A PyTorch Library of Turing-Complete Neural Networks

TL;DR

This paper introduces a PyTorch library that constructs neural networks directly from Turing machine descriptions, enabling exact simulation without training and serving as a foundation for symbolic-neural research.

Contribution

The authors develop a PyTorch package that compiles Turing machine descriptions into neural network models, demonstrating how to implement Boolean logic and tape lookup within neural architectures.

Findings

Implemented transformer and recurrent architectures based on theoretical results.

Showed how ReLU networks can realize Boolean circuits and logic gates.

Provided a runnable reference for symbolic-neural integration.

Abstract

We present a PyTorch package that compiles neural networks and their weights from Turing machine descriptions, producing models that exactly simulate the specified machine without any training. Given a transition function and a set of terminal states, the package constructs a model whose forward pass corresponds to one step of the Turing machine. Two architectures are implemented, each realizing a different theoretical result: (1) a transformer with self-attention, cross-attention, and feedforward layers based on Wei, Chen, and Ma (2021), and (2) a recurrent network based on Siegelmann and Sontag (1995) that encodes the stack in a Cantor set. We develop the constructions from first principles, showing how ReLU networks implement Boolean circuits (AND, OR, NOT, XOR gates and their composition into DNF formulas and binary adders) and how hard attention implements positional lookup on the tape. The package serves as a concrete, runnable reference for the symbolic-neural bridge, and as a foundation for future work on the stability of constructed solutions under gradient-based optimization. Code is available at https://github.com/jonrbates/turing.