Neuro-Symbolic Framework

In this paper, we study Reinforcement Learning in Parametrized Action Markov Decision Processes (PAMDP), where each decision consists of a symbolic action and numerical parameters. In such settings Reinforcement Learning algorithms typically determine parameters with one-shot estimators, which makes their training sample inefficient. Though in most PAMDP environments explicit but incomplete knowledge (e.g., rules, safety constraints, or expert heuristics) is available, it is rarely directly used to increase the sample-efficiency of training Reinforcement Learning agents. We step into this gap and propose our novel Neuro-Symbolic Knowledge- and Gradient-Guided Reinforcement Learning (KGRL) algorithm. KGRL uses domain knowledge in a Datalog knowledge base to derive the set of applicable actions and feasible parameters for a given state. This allows it to prune non-applicable actions from the decision-space and constrain the parameter spaces of the remaining actions. We then use a gradient-based parameter refinement loop to estimate the optimal parameters during training and deployment of the agent. By recording activated rules along the trajectory, KGRL additionally provides local procedural explanations on the pruning of actions and constraining of parameters. Overall, KGRL guides the agent's exploration and deployment toward feasible and constraint-aware decisions, while increasing sample efficiency during training. KGRL outperforms state-of-the-art RL baselines for PAMDPs in both, sample efficiency and episodic return.

Vision--Language Models (VLMs) have recently demonstrated promising performance on structured visual reasoning tasks, including grid-based puzzles. However, despite strong perceptual capabilities, these models lack explicit mechanisms for enforcing logical consistency and frequently generate assignments that violate underlying constraints. In this paper, we propose a neuro-symbolic approach that integrates formal constraint reasoning into the VLM solving process via a Maximum Satisfiability (MaxSAT) oracle. Rather than computing solutions directly, the symbolic component acts as a consistency validator and refinement engine. Candidate placements generated by the VLM are encoded as soft clauses in a partial MaxSAT formulation, while Sudoku constraints remain hard clauses. When inconsistencies arise, the MaxSAT solver identifies a largest mutually consistent subset of assignments, which is then translated into structured textual and visual feedback to guide subsequent refinements. We evaluate our approach on a Sudoku dataset across multiple open-source and closed-access VLMs. Results show that MaxSAT-based feedback improves logical consistency and increases the number of solved instances, particularly in full-board refinement mode. These findings demonstrate that symbolic optimisation can enhance the reliability of vision-language reasoning.

Large language models can produce superficially legal twelve-tone scores that collapse into degenerate textures. We introduce a neuro-symbolic harness that wraps a language-model proposer in a generate-verify-repair-trace loop with symbolic verification. The complete pipeline improves event-local consistency without claiming whole-piece legality. Across 40 controlled tasks and four paired models, constraint-checked delivery rises from 13.3% under raw generation to 48.1% with the harness; it abstains on the remaining 51.9% of runs. The pass rate of a narrower collision and serialisation-consistency check rises from 33.5% to 58.3%, while degeneracy remains near 0.05, including under adversarial prompting. A blinded evaluation by five experts also shows a descriptive aggregate preference for harness candidates over raw generation in adherence, perceived legality, coherence, and overall quality.

Neuro-symbolic AI based on $IFOL_B$ is a way to combine neural learning and symbolic reasoning to overcome limitations of purely neural systems (like lack of interpretability and logical structure) with formal logical machinery for self-reference. In this paper we expand the cognitive power of $IFOL_B$ by using the probability computation for the currently unknown sentences, based on Nilsson's probability structure for the $IFOL_B$. We introduce the global symmetry transformation that preserves the current knowledge database and logical deduction, and the local one used for real-time decisions about concrete (sub)problems that involve only a very strict subset of $IFOL_B$ predicates. The computation of probability density function $KI$ in both cases, based on the Shannon's maximum information entropy, is provided by neural networks of this probabilistic neuro-symbolic AGI.

Neural networks can learn algorithmic input-output mappings, but trusting a learned executor requires more than a correct final answer because the state transitions that produce it are usually hidden. To make those transitions visible, we introduce a trace-supervised symbolic neural CPU, a factorized learned execution architecture that combines recurrent control, an explicit operation router over a fixed differentiable arithmetic-logic unit bank, destination-masked register writeback, complete trajectory supervision and matched fixed-point replay. The model exposes the selected operation, source and destination registers, register trajectory, memory signals and writeback semantics at every step. On the principal 16-wide benchmark, the non-quantized executor reproduces reference execution exactly, while the eight-bit quantization-simulated executor preserves the symbolic operation path through programs of 1,000 instructions. When the same execution is evaluated against a matched fixed-point replay, the residual numerical drift disappears, showing that it comes from a mismatch between continuous and low-precision reference semantics rather than from execution failure. We compare recurrent, Transformer, temporal-convolution, temporal graph-inspired and state-space controllers, and the ablations show that operation-gate supervision is necessary for an inspectable execution path. Hidden-opcode memory-pressure tasks expose the remaining limits in delayed state use and temporal binding. We also extend the interface with ValueMemory, hybrid adaptive leaky integrate-and-fire controllers, candidate-constrained symbolic control trained through behaviour cloning and actor-critic reinforcement learning, and an RV32I base-integer semantic bridge. Together, these results establish a trace-verifiable framework for interpretable, low-precision and controllable neural execution.

As 3D software proliferates, software artifacts now extend beyond code and 2D user interfaces to include 3D assets. Among these assets, collision meshes are critical as they define the geometry used by physics engines for collision detection and physical interaction. Although existing tools can automatically generate collision meshes from visual meshes, they often fail to capture the intended interaction behavior. As a result, developers need to manually edit many heterogeneous collision meshes, a process that is time-consuming and challenging to scale. To address this problem, we present a neuro-symbolic program synthesis approach for batch-editing collision meshes. We formulate the task as a programming-by-example problem: given a family of collision meshes with the same editing intent and a small number of user demonstrations, our approach synthesizes a reusable program that captures the editing intent and applies it to non-demonstration meshes. We implement this in a tool named MeshForge, and evaluate it across 24 tasks on 600 collision meshes. MeshForge successfully synthesizes 23/24 tasks, requiring 2.2 demonstrations and 3.5 seconds of synthesis time on average.

We present a general neurosymbolic reasoning and learning methodology based on a modular integration of answer set programming with an energy based model substrate. Key contributions are: (1) supporting joint optimisation in the continuous latent space through explicit ASP-based declarative semantics fully incorporating background knowledge, constraints, non-monotonic inference; and (2) advancing recent works at the interface of answer sets, probabilistic logic, and answer set modulo theories by providing a generalised model and practical platform for ASP-centric robust, end-to-end training for applications in dynamic domains (e.g., involving perception and interaction). We provide a practical implementation, and demonstrate basic use and application (with MNIST), and evaluate with the visual question-answering benchmark Clevr and the multi-object tracking benchmark MOT.

This article offers a comprehensive overview of mechanistic interpretability, an emerging field that seeks to reverse-engineer the internal algorithms of modern neural networks. While traditional explainable AI methods often stop at surface-level input-output correlations, this approach directly addresses the opaque "black box" nature of machine learning models, which is essential for ensuring safety and auditability in high-stakes deployments. The paper provides a detailed examination of Transformer circuit analysis, exploring how internal components like the residual stream, attention mechanisms, and induction heads drive complex tasks and in-context learning. It subsequently tackles the core challenge of superposition and polysemanticity, demonstrating how tools like Sparse Autoencoders (SAEs) and transcoders can decompose tangled network activations into distinct, human-interpretable features. Furthermore, the paper explores methods for actively controlling and modifying model behavior through steering vectors and causal interventions. Finally, it connects these mechanistic insights with neurosymbolic AI frameworks designed to translate neural representations into explicit, executable logical rules.

LLM-based robot planners are fluent but cannot guarantee that their plans are executable or safe. Classical PDDL planners can guarantee these properties, but only after the problem is fully specified, and they make poor use of an LLM's ability to read context and repair plans. This paper presents a neuro-symbolic framework with three parts. All LLM calls use a locally-hosted open-weight model, so the pipeline can be deployed on-robot with no cloud dependency. First, an offline procedure that mines a single global Signal Temporal Logic (STL) constraint on mobility from demonstration data. The procedure recovers codified rules (e.g., stopping at red lights, mined from nuPlan driving logs) or population preferences (e.g., social-navigation comfort, mined from SCAND teleoperation), depending on what the demonstrations encode. Because the demonstrations are a one-class signal, we generate the missing negatives with counterfactual perturbations and an LLM violation generator and then fit the constraint by evolutionary search. We use the mined constraint to shield a vision-language driving policy on Bench2Drive and two discrete-action navigation policies on HA-VLN-CE. Second, an evolutionary PDDL planner: an LLM proposes and repairs plans, programmatic validators decide which ones survive, and the validated portion of the plan grows over iterations. We test the planner on the open-world ALFWorld Text benchmark, where it beats strong baselines and stays robust when the goal vocabulary does not match the action-model vocabulary. Third, a constrained execution loop: the planner's plan is compiled into waypoints, the waypoints are checked against the mined constraint, and the planner re-plans on a violation. We illustrate the full pipeline via demonstrations using the Gazebo simulator.

