Matrix equations and normal forms for context-free grammars

Abstract

The relationship between the set of productions of a context-free grammar and the corresponding set of defining equations is first pointed out. The closure operation on a matrix of strings is defined and this concept is used to formalize the solution to a set of linear equations. A procedure is then given for rewriting a context-free grammar in Greibach normal form, where the replacement string of each production begins with a terminal symbol. An additional procedure is given for rewriting the grammar so that each replacement string both begins and ends with a terminal symbol. Neither procedure requires the evaluation of regular expressions over the total vocabulary of the grammar, as is required by Greibach's procedure. © 2009 Springer Netherlands.