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In (Bezem 1999; Bezem 2001), M. Bezem defined an extensional semantics forpositive higher-order logic programs. Recently, it was demonstrated in(Rondogiannis and Symeonidou 2016) that Bezem's technique can be extended tohigher-order logic programs with negation, retaining its extensionalproperties, provided that it is interpreted under a logic with an infinitenumber of truth values. In (Rondogiannis and Symeonidou 2017) it was alsodemonstrated that Bezem's technique, when extended under the stable modelsemantics, does not in general lead to extensional stable models. In this paperwe consider the problem of extending Bezem's technique under the well-foundedsemantics. We demonstrate that the well-founded extension fails to retainextensionality in the general case. On the positive side, we demonstrate thatfor stratified higher-order logic programs, extensionality is indeed achieved.We analyze the reasons of the failure of extensionality in the general case,arguing that a three-valued setting can not distinguish between certainpredicates that appear to have a different behaviour inside a program context,but which happen to be identical as three-valued relations. The paper is underconsideration for acceptance in TPLP.

The Intricacies of 3-Valued Extensional Semantics for Higher-Order Logic Programs