Neuro-Symbolic Reasoning

We study structured abstraction-based reasoning for the Abstraction and Reasoning Corpus (ARC) and compare its generalization to test-time approaches. Purely neural architectures lack reliable combinatorial generalization, while strictly symbolic systems struggle with perceptual grounding. We therefore propose a neuro-symbolic architecture that extracts object-level structure from grids, uses neural priors to propose candidate transformations from a fixed domain-specific language (DSL) of atomic patterns, and filters hypotheses using cross-example consistency. Instantiated as a compositional reasoning framework based on unit patterns inspired by human visual abstraction, the system augments large language models (LLMs) with object representations and transformation proposals. On ARC-AGI-2, it improves base LLM performance from 16% to 24.4% on the public evaluation set, and to 30.8% when combined with ARC Lang Solver via a meta-classifier. These results demonstrate that separating perception, neural-guided transformation proposal, and symbolic consistency filtering improves generalization without task-specific finetuning or reinforcement learning, while reducing reliance on brute-force search and sampling-based test-time scaling. We open-source the ARC-AGI-2 Reasoner code (https://github.com/CoreThink-AI/arc-agi-2-reasoner).

Enterprise adoption of Large Language Models (LLMs) is constrained by hallucination, domain drift, and the inability to enforce regulatory compliance at the reasoning level. We present a neurosymbolic architecture implemented within the Foundation AgenticOS (FAOS) platform that addresses these limitations through ontology-constrained neural reasoning. We introduce a three-layer ontological framework--Role, Domain, and Interaction ontologies--grounding LLM-based enterprise agents. We formalize asymmetric neurosymbolic coupling: current enterprise systems constrain agent inputs (context assembly, tool discovery, governance thresholds) but not outputs, and we propose mechanisms extending this coupling to output-side validation (response checking, reasoning verification, compliance enforcement). A controlled experiment (1,800 runs across five industries and three LLMs: Claude Sonnet 4, Qwen 2.5 72B, Gemma 4 26B) finds ontology-coupled agents significantly outperform ungrounded agents on Metric Accuracy (p < .001) and Role Consistency (p < .001) across all three models with large effect sizes (Kendall's W = .46-.64). Improvements are greatest where LLM parametric knowledge is weakest--particularly in Vietnam-localized domains, where ontology lift is 2x that of English domains. Contributions: (1) a formal three-layer enterprise ontology model; (2) a taxonomy of neurosymbolic coupling patterns; (3) ontology-constrained tool discovery via SQL-pushdown scoring; (4) a proposed framework for output-side ontological validation; (5) empirical evidence for the inverse parametric knowledge effect--ontological grounding value is inversely proportional to LLM training-data coverage of the domain; (6) cross-model replication establishing model-independence; (7) a production system serving 22 industry verticals with 650+ agents.

Modern manufacturing environments demand real-time, trustworthy, and interpretable root-cause insights to sustain productivity and quality. Traditional analytics pipelines often treat anomaly detection, causal inference, and root-cause analysis as isolated stages, limiting scalability and explainability. In this work, we present CausalPulse, an industry-grade multi-agent copilot that automates causal diagnostics in smart manufacturing. It unifies anomaly detection, causal discovery, and reasoning through a neurosymbolic architecture built on standardized agentic protocols. CausalPulse is being deployed in a Robert Bosch manufacturing plant, integrating seamlessly with existing monitoring workflows and supporting real-time operation at production scale. Evaluations on both public (Future Factories) and proprietary (Planar Sensor Element) datasets show high reliability, achieving overall success rates of 98.0% and 98.73%. Per-criterion success rates reached 98.75% for planning and tool use, 97.3% for self-reflection, and 99.2% for collaboration. Runtime experiments report end-to-end latency of 50-60s per diagnostic workflow with near-linear scalability (R^2=0.97), confirming real-time readiness. Comparison with existing industrial copilots highlights distinct advantages in modularity, extensibility, and deployment maturity. These results demonstrate how CausalPulse's modular, human-in-the-loop design enables reliable, interpretable, and production-ready automation for next-generation manufacturing.

