# Categorical Construction of Logically Verifiable Neural Architectures

arXiv: https://arxiv.org/abs/2508.11647 · HTML: https://mlsanity.com/paper/2508.11647/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 9 of its 26 references that are in the map.
- First posted on arXiv 2 Aug 2025.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 9 of 9)

- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Concrete Problems in AI Safety](https://mlsanity.com/paper/1606.06565/) (2016)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [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)
- [The Next Decade in AI: Four Steps Towards Robust Artificial Intelligence](https://mlsanity.com/paper/2002.06177/) (2020)
- [Categorical Foundations of Gradient-Based Learning](https://mlsanity.com/paper/2103.01931/) (2021)

## Abstract (arXiv)

Neural networks excel at pattern recognition but struggle with reliable logical reasoning, often violating basic logical principles during inference. We address this limitation by developing a categorical framework that systematically constructs neural architectures with provable logical guarantees. Our approach treats logical theories as algebraic structures called Lawvere theories, which we transform into neural networks using categorical algebra in the 2-category of parametric maps. Unlike existing methods that impose logical constraints during training, our categorical construction embeds logical principles directly into the network's architectural structure, making logical violations mathematically impossible. We demonstrate this framework by constructing differentiable neural architectures for propositional logic that preserve boolean reasoning while remaining trainable via gradient descent. Our main theoretical result establishes a bijective correspondence between finitary logical theories and neural architectures, proving that every logically constrained network arises uniquely from our construction. This extends Categorical Deep Learning beyond geometric symmetries to semantic constraints, enabling automatic derivation of verified architectures from logical specifications. The framework provides mathematical foundations for trustworthy AI systems, with applications to theorem proving, formal verification, and safety-critical reasoning tasks requiring verifiable logical behavior.