Logic Programs under Three-Valued Łukasiewicz Semantics

Abstract

If logic programs are interpreted over a three-valued logic, then often Kleene's strong three-valued logic with complete equivalence and Fitting's associated immediate consequence operator is used. However, in such a logic the least fixed point of the Fitting operator is not necessarily a model for the program under consideration. Moreover, the model intersection property does not hold. In this paper, we consider the three-valued ***ukasiewicz semantics and show that fixed points of the Fitting operator are also models for the program under consideration and that the model intersection property holds. Moreover, we review a slightly different immediate consequence operator first introduced by Stenning and van Lambalgen and relate it to the Fitting operator under ***ukasiewicz semantics. Some examples are discussed to support the claim that ***ukasiewicz semantics and the Stenning and van Lambalgen operator is better suited to model commonsense and human reasoning.