We present a first comparative pilot study of three t-norm operators -- Lukasiewicz (T_L), Product (T_P), and Gödel (T_G) - as logical conjunction mechanisms in a neuro-symbolic reasoning system for EU AI Act compliance classification. Using the LGGT+ (Logic-Guided Graph Transformers Plus) engine and a benchmark of 1035 annotated AI system descriptions spanning four risk categories (prohibited, high_risk, limited_risk, minimal_risk), we evaluate classification accuracy, false positive and false negative rates, and operator behaviour on ambiguous cases. At n=1035, all three operators differ significantly (McNemar p<0.001). T_G achieves highest accuracy (84.5%) and best borderline recall (85%), but introduces 8 false positives (0.8%) via min-semantics over-classification. T_L and T_P maintain zero false positives, with T_P outperforming T_L (81.2% vs. 78.5%). Our principal findings are: (1) operator choice is secondary to rule base completeness; (2) T_L and T_P maintain zero false positives but miss borderline cases; (3) T_G's min-semantics achieves higher recall at cost of 0.8% false positive rate; (4) a mixed-semantics classifier is the productive next step. We release the LGGT+ core engine (201/201 tests passing) and benchmark dataset (n=1035) under Apache 2.0.

Accurate parking availability prediction is critical for intelligent transportation systems, but real-world deployments often face data sparsity, noise, and unpredictable changes. Addressing these challenges requires models that are not only accurate but also uncertainty-aware. In this work, we propose a loosely coupled neuro-symbolic framework that integrates Bayesian Neural Networks (BNNs) with symbolic reasoning to enhance robustness in uncertain environments. BNNs quantify predictive uncertainty, while symbolic knowledge extracted via decision trees and encoded using probabilistic logic programming is leveraged in two hybrid strategies: (1) using symbolic reasoning as a fallback when BNN confidence is low, and (2) refining output classes based on symbolic constraints before reapplying the BNN. We evaluate both strategies on real-world parking data under full, sparse, and noisy conditions. Results demonstrate that both hybrid methods outperform symbolic reasoning alone, and the context-refinement strategy consistently exceeds the performance of Long Short-Term Memory (LSTM) networks and BNN baselines across all prediction windows. Our findings highlight the potential of modular neuro-symbolic integration in real-world, uncertainty-prone prediction tasks.

We propose a neuro-symbolic architecture that learns four interpretable physiological concepts, oculomotor dynamics, gaze stability, prefrontal hemodynamics, and multimodal, from eye-tracking and neural hemodynamics, functional near-infrared spectroscopy, (fNIRS) windows using attention-based encoders, and combines them with differentiable approximate reasoning rules using learned weights and soft thresholds, to address both rigid hand-crafted rules and the lack of subject-level alignment diagnostics. We apply this system to fatigue classification from multimodal physiological signals, a domain that requires models that are accurate and interpretable, with internal reasoning that can be inspected for safety-critical use. In leave-one-subject-out evaluation on 18 participants (560 samples), the method achieves 72.1% +/- 12.3% accuracy, comparable to tuned baselines while exposing concept activations and rule firing strengths. Ablations indicate gains from participant-specific calibration (+5.2 pp), a modest drop without the fNIRS concept (-1.2 pp), and slightly better performance with Lukasiewicz operators than product (+0.9 pp). We also introduce concept fidelity, an offline per-subject audit metric from held-out labels, which correlates strongly with per-subject accuracy (r=0.843, p < 0.0001).

Multimodal Large Language Models (MLLMs) have demonstrated remarkable reasoning capabilities across modalities such as images and text. However, tabular data, despite being a critical real-world modality, remains relatively underexplored in multimodal learning. In this paper, we focus on the task of Tabular-Vision Multi-Modal Understanding (TVMU) and identify three core challenges: (1) high structural variability and data incompleteness in tables, (2) implicit and complex feature dependencies, and (3) significant heterogeneity in problem-solving pipelines across downstream tasks. To address these issues, we propose Thinking with Tables (TWT). TWT employs a program-aided code-based neuro-symbolic reasoning mechanism that facilitates key operations, such as information extraction and element modeling, by interacting with external environments. We evaluate TWT on eight representative datasets. Experimental results demonstrate that TWT consistently outperforms existing baselines by an average of 10% in accuracy, achieving performance comparable to, or even surpassing, proprietary commercial SOTA LLMs on TVMU tasks. Models and codes are available at https://github.com/kunyang-YU/Thinking-with-Tables

