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.