A survey on the local divisor technique

© 2015 Elsevier B.V. Local divisors allow a powerful induction scheme on the size of a monoid. We survey this technique by giving several examples of this proof method. These applications include linear temporal logic, rational expressions with Kleene stars restricted to prefix codes with bounded synchronization delay, Church-Rosser congruential languages, and Simon's Factorization Forest Theorem. We also introduce the notion of a localizable language class as a new abstract concept which unifies some of the proofs for the results above

Omega-rational expressions with bounded synchronization delay

© 2013, Springer Science+Business Media New York. In 1965 Sch ̈utzenberger published his famous result that star-free languages (SF) and aperiodic languages (AP) coincide over finite words, often written as SF = AP. Perrin generalized SF = AP to infinite words in the mid 1980s. In 1973 Sch ̈utzenberger presented another (and less known) characteri- zation of aperiodic languages in terms of rational expressions where the use of the star operation is restricted to prefix codes with bounded synchronization delay and no complementation is used. We denote this class of languages by SD. In this paper, we present a generalization of SD = AP to infinite words. This became possible via a substantial simplification of the proof for the cor- responding result for finite words. Moreover, we show that SD = AP can be viewed as more fundamental than SF = AP in the sense that the classical 1965 result of Sch ̈utzenberger and its 1980s extension to infinite words by Perrin are immediate consequences of SD = AP

Regular languages are Church-Rosser congruential

© 2015 ACM 0004-5411/2015/10-ART32 15.00. This article shows a general result about finite monoids and weight reducing string rewriting systems. As a consequence it proves a long standing conjecture in formal language theory: 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 the conjecture was that all regular languages are of this form. The article offers a stronger statement: A language is regular if and only if it is strongly Church-Rosser congruential. It is the journal version of the conference abstract which was presented at ICALP 2012