While Large Language Models (LLMs) provide semantic flexibility for robotic task planning, their susceptibility to hallucination and logical inconsistency limits their reliability in long-horizon domains. To bridge the gap between unstructured environments and rigorous plan synthesis, we propose DUPLEX, an agentic dual-system neuro-symbolic architecture that strictly confines the LLM to schema-guided information extraction rather than end-to-end planning or code generation. In our framework, a feed-forward Fast System utilizes a lightweight LLM to extract entities, relations etc. from natural language, deterministically mapping them into a Planning Domain Definition Language (PDDL) problem file for a classical symbolic planner. To resolve complex or underspecified scenarios, a Slow System is activated exclusively upon planning failure, leveraging solver diagnostics to drive a high-capacity LLM in iterative reflection and repair. Extensive evaluations across 12 classical and household planning domains demonstrate that DUPLEX significantly outperforms existing end-to-end and hybrid LLM baselines in both success rate and reliability. These results confirm that The key is not to make the LLM plan better, but to restrict the LLM to the part it is good at - structured semantic grounding - and leave logical plan synthesis to a symbolic planner.

Vision-Language Models (VLMs) have been applied to a wide range of reasoning tasks, yet it remains unclear whether they can reason robustly under distribution shifts. In this paper, we study covariate shifts in which the perceptual input distribution changes while the underlying prediction rules do not. To investigate this question, we consider visual deductive reasoning tasks, where a model is required to answer a query given an image and logical rules defined over the object concepts in the image. Empirically, we find that VLMs fine-tuned through gradient-based end-to-end training can achieve high in-distribution accuracy but fail to generalize under such shifts, suggesting that fine-tuning does not reliably induce the underlying reasoning function. This motivates a neuro-symbolic perspective that decouples perception from reasoning. However, we further observe that recent neuro-symbolic approaches that rely on black-box components for reasoning can still exhibit inconsistent robustness across tasks. To address this issue, we propose VLC, a neuro-symbolic method that combines VLM-based concept recognition with circuit-based symbolic reasoning. In particular, task rules are compiled into a symbolic program, specifically a circuit, which executes the rules exactly over the object concepts recognized by the VLM. Experiments on three simple visual deductive reasoning tasks with distinct rule sets show that VLC consistently achieves higher task accuracy on out-of-distribution data than other reasoning paradigms. Code is available at https://github.com/uiuctml/VLC.

Classroom AI systems increasingly infer high-level educational states such as engagement, confusion, collaboration, participation, and instructional quality from multimodal and linguistic signals. In multicultural and multilingual classrooms, such inferences can translate culturally situated behavior into stereotyped claims: silence may be read as disengagement, gaze aversion as inattention, code-switching as low proficiency, or indirect help-seeking as confusion. We argue that stereotype-aware classroom AI should separate observable evidence from culturally loaded interpretation and should treat unsupported construct-level claims as safety risks. We introduce NSCR, a culturally grounded neuro-symbolic framework that converts video, audio, ASR, lesson artifacts, and contextual metadata into typed facts with uncertainty, provenance, and cultural scope, then composes them through executable reasoning and policy constraints. We define a taxonomy of stereotype-prone classroom inferences and propose a benchmark agenda covering culture-conditioned state inference, evidence-grounded claim verification, multilingual and code-switched reasoning, collaboration analysis, counterfactual cultural robustness, and culture-conditioned red-teaming. We further specify metrics for stereotype leakage, unsupported attribution, cultural calibration gaps, abstention under cultural ambiguity, and evidence faithfulness. The contribution is methodological: a concrete framework and evaluation agenda for mitigating stereotyped reasoning in classroom AI, with education as a high-stakes, culturally variable deployment setting.

Logical reasoning encompasses deduction, induction, and abduction. However, while Large Language Models (LLMs) have effectively mastered the former two, abductive reasoning remains significantly underexplored. Existing frameworks, predominantly designed for static deductive tasks, fail to generalize to abductive reasoning due to unstructured state representation and lack of explicit state control. Consequently, they are inevitably prone to Evidence Fabrication, Context Drift, Failed Backtracking, and Early Stopping. To bridge this gap, we introduce Graph of States (GoS), a general-purpose neuro-symbolic framework tailored for abductive tasks. GoS grounds multi-agent collaboration in a structured belief states, utilizing a causal graph to explicitly encode logical dependencies and a state machine to govern the valid transitions of the reasoning process. By dynamically aligning the reasoning focus with these symbolic constraints, our approach transforms aimless, unconstrained exploration into a convergent, directed search. Extensive evaluations on two real-world datasets demonstrate that GoS significantly outperforms all baselines, providing a robust solution for complex abductive tasks. Code repo and all prompts: https://github.com/gaorch85/Graph-of-States.

