Non-maximal sensitivity to synchronism in periodic elementary cellular automata: exact asymptotic measures

TL;DR

This paper provides exact asymptotic measures of sensitivity to synchronism in specific elementary cellular automata rules, revealing diverse behaviors and novel proof techniques, including connections to Lucas numbers.

Contribution

It introduces precise measurements of sensitivity to synchronism for selected rules, uncovering unexpected behaviors and linking to mathematical sequences.

Findings

Diverse sensitivity values across rules

Special pairs in rule 128 analyzed

Connection to Lucas numbers in rule 8

Abstract

In [11] and [13] the authors showed that elementary cellular automata rules 0, 3, 8, 12, 15, 28, 32, 34, 44, 51, 60, 128, 136, 140, 160, 162, 170, 200 and 204 (and their conjugation, reflection, reflected-conjugation) are not maximum sensitive to synchronism, i.e. they do not have a different dynamics for each (non-equivalent) block-sequential update schedule (defined as ordered partitions of cell positions). In this work we present exact measurements of the sensitivity to synchronism for these rules, as functions of the size. These exhibit a surprising variety of values and associated proof methods, such as the special pairs of rule 128, and the connection to the bissection of Lucas numbers of rule 8.