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

A set-theoretical approach for ABox reasoning services (Extended Version)

In this paper we consider the most common ABox reasoning services for the description logic $\mathcal{DL}\langle \mathsf{4LQS^{R,\!\times}}\rangle(\mathbf{D})$ ($\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$, for short) and prove their decidability via a reduction to the satisfiability problem for the set-theoretic fragment \flqsr. The description logic $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$ is very expressive, as it admits various concept and role constructs, and data types, that allow one to represent rule-based languages such as SWRL. Decidability results are achieved by defining a generalization of the conjunctive query answering problem, called HOCQA (Higher Order Conjunctive Query Answering), that can be instantiated to the most wide\-spread ABox reasoning tasks. We also present a \ke\space based procedure for calculating the answer set from $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$ knowledge bases and higher order $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$ conjunctive queries, thus providing means for reasoning on several well-known ABox reasoning tasks. Our calculus extends a previously introduced \ke\space based decision procedure for the CQA problem.Comment: 27 pages. Extended version for RR 2017. arXiv admin note: text overlap with arXiv:1606.0733

A \textsf{C++} reasoner for the description logic \shdlssx (Extended Version)

We present an ongoing implementation of a \ke\space based reasoner for a decidable fragment of stratified elementary set theory expressing the description logic \dlssx (shortly \shdlssx). The reasoner checks the consistency of \shdlssx-knowledge bases (KBs) represented in set-theoretic terms. It is implemented in \textsf{C++} and supports \shdlssx-KBs serialized in the OWL/XML format. To the best of our knowledge, this is the first attempt to implement a reasoner for the consistency checking of a description logic represented via a fragment of set theory that can also classify standard OWL ontologies.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1702.03096, arXiv:1804.1122

A set-based reasoner for the description logic \shdlssx (Extended Version)

We present a \ke-based implementation of a reasoner for a decidable fragment of (stratified) set theory expressing the description logic \dlssx (\shdlssx, for short). Our application solves the main TBox and ABox reasoning problems for \shdlssx. In particular, it solves the consistency problem for \shdlssx-knowledge bases represented in set-theoretic terms, and a generalization of the \emph{Conjunctive Query Answering} problem in which conjunctive queries with variables of three sorts are admitted. The reasoner, which extends and optimizes a previous prototype for the consistency checking of \shdlssx-knowledge bases (see \cite{cilc17}), is implemented in \textsf{C++}. It supports \shdlssx-knowledge bases serialized in the OWL/XML format, and it admits also rules expressed in SWRL (Semantic Web Rule Language).Comment: arXiv admin note: text overlap with arXiv:1804.11222, arXiv:1707.07545, arXiv:1702.0309

Solvable Set/Hyperset Contexts: III. A Tableau System for a Fragment of Hyperset Theory

We propose a decision procedure for a fragment of the hyperset theory, HMLSS, which takes inspiration from a tableau saturation strategy presented in [3] for the fragment MLSS of well-founded set theory. The procedure alternates deduction and model checking steps, driving the correct application of otherwise very liberal rules, thus significantly speeding up the process of discovering a satisfying assignment of a given HMLSS-formula or proving that no such assignment exists

A Decidable Quantified Fragment of Set Theory Involving Ordered Pairs with Applications to Description Logics

We present a decision procedure for a quantified fragment of set theory involving ordered pairs and some operators to manipulate them. When our decision procedure is applied to formulae in this fragment whose quantifier prefixes have length bounded by a fixed constant, it runs in nondeterministic polynomial-time. Related to this fragment, we also introduce a description logic which provides an unusually large set of constructs, such as, for instance, Boolean constructs among roles. The set-theoretic nature of the description logics semantics yields a straightforward reduction of the knowledge base consistency problem to the satisfiability problem for formulae of our fragment with quantifier prefixes of length at most 2, from which the NP-completeness of reasoning in this novel description logic follows. Finally, we extend this reduction to cope also with SWRL rules