Completing circular codes in regular submonoids

AbstractLet 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. Néraud, 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 Xˆ that contains X. Actually, in [J. Néraud, Completing a code in a regular submonoid, in: Acts of MCU’2004, Lect. Notes Comput. Sci. 3354 (2005) 281–291; J. Néraud, 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 Xˆ 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 Schützenberger 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

Completing prefix codes in submonoids

AbstractLet M be a submonoid of the free monoid A*, and let X⊆M be a variable length code (for short a code). X is weakly M-complete if any word in M is a factor of some word in X* [J. Néraud, C. Selmi, Free monoid theory: maximality and completeness in arbitrary submonoids, Internat. J. Algorithms Comput. 13(5) (2003) 507–516]. Given a code X⊆M, we are interested in the construction of a weakly M-complete code that contains X, if it exists. In the case where M and X are regular sets, the existence of such a code has been established [J. Néraud, Completing a code in a regular submonoid of the free monoid, in acts of MCU’2004, Lecture Notes in Computer Sciences, Vol. 3354, Springer, Berlin, 2005, pp. 281–291; J. Néraud, On the completion of codes in submonoids with finite rank, Fund. Inform., to appear]. Actually, this result lays upon a method of construction that preserves the regularity of sets. As well known, any regular (or finite) code may be embedded into a regular (finite) prefix code that is complete in A*. In the framework of the weak completeness, we prove that the following problem is decidable:Instance: A regular submonoid M of A*, and a regular (or finite) prefix code X⊆M.Question: Does a weakly M-complete regular (finite) prefix code containing X exist