78 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 decidable quantified fragment of set theory with ordered pairs and some undecidable extensions
In this paper we address the decision problem for a fragment of set theory
with restricted quantification which extends the language studied in [4] with
pair related quantifiers and constructs, in view of possible applications in
the field of knowledge representation. We will also show that the decision
problem for our language has a non-deterministic exponential time complexity.
However, for the restricted case of formulae whose quantifier prefixes have
length bounded by a constant, the decision problem becomes NP-complete. We also
observe that in spite of such restriction, several useful set-theoretic
constructs, mostly related to maps, are expressible. Finally, we present some
undecidable extensions of our language, involving any of the operators domain,
range, image, and map composition.
[4] Michael Breban, Alfredo Ferro, Eugenio G. Omodeo and Jacob T. Schwartz
(1981): Decision procedures for elementary sublanguages of set theory. II.
Formulas involving restricted quantifiers, together with ordinal, integer, map,
and domain notions. Communications on Pure and Applied Mathematics 34, pp.
177-195Comment: In Proceedings GandALF 2012, arXiv:1210.202
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
Set-syllogistics meet combinatorics
This paper considers 03* 00* prenex sentences of pure first-order predicate calculus with equality. This is the set of formulas which Ramsey's treated in a famous article of 1930. We demonstrate that the satisfiability problem and the problem of existence of arbitrarily large models for these formulas can be reduced to the satisfiability problem for 03* 00* prenex sentences of Set Theory (in the relators 08, =). We present two satisfiability-preserving (in a broad sense) translations \u3a6 \u21a6 (Formula presented.) and \u3a6 \u21a6 \u3a6\u3c3 of 03* 00* sentences from pure logic to well-founded Set Theory, so that if (Formula presented.) is satisfiable (in the domain of Set Theory) then so is \u3a6, and if \u3a6\u3c3 is satisfiable (again, in the domain of Set Theory) then \u3a6 can be satisfied in arbitrarily large finite structures of pure logic. It turns out that |(Formula presented.)| = (Formula presented.)(|\u3a6|) and |\u3a6\u3c3| = (Formula presented.)(|\u3a6|2). Our main result makes use of the fact that 03* 00* sentences, even though constituting a decidable fragment of Set Theory, offer ways to describe infinite sets. Such a possibility is exploited to glue together infinitely many models of increasing cardinalities of a given 03* 00* logical formula, within a single pair of infinite sets
On the Convexity of a Fragment of Pure Set Theory with Applications within a Nelson-Oppen Framework
In Proceedings GandALF 2021, arXiv:2109.0779
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