  In (Bezem 1999; Bezem 2001), M. Bezem defined an extensional semantics for
positive higher-order logic programs. Recently, it was demonstrated in
(Rondogiannis and Symeonidou 2016) that Bezem's technique can be extended to
higher-order logic programs with negation, retaining its extensional
properties, provided that it is interpreted under a logic with an infinite
number of truth values. In (Rondogiannis and Symeonidou 2017) it was also
demonstrated that Bezem's technique, when extended under the stable model
semantics, does not in general lead to extensional stable models. In this paper
we consider the problem of extending Bezem's technique under the well-founded
semantics. We demonstrate that the well-founded extension fails to retain
extensionality in the general case. On the positive side, we demonstrate that
for 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 certain
predicates that appear to have a different behaviour inside a program context,
but which happen to be identical as three-valued relations. The paper is under
consideration for acceptance in TPLP.
