Minimum Model Semantics for Extensional Higher-order Logic Programming with Negation

Abstract

Extensional higher-order logic programming has been introduced as a generalization of classical logic programming. An important characteristic of this paradigm is that it preserves all the well-known properties of traditional logic programming. In this paper we consider the semantics of negation in the context of the new paradigm. Using some recent results from non-monotonic fixed-point theory, we demonstrate that every higher-order logic program with negation has a unique minimum infinite-valued model. In this way we obtain the first purely model-theoretic semantics for negation in extensional higher-order logic programming. Using our approach, we resolve an old paradox that was introduced by W. W. Wadge in order to demonstrate the semantic difficulties of higher-order logic programming.

A fixed point theorem for non-monotonic functions

“…One such case is extensional higherorder logic programming [CHRW13], which enhances classical logic programming with higherorder predicates. We have recently used the main results of the present paper in order to obtain a minimum model semantics for extensional higher-order logic programming extended with negation [CER14]. Other possible areas of logic programming that can benefit from the proposed fixed point theorem are disjunctive logic programming with negation [CPRW07] and logic programming with preferences [RT13].…”

Overview of an Abstract Fixed Point Theory for Non-Monotonic Functions and its Applications to Logic Programming

“…This abstract theory was successfully developed and is described in detail in [4]. As an application of these results, this abstract theory was used in [1] in order to obtain the first extensional semantics for higher-order logic programs with negation. Another application of the new theory to the area of non-monotonic formal grammars was proposed in [3].…”

“…We now have all we need in order to apply the main Theorem of [4], getting the following result [1]:…”

“…Such functions prove to be quite common in various investigations in logic programming and formal language theory, and may potentially have other applications. We also describe our development [1], based on the aforementioned abstract framework, of a novel denotational semantics for a very broad extension of classical logic programming. In the rest of this section we provide a short description of the beginnings of our collaboration with Zoltán that led to the above results.…”

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AbstractThe purpose of the present paper is to give an overview of our joint work with ZoltánÉsik, namely the development of an abstract fixed point theory for a class of non-monotonic functions [4] and its use in providing a novel denotational semantics for a very broad extension of classical logic programming [1]. Our purpose is to give a high-level presentation of the main developments of these two works, that avoids as much as possible the underlying technical details, and which can be used as a mild introduction to the area.

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