31 research outputs found
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 ), which admits a restricted form of quantification over
individual and set variables and the finite enumeration operator over individual variables, by showing that it
enjoys a small model property, i.e., any satisfiable formula of
has a finite model whose size depends solely on the length of
itself. Several set-theoretic constructs are expressible by
-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 \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
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
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
An optimized KE-tableau-based system for reasoning in the description logic \shdlssx (Extended Version)
We present a \ke-based procedure for the main TBox and ABox reasoning tasks
for the description logic \dlssx, in short \shdlssx. The logic \shdlssx,
representable in the decidable multi-sorted quantified set-theoretic fragment
\flqsr, combines the high scalability and efficiency of rule languages such
as the Semantic Web Rule Language (SWRL) with the expressivity of description
logics. %In fact it supports, among other features, Boolean operations on
concepts and roles, role constructs such as the product of concepts and role
chains on the left hand side of inclusion axioms, and role properties such as
transitivity, symmetry, reflexivity, and irreflexivity.
Our algorithm is based on a variant of the \ke\space system for sets of
universally quantified clauses, where the KE-elimination rule is generalized in
such a way as to incorporate the -rule. The novel system, called \keg,
turns out to be an improvement of the system introduced in \cite{RR2017} and of
standard first-order \ke x \cite{dagostino94}. Suitable benchmark test sets
executed on C++ implementations of the three mentioned systems show that the
performances of the \keg-based reasoner are often up to about 400\% better than
the ones of the other two systems. This a first step towards the construction
of efficient reasoners for expressive OWL ontologies based on fragments of
computable set-theory.Comment: arXiv admin note: text overlap with arXiv:1702.03096,
arXiv:1805.0860