Star-free languages are Church-Rosser congruential

Authors:
Volker Diekert;Manfred Kufleitner;Pascal Weil
Affiliations:
Institut für Formale Methoden der Informatik, University of Stuttgart, Universitätsstr. 38, 70569 Stuttgart, Germany;Institut für Formale Methoden der Informatik, University of Stuttgart, Universitätsstr. 38, 70569 Stuttgart, Germany;Université de Bordeaux, LaBRI, UMR 5800, F-33400 Talence, France and CNRS, LaBRI, UMR 5800, F-33400 Talence, France
Venue:
Theoretical Computer Science
Year:
2012

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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.