Some Results on Reversible Gate Classes Over Non-Binary Alphabets

TL;DR

This paper explores the structure of reversible gate classes over non-binary alphabets, revealing complex properties such as infinite generation, explicit generators, and class embeddings, advancing understanding of non-binary reversible computation.

Contribution

It provides new structural results, explicit generators, and embeddings for reversible gate classes over non-binary alphabets, expanding theoretical foundations.

Findings

A non-finitely generated reversible gate class over non-binary alphabets.

Explicit generators for various classes of gates.

Embedding of gate class posets between alphabet sizes.

Abstract

We present a collection of results concerning the structure of reversible gate classes over non-binary alphabets, including (1) a reversible gate class over non-binary alphabets that is not finitely generated (2) an explicit set of generators for the class of all gates, the class of all conservative gates, and a class of generalizations of the two (3) an embedding of the poset of reversible gate classes over an alphabet of size $k$ into that of an alphabet of size $k+1$ (4) a classification of gate classes containing the class of $(k-1,1)$-conservative gates, meaning gates that preserve the number of occurrences of a certain element in the alphabet.