Symbol-

Existing approaches to solving differential equations, such as symbolic regression, physics informed neural networks, and neural operators, typically focus on numerical approximations or blind symbolic search via fitting to numerical data. Less attention has been paid to learning structured representations of mathematical expressions that preserve commutative properties and could support mathematical reasoning in symbolic forms. Transformer models have shown strong capabilities in solving symbolic differential equations. However, standard positional embeddings in transformers are designed for sequence data. Symbolic differential equations are naturally represented by expression trees, so these positional embeddings may not efficiently capture their hierarchical structures. We investigate existing tree positional embeddings in symbolic ordinary differential equation (ODE) tasks. We systematically study their effectiveness under different settings. Our results show that tree positional embeddings aid learning in early epochs and continue to improve performance throughout, ultimately yielding consistent advantages across various data sizes and tasks. Based on learned structural representations, we apply contrastive learning to support the commutative property in mathematics. Ablation studies provide insight into how these methods interact in modelling symbolic mathematical structures.

Plane geometry remains a significant challenge in AI, requiring the integration of visual perception and mathematical reasoning. While Large Multimodal Models (LMMs) naturally handle visuo-linguistic inputs, they are often computationally intensive and opaque. We demonstrate that a pure Large Language Model (LLM), when equipped with specialized modules, can rival state-of-the-art LMMs on complex geometry problems. Our framework integrates a Geometric Vision Parser, which translates diagrams into symbolic form, with a Symbolic Solver that performs formal deductions, thereby mitigating hallucinations and promoting interpretable reasoning. To enable rigorous evaluation, we curate a benchmark of challenging problems from the 2025 Chinese Zhongkao examinations, ensuring data novelty and testing deeper deductive skills. Experiments demonstrate that our approach achieves performance comparable to Gemini 2.5 Pro while delivering clearer, human-like solutions.

A fundamental challenge in artificial intelligence is the transformation of observations into explicit symbolic representations suitable for abstraction, interpretation, and reasoning. While modern AI systems achieve remarkable perceptual capabilities through large-scale statistical learning, the resulting knowledge is typically encoded within latent parameters that are difficult to inspect or manipulate analytically. Inspired by Neuro-Symbolic AI and theories of human abstraction, this paper investigates the formation of symbolic mathematical representations from geometric observations. We propose NeuSOGA (Neuro-Symbolic Geometric Abstraction), a framework that progressively transforms observations into topological abstractions, geometric abstractions, and ultimately symbolic mathematical representations. The architecture combines topology-guided structural discovery using Euclidean Distance Transforms, foundation-model perception using Segment Anything, adaptive multi-scale geometric abstraction, and symbolic synthesis through Implicit Area Splines. The resulting representation is an analytical implicit model supporting arbitrary-order smoothness, additive composition, and closed-form evaluation. Unlike neural latent encodings, the generated representation remains interpretable, editable, and mathematically explicit. Experiments on ModelNet40 point clouds, arbitrary-view projections, and segmented optical observations demonstrate that NeuSOGA transforms diverse observations into compact symbolic representations while preserving essential geometric and topological structure across sensing modalities and viewing directions. NeuSOGA provides an interpretable and explainable pathway from observation to symbol and establishes

We present a reinforcement-learning agent that solves symbolic equations step by step, covering both nonlinear closed equations (radicals, exponentials, trigonometric) and a controlled class of restricted-open families requiring a change of variables (CoV) such as completing the square. We cast algebra as an MDP with a dynamic action space and a tree-structured policy (TreeMLP). The main policy learns from reward alone with no supervised solution traces; the CoV substitution comes from a supervised generator interchangeable with a CAS call. On closed equations the agent matches the prior best on CommonCore (0.93 greedy vs. ConPoLe's 0.925) under a single policy. On four hand-designed restricted-open families (quadratic, cubic, quartic, exponential) it reaches 0.79 beam / 0.67 greedy, exceeding the strongest non-learned search (A-star, 0.64). Learned CoV timing has content only on the exponential family, the one requiring a nested CoV, where a natural rule solves none of the held-out equations while the policy solves 75% from reward alone. At 10x scale a sharp seed-level bimodality emerges; a UCB learning-progress curriculum shows a non-significant positive trend toward mitigating it. We do not claim general open-equation solving: every open-equation result is confined to these four controlled families.

Electrical circuit analysis requires more than recognizing components in an image. A solver must ground symbols and labels, recover latent topology, select a physical model, formulate coupled equations, propagate intermediate quantities, and preserve units, signs, directions, and phase conventions. We introduce \benchmark, a benchmark of 1,000 authentic textbook problems for evaluating this complete long-horizon visual-to-symbolic reasoning process. Each problem pairs one or more circuit diagrams with a self-contained question, a typed or semantically specified answer, and a reference worked solution. An evidence-first construction pipeline aligns questions, figures, and solutions, while a reasoning-oriented taxonomy organizes problems by circuit type and dependency depth. Evaluation combines conservative typed scoring with identity-blinded multi-model semantic consensus, retaining every problem in the denominator. Across three commercial chatbot systems and six open-source multimodal large language models, the highest-scoring system reaches 84.8% accuracy. However, performance consistently deteriorates on long-horizon problems, and qualitative analysis exposes persistent failures in topology-to-target binding, physical conventions, and late-stage output propagation. \benchmark{} provides a focused testbed for measuring whether multimodal models can transform technical visual evidence into sustained, physically valid symbolic reasoning. Code are available at GitHub - CircuitReason/CircuitReason1K.

