Some Applications of General Topology To The Semantics of Logic Programms

The classical approach to the study of xed point semantics of deenite logic programs P utilises two main facts : (1) The set I J L of all interpretations of the underlying language L based on a given preinterpretation J forms a complete lattice under the partial order of set inclusion. (2) The immediate consequence map T P deened on I J L is lattice continuous. An application is then made of a theorem of Kleene to provide least xed points of T P , see 22] for a detailed account. The term classical used here reeects the fact that the underlying logic is classical two-valued and also that the resulting semantics is satisfactory in general so long as only atoms are present in clause bodies : once one permits negation symbols to be present in clause bodies, one loses monotonicity (and hence lattice continuity) of T P. Nevertheless, this approach has been extended 1] to the class of stratiied programs by adopting a diierent notion from the usual one of ordinal powers in the case of nonmonotonic operators. For various reasons, including the problems caused by extended syntax (i.e the presence of negation or other symbols in clause bodies), it has been proposed that one should work over three-valued, four-valued or even many-valued logics and much work has been done in