Extensional higher-order logic programming has been recently proposed as an interesting extension of classical logic programming. An important characteristic of the new paradigm is that it preserves all the well-known properties oftraditional logic programming. In this paper we enhance extensional higher-order logic programming with constructive negation. We argue that the main ideas underlying constructive negation are quite close to the existing proof procedurefor extensional higher-order logic programming and for this reason the two notions amalgamate quite conveniently. We demonstrate the soundness of the resulting proof procedure and describe an actual implementation of a language that embodies the above ideas. In this way we obtain the first (to our knowledge) higher-order logic programming language supporting constructive negation and offering a new style of programming that genuinely extends that of traditional logic programming.

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.