Operator Representation and Class Transitions in Elementary Cellular Automata
Muhamet Ibrahimi
Aix-Marseille Université, Université de Toulon, CNRS, CPT (UMR 7332)
Turing Centre for Living Systems, Marseille, France
Arda Güçlü
Department of Electrical and Computer Engineering
University of Illinois Urbana-Champaign, Champaign, IL, USA
Naide Jahangirov
UNAM-Institute of Materials Science and Nanotechnology
Bilkent University, 06800, Ankara, Turkey
Mecit Yaman
Department of Aeronautical Engineering
University of Turkish Aeronautical Association, 06790, Ankara, Turkey
Oguz Gülseren
Department of Physics
UNAM-Institute of Materials Science and Nanotechnology
Bilkent University, Ankara 06800, Turkey
Seymur Jahangirov
UNAM-Institute of Materials Science and Nanotechnology
Interdisciplinary Graduate Program in Neuroscience
Bilkent University, Ankara 06800, Turkey
Corresponding author: seymur@unam.bilkent.edu.tr
Abstract
We exploit the mirror and complementary symmetries of elementary cellular automata (ECAs) to rewrite their rules in terms of logical operators. The operator representation based on these fundamental symmetries enables us to construct a periodic table of ECAs that maps all unique rules in clusters of similar asymptotic behavior. We also expand the elementary cellular automaton (ECA) dynamics by introducing a parameter that scales the pace with which operators iterate the system. While tuning this parameter continuously, further emergent behavior in ECAs is unveiled as several rules undergo multiple phase transitions between periodic, chaotic and complex (class 4) behavior. This extension provides an environment for studying class transitions and complex behavior in ECAs. Moreover, the emergence of class 4 structures can potentially enlarge the capacity of many ECA rules for universal computation.
Keywords: elementary cellular automata; classes of cellular automata; deterministic transition; logistic map; Cantor set
Cite this publication as:
M. Ibrahimi, A. Güçlü, N. Jahangirov, M. Yaman, O. Gülseren and S. Jahangirov, “Operator Representation and Class Transitions in Elementary Cellular Automata,” Complex Systems, 31(4), 2022 pp. 415–432.
https://doi.org/10.25088/ComplexSystems.31.4.415
