In order to make formalization for granular computing,some kinds of formulas are constructed on a universe by a logical method. Every formula expresses a property, and can separate a semantic set which consists of all...In order to make formalization for granular computing,some kinds of formulas are constructed on a universe by a logical method. Every formula expresses a property, and can separate a semantic set which consists of all of the objects satisfying the formula.Therefore a granular space on the universe is produced based on the formulas, and the semantic sets separated by the formulas are taken as a formal definition for granules,and are called abstract granules.Furthermore,it is proved that any specific granule from an extended mathematical system can be formalized into an abstract granule,the conclusions is obtained that specific granules from approximate spaces and information systems can also be formalized into abstract granules. Based on a granular space and abstract granules,granular computing is defined,which finally realizes the goal of formalization for granular computing.展开更多
Granular computing is a very hot research field in recent years. In our previous work an algebraic quotient space model was proposed,where the quotient structure could not be deduced if the granulation was based on an...Granular computing is a very hot research field in recent years. In our previous work an algebraic quotient space model was proposed,where the quotient structure could not be deduced if the granulation was based on an equivalence relation. In this paper,definitions were given and formulas of the lower quotient congruence and upper quotient congruence were calculated to roughly represent the quotient structure. Then the accuracy and roughness were defined to measure the quotient structure in quantification. Finally,a numerical example was given to demonstrate that the rough representation and measuring methods are efficient and applicable. The work has greatly enriched the algebraic quotient space model and granular computing theory.展开更多
In this article,a real number is defined as a granulation and the real space is transformed into real granular space[1].In the entironment,solution of nonlinear equation is denoted by granulation in real granular spac...In this article,a real number is defined as a granulation and the real space is transformed into real granular space[1].In the entironment,solution of nonlinear equation is denoted by granulation in real granular space.Hence,the research of whole optimization to solve nonlinear equation based on granular computing is proposed[2].In classical case,we solve usually accurate solution of problems.If can't get accurate solution,also finding out an approximate solution to close to accurate solution.But in real space,approximate solution to close to accurate solution is very vague concept.In real granular space,all of the approximate solutions to close to accurate solution are constructed a set,it is a granulation in real granular space.Hence,this granulation is an accurate solution to solve problem in some sense,such,we avoid to say vaguely "approximate solution to close to accurate solution".We introduce the concept of granulation in one dimension real space.Any positive real number a together with moving infinite small distance ε will be constructed an interval [a-ε,a+ε],we call it as granulation in real granular space,denoted by ε(a)or [a].We will discuss related properties and operations[3] of the granulations.Let one dimension real space be R,where each real number a will be generated a granulation,hence we get a granular space R based on real space R.Obviously,R∈R.Infinite small number in real space R is only 0,and there are three infinite small granulations in real number granular space R:[0],[ε] and [-ε].As the graph in Fig.1 shows.In Fig.1,[-ε] is a negative infinite small granulation,[ε] is a positive infinite small granulation,[0] is a infinite small granulation.[a] is a granulation of real number a generating,it could be denoted by interval [a-ε,a+ε] in real space [3-5].Fig.1 Real granulations [0] and [a] Let f(x)=0 be a nonlinear equation,its graph in interval [-3,10] is showed in Fig.2.Where-3≤x≤10 Relation ρ(f| |,ε)is defined as follows:(x1,x2)∈ρ(f| |,ε)iff |f(x1)-f(x2)| < ε Where ε is any given small real number.We have five approximate solution sets on the nonlinear equation f(x)=0 by ρ(f| |,ε)∧|f(x)|[a,b]max,to denote by granulations [(xi1+xi2)/2],[(xi3+xi4)/2],[(xi5+xi6)/2],[(xi7+xi8)/2] and [(xi9+xi10)/2] respectively,where |f(x)|[a,b]max denotes local maximum on x∈[a,b].This is whole optimum on nonlinear equation in interval [-3,10].We will get best optimization solution on nonlinear equation via computing f(x)to use the five solutions denoted by granulation in one dimension real granular space[2,5].展开更多
In this paper, some important issues of granularity are discussed mainly in information systems (ISs) based on binary relation. Firstly, the vector representation method of knowledge granules is proposed in an infor-m...In this paper, some important issues of granularity are discussed mainly in information systems (ISs) based on binary relation. Firstly, the vector representation method of knowledge granules is proposed in an infor-mation system based on binary relation to eliminate limitations of set representation method. Secondly, operators among knowledge granularity are introduced and some important properties of them are studied carefully. Thirdly, distance between two knowledge granules is established and granular space is constructed based on it. Fourthly, axiomatic definition of knowledge granularity is investigated, and one can find that some existed knowledge granularities are special cases under the definition. In addition, as an application of knowledge granular space, an example is employed to validate some results in our work.展开更多
In granular computing granular structures represent knowledge on universe,in this paper several important granular structures are considered.In a general granular structure the notions of interior point, accumulation ...In granular computing granular structures represent knowledge on universe,in this paper several important granular structures are considered.In a general granular structure the notions of interior point, accumulation point and boundary point etc are proposed,by use of these notions and referring to topological method,the lower and upper approximations of a subset of universe are defined such that they are one kind of generalization of the existing approximations based on some special granular structure.Basic properties of new rough set approximations are investigated.Furthermore,granular structures on universe are characterized by the lower and upper approximation operators.展开更多
基金NaturalScienceFund ofHenan ProvinceofChina underGrant No .0611055200
文摘In order to make formalization for granular computing,some kinds of formulas are constructed on a universe by a logical method. Every formula expresses a property, and can separate a semantic set which consists of all of the objects satisfying the formula.Therefore a granular space on the universe is produced based on the formulas, and the semantic sets separated by the formulas are taken as a formal definition for granules,and are called abstract granules.Furthermore,it is proved that any specific granule from an extended mathematical system can be formalized into an abstract granule,the conclusions is obtained that specific granules from approximate spaces and information systems can also be formalized into abstract granules. Based on a granular space and abstract granules,granular computing is defined,which finally realizes the goal of formalization for granular computing.
