Probabilistic Hybrid Knowledge Bases under the Distribution Semantics

Since Logic Programming (LP) and Description Logics (DLs) are based on different assumptions (the closed and the open world assumption, respectively), combining them provides higher expressiveness in applications that require both assumptions. Several proposals have been made to combine LP and DLs. An especially successful line of research is the one based on Lifschitz’s logic of Minimal Knowledge with Negation as Failure (MKNF). Motik and Rosati introduced Hybrid knowledge bases (KBs), composed of LP rules and DL axioms, gave them an MKNF semantics and studied their complexity. Knorr et al. proposed a well-founded semantics for Hybrid KBs where the LP clause heads are non-disjunctive, which keeps querying polynomial (provided the underlying DL is polynomial) even when the LP portion is non-stratified. In this paper, we propose Probabilistic Hybrid Knowledge Bases (PHKBs), where the atom in the head of LP clauses and each DL axiom is annotated with a probability value. PHKBs are given a distribution semantics by defining a probability distribution over deterministic Hybrid KBs. The probability of a query being true is the sum of the probabilities of the deterministic KBs that entail the query. Both epistemic and statistical probability can be addressed, thanks to the integration of probabilistic LP and DLs.