Answering EL Queries in the Presence of Preferences

Conjunctive query (CQ) answering is an important reasoning task in description logics (DLs). Its goal is to retrieve the tuples of individuals that satisfy a conjunctive query; i.e., a finite set of atomic queries. These tuples are called answers. Clearly, a given CQ may have a considerable number of answers, specially if the set of individual names appearing in the ABox is large, as is the case for many existing DL ontologies. In order to manage all these answers in a structural manner, one can try to extend query answering with preference criteria, in such a way that the most preferred answers are returned first. Possibilistic networks (PNs) have arisen as a way of representing conditional preferences over a finite set of events in a compact way [1]. The general idea is to provide a possibility degree to each conditional event which is proportional to the preference given to that event. We apply this idea to model the preferences of query answers indirectly, by modeling the preferences over the contexts that entail them. In a nutshell, we divide an EL knowledge base (KB) into contexts, and use a possibilistic network to describe the joint possibility distribution over these contexts. Our formalism is based on ideas previously presented for reasoning under probabilistic uncertainty described by a Bayesian network [3]. The preference of an answer to the query is the possibility degree of the best context that entails this answer. Dually, we also compute, given a query, the most preferred source; that is, the context with the highest degree that entails this query. Similar to Bayesian networks [4], PNs are graphical models providing a compact representation of a discrete possibility distribution, through some independence assumptions [2]. A possibility distribution over a set Ω is a function Pos : Ω → [0, 1] that intuitively provides a degree of how possible is an event ω ∈ Ω to happen. This function is extended to sets Γ ⊆ Ω by defining Pos(Γ ) = supω∈Γ Pos(ω). The product conditional distribution which is defined by the equation Pos(Γ ∩Θ) = Pos(Γ | Θ) · Pos(Θ). Possibilistic networks decompose a possibility distribution into a product of conditional probability distributions that depend on the structure of a graph. A