Random Context Picture Grammars: Definitions, Theorems, Examples

Syntactic methods of picture generation have become established during the last decades. Random context picture grammars (rcpgs) generate pictures through successive refinement. These are context-free grammars with regulated rewriting: each production is context-free, but its application is regulated - permitted or forbidden - by context randomly distributed in the developing picture. We investigate three important subclasses of rcpgs, namely those that use no context, only permitting or only forbidding context. For each class we develop a necessary condition, in particular, a pumping or shrinking lemma, and show that it is properly contained in the class of rcpgs. We also introduce a broader model, so-called generalized random context picture grammars. The terminals of such a grammar are subsets of the plane, while the replacement of a variable involves the building of functions that will eventually be applied to the terminals. We show that these grammars are more powerful than iterated function systems, which are commonly used to construct fractals. This book should be of interest to researchers in formal language theory or syntactic picture generation.