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.