Abstract Forgetting is an important mechanism for logicbased agent systems. A recent interest has been in the desirable properties of forgetting in answer set programming (ASP) and their impact on the design of forgetting operators. It is known that some subsets of these properties are incompatible, i.e., they cannot be satisfied at the same time. In this paper, we are interested in the question on the largest set ? of pairs (?, V ), where ? is a logic program and V is a set of atoms, such that a forgetting operator exists that satisfies all the desirable properties for eac (?, V ) in ?. We answer this question positively by discovering the precise condition under which the knowledge forgetting, a well-established approach to forgetting in ASP, satisfies the property of strong persistence, which leads to a sufficient and necessary condition for a forgetting operator to satisfy all the desirable properties proposed in the literature. We explore computational complexities on checking the condition and present a syntactic characterization which can serve as the basis of computing knowledge forgetting in ASP.