The optimal model of a program with negation

Abstract

The aim of this paper is to provide another denotational semantics than the least fixpoint semantics for a logic program with negation.It takes place in a more general work about three-valued (partial) logic and logic programming. We have developped a logical and algebrical theory which seems well-suited to logic programs with negation. In this theory, we have extended the "consequence" operator of Van Emden and Kowalski associated with a program without negation to a "consequence" operator for programs with negation taking inconsistent programs in account. As the models of a program are exactly the post-fixpoints of this operator, the first provided denotational semantics of a program is the least fixpoint semantics.As an intersection of three-valued Herbrand models of a program is still a model of this program, this least fixpoint is also the set of all the ground literals of a program which are three-valued logical consequences of this program. Even if the least model is thus unique and defined for every consistent program, it is a partial model and may contain only few literals. A second denotational semantics is given by considering the three-valued Herbrand interpretations as partial functions from the set of ground atoms to the set of the truth values {T, F}; the theory of Manna and Shamir can be adapted in two ways: The first one is the notion of optimal fixpoint of the operator. The second one seems more interesting for our purpose, since the models of a logic program with negation are exactly the post-fixpoints of the consequence operator. It is the notion of optimal post-fixpoint of our "consequence" operator, which is the optimal model of our program. This notion is about to strengthen the three-valued logic. This optimal model is still unique and defined for every consistent program. It is still a partial model but generally strictly contains the least model, and thus really provides another denotational semantics than the least fixpoint semantics for a program with negation. It may even be more interesting since it contains more informations on the program itself. KeywordsLogic ProgramLogic ProgrammingPartial FunctionDenotational SemanticGround AtomThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.