Quantifying conflicts in propositional logic through prime implicates

Quantifying conflicts is recognized as an important issue for handling inconsistencies. Indeed, an inconsistency measure can be employed to support knowledge engineers in building a consistent and usable knowledge base or providing insights on how to repair an inconsistent one. Good measures are supposed to satisfy a set of rational properties. However, defining sound properties is sometimes problematic. In this paper, we emphasize one such property, named dominance, rarely satisfied by syntactic measures. Based on prime implicates canonical representation, we first introduce the notion of conflicting variable and use it to refine an existing inconsistency measure defined by minimally unsatisfiable sets (MUSes). Then, we provide a semantics characterization allowing us to establish relationships with multi-valued semantics. Secondly, we propose a new measure based on the notion of deduced MUSes (DMUSes), to circumscribe the internal conflicts in a given knowledge base. We also prove that this measure satisfies a new but weaker form of dominance. Finally, we show how inconsistency measures based on hitting sets of minimal inconsistent sets can be extended using hitting sets of DMUSes.