Regular languages and partial commutations

  • Authors:

  • Antonio Cano;Giovanna Guaiana;Jean-íric Pin

  • Affiliations:

  • Departamento de Sistemas Informáticos y Computación, Universidad Politécnica de Valencia, Camino de Vera s/n, P.O. Box: 22012, E-46020 Valencia, Spain;LITIS EA 4108, Université de Rouen, BP12, 76801 Saint Etienne du Rouvray, France;LIAFA, Université Paris-Diderot and CNRS, Case 7014, 75205 Paris Cedex 13, France

  • Venue:
  • Information and Computation
  • Year:
  • 2013

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Abstract

The closure of a regular language under a [partial] commutation I has been extensively studied. We present new advances on two problems of this area: (1) When is the closure of a regular language under [partial] commutation still regular? (2) Are there any robust classes of languages closed under [partial] commutation? We show that the class Pol(G) of polynomials of group languages is closed under commutation, and under partial commutation when the complement of I in A^2 is a transitive relation. We also give a sufficient graph theoretic condition on I to ensure that the closure of a language of Pol(G) under I-commutation is regular. We exhibit a very robust class of languages W which is closed under commutation. This class contains Pol(G), is decidable and can be defined as the largest positive variety of languages not containing (ab)^@?. It is also closed under intersection, union, shuffle, concatenation, quotients, length-decreasing morphisms and inverses of morphisms. If I is transitive, we show that 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.