A Connection between the Star Problem and the Finite Power Property in Trace Monoids
ICAL '99 Proceedings of the 26th International Colloquium on Automata, Languages and Programming
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This paper deals with a connection between two decision problems for recognizable trace languages: the star problem and the finite power property problem. Due to a theorem by {\sc Richomme} from 1994 [26,28], we know that both problems are decidable in trace monoids which do not contain a C4 submonoid. It is not known, whether the star problem or the finite power property are decidable in the C4 or in trace monoids containing a C4. In this paper, we show a new connection between these problems. Assume a trace monoid $\Mon(\Sigma,I)$ which is isomorphic to the Cartesian Product of two disjoint trace monoids $\Mon(\Sigma_1,I_1)$ and $\Mon(\Sigma_2,I_2)$. Assume further a recognizable language $L$ in $\Mon(\Sigma,I)$ such that every trace in $L$ contains at least one letter in $\Sigma_1$ and at least one letter in $\Sigma_2$. Then, the main theorem of this paper asserts that $L^*$ is recognizable iff $L$ has the finite power property.