FunctionS_roughsetsandlawidentification
Science in China Series F: Information Sciences
? 2008 SCIENCE IN CHINA PRESS
Received April 6, 2007; accepted June 20, 2007
doi: 10.1007/s11432-008-0050-0
?Corresponding author (email: shikq@https://www.360docs.net/doc/f316645220.html,)
Supported by the National Natural Science Foundation of China (Grant No. 60364001), the key Project of Chinese Ministry of Education (Grant
No. 206089), and the Natural Science Foundation of Shandong Province of China (Grant No. Y2007H02)
Sci China Ser F-Inf Sci | May 2008 | vol. 51 | no. 5 | 499-510
https://www.360docs.net/doc/f316645220.html, https://www.360docs.net/doc/f316645220.html,
Function S-rough sets and law identification
SHI KaiQuan 1,2? & YAO BingXue 2
1
School of Mathematics and System Science, Shandong University, Jinan 250100, China; 2 School of Mathematics Science, Liaocheng University, Liaocheng 252059, China
By introducing element equivalence class that proposes dynamic characteristic
into Pawlak Z rough sets theory, the first author of this paper improved Pawlak Z
rough sets and put forward S-rough sets (singular rough sets). S-rough sets are
defined by element equivalence class that proposes dynamic characteristic.
S-rough sets have dynamic characteristic. By introducing the function equivalence
class (law equivalence class) that proposes dynamic characteristic into S-rough
sets, the first author improved S-rough sets and put forward function S-rough sets
(function singular rough sets). Function S-rough sets have dynamic characteristic
and law characteristic, and a function is a law. By using function S-rough sets, this
paper presents law identification, law identification theorem, and law identification
criterion and applications. Function S-rough sets are a new research direction of
rough sets theory, and it is also a new tool to the research of system law identifica-
tion.
function S-rough sets, system law, law identification, identification criterion, identification theorem
Information system with multiple outputs or single output needs to identify its outputs law state
in its work time interval 1[,]j k T t t =, so that we can affirm that the output law of information
system on T j 1[,]k t t = is law stabilization or law turbulence, 1,2,,j λ= . The outputs
12, ,,m u u u in the information system with m outputs on 1[,]j k T t t = compose function
equivalence class [1, 2] (law equivalence class) 12[]{,,,}j T m u u u u = with respect to 1[,]j k T t t =,
[]j i T u u ∈is a function, and a function is a law. The outputs 12(),(),,()m u t u t u t λλλ at j t T λ∈
compose element equivalence class (law state value at t λ) 12[]{(),(),,()}t m x u t u t u t λλλλ==
12{,,}m x x x with respect to j t T λ∈.
Refs. [1, 2] presented the concepts of R -function equivalence class and function transfer. Using
these concepts, refs. [1, 2] proposed function S-rough sets (function singular rough sets), gave its
500 SHI KaiQuan et al. Sci China Ser F-Inf Sci | May 2008 | vol. 51 | no. 5 | 499-510
three forms, and proposed function rough sets (function rough sets). Function rough sets are the
reduced and static form of function S-rough. Refs. [3―9] presented further discussions on the
theory and applications of function S-rough sets.
Since function S-rough sets have dynamic characteristic and law characteristic, then there is
the crossing and pervasion point between function S-rough sets and law identification of infor-
mation system; this paper crosses and ingrafts function S-rough sets and law identification of in-
formation system and gives law identification, law identification criterion, law identification theorem, and applications of information system. The crossing, pervasion, and ingraft between
function S-rough sets and the information law identification of information system is a new re-search direction of law identification of information system.
For the convenience of accepting the discussion of this paper, the three forms of function
S-rough sets [1―9] and function rough sets [8,9] are introduced in sections 1, 2, 3 and 4 respectively.
Assumption. ()x D is a finite function universe, ()()Q x x ?D is an infinite function set,
12{,,,}m F f f f = and 12{,,,}n F f f f = are function transfer family on ()x D . [()]u x =
12{(),(),,()}n u x u x u x is R -function equivalence class, R is an equivalence relation. ()x D ,
()Q x , [()]u x , ()u x , and ()v x are denoted separately by , , [], Q u u D , and v. The concepts
in sections 1, 2, 3 and 4 can be seen in refs. [1―14].