Previous studies have shown that people can develop shared symbols, partner-specific expressions, personal idioms, inside jokes, and other parts of a relational microculture. Recent work has also examined how humans and conversational AI negotiate and revise symbolic meanings. However, long-term human-AI systems still lack a clear design model for recording how a dyad-specific expression gains meaning, checking whether both sides still accept that meaning, and safely reusing the expression in later sessions. This concept-and-prototype paper introduces Private Etymology, a machine-representable relational provenance that records how a dyad-specific symbolic expression is proposed, interpreted, negotiated, repaired, reused, revised, stabilized, contested, forgotten, or retired over time. I also propose relational reuse: reactivating a dyad-specific expression in a later session without fully explaining its meaning again. The contribution is not the invention of shared symbols or relational microcultures. Instead, this paper integrates prior ideas into persistent, revisable, and evidence-grounded symbolic units for human-AI relationships. I present a lifecycle model, an illustrative machine-readable schema, a working Apple Watch prototype, and a longitudinal research agenda. In the prototype, a language model classifies discrete conversational evidence, while deterministic local code decides whether a Shared Symbol can be updated. This prevents a free-form model confidence score or an AI proposal by itself from directly updating the persisted symbol. Private Etymology is proposed as infrastructure for conversational agents to participate in changing relational microcultures without inventing their origins or treating relational meaning as a fixed memory value.

Tool calling allows large language models (LLMs) to invoke external computation during problem solving, a useful capability in various fields including AI for mathematics. We study this setting through weighted sum-of-squares (SOS) decomposition, a machine-checkable route to proving polynomial nonnegativity and hence polynomial inequalities. A candidate decomposition can be checked exactly, but finding one requires choosing among non-unique regroupings and coordinating multiple symbolic transformations. We develop an agent that combines algebraic task training, symbolic tools, and verifier-grounded optimization for this task. Rather than training only on the composite SOS task, we construct 1.35 million synthetic examples covering eight supporting polynomial tasks together with weighted-SOS decomposition. We first apply supervised fine-tuning (SFT) to direct algebra problems and simulated symbolic traces, and then use Group Relative Policy Optimization (GRPO) with task-specific symbolic rewards. The SFT corpus contains no native tool-calling messages; at evaluation, the agent uses native SymPy calls for expansion, collection, reordering, and factorization. Every final SOS answer is checked by exact expansion and coefficient comparison. On held-out, same-generator synthetic problems, the full SFT+GRPO+tools system is the strongest of four evaluated configurations, reaching 78.96% verified success on weighted SOS, compared with 44.73% for the base model with the same tools, and 91.75% macro accuracy across nine polynomial tasks. Within this controlled setting, our work provides a case study of combining domain-specific skill training, executable tools, and verifier feedback, and may inform the design of tool-calling agents in other domains with exactly checkable outputs.

A rich and recognizable component library is the cornerstone of printed circuit board (PCB) design and generation. Traditionally, engineers manually create symbols and footprints and design PCB schematics, which is time-consuming and error-prone. Leveraging multimodal large language models (MLLMs), we develop SFgen, an agentic recognition and generation flow of symbol and footprint for electronic components. SFgen achieves 86% accuracy for symbol generation and 80% accuracy for footprint generation. We use the SFgen method to create SFnet, a database of symbols and footprints. It now has 1000 components and is expanding constantly, which lays the foundation for automatic generation of PCB designs.

In HEP data analyses, finding an adequate function to model binned data has largely relied on a manual process: guess a functional form by intuition, fit, examine, then repeat until successful. We show that this iterative process can be automated by a machine using symbolic regression, which performs a data-driven search over function space without requiring prior knowledge of what an adequate function should look like. We present the SymbolFit package, which pairs symbolic regression with uncertainty modeling to target HEP analysis use cases, and demonstrate it on the CMS and ATLAS Run 2 dijet spectra: 560 independent seeded runs across seven simple fit configurations generated over 1000 functions fitting the spectra with $χ^2/\text{NDF}\approx 1$, and 111 of the runs rediscovered the very dijet and UA2 functions used in published dijet searches.

