Handling and measuring inconsistency in non-monotonic logics

Abstract

We address the issue of quantitatively assessing the severity of inconsistencies in non-monotonic frameworks. While measuring inconsistency in classical logics has been investigated for some time now, taking the non-monotonicity into account poses new challenges. In order to tackle them, we focus on the structure of minimal strongly K-inconsistent subsets of a knowledge base K—a sound generalization of minimal inconsistent subsets to arbitrary, possibly non-monotonic, frameworks which induces a generalization of Reiter's famous hitting set duality between minimal inconsistent and maximal consistent subsets of a knowledge base. We propose measures based on this notion and investigate their behavior in a non-monotonic setting by revisiting existing rationality postulates, analyzing the compliance of the proposed measures with these postulates, and by investigating their computational complexity. Motivated by the observation that a knowledge base of a non-monotonic logic can also be repaired by adding formulas – whereas Reiter's duality is only concerned about removing –, we also investigate situations where we are given potential additional assumptions to repair a knowledge base. For this, we characterize the minimal modifications to a knowledge base in terms of a hitting set duality