The expressive power of the shuffle product

Authors:
Jean Berstel;Luc Boasson;Olivier Carton;Jean-Éric Pin;Antonio Restivo
Affiliations:
Laboratoire d'Informatique Gaspard-Monge, Université Paris-Est Marne-la-Vallée, 5 Bd Descartes, Champs-sur-Marne, F-77454 Marne-la-Vallée Cedex 2, France;LIAFA, Université Paris Diderot-Paris 7, Case 7014, 75205 Paris Cedex 13, France;LIAFA, Université Paris Diderot-Paris 7, Case 7014, 75205 Paris Cedex 13, France;LIAFA, Université Paris Diderot-Paris 7, Case 7014, 75205 Paris Cedex 13, France;Dipartimento di Matematica e Informatica, Università degli Studi du Palermo, Via Archirafi 34, 90123 Palermo, Italy
Venue:
Information and Computation
Year:
2010

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Abstract

There is an increasing interest in the shuffle product on formal languages, mainly because it is a standard tool for modeling process algebras. It still remains a mysterious operation on regular languages. Antonio Restivo proposed as a challenge to characterize the smallest class of languages containing the singletons and closed under Boolean operations, product and shuffle. This problem is still widely open, but we present some partial results on it. We also study some other smaller classes, including the smallest class containing the languages composed of a single word of length 2 which is closed under Boolean operations and shuffle by a letter (resp. shuffle by a letter and by the star of a letter). The proof techniques have both an algebraic and a combinatorial flavor.