Neuro-Symbolic Framework

Current 3D spatial reasoning methods face a fundamental trade-off: neuro-symbolic 3D (NS3D) concept learners achieve interpretable reasoning through compositional programs but are constrained to closed-set concept vocabularies and simple programs; end-to-end 3D multi-modal LLMs (3D MLLMs) could handle complex natural language and open-vocabulary concepts but suffer from black-box reasoning without explicit spatial verification. We introduce APEIRIA, a neuro-symbolic 3D MLLM to bridge two paradigms by distilling symbolic reasoning patterns into MLLMs with natural language chain-of-thought. Our three-stage curriculum progressively builds reasoning capabilities: a) 3D perception alignment grounds object visual-geometric features to the LLM, b) CoT-SFT teaches query decomposition and stepwise verification from symbolic program traces, and c) CoT-RL extends reasoning patterns to open-set concepts and deeply nested instructions. By transferring reasoning patterns rather than concept-specific knowledge, APEIRIA preserves key NS3D virtues: transparent reasoning and modular interchangeability of planning and perception components. Evaluations on grounding, question answering, and captioning show that APEIRIA surpasses prior NS3D methods and matches state-of-the-art 3D MLLMs on 3D spatial reasoning datasets, unifying symbolic methods' systematic reasoning with MLLMs' flexibility. Code is available at https://github.com/oceanflowlab/APEIRIA.

We present AXIOM, a trust-first neuro-symbolic execution architecture for natural-language mathematical reasoning. In AXIOM, the language model functions strictly as a canonicalizer: it rewrites informal problem text into a narrow schema consumed by a deterministic Computer-Algebra-System (CAS) pipeline, which derives and verifies the answer or abstains as a first-class output. Routing follows a 1:1:1 alignment between problem-shape regex, schema-specific prompt, and closed-form CAS handler, with 3,100+ such routes shipped and zero LOST_CORRECT regressions across 250+ consecutive ship commits. We report empirical results on 4 MATH categories with a cumulative correctness of 94.36% (2,592/2,747) at 100.00% trust on parseable (zero confident-wrong answers across the full 2,747-record benchmark), all four domains above the per-domain 70/90/70 floor with per-domain trust at 100.0%, and median latency of 1 ms on rule-only handlers (88% of records on the lm-eval arithmetic 20,000-record benchmark). The architecture has served ~30,000 production queries through a public deployment. The contribution we emphasize is not a final accuracy figure but the forward dynamic the architecture establishes: every logged abstain in production is a candidate correct after one ship cycle, since new tasks compose without regressing the registry. The operational discipline behind this property -- math-template bucketing, LOST_CORRECT scan as regression oracle, parseable-first onboarding, and abstain as first-class output -- constitutes a transferable framework for trustworthy neuro-symbolic systems beyond mathematics.

Generative models have emerged as a powerful paradigm for AI planning, yet their performance remains constrained by the training data distribution. One approach is to improve generated solutions during inference by scaling test-time compute. A more efficient alternative is to optimize the inference process itself. In this paper, we show that a modified version of a classical Open-Closed List (OCL) search provides just such an efficient inference procedure. Our algorithm synergizes two learned components: a generative model that performs fast rollouts from intermediate states and a heuristic model that prioritizes among candidate reasoning paths. Key contributions include novel exploration control mechanisms and integration of learned models within the OCL framework. Across multiple combinatorial planning domains, our approach outperforms both neurosymbolic search baselines and classical solvers in computational efficiency and solution quality.

In this paper, we show the possibility of a direct injection of algorithms into neural network architecture. We focus on a complex algorithm, that is, Cocke-Youger-Kasami (CYK) for parsing context-free grammars in Chomsky Normal Form and we propose CYKNN, a simple recurrent neural network architecture for encoding the CYK algorithm in trainable matrix-vector multiplications.We experimented with a very simple grammar with 4 variations showing that our approach outperforms existing LLMs with more than 20B parameters with an in-context learning setting and smaller LLMs of the Qwen family fine-tuned with LoRA. Our attempt paves the way to a different approach to neuro-symbolic methodologies.

Generating physics diagrams from text requires strict adherence to physical laws. While current generative models produce visually plausible outputs, they systematically hallucinate force vectors, ignore conservation laws, and violate geometric constraints. We present PhyDrawGen, a neuro-symbolic pipeline that decouples semantic scene understanding from physical constraint satisfaction. First, a large language model extracts a typed scene graph from the problem text. A deterministic solver then converts this graph into a Planar Straight-Line Graph (PSLG), encoding force balance, optical paths, and field topologies as exact geometric primitives. Finally, a fine-tuned Qwen-VL model implements a visually grounded propose-verify loop to iteratively correct any constraint violations. Evaluated on a benchmark of 1,449 problems spanning mechanics, optics, and electromagnetism, PhyDrawGen significantly outperforms GPT-5-image, Gemini 2.5 Flash, and Gemini 3 Pro, demonstrating robust physical accuracy even on unusual-object problems.

Many learning tasks in science and engineering are characterized by sparse datasets, which limits the effectiveness of purely data-driven approaches. At the same time, these problems are often accompanied by rich domain knowledge derived from physical laws, operational requirements, and expert heuristics. Such knowledge is frequently expressed as rules involving logical propositions and linear inequalities. Existing neuro-symbolic methods typically enforce these rules approximately through soft penalties, assume input-independent rules when designing specialized architectures, or rely on non-differentiable post-processing at inference time to achieve hard constraint satisfaction. While recent advances in differentiable optimization layers enable end-to-end feasibility enforcement within neural networks, extending these approaches to logical or mixed-integer rules remains challenging due to inherent nonconvexity. In this work, we propose a unified end-to-end framework for enforcing hard, input-dependent mixed integer linear constraints within neural networks. Our approach represents rules as disjunctive constraints and applies hierarchical convex relaxations to obtain convex hull formulations. These relaxations yield tractable linear constraints that can be embedded as differentiable optimization layers while enabling exact rule satisfaction. We demonstrate the effectiveness of the proposed framework on real-world datasets, achieving perfect rule satisfaction and strong predictive performance.

As large language models transition from bounded generative engines to agents with expansive execution privileges, AI going out of control precipitates a fundamental crisis in artificial intelligence security. Existing defense architectures heavily rely on empirical semantic guardrails and probabilistic large model adjudicators, mechanisms that fail to provide deterministic security lower bounds when facing complex semantic symbol decoupling attacks. To overcome this empirical semantic guardrail dilemma, this paper proposes a new security paradigm for agents based on the fundamental limitations of logical reasoning. Based on this paradigm, we further introduce an executable Proof-Constrained Action (ePCA) framework with a neural symbolic isolation architecture. This framework abandons semantic trust in natural language, forcing agents to losslessly formalize their intentions into first-order logical mathematical constraints before performing physical operations. Empirical evaluations of macroscopic and microscopic two-dimensional dynamic adversarial systems demonstrate that our formal verification mechanism achieves zero attack success rate and zero false positive rate across the evaluated scenarios, with extremely low computational latency. This research provides a conditional formal foundation under explicit system assumptions and an engineering paradigm for constructing the underlying defense foundation for future intelligent systems.

Recent advances in large language models have accelerated open-vocabulary EEG-to-imagined-text decoding, where non-invasive neural activity recorded during visual perception is translated into coherent natural language descriptions of viewed stimuli. However, existing systems remain highly vulnerable to biological noise, where corrupted neural projections induce hallucinated or semantically unstable generation in frozen language models. We introduce SYNAPSE (Symbolic Neural Alignment for Precise Semantic Extraction), a lightweight neuro-symbolic framework that stabilizes neural text generation through inference-time symbolic regularization. By purifying EEG-derived semantic candidates using commonsense graph structure and latent exemplars, SYNAPSE improves semantic stability without end-to-end LLM fine-tuning. Experiments across popular EEG decoding benchmarks and multiple frozen LLM backends demonstrate consistent gains over unconstrained prompting baselines, robustness under object-label ablation, and performance commensurate with substantially more resource-intensive fine-tuned systems, while preserving biometric privacy by localizing raw EEG processing entirely within the encoder stack.

