Minimum Model Semantics for Logic Programs with Negation-as-Failure

TL;DR

This paper introduces a purely model-theoretic semantics for logic programs with negation-as-failure, using an expanded uncountable truth domain and a novel interpretation of negation, resulting in a unique minimum model.

Contribution

It provides a new unifying semantics for negation-as-failure in logic programming using an expanded truth domain and a model-theoretic approach, generalizing classical semantics.

Findings

Every program has a unique minimum model M_P.

M_P can be constructed via a transfinite iteration over countable ordinals.

Collapsing M_P yields the classical well-founded model.

Abstract

We give a purely model-theoretic characterization of the semantics of logic programs with negation-as-failure allowed in clause bodies. In our semantics the meaning of a program is, as in the classical case, the unique minimum model in a program-independent ordering. We use an expanded truth domain that has an uncountable linearly ordered set of truth values between False (the minimum element) and True (the maximum), with a Zero element in the middle. The truth values below Zero are ordered like the countable ordinals. The values above Zero have exactly the reverse order. Negation is interpreted as reflection about Zero followed by a step towards Zero; the only truth value that remains unaffected by negation is Zero. We show that every program has a unique minimum model M_P, and that this model can be constructed with a T_P iteration which proceeds through the countable ordinals. Furthermore, we demonstrate that M_P can also be obtained through a model intersection construction which generalizes the well-known model intersection theorem for classical logic programming. Finally, we show that by collapsing the true and false values of the infinite-valued model M_P to (the classical) True and False, we obtain a three-valued model identical to the well-founded one.