基金Supported by the National Natural Science Foundation of China(No.61772031)the Special Energy Saving Foundation of Changsha,Hunan Province in 2017
文摘Granular computing is a very hot research field in recent years. In our previous work an algebraic quotient space model was proposed,where the quotient structure could not be deduced if the granulation was based on an equivalence relation. In this paper,definitions were given and formulas of the lower quotient congruence and upper quotient congruence were calculated to roughly represent the quotient structure. Then the accuracy and roughness were defined to measure the quotient structure in quantification. Finally,a numerical example was given to demonstrate that the rough representation and measuring methods are efficient and applicable. The work has greatly enriched the algebraic quotient space model and granular computing theory.
文摘In this article,a real number is defined as a granulation and the real space is transformed into real granular space[1].In the entironment,solution of nonlinear equation is denoted by granulation in real granular space.Hence,the research of whole optimization to solve nonlinear equation based on granular computing is proposed[2].In classical case,we solve usually accurate solution of problems.If can't get accurate solution,also finding out an approximate solution to close to accurate solution.But in real space,approximate solution to close to accurate solution is very vague concept.In real granular space,all of the approximate solutions to close to accurate solution are constructed a set,it is a granulation in real granular space.Hence,this granulation is an accurate solution to solve problem in some sense,such,we avoid to say vaguely "approximate solution to close to accurate solution".We introduce the concept of granulation in one dimension real space.Any positive real number a together with moving infinite small distance ε will be constructed an interval [a-ε,a+ε],we call it as granulation in real granular space,denoted by ε(a)or [a].We will discuss related properties and operations[3] of the granulations.Let one dimension real space be R,where each real number a will be generated a granulation,hence we get a granular space R based on real space R.Obviously,R∈R.Infinite small number in real space R is only 0,and there are three infinite small granulations in real number granular space R:[0],[ε] and [-ε].As the graph in Fig.1 shows.In Fig.1,[-ε] is a negative infinite small granulation,[ε] is a positive infinite small granulation,[0] is a infinite small granulation.[a] is a granulation of real number a generating,it could be denoted by interval [a-ε,a+ε] in real space [3-5].Fig.1 Real granulations [0] and [a] Let f(x)=0 be a nonlinear equation,its graph in interval [-3,10] is showed in Fig.2.Where-3≤x≤10 Relation ρ(f| |,ε)is defined as follows:(x1,x2)∈ρ(f| |,ε)iff |f(x1)-f(x2)| < ε Where ε is any given small real number.We have five approximate solution sets on the nonlinear equation f(x)=0 by ρ(f| |,ε)∧|f(x)|[a,b]max,to denote by granulations [(xi1+xi2)/2],[(xi3+xi4)/2],[(xi5+xi6)/2],[(xi7+xi8)/2] and [(xi9+xi10)/2] respectively,where |f(x)|[a,b]max denotes local maximum on x∈[a,b].This is whole optimum on nonlinear equation in interval [-3,10].We will get best optimization solution on nonlinear equation via computing f(x)to use the five solutions denoted by granulation in one dimension real granular space[2,5].
文摘In this paper, some important issues of granularity are discussed mainly in information systems (ISs) based on binary relation. Firstly, the vector representation method of knowledge granules is proposed in an infor-mation system based on binary relation to eliminate limitations of set representation method. Secondly, operators among knowledge granularity are introduced and some important properties of them are studied carefully. Thirdly, distance between two knowledge granules is established and granular space is constructed based on it. Fourthly, axiomatic definition of knowledge granularity is investigated, and one can find that some existed knowledge granularities are special cases under the definition. In addition, as an application of knowledge granular space, an example is employed to validate some results in our work.
基金supported by grants from the National Natural Science Foundation of China(Nos.11071284 and 61075120)the Natural Science Foundation of Zhejiang Province in China(No.Y107262).
文摘In granular computing granular structures represent knowledge on universe,in this paper several important granular structures are considered.In a general granular structure the notions of interior point, accumulation point and boundary point etc are proposed,by use of these notions and referring to topological method,the lower and upper approximations of a subset of universe are defined such that they are one kind of generalization of the existing approximations based on some special granular structure.Basic properties of new rough set approximations are investigated.Furthermore,granular structures on universe are characterized by the lower and upper approximation operators.
