When does partial commutative closure preserve regularity?

• Main Authors: Cano Gómez, Antonio, Guaiana, Giovanna, Pin, Jean-Eric
• Other Authors: Departamento de Sistemas Informáticos y Computación Valencia, Universitat Politècnica de València = Universitad Politecnica de Valencia = Polytechnic University of Valencia (UPV), Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes (LITIS), Université Le Havre Normandie (ULH), Normandie Université (NU)-Normandie Université (NU)-Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Institut national des sciences appliquées Rouen Normandie (INSA Rouen Normandie), Institut National des Sciences Appliquées (INSA)-Normandie Université (NU)-Institut National des Sciences Appliquées (INSA), Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
• Format: Conference Object
• Language: English
• Published: HAL CCSD 2008
• Subjects: [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH]; [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]; [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]; [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR]; Iceland
• Online Access: https://hal.science/hal-00340806; https://hal.science/hal-00340806/document; https://hal.science/hal-00340806/file/CommutationsWeb.pdf

International audience The closure of a regular language under commutation or partial commutation has been extensively studied. In this paper, we present new advances on two problems of this area. Problem 1. When is the closure of a regular language under [partial] commutation still regular? Problem 2. Are there any robust class of languages closed under [partial] commutation? Let A be a finite alphabet, I a partial commutation on A and D be its complement in A x A. Our main results on Problems 1 and 2 can be summarized as follows: The class Pol G (polynomials of group languages) is closed under commutation. If D is transitive, it is also closed under I-commutation. Under some simple conditions on the graph of I, the closure of a language of Pol G under I-commutation is regular. There is a very robust class of languages W which is closed under commutation. This class, which contains Pol G, is closed under intersection, union, shuffle, concatenation, residual, length preserving morphisms and inverses of morphisms. Further, it is decidable and can be defined as the largest positive variety of languages not containing (ab)*. If I is transitive, the closure of a language of W under I-commutation is regular. The proofs are nontrivial and combine several advanced techniques, including combinatorial Ramsey type arguments, algebraic properties of the syntactic monoid, finiteness conditions on semigroups and properties of insertion systems.