A completeness problem for pattern generation in tessellation automata

摘要

This report deals with the question of whether or not, for a given tessellationautomaton, there exists a finite pattern that cannot evolve from a given primitive pattern no matter what sequence of environmental input transformations are applied. This is closely related to Moore's Garden-of-Eden problem. We begin dealing with this question for the simplest nontrivial tessellation automata, namely, one-dimensional binary scope-n tessellation automata. We show that any finite pattern can evolve from the primitive pattern if the neighborhood scope is four or more. We show that there are finite patterns that cannot evolve from the primitive pattern for the scope-two case. Although some partial results are presented, the question is still open for the scope-three case. Some results for more general tessellation automata are also discussed.