A homotopy-type-theoretic generalization of neurosymbolic inference

2026-06-16

AI Summary

This work addresses the limitations of traditional neuro-symbolic systems, which rely on set theory and struggle to capture symmetries between σ-structures and the diversity of reasoning paths, thereby constraining expressiveness and efficiency. The paper introduces homotopy type theory into neuro-symbolic reasoning for the first time, replacing sets with types and employing a belief-weighted homotopy cardinality functional to naturally model symmetries while preserving proof-path information. By integrating group averaging over symmetries, the proposed approach significantly outperforms diverse ensemble training on the MNIST reasoning shortcut benchmark. It achieves superior calibration without compromising label accuracy or concept identifiability, reveals the mathematical essence of reasoning shortcuts, and yields symmetry-invariant concept posteriors.

Abstract

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. Replacing the underlying sets by types, in the sense of homotopy type theory, preserves this information, and turns this 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.

Problem

• neurosymbolic inference
• symmetry
• homotopy type theory
• reasoning shortcuts
• belief-weighted sum

Innovation

• homotopy type theory
• neurosymbolic inference
• symmetry-aware reasoning
• homotopy cardinality
• reasoning shortcuts