Daily Papers

Neural networks adapt very well to distributed and continuous representations, but struggle to generalize from small amounts of data. Symbolic systems commonly achieve data efficient generalization by exploiting modularity to benefit from local and discrete features of a representation. These features allow symbolic programs to be improved one module at a time and to experience combinatorial growth in the values they can successfully process. However, it is difficult to design a component that can be used to form symbolic abstractions and which is adequately overparametrized to learn arbitrary high-dimensional transformations. I present Graph-based Symbolically Synthesized Neural Networks (G-SSNNs), a class of neural modules that operate on representations modified with synthesized symbolic programs to include a fixed set of local and discrete features. I demonstrate that the choice of injected features within a G-SSNN module modulates the data efficiency and generalization of baseline neural models, creating predictable patterns of both heightened and curtailed generalization. By training G-SSNNs, we also derive information about desirable semantics of symbolic programs without manual engineering. This information is compact and amenable to abstraction, but can also be flexibly recontextualized for other high-dimensional settings. In future work, I will investigate data efficient generalization and the transferability of learned symbolic representations in more complex G-SSNN designs based on more complex classes of symbolic programs. Experimental code and data are available at https://github.com/shlomenu/symbolically_synthesized_networks .

Expert problem-solving is driven by powerful languages for thinking about problems and their solutions. Acquiring expertise means learning these languages -- systems of concepts, alongside the skills to use them. We present DreamCoder, a system that learns to solve problems by writing programs. It builds expertise by creating programming languages for expressing domain concepts, together with neural networks to guide the search for programs within these languages. A ``wake-sleep'' learning algorithm alternately extends the language with new symbolic abstractions and trains the neural network on imagined and replayed problems. DreamCoder solves both classic inductive programming tasks and creative tasks such as drawing pictures and building scenes. It rediscovers the basics of modern functional programming, vector algebra and classical physics, including Newton's and Coulomb's laws. Concepts are built compositionally from those learned earlier, yielding multi-layered symbolic representations that are interpretable and transferrable to new tasks, while still growing scalably and flexibly with experience.

Generalizing from individual skill executions to solving long-horizon tasks remains a core challenge in building autonomous agents. A promising direction is learning high-level, symbolic abstractions of the low-level skills of the agents, enabling reasoning and planning independent of the low-level state space. Among possible high-level representations, object-centric skill abstraction with symbolic predicates has been proven to be efficient because of its compatibility with domain-independent planners. Recent advances in foundation models have made it possible to generate symbolic predicates that operate on raw sensory inputs, a process we call generative predicate invention, to facilitate downstream abstraction learning. However, it remains unclear which formal properties the learned representations must satisfy, and how they can be learned to guarantee these properties. In this paper, we address both questions by presenting a formal theory of generative predicate invention for skill abstraction, resulting in symbolic operators that can be used for provably sound and complete planning. Within this framework, we propose SkillWrapper, a method that leverages foundation models to actively collect robot data and learn human-interpretable, plannable representations of black-box skills, using only RGB image observations. Our extensive empirical evaluation in simulation and on real robots shows that SkillWrapper learns abstract representations that enable solving unseen, long-horizon tasks in the real world with black-box skills.

Large language models (LLMs) have shown strong performance in many reasoning benchmarks. However, recent studies have pointed to memorization, rather than generalization, as one of the leading causes for such performance. LLMs, in fact, are susceptible to content variations, demonstrating a lack of robust planning or symbolic abstractions supporting their reasoning process. To improve reliability, many attempts have been made to combine LLMs with symbolic methods. Nevertheless, existing approaches fail to effectively leverage symbolic representations due to the challenges involved in developing reliable and scalable verification mechanisms. In this paper, we propose to overcome such limitations by synthesizing high-quality symbolic reasoning trajectories with stepwise pseudo-labels at scale via Monte Carlo estimation. A Process Reward Model (PRM) can be efficiently trained based on the synthesized data and then used to select more symbolic trajectories. The trajectories are then employed with Direct Preference Optimization (DPO) and Supervised Fine-Tuning (SFT) to improve logical reasoning and generalization. Our results on benchmarks (i.e., FOLIO and LogicAsker) show the effectiveness of the proposed method with gains on frontier and open-weight models. Moreover, additional experiments on claim verification data reveal that fine-tuning on the generated symbolic reasoning trajectories enhances out-of-domain generalizability, suggesting the potential impact of the proposed method in enhancing planning and logical reasoning.

DevOps programming (e.g., using CLI/API scripts or IaC frameworks) is key to cloud infrastructure management. Unlike traditional programming tasks, DevOps program testing needs provisioning and execution against actual cloud resources, which is often time-consuming, unsafe, and costly. Cloud emulators have gained popularity for easing DevOps program testing; they are generally API-level mocks that can execute DevOps programs in a local environment. Still, building these emulators remains challenging: developers must manually interpret extensive cloud documentation and handcraft logic for each service, API, and their interaction. This does not scale to the complexity of the cloud, which is further a moving target as the services and APIs evolve. CloudEmu is an automated approach that constructs emulators based on cloud documentation via neurosymbolic code synthesis. The key idea is to combine LLMs' general strengths in documentation understanding and code generation with cloud-specific symbolic abstractions that suppress hallucinations and enforce precision at scale, while using the real cloud as an oracle for automated testing, repair, and alignment. Our evaluation shows the effectiveness of CloudEmu on major cloud provider (AWS and GCP) services in both coverage and accuracy. CloudEmu outperforms the existing leading tool LocalStack, which was manually developed by a large team of engineers over a decade.

Learning deep discrete latent presentations offers a promise of better symbolic and summarized abstractions that are more useful to subsequent downstream tasks. Inspired by the seminal Vector Quantized Variational Auto-Encoder (VQ-VAE), most of work in learning deep discrete representations has mainly focused on improving the original VQ-VAE form and none of them has studied learning deep discrete representations from the generative viewpoint. In this work, we study learning deep discrete representations from the generative viewpoint. Specifically, we endow discrete distributions over sequences of codewords and learn a deterministic decoder that transports the distribution over the sequences of codewords to the data distribution via minimizing a WS distance between them. We develop further theories to connect it with the clustering viewpoint of WS distance, allowing us to have a better and more controllable clustering solution. Finally, we empirically evaluate our method on several well-known benchmarks, where it achieves better qualitative and quantitative performances than the other VQ-VAE variants in terms of the codebook utilization and image reconstruction/generation.

Unified multimodal models (UMMs) have emerged as a powerful paradigm for seamlessly unifying text and image understanding and generation. However, prevailing evaluations treat these abilities in isolation, such that tasks with multimodal inputs and outputs are scored primarily through unimodal reasoning, i.e., textual benchmarks emphasize language-based reasoning, while visual benchmarks emphasize reasoning outcomes manifested in the pixels. We introduce ROVER to address this pressing need to test reciprocal cross-modal reasoning, the use of one modality to guide, verify, or refine outputs in the other, an ability central to the vision of unified multimodal intelligence. ROVER is a human-annotated benchmark that explicitly targets reciprocal cross-modal reasoning, which contains 1312 tasks grounded in 1876 images, spanning two complementary settings. Verbally-augmented reasoning for visual generation evaluates whether models can use verbal prompts and reasoning chains to guide faithful image synthesis. Visually-augmented reasoning for verbal generation evaluates whether models can generate intermediate visualizations that strengthen their own reasoning processes for question answering. Experiments on 17 unified models reveal two key findings: (i) Cross-modal reasoning determines visual generation quality, with interleaved models significantly outperforming non-interleaved ones; notably, combining strong unimodal models fails to achieve comparable reasoning. (ii) Models show dissociation between physical and symbolic reasoning: they succeed at interpreting perceptual concepts literally but fail to construct visual abstractions for symbolic tasks, where faulty reasoning harms performance. These results highlight reciprocal cross-modal reasoning as a critical frontier for enabling true omnimodal generation.

Dynamic shape computations have become critical in modern machine learning workloads, especially in emerging large language models. The success of these models has driven demand for deploying them to a diverse set of backend environments. In this paper, we present Relax, a compiler abstraction for optimizing end-to-end dynamic machine learning workloads. Relax introduces first-class symbolic shape annotations to track dynamic shape computations globally across the program. It also introduces a cross-level abstraction that encapsulates computational graphs, loop-level tensor programs, and library calls in a single representation to enable cross-level optimizations. We build an end-to-end compilation framework using the proposed approach to optimize dynamic shape models. Experimental results on large language models show that Relax delivers performance competitive with state-of-the-art hand-optimized systems across platforms and enables deployment of emerging dynamic models to a broader set of environments, including mobile phones, embedded devices, and web browsers.

Toward combining inductive reasoning with perception abilities, we develop techniques for neurosymbolic program synthesis where perceptual input is first parsed by neural nets into a low-dimensional interpretable representation, which is then processed by a synthesized program. We explore several techniques for relaxing the problem and jointly learning all modules end-to-end with gradient descent: multitask learning; amortized inference; overparameterization; and a differentiable strategy for penalizing lengthy programs. Collectedly this toolbox improves the stability of gradient-guided program search, and suggests ways of learning both how to perceive input as discrete abstractions, and how to symbolically process those abstractions as programs.

While large language models (LLMs) now excel at code generation, a key aspect of software development is the art of refactoring: consolidating code into libraries of reusable and readable programs. In this paper, we introduce LILO, a neurosymbolic framework that iteratively synthesizes, compresses, and documents code to build libraries tailored to particular problem domains. LILO combines LLM-guided program synthesis with recent algorithmic advances in automated refactoring from Stitch: a symbolic compression system that efficiently identifies optimal lambda abstractions across large code corpora. To make these abstractions interpretable, we introduce an auto-documentation (AutoDoc) procedure that infers natural language names and docstrings based on contextual examples of usage. In addition to improving human readability, we find that AutoDoc boosts performance by helping LILO's synthesizer to interpret and deploy learned abstractions. We evaluate LILO on three inductive program synthesis benchmarks for string editing, scene reasoning, and graphics composition. Compared to existing neural and symbolic methods - including the state-of-the-art library learning algorithm DreamCoder - LILO solves more complex tasks and learns richer libraries that are grounded in linguistic knowledge.

