Truth and knowledge fixpoint semantics for many-valued logic programming

The variety of semantical approaches that have been invented for logic programs for reasoning about inconsistent databases is quite broad. Especially, we are interested in an ontological encapsulation of a many-valued logics with negation, based on bilattices, into a 2-valued logic: this new approach is a minimilistic one based on the concept of the semantic reflection. We define a Model theoretic approach for Herbrand interpretations of an encapsulated logic program and the new semantics for implication of their clauses. We introduce a trilattice of Herbrand interpretations, with the third truth dimension for this ’meta’ logic (over a bilattice of the many-valued logic), where is possible to apply the KnasterTarski theorem for a truth-monotonic ’immediate consequence operator’ instead of knowledge-monotonic Fitting’s operator for a many-valued logic programs. We extend the Fitting’s fixpoint semantics for 3-valued logic programs and show the strong connection with the fixpoint semantics of ’meta’ logic programs obtained by encapsulation of many-valued programs.