Symbolic regression (SR) seeks closed-form mathematical expressions that fit observed data. Neural SR methods amortize the search by training an encoder to map observations directly to expressions in a single pass, but this amortized inference leaves a residual amortization gap between its one-shot prediction and the true posterior. We propose Latent Equation Embedding (LEE), a framework that closes this gap through iterative amortized inference in a functionally grounded latent space. LEE learns a shared latent space Z equipped with three components: an encoder f_theta that jointly embeds symbolic tokens and numerical observations into a single latent vector z; an expression decoder g_expr that reconstructs formulas from z; and an evaluation decoder g_eval that predicts function values from z, explicitly grounding the latent space in functional behavior. At inference, LEE performs iterative refinement by re-encoding decoded expressions jointly with observations, progressively improving the latent estimate. LEE uses the encoder itself as a learned inference optimizer: each re-encoding step implicitly computes the mismatch between the candidate and the data. Because g_eval is differentiable in z, we additionally interleave continuous gradient descent with discrete re-encoding, yielding a hybrid iterative and gradient refinement procedure. On SRBench across three noise levels, against 19 baselines spanning genetic programming, symbolic-neural hybrids, and pre-trained Transformers, LEE produces expressions 2--10x simpler than the strongest accuracy-oriented baselines, including Operon, GP-GOMEA, TPSR, RAG-SR, and GenSR, with complexity 8--11 versus 20--90. These results advance the low-complexity region of the accuracy-complexity Pareto frontier and show graceful degradation as noise increases.

LLMs deployed in high-stakes domains face fundamental reliability challenges: hallucinations, inconsistencies, and privacy vulnerabilities introduce unacceptable risks where errors carry legal, financial, or safety consequences. This paper presents a hybrid verification architecture combining formal symbolic methods with neural semantic analysis to provide complementary guarantees for LLM-generated content. This architecture employs logical reasoning for input verification, leveraging completeness properties to provide decidable guarantees on structured requirements. For output validation, embedding-based semantic similarity detects contextual hallucinations where formal methods lack expressiveness. This separation is realized in a parallel, actor-based pipeline, addressing limitations of prompt-based self-verification approaches, which inherit the distributional biases that produce hallucinations. The proposed architecture and type-aware verification method are validated with HAIMEDA, a real-world medical device damage assessment reporting system developed through Action Design Research. Evaluation shows hallucination detection rates of over 83% for structured entities and 72% for semantic fabrications, with a 30% reduction in report creation time, demonstrating that neuro-symbolic architectures can provide principled safeguards for LLM deployment in data-sensitive domains.

We present RAGEAR (Retrieval-Augmented Graph-Enhanced Academic Recommender), a neurosymbolic recommender system for academic course recommendation. RAGEAR combines dense retrieval over full lecture transcripts with a symbolic Knowledge Graph modelling courses, lessons, transcript chunks, credits, study plans, and curricular information. The Knowledge Graph supports symbolic filtering and contextualisation based on structured constraints, such as credits, academic disciplines, study plans, and prerequisites. Unlike metadata-based approaches, it exploits fine-grained instructional content by retrieving transcript chunks semantically aligned with a student's query. The main contribution is a graph-aware aggregation function that propagates chunk-level evidence to course-level recommendations. The score combines three factors: the share of retrieved similarity associated with a course, the rank-based strength of its relevant chunks, and the distribution of evidence across lessons. We evaluate RAGEAR on 152 student-like queries through a human evaluation sample and a large-scale LLM-based relevance assessment. Results show that lecture transcripts improve over metadata-only retrieval, and that RAGEAR further improves ranking quality over a transcript-based normalized SumP baseline, especially for top-ranked recommendations.

Objective. Outpatient notes carry follow-up instructions pairing actions with future times ("MRI brain in two weeks"). Extracting (action, date) pairs supports scheduling and audit, but generative extractors miss the date because linking and arithmetic are implicit in decoding. We test a hybrid neural-symbolic pipeline against direct generation. Methods. We define TestSpecification and TimeSpecification entities and a ScheduledFor relation. BioBERT feeds BIO tagging and a biaffine linker; entities are canonicalized via a 28-action ontology and times normalized to day offsets deterministically. We evaluate on a 2,000-note synthetic outpatient corpus with action-disjoint splits (18 train, 6 OOV-test) against zero-shot GPT-4o-mini and LoRA-fine-tuned LLaMA-3 8B with note-level bootstrap 95% CIs. Results. On 259-note seen and OOV splits the hybrid pipeline achieves Test-Time Pair F1 of 0.997 and 0.986 with 0.00-day MAE. Baselines reach high action F1 (LLaMA-3 0.992; GPT-4o-mini 0.963 seen) but Pair F1 stays at 0.51-0.57 (LLaMA-3) and 0.53 (GPT-4o-mini), CIs non-overlapping with the hybrid. Conclusion. Separating learned entity extraction from deterministic date arithmetic outperforms generation on this benchmark, generalizes to held-out actions, and exposes failure modes. Transfer to real EHR notes is the next validation; a first-pass realism check is in Limitations.

Complex Query Answering (CQA) is a fundamental knowledge representation and reasoning task over incomplete knowledge graphs (KGs). Answering existential first-order queries with $k$ free variables (i.e., $\text{EFO}_k$ queries) is a crucial yet challenging problem, as it requires ranking answer tuples in $\mathcal{E}^k$, where $\mathcal{E}$ denotes the entity set of a KG. This quickly becomes intractable as $k$ grows. Consequently, existing benchmarks and methods rely on marginal rankings over individual variables; however, marginal rankings are a poor proxy for the true joint ranking of tuples. Building on neural symbolic search for $\text{EFO}_1$ queries, we propose Neural Scalable Symbolic Search (NS3), a budgeted framework that approximates joint ranking without enumerating $\mathcal{E}^k$. NS3 (i) answers marginalized sub-queries to obtain necessary candidate sets, (ii) merges multiple free variables into hypernodes whose domains are pruned and controlled by a dynamic budget $B$, and (iii) progressively reduces an $\text{EFO}_k$ query to an $\text{EFO}_{k-1}$ query over a budgeted reduced domain. Across three standard KG datasets, NS3 substantially improves joint ranking performance while retaining strong marginal accuracy. We further release a joint-ranking benchmark that extends existing $\text{EFO}_1$ datasets to $k=3$, enabling systematic evaluation of multi-variable queries. Our code is provided in https://github.com/HKUST-KnowComp/NS3_KDD2026.

Clinical decision-making requires reasoning over incomplete, imprecise, and linguistically expressed patient narratives. While large language models (LLMs) excel at extracting latent information from natural language, they lack the verifiability and interpretability essential for trustworthy medical AI. We propose a neuro-symbolic reasoning framework that aligns LLMs with formal logic to enable explainable and formally verifiable medical diagnosis. Patient descriptions and clinical guidelines are embedded into a neural knowledge base, where LLMs extract structured medical entities, temporal relations, and fuzzy symptom patterns, which are decoded into a symbolic knowledge base expressed in fuzzy logic and declarative rules. We perform two-stage reasoning: (1) inductive symbolic generalization to capture diagnostic patterns from encoded narratives, and (2) inference verification via a logic programming engine to derive and validate diagnoses consistent with clinical standards. Each symptom is treated as a fuzzy predicate with probabilistic weights, and inference paths are auditable, adjustable, and compatible with physician feedback. Unlike purely statistical methods, our system supports iterative refinement: misalignment between LLM-generated diagnoses and ground truth can be traced, explained, and corrected through formal rules. By combining logic-based transparency, LLM adaptability, and probabilistic robustness, the framework enables human-aligned healthcare inference with strong generalization and verifiable, step-by-step reasoning chains. We validate our framework on public benchmarks, demonstrating effective reconciliation of symbolic reasoning and LLMs with real-world clinical narratives. Results show performance comparable to state-of-the-art LLMs, while additionally providing interpretable reasoning paths and formally verifiable diagnostic conclusions.

