The Study of One-Dimensional Cellular Automata with Nearest-Nearest Neighborhoods

Abstract

In this paper, we deal with the global equivalence classification of one-dimensional cellular automata (CA) with nearest-nearest neighborhoods. We present a theoretical classification of all rules through the two homeomorphisms identified in [1]. we demonstrate the 30 global equivalence classes of 64 additive rules, and show that in the same equivalence class they have the same number of independent connected component(s) for any L (number of cells). Furthermore, basing on this platform, we also explore the global equivalence class of totalistic rule 20 and 52, which are reputedly capable of highly complex dynamical behaviors and belong to Wolfram’s class IV. It is worth mentioning that the classification method we have presented are actually applicable to one-dimensional CA with any neighborhood radius.