The extended stable models of contradiction removal semantics

Our purpose is to define a semantics that extends Contradiction Removal Semantics just as Extended Stable Model Semantics extends Well Founded Semantics, thus providing the notion of Contradiction Free Extended Stable Models. Contradiction Removal Semantics extends Well Founded Semantics to deal with contradictions arising from the introduction of classical negation. Because the Extended Stable Models structure of a program is useful for expressing defaults and abduction, it is important to study in what way the structure of the Extended Stable Models is affected by Contradiction Removal Semantics when the Well Founded Model is contradictory. Given that the Contradiction Removal Semantics is useful for expressing belief revision and counterfactual reasoning, dealing with the structure of such models is expected to be useful for mixing together all four mentioned kinds of reasoning within a single common framework. The Contradiction Free Extended Stable Semantics is an extension to the definition of Contradiction Removal Semantics, provided here in a form independent from the Well Founded Semantics, in terms of a fixpoint operator, instead of in terms of the Well Founded Semantics of transformed programs as before. This definition also clarifies how the semantics is useful for Belief Revision. Nevertheless, we also define a single program transformation such that the Extended Stable Models of a transformed program correspond exactly to the Contradiction Free Extended Stable Models of the original one. By relying on the Well Founded Semantics of the transformed program no new model determining procedures are needed.