Symbolic expressions can effectively characterize and predict circuit behavior, but deriving them directly from circuit schematics is challenging. This process requires accurate visual-to-symbolic construction of circuit structure from images and correct multi-step symbolic derivation, both of which impose strict correctness requirements. This work proposes AutoVSR, an automated framework for visual-to-symbolic generation of circuit expressions using Vision Language Models (VLMs). By reconstructing circuit diagrams into an executable intermediate representation (Executable IR) and leveraging a symbolic solver for reasoning, AutoVSR significantly improves the accuracy of symbolic expression generation. AutoVSR introduces two key innovations: an IR construction method guided by component rule retrieval and verification-based feedback, and a symbolic solver implemented as a planning agent equipped with a symbolic tool library for reliable multi-step derivation. Compared with end-to-end VLM approaches and specialized methods on the main symbolic expression generation task, AutoVSR achieves accuracy improvements of 30.01--59.45% and 41.96--51.84%, respectively. Moreover, AutoVSR surpasses closed-source state-of-the-art VLMs in inference cost and computational efficiency. Code is available at https://github.com/LongfeiLi1/AutoVSR.

Why do intelligent systems need to perform explicit symbolic reasoning? Computer science has traditionally regarded symbolic reasoning as a defining component of intelligence. Yet the remarkable success of modern foundation models raises a fundamental question: if increasingly capable AI systems can operate with little explicit symbolic reasoning, what role do symbolic methods actually play? This article argues that explicit symbolic reasoning is not a fundamental property of intelligence, but a computational consequence of operating on simplified models of reality. We propose the Compression Principle: every computational model is a simplified representation of reality, and explicit symbolic reasoning compensates for information omitted during model construction. From this principle, we derive the Modeling--Reasoning Trade-off: as computational models preserve richer representations of the world, the need for explicit symbolic reasoning correspondingly decreases. This perspective provides a unified explanation for both the historical success of symbolic methods and the remarkable effectiveness of modern foundation models. Paradoxically, the same development makes symbolic methods increasingly important for humans. As intelligent systems become more capable and more opaque, symbolic representations increasingly serve as interfaces through which humans specify requirements, verify behavior, regulate autonomous systems, and establish trust. We therefore argue that the future of symbolic methods lies not primarily as the computational engine of intelligent systems, but as the symbolic interface between increasingly capable AI systems and the humans who build, govern, and depend upon them.

Copernicus, the European Union's Earth observation program, produces petabytes of Earth observation and climate data, offering immense potential for research, policy, and applications. However, access to these datasets requires advanced programming skills and familiarity with domain-specific formats such as NetCDF or GRIB. Moreover, general-purpose Text-to-SQL systems fail when applied naively to the meteorological domain due to a profound ``Symbolic-to-Numeric'' gap. To overcome these limitations, we present an end-to-end Text-to-SQL framework specifically engineered for real-world, scalable meteorological data exploration. Our system intercepts natural language to resolve spatial and semantic ambiguities \textit{before} SQL generation. We design STRATOS, a Spatio-Temporal Resolution Agent for Text-to-SQL to dynamically bridge the symbolic-to-numeric gap by mapping fuzzy concepts to a localized ontology and resolving spatial entities via external knowledge bases. Further, our complexity-aware query rewriter rewrites expensive spatial predicates, reducing execution times from hours to seconds. Last, we introduce the STRATOS Evaluation Workload, comprising 7,520 complex query pairs explicitly designed by domain experts to test scalability and symbolic-to-numeric translation across challenging spatio-temporal dimensions previously unexplored by Text-to-SQL systems.

Human reasoning is inherently multimodal: when problems become difficult, we rarely think in words alone. We often externalize our reasoning by sketching diagrams or drawing grids to understand the underlying conceptual structure and avoid mistakes. Building on this premise, our research investigates: (a) whether grounding multi-hop textual-spatial stories into geometry-aware modalities, such as layouts or grids, improves reasoning compared to natural language-based inference; and (b) whether a model can decide when to rely on natural language reasoning and when to switch to a structured modality. We address these questions by introducing a switching metric based on trustworthiness and complexity signals, which estimates when grounding a spatial story into structure is likely to improve performance. This takes a first step toward principled modality selection in Large Language Model (LLM) reasoning. Across our settings, switching from natural language-based reasoning to a grid-based representation improves LLM performance by up to 42%, highlighting the importance of modality choice in shaping reasoning outcomes.

The automated design of agentic systems offers a promising pathway for scaling large language models (LLMs) beyond single-agent reasoning. While prior work has advanced task performance through handcrafted or automatically generated multi-agent workflows, robustness is often treated as an afterthought, leaving systems vulnerable to external adversaries and internal failures. We propose AutoRAS, a framework for the Automated design of Robust Agentic Systems. AutoRAS formulates system design as generating a sequence of symbolic primitives that jointly encode structural connectivity and behavioral actions, and learns to optimize this sequence using execution-derived safety signals and flow-based sequence-level objectives. Extensive experiments show that AutoRAS achieves the best performance in both vanilla and adversarial settings, with the smallest performance degradation under attacks. Further analyses demonstrate strong transferability, stable optimization behavior, stability across primitive sets, and favorable cost trade-offs. Our code is available at $\href{https://github.com/guohezuy/AutoRAS}{\text{this https URL}}$.

