Regular Languages Are Church-Rosser Congruential

  • Volker Diekert
  • Manfred Kufleitner
  • Klaus Reinhardt
  • Tobias Walter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7392)


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 languages are of this form. This paper solves the conjecture affirmatively by actually proving a more general result.


String rewriting Church-Rosser system regular language finite monoid finite semigroup local divisor 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Volker Diekert
    • 1
  • Manfred Kufleitner
    • 1
  • Klaus Reinhardt
    • 2
  • Tobias Walter
    • 1
  1. 1.Institut für Formale Methoden der InformatikUniversity of StuttgartGermany
  2. 2.Wilhelm-Schickard-Institut für InformatikUniversity of TübingenGermany

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