The group of reversible Turing machines

TL;DR

This paper explores the algebraic structure of reversible Turing machines, analyzing their groups and subgroups, and establishing properties like non-amenability, residual finiteness, and decidability of torsion problems in different dimensions.

Contribution

It introduces the group of reversible Turing machines, studies its properties, and connects it to automata, lamplighter groups, and logical gates, providing new insights into their algebraic and computational characteristics.

Findings

The group of Turing machines in one dimension is neither amenable nor residually finite.

The group is locally embeddable in finite groups.

The torsion problem is decidable for finite-state automata in one dimension, but not in two.

Abstract

We consider Turing machines as actions over configurations in $\Sigma^{\mathbb{Z}^d}$ which only change them locally around a marked position that can move and carry a particular state. In this setting we study the monoid of Turing machines and the group of reversible Turing machines. We also study two natural subgroups, namely the group of finite-state automata, which generalizes the topological full groups studied in the theory of orbit-equivalence, and the group of oblivious Turing machines whose movement is independent of tape contents, which generalizes lamplighter groups and has connections to the study of universal reversible logical gates. Our main results are that the group of Turing machines in one dimension is neither amenable nor residually finite, but is locally embeddable in finite groups, and that the torsion problem is decidable for finite-state automata in dimension one, but not in dimension two.