The transition of Large Language Models (LLMs) from passive generators to autonomous agents has introduced significant challenges in reliability, security, and state management. Current agentic architectures are often constructed ad-hoc, prone to hallucination cascades, infinite loops, and prompt injection attacks. This paper argues that many of these failure modes can be analyzed using control motifs long studied in systems biology, provided the comparison is made at the level of typed interfaces and coordination structure rather than literal biological mechanism. We develop a typed interface correspondence between Gene Regulatory Networks and agentic software systems using polynomial functors and wiring diagrams. Five biological motifs are mapped to composable software design patterns: Coherent Feed-Forward Loops for noise suppression, Adaptive Immunity for layered security, Mitochondrial Signaling for resource governance, Endosymbiosis for neuro-symbolic integration, and Morphogen Diffusion for spatially varying coordination. An epistemic topology layer derives Kripke-style knowledge operators from the wiring diagram's observation structure and proves four predictive theorems for multi-agent scaling. The core contributions are: (1) the Agentic Operad, a typed syntax for agent composition with provable error suppression bounds for feed-forward topologies; (2) an epistemic topology with four theorems (error amplification, sequential penalty, parallel acceleration, and tool density scaling) whose qualitative predictions are consistent with published multi-agent benchmarks; and (3) a six-layer progression from structure through development, grounded in autonomous learning frameworks and convergence proxies from the empirical literature. A reference implementation with 1,813 tests and 116 examples illustrates practical feasibility.

Current agentic workflows usually involve decomposing user requests into sequences of tool calls with correctly resolved parameters, the results of which are processed through reasoning traces in the language model's context window. The prevailing route to improve such reasoning is test-time scaling, which trains models to search over long chains of thought; but the resulting capability is entangled in model weights, is not verifiable step-by-step, and is costly at inference. We present Forethought, a neurosymbolic reasoning system that instead treats reasoning as an explicit, verifiable program, that builds from a library of symbolic and neural primitives which are composed through a domain-specific language. The result are reasoning programs, which are concrete representations of the model's work, and as such can be inspected and modified before deployment. Instantiated as a tool-calling execution kernel and evaluated across five benchmarks, Forethought improves base-model accuracy by about 30% relative and outperforms vanilla prompting, reinforcement learning scaffolds, and prompt-evolution methods, enabling small models to match or exceed frontier models capabilities. In a direct comparison, a non-reasoning model augmented with Forethought competes with a dedicated reasoning model while requiring roughly three orders of magnitude less post-training investment, and remains model-agnostic and auditable.

Understanding why discovered scenarios become critical in scenario-based testing is essential for effectively leveraging them in decision-making systems. Reasoning about such criticality can be formulated as an attribution problem. However, across different decision-making tasks, the causes of criticality may involve diverse state variables, interaction patterns, and failure mechanisms, making attribution an inherently open-ended problem beyond predefined explanation spaces. Existing attribution methods still struggle to balance open-ended reasoning flexibility with the interpretability and traceability required for critical scenario reasoning. To address this limitation, we propose NeSy-CSA, a neuro-symbolic framework that transforms open-ended critical scenario attribution from unconstrained explanation generation into structured and traceable reasoning. NeSy-CSA narrows the attribution space by selecting relevant factors, makes the reasoning process traceable through a dependency-aware evidence graph, and executes symbolic reasoning procedures derived from atomic operations, coordinated with evidence-constrained neural inference to support flexible open-ended attribution. We further introduce a process-level and result-level assessment module to evaluate the structural validity of the attribution process and the behavioral effectiveness of the attribution results under controlled interventions. Experiments across four decision-making environments show that NeSy-CSA improves two intervention-based measures of attribution effectiveness by 18.32% and 13.67% over LLM-based baselines. These results demonstrate its potential to transform discovered critical scenarios into reusable knowledge for subsequent testing and safety analysis.

AI systems increasingly propose executable scientific models whose value depends on both their symbolic structure and their fitted continuous parameters. This makes parameter calibration the bottleneck of program-and-parameter co-search: an outer loop can generate thousands of candidate programs, but each needs an inner gradient-based optimization before it can be assessed. Staging each candidate into its own differentiable graph makes individual models fast but sacrifices the program-as-data property that keeps search fluid; interpreter-based approaches preserve programs as runtime data but pay interpreter overhead that dominates the numerical work. We present the Native Differentiable Virtual Machine (NDVM), a runtime representation that differentiates executable programs without compiling each candidate into a separate graph. NDVM separates symbolic structure from differentiable numeric state: tags, symbols, environments, and control remain native runtime data, while numeric payloads live in dense batched buffers with exact reverse-mode gradients recorded along the realized execution trace, so one evaluator walk is amortized across large populations of parameter vectors. A locked cost model of a real differentiable self-hosted Scheme interpreter motivates the design. We realize NDVM as a native runtime with forward and gradient equivalence to the reference backend, about 60x per-lane batch amortization, near-linear multicore scaling, and two independent front ends. In fixed-budget co-search over LLM-proposed programs, NDVM reaches high-quality solutions about 24x sooner in wall-clock time, suggesting runtime differentiation as a practical systems foundation for scientific discovery workflows.

High-throughput plant phenotyping generates valuable data that often remains trapped in unstructured text and isolated RGB images. To bridge this semantic gap, we propose a framework for constructing a multimodal granular Knowledge Graph (KG) to monitor genotype-phenotype interactions across time and experiments. In this work, we focus on wheat Triticum aestivum as a representative target crop to validate our methodology across complex canopy environments. Our pipeline first distills noisy field notes to extract entities and relations, dynamically constructing the KG by converting unique instances into hierarchical class entities via RDF-typing. These graph nodes are then aligned with standardized ontologies (PO, RO, WTO) using PlantDeBERTa. To visually ground the constructed graph, a Vision-Language Model paired with a wheat-segmentation ViT generates attention-based softmaps, linking specific KG entities directly to image pixels. We introduce a central observation node Plant_Obs_Id to connect these multimodal subgraphs temporally. Evaluated on 500 curated WisWheat samples using Pointing Game accuracy, Visual Word Sense Disambiguation (VWSD), and rank-based metrics, our neuro-symbolic approach successfully maps complex field observations to a structured graph. This enables automated field note auditing, temporal stress monitoring, and precise spatial trait localization for wheat breeders.

In this work, we focus on SE-RRMs, a symbol-equivariant instantiation of RRMs that exhibits improved extrapolation to larger problem sizes. We propose a neuro-symbolic approach, ``Guiding with Recurrent Reasoning Models'' (G-RRM), which integrates SE-RRMs with symbolic solvers for constraint satisfaction problems. SE-RRMs act as neural solvers that generate full solution proposals and guide classical symbolic solvers, such as backtracking or SAT-based methods like Glucose 4.1 and CaDiCaL 3.0.0, that produce globally correct solutions. Centrally, we investigate when neural guidance with G-RRM improves the search efficiency of symbolic solvers. % Our experiments show that the efficacy of G-RRM depends on two conditions: first, the problem instances must have an expansive combinatorial search space to expose potential gains, and second, the solver architecture must be capable of dynamically overwriting its branching choices to recover when neural hints are imperfect. When these conditions hold, guidance drives median conflict counts to zero and yields significant wall-clock speedups: on $9\times9$ Sudoku, where the SE-RRM correctly solves $91.1\%$ of instances, backtracking accelerates by $33.3\times$ and Glucose 4.1 by $1.70\times$ (median, $p<0.001$), with Glucose 4.1 retaining a $1.17\times$ speedup on perfect-hint $25\times25$ grids. In contrast, CaDiCaL 3.0.0, whose runtime is overhead-dominated and which always respects the injected branching hints rather than overwriting them, shows no significant speedup (median $1.02\times$, n.s.) and even a small significant mean slowdown ($0.90\times$) on $9\times9$. These results delineate the regimes in which neural guidance translates into practical speedups.