The computational demands of modern AI services are increasingly shifting execution beyond centralized clouds toward a computing continuum spanning edge and end devices. However, the scale, heterogeneity, and cross-layer dependencies of these environments make resilience difficult to maintain. Existing fault-management methods are often too static, fragmented, or heavy to support timely self-healing, especially under noisy logs and edge resource constraints. To address these limitations, this paper presents NeSy-Edge, a neuro-symbolic framework for trustworthy self-healing in the computing continuum. The framework follows an edge-first design, where a resource-constrained edge node performs local perception and reasoning, while a cloud model is invoked only at the final diagnosis stage. Specifically, NeSy-Edge converts raw runtime logs into structured event representations, builds a prior-constrained sparse symbolic causal graph, and integrates causal evidence with historical troubleshooting knowledge for root-cause analysis and recovery recommendation. We evaluate our work on representative Loghub datasets under multiple levels of semantic noise, considering parsing quality, causal reasoning, end-to-end diagnosis, and edge-side resource usage. The results show that NeSy-Edge remains robust even at the highest noise level, achieving up to 75% root-cause analysis accuracy and 65% end-to-end accuracy while operating within about 1500 MB of local memory.

Conservation laws are fundamental to understanding dynamical systems, but discovering them from data remains challenging due to parameter variation, non-polynomial invariants, local minima, and false positives on chaotic systems. We introduce NGCG, a neural-symbolic pipeline that decouples dynamics learning from invariant discovery and systematically addresses these challenges. A multi-restart variance minimiser learns a near-constant latent representation; system-specific symbolic extraction (polynomial Lasso, log-basis Lasso, explicit PDE candidates, and PySR) yields closed-form expressions; a strict constancy gate and diversity filter eliminate spurious laws. On a benchmark of nine diverse systems including Hamiltonian and dissipative ODEs, chaos, and PDEs, NGCG achieves consistent discovery (DR=1.0, FDR=0.0, F1=1.0) on all four systems with true conservation laws, with constancy two to three orders of magnitude lower than the best baseline. It is the only method that succeeds on the Lotka--Volterra system, and it correctly outputs no law on all five systems without invariants. Extensive experiments demonstrate robustness to noise ($σ= 0.1$), sample efficiency (50--100 trajectories), insensitivity to hyperparameters, and runtime under one minute per system. A Pareto analysis shows that the method provides a range of candidate expressions, allowing users to trade complexity for constancy. NGCG achieves strong performance relative to prior methods for data-driven conservation-law discovery, combining high accuracy with interpretability.

In high-complexity abstract reasoning, a system must infer a latent rule from a few examples or structured observations and apply it to unseen instances. LLMs can express such rules as programs, but ordinary conversation-based refinement is largely outcome-level: it observes that an answer or output is wrong without formally re-checking which abstraction, relation, or transformation justified that outcome. We propose Abduction-Based Procedural Refinement (ABPR), a neuro-symbolic refinement approach that couples an LLM with a Prolog meta-interpreter. ABPR treats each candidate program as an executable declarative hypothesis of the latent rule and reifies its SLD goal--subgoal resolution into compact proof-tree-style derivations, following Shapiro's algorithmic program debugging (APD). In this view, refinement is not merely code-level debugging, but semantic re-checking of the model's hypothesised rule. We evaluate ABPR primarily on ARC-AGI-2, a challenging few-shot abstract rule induction benchmark over grid transformations. ABPR with Gemini-3-Flash achieves 56.67% Pass@2, while GPT-5.5 xHigh with ABPR reaches 98.33% Pass@2 on the public evaluation set. Supplementary experiments on fill-in-the-blank I-RAVEN-X and A-I-RAVEN adaptations provide evidence that the same trace-guided framework extends beyond ARC-specific grid tasks to RAVEN-style relational and analogical abstraction. Repeated-run and sensitivity analyses show that parallel trace-guided search reduces stochastic variance as search breadth and total search depth increase.

