Completing prefix codes in submonoids

Let 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?