Safe landing-site assessment in unstructured environments remains a key challenge for autonomous UAV deployment, as vision-only learning approaches often degrade under terrain variability and provide limited transparency in safety decisions. We present NEUROSYMLAND, a neuro-symbolic landing-site assessment system that integrates lightweight perception with explicit safety reasoning. The framework constructs a probabilistic semantic scene graph from onboard visual input and evaluates candidate landing regions using symbolic constraints capturing terrain flatness, obstacle clearance, and spatial consistency, enabling structured reasoning under perceptual uncertainty while maintaining edge-feasible execution. Across 72 simulated landing scenarios spanning diverse terrains, NEUROSYMLAND achieves 61 successful assessments, outperforming four competitive baselines (37-57 successes). To evaluate deployability, we further conduct 100 hardware-in-the-loop trials with randomized initial poses, profiling end-to-end latency, stage-wise execution time, and system-level metrics including CPU/GPU utilization, memory footprint, and power consumption. Results demonstrate improved robustness and interpretability with bounded edge-resource usage. Profiling shows that symbolic reasoning contributes only a small fraction of end-to-end latency, while the main computational cost arises from perception and PSSG construction. These results demonstrate the feasibility of deploying the landing-site assessment stack on edge-constrained UAV hardware, and all source code, datasets, prompts, and symbolic rule refinement examples are released in an open-source repository

As the scale and complexity of cloud-based AI systems continue to escalate, ensuring service reliability through rapid fault detection and adaptive recovery has become a critical challenge. While existing approaches integrate Large Language Models (LLMs) for semantic understanding and Deep Reinforcement Learning (DRL) for policy optimization, they often rely on sequential, loosely coupled architectures that underutilize the generative and reasoning capabilities of LLMs. In this paper, we propose a paradigm shift with PASE, a Planning-Aware Semantic self-healing engine, a novel fault self-healing framework that reconceptualizes recovery as a neuro-symbolic program synthesis task. PASE employs an LLM as a core Plan Synthesis Engine to generate structured recovery plans from a library of semantic primitives. A Neural-Symbolic World Model verifies plan feasibility through simulation, while a Meta-Prompt Optimizer, trained via DRL, learns to generate optimal prompts that guide the LLM's planning process. This tight reason-plan-verify-adapt loop enables dynamic, context-aware recovery strategy generation beyond predefined action spaces. Experiments on a real-world cloud fault injection dataset demonstrate that PASE significantly outperforms state-of-the-art methods, reducing average system recovery time by over 40% and improving fault detection accuracy in unknown fault scenarios. Our framework advances autonomous system management by unifying LLM-based reasoning with model-assisted verification and meta-learned guidance.

Vision-Language-Action (VLA) models have demonstrated promising generalization capabilities across robotic manipulation tasks, yet their real-world deployment remains limited by the lack of effective safety measures. Specifically, existing safety measures only prevent collisions caused by the robot's next action. In this paper, we propose a neuro-symbolic safety guidance mechanism for flow matching based VLAs that enables predictive collision avoidance. Flow matching based VLAs determine the next actions by predicting a trajectory (a sequence of actions) through an iterative neural flow matching process. Our method formulates safety enforcement as a minimum-norm constrained optimization problem that corrects safety violations during the denoising process of noisy intermediate trajectory predictions. By analyzing predicted trajectories and applying corrections during iterative denoising, our approach anticipates collisions before they become unavoidable. This interleaving of symbolic constraint satisfaction with neural trajectory generation enables predictive collision avoidance rather than reactive intervention. On the SafeLIBERO benchmark, our method achieves 82.8% collision avoidance and 81.6% task success, a 6.3% and 19.8% improvement respectively over single-step methods, with the largest gains on long-horizon tasks where compounding distribution shift is most pronounced. Video demonstrations of our approach are included on our project page at https://willenglish.tech/SafetyGuidedFlowMatching/.

Simulating two-dimensional frustrated quantum matter is a grand challenge due to the sign problem and exponential Hilbert space complexity. In this work, we introduce the Holographic Quantum Transformer (HQT), a physics-inspired generative architecture that leverages global self-attention to resolve non-local entanglement patterns. We validate HQT on the square lattice $J_1-J_2$ Heisenberg model. On the heavily frustrated $8 \times 8$ lattice at the quantum critical point ($J_2=0.5$), HQT reaches a ground-state energy per site ($E/N$) of $\mathbf{-0.5001(1)}$, consistent with the expected finite-size scaling trend. Beyond numerical accuracy, HQT exhibits intrinsic physical awareness, autonomously recovering the underlying $J_2$ interaction geometry through interpretable attention maps. Our central contribution is ``Holographic Transfer", a zero-shot size-extrapolation protocol with rapid alignment: a model trained on $8 \times 8$ systems is directly projected onto larger $10 \times 10$ lattices via continuous positional-embedding interpolation and head re-initialization, achieving high-fidelity initialization and rapid convergence. This zero-shot protocol yields an energy of $E/N = \mathbf{-0.49782(3)}$, statistically consistent with the variational state of the art while requiring no from-scratch training on the target lattice. Our results establish generative attention as a scalable paradigm for transferable quantum simulation.

Policy-grounded document review requires determining whether a target document complies with organization-specific policies, guidelines, or playbooks. While large language models can assist with policy interpretation and document analysis, end-to-end prompting leaves the applied policy logic implicit, making compliance decisions difficult to inspect, update, and test. We present PolicyGuard, a neuro-symbolic framework for policy-grounded document compliance review. PolicyGuard converts organizational policy guidance into an executable review engine consisting of typed relational logic rules and atom-level extraction questions. During review, LLMs answer these local questions using retrieved document evidence, and a symbolic evaluator applies the formal rules to detect non-compliance. We instantiate and evaluate PolicyGuard on company-specific NDA compliance review, where contract clauses must be checked against organization-specific negotiation policies. By separating policy formalization, local document interpretation, and symbolic compliance evaluation, PolicyGuard makes document review more explicit, maintainable, and systematically testable.

We present Sequential Planning via Anchored Robotic Keypoints, SPARK, a training-free neurosymbolic manipulation system that reaches 43.7% on six LIBERO-PRO position & task cells, more than doubling CaP-Agent0 and Vision-Language-Action (VLA) baselines. CaP-Agent0, a multi-turn code-generation agent, achieves 18.2% by re-querying an LLM at every turn, but its restart-from-scratch solution proves costly against minor policy failures. Perception is the layer that fails most under position and task changes so SPARK spends its computation there. A single Gemini call composes the plan as a typed behavior tree (BT) of composable primitives, each already containing the low-level control (motion, grasping, depth geometry) a code-generation agent would otherwise regenerate on every trial. The rest of the budget goes to perception: a second Gemini call proposes three alternative text prompts per object, SAM3 evaluates each, and we keep the prompt$\to$label pair with the most confident detection and a recovery loop then retries a failed primitive against freshly detected objects, with no new LLM call. The alternative prompts add +27.7 points on the spatial suite and +10.0 on the object suite, with the recovery loop adding +5.0 overall. SPARK runs the same primitives on three robot families (UR10e, Franka FR3, bimanual Franka) across nine unique tasks at twenty trials each, averaging 68%. Since the detector, planner, and controller modules sit behind the typed plan, they swap independently without training, and each primitive's checkable post-condition traces a failure to the corresponding module or a kinematic limit. Every trial logs a verified, labeled trajectory, so a training-free planner that already beats VLAs can supply the data those policies need without teleoperation. Project page: https://cwru-aism.github.io/spark-page/

Most of the existing neuro-symbolic AI methods focus on the scenario of static knowledge where objects do not change according to a temporal dimension. Temporal neuro-symbolic works are still under explored and are mainly developed for time-interval logic or propositional linear temporal logic. There is a lack of models studying linear temporal logics with predicates that deal with objects whose properties and relations change through the time. We present First-Order Temporal Logic Tensor Networks (FOT-LTN) that is an extension of Logic Tensor Networks (LTN) that fills this gap by considering a linear-temporal dimension. In particular, FOT-LTN joins the syntax of First-Order Linear Temporal Logic with the fuzzy (and real-valued) semantics of LTN obtaining a framework that supports both temporal operators and quantifiers and is totally differentiable. A first evaluation regards a temporal knowledge graph completion task on two synthetic datasets showing better performance of FOT-LTN with respect to dedicated (purely neural) methods.

