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In the conventional supervisory control frame-work for discrete event systems with partial event observation, it is assumed that, for each event, the corresponding output symbol is determined deterministically. However, this assumption does not hold in discrete event systems such as a system with sensor errors and a mobile system, where an output symbol depends on not only an event but also a state at which the event occurs. In this paper, we model such a discrete event system by a Mealy automaton with a nondeterministic output function. We consider two kinds of supervisors: one assigns its control action based on a permissive policy and the other based on an anti-permissive one. We present necessary and sufficient conditions for each of them to achieve a given specification. We then present algorithms for verifying these conditions. Moreover, we discuss the relationship between the two supervisors in the case that an output function is deterministic.