Matrix Equations and Normal Forms for Context-Free Grammars

  • Authors:
  • Daniel J. Rosenkrantz

  • Affiliations:
  • Department of Electrical Engineering, Columbia University, New York, N. Y.

  • Venue:
  • Journal of the ACM (JACM)
  • Year:
  • 1967

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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 replacements 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 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.