The stable model semantics under the any-world assumption

The stable model semantics has become a dominating approach to complete the knowledge provided by a logic program by means of the Closed World Assumption (CWA). The CWA asserts that any atom whose truth-value cannot be inferred from the facts and rules is supposed to be false. This assumption is orthogonal to the so-called the Open World Assumption (OWA), which asserts that every such atom's truth is supposed to be unknown. The topic of this paper is to be more fine-grained. Indeed, the objective is to allow any assignment (ie. interpretation) over a truth space, to be a default assumption. Informally, rather than to rely on the same default value for all atoms (false under the CWA, unknown under the OWA), we allow arbitrary assignments to complete the information provided a logic program. It turns out that thus the CWA and the OWA are just two particular, yet important, cases. Indeed, our extension is conservative in the following sense: (i) if we restrict our attention to the usual uniform OWA, then the semantics reduces to the Kripke-Kleene semantics, and (ii) if we restrict our attention to the uniform CWA, then our semantics reduces to the stable model semantics.