Star-Free Languages are Church-Rosser Congruential

TL;DR

This paper proves that all star-free languages can be characterized as Church-Rosser congruential languages using finite, confluent, and subword-reducing semi-Thue systems, advancing understanding of language classification.

Contribution

It establishes that every star-free language belongs to CRCL and provides an effective construction for the corresponding semi-Thue system.

Findings

All star-free languages are in CRCL.

Existence of finite, confluent, subword-reducing semi-Thue systems for star-free languages.

Construction of such systems is effective.

Abstract

The class of Church-Rosser congruential languages has been introduced by McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser congruential (belongs to CRCL), if there is a finite, confluent, and length-reducing semi-Thue system S such that L is a finite union of congruence classes modulo S. To date, it is still open whether every regular language is in CRCL. In this paper, we show that every star-free language is in CRCL. In fact, we prove a stronger statement: For every star-free language L there exists a finite, confluent, and subword-reducing semi-Thue system S such that the total number of congruence classes modulo S is finite and such that L is a union of congruence classes modulo S. The construction turns out to be effective.