Semantic Bridges Between First Order c-Representations and Cost-Based Semantics: An Initial Perspective
Nicholas Leisegang, Giovanni Casini, Thomas Meyer
TL;DR
This work analyzes semantic bridges between cost-based semantics for inconsistent DLs and c-representation semantics for defeasible reasoning in $ALCO$. It formalizes both frameworks, constructs translations $K^{\kappa}_{\omega}$ and $\kappa_{\omega}$, and proves that, under certain conditions, the interpretation ordering and select entailments align across frameworks. The contributions include precise compatibility conditions (e.g., $c$-compatibility, strong $c$-compatibility), and results linking skeptical/credulous inferences to opt-entailment on translated KBs. The findings open avenues for unifying inconsistent DL querying and defeasible FOL reasoning, enabling cross-pollination of techniques and complexity results.
Abstract
Weighted-knowledge bases and cost-based semantics represent a recent formalism introduced by Bienvenu et al. for Ontology Mediated Data Querying in the case where a given knowledge base is inconsistent. This is done by adding a weight to each statement in the knowledge base (KB), and then giving each DL interpretation a cost based on how often it breaks rules in the KB. In this paper we compare this approach with c-representations, a form of non-monotonic reasoning originally introduced by Kern-Isberner. c-Representations describe a means to interpret defeasible concept inclusions in the first-order case. This is done by assigning a numerical ranking to each interpretations via penalties for each violated conditional. We compare these two approaches on a semantic level. In particular, we show that under certain conditions a weighted knowledge base and a set of defeasible conditionals can generate the same ordering on interpretations, and therefore an equivalence of semantic structures up to relative cost. Moreover, we compare entailment described in both cases, where certain notions are equivalently expressible in both formalisms. Our results have the potential to benefit further work on both cost-based semantics and c-representations