Necessary conditions for subclasses of random context languages

Random context grammars belong to the class of context-free grammars with regulated rewriting. Their productions depend on context that may be randomly distributed in a sentential form. Context is classified as either permitting or forbidding, where permitting context enables the application of a production and forbidding context inhibits it. We have proven a pumping lemma for random permitting context languages and a shrinking lemma for random forbidding context languages. We now present new necessary conditions for both these classes of languages and illustrate them with examples. We also present and illustrate a new necessary condition for context-free languages.