# A homotopy-type-theoretic generalization of neurosymbolic inference

arXiv: https://arxiv.org/abs/2606.17851 · HTML: https://mlsanity.com/paper/2606.17851/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 25 references that are in the map.
- First posted on arXiv 16 Jun 2026.

## Builds on (most cited first, 13 of 13)

- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017)
- [Logic Tensor Networks](https://mlsanity.com/paper/2012.13635/) (2020)
- [Not All Neuro-Symbolic Concepts Are Created Equal: Analysis and Mitigation of Reasoning Shortcuts](https://mlsanity.com/paper/2305.19951/) (2023)
- [BEARS Make Neuro-Symbolic Models Aware of their Reasoning Shortcuts](https://mlsanity.com/paper/2402.12240/) (2024)
- [Position: Categorical Deep Learning is an Algebraic Theory of All Architectures](https://mlsanity.com/paper/2402.15332/) (2024)
- [Tractability through Exchangeability: A New Perspective on Efficient Probabilistic Inference](https://mlsanity.com/paper/1401.1247/) (2014)
- [A Neuro-Symbolic Benchmark Suite for Concept Quality and Reasoning Shortcuts](https://mlsanity.com/paper/2406.10368/) (2024)
- [On the Independence Assumption in Neurosymbolic Learning](https://mlsanity.com/paper/2404.08458/) (2024)
- [Shortcuts and Identifiability in Concept-based Models from a Neuro-Symbolic Lens](https://mlsanity.com/paper/2502.11245/) (2025)
- [Defining neurosymbolic AI](https://mlsanity.com/paper/2507.11127/) (2025)
- [Homotopy Theoretic and Categorical Models of Neural Information Networks](https://mlsanity.com/paper/2006.15136/) (2020)
- [Neurosymbolic Reasoning Shortcuts under the Independence Assumption](https://mlsanity.com/paper/2507.11357/) (2025)

## Abstract (arXiv)

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.