Approximation Fixpoint Theory and the Well-Founded Semantics of Higher-Order Logic Programs

TL;DR

This paper introduces a new three-valued extensional semantics for higher-order logic programs with negation, extending the well-founded semantics to a higher-order setting using approximation fixpoint theory.

Contribution

It develops a novel semantics based on Fitting-monotonic functions and establishes a minimal, well-founded model for higher-order logic programs with negation.

Findings

Defines a bijection between Fitting-monotonic functions and pairs of two-valued functions.

Extends approximation fixpoint theory to higher-order logic programming.

Generalizes the classical well-founded semantics to higher-order logic programs.

Abstract

We define a novel, extensional, three-valued semantics for higher-order logic programs with negation. The new semantics is based on interpreting the types of the source language as three-valued Fitting-monotonic functions at all levels of the type hierarchy. We prove that there exists a bijection between such Fitting-monotonic functions and pairs of two-valued-result functions where the first member of the pair is monotone-antimonotone and the second member is antimonotone-monotone. By deriving an extension of consistent approximation fixpoint theory (Denecker et al. 2004) and utilizing the above bijection, we define an iterative procedure that produces for any given higher-order logic program a distinguished extensional model. We demonstrate that this model is actually a minimal one. Moreover, we prove that our construction generalizes the familiar well-founded semantics for classical logic programs, making in this way our proposal an appealing formulation for capturing the well-founded semantics for higher-order logic programs. This paper is under consideration for acceptance in TPLP.