How Powerful are Decoder-Only Transformer Neural Models?

TL;DR

This paper proves that decoder-only transformer models used in large language models are Turing complete under certain assumptions, highlighting the importance of embedding sparsity and relating them to known computational models.

Contribution

It is the first work to establish Turing completeness of decoder-only transformers, focusing on their theoretical capabilities and the role of embedding sparsity.

Findings

Transformers are Turing complete under reasonable assumptions.

Sparsity of word embeddings is crucial for Turing completeness.

Transformers relate to Hao Wang's B machines.

Abstract

In this article we prove that the general transformer neural model undergirding modern large language models (LLMs) is Turing complete under reasonable assumptions. This is the first work to directly address the Turing completeness of the underlying technology employed in GPT-x as past work has focused on the more expressive, full auto-encoder transformer architecture. From this theoretical analysis, we show that the sparsity/compressibility of the word embedding is an important consideration for Turing completeness to hold. We also show that Transformers are are a variant of B machines studied by Hao Wang.