Large Language Models (LLMs) have changed the possibilities of Interactive Storytelling systems that process free-text user input. However, as more of these systems are built, evidence continues to mount regarding the story coherence problems that arise when relying solely on them. Recent research suggests that LLMs can effectively predict state changes within rule-based Interactive Storytelling systems, triggering pre-programmed world-state transformations. In this paper, we conduct an exploratory evaluation of whether such transformations can serve as a catalyst for player expression while aiming to address the incoherence issues typical of purely LLM-based approaches. Building upon a neuro-symbolic architecture, we conducted experiments using an open-source model (Llama 3 70B) and a closed-source model (Gemini 1.5 Flash), with testing conducted in both English and Spanish. Eight participants played two scenarios, carefully designed to assess different evaluation objectives. Our observations suggest that transformations offer a way to maintain world-state consistency while encouraging players to interact creatively through their written inputs.

Scientific machine learning often requires combining known physics with unknown parameters or correction terms learned from data. Existing approaches either ignore known structure, encode it as a soft penalty, or require hand-written PyTorch code for each equation. We present The Neural Compiler, a system that translates programs written in a first-order Scheme-like expression language into frozen, differentiable PyTorch modules. These modules match the source program to floating-point precision and provide gradients through autograd. In hybrid models, the compiled module encodes known physics exactly while learned components model the unknown remainder. We evaluate the compiler across six experiment domains: Feynman physics equations, Lotka-Volterra dynamics, a damped pendulum, a one-dimensional heat equation, three-dimensional vector mechanics, and compositional generalization. Compiled modules match hand-coded PyTorch implementations numerically for single equations, showing no accuracy loss from compilation. With only 1 to 4 trainable parameters, compiled models recover physical constants to less than 1 percent error in most cases, while standard PINN baselines with more than 8500 parameters show 7 to 93 percent error. Compiled modules also compose with zero error, while neural approximations can accumulate large errors in deep composition chains. The main value of the compiler is not improved accuracy over hand-coded equations, but systematic composability: it generates correct, differentiable modules from symbolic specifications without rewriting each equation by hand. The system supports 51 primitive operations, including vector and matrix algebra, enabling PDE discretizations and hybrid scientific models. This string-in, module-out interface also provides a natural target for large language models that translate scientific descriptions into executable differentiable modules.

Reinforcement learning under partial observability requires deciding what information to retain, yet most memory-based approaches do not explicitly model short-term-to-long-term transfer of symbolic observations. We study this transfer process in a temporal knowledge-graph memory setting and cast it as a neuro-symbolic value-based decision problem: for each observed triple, the agent chooses whether to keep or drop it before long-term insertion. To handle variable-sized short-term buffers, we use a per-item Q-learning design with shared parameters and a practical temporal-difference update over matched items across consecutive steps. On the RoomKG benchmark at long-term memory capacity 128, learned transfer decisions outperform symbolic and neural baselines, including symbolic baselines with temporal annotations and history-based LSTM/Transformer baselines. Across transfer-policy ablations, a lightweight local short-term-only variant performs best, and step-level behavior shows that the policy keeps navigation- and query-relevant facts while discarding lower-value candidate facts, supporting explicit and interpretable memory decisions under memory constraints.

Knowledge graphs have become the primary vehicle for data integration and are critical to the success of modern AI, but the diversity of KG modelling practices, from lightweight vocabularies to richly axiomatised ontologies, makes integration and reuse expensive and brittle. This challenge is particularly acute in neuro-symbolic AI, where bridging neural and symbolic components depends on the ability to reengineer KGs to fit new requirements; GenAI now offers unprecedented automation capability, but without a principled understanding of the KG space, such automation remains conceptually ungrounded. We introduce the ontological continuum as that missing conceptualisation, a theoretical construct a theoretical construct whose characterisation framework is defined by two orthogonal distinctions: semantics vs pragmatics, and properties vs affordances; together these define a vocabulary to describe, compare, navigate, and transform KGs across the full range of modelling practices. The methodological stance is empirical: rather than prescribing how KGs should be modelled, the continuum aims to define a theory of the existent, derived from observation of real-world KG engineering practices and whose structure can be made formally explicit, for example, through Formal Concept Analysis (FCA). We ground the vision through a case study on provenance knowledge, showing how a single concern manifests differently across the continuum. We articulate five open research challenges and invite the community to develop the ontological continuum as a shared research agenda.

Formal mathematics libraries are rapidly expanding, creating a growing need to refactor verified proofs for maintainability and to improve training data quality for neural provers. However, scalable proof optimization is hindered by heterogeneous and heuristically specified objectives, scarce data, and high training and inference costs. To overcome these challenges, we introduce ImProver 2, a neurosymbolic framework for automated proof optimization in Lean 4. ImProver 2 combines a data-efficient expert-iteration pipeline with a scaffold that exposes formal structure alongside lightweight informal abstractions. We further introduce a suite of metrics capturing structural proof properties. Using ImProver 2, we train a 7B-parameter model that outperforms orders-of-magnitude larger models within the same model family, and is competitive with mid-tier frontier models across metrics. We additionally demonstrate that our neurosymbolic scaffold significantly improves performance across both small and frontier models. We show that with proper scaffolding and training, small models can effectively restructure research-level proofs over complex and varied metrics, matching substantially larger systems and establishing proof optimization as a scalable, learnable task.

The exponential growth of global academic output has confronted researchers and AI agents with an unprecedented information explosion,'' where fragmented and unstructured knowledge organization impedes deep interdisciplinary integration. Current academic retrieval tools predominantly rely on superficial keyword matching or vector-space semantic retrieval, which lack the topological reasoning capabilities required to navigate complex logical connections. Agentic deep-research-based frameworks are often prone to logical hallucinations and consuming high inference costs. To bridge this gap, in this report, we introduce SciAtlas, a large-scale, multi-disciplinary, heterogeneous academic resource knowledge graph designed as a panoramic scientific evolution network. By integrating over 43M papers from 26 disciplines, and a total of 157M entities and 3B triplets, SciAtlas provides a structured topological cognitive substrate that dismantles disciplinary barriers and furnishes AI agents with a global perspective. Furthermore, we develop a neuro-symbolic retrieval algorithm featuring tri-path collaborative recall and graph reranking, achieving a seamless transition from simple semantic matching to deterministic association discovery. We also present key application directions of SciAtlas, including literature review, automated research trend synthesis, idea positioning, and academic trajectory exploration, to demonstrate that SciAtlas can serve as an effective cognitive map'' to empower the full loop of automated scientific research while significantly reducing reasoning costs. We have released the interfaces for KG retrieval and various downstream tasks in our GitHub repo.

Mechanistic interpretability produces circuit-level causal analyses of neural network behaviour, but discovered circuits often remain isolated experimental artefacts: there is no shared formal representation for what circuits compute, how they relate, or when two findings provide evidence for the same mechanism. This work provides a formal infrastructure for cumulative mechanistic science by treating circuit interpretation as inductive theory construction. Each circuit is characterised at two levels: a Causal Functional Signature (CFS), which grounds component behaviour in causal attribution evidence and token role profiles, and an architectural signature $τ_{\mathrm{arch}}$, learned by inductive logic programming (ILP) from scale-invariant structural predicates. Together, these constitute a formal coherence layer that makes mechanistic claims explicit, comparable via $θ$-subsumption, and portable across model scales. CFS reveals qualitatively distinct computational strategies across task types, including attention-mediated copying versus MLP-mediated binding. ILP signatures achieve substantially better structural separation than graph kernel and feature-vector baselines, and support principled transfer across model scales and architecture families.

