Neurosymbolic Learning

Integrating domain knowledge into deep neural networks is a promising way to improve generalization. Existing methods either encode prior knowledge in the loss function or apply post-processing modules, but both depend on identifying useful symbolic knowledge to integrate. Since such rules are often unavailable in real-world vision tasks, we propose a method for targeted knowledge discovery. We propose a Differentiable Knowledge Unit (DKU) that enables modulating the classifier logits, yielding refined class probabilities. The DKU uses implication rules to represent relationships between task classes and implicit concepts learned entirely from the main task supervision, without requiring concept labels. Concepts are identified by dedicated classifiers, whose probabilities are passed to DKU alongside the primary class probabilities. DKU computes a logic-based adjustment vector via fuzzy inference, which modulates the primary class logits to yield refined class probabilities. When concept classifiers represent concepts that do not support the logical rule structure, the resulting adjustments to the class probabilities do not directly minimize the supervision loss. Consequently, optimizing the supervision loss on these adjusted class probabilities implicitly trains the concept classifiers. We construct the rule base so that bidirectional logical relations connect concepts and classes. We enforce the concepts to be distinct from each other and with respect to the classes. This design enforces a clean supervision signal for concept learning. We evaluate our methods on the PASCAL-VOC, COCO, and MedMNIST datasets. We demonstrate improvement through our knowledge integration across these datasets. We conduct domain generalization and hard-sample ablation studies and find that our implicit knowledge discovery and integration outperforms the baseline.

Neurosymbolic systems can satisfy logical constraints during learning without achieving the intended concept-label correspondence; this is a problem known as reasoning shortcuts. We formalize reasoning shortcuts as a constraint satisfaction problem and investigate under which conditions concept mappings are uniquely determined by the constraints. We prove that a discrimination property (requiring that no valid concept mapping can be transformed into another valid mapping by swapping two concept values) is necessary for shortcut-freeness under bijective mappings, but demonstrate via a counterexample that it is insufficient even when the constraint graph is connected. We develop an ASP-based algorithm that verifies whether a given constraint set uniquely determines the intended concept mapping, with proven soundness and completeness. When shortcuts are detected, a greedy repair algorithm eliminates them by augmenting the constraint set, converging in at most $k$ iterations, where $k$ is the number of alternative valid mappings. We further provide a complexity classification: deciding shortcut-freeness is coNP-complete, counting shortcuts is #P-complete, and finding minimal repairs is NP-hard. We also establish sample complexity bounds showing that logarithmically many label queries suffice for disambiguation in favorable cases, while querying all ambiguous positions suffices in the worst case. Experiments across eight benchmark domains validate our approach.

Understanding natural and engineered systems often relies on symbolic formulations, such as differential equations, which provide interpretability and transferability beyond black-box models. We introduce Latent Grammar Flow (LGF), a neuro-symbolic generative framework for discovering ordinary differential equations from data. LGF embeds equations as grammar-based representations into a discrete latent space and forces semantically similar equations to be positioned closer together with a behavioural loss. Then, a discrete flow model guides the sampling process to recursively generate candidate equations that best fit the observed data. Domain knowledge and constraints, such as stability, can be either embedded into the rules or used as conditional predictors.

Recent works use a neuro-symbolic framework for general manipulation policies. The advantage of this framework is that -- by applying off-the-shelf vision and language models -- the robot can break complex tasks down into semantic subtasks. However, the fundamental bottleneck is that the robot needs skills to ground these subtasks into embodied motions. Skills can take many forms (e.g., trajectory snippets, motion primitives, coded functions), but regardless of their form skills act as a constraint. The high-level policy can only ground its language reasoning through the available skills; if the robot cannot generate the right skill for the current task, its policy will fail. We propose to address this limitation -- and dynamically expand the robot's skills -- by leveraging user feedback. When a robot fails, humans can intuitively explain what went wrong (e.g., ``no, go higher''). While a simple approach is to recall this exact text the next time the robot faces a similar situation, we hypothesize that by collecting, clustering, and re-phrasing natural language corrections across multiple users and tasks, we can synthesize more general text guidance and coded skill templates. Applying this hypothesis we develop Memory Enhanced Manipulation (MEMO). MEMO builds and maintains a retrieval-augmented skillbook gathered from human feedback and task successes. At run time, MEMO retrieves relevant text and code from this skillbook, enabling the robot's policy to generate new skills while reasoning over multi-task human feedback. Our experiments demonstrate that using MEMO to aggregate local feedback into general skill templates enables generalization to novel tasks where existing baselines fall short. See supplemental material here: https://collab.me.vt.edu/memo

Adaptive Neuro-Fuzzy Inference System (ANFIS) was designed to combine the learning capabilities of neural network with the reasoning transparency of fuzzy logic. However, conventional ANFIS architectures suffer from structural complexity, where the product-based inference mechanism causes an exponential explosion of rules in high-dimensional spaces. We herein propose the Kolmogorov-Arnold Neuro-Fuzzy Inference System (KANFIS), a compact neuro-symbolic architecture that unifies fuzzy reasoning with additive function decomposition. KANFIS employs an additive aggregation mechanism, under which both model parameters and rule complexity scale linearly with input dimensionality rather than exponentially. Furthermore, KANFIS is compatible with both Type-1 (T1) and Interval Type-2 (IT2) fuzzy logic systems, enabling explicit modeling of uncertainty and ambiguity in fuzzy representations. By using sparse masking mechanisms, KANFIS generates compact and structured rule sets, resulting in an intrinsically interpretable model with clear rule semantics and transparent inference processes. Empirical results demonstrate that KANFIS achieves competitive performance against representative neural and neuro-fuzzy baselines.

Traditional neural networks have an impressive classification performance, but what they learn cannot be inspected, verified or extracted. Neural Logic Networks on the other hand have an interpretable structure that enables them to learn a logical mechanism relating the inputs and outputs with AND and OR operations. We generalize these networks with NOT operations and biases that take into account unobserved data and develop a rigorous logical and probabilistic modeling in terms of concept combinations to motivate their use. We also propose a novel factorized IF-THEN rule structure for the model as well as a modified learning algorithm. Our method improves the state-of-the-art in Boolean networks discovery and is able to learn relevant, interpretable rules in tabular classification, notably on examples from the medical and industrial fields where interpretability has tangible value.

Vision-Language-Action (VLA) models are formulated to ground instructions in visual context and generate action sequences for robotic manipulation. Despite recent progress, VLA models still face structure-blind backbones, backbone-bound generalization, and flat single-objective optimization. To address these challenges, we propose a novel Neuro-Symbolic Vision-Language-Action (NS-VLA) framework. It introduces a Neuro-Symbolic Encoder for plan-constrained primitive inference, a Neuro-Symbolic Solver that conditions a backbone-agnostic policy on the active primitive, and Hierarchical Joint Policy Optimization with reward-granularity matching. Experiments on robotic manipulation benchmarks demonstrate that NS-VLA outperforms previous methods in both one-shot training and data-perturbed settings, while simultaneously exhibiting superior zero-shot generalizability and expanded exploration space. Our code is publicly available.