Distance Functions, Instance Functions, and Preference Relations

A standard way of presenting the semantics of default information is via preference relations on models. Under this view, the default information gives rise to a preference relation which orders models according to how well they satisfy it. Diierent ways of deening the relation in terms of the default information can be used to give diierent granularities to the default status the information has. Two other important ways of specifying the semantics of defaults (and controlling gran-ularity) are distance functions and instances. On the distance function view, we deene a function which, given two models, returns some notion of how diierent they are from one another. This distance function can be used to describe the semantics of defaults, since one selects models of a theory which are closest to complying with the defaults. On the instances view, we deene a function which takes a default and returns a set of formulas (its \instances") which can be accepted or rejected independently. Our aim in this paper is to interrelate these three diierent frameworks.