Comparison of Two Context-Free Rewriting Systems with Simple Context-Checking Mechanisms

TL;DR

This paper establishes that semi-conditional grammars of degree (1,1) generate exactly the same languages as random context languages, and extends known normal forms to non-erasing cases, clarifying derivation relations.

Contribution

It proves the equivalence between semi-conditional grammars of degree (1,1) and random context languages, and extends normal form results to non-erasing grammars.

Findings

Semi-conditional grammars of degree (1,1) generate the same language family as random context languages.

Normal form results for erasing grammars also hold for non-erasing grammars.

Clarifies the definition of the direct derivation step in the literature.

Abstract

This paper solves an open problem concerning the generative power of nonerasing context-free rewriting systems using a simple mechanism for checking for context dependencies, in the literature known as semi-conditional grammars of degree (1,1). In these grammars, two nonterminal symbols are attached to each context-free production, and such a production is applicable if one of the two attached symbols occurs in the current sentential form, while the other does not. Specifically, this paper demonstrates that the family of languages generated by semi-conditional grammars of degree (1,1) coincides with the family of random context languages. In addition, it shows that the normal form proved by Mayer for random context grammars with erasing productions holds for random context grammars without erasing productions, too. It also discusses two possible definitions of the relation of the direct derivation step used in the literature.