Wyniki wyszukiwania - Biblioteka Nauki

Ten serwis zostanie wyłączony 2025-02-11.

Nowa wersja platformy, zawierająca wyłącznie zasoby pełnotekstowe, jest już dostępna.
Przejdź na https://bibliotekanauki.pl

Ograniczanie wyników

Znaleziono wyników: 3

Liczba wyników na stronie

Wyniki wyszukiwania

Wyszukiwano:
w słowach kluczowych: lower approximation

Sortuj według:

Ogranicz wyniki do:

Binary Relations-based Rough Sets – an Automated Approach

100%

Grabowski A.

Formalized Mathematics

2016

tom 24

nr 2

143-155

Rough sets, developed by Zdzisław Pawlak [12], are an important tool to describe the state of incomplete or partially unknown information. In this article, which is essentially the continuation of [8], we try to give the characterization of approximation operators in terms of ordinary properties of underlying relations (some of them, as serial and mediate relations, were not available in the Mizar Mathematical Library [11]). Here we drop the classical equivalence- and tolerance-based models of rough sets trying to formalize some parts of [18]. The main aim of this Mizar article is to provide a formal counterpart for the rest of the paper of William Zhu [18]. In order to do this, we recall also Theorem 3 from Y.Y. Yao’s paper [17]. The first part of our formalization (covering first seven pages) is contained in [8]. Now we start from page 5003, sec. 3.4. [18]. We formalized almost all numbered items (definitions, propositions, theorems, and corollaries), with the exception of Proposition 7, where we stated our theorem only in terms of singletons. We provided more thorough discussion of the property positive alliance and its connection with seriality and reflexivity (and also transitivity). Examples were not covered as a rule as we tried to construct a more general mechanism of finding appropriate models for approximation spaces in Mizar providing more automatization than it is now [10]. Of course, we can see some more general applications of some registrations of clusters, essentially not dealing with the notion of an approximation: the notions of an alliance binary relation were not defined in the Mizar Mathematical Library before, and we should think about other properties which are also absent but needed in the context of rough approximations [9], [5]. Via theory merging, using mechanisms described in [6] and [7], such elementary constructions can be extended to other frameworks.

Rough Approximation Operators in Group Mapping and Their Applications to Coding Theory

88%

Xu J. , Qiu R. , Tao Z.

Fundamenta Informaticae

2016

tom Vol. 145, nr 1

93--109

This paper focuses on rough approximation operators in group mapping. The relationships between rough set theory and group theory are considered from a novel perspective. The necessary and sufficient conditions for the upper approximation and lower approximation of a group to be groups are analyzed. In addition, the homomorphism and isomorphism between two groups which have related upper or lower approximations are investigated. Finally, the applications of rough approximation operators in group mapping to coding theory are developed.

Interval regression analysis with polynomial and its similarity to rough sets concept

63%

Tanaka H. , Lee H.

Fundamenta Informaticae

1999

tom Vol. 37, Nr 1,2

71-87

This paper proposes interval regression analysis with polynomials. For data sets with crisp inputs and interval outputs, three estimation models called as an upper, a lower, and a possi-bility estimation models can be formulated from the concepts of the possibility and necessity measures. Always there exists an upper and a possibility estimation model when a linear sys-tem with interval coefficients is considered, but it is not assured to attain a solution for a lower estimation model in an interval linear system. If we can not obtain the lower estimation model, it might be caused by adopting a model not fitting to the given data. Thus we consider polynomials to find a regression model which fits well to the given observations. The possibility model is used to check the existence of the lower model. If we can find a proper lower model, the estimated upper and lower models deserve more credit than the previous models in the former studies. We also introduce the measure of fitness to gauge the degree of approximation of the obtained models to the given data. The upper and lower estimation models in interval regression analysis can be considered as the upper and lower approximation in rough sets. The similarity between the interval estimation models and the rough sets concept is also discussed. In order to illustrate our approach, numerical examples are shown.