Formal verification via interactive theorem proving is increasingly used to ensure the correctness of critical systems, yet constructing large proof scripts remains highly manual and limits scalability. Advances in large language models (LLMs), especially in mathematical reasoning, make their integration into software verification increasingly promising. This paper introduces a neuro-symbolic proof generation framework designed to automate proof search for system-level verification projects. The framework performs a best-first tree search over proof states, repeatedly querying an LLM for the next candidate proof step. On the neural side, we fine-tune LLMs using datasets of proof state-step pairs; on the symbolic side, we incorporate a range of ITP tools to repair rejected steps, filter and rank proof states, and automatically discharge subgoals when search progress stalls. This synergy enables data-efficient LLM adaptation and semantics-informed pruning of the search space. We implement the framework on a new Isabelle REPL that exposes fine-grained proof states and automation tools, and evaluate it on the FVEL seL4 benchmark and additional Isabelle developments. On seL4, the system proves up to 77.6% of the theorems, substantially surpassing previous LLM-based approaches and standalone Sledgehammer, while solving significantly more multi-step proofs. Results across further Isabelle benchmarks demonstrate strong generalization, indicating a viable path toward scalable automated software verification.

We present a formal and constructive framework for simulating Alternating Finite Automata (AFAs) using Logic-Gated Time-Shared Feedforward Networks (LG-TS-FFNs). Unlike prior neural automata models limited to Nondeterministic Finite Automata (NFAs) and existential reachability, our architecture integrates learnable, state-dependent biases that function as differentiable logic gates, enabling the representation of both Existential OR and Universal AND aggregation within a shared-parameter linear recurrence. We prove that this architectural modification upgrades the network's computational class to be structurally isomorphic to AFAs, thereby inheriting their exponential succinctness: the network can represent regular languages requiring $2^n$ states in an NFA with only $n$ neurons. We rigorously establish that the forward pass of an LG-TS-FFN exactly simulates the reachability dynamics of an AFA, including instantaneous $\varepsilon$-closures. Furthermore, we demonstrate empirical learnability: a continuous relaxation of the logic gates allows the network to simultaneously recover the automaton's topology and logical semantics from binary labels via standard gradient descent. Extensive experiments confirm that our model achieves perfect recovery of ground-truth automata, bridging the gap between statistical learning and succinct, universal logical reasoning.

Recent advances in Vision-Language Models (VLMs) have enabled video-instructed robotic programming, allowing agents to interpret video demonstrations and generate executable control code. We formulate video-instructed robotic programming as a cross-domain adaptation problem, where perceptual and physical differences between demonstration and deployment induce procedural mismatches. However, current VLMs lack the procedural understanding needed to reformulate causal dependencies and achieve task-compatible behavior under such domain shifts. We introduce NeSyCR, a neurosymbolic counterfactual reasoning framework that enables verifiable adaptation of task procedures, providing a reliable synthesis of code policies. NeSyCR abstracts video demonstrations into symbolic trajectories that capture the underlying task procedure. Given deployment observations, it derives counterfactual states that reveal cross-domain incompatibilities. By exploring the symbolic state space with verifiable checks, NeSyCR proposes procedural revisions that restore compatibility with the demonstrated procedure. NeSyCR achieves a 31.14% improvement in task success over the strongest baseline Statler, showing robust cross-domain adaptation across both simulated and real-world manipulation tasks.

Neuro-symbolic artificial intelligence (AI) systems typically couple a neural perception module to a discrete symbolic solver through a non-differentiable boundary, preventing constraint-satisfaction feedback from reaching the perception encoder during training. We introduce AS2 (Attention-Based Soft Answer Sets), a fully differentiable neuro-symbolic architecture that replaces the discrete solver with a soft, continuous approximation of the Answer Set Programming (ASP) immediate consequence operator $T_P$. AS2 maintains per-position probability distributions over a finite symbol domain throughout the forward pass and trains end-to-end by minimizing the fixed-point residual of a probabilistic lift of $T_P$, thereby differentiating through the constraint check without invoking an external solver at either training or inference time. The architecture is entirely free of conventional positional embeddings. Instead, it encodes problem structure through constraint-group membership embeddings that directly reflect the declarative ASP specification, making the model agnostic to arbitrary position indexing. On Visual Sudoku, AS2 achieves 99.89% cell accuracy and 100% constraint satisfaction (verified by Clingo) across 1,000 test boards, using a greedy constrained decoding procedure that requires no external solver. On MNIST Addition with $N \in \{2, 4, 8\}$ addends, AS2 achieves digit accuracy above 99.7% across all scales. These results demonstrate that a soft differentiable fixpoint operator, combined with constraint-aware attention and declarative constraint specification, can match or exceed pipeline and solver-based neuro-symbolic systems while maintaining full end-to-end differentiability.