This project introduces the CRISTAL Method (Coherent Reliable Intentional Synthesis of Truthful Analysis Logic), a neurosymbolic framework for automating complex analysis workflows, with fundamental investment analysis as a primary use case. This domain poses major challenges: high structural uncertainty, noisy and subjective data, tight attention budgets, and the need for justified, reproducible decisions. Human analysts often struggle in this domain due to cognitive biases and limitations, suggesting significant value in automation. But while LLM-based agents have been proposed as analytical aids, their limitations -- poor numerical reasoning, unawareness of uncertainty, and lack of reproducibility -- hinder their effectiveness in this context. CRISTAL addresses these gaps through a principled blend of statistical model synthesis, continuous learning, and active learning. Starting from a natural-language prior knowledge curriculum, CRISTAL builds a dynamic, interpretable probabilistic program that enables full Bayesian inference, including uncertainty quantification and budget-aware data acquisition. CRISTAL continually refines its world model during analysis, leveraging LLMs for code synthesis and learning. We validate CRISTAL on a novel benchmark of synthetic equities with rich financial and textual data. On a company classification task, CRISTAL achieves Bayes-optimal accuracy with just 5 examples and a 5-second budget, outperforming state-of-the-art LLMs that plateau around 40% accuracy even with order-of-magnitude more input data and compute.

Geometry Problem Solving have increasingly adopt the neuro-symbolic paradigm, combining neural intuition with symbolic rigor. However, current frameworks suffer from severe bottlenecks in two core stages: autoformalization, which treats multimodal translation as a static task decoupled from downstream solver compatibility, and theorem prediction, where solvers frequently hit a deductive impasse due to fixed rule libraries. To address these, we propose SD-GPS, a solver-driven framework that treats the symbolic solver as an execution oracle throughout both formalization and deduction. First, Solver-Driven Autoformalization unifies supervised formal-language adaptation and solvability-guided reinforcement learning into a single module built on QwenVL3-2B, making executability the central training signal. Second, Verified Theorem Proposing introduces an impasse-aware agent that proposes local auxiliary lemmas from current proof states, ensuring soundness by filtering all proposals through symbolic verification. Empirical evaluations on Geometry3K and PGPS9K demonstrate that SD-GPS consistently outperforms existing MLLM, neural, and neuro-symbolic methods across standard completion, multiple-choice, and cross-modal reference regimes, proving that closing the loop between multimodal perception and symbolic execution significantly improves geometric reasoning, offering profound insights into how neural agents can be grounded by formal systems to achieve verifiable problem-solving capabilities.

Sphere neural networks have achieved symbolic level syllogistic reasoning without training data, raising the question of where the limit of the scaling law for logical reasoning lies, i.e., whether data-driven machine learning systems can achieve the same level by increasing training data and training time. We show two methodological limitations that prevent supervised deep learning from reaching the symbolic-level syllogistic reasoning: (1) training data can not distinguish all 24 types of valid syllogistic reasoning; (2) end-to-end mapping from premises to conclusion introduces contradictory training targets between neural components for pattern recognition and logical reasoning. Beside theoretical analysis, we experimentally illustrate that Euler Net cannot achieve rigorous syllogistic reasoning. We further challenge the most recent ChatGPTs (GPT-5-nano and GPT-5) to determine the satisfiability of syllogistic statements in four surface forms (patterns): words, double words, simple symbols, and long random symbols, showing that surface forms affect the reasoning performance and that ChatGPT GPT-5 may reach 100% accuracy but still provide incorrect explanations. As empirical training processes are stopped after achieving 100% accuracy, we conclude that supervised machine learning systems will not attain the rigour of symbolic logical reasoning.

While WGS-based AMR prediction has reached high accuracy, existing models lack a mechanism to ground neural attributions in established biological pathways. We present KG-TRACE, a novel neuro-symbolic framework that integrates the WHO mutation knowledge graph (KG) as a structured biological constraint on a neural genomic model. Unlike existing methods that learn statistical patterns in isolation, KG-TRACE fuses genomic features and RotatE-based KG embeddings through a learned epistemic trust gate, dynamically weighting neural evidence against symbolic biological knowledge. Evaluated on the CRyPTIC M. tuberculosis cohort, KG-TRACE achieves an AUROC of 0.9760 for isoniazid, achieving competitive accuracy while its primary value lies in symbolic grounding, not predictive uplift. More importantly, we introduce the Biological Grounding Ratio (BGR), a dataset-level metric that quantifies alignment between neural attributions and established biology. Our framework achieves a 92.5% symbolic coverage of isoniazid-resistant predictions and effectively identifies MDR co-occurrence artifacts by issuing laboratory follow-up flags for 'UNCERTAIN' cases. We demonstrate that neuro-symbolic grounding provides a verifiable audit trail for clinicians, bridging the gap between predictive accuracy and clinical trust.

Evaluating LLM agents requires dynamic environments that go beyond static reasoning and zero-sum games. Real-world economic interaction is often open-ended and mixed-motive: agents must negotiate, create positive-sum surplus, compete for scarce assets, and plan under delayed returns. We introduce SidConArena, a new benchmark framework for evaluating LLM agents in open-ended, positive-sum bargaining. SidConArena formalizes a multi-player economy as a finite-horizon partially observable stochastic game with three coupled phases: natural-language negotiation with binding trades, deterministic converter-based production, and sealed-bid auctions for long-term assets. The framework combines structured observations, phase-aware agent dispatching, a neural-symbolic action interface, and asynchronous execution, enabling free-form interaction while preserving rule-grounded evaluation. Across homogeneous and heterogeneous tournaments, stronger frontier models achieve higher economic outcomes, yet agents still misvalue resources, bargain passively, and remain limited in long-horizon investment planning.

Driving VLA models incorporating Chain-of-Thought (CoT) reasoning are attractive because they leverage pretrained VLM representations and expose intermediate decisions in natural language, yet current rationales often lack the step-by-step decision semantics needed to keep the rationale causally connected to the planned motion. We introduce Neuro-Symbolic Drive, a neuro-symbolic driving framework that supervises a driving VLA with rule-grounded reasoning traces extracted directly from classical rule-based planners. Our key observation is that rule-based planners are symbolic AI systems that already function as executable reasoning engines: they reason about active safety constraints, search over candidate maneuvers, and select a final trajectory. We instrument these planners in simulation to capture both the executed trajectory and the internal decision trace at each rule-evaluation step. Each trace is serialized into structured rule-grounded reasoning and paired with the trajectory to fine-tune Qwen3.5-4B as a driving VLA. Because these traces are derived directly from the planner states that determine the action, they ensure reasoning is structurally coupled to motion generation by construction, rather than by post-hoc alignment. On our simulator-generated benchmark, detailed rule-grounded reasoning reduces ADE@3s from 0.47 to 0.26 and miss rate from 8.30% to 6.40% under three-camera perception, and from 0.54 to 0.26 and 10.13% to 5.99% under eight-camera perception. Neuro-Symbolic Drive thus converts neuro-symbolic planning logic into structured supervision. Code base: https://github.com/XiangboGaoBarry/Neural-Symbolic-Drive.

Decision-making in real-world settings rarely follows a fixed script. Instead, it unfolds as a dynamic reasoning process in which the appropriate course of action evolves as new context and data become available. Traditional Business Process Management systems provide rigor, determinism, and auditability, yet they generally struggle to adapt their execution at runtime. Conversely, agentic systems based on Large Language Models (LLMs) bring flexibility to decision-making, but they are inherently opaque, often unreliable, and suffer from significant scalability constraints when operating over large datasets. To combine these complementary paradigms, we introduce VADAOrchestra, a neurosymbolic framework that models complex workflows as evolving reasoning processes. The framework adopts a hybrid approach: given a user query and a collection of data sources, an LLM-based orchestrator incrementally plans and adapts the workflow. This is encoded as a logic program in a fragment of Datalog+/- where predicates correspond to tool invocations and rules represent both predefined domain dependencies and logic constructs synthesized on demand to manipulate intermediate results. All logical inference tasks are then executed by a state-of-the-art Datalog+/- symbolic engine. This approach provides a verifiable reasoning trace, supporting the auditability and reproducibility of the entire process. Furthermore, by decoupling high-level orchestration from symbolic inference, it addresses scalability concerns, enabling complex reasoning over large datasets through targeted data querying. We evaluate VADAOrchestra on real-world financial use cases, demonstrating faithfulness, scalability, and explainability compared to standard agentic architectures.

