How to Demonstrate Metalinearness and Regularity by Tree-Restricted General Grammars

TL;DR

This paper introduces derivation trees for general grammars and establishes conditions under which the generated languages are linear or regular, providing a new method to demonstrate language regularity and linearity.

Contribution

It presents a novel framework using derivation trees and context-dependent pairs to prove language regularity and linearity for general grammars.

Findings

Languages generated by certain general grammars are regular under specific tree conditions.

The framework provides a powerful tool for demonstrating language regularity and linearity.

Conditions involving context-dependent pairs determine language class membership.

Abstract

This paper introduces derivation trees for general grammars. Within these trees, it defines context-dependent pairs of nodes, corresponding to rewriting two neighboring symbols using a non context-free rule. It proves that the language generated by a linear core general grammar with a slow-branching derivation tree is k-linear if there is a constant u such that every sentence w in the generated language is the frontier of a derivation tree in which any pair of neighboring paths contains u or fewer context-dependent pairs of nodes. Next, it proves that the language generated by a general grammar with a regular core is regular if there is a constant u such that every sentence w in the generated language is the frontier of a derivation tree in which any pair of neighboring paths contains u or fewer context-dependent pairs of nodes. The paper explains that this result is a powerful tool for showing that certain languages are k-linear or regular.