Programming with Generative AI (GenAI) models is a type of Neurosymbolic programming and has seen tremendous adoption across many domains. However, leveraging GenAI models in code today can be complex, counter-intuitive and often require specialized frameworks, leading to increased complexity. This is because it is currently unclear as to the right abstractions through which we should marry GenAI models with the nature of traditional programming code constructs. In this paper, we introduce a set of novel abstractions to help bridge the gap between Neuro- and symbolic programming. We introduce Meaning, a new specialized type that represents the underlying semantic value of traditional types (e.g., string). We make the case that GenAI models, LLMs in particular, should be reasoned as a meaning-type wrapped code construct at the language level. We formulate the problem of translation between meaning and traditional types and propose Automatic Meaning-Type Transformation (A-MTT), a runtime feature that abstracts this translation away from the developers by automatically converting between M eaning and types at the interface of LLM invocation. Leveraging this new set of code constructs and OTT, we demonstrate example implementation of neurosymbolic programs that seamlessly utilizes LLMs to solve problems in place of potentially complex traditional programming logic.

Many recent studies have found evidence for emergent reasoning capabilities in large language models (LLMs), but debate persists concerning the robustness of these capabilities, and the extent to which they depend on structured reasoning mechanisms. To shed light on these issues, we study the internal mechanisms that support abstract reasoning in LLMs. We identify an emergent symbolic architecture that implements abstract reasoning via a series of three computations. In early layers, symbol abstraction heads convert input tokens to abstract variables based on the relations between those tokens. In intermediate layers, symbolic induction heads perform sequence induction over these abstract variables. Finally, in later layers, retrieval heads predict the next token by retrieving the value associated with the predicted abstract variable. These results point toward a resolution of the longstanding debate between symbolic and neural network approaches, suggesting that emergent reasoning in neural networks depends on the emergence of symbolic mechanisms.

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.

Learning to manipulate dynamic and deformable objects from a single demonstration video holds great promise in terms of scalability. Previous approaches have predominantly focused on either replaying object relationships or actor trajectories. The former often struggles to generalize across diverse tasks, while the latter suffers from data inefficiency. Moreover, both methodologies encounter challenges in capturing invisible physical attributes, such as forces. In this paper, we propose to interpret video demonstrations through Parameterized Symbolic Abstraction Graphs (PSAG), where nodes represent objects and edges denote relationships between objects. We further ground geometric constraints through simulation to estimate non-geometric, visually imperceptible attributes. The augmented PSAG is then applied in real robot experiments. Our approach has been validated across a range of tasks, such as Cutting Avocado, Cutting Vegetable, Pouring Liquid, Rolling Dough, and Slicing Pizza. We demonstrate successful generalization to novel objects with distinct visual and physical properties.

Loop invariants are fundamental to reasoning about programs with loops. They establish properties about a given loop's behavior. When they additionally are inductive, they become useful for the task of formal verification that seeks to establish strong mathematical guarantees about program's runtime behavior. The inductiveness ensures that the invariants can be checked locally without consulting the entire program, thus are indispensable artifacts in a formal proof of correctness. Finding inductive loop invariants is an undecidable problem, and despite a long history of research towards practical solutions, it remains far from a solved problem. This paper investigates the capabilities of the Large Language Models (LLMs) in offering a new solution towards this old, yet important problem. To that end, we first curate a dataset of verification problems on programs with loops. Next, we design a prompt for exploiting LLMs, obtaining inductive loop invariants, that are checked for correctness using sound symbolic tools. Finally, we explore the effectiveness of using an efficient combination of a symbolic tool and an LLM on our dataset and compare it against a purely symbolic baseline. Our results demonstrate that LLMs can help improve the state-of-the-art in automated program verification.

Modern systems in artificial intelligence (AI) somehow excel in domains for which they seem poorly suited. Intelligence has traditionally been modeled as operating over structured combinations of symbols, such as logical formulas. However, the strongest modern AI systems are based on neural networks, which instead represent information in continuous vectors. Vectors seem inadequate for capturing the structure of language, logic, and other cognitive domains, yet neural networks achieve impressive performance in these areas. How do they do it? In this work, we propose a potential answer: Despite appearances, perhaps the internal representations of neural networks implicitly realize symbolic structure. In support of this hypothesis, we show that the vector representations of a variety of neural networks can be closely approximated with symbolic structures: we can replace the network's entire representation-generating process with a closed-form equation instantiating a symbolic structure, and the network's behavior remains largely unchanged. This finding holds for both small-scale neural networks trained to manipulate lists as well as large language models (LLMs) operating in four domains that are central in symbolic traditions: arithmetic, logic, computer code, and language. Further, our symbolic approximation allows us to modify an LLM's behavior in targeted ways via precise interventions on its internal representations, showing that the LLM's behavior is reliant on the symbolic structures we have identified. This work provides a potential way to reconcile longstanding symbolic conceptions of intelligence with the vector-based nature of modern AI.

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.

Abstraction is a core tenet of human cognition and communication. When composing natural language instructions, humans naturally evoke abstraction to convey complex procedures in an efficient and concise way. Yet, interpreting and grounding abstraction expressed in NL has not yet been systematically studied in NLP, with no accepted benchmarks specifically eliciting abstraction in NL. In this work, we set the foundation for a systematic study of processing and grounding abstraction in NLP. First, we deliver a novel abstraction elicitation method and present Hexagons, a 2D instruction-following game. Using Hexagons we collected over 4k naturally-occurring visually-grounded instructions rich with diverse types of abstractions. From these data, we derive an instruction-to-execution task and assess different types of neural models. Our results show that contemporary models and modeling practices are substantially inferior to human performance, and that models' performance is inversely correlated with the level of abstraction, showing less satisfying performance on higher levels of abstraction. These findings are consistent across models and setups, confirming that abstraction is a challenging phenomenon deserving further attention and study in NLP/AI research.

AI for Mathematics (AI4Math) has emerged as a distinct field that leverages machine learning to navigate mathematical landscapes historically intractable for early symbolic systems. While mid-20th-century symbolic approaches successfully automated formal logic, they faced severe scalability limitations due to the combinatorial explosion of the search space. The recent integration of data-driven approaches has revitalized this pursuit. In this review, we provide a systematic overview of AI4Math, highlighting its primary focus on developing AI models to support mathematical research. Crucially, we emphasize that this is not merely the application of AI to mathematical activities; it also encompasses the development of stronger AI systems where the rigorous nature of mathematics serves as a premier testbed for advancing general reasoning capabilities. We categorize existing research into two complementary directions: problem-specific modeling, involving the design of specialized architectures for distinct mathematical tasks, and general-purpose modeling, focusing on foundation models capable of broader reasoning, retrieval, and exploratory workflows. We conclude by discussing key challenges and prospects, advocating for AI systems that go beyond facilitating formal correctness to enabling the discovery of meaningful results and unified theories, recognizing that the true value of a proof lies in the insights and tools it offers to the broader mathematical landscape.

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.

The integration of symbolic computing with neural networks has intrigued researchers since the first theorizations of Artificial intelligence (AI). The ability of Neuro-Symbolic (NeSy) methods to infer or exploit behavioral schema has been widely considered as one of the possible proxies for human-level intelligence. However, the limited semantic generalizability and the challenges in declining complex domains with pre-defined patterns and rules hinder their practical implementation in real-world scenarios. The unprecedented results achieved by connectionist systems since the last AI breakthrough in 2017 have raised questions about the competitiveness of NeSy solutions, with particular emphasis on the Natural Language Processing and Computer Vision fields. This survey examines task-specific advancements in the NeSy domain to explore how incorporating symbolic systems can enhance explainability and reasoning capabilities. Our findings are meant to serve as a resource for researchers exploring explainable NeSy methodologies for real-life tasks and applications. Reproducibility details and in-depth comments on each surveyed research work are made available at https://github.com/disi-unibo-nlp/task-oriented-neuro-symbolic.git.

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.

Causal abstraction is a promising theoretical framework for explainable artificial intelligence that defines when an interpretable high-level causal model is a faithful simplification of a low-level deep learning system. However, existing causal abstraction methods have two major limitations: they require a brute-force search over alignments between the high-level model and the low-level one, and they presuppose that variables in the high-level model will align with disjoint sets of neurons in the low-level one. In this paper, we present distributed alignment search (DAS), which overcomes these limitations. In DAS, we find the alignment between high-level and low-level models using gradient descent rather than conducting a brute-force search, and we allow individual neurons to play multiple distinct roles by analyzing representations in non-standard bases-distributed representations. Our experiments show that DAS can discover internal structure that prior approaches miss. Overall, DAS removes previous obstacles to conducting causal abstraction analyses and allows us to find conceptual structure in trained neural nets.

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.

Reasoning requires going beyond pattern matching or memorization of solutions to identify and implement "algorithmic procedures" that can be used to deduce answers to hard problems. Doing so requires realizing the most relevant primitives, intermediate results, or shared procedures, and building upon them. While RL post-training on long chains of thought ultimately aims to uncover this kind of algorithmic behavior, most reasoning traces learned by large models fail to consistently capture or reuse procedures, instead drifting into verbose and degenerate exploration. To address more effective reasoning, we introduce reasoning abstractions: concise natural language descriptions of procedural and factual knowledge that guide the model toward learning successful reasoning. We train models to be capable of proposing multiple abstractions given a problem, followed by RL that incentivizes building a solution while using the information provided by these abstractions. This results in a two-player RL training paradigm, abbreviated as RLAD, that jointly trains an abstraction generator and a solution generator. This setup effectively enables structured exploration, decouples learning signals of abstraction proposal and solution generation, and improves generalization to harder problems. We also show that allocating more test-time compute to generating abstractions is more beneficial for performance than generating more solutions at large test budgets, illustrating the role of abstractions in guiding meaningful exploration.