1 Function one-direction S-rough sets
Q is the one direction S-function sets (one direction singular function sets) of Q ?D
; more-over,
{|,,()}.Q Q v v v Q f v u Q =∈∈=∈ ∪D (1) f Q is f -extension of Q ; moreover,
{|,,()}f Q v v v Q f v u Q =∈∈=∈D , (2) where f F ∈ is a function transfer [1,2,8,9], there is a , ,v v Q ∈∈D and the effect of f F ∈ is
to change v into ()f v u Q =∈.
(,)()R F Q and (,)()R F Q are the lower approximation and the upper approximation of
Q ? D
respectively; moreover,
(,)()[]{|, []},R F Q u u u u Q °==∈? ∪D (3) (,)()[]
{|, []}.R F Q u u u u Q φ==∈≠ ∪∩D (4)
()nR B Q is R -boundary of Q ? D
; moreover, ()nR B Q =(,)()R F Q (,)(),R F Q ? (5) where F φ≠. The set pair composed by (,)()R F Q and (,)()R F Q is function one direction S-rough
SHI KaiQuan et al. Sci China Ser F-Inf Sci | May 2008 | vol. 51 | no. 5 | 499-510 501
sets (function one direction singular rough sets) of Q ? D ; moreover,
((,)()R F Q ,(,)()R F Q ). (6) ()As Q is the assistant set generated by function one direction S-rough sets; moreover,
(){|,,() }As Q v v v Q f v u Q =∈∈=∈
D , (7) where “∈
” in ()As Q denotes the relationship between element ()f v u = and set Q satisfies the characteristic function ()01f v Q χ<<. The relationship between element ()f v u = and set Q in expression (2) satisfies the charac-
teristic function () 1.f v Q χ=
2 Function two-direction S-rough sets
Q * is the two-direction S-function sets (two-direction singular function sets) of Q ?D
; more-over,
{|,,()}Q Q v v v Q f v u Q ?′=∈∈=∈∪D , (8) {|,()}Q Q u u Q f u v Q ′=?∈=∈. (9)
Q is f -contract of Q ?D
; moreover, {|,()}f Q u u Q f u v Q =∈=∈, (10) where f F ∈ is a function transfer [1,2,8,9], there is u Q ∈, and the effect of f F ∈ is to change
u into ()f u v Q =∈.
(,)()R Q ? F and (,)()R Q ? F are the lower approximation and the upper approximation
of Q ??D , respectively; moreover,
(,)()[]
{|,[]},
R Q u u u u Q ??==∈? ∪F D (11) (,)()[]
{|,[]}.R Q u u u u Q φ??==∈≠ ∪∩F D (12)
()nR B Q ? is R -boundary of Q ??D ; moreover,
()(,)()(,)()nR B Q R Q R Q ???=? F F , where F F =∪F , F φ≠, and F φ≠.
The set pair composed by (,)())R Q ? F and (,)()R Q ? F is function two-direction
S-rough sets (function two-direction singular rough sets) of Q ??D ; moreover,
((,)(),(,)())R Q R Q ?? F F .
()As Q ? is the assistant set generated by function two-direction S-rough sets; moreover,
502 SHI KaiQuan et al. Sci China Ser F-Inf Sci | May 2008 | vol. 51 | no. 5 | 499-510 (){|,,()and ,()}As Q u v v Q f v u Q u Q f u v Q ?=∈∈=∈∈=∈ D , (13) where “∈ ” in ()As Q ? denotes the relationship between element ()f u v = and set Q satisfies characteristic function ()10f u Q
χ?<<. The relationship between element ()f u v = and set
Q in (10) satisfies characteristic function ()1f u Q χ=?.
3 Dual of function one-direction S-rough sets
Q ′?D is the dual of one-direction S-function set {|,,()}Q Q v v v Q f v u Q =∈∈=∈ ∪D ; moreover,
{|,()}Q Q u u Q f u v Q ′=?∈=∈.
