A provably stable neural network Turing Machine

TL;DR

This paper introduces a provably stable neural network architecture that can simulate Turing machines and pushdown automata, providing theoretical bounds on the computational power of bounded precision neural networks with memory.

Contribution

It presents a neural stack, neural pushdown automaton, and neural Turing machine with proven stability and the ability to simulate classical automata and Turing machines, including a universal Turing machine with minimal precision.

Findings

Neural stack architecture is stable and closely approximates discrete stacks.

Neural pushdown automaton can simulate any PDA.

Neural Turing machine can simulate any Turing machine with only seven neurons.

Abstract

We introduce a neural stack architecture, including a differentiable parametrized stack operator that approximates stack push and pop operations for suitable choices of parameters that explicitly represents a stack. We prove the stability of this stack architecture: after arbitrarily many stack operations, the state of the neural stack still closely resembles the state of the discrete stack. Using the neural stack with a recurrent neural network, we introduce a neural network Pushdown Automaton (nnPDA) and prove that nnPDA with finite/bounded neurons and time can simulate any PDA. Furthermore, we extend our construction and propose new architecture neural state Turing Machine (nnTM). We prove that differentiable nnTM with bounded neurons can simulate Turing Machine (TM) in real-time. Just like the neural stack, these architectures are also stable. Finally, we extend our construction to show that differentiable nnTM is equivalent to Universal Turing Machine (UTM) and can simulate any TM with only \textbf{seven finite/bounded precision} neurons. This work provides a new theoretical bound for the computational capability of bounded precision RNNs augmented with memory.