Over the last decades, deep neural networks based-models became the dominant paradigm in machine learning. Further, the use of artificial neural networks in symbolic learning has been seen as increasingly relevant recently. To study the capabilities of neural networks in the symbolic AI domain, researchers have explored the ability of deep neural networks to learn mathematical constructions, such as addition and multiplication, logic inference, such as theorem provers, and even the execution of computer programs. The latter is known to be too complex a task for neural networks. Therefore, the results were not always successful, and often required the introduction of biased elements in the learning process, in addition to restricting the scope of possible programs to be executed. In this work, we will analyze the ability of neural networks to learn how to execute programs as a whole. To do so, we propose a different approach. Instead of using an imperative programming language, with complex structures, we use the Lambda Calculus (λ-Calculus), a simple, but Turing-Complete mathematical formalism, which serves as the basis for modern functional programming languages and is at the heart of computability theory. We will introduce the use of integrated neural learning and lambda calculi formalization. Finally, we explore execution of a program in λ-Calculus is based on reductions, we will show that it is enough to learn how to perform these reductions so that we can execute any program. Keywords: Machine Learning, Lambda Calculus, Neurosymbolic AI, Neural Networks, Transformer Model, Sequence-to-Sequence Models, Computational Models

Human reasoning can be understood as a cooperation between the intuitive, associative "System-1" and the deliberative, logical "System-2". For existing System-1-like methods in visual activity understanding, it is crucial to integrate System-2 processing to improve explainability, generalization, and data efficiency. One possible path of activity reasoning is building a symbolic system composed of symbols and rules, where one rule connects multiple symbols, implying human knowledge and reasoning abilities. Previous methods have made progress, but are defective with limited symbols from handcraft and limited rules from visual-based annotations, failing to cover the complex patterns of activities and lacking compositional generalization. To overcome the defects, we propose a new symbolic system with two ideal important properties: broad-coverage symbols and rational rules. Collecting massive human knowledge via manual annotations is expensive to instantiate this symbolic system. Instead, we leverage the recent advancement of LLMs (Large Language Models) as an approximation of the two ideal properties, i.e., Symbols from Large Language Models (Symbol-LLM). Then, given an image, visual contents from the images are extracted and checked as symbols and activity semantics are reasoned out based on rules via fuzzy logic calculation. Our method shows superiority in extensive activity understanding tasks. Code and data are available at https://mvig-rhos.com/symbol_llm.

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.

What mathematical functions do neural network components learn? Symbolic distillation addresses this question by expressing neural network components with interpretable, closed-form mathematical expressions that expose the functional structure learned during training. We develop symbolic distillation as a systematic, architecture-agnostic methodology, and release our approach as the open-source SymTorch package - a PySR-powered library built natively for the PyTorch ecosystem. Applying this methodology across diverse architectures, we find that SymTorch is successful in the automated discovery of physical laws. Specifically, our approach (1) recovers pairwise interaction forces from graph neural networks trained on empirical n-body observations, (2) distills the exact closed-form PDE/ODE solutions of multiple physical systems, including the value of constants, from physics-informed neural networks trained on sparse data, and (3) uncovers the chaotic dynamics of the Lorenz system from high-dimensional data, ultimately outperforming the base neural network on downstream prediction tasks. We further demonstrate the utility of our framework for model interpretability by providing an optimized implementation of SLIME - a symbolic extension to the LIME explainability method. SLIME consistently outperforms LIME across predictive metrics across eight popular classification and regression benchmarks, while still providing an interpretable local symbolic model. Lastly, we investigate replacing transformer MLP layers with symbolic surrogates: replacing 1-7 layers with symbolic approximations yields 2-19\% throughput improvements and up to 18.7\% VRAM reduction, with the resulting hybrid models lying on the Pareto front of throughput versus perplexity among open-source LLMs of comparable scale.

Compositionality is a pivotal property of symbolic reasoning. However, how well recent neural models capture compositionality remains underexplored in the symbolic reasoning tasks. This study empirically addresses this question by systematically examining recently published pre-trained seq2seq models with a carefully controlled dataset of multi-hop arithmetic symbolic reasoning. We introduce a skill tree on compositionality in arithmetic symbolic reasoning that defines the hierarchical levels of complexity along with three compositionality dimensions: systematicity, productivity, and substitutivity. Our experiments revealed that among the three types of composition, the models struggled most with systematicity, performing poorly even with relatively simple compositions. That difficulty was not resolved even after training the models with intermediate reasoning steps.

We present a novel method for symbolic regression (SR), the task of searching for compact programmatic hypotheses that best explain a dataset. The problem is commonly solved using genetic algorithms; we show that we can enhance such methods by inducing a library of abstract textual concepts. Our algorithm, called LaSR, uses zero-shot queries to a large language model (LLM) to discover and evolve concepts occurring in known high-performing hypotheses. We discover new hypotheses using a mix of standard evolutionary steps and LLM-guided steps (obtained through zero-shot LLM queries) conditioned on discovered concepts. Once discovered, hypotheses are used in a new round of concept abstraction and evolution. We validate LaSR on the Feynman equations, a popular SR benchmark, as well as a set of synthetic tasks. On these benchmarks, LaSR substantially outperforms a variety of state-of-the-art SR approaches based on deep learning and evolutionary algorithms. Moreover, we show that LaSR can be used to discover a novel and powerful scaling law for LLMs.

Cognitive research indicates that abstraction ability is essential in human intelligence, which remains under-explored in language models. In this paper, we present AbsPyramid, a unified entailment graph of 221K textual descriptions of abstraction knowledge. While existing resources only touch nouns or verbs within simplified events or specific domains, AbsPyramid collects abstract knowledge for three components of diverse events to comprehensively evaluate the abstraction ability of language models in the open domain. Experimental results demonstrate that current LLMs face challenges comprehending abstraction knowledge in zero-shot and few-shot settings. By training on our rich abstraction knowledge, we find LLMs can acquire basic abstraction abilities and generalize to unseen events. In the meantime, we empirically show that our benchmark is comprehensive to enhance LLMs across two previous abstraction tasks.

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.

The concept of causal abstraction got recently popularised to demystify the opaque decision-making processes of machine learning models; in short, a neural network can be abstracted as a higher-level algorithm if there exists a function which allows us to map between them. Notably, most interpretability papers implement these maps as linear functions, motivated by the linear representation hypothesis: the idea that features are encoded linearly in a model's representations. However, this linearity constraint is not required by the definition of causal abstraction. In this work, we critically examine the concept of causal abstraction by considering arbitrarily powerful alignment maps. In particular, we prove that under reasonable assumptions, any neural network can be mapped to any algorithm, rendering this unrestricted notion of causal abstraction trivial and uninformative. We complement these theoretical findings with empirical evidence, demonstrating that it is possible to perfectly map models to algorithms even when these models are incapable of solving the actual task; e.g., on an experiment using randomly initialised language models, our alignment maps reach 100% interchange-intervention accuracy on the indirect object identification task. This raises the non-linear representation dilemma: if we lift the linearity constraint imposed to alignment maps in causal abstraction analyses, we are left with no principled way to balance the inherent trade-off between these maps' complexity and accuracy. Together, these results suggest an answer to our title's question: causal abstraction is not enough for mechanistic interpretability, as it becomes vacuous without assumptions about how models encode information. Studying the connection between this information-encoding assumption and causal abstraction should lead to exciting future work.

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.

Harnessing the statistical power of neural networks to perform language understanding and symbolic reasoning is difficult, when it requires executing efficient discrete operations against a large knowledge-base. In this work, we introduce a Neural Symbolic Machine, which contains (a) a neural "programmer", i.e., a sequence-to-sequence model that maps language utterances to programs and utilizes a key-variable memory to handle compositionality (b) a symbolic "computer", i.e., a Lisp interpreter that performs program execution, and helps find good programs by pruning the search space. We apply REINFORCE to directly optimize the task reward of this structured prediction problem. To train with weak supervision and improve the stability of REINFORCE, we augment it with an iterative maximum-likelihood training process. NSM outperforms the state-of-the-art on the WebQuestionsSP dataset when trained from question-answer pairs only, without requiring any feature engineering or domain-specific knowledge.

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.

Tokenization is the first - and often underappreciated - layer of computation in language models. While Chain-of-Thought (CoT) prompting enables transformer models to approximate recurrent computation by externalizing intermediate steps, we show that the success of such reasoning is fundamentally bounded by the structure of tokenized inputs. This work presents a theoretical and empirical investigation into how tokenization schemes, particularly subword-based methods like byte-pair encoding (BPE), impede symbolic computation by merging or obscuring atomic reasoning units. We introduce the notion of Token Awareness to formalize how poor token granularity disrupts logical alignment and prevents models from generalizing symbolic procedures. Through systematic evaluation on arithmetic and symbolic tasks, we demonstrate that token structure dramatically affect reasoning performance, causing failure even with CoT, while atomically-aligned formats unlock strong generalization, allowing small models (e.g., GPT-4o-mini) to outperform larger systems (e.g., o1) in structured reasoning. Our findings reveal that symbolic reasoning ability in LLMs is not purely architectural, but deeply conditioned on token-level representations.

While Multimodal Large Language Models (MLLMs) have achieved remarkable success in interpreting natural scenes, their ability to process discrete symbols -- the fundamental building blocks of human cognition -- remains a critical open question. Unlike continuous visual data, symbols such as mathematical formulas, chemical structures, and linguistic characters require precise, deeper interpretation. This paper introduces a comprehensive benchmark to evaluate how top-tier MLLMs navigate these "discrete semantic spaces" across five domains: language, culture, mathematics, physics, and chemistry. Our investigation uncovers a counterintuitive phenomenon: models often fail at basic symbol recognition yet succeed in complex reasoning tasks, suggesting they rely on linguistic probability rather than true visual perception. By exposing this "cognitive mismatch", we highlight a significant gap in current AI capabilities: the struggle to truly perceive and understand the symbolic languages that underpin scientific discovery and abstract thought. This work offers a roadmap for developing more rigorous, human-aligned intelligent systems.

While large language models (LLMs) have demonstrated impressive capabilities in formal theorem proving, current benchmarks fail to adequately measure library-grounded abstraction -- the ability to reason with high-level interfaces and reusable structures central to modern mathematics and software engineering. We introduce LeanCat, a challenging benchmark comprising 100 fully formalized category-theory tasks in Lean. Unlike algebra or arithmetic, category theory serves as a rigorous stress test for structural, interface-level reasoning. Our evaluation reveals a severe abstraction gap: the best state-of-the-art model solves only 12.0% of tasks at pass@4, with performance collapsing from 55.0% on Easy tasks to 0.0% on High-difficulty tasks, highlighting a failure in compositional generalization. To overcome this, we evaluate LeanBridge, a retrieval-augmented agent that employs a retrieve-generate-verify loop. LeanBridge achieves a peak success rate of 24.0% -- doubling the performance of the best static baseline. These results empirically demonstrate that iterative refinement and dynamic library retrieval are not merely optimizations but strict necessities for neuro-symbolic reasoning in abstract domains. LeanCat offers a compact, reusable testbed for tracking progress toward reliable, research-level formalization.

