Web ontology representation and reasoning via fragments of set theory

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(\D)--admitting features such as min/max cardinality constructs on the left-hand/right-hand side of inclusion axioms, role chain axioms, and datatypes--which turns out to be quite expressive if compared with \mathcal{SROIQ}(\D), the description logic underpinning the Web Ontology Language OWL. Then we show that the consistency problem for \mathcal{DL}\langle 4LQS^R\rangle(\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(\D)-knowledge bases is \textbf{NP}-complete. Finally, we provide a $4LQS^R$-translation of rules belonging to the Semantic Web Rule Language (SWRL)

The decision problem for a three-sorted fragment of set theory with restricted quantification and finite enumerations

We solve the satisfiability problem for a three-sorted fragment of set theory (denoted $3LQST_0^R$), which admits a restricted form of quantification over individual and set variables and the finite enumeration operator $\{\text{-}, \text{-}, \ldots, \text{-}\}$ over individual variables, by showing that it enjoys a small model property, i.e., any satisfiable formula $\psi$ of $3LQST_0^R$ has a finite model whose size depends solely on the length of $\psi$ itself. Several set-theoretic constructs are expressible by $3LQST_0^R$-formulae, such as some variants of the power set operator and the unordered Cartesian product. In particular, concerning the unordered Cartesian product, we show that when finite enumerations are used to represent the construct, the resulting formula is exponentially shorter than the one that can be constructed without resorting to such terms