Neuro-Symbolic Concept-based Models (NeSy-CBMs) are a family of architectures that integrate neural networks with symbolic reasoning for enhanced reliability in high-stakes applications. They work by first extracting high-level concepts from the input and then inferring a task label from these compatibly with given logical constraints. Yet, their label and concept predictions can be overconfident, making it difficult for stakeholders to gauge when the model's decisions can be trusted. We address this issue by integrating ideas from Conformal Prediction (CP), a framework providing rigorous, distribution-free coverage guarantees. We formalize three desiderata -- consistency, coverage, and conciseness -- that any conformal method for NeSy-CBMs should satisfy, and show that existing approaches fall short of at least one. We then introduce COCOCO, a post-hoc framework that conformalizes concepts and labels jointly and reconciles them via a single deduction-abduction revision step. COCOCO satisfies all three desiderata, retains distribution-free coverage, is robust to imperfect knowledge and supports user-specified size budgets. Our experiments on 8 data sets highlight how COCOCO compares favorably against competitors and natural baselines in terms of performance and set size.
Concept-based (CB) models provide interpretability and support test-time human intervention, while standard neural networks (NN) offer strong task performance but little transparency. Prior work has explored hybrid formulations that integrate concepts and additional representations to improve accuracy, often at the cost of human interventions. We introduce the \emph{Synergy Concept-Based Model (SynCB)} framework, that combines a CB branch with a complementary neural branch, and a trainable routing module that dynamically selects which branch to use for each input. Unlike prior models, which fuse residual and concept-based predictions, SynCB keeps the two branches distinct and coordinates them through the routing module. Moreover, both branches are learned jointly, allowing information sharing between the complementary neural branch and CB branches through their common backbone. To improve responsiveness to interventions, we further introduce a test-time intervention policy and a corresponding loss. Across five datasets and CB benchmarks, SynCB consistently achieves higher task accuracy while remaining more responsive to human interventions, surpassing the full neural baseline by up to 3.9 percentage points and exceeding the strongest competitor in intervention performance by up to 6.43 percentage points.
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.
Learning user-defined concepts as constraint networks has been extensively studied in the constraint acquisition (CA) literature. However, existing approaches typically rely on intensive interactions with a human oracle, making the learning process costly in terms of time and number of queries. In this paper, we propose a neuro-symbolic framework for automatic CA that significantly reduces user involvement by introducing neural Oracle Transformer models which learn to emulate user responses and to generalize conceptual knowledge. Trained on previously available examples, the learned oracle interacts with a dedicated CA engine, FastCA, which systematically refines the oracle's responses into a sound, consistent, and interpretable constraint network. This neuro-symbolic interaction enables the recovery of structured symbolic models from data without prior domain knowledge. Our results demonstrate that this neuro-symbolic interplay effectively aligns data-driven pattern recognition with symbolic reasoning, offering a robust approach to automating model construction in combinatorial domains.