Approximating Horn Knowledge Bases in Regular Description Logics to Have PTIME Data Complexity

Abstract

This work is a continuation of our previous works [4,5]. We assume that the reader is familiar with description logics (DLs). A knowledge base in a description logic is a tuple $({\mathcal{R}},{\mathcal{T}},{\mathcal{A}})$ consisting of an RBox ${\mathcal{R}}$ of assertions about roles, a TBox ${\mathcal{T}}$ of global assumptions about concepts, and an ABox ${\mathcal{A}}$ of facts about individuals (objects) and roles. The instance checking problem in a DL is to check whether a given individual a is an instance of a concept C w.r.t. a knowledge base $({\mathcal{R}},{\mathcal{T}},{\mathcal{A}})$, written as $({\mathcal{R}},{\mathcal{T}},{\mathcal{A}}) \models C(a)$. This problem in DLs including the basic description logic $\mathcal{ALC}$ (with ${\mathcal{R}} = \emptyset$) is EXPTIME-hard.From the point of view of deductive databases, ${\mathcal{A}}$ is assumed to be much larger than ${\mathcal{R}}$ and ${\mathcal{T}}$, and it makes sense to consider the data complexity, which is measured when the query consisting of ${\mathcal{R}}$, ${\mathcal{T}}$, C, a is fixed while ${\mathcal{A}}$ varies as input data.It is desirable to find and study fragments of DLs with PTIME data complexity. Several authors have recently introduced a number of Horn fragments of DLs with PTIME data complexity [2,1,3]. The most expressive fragment from those is Horn-$\mathcal{SHIQ}$ introduced by Hustadt et al. [3]. It assumes, however, that the constructor ∀ R.C does not occur in bodies of program clauses and goals.The data complexity of the “general Horn fragment of $\mathcal{ALC}$” is coNP-hard [6]. So, to obtain PTIME data complexity one has to adopt some restrictions for the “general Horn fragments of DLs”. The goal is to find as less restrictive conditions as possible.