(,)()R F Q ′ and (,)()R F Q ′ are the lower approximation and the upper approximation of Q ′?D , respectively; moreover,
(,)()[]
{|,[]},R F Q u u u u Q ′=′=∈? ∪D
(,)()[]{|,[]}.
R F Q u u u u Q φ′=′=∈≠ ∪∩D ()nR B Q ′ is R -boundary of Q ′?D ; moreover,
()(,)()(,)()nR B Q R F Q R F Q ′′′=? , where F φ≠. The set pair composed by (,)()R F Q ′ and (,)()R F Q ′ are dual of function one-direction S-rough sets (dual of function one-direction singular rough sets) of Q ′?D ; moreover,
((,)(),(,)())R F Q R F Q ′′ .
()As Q ′ is the assistant set generated by dual of function one-direction S-rough sets; more-over,
(){|,()}As Q u u Q f u v Q ′=∈=∈
. 4 Function rough sets
According to the result in section 2, if function transfer family F φ= and F φ=, namely ,F F φ==∪F then there are {|,, ()}v v v Q f v u Q φ∈∈=∈=D and {|, ()u u Q f v ∈= }v Q ∈φ= in expressions (8) and (9), respectively, thus there is Q Q Q ?′==, and there are the lower approximation ()R Q ? and the upper approximation ()R Q ? of Q ?D
; moreover,
()[]
{|,[]},R Q u u u u Q ?==∈?∪D
SHI KaiQuan et al. Sci China Ser F-Inf Sci | May 2008 | vol. 51 | no. 5 | 499-510 503
()[]{|,[]}.
R Q u u u u Q φ?==∈≠∪∩D ()nR B Q is R -boundary of Q ?D ; moreover,
()()()nR B Q R Q R Q ??=?.
The set pair composed by ()R Q ? and ()R Q ? are function rough set of Q ?D
, moreover ((),())R Q R Q ??.
By using the results of sections 1―4, it is easy to prove Theorems 1―5 and Corollaries 1―3. Theorem 1. Function two-direction S-rough sets and function one-direction S-rough sets satisfy ((,)(),(,)())F R Q R Q φ??= F F ((,)(),(,)())R F Q R F Q = . (14) Proof. If ,F φ= namely F F =∪F becomes F =F , then there is f Q ={|,u u Q ∈ }v Q φ=∈= in expression (10); expressions (8) and (9) become {|,Q Q v v ?′=∈∪D ,v Q ∈ ()}({|,()}){|,,()}{|,f v u Q Q u u Q f u v Q v v v Q f v u Q Q v v =∈=?∈=∈∈∈=∈=∈∪∪D D ,v Q ∈ ()}f v u Q Q =∈= ; expressions (11) and (12) respectively become (,)()R Q ?= F []{|, []}{|, []}[](,)()u u u u Q u u u Q u R F Q ?=∈?=∈?== ∪∪D D and (,)()R Q ?= F []{|,u u u =∈∪D []}{|,[]}[]u Q u u u Q u φφ?≠=∈≠== ∩∩∪D (,)().R F Q Thus, there is
((,)(),R Q ? F (,)())F R Q φ?= F ((,)(),(,)())R F Q R F Q = .
Theorem 2. Function two-direction S-rough sets and dual of function one-direction S-rough sets satisfy
((,)(),(,)())F R Q R Q φ??= F F ((,)(),(,)())R F Q R F Q °°′′=.
Theorem 3. Function two-direction S-rough sets and function rough sets satisfy
((,)(),(,)())R Q R Q φ??= F F F ((),())R Q R Q ??=.
Theorem 4. Function one-direction S-rough sets and function rough sets satisfy
((,)(),(,)())F R F Q R F Q φ== ((),())R Q R Q ??.
Theorem 5. Dual of function one direction S-rough sets and function rough sets satisfy
((,)(),(,)())F R F Q R F Q φ=′′= ((),())R Q R Q ??.
