Co-NP-Hardness of the Soundness Problem for Asymmetric-Choice Workflow Nets

van der Aalst et al. proved that the soundness problem is solvable in polynomial time for free-choice workflow nets (FCWF-nets). However, FCWF-nets cannot model most web services composition and interorganizational business processes because the interaction among processes does not usually satisfy the free-choice requirement. Asymmetric-choice workflow nets (ACWF-nets) as a larger class than FCWF-nets can model lots of such cases. Our previous work showed that the (weak) soundness problem is co-NP-hard for three-bounded ACWF-nets. Later, Tiplea et al. proved that for three-bounded acyclic ACWF-nets, the weak soundness problem is co-NP-complete. We sharp these results in this paper. First, we prove that for ACWF-nets, whether they are one-bounded or k-bounded (k > 1), the soundness problem is co-NP-hard. Second, it is proven that the soundness is equivalent to the weak soundness for any acyclic ACWF-nets, i.e., an acyclic ACWF-net is sound if and only if it is weakly sound.