Video-based seizure detection is essential for the management of epilepsy patients, offering a non-invasive complement to electroencephalography. While several deep learning approaches have been developed for video-based seizure detection, none are inherently interpretable, limiting their adoption and translation into clinical practice. We present, to our knowledge, the first exploration of a neurosymbolic framework for video-based seizure detection that directly addresses this gap. Our approach (1) extracts patient-centric skeleton sequences from epilepsy monitoring units via a prompt-guided foundation model, (2) predicts binary spatio-temporal concept activations grounded in clinical motor semiology guidelines, and (3) composes them via differentiable logic into interpretable Boolean rules with auditable contributions. Furthermore, to mitigate false positives arising from the traditional binary formulation (seizure vs.\ non-seizure), we sub-classify non-seizure segments into clinically relevant normal activities, providing the model with fine-grained discriminative supervision. Evaluated on two public seizure video benchmarks, our framework achieves 89.78% sensitivity with 0.06 false detections per hour on SAHZU and 85.27%,0.09 on IEEE, while producing complete three-level interpretability: every prediction decomposes into which motor primitives were detected, how they were logically composed, and how much each rule contributed to the clinical decision. We publicly release all annotations, extracted pose sequences, our data pipeline and code, https://github.com/Mr-TalhaIlyas/CDSD/.

Large Language Models (LLMs) offer a promising path to automate Clinical Trial Matching (CTM), but still struggle with the deterministic verification required for complex eligibility criteria. Conversely, purely symbolic methods provide formal rigour but break down when faced with incomplete patient records and noisy clinical evidence. To bridge this gap, we investigate a hybrid framework for CTM combining LLMs with logical verification. In particular, we introduce an abductive neurosymbolic CTM framework (αNeSy-CTM), which leverages the linguistic and world knowledge in LLMs to support reasoning over noisy and underspecified clinical text. Extensive evaluation demonstrates that αNeSy-CTM substantially outperforms standalone LLM baselines, achieving up to 30% relative improvement over zero-shot baselines. In addition, our analyses confirm the impact of abductive reasoning on CTM, with αNeSy-CTM exhibiting improved accuracy, specificity, and robustness over a non-abductive neurosymbolic setting. Furthermore, αNeSy-CTM and Chain-of-Thought (CoT) reasoning prove highly complementary, highlighting the potential for a hybrid routing policy. Ultimately, this paper demonstrates the impact of neurosymbolic methods for automating CTM, providing a path toward the next generation of auditable, LLM-driven clinical applications.

Machine learning models are predominantly evaluated through predictive performance metrics such as ranking quality, prediction error, or classification accuracy. While these metrics effectively quantify how closely predictions match the ground truth, they do not assess whether model outputs respect predefined logical or domain-specific constraints. In high-stakes applications, including healthcare, finance, and autonomous systems, logical consistency can be as critical as predictive accuracy, yet no standard metric captures this dimension. We introduce the Rule Violation Score (RVS), a complementary evaluation metric that quantifies the extent to which a predictive model respects a given set of logical rules, independently of predictive accuracy. RVS treats hard rules (strict constraints) and soft rules (statistical regularities) differently, can be evaluated on any dataset and on any predictive model expressed over a relational vocabulary, and can be computed using SQL queries that are automatically generated for Horn rules. Beyond evaluating models, RVS can also evaluate the logical consistency of training datasets and help identify poorly defined rules. We evaluate RVS on three benchmarks covering knowledge graph link prediction and relational regression, including rule-based, embedding-based, and neuro-symbolic predictive models. Our results demonstrate that two models achieving comparable predictive accuracy can exhibit substantially different levels of logical compliance, revealing differences in model behavior that standard metrics fail to capture.

Neurosymbolic semantics is fragmented: classical, fuzzy, probabilistic and neural systems each define truth by their own inductive rules. NeSyCat, extending ULLER, subsumes them under a single inductive definition of truth, parametric in a strong monad and an aggregation structure on truth-values. NeSyCat has so far lacked an account of predicates and functions learned by neural networks. We provide NeSyCat Torch as the missing link and interpret computational symbols via neural networks, implementing the framework in probabilistic programming and tensor-based backends. We use the distribution monad for reference semantics and metric evaluation, and complement it by a monad for numerically stable, differentiable training: the lazy log-tensor monad over the log-semiring. For efficient training in batches, we furthermore employ a batch monad. The axioms are the source code: written once in monad-based do-notation, monadic bind performs marginalisation, lazily pruning unneeded branches. On MNIST addition, our HaskTorch, JAX, and PyTorch implementations outperform LTN and DeepProbLog in speed and accuracy, while achieving nearly the accuracy of DeepStochLog. However, unlike DeepStochLog, we stay in a uniform framework that applies to many first-order NeSy approaches. Namely, the construction is parametric in the monad; instantiating it with, e.g., the Giry monad extends the approach to continuous probability (working out a neural representation here is left for future work).

Dynamical systems are fundamental to modeling the natural world, yet modeling them involves a persistent trade-off: manually prescribed mechanistic models are interpretable by design but often overly simplistic and misspecified; in contrast, flexible data-driven neural methods lack physical insight. Hybrid modeling aims for the best of both worlds by combining a prescribed or symbolic, physics-based component with a flexible neural network. A critical challenge, however, is that the neural component may relearn mechanistic parts, yielding redundant and uninterpretable models, especially when the symbolic structure itself is discovered from data. Existing methods based on standard $L^2$ regularization rely on a projection argument that breaks when the symbolic component is learned through sparse discovery, allowing the neural augmentation to overlap with symbolic structure. We introduce \textbf{OrthoReg} (Orthogonal Regularization), which directly penalizes overlap between the symbolic and neural components, preventing symbolic structure from being absorbed by the neural residual. This yields a complementary decomposition: the symbolic part captures what the library can express, and the neural part captures what remains. On benchmark dynamical systems with partial library mismatch, OrthoReg improves symbolic recovery and out-of-distribution behavior.

With sophisticated cyber-attacks becoming increasingly prevalent, modern networks require intelligent autonomous cyber-defense agents trained via Reinforcement Learning (RL). These agents employ neurosymbolic approaches such as behavior trees with learning-enabled components (LECs) to learn, reason, adapt, and implement security rules while maintaining critical operations. However, these autonomous networks are partially observable systems, i.e., the cyber-attacker's (red agent's) actions are not observable, making it difficult for the defender to predict red actions, learn red policies, or assess the attacker's intrusion levels. To address this, we propose a Policy Learning Technique using imitation learning to learn policies for partially observable RL agents with discrete states and discrete actions. We apply this technique in an autonomous cyber environment to predict red agent's actions from network observations and defender actions. Integrated with a neurosymbolic cyber-defense agent, our method effectively handles different red policies and achieves high prediction accuracy across diverse simulated scenarios.

Recent advances in Large Language Models (LLMs) and multi-agent systems have driven the rise of Agentic AI, showing promise for medical reasoning. However, open-ended conversational agents remain prone to two critical failure modes: premature diagnostic handoff and silent clinical hallucinations that may go undetected before reaching the patient. In this work, we propose a multi-agent framework that addresses both issues by replacing ``LLM-as-a-judge'' routing with deterministic orchestration constraints. The framework incorporates two safety mechanisms. First, a neuro-symbolic state-tracking gate enforces completeness of the OLDCARTS clinical protocol (Onset, Location, Duration, Character, Aggravating/Alleviating factors, Radiation, Timing, and Severity) by blocking diagnostic transitions until all required dimensions are collected. Second, an epistemic uncertainty quantification (UQ) gate computes semantic entropy (H) across K=5 independent diagnostic samples to identify and intercept divergent outputs before delivery. We evaluate the system using simulated patient agents powered by the llama-3.1-70b-instruct model on 150 test cases. The full architecture achieves 49.3% diagnostic precision, representing an absolute improvement of 11.3 percentage points over an unconstrained baseline. Additionally, we observe a statistically significant negative correlation (r = -0.181, p < 0.05) between OLDCARTS completeness (σ) and semantic entropy (H), suggesting that structured information gathering is associated with reduced diagnostic uncertainty.

Reasoning about what agents can achieve through strategic interaction is a core challenge in Multi-Agent Systems (MAS). Logics for strategic ability, such as ATL, provide rigorous methods, but their adoption is often hindered by the computational cost of strategy synthesis. We introduce a neuro-symbolic framework that integrates large language models (LLMs) into the model-checking pipeline for MAS. The LLM acts as a strategy-generation oracle, proposing candidate strategies that are then formally validated by a standard MAS model checker. This generate-and-certify architecture uses LLM guidance to navigate large combinatorial strategy spaces while preserving formal soundness: generated strategies are accepted only when certified by the verifier. We instantiate the framework for bounded strategic reasoning in NatATL and introduce the first NatATL strategy-synthesis dataset, consisting of 4211 instances. Experiments with an open-weight Qwen3-32B model show that our certified pipeline achieves 92% accuracy on strategy-synthesis outcomes.