Corollary 1. If function two-direction S-rough sets degenerates into function rough sets, then there must be
()As Q φ?=. (15) In fact, if function two-direction S-rough sets degenerates into function rough sets, or ((,)(),(,)())R Q R Q ?? F F ((),())R Q R Q ??=, then there must be φ=F or F φ= and
F φ=. Thus, expression (15) becomes *(){|, , ()As Q u v v Q f v u Q =∈∈=∈
D and ()f u = }v Q ∈ φ=. Corollary 2. If function one direction S-rough sets degenerates into function rough sets, then there must be
504 SHI KaiQuan et al. Sci China Ser F-Inf Sci | May 2008 | vol. 51 | no. 5 | 499-510
()As Q φ= .
Corollary 3. If dual of function one-direction S-rough sets degenerates into function rough sets, then there must be
()As Q φ′=.
By Theorems 1―5 and Corollaries 1―3, it is easy to prove Propositions 1―4.
Proposition 1. Function two-direction S-rough sets are the general form of function one- direction S-rough sets, and function one-direction S-rough sets are the special case of function two- direction S-rough sets.
Proposition 2. Function two-direction S-rough sets are the general form of dual of function one-direction S-rough sets, and dual of function one-direction S-rough sets is the special case of function two-direction S-rough sets.
Proposition 3. Function one-direction S-rough sets are the general form of function rough sets, and function rough sets are the special case of function one direction S-rough sets. Proposition 4. Dual of function one-direction S-rough sets is the general form of function rough sets, and function rough sets are the special case of dual of function one direction S-rough sets.
More concepts and characteristics of function S-rough sets can be found in refs. [1―9]. In the discussion of section 5, the concepts of R -function equivalence class [u ] and law [u ] are not dis-tinguished and can be straightly used.
5 The attribute characteristic of law and law generation
[u ] is the law with the attribute set 12{,,,}k V αααα=? . If there is an attribute ,V β∈ βα∈, and an element transfer [8―14] f F ∈ which changes βinto ()f βα′=α∈, then []f u is called f -generation law of [u ]. Obviously, there is
card([])card([])f u u ≤,
card()card()f αα≤,
where {
()}f f ααβα′==∪ is the attribute set of []f u , card([u ]) is the cardinal number of [u ], and V is the attribute universe.
[]u is the law with the attribute set 12{,,,}k V αααα=? . If there is an attribute ,i αα∈ (1,2,,,)i k λλ∈< , and an element transfer [8―14] f F ∈ which changes i α into ()i f α= i βα∈, then []f u is called f -generation law of [u ]. Obviously, there is
card([])card([])f u u ≤,
card()card()f αα≤, where {()}f i i f αααβ=?= is the attribute set of []f u .
In the following, we discuss the law generation of [u ].
Given the law 12[]{,,,}m u u u u = , ?i u
is the discrete data distributing of []i u u ∈; moreover, ,1,2,?{,,,}i i i i n u
u u u = ,
SHI KaiQuan et al. Sci China Ser F-Inf Sci | May 2008 | vol. 51 | no. 5 | 499-510 505 where ,,1,2,,,1,2,,i k u R i m k n ∈== and R is real numbers set.
By using ,,1???,,1,2,,m
k i k i k i i u u
u u k n ?==∈=∑ , we obtain the composite law []u ? of []u . []u ? is denoted by [u ] briefly. The data point form of the discrete data distributing of [u ]([u ]=[u ]*) is given in expression (16):
1122(,),(,),,(,)n n x y x y x y . (16) By using expression (16) and Lagrange interpolation formula 1,1()n n i j j j i i j i j x x p x y x x ==≠???=???????
∑∏, (17) we obtain the polynomial law ()p x generated by []u ([][])u u ?=; moreover,
121210()n n n n p x a x a x a x a ????=++++ . (18) In fact, p (x ) in expression (18) is one of the forms generated by [u ]([u ]=[u ]*), and the other forms generated by [u ] are omitted here.
By using the above concepts, it is easy to prove Theorems 6, 7 and Corollaries 4, 5.
Theorem 6. If ()p x is the law generated by [u ] and ()f p x is the law generated by []f u , then
DIS ((),())f p x p x .
Theorem 7. If ()p x is the law generated by [u ] and ()f p x is the law generated by []f u , then DIS ((),())f p x p x ,
where DIS=discernibility [15, 16].
Corollary 4. If the attribute set α of [u ] and the attribute set f α of []f u satisfy α= {()}f f αα′?, then there must be
IND((),())f p x p x .