Variable-order Markov models preserve local symbolic syntax by adapting context length, but long continuations can enter recurring high-order "tunnels": repeated suffixes, locally periodic passages, or copied fragments longer than the formal Markov order. This paper introduces signed pattern control for variable-order Markov generation with BP-Regular sampling. A weighted recurrence automaton computes an activation R for a chosen family of target patterns, and belief propagation samples exactly from P_beta(x) proportional to P_0(x) exp(beta R(x)). Negative coupling makes the target patterns costly during sampling; positive coupling rewards the same patterns and turns them into controlled attractors. The target family may be mined online from overactive generated material, supplied by a score or style vocabulary, or designed as an experimental probe. The main experiments use the online homeostatic case, choosing patterns that become overactive in the sampling history. On six duration-bearing monophonic sources, including Bach and Telemann material, the negative branch reduces generated 8-gram self-reuse, increases the effective number of generated 8-grams, and increases coverage of training-supported 4-gram contexts while preserving substantial lower-order support. A pitch-sequence replication on five Weimar Jazz Database solos gives the same anti-reuse signature outside Baroque material. The same signed mechanism also provides a positive branch for probing attractor basins, phase transitions, and hysteresis in the underlying variable-order model.

Handwritten Mathematical Expression Generation (HMEG) is challenging due to the complex two-dimensional layouts and long-range structural dependencies of mathematical expressions. Existing methods typically rely on explicit spatial supervision, such as symbol-level bounding boxes, which incurs high annotation costs and limits scalability. In this work, we propose DiffMath, a symbol- and graph-aware latent diffusion framework that leverages the hierarchical structure inherent in LaTeX as a structural prior, eliminating the need for positional supervision. First, we design a Relational Abstract Syntax Tree (RelAST), a generation-oriented representation that distills MathML trees into compact triplet sequences [S, R, D], where each token directly encodes a symbol identity, spatial relation, or nesting depth. Second, we introduce MathVAE, which learns structure-preserving latent representations through symbol-aware and relation-aware perceptual regularization, ensuring that the latent space captures both character semantics and spatial topology. Third, MathDiT performs conditional denoising in this structured latent space, further guided by a global symbol-count prior via Adaptive Layer Normalization (AdaLN) to improve structural coherence. Experiments show that DiffMath produces structurally consistent handwritten expressions, achieves superior performance over existing methods, and improves the accuracy of downstream OCR models through synthetic data augmentation.

In this paper, we present a procedure for numeric planning based on Symbolic Pattern Planning (SPP). Given a numeric planning problem $Π$, a pattern $\prec$ is a sequence of actions used to define a formula encoding the subsequences of $\prec$ executable from a starting state $S$. Cardellini, Giunchiglia, and Maratea (2024a) follow the Planning as Satisfiability approach by defining, at each step $n \ge 0$, a formula $Π^\prec_n$ in which $(i)$ the pattern $\prec$ is computed only for $n=0$ in the initial state $I$ of $Π$, and then exploited at each step $n$, $(ii)$ the starting state $S$ is set to $I$, and $(iii)$ the set $G$ of goals is required to hold in the last state that can be reached by one of the subsequences of $\prec$ concatenated $n$ times. The procedure begins with $n=0$, terminates as soon as $Π^\prec_n$ is satisfiable, and otherwise proceeds by incrementing $n$. In this paper, possibly at each step, $(i)$ we symbolically search for an intermediate state $P$ reachable from $I$, closer to a goal state, $(ii)$ dynamically recompute the pattern $\prec_h$ -- to be used in the next step -- in $P$, $(iii)$ refine the pattern $\prec_g$ used to reach $P$, and $(iv)$ start the new search from the state $S$ which can be either the initial state $I$ or the last computed intermediate state $P$, exploiting the computed patterns $\prec_g$ and $\prec_h$ to define the pattern $\prec$ to be used in the search. In particular, at each step, we define a formula $Π^{\prec}_{S,P}$ encoding the existence of a state $P'$ closer than $P$ to a goal state, with $P'$ reachable from the starting state $S$ when using the pattern $\prec$. We present different techniques for producing such formulas, each corresponding to a different strategy for exploring the search space. We prove their correctness and completeness, the latter under certain conditions.

We study context compression for multi-hop question answering with small language models. We propose Telegraph English, a readable symbolic format that rewrites retrieved passages into structured entity-relation statements, preserving reasoning evidence at lower token cost. In controlled experiments on MuSiQue, TwoWiki, and HotpotQA, Telegraph English outperforms three matched-budget compression baselines (character-level deletion, truncation, and random sub-sampling) on every dataset, with gains of 13 to 20 F1 percentage point. It also outperforms a coherent prose summary produced by the same encoder on the hardest dataset. A pre-registered depth-interaction hypothesis is null: the advantage does not grow with reasoning depth within datasets. We interpret these results as evidence that readable symbolic re-expression preserves entity content more densely than either natural language or coherent summarization at matched token budget.

