Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Martin Havel, Zbyněk Křivka, Alexander Meduna

TL;DR

It is proved 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.

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.

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

TL;DR

It is proved 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.

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.
Paper Structure (6 sections, 6 theorems, 9 equations, 3 figures)

This paper contains 6 sections, 6 theorems, 9 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and Examples
  4. Results
  5. Use
  6. Final Remarks and Open Problems

Key Result

Lemma 1

A language $L$ is recursively enumerable iff $L = L(G)$, where $G$ is a linear core general grammar.

Figures (3)

  • Figure 1: Illustration of context dependency in $t$
  • Figure 2: Labeled ordered tree $t$
  • Figure 3: $\prescript{}{G}\triangle_{aaa0011a0011b}$

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 3
  • Definition 4
  • Example 1
  • Definition 5
  • ...and 19 more