The Block Neighborhood

TL;DR

This paper introduces the concept of block neighborhood for reversible cellular automata, providing a combinatorial characterization that simplifies computation and enables bounds on neighborhood size, impacting quantum computing and CA decomposition.

Contribution

It offers a new combinatorial framework for understanding block neighborhoods in reversible CA, facilitating easier computation and bounding of neighborhood sizes.

Findings

Provided a combinatorial characterization of block neighborhoods.

Derived upper bounds on block neighborhoods based on classical and inverse neighborhoods.

Identified a class of elementary CAs that cannot be decomposed into simpler parts.

Abstract

We define the block neighborhood of a reversible CA, which is related both to its decomposition into a product of block permutations and to quantum computing. We give a purely combinatorial characterization of the block neighborhood, which helps in two ways. First, it makes the computation of the block neighborhood of a given CA relatively easy. Second, it allows us to derive upper bounds on the block neighborhood: for a single CA as function of the classical and inverse neighborhoods, and for the composition of several CAs. One consequence of that is a characterization of a class of "elementary" CAs that cannot be written as the composition of two simpler parts whose neighborhoods and inverse neighborhoods would be reduced by one half.