Knowledge extraction from symbolic data often produces abstractions that are formally defined but not immediately interpretable by users. Formal Concept Analysis (FCA) and Relational Concept Analysis (RCA) provide representative settings for this issue: they generate explicit conceptual structures, implications, and relational dependencies from object descriptions and relations. Although these structures are explainable by design, their concepts are often identified by technical labels, which limits their use as human-interpretable knowledge units. Assigning meaningful names to such concepts is therefore a key issue for interpretation, navigation, validation, and reuse by domain experts. This paper investigates concept naming in FCA and RCA from a symbolic knowledge representation perspective. We first characterize the linguistic and terminological challenges involved in naming generated symbolic abstractions, including ambiguity, discrimination, concision, and consistency across related concepts. We then propose a configurable framework for LLM-assisted concept naming. The framework relies on a variability model that controls which sources of information are exposed during naming, such as intent, extent, inherited information, neighboring concepts, implications, and relational attributes. It thereby makes explicit the semantic choices involved in moving from formal concept descriptions to human-readable names. The approach is illustrated as a proof of concept on a small relational dataset in the pizzeria domain. This illustration shows how different configurations influence the names suggested by an LLM, and how naming variability can reveal interpretation choices, relational dependencies, and possible modeling issues in the underlying symbolic data.

Recent large language models (LLMs) often appear to exhibit spatial reasoning ability; however, this capability is largely \emph{symbolic}, arising from pattern matching over spatial language rather than true \emph{geometric} reasoning over space. Because LLMs operate on discrete tokens, they lack native support for continuous spatial representations, explicit geometric computation, and structured spatial operators. To address this limitation, we introduce the \emph{Spatial Language Model (SLM)}, the first multimodal LLM that treats location information as a first-class modality and enables geometric spatial reasoning within the model's inference process. SLM directly operates on learned spatial representations rather than textual descriptions of spatial relations. To support effective training, we construct a \emph{Spatial Instruction Dataset} that aligns spatial representations, atomic geometric operations, and natural language instructions. We further propose a new benchmark named \emph{SpatialEval}, which is designed to evaluate spatial reasoning across attributes, distance, topology, and relative-position tasks. Extensive experiments show that SLM significantly outperforms existing LLM-based approaches that rely on symbolic reasoning via prompt engineering or textual abstraction, demonstrating the benefits of integrating geometric spatial representations for robust spatial reasoning. Our instruction dataset, evaluation benchmark, model training codes, and models' checkpoints can be found at: \hyperlink{https://github.com/chuchen2017/SLM}{https://github.com/chuchen2017/SLM}.

Transformer-based language models are widespread in today's society. As such, understanding the mechanisms by which they solve structured tasks and predicting how they may behave in novel scenarios is of great importance for safe deployment. We study the learning dynamics of attention heads in a controlled setting by training a decoder-only Transformer (GPT-J) on two structurally equivalent multi-hop reasoning tasks: a number task requiring positional reasoning and a letter task requiring symbolic reasoning. Using a recently introduced metric that classifies attention-head behavior as positional or symbolic for a given prompt, we show that successful learning is associated with the emergence of pure heads, i.e., heads that express themselves as either positional or symbolic. Despite the tasks' structural equivalence, they impose different mechanistic demands: the number task requires both positional and symbolic heads, whereas the letter task requires only symbolic heads. We then identify the computational roles of these heads, characterize the basic functions they implement, and give theoretical constructions showing how single-layer RoPE-based attention can realize these functions through geometrically interpretable query, key, and value operations. This analysis yields a quantitative separation between positional and symbolic mechanisms in their robustness to longer sequences, formalized through a novel notion of discrepancy. We empirically validate the resulting predictions in both controlled and real-world models, showing that symbolic mechanisms extrapolate more reliably to longer sequences while positional mechanisms face sharper limitations.

When generating outputs for domains with specific validity constraints (e.g., a program should compile), LLMs often fail in a small number of focused ways: for example, by using Python function names when generating TypeScript. We observe that these error patterns can be represented using a small number of constraints that can be learned in practice. We propose \emph{prefix filters}, which are per-domain-and-LLM symbolic functions, as objects to capture the error patterns, Palla as an algorithm to learn prefix filters efficiently in practice, and implement Palla. Prefix filters learned by Palla i) help us quantitatively analyze the error patterns of LLMs, and ii) can be used to constrain the outputs of a model via constrained sampling algorithms. For example, Palla boosts compile rates for Qwen2.5-1.5B on TypeScript generation, by over 60%, allowing Qwen2.5-1.5B to achieve similar performance to Llama3.1-8B unconstrained.

Retrieval-Augmented Generation (RAG) systems for question answering typically retrieve evidence by semantic similarity between the query and document chunks. While effective for unstructured text, this approach is less reliable on semi-structured corpora where answering may require exact filtering, aggregation, or exhaustive retrieval over structured attributes across multiple documents. Symbolic approaches support such operations, but they are often brittle on noisy natural-language corpora. We address this gap with DualGraph, a RAG framework that represents documents through two complementary views: a Textual Knowledge Graph for semantic retrieval and a Symbolic Knowledge Graph for symbolic querying over typed subject--predicate--object triples. Building on these two components, we provide multiple strategies for selecting or combining semantic and symbolic evidence.We also introduce SpecsQA, a benchmark from a commercial shopping website with semi-structured product documents and manually curated questions spanning open-ended and specification-oriented retrieval. Experiments show that DualGraph consistently outperforms state-of-the-art dense-retrieval, GraphRAG, symbolic, and table-oriented baselines across question types.Code and data are available at https://github.com/corneliocristina/DualGraphRAG.

