The well-founded semantics of logic programs over bilattices: an alternative Characterisation

The well-founded semantics is one of the most widely studied and used semantics of logic programs with negation. It is well-known that this semantics can be defined in terms of the ``least' (according to the knowledge order) stable-model semantics, where this latter is based on the Gelfond-Lifschitz transformation. Fitting has generalized the Gelfond-Lifschitz transformation to the case where the underlying space of truth is that of bilattices and, thus, has extended the well-founded and the stable model semantics to bilattices. The main idea of the transformation is to separate the roles of positive and negative information, which, on the other hand, avoids the kind of symmetry and natural management of negation that is present in the well-known Kripke-Kleene semantics of logic programs with negation. In this paper, we show that this separation is not necessary. In particular, we show that the well-founded semantics over bilattices can be defined as an extension, based on the closed world assumption, of the Kripke-Kleene semantics. In particular, we define an immediate consequence operator $ecop$, which relies on Kripke-Kleene's $phip$ operator only, whose least fixed-point according to the knowledge order coincides with the well-founded semantics over bilattices. As a consequence, we neither require any separation of roles of positive and negative information nor any program transformation, but rely on a natural management of negation only, making clearer the role of the closed world assumption in the well-founded semantics.