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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 antimonotonemonotone. 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.

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.A. Charalambidis et al.The above line of research started many years ago by W. W. Wadge (Wadge 1991) who considered positive higher-order logic programs (i.e., programs without negation in clause bodies). Wadge argued that if such a program obeys some simple and natural syntactic rules, then it has a unique minimum Herbrand model. It is well-known that the minimum model property is a cornerstone of the theory of first-order logic programming (van Emden and Kowalski 1976). In this respect, Wadge's result suggested that it might be possible to extend all the elegant theory of classical logic programming to the higher-order case. The results in (Wadge 1991) were obtained using standard techniques from denotational semantics involving continuous interpretations and Kleene's least fixpoint theorem. A few years after Wadge's initial result, M. Bezem came to similar conclusions (Bezem 1999) but from a different direction. In particular, Bezem demonstrated that by using a fixpoint construction on the ground instantiation of the source higher-order program, one can obtain a model of the original program that satisfies an extensionality condition defined in (Bezem 1999). Despite their different philosophies, Wadge's and Bezem's approaches have recently been shown (Charalambidis et al. 2017) to have close connections. Apart from the above results, recent work (Charalambidis et al. 2013) has also shown that we can define a sound and complete proof procedure for positive higher-order logic programs, which generalizes classical SLD-resolution. In other words, the central results for positive first-order logic programs, generalize to the higher-order case.A natural question that arises is whether one can still obtain an extensional semantics if negation is added to programs. Surprisingly, this question proved harder to resolve. The first result in this direction was reported in (Charalambidis et al. 2014), where it was demonstrated that every higher-order logic program with negation has a minimum extens...

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In this paper we examine the problem of providing a purely extensional three-valued semantics for higher-order logic programs with negation. We demonstrate that a technique that was proposed by M. Bezem for providing extensional semantics to positive higher-order logic programs, fails when applied to higher-order logic programs with negation. On the positive side, we demonstrate that for stratified higher-order logic programs, extensionality is indeed achieved by the technique. We analyze the reasons of the failure of extensionality in the general case, arguing that a three-valued setting can not distinguish between certain predicates that appear to have a different behaviour inside a program context, but which happen to be identical as three-valued relations.