Sutra is a typed, purely functional programming language whose compiled forward pass is a PyTorch neural network. The compiler beta-reduces the whole program -- primitives, control flow, string I/O -- to one fused tensor-op graph over a frozen embedding substrate. Rotation binding, unbind, bundle, polynomial Kleene three-valued logic, and tail-recursive loops all lower to tensor operations; the Kleene connectives are Lagrange-interpolated polynomials exact on the {-1, 0, +1} truth grid. Validation is one fact tested two ways. (1) The same program runs on four frozen embeddings spanning two modalities -- three text encoders (nomic-embed-text, all-minilm, mxbai-embed-large) and one protein language model (ESM-2) -- and decodes bundles at 100% accuracy through width k=8 on every substrate, where the textbook Hadamard product has already collapsed (2.5% on mxbai-embed-large, 7.5% on all-minilm). (2) PyTorch autograd flows through the actually compiled graph: a fuzzy-rule classifier written in .su trains from random init (18.7 +/- 9.5%; chance = 20%, five classes) to 100.0 +/- 0.0% (three seeds) by backpropagating through the emitted graph, the symbolic source unmodified. A weighted variant additionally trains a scalar cosine gain and writes it back into the .su source as a numeric literal; recompiling reproduces the trained behaviour to ~2e-7 per logit, so the trained model is itself legible, recompilable code. The same artifact is therefore both a logic program and a trainable neural network.

Concept-based (CB) models provide interpretability and support test-time human intervention, while standard neural networks (NN) offer strong task performance but little transparency. Prior work has explored hybrid formulations that integrate concepts and additional representations to improve accuracy, often at the cost of human interventions. We introduce the \emph{Synergy Concept-Based Model (SynCB)} framework, that combines a CB branch with a complementary neural branch, and a trainable routing module that dynamically selects which branch to use for each input. Unlike prior models, which fuse residual and concept-based predictions, SynCB keeps the two branches distinct and coordinates them through the routing module. Moreover, both branches are learned jointly, allowing information sharing between the complementary neural branch and CB branches through their common backbone. To improve responsiveness to interventions, we further introduce a test-time intervention policy and a corresponding loss. Across five datasets and CB benchmarks, SynCB consistently achieves higher task accuracy while remaining more responsive to human interventions, surpassing the full neural baseline by up to 3.9 percentage points and exceeding the strongest competitor in intervention performance by up to 6.43 percentage points.

Effectively translating between natural language (NL) and formal logics like Linear Temporal Logic (LTL) requires expertise that limits formal verification's reach in safety-critical development. Template-based approaches sacrifice expressiveness for reliability; neural methods achieve fluency but provide no correctness guarantees. We present NeuroNL2LTL, a neurosymbolic architecture unifying learned translation with formal verification. NeuroNL2LTL routes translation through an intermediate representation whose mapping to LTL is structure-preserving by construction. Generated specifications undergo satisfiability and non-triviality checking; a minimal-edit repair mechanism corrects near-miss outputs before they reach downstream tools. The central innovation is verifier-in-the-loop training: verification outcomes serve as reward signals for reinforcement learning, producing neural components that optimize directly for formal correctness. On 200,000+ requirements spanning aerospace, robotics, autonomous vehicles, and ten additional domains, NeuroNL2LTL achieves 28% semantic equivalence with reference specifications while ensuring 86% of outputs are verified satisfiable. The system also generates contextually grounded explanations from LTL, enabling domain experts to validate specifications without specialized training. This work demonstrates that formal verification can function as both training objective and runtime filter for neural specification systems, allowing us to build neural-based tools whose reliability derives from logical guarantees rather than statistical confidence.

Claim verification is an important problem in high-stakes settings, including health and finance. When information underpinning claims is incomplete or conflicting, uncertain answers may be more appropriate than binary true or false classifications. In all cases, faithful explanations of the considerations determining the final verdict are crucial. We introduce inference-time argumentation (ITA), a trainable neurosymbolic framework for ternary claim verification in which a formal argumentation semantics giving the strength of claims is used both (i) to guide LLM training as models learn to generate arguments and assign them base scores (representing intrinsic strengths) and (ii) to compute ternary (true/false/uncertain) predictions from generated, scored arguments. As a result, at training time, argument generation and scoring can be optimised according to the quality of the induced argumentative predictions. Moreover, at inference time, the final prediction is faithful, by construction, to the arguments and scores determining the verdict, rather than being justified by a potentially unfaithful post-hoc reasoning trace as in conventional reasoning models. We finally show that, on two datasets for ternary claim verification, ITA improves upon argumentative baselines and can perform competitively against non-argumentative direct-prediction baselines, while providing verdicts that are computed deterministically from explicit, inspectable argumentative structures.

Task-oriented dialogue systems -- handling transactions, reservations, and service requests -- require predictable behavior, yet the moderately-sized LLMs needed for practical latency are prone to hallucination and format errors that cascade into incorrect actions (e.g., a hotel booked for the wrong date). We propose ReacTOD, a bounded neuro-symbolic architecture that reformulates NLU as discrete tool calls within a self-correcting ReAct loop governed by deterministic validation. A bounded ReAct loop enables iterative self-correction, improving accuracy by up to 9.3 percentage points over single-pass inference on MultiWOZ. A symbolic validator enforces action compliance, schema conformance, and coreference consistency on every dialogue state update, achieving a 93.1% self-correction rate on intercepted errors and producing structured execution traces. Incremental state prediction and on-demand history retrieval keep prompts compact, empirically improving instruction adherence in parameter-constrained models. On MultiWOZ 2.1, ReacTOD achieves a new zero-shot state-of-the-art: gpt-oss-20B reaches 52.71% joint goal accuracy, surpassing the previous best by 14 percentage points, while Qwen3-8B achieves 47.34% with only 8B parameters. On the Schema-Guided Dialogue (SGD) benchmark, ReacTOD with Claude-Opus-4.6 achieves 80.68% JGA under fully end-to-end evaluation with predicted domains, and Qwen3-32B reaches 64.09% -- demonstrating cross-benchmark generalization without task-specific training data.

The rapid advancement of autonomous driving (AD) technologies has outpaced the development of robust safety evaluation methods. Conventional testing relies on exposing AD systems to vast numbers of real-world traffic scenes -- a brute-force approach that is prohibitively expensive and statistically ineffective at capturing the rare, safety-critical edge cases essential for validating real-world robustness. To address this fundamental limitation, we introduce STRELGen, a scalable framework for the targeted generation of safety-critical driving scenarios. STRELGen synergistically combines a multi-agent trajectory-generation diffusion model (DM) with Spatio-Temporal Logic (STREL) specifications that encode complex safety and realism properties through a highly interpretable formalism. Crucially, monitoring satisfaction levels of these specifications is differentiable, enabling gradient-based search. At inference time, we optimize directly over the DM latent space to maximize STREL formula satisfaction. The result is efficient generation of highly plausible yet safety-critical multi-agent scenarios that lie within the learned data distribution. STRELGen thus provides a flexible, interpretable, and powerful tool for stress-testing autonomous driving systems, moving beyond the limitations of brute-force data collection.

Neuro-Symbolic Concept-based Models (NeSy-CBMs) are a family of architectures that integrate neural networks with symbolic reasoning for enhanced reliability in high-stakes applications. They work by first extracting high-level concepts from the input and then inferring a task label from these compatibly with given logical constraints. Yet, their label and concept predictions can be overconfident, making it difficult for stakeholders to gauge when the model's decisions can be trusted. We address this issue by integrating ideas from Conformal Prediction (CP), a framework providing rigorous, distribution-free coverage guarantees. We formalize three desiderata -- consistency, coverage, and conciseness -- that any conformal method for NeSy-CBMs should satisfy, and show that existing approaches fall short of at least one. We then introduce COCOCO, a post-hoc framework that conformalizes concepts and labels jointly and reconciles them via a single deduction-abduction revision step. COCOCO satisfies all three desiderata, retains distribution-free coverage, is robust to imperfect knowledge and supports user-specified size budgets. Our experiments on 8 data sets highlight how COCOCO compares favorably against competitors and natural baselines in terms of performance and set size.

