Decidability Equivalence between the Star Problem and the Finite Power Problem in Trace Monoids

Abstract

In the last decade, some researches on the star problem in trace monoids (is the iteration of a recognizable language also recognizable?) has pointed out the interest of the finite power property to achieve partial solutions of this problem. We prove that the star problem is decidable in some trace monoid if and only if in the same monoid, it is decidable whether a recognizable language has the finite power property. Intermediary results allow us to give a shorter proof for the decidability of the two previous problems in every trace monoid without C4-submonoid. We also deal with some earlier ideas, conjectures, and questions which have been raised in the research on the star problem and the finite power property, e.g., we show the decidability of these problems for recognizable languages which contain at most one non-connected trace. This article is published as Technical Report both at the Department of Computer Science at Dresden University of Technology and at the Laboratoire de ..