We investigate a relatively under-explored class of hybrid neurosymbolic models that integrate symbolic learning with neural reasoning to construct data generators meeting formal correctness criteria. In Symbolic Neural Generators (SNGs), symbolic learners examine logical specifications of feasible data from a small set of instances -- sometimes just one. Each specification in turn constrains the conditional information supplied to a neural-based generator, which rejects any instance violating the symbolic specification. Like other neurosymbolic approaches, SNG exploits the complementary strengths of symbolic and neural methods. The outcome of an SNG is a pair (H, X), where H is a symbolic description of feasible instances constructed from data, and X a set of generated new instances that satisfy the description. We introduce a semantics for such systems, based on the construction of appropriate base and fibre partially-ordered sets combined into an overall partial order. We implement an SNG combining a restricted form of Inductive Logic Programming (ILP) with a large language model (LLM) and evaluate it on early-stage drug design. Our main interest is the description and the set of potential inhibitor molecules generated by the SNG. On benchmark problems -- where drug targets are well understood -- SNG performance is statistically comparable to state-of-the-art methods. On exploratory problems with poorly understood targets, generated molecules exhibit binding affinities on par with leading clinical candidates. Experts further find the symbolic specifications useful as preliminary filters, with several generated molecules identified as viable for synthesis and wet-lab testing.

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.

We study syllogistic reasoning in LLMs from the logical and natural language perspectives. In process, we explore fundamental reasoning capabilities of the LLMs and the direction this research is moving forward. To aid in our studies, we use 14 large language models and investigate their syllogistic reasoning capabilities in terms of symbolic inferences as well as natural language understanding. Even though this reasoning mechanism is not a uniform emergent property across LLMs, the perfect symbolic performances in certain models make us wonder whether LLMs are becoming more and more formal reasoning mechanisms, rather than making explicit the nuances of human reasoning.

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.

Small Language Models (SLMs) are attracting attention due to the high computational demands and privacy concerns of Large Language Models (LLMs). Some studies fine-tune SLMs using Chains of Thought (CoT) data distilled from LLMs, aiming to enhance their reasoning ability. Furthermore, Some CoT distillation methods introduce external symbolic knowledge into the generation process to improve the limited knowledge memory, reasoning ability and out-of-domain (OOD) generalization of SLMs. However, the introduction of symbolic knowledge increases computational overhead and introduces potential noise. In this paper, we introduce SKIntern, an innovative approach that empowers SLMs to internalize symbolic knowledge and few-shot examples gradually through a progressive fine-tuning process, guided by a predefined linear decay schedule under curriculum learning. By efficiently internalizing knowledge, SKIntern reduces computational overhead and speeds up the reasoning process by focusing solely on the question during inference. It outperforms state-of-the-art baselines by over 5\%, while reducing inference costs (measured in FLOPs) by up to 4times across a wide range of SLMs in both in-domain (ID) and out-of-domain (OOD) tasks. Our code will be available at https://github.com/Xnhyacinth/SKIntern.

Scientists often infer abstract procedures from specific instances of problems and use the abstractions to generate new, related instances. For example, programs encoding the formal rules and properties of a system have been useful in fields ranging from RL (procedural environments) to physics (simulation engines). These programs can be seen as functions which execute to different outputs based on their parameterizations (e.g., gridworld configuration or initial physical conditions). We introduce the term EFA (Executable Functional Abstraction) to denote such programs for math problems. EFA-like constructs have been shown to be useful for math reasoning as problem generators for stress-testing models. However, prior work has been limited to abstractions for grade-school math (whose simple rules are easy to encode in programs), while generating EFAs for advanced math has thus far required human engineering. We explore the automatic construction of EFAs for advanced math problems. We operationalize the task of automatically constructing EFAs as a program synthesis task, and develop EFAGen, which conditions an LLM on a seed math problem and its step-by-step solution to generate candidate EFA programs that are faithful to the generalized problem and solution class underlying the seed problem. Furthermore, we formalize properties any valid EFA must possess in terms of executable unit tests, and show how the tests can be used as verifiable rewards to train LLMs to become better writers of EFAs. We demonstrate that EFAs constructed by EFAGen behave rationally by remaining faithful to seed problems, produce learnable problem variations, and that EFAGen can infer EFAs across multiple diverse sources of competition-level math problems. Finally, we show downstream uses of model-written EFAs e.g. finding problem variations that are harder or easier for a learner to solve, as well as data generation.

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

In 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.

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.

Current advances in Artificial Intelligence (AI) and Machine Learning (ML) have achieved unprecedented impact across research communities and industry. Nevertheless, concerns about trust, safety, interpretability and accountability of AI were raised by influential thinkers. Many have identified the need for well-founded knowledge representation and reasoning to be integrated with deep learning and for sound explainability. Neural-symbolic computing has been an active area of research for many years seeking to bring together robust learning in neural networks with reasoning and explainability via symbolic representations for network models. In this paper, we relate recent and early research results in neurosymbolic AI with the objective of identifying the key ingredients of the next wave of AI systems. We focus on research that integrates in a principled way neural network-based learning with symbolic knowledge representation and logical reasoning. The insights provided by 20 years of neural-symbolic computing are shown to shed new light onto the increasingly prominent role of trust, safety, interpretability and accountability of AI. We also identify promising directions and challenges for the next decade of AI research from the perspective of neural-symbolic systems.

Automated proving of polynomial inequalities is a fundamental challenge in automated mathematical reasoning, where rich algebraic structure and a rapidly growing certificate search space hinder scalability. Purely symbolic approaches provide strong guarantees but often scale poorly as the number of variables or the degree increases, due to expensive algebraic manipulations and rapidly growing intermediate expressions. In parallel, LLM-guided methods have made notable progress, particularly on competition-style inequalities with a small number of variables. To address the remaining scalability challenges, we propose NSPI, a neuro-symbolic framework that combines the complementary strengths of LLMs and symbolic computation for polynomial-inequality proving. Concretely, an LLM proposes a conjecture in the form of an approximate polynomial Sum-Of-Squares (SOS) decomposition; we refine it via symbolic computation to obtain an exact polynomial SOS representation, which directly proves the target inequality, and we further certify the proof in Lean, yielding an end-to-end pipeline from heuristic discovery to machine-checked proof. Experiments on challenging benchmarks involving polynomials with up to 10 variables demonstrate the effectiveness and scalability of the proposed method.

Scientific discovery via symbolic regression is often viewed as statistically and computationally intractable because the hypothesis space of expressions grows combinatorially with depth. This paper revisits the statistical side through the lens of PAC learning, focusing on compositional function trees built from a finite vocabulary of smooth operators (e.g., {+,times,sin,exp} and affine maps). We prove that the relevant generalization quantity, Rademacher complexity, hence the excess risk, does not necessarily blow up exponentially with the number of distinct symbolic structures, but is controlled by (i) the depth d and (ii) the Lipschitz constants of the base operators along the composed computation graph. Concretely, under mild Lipschitz conditions on operators and bounded affine leaves, a finite-union bound over a vocabulary of size K=|H_{base}| together with Maurer-type vector contraction yields R_n(H_{comp}^{d}) leq (Kb2L)^{d-1}R_n(H_{comp}^{1}) with arity bound b; corresponding high-probability risk bounds scale as O(L^{d}/n) when K,b=O(1) and R_n(H_{comp}^{1})=O(n^{-1/2}). We complement the theory with a modular codebase that trains differentiable operator trees (not MLPs) on synthetic "physics-like" targets of controlled depth and shows that the empirical generalization gap correlates positively with the predicted complexity term (L^{d})/n.

Humans interact with the environment using a combination of perception - transforming sensory inputs from their environment into symbols, and cognition - mapping symbols to knowledge about the environment for supporting abstraction, reasoning by analogy, and long-term planning. Human perception-inspired machine perception, in the context of AI, refers to large-scale pattern recognition from raw data using neural networks trained using self-supervised learning objectives such as next-word prediction or object recognition. On the other hand, machine cognition encompasses more complex computations, such as using knowledge of the environment to guide reasoning, analogy, and long-term planning. Humans can also control and explain their cognitive functions. This seems to require the retention of symbolic mappings from perception outputs to knowledge about their environment. For example, humans can follow and explain the guidelines and safety constraints driving their decision-making in safety-critical applications such as healthcare, criminal justice, and autonomous driving. This article introduces the rapidly emerging paradigm of Neurosymbolic AI combines neural networks and knowledge-guided symbolic approaches to create more capable and flexible AI systems. These systems have immense potential to advance both algorithm-level (e.g., abstraction, analogy, reasoning) and application-level (e.g., explainable and safety-constrained decision-making) capabilities of AI systems.

This paper presents the Functional Machine Calculus (FMC) as a simple model of higher-order computation with "reader/writer" effects: higher-order mutable store, input/output, and probabilistic and non-deterministic computation. The FMC derives from the lambda-calculus by taking the standard operational perspective of a call-by-name stack machine as primary, and introducing two natural generalizations. One, "locations", introduces multiple stacks, which each may represent an effect and so enable effect operators to be encoded into the abstraction and application constructs of the calculus. The second, "sequencing", is known from kappa-calculus and concatenative programming languages, and introduces the imperative notions of "skip" and "sequence". This enables the encoding of reduction strategies, including call-by-value lambda-calculus and monadic constructs. The encoding of effects into generalized abstraction and application means that standard results from the lambda-calculus may carry over to effects. The main result is confluence, which is possible because encoded effects reduce algebraically rather than operationally. Reduction generates the familiar algebraic laws for state, and unlike in the monadic setting, reader/writer effects combine seamlessly. A system of simple types confers termination of the machine.

We develop a general theory of semantic dynamics for large language models by formalizing them as Continuous State Machines (CSMs): smooth dynamical systems whose latent manifolds evolve under probabilistic transition operators. The associated transfer operator P: L^2(M,μ) to L^2(M,μ) encodes the propagation of semantic mass. Under mild regularity assumptions (compactness, ergodicity, bounded Jacobian), P is compact with discrete spectrum. Within this setting, we prove the Semantic Characterization Theorem (SCT): the leading eigenfunctions of P induce finitely many spectral basins of invariant meaning, each definable in an o-minimal structure over R. Thus spectral lumpability and logical tameness coincide. This explains how discrete symbolic semantics can emerge from continuous computation: the continuous activation manifold collapses into a finite, logically interpretable ontology. We further extend the SCT to stochastic and adiabatic (time-inhomogeneous) settings, showing that slowly drifting kernels preserve compactness, spectral coherence, and basin structure.

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.

