A Kripke-like Model for Negation as Failure

We extend the Kripke-like model theory given in 10] for a fragment of rst-order hereditary Harrop formulae to include negated atoms in goals. This gives us a formal framework in which to study the role of Negation as Failure rule. The class of predicates for which Negation As Failure is applicable is discussed, as well as the predicates for which some other form of negation will need to be used. We show how the former class may be incorporated into the model theory, giving a generalisation of the usual T ! construction. No restriction on the class of programs is needed for this approach; the construction may be used for programs which are not locally stratiiedd14]. This is accomplished by the use of a success level and a failure level of a goal, either or both of which may be innnite. The resulting T operator is not monotonic, which necessitates a slight departure from the standard procedure, but the important properties of the construction still hold.