A wide range of neurosymbolic (NeSy) systems compute one functional: a belief-weighted sum of a logical quantity over a space of $σ$-structures, of which weighted model counting, fuzzy logic, and probabilistic logic are special cases. This account is built on sets, and a set deliberately forgets two things that are important for NeSy: when two $σ$-structures are the same up to a symmetry of the theory, and how many distinct proofs witness a query. Types, in the sense of homotopy type theory, preserve this information and turn the functional into a belief-weighted homotopy cardinality, a notion of size that counts each object in inverse proportion to its symmetries. We develop the framework from scratch for NeSy systems, prove a conservativity theorem that recovers the classical functional when symmetries are trivial, and show that the symmetry our framework exposes is exactly the one behind reasoning shortcuts. The payoff is concrete: the shortcut-aware concept posterior that recent methods reach by ensembling or expressive density estimation is the only symmetry-invariant point of the confusion-set simplex, computable in closed form by averaging a single model over the symmetry group. On MNIST reasoning-shortcut benchmarks this single-model wrapper is better calibrated than a diversity-trained ensemble, while leaving label accuracy and identifiable concepts untouched. Code is freely available at https://github.com/bio-ontology-research-group/hott-nesy.

Small-scale Large Language Models (LLMs) natively default to literal semantic interpretations, making few-shot irony detection a persistent challenge in noisy, user-generated text. In this study, we show that, despite this limitation, inference from compact LLMs can still be effectively harnessed for this task. We introduce the Robust Dual-Signal (RDS) Fusion framework, a hybrid neuro-symbolic architecture that utilizes a compressed Chain-of-Thought (CoT) of an LLM, alongside a static, pretrained RoBERTa and a symbolic prior module in two downstream fusions. RDS Fusion demonstrates better performance than an unrestricted reasoning ablation, while requiring substantially lower inference time. We also find that in our experiments, unrestricted reasoning does not yield better results than restricted reasoning. Evaluated on a strictly held-out TweetEval test set ($N=734$), RDS Fusion achieves $78.1\%$ accuracy and a Macro F1 of $0.777$, which is comparable to a finetuned BERTweet-base. On the heavily imbalanced iSarcasm dataset, the frozen CoT pipeline suppresses $22.5\%$ of baseline false positives, yielding a Macro F1 of $0.6726$ and Ironic F1 of $0.4821$, outperforming multiple heavily supervised SemEval transformer ensembles. Paired McNemar's tests show that, while adding the symbolic prior to the neural baseline yields an insignificant gain, and the RDS Fusion is statistically insignificant compared to the combined RoBERTa and symbolic prior ablation; the concurrent fusion achieves a statistically significant improvement over the standalone baseline ($p=0.005$).

LLM-based agents have demonstrated strong capabilities in data-intensive analytical tasks, yet their outputs are rarely verifiable: a reliance on linear text trajectories makes their reasoning difficult to audit. In particular, deterministic computations over raw data and semantic deductions over natural-language claims are often entangled in an unstructured stream, leaving numerical conclusions hard to reproduce and qualitative judgments hard to inspect. To address this, we propose VeriGraph, a traceable neuro-symbolic reasoning framework that enables agents to construct an explicit heterogeneous evidence directed acyclic graph (DAG) during execution. VeriGraph introduces three evidence-expansion primitives, namely computational, grounding, and derivational expansion, to connect raw data, interpreter variables, computed results, and natural-language claims in a unified graph. Under this formulation, structural traceability is reduced to graph reachability from raw data sources to terminal claims, while semantic support is measured by claim-level evidence evaluation. To improve graph construction, we further design a graph-based policy optimization strategy with a composite reward that jointly supervises answer correctness, computational integrity, and derivational coherence. Experiments on four benchmarks show that VeriGraph-8B achieves the highest overall score among all baselines. More importantly, VeriGraph produces auditable evidence graphs with substantially stronger claim grounding, achieving a 87.61% Grounding Rate under our claim-level evidence support evaluation. These results suggest that explicit evidence-graph construction is a promising path toward verifiable data-analytic agents. Our code is available at https://github.com/ignorejjj/VeriGraph.

Code-as-Policies (CaP) has shown that large language models (LLMs) can write code to solve robotics tasks by composing perception, planning, and control primitives. Recent CaP systems, however, rely on multi-turn code-generation loops at test time, which is often infeasible for real-time robot control. We introduce Robotics Harness Optimization (RHO), a novel paradigm in which tool-enabled coding agents, at training time, propose and search for interpretable, neurosymbolic multi-file policy repositories (Repositories-as-Policies) that compose these primitives rather than a single prompt, function, or file. RHO searches with reflective feedback from environment reward and execution rather than teleoperation demonstrations. It generalizes to perturbed pick-and-place settings like LIBERO-PRO, where OpenVLA scores 0.0% and $π_{0.5}$ averages 12.83%. Using the same low-level primitives, RHO reaches a 45.0% success rate, 2.5x higher than the strongest multi-turn agentic system, and 3.5x higher than $π_{0.5}$. On Robosuite, RHO sets a new state-of-the-art of 70.0%, exceeding the prior multi-turn record of 68.29% using single-turn execution with no corrective LLM code edits at deployment. When an LLM is used in the control loop, as on RAI's O3DE benchmark, RHO optimizes the deployed agent's multi-file harness of prompts, tools, and control code, improving held-out success from 23.5% to 44.3% with 20% less wall-clock time and 27% fewer tool calls.

Interpretability is essential for trustworthy medical image diagnosis. However, existing concept-driven interpretable methods have key limitations: Concept Bottleneck Models (CBMs) require scoring all predefined concepts at inference time and for manual intervention, imposing a substantial burden on clinicians, while rationale-based generative approaches often select concepts by class discriminability, which can drift from diagnostic ontologies. To address these issues, we propose Neuro-Symbolic Rule Distillation (NeRD), a framework that produces efficient, ontology-grounded reasoning chains that are sufficient yet non-redundant, without manually crafting diagnostic rules. Experiments on two skin datasets demonstrate strong diagnostic performance and interpretability, and blinded expert evaluation confirms the clinical plausibility of NeRD rationales. Our method further enables a first expert-in-the-loop study for Multimodal Chain-of-Thought-based diagnosis, achieving efficient and effective concept-level intervention.

Accurate and interpretable motion prediction for heterogeneous traffic spaces, including pedestrians, bicycles, cars, and trucks, is essential for safe autonomous navigation. Nevertheless, state-of-the-art approaches remain predominantly black-box, lacking explicit encoding of the regulatory and behavioral constraints of real-world mobility. We propose Trajectory Compliance-Shaping (TraCS), a neuro-symbolic framework that augments existing black-box motion prediction backbones with interpretable and probabilistic first-order logic. To do so, TraCS employs an agentic code-generation pipeline to bridge the gap between natural-language descriptions of traffic regulations and probabilistic motion prediction. Furthermore, TraCS employs a reactive data-streaming inference engine that maintains and efficiently updates compliance landscapes as scenes evolve. To prevent TraCS from overconfidently steering the backbone's predictions in the wrong direction, we propose a neural confidence rating learned as a context-aware attenuation of the compliance signal. We demonstrate on the Argoverse 2 benchmark how TraCS consistently improves state-of-the-art prediction backbones, showing that probabilistic and symbolic compliance reasoning is a broadly applicable and computationally efficient complement to purely neural motion predictors.

Neurosymbolic (NeSy) models integrate neural networks and symbolic reasoning for robust and interpretable AI. State-of-the-art NeSy models require that the symbolic component is expressed in a differentiable way, often complicating the use of approximate inference. We propose EM-NeSy which casts probabilistic NeSy learning as an instance of the Expectation-Maximization (EM) algorithm. In the expectation step, we compute the posterior over the neurally predicted symbols conditioned on the label via probabilistic inference. In the maximization step, we update the neural parameters based on this posterior using gradient descent only through the neural component. This formulation unlocks the full potential of the EM algorithm for NeSy learning. It allows NeSy to extend naturally to approximate reasoning without any additional modifications or differentiability requirements of the symbolic component. Furthermore, it recovers the standard end-to-end gradient-based NeSy setting under exact inference. Our experimental results demonstrate the scalability and computational efficiency of EM-NeSy.

