The Intricacies of Three-Valued Extensional Semantics for Higher-Order Logic Programs

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.