Equivalence of two Fixed-Point Semantics for Definitional Higher-Order Logic Programs

TL;DR

This paper compares two extensional semantics approaches for higher-order logic programming, showing their equivalence for definitional programs but differences when existential variables are involved, thus clarifying semantic foundations.

Contribution

It establishes the conditions under which the two fixed-point semantics for higher-order logic programs coincide or differ, enhancing understanding of their relationship.

Findings

Semantics coincide for definitional programs.

Semantics differ when existential higher-order variables are present.

Provides insights into the semantic foundations of higher-order logic programming.

Abstract

Two distinct research approaches have been proposed for assigning a purely extensional semantics to higher-order logic programming. The former approach uses classical domain theoretic tools while the latter builds on a fixed-point construction defined on a syntactic instantiation of the source program. The relationships between these two approaches had not been investigated until now. In this paper we demonstrate that for a very broad class of programs, namely the class of definitional programs introduced by W. W. Wadge, the two approaches coincide (with respect to ground atoms that involve symbols of the program). On the other hand, we argue that if existential higher-order variables are allowed to appear in the bodies of program rules, the two approaches are in general different. The results of the paper contribute to a better understanding of the semantics of higher-order logic programming.