Maximum sensitivity to update schedules of elementary cellular automata over periodic configurations

Abstract

This work is a thoughtful extension of the ideas sketched in [Montalva et al. 2017], aiming at classifying elementary cellular automata (ECA) according to their maximal one-step sensitivity to changes in the schedule of cells update. It provides a complete classification of the ECA rule space for all period sizes n > 9 and, together with the classification for all period sizes n ≤ 9 presented in [Montalva et al. 2018], closes this problem and opens further questionings. Most of the 256 ECA rule's sensitivity is proved or disproved to be maximum thanks to an automatic application of basic methods. We formalize meticulous case disjunctions that lead to the results, and patch failing cases for some rules with simple arguments. This gives new insights on the dynamics of ECA rules depending on the proof method employed, as for the last rules 45 and 105 requiring (0011) * induction patterns.