Winner-take-all (WTA) networks constitute a central circuit motif in cortical networks of the brain. In addition, WTA-like activations are abundant in modern deep learning models in the form of the softmax activation for example in attention layers of transformers. While their role in the extraction of latent factors has been studied for relatively simple generative models, their role in the context of highly non-linearly entangled latent factors has remained elusive. In this article, we show that a WTA bottleneck within a deep neural network can enforce under certain well-defined conditions the extraction of categorical latent factors of the data in a multi-task learning setup. In particular, we prove that the representation that emerges in the WTA bottleneck is highly symbolic, where a single neuron or a population of neurons encodes the presence of a single abstract feature such as a specific object, color, or position. We furthermore show empirically on two datasets, that this also holds for architectures and setups that do not fully comply with the assumptions of our theorem and demonstrate the advantages of the acquired symbolic representation for generalization. Our proposed model provides insights into the generalization capabilities of deep neural networks with WTA-like components and may serve as an interface between symbolic and subsymbolic AI systems.

LLM prompting is widely used for naturally stated tasks, yet it is unreliable it may succeed on a few test cases but fail at deployment time. We study performance prediction: given a program, either symbolic (e.g. Python) or a prompt executed on an LLM, and a few in-domain examples, predict its performance on unseen tasks from the same domain. We use a simple coin-flip model, treating each pass/fail program execution as a Bernoulli random variable, whose success probability is the programs unknown performance. In this model, performance depends entirely on: 1) the observed execution outcomes on test cases, and 2) a prior over performances. We compile empirical performance priors from a corpus of diverse programs and tasks, and find that performance for symbolic programs (e.g., Python) are all or nothing, while prompt programs have a diffuse prior with many nearly-correct programs. This difference explains why a few passing tests can certify symbolic programs but not prompt programs. Building on this insight, we develop RAP (Retrieved Approximate Prior), which retrieves similar tasks and prompt programs from an existing corpus to construct a proxy prior, which is then used to predict performance. We show RAP achieves solid performances.

Semantic communication systems for goal-oriented transmission must protect task-relevant information not only through source compression but also via physical layer mapping. Existing approaches decouple constellation design and semantic encoding, exposing critical symbols to channel errors at the same rate as irrelevant ones. Contrary to this, in this paper, a joint semantic-physical layer framework is proposed, which is composed of a vector quantized-variational autoencoder that extracts discrete latent concepts, a semantic criticality indicator (SCI) that scores each concept by task relevance, and a deep reinforcement learning agent that dynamically selects the transmission subset based on instantaneous channel conditions. At the physical layer, a learned semantic-aware M -QAM constellation assigns symbol positions according to joint co-occurrence statistics and SCI scores, departing from the uniform spacing and Gray coding of standard M -QAM which minimizes average BER without regard for semantic content. We introduce a novel semantic symbol vulnerability (SSV) metric and a semantic protection probability (SPP) to quantify the exposure of task-critical symbols to decoding errors, and prove that any Gray-coded constellation is strictly suboptimal in SCI-Weighted SSV whenever the source exhibits non-uniform semantic importance and co-occurrence statistics. Simulation results demonstrate that the proposed constellation achieves near 100% SPP across modulation orders from 4-QAM to 1024-QAM versus 50% for standard constellations at high spectral efficiency, a 21:1 compression ratio with semantic quality above 0.9, generalizing across MNIST, Fashion-MNIST, and FSDD without modification.

Although natural language is the default medium for Large Language Models (LLMs), its limited expressive capacity creates a profound bottleneck for complex problem-solving. While recent advancements in AI have relied heavily on scaling, merely internalizing knowledge does not guarantee its effective application. Defining language representation as the linguistic and symbolic constructs used to map and model the real world, this paper argues that shaping schemas through advanced language representation is the next frontier for expanding LLM intelligence. We posit that an LLM's knowledge activation and organization -- its schema -- depends heavily on the structural and symbolic sophistication of the language used to represent a given task. This paper contributes both a formalization of this claim and the empirical evidence to support it. With a new formalization, we present multiple lines of evidence to support our position: Firstly, we review recent empirical practices and emerging methodologies that demonstrate the substantial performance gains achievable through deliberate language representation design, even without modifying model parameters or scale. Secondly, we conduct controlled experiments showing that LLM performance and its internal feature activations vary under different language representations of the same underlying task. Together, these findings highlight language representation design as a promising direction for future research.

