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Abstract: By grouping several sites together into one, a cellular automaton can be transformed into another with more states and a smaller neighborhood; if the neighborhood has just two sites, we can think of the resulting CA rule as a binary operation. We show that if the blocked rule satisfies an identity which holds for a broad class of algebras, then the underlying rule must have essentially the same structure, and must depend only on its leftmost and rightmost inputs; roughly speaking, that... (Update)
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...structure cannot have a block algebra with any of these simple algebraic properties. Preliminary versions of these results appeared in [6], where Theorem 1 was proved for a much smaller class of algebras we called special medial. 2 Preliminaries A binary operation or...
.... one with r 0 = 1=2 where each site has only two predecessors in a staggered space time: Phi(a) i = OE(a i Gamma1=2 ; a i 1=2 ) See [24] for a study of this transformation s algebraic properties. We can then think of the CA rule as a binary algebra, where OE(a; b) a...
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4: Cellular automata as algebraic systems (context) - Pedersen - 1992
4: Statistical mechanics of cellular automata (context) - Wolfram - 1983
3: Quasi-linear cellular automata - Moore - 1997
BibTeX entry: (Update)
C. Moore and A. Drisko, "Algebraic properties of the block transformation on cellular automata." Santa Fe Institute Working Paper 95-09-080. http://citeseer.ist.psu.edu/moore95algebraic.html More
@article{ moore96algebraic,
author = "Moore and Drisko",
title = "Algebraic Properties of the Block Transformation on Cellular Automata",
journal = "COMPSYSTS: Complex Systems",
volume = "10",
year = "1996",
url = "citeseer.ist.psu.edu/moore95algebraic.html" }
Citations (may not include all citations):70 Statistical Mechanics of Cellular Automata (context) - Wolfram - 1983
52 Latin Squares and their Applications (context) - D'enes, Keedwell - 1974
16 Quasi-linear Cellular Automata - Moore
10 Quasigroups and Loops: An Introduction (context) - Pflugfelder - 1990
9 Non-Abelian Cellular Automata - Moore
9 Cellular Automata as Algebraic Systems (context) - Pedersen - 1992
6 Partially Permutive Cellular Automata - Eloranta - 1993
5 Combinatorial Construction of One-Dimensional Reversible Cel.. (context) - Boykett - 1995
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