Despite the tremendous success, existing machine learning models still fall short of human-like systematic generalization -- learning compositional rules from limited data and applying them to unseen combinations in various domains. We propose Neural-Symbolic Recursive Machine (NSR) to tackle this deficiency. The core representation of NSR is a Grounded Symbol System (GSS) with combinatorial syntax and semantics, which entirely emerges from training data. Akin to the neuroscience studies suggesting separate brain systems for perceptual, syntactic, and semantic processing, NSR implements analogous separate modules of neural perception, syntactic parsing, and semantic reasoning, which are jointly learned by a deduction-abduction algorithm. We prove that NSR is expressive enough to model various sequence-to-sequence tasks. Superior systematic generalization is achieved via the inductive biases of equivariance and recursiveness embedded in NSR. In experiments, NSR achieves state-of-the-art performance in three benchmarks from different domains: SCAN for semantic parsing, PCFG for string manipulation, and HINT for arithmetic reasoning. Specifically, NSR achieves 100% generalization accuracy on SCAN and PCFG and outperforms state-of-the-art models on HINT by about 23%. Our NSR demonstrates stronger generalization than pure neural networks due to its symbolic representation and inductive biases. NSR also demonstrates better transferability than existing neural-symbolic approaches due to less domain-specific knowledge required.

Modern Vision-Language Models (VLMs) have achieved impressive performance in various tasks, yet they often struggle with compositional reasoning, the ability to decompose and recombine concepts to solve novel problems. While neuro-symbolic approaches offer a promising direction, they are typically constrained by crisp logical execution or predefined predicates, which limit flexibility. In this work, we introduce NePTune, a neuro-symbolic framework that overcomes these limitations through a hybrid execution model that integrates the perception capabilities of foundation vision models with the compositional expressiveness of symbolic reasoning. NePTune dynamically translates natural language queries into executable Python programs that blend imperative control flow with soft logic operators capable of reasoning over VLM-generated uncertainty. Operating in a training-free manner, NePTune, with a modular design, decouples perception from reasoning, yet its differentiable operations support fine-tuning. We evaluate NePTune on multiple visual reasoning benchmarks and various domains, utilizing adversarial tests, and demonstrate a significant improvement over strong base models, as well as its effective compositional generalization and adaptation capabilities in novel environments.

Programs are an increasingly popular representation for visual data, exposing compact, interpretable structure that supports manipulation. Visual programs are usually written in domain-specific languages (DSLs). Finding "good" programs, that only expose meaningful degrees of freedom, requires access to a DSL with a "good" library of functions, both of which are typically authored by domain experts. We present ShapeCoder, the first system capable of taking a dataset of shapes, represented with unstructured primitives, and jointly discovering (i) useful abstraction functions and (ii) programs that use these abstractions to explain the input shapes. The discovered abstractions capture common patterns (both structural and parametric) across the dataset, so that programs rewritten with these abstractions are more compact, and expose fewer degrees of freedom. ShapeCoder improves upon previous abstraction discovery methods, finding better abstractions, for more complex inputs, under less stringent input assumptions. This is principally made possible by two methodological advancements: (a) a shape to program recognition network that learns to solve sub-problems and (b) the use of e-graphs, augmented with a conditional rewrite scheme, to determine when abstractions with complex parametric expressions can be applied, in a tractable manner. We evaluate ShapeCoder on multiple datasets of 3D shapes, where primitive decompositions are either parsed from manual annotations or produced by an unsupervised cuboid abstraction method. In all domains, ShapeCoder discovers a library of abstractions that capture high-level relationships, remove extraneous degrees of freedom, and achieve better dataset compression compared with alternative approaches. Finally, we investigate how programs rewritten to use discovered abstractions prove useful for downstream tasks.

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.

In our opinion the exuberance surrounding the relative success of data-driven large language models (LLMs) is slightly misguided and for several reasons (i) LLMs cannot be relied upon for factual information since for LLMs all ingested text (factual or non-factual) was created equal; (ii) due to their subsymbolic na-ture, whatever 'knowledge' these models acquire about language will always be buried in billions of microfeatures (weights), none of which is meaningful on its own; and (iii) LLMs will often fail to make the correct inferences in several linguistic contexts (e.g., nominal compounds, copredication, quantifier scope ambi-guities, intensional contexts. Since we believe the relative success of data-driven large language models (LLMs) is not a reflection on the symbolic vs. subsymbol-ic debate but a reflection on applying the successful strategy of a bottom-up reverse engineering of language at scale, we suggest in this paper applying the effective bottom-up strategy in a symbolic setting resulting in symbolic, explainable, and ontologically grounded language models.

This work introduces Symbolic-Aided Chain-of-Thought (CoT), an improved approach to standard CoT, for logical reasoning in large language models (LLMs). The key idea is to integrate lightweight symbolic representations into few-shot prompts, structuring the inference steps with a consistent strategy to make reasoning patterns more explicit within a non-iterative reasoning process. By incorporating these symbolic structures, our method preserves the generalizability of standard prompting techniques while enhancing the transparency, interpretability, and analyzability of LLM logical reasoning. Extensive experiments on four well-known logical reasoning benchmarks -- ProofWriter, FOLIO, ProntoQA, and LogicalDeduction, which cover diverse reasoning scenarios -- demonstrate the effectiveness of the proposed approach, particularly in complex reasoning tasks that require navigating multiple constraints or rules. Notably, Symbolic-Aided CoT consistently improves LLMs' reasoning capabilities across various model sizes and significantly outperforms conventional CoT on three out of four datasets, ProofWriter, ProntoQA, and LogicalDeduction.

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.

Neural-symbolic methods have demonstrated efficiency in enhancing the reasoning abilities of large language models (LLMs). However, existing methods mainly rely on syntactically mapping natural languages to complete formal languages like Python and SQL. Those methods require that reasoning tasks be convertible into programs, which cater to the computer execution mindset and deviate from human reasoning habits. To broaden symbolic methods' applicability and adaptability in the real world, we propose the Meta-Reasoning from a linguistic perspective. This method empowers LLMs to deconstruct reasoning-independent semantic information into generic symbolic representations, thereby efficiently capturing more generalized reasoning knowledge. We conduct extensive experiments on more than ten datasets encompassing conventional reasoning tasks like arithmetic, symbolic, and logical reasoning, and the more complex interactive reasoning tasks like theory-of-mind reasoning. Experimental results demonstrate that Meta-Reasoning significantly enhances in-context reasoning accuracy, learning efficiency, out-of-domain generalization, and output stability compared to the Chain-of-Thought technique. Code and data are publicly available at https://github.com/Alsace08/Meta-Reasoning.

Discovering the underlying mathematical expressions describing a dataset is a core challenge for artificial intelligence. This is the problem of symbolic regression. Despite recent advances in training neural networks to solve complex tasks, deep learning approaches to symbolic regression are underexplored. We propose a framework that leverages deep learning for symbolic regression via a simple idea: use a large model to search the space of small models. Specifically, we use a recurrent neural network to emit a distribution over tractable mathematical expressions and employ a novel risk-seeking policy gradient to train the network to generate better-fitting expressions. Our algorithm outperforms several baseline methods (including Eureqa, the gold standard for symbolic regression) in its ability to exactly recover symbolic expressions on a series of benchmark problems, both with and without added noise. More broadly, our contributions include a framework that can be applied to optimize hierarchical, variable-length objects under a black-box performance metric, with the ability to incorporate constraints in situ, and a risk-seeking policy gradient formulation that optimizes for best-case performance instead of expected performance.

Language models exhibit fundamental limitations -- hallucination, brittleness, and lack of formal grounding -- that are particularly problematic in high-stakes specialist fields requiring verifiable reasoning. I investigate whether formal domain ontologies can enhance language model reliability through retrieval-augmented generation. Using mathematics as proof of concept, I implement a neuro-symbolic pipeline leveraging the OpenMath ontology with hybrid retrieval and cross-encoder reranking to inject relevant definitions into model prompts. Evaluation on the MATH benchmark with three open-source models reveals that ontology-guided context improves performance when retrieval quality is high, but irrelevant context actively degrades it -- highlighting both the promise and challenges of neuro-symbolic approaches.

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.

This paper presents the Relational Machine Calculus (RMC): a simple, foundational model of first-order relational programming. The RMC originates from the Functional Machine Calculus (FMC), which generalizes the lambda-calculus and its standard call-by-name stack machine in two directions. One, "locations", introduces multiple stacks, which enable effect operators to be encoded into the abstraction and application constructs. The second, "sequencing", introduces the imperative notions of "skip" and "sequence", similar to kappa-calculus and concatenative programming languages. The key observation of the RMC is that the first-order fragment of the FMC exhibits a latent duality which, given a simple decomposition of the relevant constructors, can be concretely expressed as an involution on syntax. Semantically, this gives rise to a sound and complete calculus for string diagrams of Frobenius monoids. We consider unification as the corresponding symmetric generalization of beta-reduction. By further including standard operators of Kleene algebra, the RMC embeds a range of computational models: the kappa-calculus, logic programming, automata, Interaction Nets, and Petri Nets, among others. These embeddings preserve operational semantics, which for the RMC is again given by a generalization of the standard stack machine for the lambda-calculus. The equational theory of the RMC (which supports reasoning about its operational semantics) is conservative over both the first-order lambda-calculus and Kleene algebra, and can be oriented to give a confluent reduction relation.

The pursuit of automated scientific discovery has fueled progress from symbolic logic to modern AI, forging new frontiers in reasoning and pattern recognition. Transformers function as potential systems, where every possible relationship remains latent potentiality until tasks impose constraints, akin to measurement. Yet, refining their sampling requires more than probabilistic selection: solutions must conform to specific structures or rules, ensuring consistency and the invocation of general principles. We present Graph-PReFLexOR (Graph-based Preference-based Recursive Language Modeling for Exploratory Optimization of Reasoning), a framework that combines graph reasoning with symbolic abstraction to dynamically expand domain knowledge. Inspired by reinforcement learning, Graph-PReFLexOR defines reasoning as a structured mapping, where tasks yield knowledge graphs, abstract patterns, and ultimately, final answers. Inspired by category theory, it encodes concepts as nodes and their relationships as edges, supporting hierarchical inference and adaptive learning through isomorphic representations. Demonstrations include hypothesis generation, materials design, and creative reasoning, such as discovering relationships between mythological concepts like 'thin places' with materials science. We propose a 'knowledge garden growth' strategy that integrates insights across domains, promoting interdisciplinary connections. Results with a 3-billion-parameter Graph-PReFLexOR model show superior reasoning depth and adaptability, underscoring the potential for transparent, multidisciplinary AI-driven discovery. It lays the groundwork for general autonomous reasoning solutions.

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.

