Smallest equivalent sets for finite propositional formula circumscription

Abstract

Circumscription uses classical logic in order to modelize rules with exceptions and implicit knowledge. Formula circumscription is known to be easier to use in order to modelize a situation. We describe when two sets of formulas give the same result, when circumscribed. Two kinds of such equivalence are interesting: the ordinary one (two sets give the same circumscription) and the strong one (when completed by any arbitrary set, the two sets give the same circumscription) which corresponds to having the same closure for logical "and" and "or". In this paper, we consider only the finite case, focusing on looking for the smallest possible sets equivalent to a given set, for the two kinds of equivalence.We need to revisit a characterization result of formula circumscription. Then, we are able to describe a way to get all the sets equivalent to a given set, and also a way to get the smallest such sets. These results should help the automatic computation, and also the translation in terms of circumscription of complex situations.