Monitoring complex industrial assets relies on engineer-authored symbolic rules that trigger based on sensor conditions and prompt technicians to perform corrective actions. The bottleneck is not detection but response: translating rules into maintenance steps requires asset-specific knowledge gained through years of practice. We investigate whether LLMs can serve as decision support for this rule-to-action step and introduce \ours{}, a benchmark of 6{,}690 expert-validated multiple-choice questions from 118 rule-action pairs across 16 asset types. We contribute (i) a symbolic-to-MCQA pipeline normalizing rules to Disjunctive Normal Form with embedding-based distractor sampling, (ii) five variants probing distinct failure modes (Pro, Pert, Verbose, Aug, Rationale), and (iii) a benchmark of 29 LLMs and 4 embedding baselines. A human evaluation (9 practitioners, mean 45.0%) confirms \ours{} requires specialist knowledge beyond operational experience. Three findings stand out. The frontier has closed: the top three LLMs lie within one Macro point, with Bradley-Terry Elo placing claude-opus-4-6 30 points above the next model. Yet \ours{},Pro exposes brittleness, with every model losing 13--60% relative accuracy under distractor expansion. \ours{},Aug exposes pattern-matching: under condition inversion, frontier models still select the original answer 49--63% of the time. The deployment bottleneck is not capability but calibration: frontier models handle template-style fault detection but break under structural perturbation.

The theory of constructed emotion defines social reality as the community-level consensus on emotion concepts assigned to interoceptive sensations arising from bodily allostasis and social interaction. In this study, we simulate this emergence process using a computational model that integrates symbol emergence with degrees of freedom in symbol interpretation and active inference. Two agents receive interoceptive signals, exchange inferred symbols, and simultaneously adapt their bodily control goals and symbol interpretations to each other. Experimental results show that the interoceptive prior preferences and symbol probability distributions of the two agents converge, confirming the emergence of social reality grounded in social consensus.

Template tasks have emerged as a clean testbed for asking whether transformers reason with abstract symbols rather than concrete token names. We study the fixed-label classification version of this problem, where train and test examples share latent templates but may use disjoint vocabularies. Unlike next-token prediction, the model need not emit unseen symbols; it must learn a decision rule invariant to symbol renaming. We analyze regularized kernel logistic classification in the transformer-kernel regime. Our main result decomposes the learned predictor into an ideal template-level classifier and a finite-sample perturbation caused by accidental token overlaps in the training data. We encode these overlaps by a colored collision graph and prove high-probability margin-transfer guarantees for fresh-symbol classification. This perspective extends template-based analyses to logistic classification and refines scalar diversity conditions: vocabulary size controls the average rate of collisions, but collision geometry controls whether the ideal classification margin is preserved. More broadly, the same perturbation framework applies to abstraction-augmented inputs, yielding a general margin-versus-collision criterion for identifying when prompting strategies improve fresh-symbol generalization. Synthetic template experiments illustrate the predicted roles of regularization, sample size, and transformer-kernel structure.

We describe a dynamical system in which a symbolic field is coupled to a geometric field via a bipartite Hilbert-Schmidt kernel. The system is fully described by a retarded functional differential equation (RFDE) on the history space, subject to Lipschitz and small gain conditions. We show that the RFDE is well-posed under constant input and that it admits a compact global attractor. The principal subsystem $(H_L, X_R, P)$, which is comprised of the two primary fields as well as an executive field, is shown to be globally stable independent of delay, provided that the interfield coupling satisfies $C_{\mathcal{K}}^2<μ_Lμ_R$. In addition, we describe design specifications that fulfill the hypotheses of the main Theorem.

Symbolic regression discovers explicit, interpretable equations without assuming a functional form in advance. A Bayesian approach strengthens this through probability distributions over candidate expressions, thus quantifying uncertainty in the presence of noisy and limited data. Deep Symbolic Regression (DSR) uses a neural network to generate symbolic expressions, but it is designed to identify a single best-fitting expression rather than infer a posterior distribution over models. We introduce Deep Variational Inference Symbolic Regression (DVISR), a variational Bayesian extension of DSR. DVISR replaces the original reward with the integrand of the evidence lower bound. It also extends the network architecture to output distributions over constants within expressions, enabling posterior inference over both expression trees and their associated constants. We show that DVISR can recover the true posterior in simple settings, both with and without constant tokens, and we examine how its performance changes as the size of the expression space increases. These results position DVISR as a step toward scalable Bayesian symbolic regression with uncertainty over full symbolic models.

We present a framework for the simultaneous symbolic recovery of a function and its antiderivative from data. The framework rests on three ideas. First, a derivative algebra: the observation that the product rule $\frac{d}{dx}[f \cdot g] = f'g + fg'$ and the chain rule, applied to a seed set of elementary functions, generate a self-expanding system of function-derivative pairs -- a living library that grows each time a new function is discovered. Second, two complementary primitives -- EML$\,(e^u - \ln v)$, which is theoretically complete for all elementary functions, and SOL$\,(\sin u - \cos v)$, introduced here, which makes trigonometric atoms available at depth1 instead of depth$\sim$8 -- that seed the library with core atoms cheaply. Third, additive atomic forests: finite sums of primitive trees, optionally composed via multiplicative nodes, whose derivatives are fitted to data by continuous optimisation or by exhaustive search over the library. Because differentiation of each atom is determined by construction, the forest simultaneously encodes a symbolic expression $F$ and its derivative $F'$; no symbolic integration step is required. The library is not a fixed object: it self-constructs from a small seed set by recursive application of the product rule, chain rule, and the two primitives, and it can grow as newly discovered functions are folded back in. The larger the library, the richer the expressible class of candidate functions. We give conditional completeness, additive-depth, and analytic simultaneous-recovery results for the framework. Empirically, in our reported runs on 17 classification benchmarks, sparse atom combinations match or exceed XGBoost on 13 datasets while producing interpretable formulas.