Translating formal language into natural language is a foundational challenge in NLP, driving various downstream applications in semantic parsing, theorem validation, and question answering. In this study, we introduce First-Order Logic to Natural Sentence (FOL2NS), a neurosymbolic framework designed to generate synthetic FOL formulas and convert them into natural human expressions. It handles deeply nested structures with varying quantifier depths (QD), which are rarely captured by existing corpora. By combining rule-driven modules with fine-tuned language models, FOL2NS enhances the diversity and coverage of the generated samples. In our experiments, we systematically evaluate the framework's capabilities through both character-level analysis and overall performance metrics. Experimental results show that FOL2NS can reliably produce well-formed templates and fluent statements, but it faces challenges in achieving precise semantic representations and natural generation as structural complexity increases.

We present NeuSymMS, an adaptive memory system that enables large language model (LLM) agents to learn, remember, and reason about users across sessions via a hybrid neuro-symbolic architecture. NeuSymMS couples neural fact extraction from unstructured dialogue using LLMs and a CLIPS-based expert system that classifies, deduplicates, and reconciles facts under explicit lifecycle rules. The system represents knowledge as subject-relation-value triples stored in relational database management system. It supports user/agents/agent-to-agent scoping, and implements a dual-horizon (short-term and long-term) memory model. IT leverages access-based promotion and time-based pruning of the memory on both horizpons. NeuSymMS maintains continuity of memory while avoiding context-window bloat and cross-entity contamination. We argue that this architecture offers a practical path to trustworthy, auditable memory for production agentic systems and discuss its novelty relative to log retrieval, summarization, and key-value approaches.

Discrete diffusion models are a powerful, emerging paradigm for code generation. They construct programs through iterative refinement of partially corrupted token sequences and enable parallel token refinement. Importantly, this paradigm exposes a global program state at each denoising step, which provides a natural intervention point for enforcing program-level functionality and security constraints, guiding the generation before the final code is committed. Building on this observation, the paper introduces Constrained Diffusion for Code (CDC), a training-free neurosymbolic inference framework that integrates constraint satisfaction directly into the reverse denoising process. CDC augments the base discrete diffusion sampler with constraint-aware denoising operators that combine mathematical optimization with program analysis to identify constraint-relevant regions of the intermediate program state and locally adjust the denoising trajectory, steering generation toward feasible programs while remaining close to the base model. Across code generation benchmarks, CDC consistently improves constraint satisfaction in functional correctness, security, and even syntax, outperforming discrete diffusion and autoregressive baselines with less corrective computation and more localized edits.

Despite the effectiveness of Cube-and-Conquer (C&C) for solving challenging Boolean Satisfiability (SAT) problems, no prior work has shown that transformer-based models can learn effective cubing heuristics. We introduce a neuro-symbolic post-training framework for this task. We design an MCTS-based data curation pipeline that uses symbolic heuristics to explore splitting decisions over SAT competition formulas, producing preference data grounded in solver statistics and augmented with reasoning traces from a teacher model. Our two-stage post-training, supervised fine-tuning (SFT) followed by direct preference optimization (DPO), enables a 4B-parameter model to achieve a pass@5 score of 53 on 100 SAT competition benchmarks, surpassing frontier LLMs such as Claude-Sonnet-4 (50) and matching the best symbolic heuristic (53). Ablations show that SFT alone improves pass@5 from 46 to 51, with DPO adding 2 additional benchmarks; an entropy/agreement ablation on realized first-cube decisions further shows that SFT, not DPO, accounts for the root-level decision diversity that produces complementary per-run coverage over deterministic symbolic methods. This demonstrates that transformers can be trained to make effective cubing decisions in a domain traditionally dominated by symbolic methods.

Recent advances in large language models (LLMs) have significantly enhanced automated legal reasoning. Yet, it remains unclear whether their performance reflects genuine legal reasoning ability or artifacts of data contamination. We present a comprehensive empirical study of tax law reasoning approaches and implement a contamination detection protocol to rigorously assess LLM reliability. We show that performance can be inflated by contamination. Building on this analysis, we conduct a systematic evaluation, comparing monolithic LLMs with hybrid systems that translate statutory text into formal representations and delegate inference to symbolic solvers. We build a novel test suite designed to probe generalization to unseen documents via case and rule variations. Our findings indicate that legal reasoning is inherently compositional and that neuro-symbolic frameworks offer a more reliable and robust foundation for legal AI, as well as improved generalization to unobserved situations.

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.

Reactive synthesis, the problem of automatically constructing a hardware circuit from a logical specification, is a long-standing challenge in formal verification. It is elusive for two reasons: It is algorithmically hard, and writing formal specifications by hand is notoriously difficult. In this paper, we tackle both sides of the problem. For the algorithmic side, we present a neuro-symbolic approach to reactive synthesis that couples large reasoning models with model checkers to iteratively repair a synthesized Verilog implementation via sound symbolic feedback. Our approach solves more benchmarks than the best dedicated tools in the annual synthesis competition and extends to constructing parameterized systems, a problem known to be undecidable. On the specification side, we introduce an autoformalization step that shifts the specification task from temporal logic to natural language by introducing a hand-authored dataset of natural-language specifications for evaluation. We demonstrate performance comparable to that of starting from formal specifications, establishing natural synthesis as a viable end-to-end workflow.

The reliability of Healthcare Information Systems (HIS) is frequently compromised by human-induced data entry errors, which existing statistical anomaly detection methods fail to distinguish from legitimate clinical extremes. This paper proposes Logic-GNN, a novel neuro-symbolic framework that treats clinical records as a structured private language'' governed by latent logical games. By integrating Temporal Graph Neural Networks (TGNN) with Graph Kolmogorov Complexity, we induce a symbolic grammar that represents the underlying logic of medical interactions. We define anomalies as grammatical violations'' that cause a significant expansion in the Minimum Description Length (MDL) of the clinical graph. Evaluated on the Sina System dataset (2M+ records), Logic-GNN achieves an F1-score of 0.94, outperforming state-of-the-art baselines by 12% in distinguishing between life-threatening medical outliers and data corruption. Our approach introduces a self-healing mechanism that suggests logical corrections to maintain data integrity in real-time HIS environments.

The growing adoption of large language models in legal practice brings both significant promise and serious risk. Legal professionals stand to benefit from AI that can reason over contracts, draft documents, and analyze sources at scale, yet the high-stakes nature of legal work demands a level of rigor that current AI systems do not provide. The central problem is not simply that LLMs hallucinate facts and references; it is that they systematically draw inferences that go beyond what the source text actually supports, presenting assumption-laden conclusions as if they were logically grounded. This proposal presents a neuro-symbolic approach to legal AI that combines the expressive power of large language models with the rigor of formal verification, aiming to make AI-assisted legal reasoning both capable and trustworthy, thus reducing the burden of manual verification without sacrificing the accountability that legal practice demands.

Differentiable Logics are deployed in neuro-symbolic learning tasks as a way of embedding logical constraints in the training objective of neural networks. A differentiable logic consists of a syntax to write logical properties and a semantics to interpret them as real-valued functions to be folded in the loss function. A defining trade-off of the field is that between logical properties of the connectives, and analytic concerns for the semantics, with both aspects being relevant in applications. At one extreme we find fuzzy logics, that have well-established algebraic and proof-theoretic foundations, and at the other ad-hoc differentiable logics like Fischer's DL2, conceived for deep learning applications. However, no satisfactory foundation has emerged yet. We propose a resolution to this long-standing tension via a novel logic, Quantitative Linear Logic (QLL), with foundational ambitions. Our design is driven by naturality -- the idea that, since logical constraints are translated to losses, the semantics of the connectives should be pertinent operations used in ML practice (that is, sum and log-sum-exp) on additive quantities (like logits). We then judge the result on two aspects: logical adequacy -- that they satisfy most of the standard logical laws of Linear Logic; and empirical effectiveness -- test-time performance (as measured by adversarial attacks) is well-correlated to the actual verification of the logical constraints (as measured by off-the-shelf neural network verifiers), which makes QLL stand out among SoTA techniques.