In fact, the attribute set of ()f p x is {()}f f ααβα′==∪, where ,V ββα∈∈. If element transfer f F ∈changes β into ()f βαα′=∈, and element transfer f F ∈ changes α′ into ()f αβα′=∈, then f αα=?{()}f αβ′=. Thus, ()()f p x p x =, namely ()p x and ()f p x are indiscernible [15,16] .
Corollary 5. If the attribute set α of []u and the attribute set f α of []f u satisfy α= {()}f i f αβ∪, then there must be
IND((),())f p x p x . In fact, the attribute set of ()f p x is {()}f i i f αααβ=?=, where i αα∈. If element transfer f F ∈changes i α into ()i i f αβα=∈, and element transfer f F ∈ changes i β
506 SHI KaiQuan et al. Sci China Ser F-Inf Sci | May 2008 | vol. 51 | no. 5 | 499-510
into ()i i f βαα=∈, then {()}f i i f ααβα==∪. Thus,()()f p x p x =, namely ()p x and ()f p x are indiscernible [15,16].
Here IND = indiscernibility [15,16].
6 Law identification criterion and law identification application
6.1 Law identification criterion
For any two laws []i u and []j u on []U , i j ≠, if their law distance D([],[])i j i j u u ≠ satisfies D([],[])0i j i j u u ≠≠, (19) then []j u is identifiable with respect to []i u .
Here D([],[])i j i j u u ≠is the distance [14] between []i u and []j u , and 1D([],[])i j i j u u n ≠= 1122,0(())
n j i i j b a ?=?∑; j b is the coefficient of law ()j p x =121210n n n n b x b x b x b ????++++ gen-
erated by []j u and 0,1,2,,1j n =? ; i a is the coefficient of law 11()n i n p x a x ??=+ 2210n n a x a x a ??+++ generated by []i u ,0,1,2,i = ,1n ? .
Obviously, if D([], [])0i j i j u u ≠=, then []j u is not identifiable with respect to []i u . []i u ,
[]j u []∈U , and i j ≠.
6.2 Law identification theorem
Theorem 8 (law F -identification theorem). Law [u ]j is identifiable with respect to law u ]i if and only if function transfer family F satisfies
F φ≠, (20) where []i u ,[]j u []∈U , and i j ≠.
Proof. 1) Given the law [][]i u ∈U and ,1,2,[]{,,,}i i i i m u u u u = . Since F φ≠ in (20) and there is ,[]i v v u ∈∈D and f F ∈ changes v into (
)[]i f v u u ′=∈, then []i u becomes []j u = ,1,2,[]{()}{,,,,}i i i i m u f v u u u u u ′′==∪ ,1,2,,1{,,,,}i i i m i m u u u u += , and then there are the composite discrete data distributing []i u ?and []j u ? of []i u and []j u , respectively. Suppose
11()n i n p x a x
???= 22n n a x ??++ 10a x a ++ is the polynomial law generated by []i u ?, 1212()n n j n n p x b x b x ?????=++ 1b x ++ 0b is the polynomial law generated by []j u ?. Accord-ing to section 5, we obtain 1n a ?≤ 1221100,,,,n n n b a b a b a b ??? ≤≤≤, that is, ()i p x ? and ()j p x ? satisfy the law identification criterion, or
D([],[])0i j i j u u ≠≠,
then []j u is identifiable with respect to []i u .
2) If []j u is identifiable with respect to []i u , or
SHI KaiQuan et al. Sci China Ser F-Inf Sci | May 2008 | vol. 51 | no. 5 | 499-510 507 D([],[])0i j i j u u ≠≠,
then there must be F φ≠.
Theorem 9 (law F -identification theorem). Law [u ]j is identifiable with respect to law [u ]i if and only if function transfer family F satisfies
F φ≠, (21)
where []i u ,[]j u []∈U , and i j ≠.
