On the Algebraic Representation of One-Tape Deterministic Turing Machine

TL;DR

This paper introduces an algebraic tensor-based framework for representing one-tape deterministic Turing machines, capturing configurations and transitions through high-order tensors and their products.

Contribution

It presents a novel algebraic tensor representation of Turing machine configurations and transitions, demonstrating harmonic tensor products that obey associativity.

Findings

Configurations modeled by 4th order tensors

Transition functions represented by 8th order tensors

Tensor products model evolution and composition of Turing machines

Abstract

An algebraic representation of the Turing machines is given, where the configurations of Turing machines are represented by 4 order tensors, and the transition functions by 8 order tensors. Two types of tensor product are defined, one is to model the evolution of the Turing machines, and the other is to model the compositions of transition functions. It is shown that the two types of tensor product are harmonic in the sense that the associate law is obeyed.