Natural-language software requirements are often ambiguous, inconsistent, and underspecified; in safety-critical domains, these defects propagate into formal models that verify the wrong specification and into implementations that ship unsafe behavior. We show that large language models, equipped with an SMT solver, can audit such requirements: translating them into formal logic, detecting ambiguity through stochastic variation in the generated formalization, and exposing inconsistency, vacuousness, and safety violations through solver queries on the resulting specification. We present VERIMED, a neurosymbolic pipeline that operationalizes this idea for medical-device software requirements, and report two findings. First, stochastic variation across independent formalizations is a signal of ambiguity: requirements that admit multiple plausible interpretations produce SMT-inequivalent formalizations, and bidirectional SMT equivalence checking turns this disagreement into a solver-checkable test. Second, the usefulness of symbolic feedback depends on its granularity: in counterexample-guided repair on a hemodialysis question-answering benchmark, concrete SMT counterexamples raise verified accuracy from 55.4% to 98.5%. Over an extensive experimental evaluation on open-source hemodialysis safety requirements, we show that the LLM-based approach in VERIMED successfully reduces ambiguity-sensitive requirements and enables rigorous auditing of software requirements through SMT-based queries.

Weakly supervised semantic segmentation (WSSS) trains dense pixel-level segmentation models from partial or coarse annotations such as bounding boxes, scribbles, or image-level tags. While recent work leverages foundation models such as the Segment Anything Model (SAM) to generate pseudo-labels, these approaches typically depend on heuristic prompt choices and offer limited ways to incorporate prior knowledge or heterogeneous labels. We address this gap by taking a neurosymbolic perspective: integrating differentiable fuzzy logic with deep segmentation models. Weak annotations and domain-specific priors are unified as continuous logical constraints that fine-tune SAM under weak supervision. The refined foundation model then produces improved pseudo-labels, from which we train a second-stage prompt-free segmentation model. Experiments on Pascal VOC 2012 and the REFUGE2 optic disc/cup segmentation dataset show that our logic-guided fine-tuning yields higher-quality pseudo-labels, leading to state-of-the-art segmentation accuracy that often exceeds densely supervised baselines.

Geometric problem solving, as a typical multimodal reasoning problem, has attracted much attention and made great progress recently, however most of works focus on plane geometry while usually fail in solid geometry due to 3D spatial diagrams and complex reasoning. To bridge this gap, we introduce Hilbert-Geo, the first unified formal language framework for solid geometry, including an extensive predicate library and a dedicated theorem bank. Based on this framework, we propose a Parse2Reason method containing two steps of first parsing then reasoning. In the parsing step, we utilize conditional description language (CDL), a formalized language composed of predicates specifically designed to construct geometric conditions, to represent both problem description (natural text) and solid diagrams (visual image). In the reasoning step, we leverage those formal CDL and the theorem bank to perform relational inference and algebraic computation, generating strictly correct, verifiable, and human-readable reasoning processes. Notably, our proposed Hilbert-Geo is also applicable to plane geometry. To advance geometric reasoning, we curate two expert-annotated dataset SolidFGeo2k and PlaneFGeo3k, which are furnished with geometric formal language annotations, solutions and answers. Extensive experiments show that our proposed method achieves the state-of-the-art (SOTA) performance 77.3% in SolidFGeo2k and 84.1% in MathVerse-Solid (one small subset in MathVerse dedicated to solid geometry), substantially outperforming leading MLLMs, such as Gemini-2.5-pro (54.2% on SolidFGeo2k) and GPT-5 (62.9% on MathVerse-Solid). In addition, our method achieves the SOTA accuracy 80.2% in PlaneFGeo3k, demonstrating the generality of the Hilbert-Geo in geometric reasoning. Our code and datasets are released at https://github.com/PremiLab-Math/Hilbert-Geo.

Deep learning has become the dominant approach for creating high capacity, scalable models across diverse data modalities. However, because these models rely on a large number of learned parameters, tightly couple feature extraction with task objectives, and often lack explicit reasoning mechanisms, it is difficult for humans to understand how they arrive at their predictions. Understanding what representations emerge and why they arise from the training data remains an open challenge. We introduce Deep Arguing, a novel neurosymbolic approach that integrates deep learning with argumentation construction and reasoning for interpretable classification with different data modalities. In our approach deep neural networks construct an argumentation structure wherein data points support their assigned label and attack different ones. Using differentiable argumentation semantics for reasoning, the model is trained end-to-end to jointly learn feature representation and argumentative interactions. This results in argumentation structures providing faithful case-based explanations for predictions. Structure constraints over the argumentation graph guide learning, improving both interpretability and predictive performance. Experiments with tabular and imaging datasets show that Deep Arguing achieves performance competitive with standard baselines whilst offering interpretable argumentative reasoning.

Multimodal large language models (MLLMs) are increasingly integrated into autonomous driving (AD) systems; however, they remain vulnerable to diverse safety threats, particularly in accident-prone scenarios. Recent safeguard mechanisms have shown promise by incorporating logical constraints, yet most rely on static formulations that lack temporally grounded safety reasoning over evolving traffic interactions, resulting in limited robustness in dynamic driving environments. To address these limitations, we propose GuardAD, a model-agnostic safeguard that formulates AD safety as an evolving Markovian logical state. GuardAD introduces Neuro-Symbolic Logic Formalization, which represents safety predicates over heterogeneous traffic participants and continuously induces them via n-th order Markovian Logic Induction. This design enables the inference of emerging and latent hazards beyond single-step observations. Rather than simply vetoing unsafe actions, GuardAD performs Logic-Driven Action Revision, where inferred safety states actively guide action refinement without modifying the underlying MLLM. Extensive experiments on multiple benchmarks and AD-MLLMs demonstrate that GuardAD substantially reduces accident rates (-32.07%) while slightly improving task performance (+6.85%). Moreover, closed-loop simulation evaluations, together with physical-world vehicle studies, further validate the effectiveness and potential of GuardAD.

In this paper, we present SCALAR (Symbolic Conjecture and LLM-Assisted Reasoning), a neurosymbolic framework for automated conjecture generation in quantum circuit analysis built on top of the CUDA-Q open source framework. The system integrates quantum simulation, symbolic conjecture generation, and LLM-based interpretation. We evaluate SCALAR on 82 MaxCut instances from the MQLib benchmark dataset and extend the analysis to 2,000 randomly generated graphs across four topologies: regular, Erdos-Renyi, Barabasi-Albert, and Watts-Strogatz. The framework generates conjectured bounds relating optimal QAOA parameters to graph invariants, including known relationships such as periodicity constraints on the phase separation parameter $γ$. SCALAR also recovers previously reported parameter transfer phenomena across structurally similar instances. Additionally, the system identifies correlations between graph structural features and optimization landscape properties, which we characterize through invariant-based descriptors. Using CUDA-Q tensor network simulator, we scale experiments to instances of up to 77 qubits. We discuss the accuracy, generality, and limitations of the generated conjectures, including sensitivity to graph class and quantum circuit depth.

Deep neural networks achieve high accuracy on image classification tasks. Yet, they often produce overconfident predictions as which fail to express epistemic uncertainty, and frequently violate logical or structural constraints present in the data. These limitations are particularly pronounced in hierarchical classification, where predictions across fine and coarse levels must remain coherent. We propose, for the first time, a unified neurosymbolic and epistemic modelling framework that augments Swin Transformers with focal set reasoning and differentiable fuzzy logic. Rather than treating labels as isolated categories, our method induces data-driven focal sets within the learnt embedding space, which helps capture epistemic uncertainty over multiple plausible fine-grained classes. These focal sets form the basis of a belief-theoretic layer that uses fuzzy membership functions and t-norm conjunctions to encourage consistency between fine- and coarse-grained predictions. A learnable loss further balances calibration, mass regularisation, and logical consistency, allowing the model to adaptively trade off symbolic structure with data-driven evidence. In experiments on hierarchical image classification, our framework maintains accuracy on par with transformer baselines while providing more calibrated and interpretable predictions, reducing overconfidence and enforcing high logical consistency across hierarchical outputs. Our experimental results show that combining focal set reasoning with fuzzy logic provides a practical step toward deep learning models that are both accurate and epistemically aware.

