Weak extensions for default theories

Abstract

We propose a notion of weak extension of a default theory, motivated by a kind of paraconsistent view of default reasoning, and which coincides with Reiter's extensions when they exist. When the default theory is inconsistent, in the sense that it has no extension, we determine defaults which are not involved in the inconsistency, preserving the possibility to draw some default conclusions. Moreover, any finite default theory has weak extensions, though the question remains open for infinite default theories. Weak extensions correspond to some consistent labelings of the defaults, according to a notion of consistency close to the standard one in the TMS, so computing weak extensions may adapt standard methods of the TMS. We also compare weak extensions to propositions based on a maximal consistent set of defaults, and show that maximal consistency generate far more extensions. Finally, we briefly sketch how our results can be extended to Autoepistemic Logic and to Logic Programming.