We prove that Moore machines can be exactly realized as state-space models (SSMs), establishing a formal correspondence between symbolic automata and these continuous machine learning architectures. These Moore-SSMs preserve both the complete symbolic structure and input-output behavior of the original Moore machine, but operate in Euclidean space. With this correspondence, we compare the training of SSMs with both passive and active automata learning. In recovering automata from the SYNTCOMP benchmark, we show that SSMs require orders of magnitude more data than symbolic methods and fail to learn state structure. This suggests that symbolic structure provides a strong inductive bias for learning these systems. We leverage this insight to combine the strengths of both automata learning and SSMs in order to learn complex systems efficiently. We learn an adaptive arbitration policy on a suite of arbiters from SYNTCOMP and show that initializing SSMs with symbolically-learned approximations learn both faster and better. We see 2-5 times faster convergence compared to randomly initialized models and better overall model accuracies on test data. Our work lifts automata learning out of purely discrete spaces, enabling principled exploitation of symbolic structure in continuous domains for efficiently learning in complex settings.

Mathematical reasoning is a hallmark of human intelligence, requiring logical deduction, symbolic manipulation, and abstract thinking. Recent multimodal large language models (MLLMs) have demonstrated strong performance on geometry problems through multi-step reasoning. To better emulate human problem-solving, intermediate steps can incorporate auxiliary visual constructions, such as additional lines or points, which improve geometric interpretation and educational clarity. In this work, we introduce the GeoMathCode, where programmatic representations serve as intermediate visual outputs. We further conduct an in-depth analysis of the underlying reasoning geometry. Experimental results show that reasoning and code generation steps can be disentangled in the latent space, while supervised fine-tuning (SFT) makes the reasoning manifold more structured and informative. Moreover, hierarchical syntactic code structures emerge as disentangled latent subspaces, and contain more mathematical symbolic information than visual representations.

Analytical solutions to differential equations offer exact, interpretable insight but are rarely available because discovering them requires expert intuition or exhaustive search of combinatorial spaces. We introduce SIGS, a neuro-symbolic framework for equation-driven closed-form solution discovery. SIGS uses a context-free grammar to generate mathematically valid and physically meaningful building blocks, with a user-specified Ansatz prescribing how these blocks combine, embeds them into a topology-regularised continuous latent manifold, and searches this manifold in two stages: structure selection followed by coefficient refinement using gradient descent, scoring candidates only against the PDE residual and prescribed boundary and initial conditions. This design unifies symbolic reasoning with numerical optimization; the grammar constrains candidate solution blocks to be proper by construction, while the latent search makes exploration tractable and data-free. SIGS is the first neuro-symbolic method to (i) recover analytical solutions for coupled nonlinear PDE systems, (ii) discover equivalent symbolic forms when the grammar lacks the natural primitives, and (iii) produce accurate symbolic approximations for PDEs lacking known closed-form solutions. Overall, SIGS improves over existing symbolic methods by orders of magnitude in both accuracy and runtime across standard PDE benchmarks.

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.

Chain-of-Thought (CoT) reasoning has become a powerful driver of trajectory prediction in VLA-based autonomous driving, yet its autoregressive nature imposes a latency cost that is prohibitive for real-time deployment. Latent CoT methods attempt to close this gap by compressing reasoning into continuous hidden states, but consistently fall short of their explicit counterparts. We suggest that this is due to purely linguistic latent representations compressing a symbolic abstraction of the world, rather than the causal dynamics that actually govern driving. Thus, we present OneVL (One-step latent reasoning and planning with Vision-Language explanations), a unified VLA and World Model framework that routes reasoning through compact latent tokens supervised by dual auxiliary decoders. Alongside a language decoder that reconstructs text CoT, we introduce a visual world model decoder that predicts future-frame tokens, forcing the latent space to internalize the causal dynamics of road geometry, agent motion, and environmental change. A three-stage training pipeline progressively aligns these latents with trajectory, language, and visual objectives, ensuring stable joint optimization. At inference, the auxiliary decoders are discarded and all latent tokens are prefilled in a single parallel pass, matching the speed of answer-only prediction. Across four benchmarks, OneVL becomes the first latent CoT method to surpass explicit CoT, delivering state-of-the-art accuracy at answer-only latency, and providing direct evidence that tighter compression, when guided in both language and world-model supervision, produces more generalizable representations than verbose token-by-token reasoning. Project Page: https://xiaomi-embodied-intelligence.github.io/OneVL

A faithful and interpretable explanation of an AI model's behavior and internal structure is a high-level explanation that is human-intelligible but also consistent with the known, but often opaque low-level causal details of the model. We argue that the theory of causal abstraction provides the mathematical foundations for the desired kinds of model explanations. In causal abstraction analysis, we use interventions on model-internal states to rigorously assess whether an interpretable high-level causal model is a faithful description of an AI model. Our contributions in this area are: (1) We generalize causal abstraction to cyclic causal structures and typed high-level variables. (2) We show how multi-source interchange interventions can be used to conduct causal abstraction analyses. (3) We define a notion of approximate causal abstraction that allows us to assess the degree to which a high-level causal model is a causal abstraction of a lower-level one. (4) We prove constructive causal abstraction can be decomposed into three operations we refer to as marginalization, variable-merge, and value-merge. (5) We formalize the XAI methods of LIME, causal effect estimation, causal mediation analysis, iterated nullspace projection, and circuit-based explanations as special cases of causal abstraction analysis.

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.

We introduce MathSticks, a benchmark for Visual Symbolic Compositional Reasoning (VSCR), which unifies visual perception, symbolic manipulation, and arithmetic consistency. Each task presents an incorrect matchstick equation that must be corrected by moving one or two sticks under strict conservation rules. The benchmark includes both text-guided and purely visual settings, systematically covering digit scale, move complexity, solution multiplicity, and operator variation, with 1.4M generated instances and a curated test set. Evaluations of 14 vision--language models reveal substantial limitations: closed-source models succeed only on simple cases, open-source models fail in the visual regime, while humans exceed 90\% accuracy. These findings establish MathSticks as a rigorous testbed for advancing compositional reasoning across vision and symbols. Our code and dataset are publicly available at https://github.com/Yuheng2000/MathSticks.

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.

Geometric Problem Solving (GPS) remains at the heart of enhancing mathematical reasoning in large language models because it requires the combination of diagrammatic understanding, symbolic manipulation and logical inference. In existing literature, researchers have chiefly focused on synchronising the diagram descriptions with text literals and solving the problem. In this vein, they have either taken a neural, symbolic or neuro-symbolic approach. But this solves only the first two of the requirements, namely diagrammatic understanding and symbolic manipulation, while leaving logical inference underdeveloped. The logical inference is often limited to one chain-of-thought (CoT). To address this weakness in hitherto existing models, this paper proposes MARS-GPS, that generates multiple parallel reasoning rollouts augmented with Python code execution for numerical verification, ranks them using token-level entropy as a confidence signal, and aggregates answers through a multi-stage voting and self-verification pipeline. Empirical results show that MARS-GPS with 8 parallel rollouts achieves 88.8% on Geometry3K, a nearly +11% improvement over the prior state-of-the-art, with accuracy scaling consistently as the number of rollouts increases from 1 to 16 (+6.0% on ablation subset). We provide our code and data in an anonymous repository: https://anonymous.4open.science/r/MARS-GPS-DE55.

The symbolic discovery of Ordinary Differential Equations (ODEs) from trajectory data plays a pivotal role in AI-driven scientific discovery. Existing symbolic methods predominantly rely on fixed, pre-collected training datasets, which often result in suboptimal performance, as demonstrated in our case study in Figure 1. Drawing inspiration from active learning, we investigate strategies to query informative trajectory data that can enhance the evaluation of predicted ODEs. However, the butterfly effect in dynamical systems reveals that small variations in initial conditions can lead to drastically different trajectories, necessitating the storage of vast quantities of trajectory data using conventional active learning. To address this, we introduce Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching (APPS). Instead of directly selecting individual initial conditions, our APPS first identifies an informative region within the phase space and then samples a batch of initial conditions from this region. Compared to traditional active learning methods, APPS mitigates the gap of maintaining a large amount of data. Extensive experiments demonstrate that APPS consistently discovers more accurate ODE expressions than baseline methods using passively collected datasets.

We design a suite of minimal algorithmic tasks that are a loose abstraction of open-ended real-world tasks. This allows us to cleanly and controllably quantify the creative limits of the present-day language model. Much like real-world tasks that require a creative, far-sighted leap of thought, our tasks require an implicit, open-ended stochastic planning step that either (a) discovers new connections in an abstract knowledge graph (like in wordplay, drawing analogies, or research) or (b) constructs new patterns (like in designing math problems or new proteins). In these tasks, we empirically and conceptually argue how next-token learning is myopic and memorizes excessively; comparatively, multi-token approaches, namely teacherless training and diffusion models, excel in producing diverse and original output. Secondly, in our tasks, we find that to elicit randomness from the Transformer without hurting coherence, it is better to inject noise right at the input layer (via a method we dub hash-conditioning) rather than defer to temperature sampling from the output layer. Thus, our work offers a principled, minimal test-bed for analyzing open-ended creative skills, and offers new arguments for going beyond next-token learning and softmax-based sampling. We make part of the code available under https://github.com/chenwu98/algorithmic-creativity

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.

Traditional Task and Motion Planning (TAMP) systems depend on physics models for motion planning and discrete symbolic models for task planning. Although physics model are often available, symbolic models (consisting of symbolic state interpretation and action models) must be meticulously handcrafted or learned from labeled data. This process is both resource-intensive and constrains the solution to the specific domain, limiting scalability and adaptability. On the other hand, Visual Language Models (VLMs) show desirable zero-shot visual understanding (due to their extensive training on heterogeneous data), but still achieve limited planning capabilities. Therefore, integrating VLMs with classical planning for long-horizon reasoning in TAMP problems offers high potential. Recent works in this direction still lack generality and depend on handcrafted, task-specific solutions, e.g. describing all possible objects in advance, or using symbolic action models. We propose a framework that generalizes well to unseen problem instances. The method requires only lifted predicates describing relations among objects and uses VLMs to ground them from images to obtain the symbolic state. Planning is performed with domain-independent heuristic search using goal-count and width-based heuristics, without need for action models. Symbolic search over VLM-grounded state-space outperforms direct VLM-based planning and performs on par with approaches that use a VLM-derived heuristic. This shows that domain-independent search can effectively solve problems across domains with large combinatorial state spaces. We extensively evaluate on extensively evaluate our method and achieve state-of-the-art results on the ProDG and ViPlan benchmarks.