Register-Transfer Level (RTL) synthesis and summarization are central to hardware design automation but remain challenging for Large Language Models (LLMs) due to rigid HDL syntax, limited supervision, and weak alignment with natural language. Existing prompting and retrieval-augmented generation (RAG) methods have not incorporated symbolic planning, limiting their structural precision. We introduce SYMDIREC, a neuro-symbolic framework that decomposes RTL tasks into symbolic subgoals, retrieves relevant code via a fine-tuned retriever, and assembles verified outputs through LLM reasoning. Supporting both Verilog and VHDL without LLM fine-tuning, SYMDIREC achieves ~20% higher Pass@1 rates for synthesis and 15-20% ROUGE-L improvements for summarization over prompting and RAG baselines, demonstrating the benefits of symbolic guidance in RTL tasks.

Remote sensing underpins crucial applications such as disaster relief and ecological field surveys, where systems must understand complex scenes and constraints and make reliable decisions. Current remote-sensing benchmarks mainly focus on evaluating perception and reasoning capabilities of multimodal large language models (MLLMs). They fail to assess planning capability, stemming either from the difficulty of curating and validating planning tasks at scale or from evaluation protocols that are inaccurate and inadequate. To address these limitations, we introduce NeSy-Route, a large-scale neuro-symbolic benchmark for constrained route planning in remote sensing. Within this benchmark, we introduce an automated data-generation framework that integrates high-fidelity semantic masks with heuristic search to produce diverse route-planning tasks with provably optimal solutions. This allows NeSy-Route to comprehensively evaluate planning across 10,821 route-planning samples, nearly 10 times larger than the largest prior benchmark. Furthermore, a three-level hierarchical neuro-symbolic evaluation protocol is developed to enable accurate assessment and support fine-grained analysis on perception, reasoning, and planning simultaneously. Our comprehensive evaluation of various state-of-the-art MLLMs demonstrates that existing MLLMs show significant deficiencies in perception and planning capabilities. We hope NeSy-Route can support further research and development of more powerful MLLMs for remote sensing. The dataset and code are available at https://mingyang1010.github.io/NeSy-Route/.

Neuro-symbolic reasoning increasingly demands frameworks that unite the formal rigor of logic with the interpretability of large language models (LLMs). We introduce an end to end explainability by construction pipeline integrating the Automated Theorem Generator Delta1 based on the full triangular standard contradiction (FTSC) with LLMs. Delta1 deterministically constructs minimal unsatisfiable clause sets and complete theorems in polynomial time, ensuring both soundness and minimality by construction. The LLM layer verbalizes each theorem and proof trace into coherent natural language explanations and actionable insights. Empirical studies across health care, compliance, and regulatory domains show that Delta1 and LLM enables interpretable, auditable, and domain aligned reasoning. This work advances the convergence of logic, language, and learning, positioning constructive theorem generation as a principled foundation for neuro-symbolic explainable AI.

Existing end-to-end autonomous driving models rely heavily on purely data-driven inductive reasoning. This "black-box" nature leads to a lack of interpretability and absolute safety guarantees in complex, long-tail scenarios. To overcome this bottleneck, we propose a novel neuro-symbolic trajectory planning framework that seamlessly integrates rigorous deductive reasoning into end-to-end neural networks. Specifically, our framework utilizes a Large Language Model (LLM) to dynamically extract scene rules and employs an Answer Set Programming (ASP) solver for deterministic logical arbitration, generating safe and traceable discrete driving decisions. To bridge the gap between discrete symbols and continuous trajectories, we introduce a decision-conditioned decoding mechanism that transforms high-level logical decisions into learnable embedding vectors, simultaneously constraining the planning query and the physical initial velocity of a differentiable Kinematic Bicycle Model (KBM). By combining KBM-generated physical baseline trajectories with neural residual corrections, our approach inherently guarantees kinematic feasibility while ensuring a high degree of transparency. On the nuScenes benchmark, our method comprehensively outperforms the state-of-the-art baseline MomAD, reducing the L2 mean error to 0.57 m, decreasing the collision rate to 0.075%, and optimizing trajectory prediction consistency (TPC) to 0.47 m.

