Reasoning with maximal consistent signatures
Matthias Thimm, Jandson Santos Ribeiro Santos
TL;DR
The paper addresses reasoning with inconsistent knowledge by replacing subset-based maximal consistent subsets with maximal consistent subsignatures defined via forgetting the remaining propositions. It formalizes forgetting operators $\boxminus$ and projection $K|^{\boxminus}_S$, defines semantic notions $\mathsf{MISig}^{\boxminus}(K)$ and $\mathsf{MCSig}^{\boxminus}(K)$, and proves that hitting set duality persists in this signature-based setting. It then introduces inevitable and weak inference ${|\sim}^{\boxminus}_{i}$ and ${|\sim}^{\boxminus}_{w}$, analyzes their rationality properties, and provides detailed complexity results (DP, $\Sigma^{P}_{2}$, $\Pi^{P}_{2}$) for various decision tasks, while connecting the framework to inconsistency measurement and Priest's three-valued paraconsistent logic. The work offers a unified, semantics-driven approach to inconsistency-tolerant reasoning with practical implications for knowledge bases and paraconsistent reasoning research.
Abstract
We analyse a specific instance of the general approach of reasoning based on forgetting by Lang and Marquis. More precisely, we discuss an approach for reasoning with inconsistent information using maximal consistent subsignatures, where a maximal consistent subsignature is a maximal set of propositions such that forgetting the remaining propositions restores consistency. We analyse maximal consistent subsignatures and the corresponding minimal inconsistent subsignatures in-depth and show, among others, that the hitting set duality applies for them as well. We further analyse inference relations based on maximal consistent subsignatures wrt. rationality postulates from non-monotonic reasoning and computational complexity. We also consider the relationship of our approach with inconsistency measurement and paraconsistent reasoning.