While tool-augmented Large Language Models have significantly improved multi-step reasoning in quantitative STEM tasks, a critical residual failure mode remains: intermediate reasoning steps that are syntactically well-formed, mathematically executable, and unit-consistent, yet contextually ungrounded. Current approaches either rely on formal verifiers that cannot assess semantic intent, or burden Process Reward Models (PRMs) with the dual task of checking both arithmetic and logic. In this paper, we propose a neuro-symbolic framework that cleanly decouples reasoning into two formal dimensions: Symbolic Validity (V) and Semantic Groundedness (G). We guarantee V by construction using a deterministic symbolic verifier acting as a hard filter. To assess G, we train a PRM conditionally on the verifier-accepted manifold. To train this PRM efficiently, we introduce Counterfactual Symbolic Perturbation (CSP), a novel data synthesis strategy that algorithmically generates constraint-preserving hard negatives (steps that perfectly pass the verifier but are logically flawed). At inference, we deploy a verifier-first constrained search that guarantees execution consistency for verifier-covered operations while relying on the PRM solely to rank semantic grounding. By targeting the exact residual error class of strong tool-using LLMs, our method significantly improves reasoning reliability without the sprawling heuristics of prior frameworks.

Fine-tuning language models on tasks with instructions has demonstrated potential in facilitating zero-shot generalization to unseen tasks. In this paper, we introduce a straightforward yet effective method for enhancing instruction tuning by employing symbolic tasks. Compared to crowdsourced human tasks or model-generated tasks, symbolic tasks present a unique advantage as they can be easily generated in vast quantities, theoretically providing an infinite supply of high-quality training instances. To explore the potential of symbolic tasks, we carry out an extensive case study on the representative symbolic task of SQL execution. Empirical results on various benchmarks validate that the integration of SQL execution leads to significant improvements in zero-shot scenarios, particularly in table reasoning. Notably, our 3B model surpasses both the 175B GPT-3 and ChatGPT in zero-shot table reasoning across four benchmarks. Furthermore, experimental results on BBH (27 tasks) and MMLU (57 tasks) reveal that language models can be enhanced through symbolic tasks without compromising their generality. We hope that our paper serves as a catalyst, inspiring increased efforts to incorporate symbolic tasks in instruction tuning.

This paper revisits Ramon Llull's Ars combinatoria - a medieval framework for generating knowledge through symbolic recombination - as a conceptual foundation for building a modern Llull's thinking machine for research ideation. Our approach defines three compositional axes: Theme (e.g., efficiency, adaptivity), Domain (e.g., question answering, machine translation), and Method (e.g., adversarial training, linear attention). These elements represent high-level abstractions common in scientific work - motivations, problem settings, and technical approaches - and serve as building blocks for LLM-driven exploration. We mine elements from human experts or conference papers and show that prompting LLMs with curated combinations produces research ideas that are diverse, relevant, and grounded in current literature. This modern thinking machine offers a lightweight, interpretable tool for augmenting scientific creativity and suggests a path toward collaborative ideation between humans and AI.

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.

We introduce the AutoGRAMS framework for programming multi-step interactions with language models. AutoGRAMS represents AI agents as a graph, where each node can execute either a language modeling instruction or traditional code. Likewise, transitions in the graph can be governed by either language modeling decisions or traditional branch logic. AutoGRAMS supports using variables as memory and allows nodes to call other AutoGRAMS graphs as functions. We show how AutoGRAMS can be used to design highly sophisticated agents, including self-referential agents that can modify their own graph. AutoGRAMS's graph-centric approach aids interpretability, controllability, and safety during the design, development, and deployment of AI agents. We provide our framework as open source at https://github.com/autograms/autograms .

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 introduce Exhaustive Symbolic Integration (ESI), a method that enumerates all symbolic functions up to a given complexity k within a specified operator basis and determines which admit closed-form antiderivatives within the same class. This allows us to compute the "integrability fraction" ρ(k) (the fraction of functions whose derivatives lie within the same class), which we do for five operator bases including combinations of rational functions, powers, exponentials, logarithms and trigonometric functions. We find that ρ(k) declines at high complexity and that the operator basis has a dramatic effect -- in particular, adding the logarithm boosts ρ(k) by a factor of sim3 and produces or exacerbates a clear peak at k=6. We also deploy ESI as a novel integration algorithm, identifying three integrals that resist SymPy, Mathematica, RUBI, FriCAS, Maxima and Giac under all tested strategies. When an antiderivative can be found by multiple methods, ESI often returns the simplest form. These results reveal that the landscape of symbolic integrability is shaped primarily by the choice of operators, and that exhaustive enumeration can systematically discover integrable forms -- including novel ones -- that elude computer albegra systems.

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.

Compositional generalization is crucial for artificial intelligence agents to solve complex vision-language reasoning tasks. Neuro-symbolic approaches have demonstrated promise in capturing compositional structures, but they face critical challenges: (a) reliance on predefined predicates for symbolic representations that limit adaptability, (b) difficulty in extracting predicates from raw data, and (c) using non-differentiable operations for combining primitive concepts. To address these issues, we propose NeSyCoCo, a neuro-symbolic framework that leverages large language models (LLMs) to generate symbolic representations and map them to differentiable neural computations. NeSyCoCo introduces three innovations: (a) augmenting natural language inputs with dependency structures to enhance the alignment with symbolic representations, (b) employing distributed word representations to link diverse, linguistically motivated logical predicates to neural modules, and (c) using the soft composition of normalized predicate scores to align symbolic and differentiable reasoning. Our framework achieves state-of-the-art results on the ReaSCAN and CLEVR-CoGenT compositional generalization benchmarks and demonstrates robust performance with novel concepts in the CLEVR-SYN benchmark.

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".

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.

We present a framework where neural models develop an AI Mother Tongue, a native symbolic language that simultaneously supports intuitive reasoning, compositional symbol chains, and inherent interpretability. Unlike post-hoc explanation methods, our approach embeds reasoning directly into the model's representations: symbols capture meaningful semantic patterns, chains trace decision paths, and gated induction mechanisms guide selective focus, yielding transparent yet flexible reasoning. We introduce complementary training objectives to enhance symbol purity and decision sparsity, and employ a sequential specialization strategy to first build broad symbolic competence and then refine intuitive judgments. Experiments on AI tasks demonstrate competitive accuracy alongside verifiable reasoning traces, showing that AI Mother Tongue can serve as a unified mechanism for interpretability, intuition, and symbolic reasoning in neural models.

The formalization of existing mathematical proofs is a notoriously difficult process. Despite decades of research on automation and proof assistants, writing formal proofs remains arduous and only accessible to a few experts. While previous studies to automate formalization focused on powerful search algorithms, no attempts were made to take advantage of available informal proofs. In this work, we introduce Draft, Sketch, and Prove (DSP), a method that maps informal proofs to formal proof sketches, and uses the sketches to guide an automated prover by directing its search to easier sub-problems. We investigate two relevant setups where informal proofs are either written by humans or generated by a language model. Our experiments and ablation studies show that large language models are able to produce well-structured formal sketches that follow the same reasoning steps as the informal proofs. Guiding an automated prover with these sketches enhances its performance from 20.9% to 39.3% on a collection of mathematical competition problems.

Large language models (LLMs) are trained and tested extensively on symbolic representations such as code and graphs, yet real-world user tasks are often specified in natural language. To what extent can LLMs generalize across these representations? Here, we approach this question by studying isomorphic tasks involving procedures represented in code, graphs, and natural language (e.g., scheduling steps in planning). We find that training LLMs with popular post-training methods on graphs or code data alone does not reliably generalize to corresponding natural language tasks, while training solely on natural language can lead to inefficient performance gains. To address this gap, we propose a two-stage reinforcement learning curriculum that first trains on symbolic, then natural language data. The curriculum substantially improves model performance across model families and tasks. Remarkably, a 1.5B Qwen model trained by our method can closely match zero-shot GPT-4o in naturalistic planning. Finally, our analysis suggests that successful cross-representation generalization can be interpreted as a form of generative analogy, which our curriculum effectively encourages. The dataset and code used in this paper can be found https://github.com/fangru-lin/procedure_generalization_llm{here}.

This paper addresses the problem of formalizing and quantifying the concept of "intensional inheritance" between two concepts. We begin by conceiving the intensional inheritance of W from F as the amount of information the proposition "x is F " provides about the proposition "x is W. To flesh this out, we consider concepts F and W defined by sets of properties left{F_{1}, F_{2}, ldots, F_{n}right} and left{W_{1}, W_{2}, ldots, W_{m}right} with associated degrees left{d_{1}, d_{2}, ldots, d_{n}right} and left{e_{1}, e_{2}, ldots, e_{m}right}, respectively, where the properties may overlap. We then derive formulas for the intensional inheritance using both Shannon information theory and algorithmic information theory, incorporating interaction information among properties. We examine a special case where all properties are mutually exclusive and calculate the intensional inheritance in this case in both frameworks. We also derive expressions for P(W mid F) based on the mutual information formula. Finally we consider the relationship between intensional inheritance and conventional set-theoretic "extensional" inheritance, concluding that in our information-theoretic framework, extensional inheritance emerges as a special case of intensional inheritance.

Background: The field of Artificial Intelligence has undergone cyclical periods of growth and decline, known as AI summers and winters. Currently, we are in the third AI summer, characterized by significant advancements and commercialization, particularly in the integration of Symbolic AI and Sub-Symbolic AI, leading to the emergence of Neuro-Symbolic AI. Methods: The review followed the PRISMA methodology, utilizing databases such as IEEE Explore, Google Scholar, arXiv, ACM, and SpringerLink. The inclusion criteria targeted peer-reviewed papers published between 2020 and 2024. Papers were screened for relevance to Neuro-Symbolic AI, with further inclusion based on the availability of associated codebases to ensure reproducibility. Results: From an initial pool of 1,428 papers, 167 met the inclusion criteria and were analyzed in detail. The majority of research efforts are concentrated in the areas of learning and inference (63%), logic and reasoning (35%), and knowledge representation (44%). Explainability and trustworthiness are less represented (28%), with Meta-Cognition being the least explored area (5%). The review identifies significant interdisciplinary opportunities, particularly in integrating explainability and trustworthiness with other research areas. Conclusion: Neuro-Symbolic AI research has seen rapid growth since 2020, with concentrated efforts in learning and inference. Significant gaps remain in explainability, trustworthiness, and Meta-Cognition. Addressing these gaps through interdisciplinary research will be crucial for advancing the field towards more intelligent, reliable, and context-aware AI systems.

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.

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.