LLM-based agents are entering regulated industries where they automate judgment intensive quality management processes. We argue that symbolic structures already embedded in these domains, including regulations, typed process models, and compliance constraints, should be treated not merely as external monitoring mechanisms but as core architectural components that shape the agent's decision-making and behavior. We propose compliance-by-construction as a complementary paradigm to guardrail-based monitoring: a structural foundation that prevents control-flow violations, while guardrails remain essential for catching semantic errors. We identify a structured set of neuro-symbolic research challenges on foundational and capability level and show that addressing them jointly enables compliance-by-construction. We call on the neuro-symbolic community to engage with regulated process automation as a high impact research domain.

Computational creativity in Interactive Fiction faces a fundamental tension: Large Language Models (LLM) may produce creative narratives but struggle with world coherence, while symbolic systems ensure consistency but lack creative flexibility. We present IVIE (Incremental & Validated Interactive Experiences), a neuro-symbolic approach to generating complete and playable interactive fiction worlds from scratch. Building upon PAYADOR's neuro-symbolic framework, IVIE implements a four-stage incremental generation pipeline that delegates creative decisions--setting and character creation, puzzle design--to LLMs while grounding the world state through symbolic validation. The system generates worlds with interconnected locations, functional items, non-player characters, and coherent puzzles, all structured around a central goal-oriented architecture. Human evaluation shows the approach generates immersive, thematically coherent worlds with high player engagement. Results seem to indicate that the neuro-symbolic approach successfully balances flexibility with narrative coherence: symbolic validation grounds LLM generation without eliminating generative freedom. However, challenges remain: LLM inconsistencies occasionally bypass puzzle constraints, and objective validation gaps allow some structurally impossible goals. We identify key design considerations for future neurosymbolic interactive storytelling systems, particularly regarding LLM capabilities and their limitations.

Information lattice learning (ILL) learns interpretable rules of a signal by alternately projecting the signal onto a partition lattice that encodes a hierarchy of abstractions and lifting selected rules back to the signal domain. When the signal is a probability mass function, we show the probabilistic rules learned by ILL admit a natural probabilistic graphical model (PGM) interpretation and develop this interpretation in detail. A partition in ILL induces a deterministic quotient variable, and a rule is the marginal law of that quotient variable. A rule set is therefore a collection of marginal constraints over interpretable abstractions. General lifting is the feasible family of all joint distributions satisfying those constraints, while special lifting chooses a maximum-ignorance reconstruction, implemented in ILL by an L2 uniformity principle closely related to maximum entropy. Under a Shannon-entropy lifting, the same constraints yield a log-linear factor graph whose factors are indexed by learned abstractions. The information lattice itself, however, is not a Bayesian network: its edges encode refinement and coarsening of abstractions, not conditional dependence. Thus ILL is best viewed as structure learning for interpretable constraint-based factor graphs over quotient variables. This view clarifies how ILL relates to graphical models and maximum entropy models, while suggesting new directions for inference, identifiability, and hybrid symbolic-probabilistic learning.

Environments serve as interactive systems for large language model (LLM) based agents across diverse scenarios and play a crucial role in driving the continual evolution of model capabilities. Despite this importance, existing work lacks a systematic categorization and deep analysis. This paper systematically studies current researches on agentic environments from the perspective of the environment engineering lifecycle, covering their modeling, synthesis, evaluation and application. Specifically, the paper first introduces representative environments from the perspectives of eight attributes and eight domains, providing detailed analyses of their development paths and highlighting their core capabilities. Second, for automated environment synthesis, two paradigms are introduced, such as symbolic synthesis and neural synthesis. This paper also shows different environment evaluation methods in each paradigm. Thirdly, the corresponding environment applications from the perspective of agent-environment co-evolution are discussed. In specific, the paper characterizes the primary pathways for agent evolution in dynamic environments from four complementary perspectives: memory-centric experience evolution, orchestration-centric workflow evolution, trajectory-centric offline evolution, and exploration-centric online evolution. And three paradigms of environment evolution are identified, namely neural-driven, difficulty-driven, and scaling-driven approaches. At last, several promising future directions are discussed, including Environment-as-a-Service, Multi-agent Environments, and Neural-Symbolic Environments.

Tensor Product Representations provide the structural fidelity required for symbolic reasoning in models but suffer from exponential dimensionality growth when encoding deep recursive structures. Conversely, Vector Symbolic Architectures maintain constant dimensionality but sacrifice capacity and fidelity due to noisy compression via superposition. In this work, we propose Orthogonal Subspace Carving (OSC), a memory architecture that binds fillers to roles by projecting onto the null space of the role basis before aggregating into a fixed order-p tensor. OSC uses projections to enforce geometric orthogonality between bound structures within a static memory trace. We show that this mechanism decouples the tensor order from the structural depth, enabling deep recursive binding within a constant memory footprint. By performing retrieval via recognition, this construction allows for component vectors that are orders of magnitude smaller than the memory tensor, giving superior memory efficiency in settings involving high superposition. We also show that TPR is a special case of binding in Clifford algebra, and give a Clifford formulation of OSC.

Multi-pair robotic assembly in unstructured environments faces spatial interference and contact uncertainties. Existing paradigms fail to bridge cognitive decision-making and physical execution, as they either encounter state-space explosion and knowledge bottlenecks or suffer from logical hallucinations and topological conflicts. We propose an end-to-end neuro-symbolic framework that solves the challenge hierarchically: generating optimal subgraphs for each pair, decoupling generality from edge cases, and then resolving cross-pair interferences. Given an eye-on-hand RGB-D assembly scene, the framework extracts semantic instance identity and state while quantifying the scene for divergence calculation. For each pair, optimal subgraph is generated via LLM using barely basic actions to mitigate hallucinations. Supportive actions for edge cases are reasoned and inserted with a lightweight discriminator. Driven by the divergence between the quantified baseline and current scene, it is easily extensible at low cost. Augmented subgraphs are topologically coordinated into global sequences while preserving internal behavioral coherence. Dynamic behavior trees embedding atomic skills close the force-aware execution loop. Offline evaluation on 100 real-world scenes achieves 97.00% global executability, outperforming classical and state-of-the-art planners. Real-robot deployment on a UR3 arm attains 90% success rate with 0.5 mm tolerance under strong interference, demonstrating a unified and verifiable solution for complex autonomous assembly.

Neurosymbolic AI combines neural networks with symbolic programs to create robust and explainable predictions. One such framework is NeurASP, which trains a neural network to predict concepts and reasons over them using rules written in answer set programming (ASP) to solve downstream tasks. Crucially, labels are only provided for the downstream prediction produced by the symbolic rules, not for the latent concepts themselves. Backpropagation through the non-differentiable ASP component requires expensive probability and gradient calculations, which has hindered scalability to more sophisticated tasks. In this paper, we address the current limitations of NeurASP by improving its computational performance through vectorization, batch processing and caching of intermediate computations during training. We compare computation speeds between the original and our new implementation of NeurASP and report speedups of multiple orders of magnitude for larger tasks. To this end, we propose a new dataset of difficult tasks involving playing cards, which we use to test the capabilities of NeurASP's enhanced learning function.

Mathematical reasoning has long served as a stringent test of machine intelligence; over the past decade, it has moved from a niche problem within NLP to one of the most consequential AI frontiers. This survey provides a unified account of the field's evolution, from early rule-based math word problem (MWP) solvers and template-driven geometry systems, through neural expression generation and LLM prompting, to contemporary reasoning models, multi-agent systems, neuro-symbolic theorem provers, and verified discovery workflows. We organize the landscape along four axes: (i) informal reasoning over text and diagrams, spanning MWP solving, multimodal geometry, and VLMs; (ii) formal reasoning in proof assistants, including autoformalization, tactic prediction, compiler-guided repair, and proof search; (iii) mathematical discovery, where systems propose constructions, improve bounds, or assist attacks on open problems; and (iv) the inference and training-time techniques, including CoT prompting, tool use, process reward models, and RLVR, that increasingly connect generation with verification. We catalog major benchmarks across grade-school arithmetic, competition mathematics, geometry, formal proving, multimodal and multilingual reasoning, and expert evaluation, and we examine benchmark saturation, contamination, reporting mismatches, and the distinction between pass@1, majority voting, and verifier-assisted pass@$k$. We critically assess failure modes: brittleness under perturbation, reward hacking, multimodal grounding failures, fragile formalization, and the energy cost of reasoning-scale inference. Drawing on recent perspectives from working mathematicians, we identify future directions centered on verified-discovery workflows, reasoning efficiency, and infrastructure to make AI-assisted formalization broadly usable. Companion materials: https://github.com/Starscream-11813/awesome-AI4Math.

