On Some Complexity Results for Even Linear Languages

TL;DR

This paper introduces Dyck normal form for context-free grammars, demonstrating a homomorphic relationship with Dyck languages, and proves that even linear languages are contained within the complexity class AC1.

Contribution

It presents a new normal form for context-free grammars and establishes a connection between even linear languages and Dyck languages, providing complexity insights.

Findings

Dyck normal form enables a homomorphic mapping from Dyck words to context-free languages.

Every context-free language can be represented as a homomorphic image of a subset of a Dyck language.

Proves that even linear languages are included in the class AC1.

Abstract

We deal with a normal form for context-free grammars, called Dyck normal form. This normal form is a syntactical restriction of the Chomsky normal form, in which the two nonterminals occurring on the right-hand side of a rule are paired nonterminals. This pairwise property, along with several other terminal rewriting conditions, makes it possible to define a homomorphism from Dyck words to words generated by a grammar in Dyck normal form. We prove that for each context-free language L, there exist an integer K and a homomorphism phi such that L=phi(D'_K), where D'_K is a subset of D_K and D_K is the one-sided Dyck language over K letters. As an application we give an alternative proof of the inclusion of the class of even linear languages in AC1.