Abstract
In this paper we use results from Computable Set Theory as a means to represent and reason about description logics and rule languages for the semantic web.
Specifically, we introduce the description logic \(\mathcal {DL}\langle 4LQS^R\rangle (\mathbf {D})\)–allowing features such as min/max cardinality constructs on the left-hand/right-hand side of inclusion axioms, role chain axioms, and datatypes–which turn out to be quite expressive if compared with \(\mathcal {SROIQ}(\mathbf {D})\), the description logic underpinning the Web Ontology Language OWL. Then we show that the consistency problem for \(\mathcal {DL}\langle 4LQS^R\rangle (\mathbf {D})\)-knowledge bases is decidable by reducing it, through a suitable translation process, to the satisfiability problem of the stratified fragment \(4LQS^R\) of set theory, involving variables of four sorts and a restricted form of quantification. We prove also that, under suitable not very restrictive constraints, the consistency problem for \(\mathcal {DL}\langle 4LQS^R\rangle (\mathbf {D})\)-knowledge bases is NP-complete. Finally, we provide a \(4LQS^R\)-translation of rules belonging to the Semantic Web Rule Language (SWRL).
Work partially supported by the FIR project COMPACT: Computazione affidabile su testi firmati, code: D84C46.
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Notes
- 1.
The notion of syntax tree for 4LQS-formulae is similar to the notion of syntax tree for formulae of first-order logic. A precise definition of the latter can be found in [11].
- 2.
The use of level 3 variables to model abstract and concrete role terms is motivated by the fact that their elements, that is ordered pairs \(\langle x, y \rangle \), are encoded in Kuratowski’s style as \(\{\{x\}, \{x,y\}\}\), namely as collections of sets of objects. Variables of level 2 are used in the formulae \(\psi _8\) and \(\psi _9\) of the construction to model the fact that level 3 variables representing role terms are binary relations.
- 3.
We recall that a logic is non-monotonic if some conclusions can be invalidated when more knowledge is added.
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Cantone, D., Longo, C., Nicolosi-Asmundo, M., Santamaria, D.F. (2015). Web Ontology Representation and Reasoning via Fragments of Set Theory. In: ten Cate, B., Mileo, A. (eds) Web Reasoning and Rule Systems. RR 2015. Lecture Notes in Computer Science(), vol 9209. Springer, Cham. https://doi.org/10.1007/978-3-319-22002-4_6
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