Theorem 10 (invalid theorem of law identification). Law [u ]j is not identifiable with respect to law [u ]i if and only if the distance D([],[])i j i j u u ≠ between [u ]j and [u ]i satisfies D([],[])0i j i j u u ≠=. (22) In fact, if [u ]j and [u ]i satisfies D([],[])0i j i j u u ≠=, then the polynomial law ()j p x generated
by [u ]j and the polynomial law ()i p x
generated by [u ]i satisfy 11()n j n p x b x ??= 2122101210(),n n n n n n i b x b x b a x a x a x a p x ??????++++=++++= or 11,n n a b ??= 2n a ?= 2,n b ?1100,,a b a b == , and obviously there is
IND([],[])j i i j u u ≠.
6.3 The application of law identification
Here, a simple example that comes from a part of an information transfer system is given. This example, which is an application of f -law identification, is discussed based on function one- direction S-rough sets. The application of f -law identification is omitted.
Suppose [u ] is a subsystem of system Ω, 12[]{,}u u u =, 12,u u is two output laws (functions) of [u ], by the concept of composite law in section 5, we obtain the composite output law p (x ) of
[u ] which is a 7th polynomial; moreover,
765432176543210
765432()0.00550.1578 1.841911.202838.242473.189471.360229.0000.
p x a x a x a x a x a x a x a x a x x x x x x x =+++++++=?+?+?+?+ (23) Expression (23) denotes system [u ] is under normal work, law p (x ) of subsystem [u ] consists of given law p (x )0; moreover, ()p x =0()p x , or law distance 0D((),())0p x p x =. The form of given law p (x )0 is omitted.
At time j t T ∈, subsystem [u ] encounters the disturbance or attack of unknown law []i v u ∈, and that makes the law ()p x of [u ] become ()f p x ; moreover,
765432176543210
765432()0.00140.02750.17080.0792 3.1496 12.706716.822610.8000.
f p x b x b x b x b x b x b x b x b x x x x x x x =+++++++=?+??+?+
(24)
At time j t T ∈, expressions (23) and (24) satisfy law identification criterion, or
D((),())0f p x p x ≠,
508 SHI KaiQuan et al. Sci China Ser F-Inf Sci | May 2008 | vol. 51 | no. 5 | 499-510
the law of subsystem [u ] is identified at time j t T ∈,or ()()f p x p x ≠ at j t T ∈. The real presentation of subsystem [u ] is that the deferent image is confusion or distortion.
Using expressions (23) and (24), we can find out the unknown law v i . Discern expressions (23) and (24) separately and denote them by the form of data points
12345678():(1,),(2,),(3,),(4,),(5,),(6,),(7,),(8,)p x y y y y y y y y , (25) 57123468():(1,),(2,),(3,),(4,),(5,),(6,),(7,),(8,)f f f f f f f f f p x y y y y y y y y , (26) where the concrete data of expressions (25) and (26) are omitted. By expressions (25) and (26), and ,1,2,8,f i i i y y y i Δ=?= we obtain the data points ():(1,1.5),(2,2.6),(3,3.7),(4,4),(5,6),(6,8),(7,0.8),(8,3)v x . (27)
By expressions (27) and (17), we obtain
765432176543210765432()0.00680.1853 2.012811.281935.092860.482854.537618.2000.
v x c x c x c x c x c x c x c x c x x x x x x x =+++++++=?+?+?+? (28)
Expression (28) is the unknown disturbance or attack law v that invades subsystem [u ]. Using the intelligence recognition and tracking filter equipment of subsystem [u ], v (x ) can be isolated from [u ]. For some reason, the structure of intelligence recognition and tracking filter equipment are omitted. This simple example indicates that if there is ()[()]v x u x ∈ and f F ∈ which changes v (x ) into (())()[()]f v x u x u x ′=∈, then subsystem 12[]{,}u u u =become [][]{(())}f u u f v x =∪ 12123{,,(())}{,,}u u f v x u u u ==, and
D([],[])0f u u ≠. (29) Expression (29) shows the output law p (x ) of subsystem is attacked, and p (x ) departs the given law, so it is identified. Function S-rough sets have many applications in law identification of in-formation transfer systems, which are omitted here.
7 Element transfer and function transfer
Here, the concepts of element transfer and function transfer are given, which are important for people to accept the discussions in sections 1, 2, 3, 5.
