Star-free languages are Church–Rosser congruential

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.