On the complexity of synthesizing a minimum-weighted supervisor under partial observation

With full observation the problem of synthesizing a minimum-weighted supervisor has been shown polynomial-time solvable. With partial observation the problem becomes NP-hard. In this paper we will show that the decision version of the problem becomes NP-complete when the natural projection is a natural observer. This NP-completeness result is unexpected, considering that the logic supervisor synthesis problem under the same assumption becomes polynomial-time solvable.