On the Generative Power of Cancel Minimal Linear Grammars with Single Nonterminal Symbol except the Start Symbol

Abstract

This paper concerns cancel minimal linear grammars ([5]) that was introduced to generalize Geffert normal forms for phrase structure grammars. We consider the generative power of restricted cancel minimal linear grammars: the grammars have only one nonterminal symbol C except the start symbol S, and their productions consist of context-free type productions, the left-hand side of which is S and the right-hand side contains at most one occurrence of S, and a unique cancellation production C^m → ε that replaces the string C^m by the empty string ε. We show that, for any given positive integer m, the class of languages generated by cancel minimal linear grammars with C^m → ε, is properly included in the class of linear languages. Conversely, we show that for any linear language L, there exists some positive integer m such that a cancel minimal linear grammar with C^m → ε generates L. We also show how the generative power of cancel minimal linear grammars with a unique cancellation production C^m → ε vary according to changes of m and restrictions imposed on occurrences of terminal symbols in the right-hand side of productions.