26,586 research outputs found
Bipolar querying of valid-time intervals subject to uncertainty
Databases model parts of reality by containing data representing properties of real-world objects or concepts. Often, some of these properties are time-related. Thus, databases often contain data representing time-related information. However, as they may be produced by humans, such data or information may contain imperfections like uncertainties. An important purpose of databases is to allow their data to be queried, to allow access to the information these data represent. Users may do this using queries, in which they describe their preferences concerning the data they are (not) interested in. Because users may have both positive and negative such preferences, they may want to query databases in a bipolar way. Such preferences may also have a temporal nature, but, traditionally, temporal query conditions are handled specifically. In this paper, a novel technique is presented to query a valid-time relation containing uncertain valid-time data in a bipolar way, which allows the query to have a single bipolar temporal query condition
-Generic Computability, Turing Reducibility and Asymptotic Density
Generic computability has been studied in group theory and we now study it in
the context of classical computability theory. A set A of natural numbers is
generically computable if there is a partial computable function f whose domain
has density 1 and which agrees with the characteristic function of A on its
domain. A set A is coarsely computable if there is a computable set C such that
the symmetric difference of A and C has density 0. We prove that there is a
c.e. set which is generically computable but not coarsely computable and vice
versa. We show that every nonzero Turing degree contains a set which is not
coarsely computable. We prove that there is a c.e. set of density 1 which has
no computable subset of density 1. As a corollary, there is a generically
computable set A such that no generic algorithm for A has computable domain. We
define a general notion of generic reducibility in the spirt of Turing
reducibility and show that there is a natural order-preserving embedding of the
Turing degrees into the generic degrees which is not surjective
Effective Choice and Boundedness Principles in Computable Analysis
In this paper we study a new approach to classify mathematical theorems
according to their computational content. Basically, we are asking the question
which theorems can be continuously or computably transferred into each other?
For this purpose theorems are considered via their realizers which are
operations with certain input and output data. The technical tool to express
continuous or computable relations between such operations is Weihrauch
reducibility and the partially ordered degree structure induced by it. We have
identified certain choice principles which are cornerstones among Weihrauch
degrees and it turns out that certain core theorems in analysis can be
classified naturally in this structure. In particular, we study theorems such
as the Intermediate Value Theorem, the Baire Category Theorem, the Banach
Inverse Mapping Theorem and others. We also explore how existing
classifications of the Hahn-Banach Theorem and Weak K"onig's Lemma fit into
this picture. We compare the results of our classification with existing
classifications in constructive and reverse mathematics and we claim that in a
certain sense our classification is finer and sheds some new light on the
computational content of the respective theorems. We develop a number of
separation techniques based on a new parallelization principle, on certain
invariance properties of Weihrauch reducibility, on the Low Basis Theorem of
Jockusch and Soare and based on the Baire Category Theorem. Finally, we present
a number of metatheorems that allow to derive upper bounds for the
classification of the Weihrauch degree of many theorems and we discuss the
Brouwer Fixed Point Theorem as an example
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