Designing mechanical linkages involves combinatorial topology selection and continuous parameter fitting. We show that language models can systematically improve linkage designs through symbolic representations. Language model agents explore discrete topologies while numerical optimisers fit continuous parameters. A symbolic lifting operator translates simulator trajectories into qualitative descriptors, motion labels, temporal predicates, and structural diagnostics that models interpret across iterative design cycles. Across six engineering-relevant motion targets and three open-source models (Llama 3.3 70B, Qwen3 4B, Qwen3 MoE 30B-A3B), the modular architecture reduces geometric error by up to 68% and improves structural validity by up to 134% over monolithic baselines. Critically, 78.6% of iterative refinement trajectories show measurable improvement, with the system correctly diagnosing overconstraint (56.3%) and underconstraint (35.6%) failure modes and proposing grounded corrections. Models across all three families acquire interpretable mechanical reasoning strategies without fine-tuning, demonstrating that principled symbolic abstraction bridges generative AI and the numerical precision required for engineering design.

Recent advancements in large language models have led to significant improvements across various tasks, including mathematical reasoning, which is used to assess models' intelligence in logical reasoning and problem-solving. Models are evaluated on mathematical reasoning benchmarks by verifying the correctness of the final answer against a ground truth answer. A common approach for this verification is based on symbolic mathematics comparison, which fails to generalize across diverse mathematical representations and solution formats. In this work, we offer a robust and flexible alternative to rule-based symbolic mathematics comparison. We propose an LLM-based evaluation framework for evaluating model-generated answers, enabling accurate evaluation across diverse mathematical representations and answer formats. We present failure cases of symbolic evaluation in two popular frameworks, Lighteval and SimpleRL, and compare them to our approach, demonstrating clear improvements over commonly used methods. Our framework enables more reliable evaluation and benchmarking, leading to more accurate performance monitoring, which is important for advancing mathematical problem-solving and intelligent systems.

Determining identifiability of causal effects from observational data under latent confounding is a central challenge in causal inference. For linear structural causal models, identifiability of causal effects is decidable through symbolic computation. However, standard approaches based on Gröbner bases become computationally infeasible beyond small settings due to their doubly exponential complexity. In this work, we study how to practically use symbolic computation for deciding rational identifiability. In particular, we present an efficient algorithm that provably finds the lowest degree identifying formulas. For a causal effect of interest, if there exists an identification formula of a prespecified maximal degree, our algorithm returns such a formula in quasi-polynomial time.

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 natural language is the de facto communication medium for LLM-based agents, it presents a fundamental constraint. The process of downsampling rich, internal latent states into discrete tokens inherently limits the depth and nuance of information that can be transmitted, thereby hindering collaborative problem-solving. Inspired by telepathy, which bypasses symbolic language in communication, we propose Interlat (Inter-agent Latent Space Communication), a paradigm that leverages the continuous last hidden states of an LLM as a representation of its thought for direct communication (termed latent communication). An additional learned compression process further compresses latent communication via latent space reasoning. Experiments demonstrate that Interlat outperforms both fine-tuned chain-of-thought (CoT) prompting and single-agent baselines, even across heterogeneous models, promoting more exploratory behavior and enabling genuine utilization of latent information. Further compression not only substantially accelerates inference by up to 24 times but also maintains competitive performance through an efficient information-preserving mechanism. We position this work as a feasibility study of entirely latent space inter-agent communication, and our results highlight its potential, offering valuable insights for future research. Our code is available at https://github.com/XiaoDu-flying/Interlat.

The expansion of time-series data from sensors and monitoring systems has made compact representations increasingly important. Such representations should retain signal structure while cutting storage, transmission and computation costs. Adaptive Brownian Bridge-based Aggregation (ABBA) addresses this need by converting long numerical series into short symbolic sequences, but reductions in parameter storage and computational precision remain desirable. We propose Quantized ABBA (QABBA), a quantized version of ABBA. By quantizing the symbolic centers, QABBA reduces the parameter footprint and enables integer arithmetic while maintaining high reconstruction quality. We establish several error bounds for the additional approximation introduced by quantization: a dimension-free bound on the excess error of each segment, a time-domain reconstruction-error bound, a stability condition for symbolic assignment, and a rule for allocating bits between segment lengths and increments. The resulting symbolic strings can be passed directly to a pretrained large language model (LLM) without any extra time-series embedding layer. Experiments on the Monash regression archive, UCR Time Series Classification Archive, and UEA Multivariate Time Series Classification Archive demonstrate a practical trade-off among storage, reconstruction accuracy and downstream predictive performance. QABBA therefore provides an error-controlled, low-precision symbolic representation for time-series compression and LLM-based analysis.