In this work we study offline reinforcement learning (RL) under temporally extended task constraints expressed in Linear Temporal Logic over finite traces (LTLf). Recently, transformer-based approaches such as Trajectory Transformers and Decision Transformers have been adopted to address RL as a sequence modeling problem. However, these methods optimize purely for reward and do not account for high-level temporal requirements. Here, we introduce a neurosymbolic framework that injects LTLf background knowledge into such transformer-based RL policies. Our approach compiles LTLf formulas into deterministic finite automata (DFAs) and integrates them into the learning process through a differentiable representation and a logic-based loss function. In particular, we derive differentiable satisfaction signals from DFA progression and use them as a regularization term during training. The resulting method is architecture-agnostic across different models. We evaluate the proposed framework on navigation environments with specification suites covering combinations of safety and reachability temporal properties. Experimental results show that incorporating background knowledge not only improves constraint satisfaction, but also maintains competitive return compared to vanilla baselines.

Flexible robotic automation requires systems that interpret operator intent, verify physical feasibility, and recover from execution failures across both the planning and execution stages. This paper proposes an agentic neuro-symbolic framework for human-in-the-loop industrial robotics, in which LLMs are used for tasks that require language understanding or contextual reasoning, while all verification, sequencing, and execution remain deterministic. The framework adapts the Planner-Generator-Evaluator (PGE) harness pattern from software engineering into a Specifier-Designer-Inspector (SDI) architecture for industrial robotics, combined with LangGraph-based dynamic routing for failure recovery. A two-tier recovery mechanism addresses structure-level replanning through context-aware orchestration and execution-level geometric failures through deterministic recovery skills. A Unity3D digital twin supports human inspection, modification, and re-verification prior to physical execution. Evaluated on natural-language commands across multiple difficulty levels against ten baselines, the proposed method achieves the highest task success. Ablation results confirm that structured command expansion, symbolic verification, selective LLM routing, and recovery skills are each individually necessary.

Neural symbolic regression models improve inference efficiency by shifting structural search to pretraining, but their one-pass autoregressive decoding is prone to error accumulation, which may lead to generating structurally incorrect expressions, especially in complex expression generation scenarios. Existing rectification strategies can alleviate this issue, but they often depend on restarting global search, thereby weakening the efficiency advantage of neural models, and remain susceptible to error accumulation. In this paper, we propose EditSR, a two-layer framework that combines a neural symbolic regression model in the first layer with an edit-based Rectifier in the second layer to achieve efficient prediction and post-hoc rectification. Instead of restarting the global search, we maintain rectification efficiency by pretraining the Rectifier. Specifically, we formulate the rectification process as a step-by-step state-transition chain starting from an incorrect expression, and develop a state-transition algorithm to construct supervised rectification chains for training the Rectifier. To ensure syntactic validity throughout rectification, each edit action is restricted to a syntactically valid space so that every edited expression remains parseable. In addition, because each edit decision is conditioned on the current state rather than the history, the Rectifier allows errors made in earlier steps to be rectified by subsequent edits, thereby reducing the risk of error accumulation. Extensive experiments and ablation studies show that EditSR substantially improves symbolic structure recovery with limited extra cost, with more pronounced gains on complex expressions, where one-pass autoregressive decoding is more susceptible to error accumulation.

Task planning often suffers from severe efficiency bottlenecks when robots must reason over long-horizon action sequences under complex logical constraints, including object affordances, spatial relationships, and sequential action dependencies. Recent neuro-symbolic methods improve planning efficiency by learning object-importance scores to prune task-irrelevant objects, but they typically rely on fixed offline supervision generated from full search spaces. This creates a train-test mismatch: at deployment, the planner operates in pruned search spaces induced by the model's own imperfect predictions, leading to exposure bias and degraded planning performance. To address this challenge, we formulate object-importance learning for task planning as an imperative learning-based bilevel optimization problem. The upper level optimizes a neural scorer, while the lower level solves a symbolic planning problem in the score-pruned search space. To stabilize this learning process, we introduce a 3R strategy into the lower-level planning, using parallel Repair, Restart, and Rollback recovery to provide reliable and adaptive feedback for upper-level learning. Experiments on three challenging benchmarks demonstrate state-of-the-art performance, including an 80.04% reduction in failure rate and a 57.14% reduction in planning time. We further validate the framework on a quadruped-based mobile manipulator in simulation and the real world, demonstrating its potential for efficient and deployable neuro-symbolic task planning.

While deep learning has excelled in various domains, its application to sequential decision-making in finance remains challenging due to the low Signal-to-Noise Ratio (SNR) and non-stationarity of financial data. Leveraging the reasoning capabilities of Large Language Models (LLMs), we propose \textbf{PandaAI}, a closed-loop neuro-symbolic LLM agent with market regime modeling and constrained alpha generation, which bridges general LLM reasoning with financial rigor and suppresses the financial toxicity of LLM-generated outputs. To bridge the gap between general linguistic capability and financial rigor, we fine-tune a domain-specific LLM. Furthermore, we integrate this LLM into a modular architecture and form a closed-loop system. Unlike traditional models that optimize isolated prediction metrics, \textbf{PandaAI} is designed as a neuro-symbolic agent that navigates the complex, real-world financial environment with explicit risk awareness. Extensive experiments on CSI 300 stock data show that \textbf{PandaAI} achieves a $18.2\%$ higher Rank IC and $25.7\%$ lower maximum drawdown than state-of-the-art time-series models. Our constrained LLM generation and dual-channel adaptation method provide a general paradigm for LLM deployment in high-stakes sequential decision-making scenarios.

Geometry problem solving poses distinct challenges in artificial intelligence. Existing approaches typically fall into two paradigms: symbolic methods, which exhibit limited adaptability, and neural methods, which are prone to hallucinations. Recent neuro-symbolic hybrids predominantly rely on a unidirectional pipeline where neural outputs are fed into solvers without feedback, making system brittle to early-stage errors. To break this unidirectional bottleneck, we propose BiNSGPS, a framework that establishes Bidirectional Neuro-Symbolic Interaction (BiNS) between a MLLM Adviser and a Symbolic Solver. MLLM Adviser actively incorporates feedback from the symbolic solver to dynamically rectify inconsistent formal representations or propose auxiliary hypotheses, resolving symbolic conflicts and facilitating complex deductions.

Scientific simulators are increasingly being integrated into LLM-driven systems for high-stakes simulation-driven decision-making. However, existing frameworks primarily use LLMs to generate, calibrate, or execute simulators, treating them as black-box interfaces rather than as structured mechanistic systems that can be reasoned about. As a result, current approaches lack the ability to identify, represent, and reason about the assumptions and mechanisms underlying simulator behavior, limiting transparency, auditability, and decision justification. We introduce MechSim, a mechanism-grounded neuro-symbolic reasoning framework for executable scientific simulators. Unlike prior neuro-symbolic approaches that primarily reason over static symbolic structures, MechSim enables LLM agents to reason about the mechanisms, assumptions, and execution behavior of scientific simulators. Our framework represents simulators through a shared structured schema capturing assumptions, variables, mechanism dependencies, and execution traces. On top of this representation, LLM agents operate as constrained reasoning engines that generate structured, evidence-grounded explanations linking simulator outcomes to their underlying mechanisms. We evaluate our approach across multiple high-stakes domains and show that it improves mechanism-level explanation quality, simulator analysis, and downstream decision-making reliability.

Visual Question Answering (VQA) is the task of answering questions about images, requiring the integration of multimodal input and reasoning. Modular approaches that incorporate logic-based representations into the reasoning component offer clear advantages over end-to-end trained systems, particularly in terms of interpretability. However, adapting or extending these representations when task requirements change can place a significant burden on developers. To address this challenge, we present an approach for distilling rules from Large Language Models (LLMs). Our method prompts an LLM to extend an initial VQA reasoning theory, expressed as an answer-set program, to meet new requirements of the task. Examples from VQA datasets guide the LLM, validate the results, and help correct erroneous rules by leveraging feedback from the ASP solver. We demonstrate that our approach is effective across diverse VQA datasets. Notably, only a few examples are needed to elicit correct rules from LLMs. Our experiments suggest that rule distillation from LLMs is a promising alternative to traditional data-driven rule learning approaches. Under consideration in Theory and Practice of Logic Programming (TPLP).