Membership and finiteness problems for rational sets of regular languages

  • Authors:

  • Sergey Afonin;Elena Hazova

  • Affiliations:

  • Institute of Mechanics, Lomonosov Moscow State University, Moscow, Russia;Institute of Mechanics, Lomonosov Moscow State University, Moscow, Russia

  • Venue:
  • DLT'05 Proceedings of the 9th international conference on Developments in Language Theory
  • Year:
  • 2005

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Abstract

Let Σ be a finite alphabet. A set $\mathcal{R}$ of regular languages over Σ is called rational if there exists a finite set $\mathcal E$ of regular languages over Σ, such that $\mathcal{R}$ is a rational subset of the finitely generated semigroup $(\mathcal{S},\cdot)=\langle\mathcal E\rangle$ with $\mathcal E$ as the set of generators and language concatenation as a product. We prove that for any rational set $\mathcal{R}$ and any regular language R⊆Σ* it is decidable (1) whether $R\in\mathcal{R}$ or not, and (2) whether $\mathcal{R}$ is finite or not.