An Elementary Proof of Double Greibach Normal Form

In [1,6,7] elementary proofs are given of Greibach Normal Form of context-free grammars [4]. Here we do the same for Double Greibach Normal Form, where it is required that each right-hand side of a production both starts and ends with a terminal symbol [5]. In [5], first a grammar in Greibach Normal Form is constructed, which is then transformed into Double Greibach Normal Form; both transformations use matrix manipulation. Here we give a direct construction, with an elementary correctness proof. We first discuss the construction of Urbanek in 161. Let G = (N, T, S, P> be a context-free grammar in Chomsky Normal Form. N is the set of nonterminals, T the set of terminals, S is the initial nonterminal, and P is the set of productions, of the form A --) BC or A + a (where A,B,C E N and a E T). First a grammar H = (N’, T, S, P’) is constructed that is not yet in Greibach NF: N’ = N u (N x N) and P’ consists of the following productions (where capitals are