The Topos of Transformer Networks

TL;DR

This paper offers a theoretical analysis of transformer networks using topos theory, revealing their higher-order reasoning capabilities and embedding in a logical framework, contrasting them with other neural architectures.

Contribution

It introduces a topos-theoretic perspective on transformers, showing they operate in a higher-order logic fragment unlike other neural networks.

Findings

Transformers reside in their topos completion, indicating higher-order reasoning.

Other architectures embed in a pretopos of piecewise-linear functions, representing first-order logic.

The analysis connects neural network architecture with logical fragments and cybernetic frameworks.

Abstract

The transformer neural network has significantly out-shined all other neural network architectures as the engine behind large language models. We provide a theoretical analysis of the expressivity of the transformer architecture through the lens of topos theory. From this viewpoint, we show that many common neural network architectures, such as the convolutional, recurrent and graph convolutional networks, can be embedded in a pretopos of piecewise-linear functions, but that the transformer necessarily lives in its topos completion. In particular, this suggests that the two network families instantiate different fragments of logic: the former are first order, whereas transformers are higher-order reasoners. Furthermore, we draw parallels with architecture search and gradient descent, integrating our analysis in the framework of cybernetic agents.