DeepLog is an operational neurosymbolic framework that unifies logic and deep learning within standard PyTorch workflows. While existing neurosymbolic systems focus on a particular paradigm and semantics, DeepLog serves as a universal backend that can emulate many systems in the neurosymbolic alphabet soup. By treating diverse neurosymbolic languages as high-level specifications, the DeepLog software automatically compiles them into optimized arithmetic circuits. This design lowers the barrier for machine learning practitioners by treating logic as composable modules, while providing neurosymbolic developers with a shared, high-performance basis for prototyping new integration strategies. The code is available here: https://github.com/ML-KULeuven/deeplog

Extracting multi-step explanations from knowledge graphs poses a combinatorial challenge requiring both heuristic guidance (as candidates proliferate with depth) and credit assignment (as path quality emerges over extended sequences). Frontier LLMs, strong on knowledge/reasoning benchmarks, offer a compelling source of such heuristics, yet their knowledge comes sans guarantees and compositional performance degrades as chains lengthen. We thus present TESSERA, a 3-part neuro-symbolic framework that uses LLMs in a circumscribed role: for local discriminative judgement rather than autonomous multi-step generation; the knowledge graph then defines the hypothesis space enforcing hard structural constraints, and MCTS coordinates the long-horizon search with principled credit assignment via backpropagation. LLMs perform dual roles as a prior policy biasing exploration and a comparative state evaluator supplying reward signals. Evaluation on drug mechanism elucidation across two complementary knowledge graphs demonstrates fidelity to curated biology while surfacing coherent alternative mechanisms, with ablations confirming discriminative contribution from both LLM components. Beyond its current application, our framework offers a general paradigm for compositional reasoning over structured knowledge.

Large Multimodal Models (LMMs) often struggle with geometric reasoning due to visual hallucinations and a lack of mathematically precise Chain-of-Thought (CoT) data. To address this, we propose the GeoSym Engine, an automated and scalable neuro-symbolic framework. By leveraging a type-conditional grammar and an analytic SymGT Solver, it derives exact symbolic ground truths and seamlessly integrates with a robust rendering pipeline to produce high-precision geometric diagrams. Using this engine, we construct GeoSym127K, a difficulty-stratified dataset featuring 51K high-resolution images, 127K questions with symbolic ground truths, and 55K answer-verified CoT QA pairs. We also introduce GeoSym-Bench, an expert-curated suite of 511 complex samples for rigorous evaluation. Through extensive supervised fine-tuning (SFT), we demonstrate that GeoSym drives concentrated improvements specifically on diagram-dependent and multi-step geometry tasks. Our Qwen3-VL-8B model gains an absolute +22.21% on the MathVerse Vision-Only subset and reaches 61.52% (+6.19% improvement) on WeMath, mitigating long-horizon logic fragmentation and outperforming advanced closed-source models like Doubao-1.8. Furthermore, applying Reinforcement Learning with Verifiable Rewards (RLVR) via GRPO reveals that initializing from structural SFT checkpoints substantially elevates the performance ceiling over zero-shot RL. Driven by deterministic exact-match signals, this showcases the robust scaling potential of our verifiable reasoning synthesis. Datasets and code are available at https://huggingface.co/datasets/Tomie0506/GeoSym127K and https://github.com/Tomie56/GeoSym127K.

Coherent, continuous spatial representations are critical for synthesizing physical and perceptual phenomena into a single representational space. Radial basis kernels provide a path forward for this type of distributed representation. In this work, we aim to characterize and analyze common radial basis kernels realizable in the neurally-plausible framework of spatial semantic pointers. Further, we analyze previous radial basis kernel work based on grid cell-like representations and demonstrate that such representations are both capable of and optimal for realizing radial basis kernels.

Recent efforts to improve the reasoning abilities of Large Language Models (LLMs) have focused on integrating formal logic solvers within neurosymbolic frameworks. A key challenge is that formal solvers lack commonsense world knowledge, preventing them from making reasoning steps that humans find obvious. Prior methods address this by using LLMs to supply missing commonsense assumptions, but these approaches implicitly assume universal agreement on such commonsense facts. In reality, commonsense beliefs vary across individuals. We propose a probabilistic framework for abductive commonsense reasoning that explicitly models this variation, aiming to determine whether most people would judge a statement as true or false. We introduce Probabilistic Abductive CommonSense (PACS), a novel algorithm that uses an LLM and a formal solver to sample proofs as observations of individuals' distinct commonsense beliefs, and aggregates conclusions across these samples. Empirically, PACS outperforms chain-of-thought reasoning, prior neurosymbolic methods, and search-based approaches across multiple benchmarks.

Robot swarms promise scalable assistance in complex and hazardous environments. Task planning lies at the core of human-swarm collaboration, translating the operator's intent into coordinated swarm actions and helping determine when validation or intervention is required during execution. In long-horizon missions under dynamic scenarios, however, reliable task planning becomes difficult to maintain: emerging events and changing conditions demand continual adaptation, and sustained operator oversight imposes substantial cognitive burden. Existing LLM-based planning tools can support plan generation, yet they remain susceptible to invalid task orderings and infeasible robot actions, resulting in frequent manual adjustment. Here we introduce a neuro-symbolic framework for long-horizon human-swarm collaboration that tightly melds verifiable task planning with context-grounded LLM reasoning. We formalize mission goals and operational rules as temporal logic formulas and admissible task orderings as task automata. Conditioned on these formal constraints and live perceptual context, LLMs generate executable subtask sequences that satisfy mission rules and remain grounded in the current scene. An uncertainty-aware scheduler then assigns subtasks across the heterogeneous swarm to maximize parallelisms while remaining resilient to disruptions. An event-triggered interaction protocol further limits operator involvement to sparse, high-level confirmation and guidance. Deployment on a heterogeneous robotic fleet yields similar results while remaining robust to hardware-specific actuation and communication uncertainties. Together, these results support a formal and scalable paradigm for reliable and low-overhead human-swarm collaboration in dynamic environments

Tacit knowledge plays a central role in human expertise, yet it remains difficult to capture, formalize, and reuse in machine-interpretable form. This challenge is especially relevant in procedural domains, where successful execution depends not only on explicit instructions, but also on implicit assumptions, contextual constraints, embodied skills, and experience-based judgments rarely documented. As a result, current knowledge engineering pipelines struggle to transform tacit and process-centric knowledge into formally specified, machine-interpretable representations that can be queried, validated, reasoned over, and reused. In this paper, we introduce a neuro-symbolic framework that combines Logic-Augmented Generation and an Active-Inference-inspired approach for ontology-grounded Knowledge Graph construction. We evaluate the approach in a knowledge transfer case study in manufacturing, using assembly-like repair procedures from instructional videos as a reproducible proxy domain. Results show that the proposed solution improves completeness and semantic quality, advancing neuro-symbolic knowledge engineering for industrial domains.

Neural network verification is an active and rapidly maturing research area, with a growing ecosystem of solvers and tools. The VNN-LIB standard was introduced to support interoperability in this ecosystem, but Version1.0 has several serious short-comings as a formal foundation: it lacks a precise syntax, semantics, and type system, offers limited expressivity, and relies on externally defined ONNX models whose semantics are informal and constantly evolving. The latter distinguishes VNN-LIB from established standards such as SMT-LIB, where queries are self-contained and have fixed semantics. In this paper we address these challenges by developing the theoretical foundations of VNN-LIB2.0. Our key contribution is the introduction of the notion of a \emph{network theory}, which abstractly characterises the minimal semantic interface required from a neural network model format. This abstraction enables VNN-LIB to be defined independently of any specific ONNX version while remaining compatible with evolving model representations. Building on this foundation, we present a formal syntax for a more expressive query language, a type system for it over the numeric domains provided by the network theory, and finally a formal semantics. To ensure internal consistency, the standard is mechanised in the Agda theorem prover. VNN-LIB~2.0 therefore provides robust and rigorous foundations for trustworthy neural network verification.

