International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 1985 / Article

Open Access

Volume 8 |Article ID 453232 | https://doi.org/10.1155/S0161171285000631

Michael D. Taylor, "Weak gardens of Eden for 1-dimensional tessellation automata", International Journal of Mathematics and Mathematical Sciences, vol. 8, Article ID 453232, 9 pages, 1985. https://doi.org/10.1155/S0161171285000631

Weak gardens of Eden for 1-dimensional tessellation automata

Received16 Jul 1984

Abstract

If T is the parallel map associated with a 1-dimensional tessellation automaton, then we say a configuration f is a weak Garden of Eden for T if f has no pre-image under T other than a shift of itself. Let WG(T)= the set of weak Gardens of Eden for T and G(T)= the set of Gardens of Eden (i.e., the set of configurations not in the range of T). Typically members of WG(T)G(T) satisfy an equation of the form Tf=Smf where Sm is the shift defined by (Smf)(j)=f(j+m). Subject to a mild restriction on m, the equation Tf=Smf always has a solution f, and all such solutions are periodic. We present a few other properties of weak Gardens of Eden and a characterization of WG(T) for a class of parallel maps we call (0,1)-characteristic transformations in the case where there are at least three cell states.

Copyright © 1985 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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