Categorical Construction of Logically Verifiable Neural Architectures

TL;DR

This paper introduces a categorical framework that constructs neural networks with built-in logical guarantees, enabling verifiable reasoning by embedding logical principles directly into network architecture.

Contribution

It develops a novel categorical method to systematically create neural architectures from logical theories, ensuring logical correctness by design.

Findings

Establishes a bijective correspondence between logical theories and neural architectures.

Demonstrates differentiable architectures that preserve boolean logic.

Provides a foundation for trustworthy AI with verifiable logical behavior.

Abstract

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.