Minimum Model Semantics for Extensional Higher-order Logic Programming with Negation
This work addresses semantic difficulties in higher-order logic programming for researchers in logic and programming languages, but it is incremental as it builds on existing paradigms.
The paper tackles the semantics of negation in extensional higher-order logic programming, demonstrating that every such program has a unique minimum infinite-valued model, thereby resolving an old paradox introduced by W. W. Wadge.
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.