Context-Sensitive Bayesian Description Logics

Research embracing context-sensitivity in the domain of knowledge representation (KR) has been scarce, especially when the context is uncertain. The current study deals with this problem from a logic perspective and provides a framework combining Description Logics (DLs) with Bayesian Networks (BNs). In this framework, we use BNs to describe contexts that are semantically linked to classical DL axioms. As an application scenario, we consider the Bayesian extension BEL of the lightweight DL EL. We define four reasoning problems; namely, precise subsumption, positive subsumption, certain subsumption and finding the most likely context for a subsumption. We provide an algorithm that solves the precise subsumption in PSPACE. Positive subsumption is shown to be NP-complete and certain subsumption coNP-complete. We present a completion-like algorithm, which is in EXPTIME, to find the most likely context for a subsumption. The scenario is then generalised to Bayesian extensions of classic-valued, monotonic DLs, where precise entailment, positive entailment, certain entailment and finding the most likely context for an entailment are defined as lifted reasoning problems. It is shown that precise entailment, positive entailment and certain entailment can be solved by generalising the algorithms developed for the corresponding reasoning problems in BEL. Lastly, the complexities of these problems are shown to be bound with the complexity of entailment checking in the underlying DL, provided this is PSPACE-hard.