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 Łukasiewicz 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 Łukasiewicz semantics. Some examples are discussed to support the claim that Łukasiewicz semantics and the Stenning and van Lambalgen operator is better suited to model commonsense and human reasoning. © 2009 Springer Berlin Heidelberg.