Complexity of Pattern Generation by Map-L-Systems

A certain type of two-dimensional generative grammars has been introduced in [I] intended as a theoretical model for the development of organisms with twodimensional characteristics. Such a grammar will generate out of a single cell (egg) a sequence of plannar topological patterns (maps) according to a preassigned finite set of context sensitive rules• At each stage of the generation the rules are applied simultaneously to the existing pattern in order to create the next pattern by a binary (at most) splitting of its "cells" (a process which is similar to the generation process of an actual living organism)• In order to account for the environmental context governing the splitting process, the cells are assumed to be in one out of a finite set of "states" (represented by letters or colors belonging to a given alphabet) at each stage of the generation• The pattern is thus assumed to be "colored", and the coloration changes from generation to generation according to the rules of the two-dimensional grammar. The model described intuitively above allows one to define and study the complexity of pattern generation by two-dimensional grammars in several aspects such as: (a) Is it possible to find a number m such that for any given specific uncolored pattern in the plane, there exists a grammar with no more than m colors which will generate the given pattern among all the patterns generated by it? (b) What is the relation between the number