# Characterizing configuration spaces of simple threshold cellular automata

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## Abstract

We study herewith *the simple threshold cellular automata (CA), * as perhaps the simplest broad class of *CA * with non-additive (that is, non-linear and non-affine) local update rules. We characterize all possible computations of the most interesting rule for such *CA*, namely, the *Majority (MAJ) * rule, both in the classical, parallel CA case, and in case of the corresponding sequential CA where the nodes update sequentially, one at a time. We compare and contrast the configuration spaces of arbitrary simple threshold automata in those two cases, and point out that some parallel threshold CA cannot be simulated by any of their sequential equivalents. We show that the temporal cycles exist only in case of (some) parallel simple threshold CA, but can never take place in sequential threshold CA. We also show that most threshold *CA * have very few fixed point configurations and few (if any) cycle configurations, and that, while the *MAJ * sequential and parallel *CA * may have many fixed points, nonetheless “almost all” configurations, in both parallel and sequential cases, are transient states. Finally, motivated by the contrasts between parallel and sequential simple threshold CA, we try to motivate the study of *genuinely asynchronous CA.*

## BibTeX

@inproceedings{conf/acri/TosicA04, author = "Tosic, Predrag T. and Agha, Gul", editor = "Sloot, Peter M. A. and Chopard, Bastien and Hoekstra, Alfons G.", title = "Characterizing Configuration Spaces of Simple Threshold Cellular Automata", booktitle = "ACRI", crossref = "conf/acri/2004", ee = "http://dx.doi.org/10.1007/978-3-540-30479-1_89", keywords = "complex systems, multi-agent systems", pages = "861-870", year = "2004", } @proceedings{conf/acri/2004, editor = "Sloot, Peter M. A. and Chopard, Bastien and Hoekstra, Alfons G.", title = "Cellular Automata, 6th International Conference on Cellular Automata for Research and Industry, ACRI 2004, Amsterdam, The Netherlands, October 25-28, 2004, Proceedings", isbn = "3-540-23596-5", publisher = "Springer", series = "Lecture Notes in Computer Science", volume = "3305", year = "2004", }