Logic in the Time of WWW: An OWL View

Abstract

This paper analyses the classical automated reasoning problem-given a theory \(\mathcal{T}\), a set \(\mathcal{A}\) of ground atoms and a formula ϕ, decide whether \((\mathcal{T}, \mathcal{A}) \models \varphi\)–in the context of the OWL 2 Web Ontology Language and ontology-based data access (OBDA). In a typical OBDA scenario, \(\mathcal{T}\) is an OWL 2 ‘ontology’ providing a user-oriented view of raw data \(\mathcal{A}\) and ϕ(x) is a query with answer variables x. Unlike classical automated reasoning, an important requirement for OBDA is that it should scale to large amounts of data and preferably be as efficient as standard relational database management systems. There are various ways of formalising this requirement, which give rise to different fragments of firstorder logic suitable for OBDA. For example, according to the query-rewriting approach of [1], given \(\mathcal{T}, \mathcal{A}\) and ϕ(x), one has to compute a new query ϕ′(x), independently of \(\mathcal{A}\), such that, for any tuple a, \((\mathcal{T}, \mathcal{A}) \models \varphi(a)\) iff \(\mathcal{A} \models \varphi'(a)\). As a result, this approach can only be applicable to the languages for which query-answering belongs to the class AC0 for data complexity (that is, if only \(\mathcal{A}\) is regarded as input, whereas both \(\mathcal{T}\) and ϕ are regarded as fixed). In the combined approach to OBDA [4,2], given \(\mathcal{T}\) , \(\mathcal{A}\) and ϕ(x), one has to compute new (i) \(\mathcal{A}' \supseteq \mathcal{A}\) in polynomial time in \(\mathcal{T}\) and \(\mathcal{A}\), and (ii) ϕ′(x) in polynomial tome in \(\mathcal{T}\) and ϕ such that, for any tuple a, \((\mathcal{T}, \mathcal{A}) \models \varphi(a)\) iff \(\mathcal{A}' \models \varphi'(a)\). The combined approach can be used for languages with polynomial queryanswering.