Formal verification of memory-manipulating programs critically depends on precise function specifications that capture memory states written by experts. This requirement has become a major bottleneck as large language models (LLMs) increasingly generate low-level systems code whose correctness cannot be assumed. To enable scalable formal verification, we focus exclusively on function specification generation, deliberately avoiding the synthesis of complex loop invariants that are central to traditional verification pipelines. We propose a neuro-symbolic framework for automatically generating memory-aware formal function specifications for C programs from natural language problem descriptions and function signatures. The pipeline first produces candidate specifications via in-context learning, and then iteratively refines them using compiler diagnostics from symbolic provers and the verification toolchain. In particular, we validate candidate specifications by constructing a proof for the negation of the specification with concrete examples, enabling machine-checked rejection of plausible-but-incorrect specifications. To support systematic evaluation, we introduce LeetCode-C-Spec, a new benchmark of 200 C programming problems for generating memory-aware formal function specifications. Experiments show that iterative refinement substantially improves syntactic validity, while symbolic prover-based refutation significantly enhances correctness assessment by filtering false positives that LLM-only judges frequently accept. Our results demonstrate that combining neural generation with symbolic feedback provides an effective approach to formal specification synthesis for memory-safe systems software.

In dynamic open-world environments, autonomous agents often encounter novelties that hinder their ability to find plans to achieve their goals. Specifically, traditional symbolic planners fail to generate plans when the robot's planning domain lacks the operators that enable it to interact appropriately with novel objects in the environment. We propose a neuro-symbolic architecture that integrates symbolic planning, reinforcement learning, and a large language model (LLM) to learn how to handle novel objects. In particular, we leverage the common sense reasoning capability of the LLM to identify missing operators, generate plans with the symbolic AI planner, and write reward functions to guide the reinforcement learning agent in learning control policies for newly identified operators. Our method outperforms the state-of-the-art methods in operator discovery as well as operator learning in continuous robotic domains.Our webpage and code can be access here: https://helenlu66.github.io/hybridLLMguided/

We present a new self-supervised machine learning approach for symbolic simplification of complex mathematical expressions. Training data is generated by scrambling simple expressions and recording the inverse operations, creating oracle trajectories that provide both goal states and explicit paths to reach them. A permutation-equivariant, transformer-based policy network is then trained on this data step-wise to predict the oracle action given the input expression. We demonstrate this approach on two problems in high-energy physics: dilogarithm reduction and spinor-helicity scattering amplitude simplification. In both cases, our trained policy network achieves near perfect solve rates across a wide range of difficulty levels, substantially outperforming prior approaches based on reinforcement learning and end-to-end regression. When combined with contrastive grouping and beam search, our model achieves a 100% full simplification rate on a representative selection of 5-point gluon tree-level amplitudes in Yang-Mills theory, including expressions with over 200 initial terms.

While enterprises amass vast quantities of data, much of it remains chaotic and effectively dormant, preventing decision-making based on comprehensive information. Existing neuro-symbolic approaches rely on disjoint pipelines and struggle with error propagation. We introduce the large ontology model (LOM), a unified framework that seamlessly integrates ontology construction, semantic alignment, and logical reasoning into a single end-to-end architecture. LOM employs a construct-align-reason (CAR) pipeline, leveraging its unified architecture across all three stages: it first autonomously constructs a domain-specific ontological universe from raw data, then aligns neural generation with this structural reality using a graph-aware encoder and reinforcement learning, and finally executes deterministic reasoning over the constructed topology, node attributes and relation types. We evaluate LOM on a comprehensive benchmark constructed from diverse real-world enterprise datasets. Experimental results demonstrate that LOM-4B achieves 88.8% accuracy in ontology completion and 94% in complex graph reasoning tasks, significantly outperforming state-of-the-art LLMs. These findings validate that autonomous logical construction is essential for achieving deterministic, enterprise-grade intelligence.

