A representation of context-free grammars with the help of finite digraphs

TL;DR

This paper introduces a novel finite digraph-based transition diagram for context-free grammars, providing a new graphical representation that characterizes language membership through proper walks.

Contribution

It presents a new graph model for context-free grammars and proves its equivalence to language membership, offering a fresh perspective on grammar representation.

Findings

Transition diagram accurately describes context-free grammar behavior.

A word belongs to the language iff it can be generated by a proper walk.

The approach differs from existing graph models for grammars.

Abstract

For any context-free grammar, we build a transition diagram, that is, a finite directed graph with labeled arcs, which describes the work of the grammar. This approach is new, and it is different from previously known graph models. We define the concept of proper walk in this transition diagram and we prove that a word belongs to a given context-free language if and only if this word can be obtained with the help of a proper walk.