Regular Languages are Church-Rosser Congruential

TL;DR

This paper proves that every regular language can be characterized as Church-Rosser congruential, confirming a long-standing conjecture in formal language theory.

Contribution

It establishes that all regular languages are Church-Rosser congruential, expanding the understanding of their algebraic and rewriting system properties.

Findings

All regular languages are Church-Rosser congruential.

The class of Church-Rosser congruential languages includes all regular languages.

The result confirms a long-standing conjecture in formal language theory.

Abstract

This paper proves a long standing conjecture in formal language theory. It shows that all regular languages are Church-Rosser congruential. The class of Church-Rosser congruential languages was introduced by McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser congruential, if there exists a finite confluent, and length-reducing semi-Thue system S such that L is a finite union of congruence classes modulo S. It was known that there are deterministic linear context-free languages which are not Church-Rosser congruential, but on the other hand it was strongly believed that all regular language are of this form. Actually, this paper proves a more general result.