Intelligent agents must reason over both continuous dynamics and discrete representations to generate effective plans in complex environments. Previous studies have shown that symbolic abstractions can emerge from neural effect predictors trained with a robot's unsupervised exploration. However, these methods rely on deterministic symbolic domains, lack mechanisms to verify the generated symbolic plans, and operate only at the abstract level, often failing to capture the continuous dynamics of the environment. To overcome these limitations, we propose a bilevel neuro-symbolic framework in which learned probabilistic symbolic rules generate candidate plans rapidly at the high level, and learned continuous effect models verify these plans and perform forward search when necessary at the low level. Our experiments on multi-object manipulation tasks demonstrate that the proposed bilevel method outperforms symbolic-only approaches, reliably identifying failing plans through verification, and achieves planning performance statistically comparable to continuous forward search while resolving most problems via efficient symbolic reasoning.

We study mathematical discovery through the lens of neurosymbolic reasoning, where an AI agent powered by a large language model (LLM), coupled with symbolic computation tools, and human strategic direction, jointly produced a new result in combinatorial design theory. The main result of this human-AI collaboration is a tight lower bound on the imbalance of Latin squares for the notoriously difficult case $n \equiv 1 \pmod{3}$. We reconstruct the discovery process from detailed interaction logs spanning multiple sessions over several days and identify the distinct cognitive contributions of each component. The AI agent proved effective at uncovering hidden structure and generating hypotheses. The symbolic component consists of computer algebra, constraint solvers, and simulated annealing, which provides rigorous verification and exhaustive enumeration. Human steering supplied the critical research pivot that transformed a dead end into a productive inquiry. Our analysis reveals that multi-model deliberation among frontier LLMs proved reliable for criticism and error detection but unreliable for constructive claims. The resulting human-AI mathematical contribution, a tight lower bound of $4n(n{-}1)/9$, is achieved via a novel class of near-perfect permutations. The bound was formally verified in Lean 4. Our experiments show that neurosymbolic systems can indeed produce genuine discoveries in pure mathematics.

Large language models (LLMs) show significant potential for clinical decision support (CDS), yet their black-box nature -- characterized by untraceable reasoning and probabilistic hallucinations -- poses severe challenges in acupuncture, a field demanding rigorous interpretability and safety. To address this, we propose CORE-Acu, a neuro-symbolic framework for acupuncture clinical decision support that integrates Structured Chain-of-Thought (S-CoT) with knowledge graph (KG) safety verification. First, we construct the first acupuncture Structured Reasoning Trace dataset and a schema-constrained fine-tuning framework. By enforcing an explicit causal chain from pattern identification to treatment principles, treatment plans, and acupoint selection, we transform implicit Traditional Chinese Medicine (TCM) reasoning into interpretable generation constraints, mitigating the opacity of LLM-based CDS. Furthermore, we construct a TCM safety knowledge graph and establish a ``Generate--Verify--Revise'' closed-loop inference system based on a Symbolic Veto Mechanism, employing deterministic rules to intercept hallucinations and enforce hard safety boundaries. Finally, we introduce the Lexicon-Matched Entity-Reweighted Loss (LMERL), which corrects terminology drift caused by the frequency--importance mismatch in general optimization by adaptively amplifying gradient contributions of high-risk entities during fine-tuning. Experiments on 1,000 held-out cases demonstrate CORE-Acu's superior entity fidelity and reasoning quality. Crucially, CORE-Acu achieved 0/1,000 observed safety violations (95% CI: 0--0.37%), whereas GPT-4o exhibited an 8.5% violation rate under identical rules. These results establish CORE-Acu as a robust neuro-symbolic framework for acupuncture clinical decision support, guaranteeing both reasoning auditability and strict safety compliance.

Interpretable machine learning is essential in high-stakes domains where decision-making requires accountability, transparency, and trust. While rule-based models offer global and exact interpretability, learning rule sets that simultaneously achieve high predictive performance and low, human-understandable complexity remains challenging. To address this, we introduce TT-Sparse, a flexible neural building block that leverages differentiable truth tables as nodes to learn sparse, effective connections. A key contribution of our approach is a new soft TopK operator with straight-through estimation for learning discrete, cardinality-constrained feature selection in an end-to-end differentiable manner. Crucially, the forward pass remains sparse, enabling efficient computation and exact symbolic rule extraction. As a result, each node (and the entire model) can be transformed exactly into compact, globally interpretable DNF/CNF Boolean formulas via Quine-McCluskey minimization. Extensive empirical results across 28 datasets spanning binary, multiclass, and regression tasks show that the learned sparse rules exhibit superior predictive performance with lower complexity compared to existing state-of-the-art methods.