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

Abstract.

In the last decade, research on the star problem in trace monoids (is the iteration of a recognizable language also recognizable?) has pointed out the importance of the finite power property to achieve partial solutions to 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. Intermediate results allow us to give a shorter proof for the decidability of the two previous problems in every trace monoid without a 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.