DTDs are continuously updated according to changes of the real world. Let t be an XML document valid against a DTD D, and suppose that D is updated by an update script s. In general, we cannot uniquely "infer" a transformation of t from s, i.e., we cannot uniquely determine the elements in t that should be deleted and the positions in t that new elements should be inserted into. In this paper, we consider inferring K optimum transformations of t from s so that a user finds the most desirable transformation among them. We first show that the problem of inferring K optimum transformations of an XML document from an update