In this paper we present a comparative study of path feasibility queries generated during path exploration based software engineering methods. Symbolic execution based methods are gaining importance in different aspects of software engineering e.g. proving properties about programs, test case generation, comparing different executions of programs. These methods use SMT solvers to check the satisfiability of path feasibility queries written as a formula in the supported theories. We study the performance of solving such path feasibility queries using SMT solvers for real world programs. Our path condition formulas are generated in a theory of quantifier free bit vectors with arrays (QF_ABV). We show that among the different SMT solvers, STP is better than Z3 by an order of magnitude for such kind of queries. As an application we design a new program analysis (Change Value Analysis) based on our study which exploits undefined behaviors in programs. We have implemented our analysis in LLVM and tested it with the benchmark of SIR programs. It reduces the time taken for solving path feasibility queries by 48%. The study can serve as guidance to practitioners using path feasibility queries to create scalable software engineering methods based on symbolic execution.

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.

Control systems are ubiquitous in modern technology, comprising an engineered plant to be kept within specific, often fine-tuned, limits, and a separate controller that ensures this is the case. While modern controllers often employ digital computers, other examples are purely mechanical, or even biological. It is an open question whether computation is happening within all controllers by virtue of them being part of a control system. Abstraction/ Representation theory (ART) has been developed to tackle just this question of whether a physical system is computing. Here, we demonstrate how to use ART to model control systems, and analyse them for computational properties. We determine that the plant of a control system is (a proxy for) the representational entity necessary in ART for the existence of any computation: the plant is the user of the controller. We consider specific systems: a digital thermostat, an electro-mechanical thermostat, the purely mechanical centrifugal governor, and an open-loop human-controlled heating system. We show that all these systems, and control systems in general, are performing some degree of computation. As an initial use of these results, we apply them to computationalism within cognitive theory: we show the governor is computing, so it cannot play its role of counter-example in the question of whether the brain is too.

Abstraction ability is crucial in human intelligence, which can also benefit various tasks in NLP study. Existing work shows that LLMs are deficient in abstract ability, and how to improve it remains unexplored. In this work, we design the framework AbsInstruct to enhance LLMs' abstraction ability through instruction tuning. The framework builds instructions with in-depth explanations to assist LLMs in capturing the underlying rationale of abstraction. Meanwhile, we introduce a plausibility estimator to select instructions that are more consistent with the abstraction knowledge of LLMs to be aligned. Then, our framework combines abstraction instructions with general-purpose ones to build a hybrid dataset. Extensive experiments and analyses demonstrate that our framework can considerably enhance LLMs' abstraction ability with strong generalization performance while maintaining their general instruction-following abilities.

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.

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.

Informally, the 'linear representation hypothesis' is the idea that high-level concepts are represented linearly as directions in some representation space. In this paper, we address two closely related questions: What does "linear representation" actually mean? And, how do we make sense of geometric notions (e.g., cosine similarity or projection) in the representation space? To answer these, we use the language of counterfactuals to give two formalizations of "linear representation", one in the output (word) representation space, and one in the input (sentence) space. We then prove these connect to linear probing and model steering, respectively. To make sense of geometric notions, we use the formalization to identify a particular (non-Euclidean) inner product that respects language structure in a sense we make precise. Using this causal inner product, we show how to unify all notions of linear representation. In particular, this allows the construction of probes and steering vectors using counterfactual pairs. Experiments with LLaMA-2 demonstrate the existence of linear representations of concepts, the connection to interpretation and control, and the fundamental role of the choice of inner product.

Reasoning methods such as chain-of-thought prompting and self-consistency have shown immense potential to improve the accuracy of large language models across various reasoning tasks. However such methods involve generation of lengthy reasoning chains, which substantially increases token consumption, computational cost, and latency. To address this inefficiency, we propose ProofSketch, a verification-guided reasoning framework that integrates symbolic closure computation, lexicographic verification and adaptive sketch generation. Our experiments show that ProofSketch consistently reduces token usage while improving accuracy, demonstrating that this approach offers a promising path for efficient and trustworthy reasoning.

The demonstrated code-understanding capability of LLMs raises the question of whether they can be used for automated program verification, a task that often demands high-level abstract reasoning about program properties, which is challenging for verification tools. We propose a general methodology to combine the power of LLMs and automated reasoners for automated program verification. We formally describe this methodology as a set of derivation rules and prove its soundness. We instantiate the calculus as a sound automated verification procedure, which led to practical improvements on a set of synthetic and competition benchmarks.

The ongoing development of Lean 4's Mathlib has produced a macroscopic structural complexity that interweaves logical, mathematical, and infrastructural dependencies. We present a network analysis of this library, extracting its dependency structure into a multilayer graph of 308,129 declarations, 8.4 million edges, and 7,563 modules. By introducing graph decompositions that isolate explicit edges from those synthesized by the compiler or driven by proofs, we quantify the structural properties of formalized mathematics. Our analysis reveals three findings. First, taxonomies designed by humans diverge from logical structures, exhibiting a 50.9% coupling across namespaces. Second, developers utilize a median of 1.6% of the imported scope. Third, formalization compresses semantic hierarchies, with network centrality capturing language infrastructure rather than mathematical relevance.

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

To improve the performance and explainability of LLM-based natural language reasoning, structured reasoning can be applied to generate explicitly structured proofs. Among different methods for structured reasoning, we specifically focus on backward chaining, where the proof goal is recursively decomposed to subgoals by searching and applying rules. We argue that current LLM-based backward chaining systems (e.g. Least-to-most prompting and LAMBADA) are incomplete, as they omit crucial algorithmic components identified from the classic backward chaining algorithm in computational logic (SLD Resolution). To this end, we propose a novel backward chaining system, SymBa (Symbolic Backward Chaining), which integrates a symbolic solver and an LLM. In SymBa, the solver controls the proof process, and the LLM is only called when the solver requires new information to complete the proof. Empowered by completeness, SymBa achieves a significant improvement in seven deductive, relational, and arithmetic reasoning benchmarks compared to the baselines.

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.

Large language models (LLMs) with memory are computationally universal. However, mainstream LLMs are not taking full advantage of memory, and the designs are heavily influenced by biological brains. Due to their approximate nature and proneness to the accumulation of errors, conventional neural memory mechanisms cannot support LLMs to simulate complex reasoning. In this paper, we seek inspiration from modern computer architectures to augment LLMs with symbolic memory for complex multi-hop reasoning. Such a symbolic memory framework is instantiated as an LLM and a set of SQL databases, where the LLM generates SQL instructions to manipulate the SQL databases. We validate the effectiveness of the proposed memory framework on a synthetic dataset requiring complex reasoning. The project website is available at https://chatdatabase.github.io/ .

There is considerable confusion about the role of Large Language Models (LLMs) in planning and reasoning tasks. On one side are over-optimistic claims that LLMs can indeed do these tasks with just the right prompting or self-verification strategies. On the other side are perhaps over-pessimistic claims that all that LLMs are good for in planning/reasoning tasks are as mere translators of the problem specification from one syntactic format to another, and ship the problem off to external symbolic solvers. In this position paper, we take the view that both these extremes are misguided. We argue that auto-regressive LLMs cannot, by themselves, do planning or self-verification (which is after all a form of reasoning), and shed some light on the reasons for misunderstandings in the literature. We will also argue that LLMs should be viewed as universal approximate knowledge sources that have much more meaningful roles to play in planning/reasoning tasks beyond simple front-end/back-end format translators. We present a vision of {\bf LLM-Modulo Frameworks} that combine the strengths of LLMs with external model-based verifiers in a tighter bi-directional interaction regime. We will show how the models driving the external verifiers themselves can be acquired with the help of LLMs. We will also argue that rather than simply pipelining LLMs and symbolic components, this LLM-Modulo Framework provides a better neuro-symbolic approach that offers tighter integration between LLMs and symbolic components, and allows extending the scope of model-based planning/reasoning regimes towards more flexible knowledge, problem and preference specifications.

The Functional Machine Calculus (Heijltjes 2022) is a new approach to unifying the imperative and functional programming paradigms. It extends the lambda-calculus, preserving the key features of confluent reduction and typed termination, to embed computational effects, evaluation strategies, and control flow operations. The first instalment modelled sequential higher-order computation with global store, input/output, probabilities, and non-determinism, and embedded both the call-by-name and call-by-value lambda-calculus, as well as Moggi's computational metalanguage and Levy's call-by-push-value. The present paper extends the calculus from sequential to branching and looping control flow. This allows the faithful embedding of a minimal but complete imperative language, including conditionals, exception handling, and iteration, as well as constants and algebraic data types. The calculus is defined through a simple operational semantics, extending the (simplified) Krivine machine for the lambda-calculus with multiple operand stacks to model effects and a continuation stack to model sequential, branching, and looping computation. It features a confluent reduction relation and a system of simple types that guarantees termination of the machine and strong normalization of reduction (in the absence of iteration). These properties carry over to the embedded imperative language, providing a unified functional-imperative model of computation that supports simple types, a direct and intuitive operational semantics, and a confluent reduction semantics.

Large language models (LLMs) have demonstrated impressive capabilities in natural language understanding and generation, but they exhibit problems with logical consistency in the output they generate. How can we harness LLMs' broad-coverage parametric knowledge in formal reasoning despite their inconsistency? We present a method for directly integrating an LLM into the interpretation function of the formal semantics for a paraconsistent logic. We provide experimental evidence for the feasibility of the method by evaluating the function using datasets created from several short-form factuality benchmarks. Unlike prior work, our method offers a theoretical framework for neuro-symbolic reasoning that leverages an LLM's knowledge while preserving the underlying logic's soundness and completeness properties.

While large language models (LLMs) are increasingly being used for program synthesis, they lack the global view needed to develop useful abstractions; they generally predict programs one at a time, often repeating the same functionality. Generating redundant code from scratch is both inefficient and error-prone. To address this, we propose Refactoring for Generalizable Abstraction Learning (ReGAL), a gradient-free method for learning a library of reusable functions via code refactorization, i.e. restructuring code without changing its execution output. ReGAL learns from a small set of existing programs, iteratively verifying and refining its abstractions via execution. We find that the shared function libraries discovered by ReGAL make programs easier to predict across diverse domains. On three datasets (LOGO graphics generation, Date reasoning, and TextCraft, a Minecraft-based text game), both open-source and proprietary LLMs improve in accuracy when predicting programs with ReGAL functions. For CodeLlama-13B, ReGAL results in absolute accuracy increases of 11.5% on graphics, 26.1% on date understanding, and 8.1% on TextCraft, outperforming GPT-3.5 in two of three domains. Our analysis reveals ReGAL's abstractions encapsulate frequently-used subroutines as well as environment dynamics.