete to make sure that the whole complex plane is covered as conjectured in Riemann Hypothesis. We also discuss the method of enhanced top-p sampling with large language models (LLMs) for reasoning, where next token prediction is not just based on the estimated probabilities of each possible token in the current round but also based on accumulated path probabilities among multiple top-k chain of thoughts (CoTs) paths. The probabilistic modeling of cross entropy optimization and reasoning may suit well with the analysis of Riemann Hypothesis as Riemann Zeta functions are inherently dealing with the sums of infinite components of a complex number series. We hope that our analysis in this paper could shed some light on some of the insights of Riemann Hypothesis. The framework and techniques presented in this paper, coupled with recent developments with chain of thought (CoT) or diagram of thought (DoT) reasoning in large language models (LLMs) with reinforcement learning (RL) [1, 7, 18, 21, 24, 34, 39-41], could pave the way for eventual proof of Riemann Hypothesis [27].