22,411 research outputs found
Characteristic of partition-circuit matroid through approximation number
Rough set theory is a useful tool to deal with uncertain, granular and
incomplete knowledge in information systems. And it is based on equivalence
relations or partitions. Matroid theory is a structure that generalizes linear
independence in vector spaces, and has a variety of applications in many
fields. In this paper, we propose a new type of matroids, namely,
partition-circuit matroids, which are induced by partitions. Firstly, a
partition satisfies circuit axioms in matroid theory, then it can induce a
matroid which is called a partition-circuit matroid. A partition and an
equivalence relation on the same universe are one-to-one corresponding, then
some characteristics of partition-circuit matroids are studied through rough
sets. Secondly, similar to the upper approximation number which is proposed by
Wang and Zhu, we define the lower approximation number. Some characteristics of
partition-circuit matroids and the dual matroids of them are investigated
through the lower approximation number and the upper approximation number.Comment: 12 page
Knowledge Engineering from Data Perspective: Granular Computing Approach
The concept of rough set theory is a mathematical approach to uncertainly and vagueness in data analysis, introduced by Zdzislaw Pawlak in 1980s. Rough set theory assumes the underlying structure of knowledge is a partition. We have extended Pawlak’s concept of knowledge to coverings. We have taken a soft approach regarding any generalized subset as a basic knowledge. We regard a covering as basic knowledge from which the theory of knowledge approximations and learning, knowledge dependency and reduct are developed
Parametric matroid of rough set
Rough set is mainly concerned with the approximations of objects through an
equivalence relation on a universe. Matroid is a combinatorial generalization
of linear independence in vector spaces. In this paper, we define a parametric
set family, with any subset of a universe as its parameter, to connect rough
sets and matroids. On the one hand, for a universe and an equivalence relation
on the universe, a parametric set family is defined through the lower
approximation operator. This parametric set family is proved to satisfy the
independent set axiom of matroids, therefore it can generate a matroid, called
a parametric matroid of the rough set. Three equivalent representations of the
parametric set family are obtained. Moreover, the parametric matroid of the
rough set is proved to be the direct sum of a partition-circuit matroid and a
free matroid. On the other hand, since partition-circuit matroids were well
studied through the lower approximation number, we use it to investigate the
parametric matroid of the rough set. Several characteristics of the parametric
matroid of the rough set, such as independent sets, bases, circuits, the rank
function and the closure operator, are expressed by the lower approximation
number.Comment: 15 page
Covering matroid
In this paper, we propose a new type of matroids, namely covering matroids,
and investigate the connections with the second type of covering-based rough
sets and some existing special matroids. Firstly, as an extension of
partitions, coverings are more natural combinatorial objects and can sometimes
be more efficient to deal with problems in the real world. Through extending
partitions to coverings, we propose a new type of matroids called covering
matroids and prove them to be an extension of partition matroids. Secondly,
since some researchers have successfully applied partition matroids to
classical rough sets, we study the relationships between covering matroids and
covering-based rough sets which are an extension of classical rough sets.
Thirdly, in matroid theory, there are many special matroids, such as
transversal matroids, partition matroids, 2-circuit matroid and
partition-circuit matroids. The relationships among several special matroids
and covering matroids are studied.Comment: 15 page
Some characteristics of matroids through rough sets
At present, practical application and theoretical discussion of rough sets
are two hot problems in computer science. The core concepts of rough set theory
are upper and lower approximation operators based on equivalence relations.
Matroid, as a branch of mathematics, is a structure that generalizes linear
independence in vector spaces. Further, matroid theory borrows extensively from
the terminology of linear algebra and graph theory. We can combine rough set
theory with matroid theory through using rough sets to study some
characteristics of matroids. In this paper, we apply rough sets to matroids
through defining a family of sets which are constructed from the upper
approximation operator with respect to an equivalence relation. First, we prove
the family of sets satisfies the support set axioms of matroids, and then we
obtain a matroid. We say the matroids induced by the equivalence relation and a
type of matroid, namely support matroid, is induced. Second, through rough
sets, some characteristics of matroids such as independent sets, support sets,
bases, hyperplanes and closed sets are investigated.Comment: 13 page
Approximations from Anywhere and General Rough Sets
Not all approximations arise from information systems. The problem of fitting
approximations, subjected to some rules (and related data), to information
systems in a rough scheme of things is known as the \emph{inverse problem}. The
inverse problem is more general than the duality (or abstract representation)
problems and was introduced by the present author in her earlier papers. From
the practical perspective, a few (as opposed to one) theoretical frameworks may
be suitable for formulating the problem itself. \emph{Granular operator spaces}
have been recently introduced and investigated by the present author in her
recent work in the context of antichain based and dialectical semantics for
general rough sets. The nature of the inverse problem is examined from
number-theoretic and combinatorial perspectives in a higher order variant of
granular operator spaces and some necessary conditions are proved. The results
and the novel approach would be useful in a number of unsupervised and semi
supervised learning contexts and algorithms.Comment: 20 Pages. Scheduled to appear in IJCRS'2017 LNCS Proceedings,
Springe
- …