Universal gates with wires in a row

TL;DR

This paper demonstrates that a single reversible cellular automaton combined with shifts can generate all local permutations on an infinite row of binary wires, providing optimal and minimal generating sets.

Contribution

It introduces minimal generating sets for the group of reversible gates on binary wires, including a single automaton and shifts, and classifies pairs of words generating all local permutations.

Findings

A single automaton (ECA 57 or 99) plus shift generates all reversible gates.

Classified pairs of words that, with shift and bit flip, generate all local permutations.

Results extend to wires on a cycle, confirming a conjecture.

Abstract

We give some optimal size generating sets for the group generated by shifts and local permutations on the binary full shift. We show that a single generator, namely the fully asynchronous application of the elementary cellular automaton 57 (or, by symmetry, ECA 99), suffices in addition to the shift. In the terminology of logical gates, we have a single reversible gate whose shifts generate all (finitary) reversible gates on infinitely many binary-valued wires that lie in a row and cannot (a priori) be rearranged. We classify pairs of words $u, v$ such that the gate swapping these two words, together with the shift and the bit flip, generates all local permutations. As a corollary, we obtain analogous results in the case where the wires are arranged on a cycle, confirming a conjecture of Macauley-McCammond-Mortveit and Vielhaber.