Imitation learning is widely used for learning to act in complex environments. While pure neural-based methods handle high dimensional data effectively, they suffer from the requirement of large number of samples and are prone to overfitting. Pure symbolic approaches, while generalize well, do not handle high-dimensional data effectively. We propose a neurosymbolic approach that achieves the best of both worlds, i.e, handling high-dimensional data while achieving generalization. The key advantage of our approach is that it can effectively exploit additional privileged information that is available only during training (in our case, gaze data). Our empirical evaluations demonstrate the effectiveness, efficiency and the generalization capability of our proposed approach.

Skeleton-based human activity recognition has achieved strong empirical performance, yet most existing models remain black boxes and difficult to interpret. In this work, we introduce a neurosymbolic formulation of skeleton-based HAR that reframes action recognition as concept-driven first-order logical reasoning over motion primitives. Our framework bridges representation learning and symbolic inference by grounding first-order logic predicates in learnable spatial and temporal motion concepts. Specifically, we employ a standard spatio-temporal skeleton encoder to extract latent motion representations, which are then mapped to interpretable concept predicates via a spatio-temporal concept decoder that explicitly separates pose-centric and dynamics-centric abstractions. These concept predicates are composed through differentiable first-order logic layers, enabling the model to learn human-readable logical rules that govern action semantics. To impose semantic structure on the learned concepts, we align skeleton representations with LLM-derived descriptions of atomic motion primitives, establishing a shared conceptual space for perception and reasoning. Extensive experiments on NTU RGB+D 60/120 and NW-UCLA demonstrate that our approach achieves competitive recognition performance while providing explicit, interpretable explanations grounded in logical structure. Our results highlight neurosymbolic reasoning as an effective paradigm for interpretable spatio-temporal action understanding. Code: https://github.com/Mr-TalhaIlyas/REASON

Autonomous systems that build structures from natural-language instructions need reliable spatial reasoning, yet large language models (LLMs) make systematic coordinate errors when generating three-dimensional block placements. We present a neuro-symbolic pipeline based on 2.5-D decomposition: the LLM plans in the two-dimensional horizontal plane while a deterministic executor computes all vertical placements from column occupancy, eliminating an entire class of errors. On the Build What I Mean benchmark (160 rounds), GPT-4o-mini with this pipeline achieves 94.6% mean structural accuracy across 12 independent runs, within 3.0 percentage points of the 97.6% ceiling imposed by architect-agent errors that no builder-side improvement can address. This outperforms both GPT-4o at 90.3% and the best competing system at 76.3%. A controlled ablation confirms that 2.5-D decomposition is the dominant contributor, accounting for 28.7 percentage points of accuracy. The pipeline transfers directly to edge hardware: Nemotron-3 120B on an NVIDIA Jetson Thor AGX achieves 96.0% mean structural accuracy with the identical pipeline, slightly exceeding the cloud result. Expanding the system prompt by four targeted examples to exceed the model's 8,320-token prefix cache page size, combined with low-effort reasoning, reduces mean per-request latency by 3X to 19.7 seconds at 95.6% accuracy. The underlying principle, removing deterministic dimensions from the LLM's output space, applies to any autonomous construction or assembly task where gravity or other physical constraints fix one or more degrees of freedom. A transfer experiment on 500 IGLU collaborative building tasks confirms the effect generalizes beyond the primary benchmark.

GCC and LLVM underpin much of modern software infrastructure, relying on distinct Intermediate Representations (IRs) to drive optimizations and code generation. However, the semantic and structural differences between these IRs create significant barriers for cross-toolchain interaction, limiting the reuse of compiler frontends, backends, and optimization pipelines across programming languages and compilation ecosystems. Traditional rule-based translators have attempted to bridge this gap, but their complexity and maintenance cost have hindered practical adoption. In this context, Large Language Models (LLMs) appear to be an emerging technology that offers a data-driven alternative, capable of learning complex mappings between heterogeneous compiler IRs directly from sufficiently representative examples. To explore this approach, this paper presents IRIS-14B, a 14-billion-parameter transformer model fine-tuned to translate GIMPLE (as emitted by GCC) to LLVM IR (as emitted by LLVM). The model is trained on paired IRs extracted from C sources and evaluated on the GIMPLE-to-LLVM IR transformation applied to IRs derived from real-world C code and competitive programming problems. To the best of our knowledge, IRIS-14B is the first model trained explicitly for IR-to-IR translation. It outperforms the accuracy of widely used models, including the largest state-of-the-art open models available today, ranging from 13 to 1,000 billion parameters, by up to 44 percentage points. The proposed transformation supports the integration of LLMs as complementary components within hybrid neuro-symbolic compiler architectures, where models such as IRIS-14B act as interoperability layers enabling cross-toolchain workflows without modifying existing compiler passes, while traditional compiler infrastructure continues to perform deterministic compilation and optimization.

Transfer learning in reinforcement learning (RL) seeks to accelerate learning in new tasks by leveraging knowledge from related sources. Existing neurosymbolic transfer methods, however, typically rely on manually specified task automata, assume a single source task, and use fixed knowledge-integration mechanisms that cannot adapt to varying source relevance. We propose LANTERN, a unified framework for multi-source neurosymbolic transfer that addresses these limitations through three components: (i) deterministic finite automata generated from natural language task descriptions using large language models, (ii) semantic embedding-based aggregation of multiple source policies weighted by cross-task similarity, and (iii) adaptive teacher-student gating based on temporal-difference error and semantic uncertainty. Across domains spanning resource management, navigation, and control, LANTERN achieves 40-60% improvements in sample efficiency over existing baselines while remaining robust to poorly aligned sources. These results demonstrate that multi-source, adaptively weighted neurosymbolic transfer can improve scalability and robustness in symbolic RL settings.

This paper describes our system submitted to SemEval-2026 Task 11: Disentangling Content and Formal Reasoning in Large Language Models. We present an efficient modular neuro-symbolic approach, combining a symbolic prover with small reasoning LLMs (4B parameters). The system consists of an LLM-based parser that translates natural language syllogisms to a first-order logic (FOL) representation, an automated theorem prover, and two optional modules: machine translation for multilingual inputs and a symbolic retrieval component for the identification of relevant premises. The system achieves competitive accuracy and relatively low content effect on most subtasks. Our ablations show that this approach outperforms LLM-based zero-shot baselines in this parameter size range, but also reveal limited multilingual capabilities of small LLMs. Finally, we include a discussion of the task's main ranking metric and analyze its limitations.

Transformer based pre-trained large language models have become ubiquitous. There is increasing evidence to suggest that even with large scale pre-training, these models do not capture complete compositional context and certainly not, the full human analogous context. Besides, by the very nature of the architecture, these models hallucinate, are difficult to maintain, are not easily interpretable and require enormous compute resources for training and inference. Here, we describe Gyan, an explainable language model based on a novel non-transformer architecture, without any of these limitations. Gyan achieves SOTA performance on 3 widely cited data sets and superior performance on two proprietary data sets. The novel architecture decouples the language model from knowledge acquisition and representation. The model draws on rhetorical structure theory, semantic role theory and knowledge-based computational linguistics. Gyan's meaning representation structure captures the complete compositional context and attempts to mimic humans by expanding the context to a 'world model'. AI model adoption critically depends on trust and transparency especially in mission critical use cases. Collectively, our results demonstrate that it is possible to create models which are trustable and reliable for mission critical tasks. We believe our work has tremendous potential for guiding the development of transparent and trusted architectures for language models.