Completing circular codes in regular submonoids

  • Authors:

  • Jean Néraud

  • Affiliations:

  • Université de ROUEN, Faculté des Sciences, Campus du Madrillet, Département Informatique, Avenue de lUniversité, BP 12, 76801 SAINT ETIENNE DU ROUVRAY cedex, France

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2008

Quantified Score

Hi-index 5.23

Visualization

Abstract

Let M be an arbitrary submonoid of the free monoid A^*, and let X@?M be a variable length code (for short a code). X is weakly M-complete iff any word in M is a factor of some word in X^* [J. Neraud, C. Selmi, Free monoid theory: Maximality and completeness in arbitrary submonoids, Internat. J. Algebra Comput. 13 (5) (2003) 507-516]. Given a regular submonoid M, and given an arbitrary code X@?M, we are interested in the existence of a weakly M-complete code X@? that contains X. Actually, in [J. Neraud, Completing a code in a regular submonoid, in: Acts of MCU'2004, Lect. Notes Comput. Sci. 3354 (2005) 281-291; J. Neraud, Completing a code in a submonoid of finite rank, Fund. Inform. 74 (2006) 549-562], by presenting a general formula, we have established that, in any case, such a code X@? exists. In the present paper, we prove that any regular circular code X@?M may be embedded into a weakly M-complete one iff the minimal automaton with behavior M has a synchronizing word. As a consequence of our result an extension of the famous theorem of Schutzenberger is stated for regular circular codes in the framework of regular submonoids. We study also the behaviour of the subclass of uniformly synchronous codes in connection with these questions.