Extensional Semantics for Higher-Order Logic Programs with Negation

TL;DR

This paper introduces an extensional semantics for higher-order logic programs with negation, generalizing previous work and providing new notions of stratification, with implications for stable model semantics.

Contribution

It develops a novel extensional semantics for higher-order logic programs with negation and introduces generalized stratification concepts.

Findings

Semantics never assigns unknown truth to stratified programs

Generalized stratification extends classical notions

Stable model semantics do not always produce extensional models

Abstract

We develop an extensional semantics for higher-order logic programs with negation, generalizing the technique that was introduced in [Bezem99,Bezem01] for positive higher-order programs. In this way we provide an alternative extensional semantics for higher-order logic programs with negation to the one proposed in [CharalambidisER14]. As an immediate useful consequence of our developments, we define for the language we consider the notions of stratification and local stratification, which generalize the familiar such notions from classical logic programming. We demonstrate that for stratified and locally stratified higher-order logic programs, the proposed semantics never assigns the unknown truth value. We conclude the paper by providing a negative result: we demonstrate that the well-known stable model semantics of classical logic programming, if extended according to the technique of [Bezem99,Bezem01] to higher-order logic programs, does not in general lead to extensional stable models.