International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 3, Pages 579-587
doi:10.1155/S0161171285000631

Weak gardens of Eden for 1-dimensional tessellation automata

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.