59 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
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 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 -translation of rules
belonging to the Semantic Web Rule Language (SWRL)
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
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
The Satisfiability Problem for Boolean Set Theory with a Choice Correspondence
Given a set U of alternatives, a choice (correspondence) on U is a
contractive map c defined on a family Omega of nonempty subsets of U.
Semantically, a choice c associates to each menu A in Omega a nonempty subset
c(A) of A comprising all elements of A that are deemed selectable by an agent.
A choice on U is total if its domain is the powerset of U minus the empty set,
and partial otherwise. According to the theory of revealed preferences, a
choice is rationalizable if it can be retrieved from a binary relation on U by
taking all maximal elements of each menu. It is well-known that rationalizable
choices are characterized by the satisfaction of suitable axioms of
consistency, which codify logical rules of selection within menus. For
instance, WARP (Weak Axiom of Revealed Preference) characterizes choices
rationalizable by a transitive relation. Here we study the satisfiability
problem for unquantified formulae of an elementary fragment of set theory
involving a choice function symbol c, the Boolean set operators and the
singleton, the equality and inclusion predicates, and the propositional
connectives. In particular, we consider the cases in which the interpretation
of c satisfies any combination of two specific axioms of consistency, whose
conjunction is equivalent to WARP. In two cases we prove that the related
satisfiability problem is NP-complete, whereas in the remaining cases we obtain
NP-completeness under the additional assumption that the number of choice terms
is constant.Comment: In Proceedings GandALF 2017, arXiv:1709.01761. "extended" version at
arXiv:1708.0612
Applying Formal Methods to Networking: Theory, Techniques and Applications
Despite its great importance, modern network infrastructure is remarkable for
the lack of rigor in its engineering. The Internet which began as a research
experiment was never designed to handle the users and applications it hosts
today. The lack of formalization of the Internet architecture meant limited
abstractions and modularity, especially for the control and management planes,
thus requiring for every new need a new protocol built from scratch. This led
to an unwieldy ossified Internet architecture resistant to any attempts at
formal verification, and an Internet culture where expediency and pragmatism
are favored over formal correctness. Fortunately, recent work in the space of
clean slate Internet design---especially, the software defined networking (SDN)
paradigm---offers the Internet community another chance to develop the right
kind of architecture and abstractions. This has also led to a great resurgence
in interest of applying formal methods to specification, verification, and
synthesis of networking protocols and applications. In this paper, we present a
self-contained tutorial of the formidable amount of work that has been done in
formal methods, and present a survey of its applications to networking.Comment: 30 pages, submitted to IEEE Communications Surveys and Tutorial
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