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arXiv · 2309.14690

On the Tensor Representation and Algebraic Homomorphism of the Neural State Turing Machine

Ankur Mali·Alexander Ororbia·Daniel Kifer·Lee Giles
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Abstract

Recurrent neural networks (RNNs) and transformers have been shown to be Turing-complete, but this result assumes infinite precision in their hidden representations, positional encodings for transformers, and unbounded computation time in general. In practical applications, however, it is crucial to have real-time models that can recognize Turing complete grammars in a single pass. To address this issue and to better understand the true computational power of artificial neural networks (ANNs), we introduce a new class of recurrent models called the neural state Turing machine (NSTM). The NSTM has bounded weights and finite-precision connections and can simulate any Turing Machine in real-time. In contrast to prior work that assumes unbounded time and precision in weights, to demonstrate equivalence with TMs, we prove that a $13$-neuron bounded tensor RNN, coupled with third-order synapses, can model any TM class in real-time. Furthermore, under the Markov assumption, we provide a new theoretical bound for a non-recurrent network augmented with memory, showing that a tensor feedforward network with $25$th-order finite precision weights is equivalent to a universal TM.

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Ankur Mali, Alexander Ororbia, Daniel Kifer, Lee Giles. 2023-09-26. On the Tensor Representation and Algebraic Homomorphism of the Neural State Turing Machine. https://arxiv.org/abs/2309.14690

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