Final Sentential Forms

TL;DR

This paper explores the properties of final sentential forms in context-free grammars, establishing conditions under which their final languages are context-free or recursively enumerable, with implications for language classification.

Contribution

It introduces the concept of final sentential forms and characterizes the languages generated by context-free grammars finalized by regular or specific recursively enumerable languages.

Findings

Languages finalized by regular languages are context-free.

Existence of propagating grammars characterizes recursively enumerable languages.

Final forms relate to language classification and computational properties.

Abstract

Let G be a context-free grammar with a total alphabet V, and let F be a final language over an alphabet W such that W is a subset of V. A final sentential form is any sentential form of G that, after omitting symbols from V - W, it belongs to F. The string resulting from the elimination of all nonterminals from W in a final sentential form is in the language of G finalized by F if and only if it contains only terminals. The language of any context-free grammar finalized by a regular language is context-free. On the other hand, it is demonstrated that L is a recursively enumerable language if and only if there exists a propagating context-free grammar G such that L equals the language of G finalized by {w#w^R | w is a string over a binary alphabet}, where w^R is the reversal of w.