Around Context-Free Grammars - a Normal Form, a Representation Theorem, and a Regular Approximation

We introduce a normal form for context-free grammars, called Dyck normal form. This is a syntactical restriction of the Chomsky normal form, in which the two nonterminals occurring on the right-hand side of a rule are paired nonterminals. This pairwise property allows to define a homomorphism from Dyck words to words generated by a grammar in Dyck normal form. We prove that for each context-free language L, there exist an integer K and a homomorphism φ such that L = φ(D′ K), where D ′ K ⊆ DK , and DK is the one-sided Dyck language over K letters. Through a transition-like diagram for a context-free grammar in Dyck normal form, we effectively build a regular language R such that D′ K = R∩DK , which leads to the Chomsky-Schützenberger theorem. Using graphical approaches we refine R such that the Chomsky-Schützenberger theorem still holds. Based on this readjustment we sketch a transition diagram for a regular grammar that generates a regular superset approximation for the initial context-free language.