Given R -element equivalence class 12[]{,,}m x x x x U =? , 12{,}m Y y y y = is the charac-teristic value set of [x ], y i is the characteristic value of x i , and 1,2,i m = .[,]a b is the char-acteristic value interval generated by Y ; moreover,
1
1
min(),max();,m i i m i i i a y b y y R ====∈ R is real number set.
Obviously, for the elements ,[]i j x x x ∈, there are the characteristic values ,[,]i j y y a b ∈ of ,i j x x , respectively. For the element ,p q x x U ∈ and ,[]p q x x x ∈, there are the characteristic
SHI KaiQuan et al. Sci China Ser F-Inf Sci | May 2008 | vol. 51 | no. 5 | 499-510 509 values ,[,]p q y y a b ∈ of ,p q x x , respectively. If there is a transfer f F ∈ which changes y p , y q
into (
),()[,]p q f y f y a b ∈, then there are ,[]p q x x x ∈, or ,,,[]p q p q x x U x x x ∈∈(),()[].p q f x f x x ?∈
Transfer f F ∈ is element transfer.
For element []x x λ∈, y λ is the characteristic value of x λ, if there is transfer f F ∈ which changes [,]y a b λ∈ into ()[,]f y a b λ∈, then there is []x x λ∈, or
[]()[].x x f x u x λλλ∈?=∈ Transfer f F ∈ is an element transfer.
Given R -function equivalence class [()],u x ?D 12[()]{(),(),()}m u x u x u x u x = . [,]i i a b is
the definition domain of ()i u x ∈[()]u x , i i a b ≤, ,
i i a b R ∈,1,2,i m = , and R is real num-ber set. [,]i i c d is the value domain of ()[()]i u x u x ∈, i i c d ≤, and ,i i c d R ∈. The definition domain and value domain of R -function equivalence class [()]u x are [a , b ] and [c , d ], respec-
tively, a ≤b , c ≤d and ,
, , a b c d R ∈; moreover, 1
111
min(),max(),min(),max().m m i i i i m m i i i i a a b b c c d d ======== If ()[()]j u x u x ∈, then [,]x a b ∈ and ()[,]j u x c d ∈; if ()[()]p u x u x ∈, then [,]x a b ∈ and ()p u x [,]c d ∈; if there is a transfer f F ∈, for ()[()]x u x ν∈, and makes [,],(())x a b f x ν∈ [,]c d ∈ in ()x ν, then there is (())()[()]f x u x u x ν=∈, and f F ∈is a function transfer, or
(),()[()](())()[()]x x u x f x u x u x ννν?∈∈?=∈D .
On the contrary, if there is ()[()]j u x u x ∈,f F ∈, f makes [,]x a b ∈, and (())j f u x [,]c d ∈ in ()j u x , then there is (())()[()]j j f u x x u x ν=∈, and f F ∈ is a function transfer, or
()[()](())()[()]j j j u x u x f u x x u x ν?∈?=∈.
Intuitively, f F ∈ is an element transfer by which element x out of []x is transferred in
[]x , and f F ∈ is an element transfer by which element x in []x is transferred out from []x . f F ∈ is a function transfer by which function v (x ) out of [()]u x is transferred in [()]u x , and f F ∈ is a function transfer by which function v (x ) in [()]u x is transferred out form [()]u x . Both element transfers f F ∈, f F ∈ and function transfers f F ∈, f F ∈ are a kind of simple function family.
8 Discussion
Poland mathematician Pawlak Z [15,16] put forward rough sets in 1982 and gave the structure and application of rough sets. This splendid scholarship achievement makes many theory and applica-tion scholars interested in rough sets. Pawlak Z rough sets are defined by static R -element
equivalence class [x] and Pawlak Z rough sets are static rough sets, so the application domain of Pawlak Z is restrained. In 2002, Shi[10] put forward S-rough sets (singular rough sets) which im-proved Pawlak Z rough sets. S-rough sets are defined by dynamic R-element equivalence class [x], so S-rough sets have dynamic characteristic which comes from the element transfer. In refs. [8,9,11―14] Shi gave further discussion about S-rough sets. S-rough sets extended Pawlak Z rough sets and developed the application domain of it. Using Pawlak Z rough sets and S-rough sets to identify and mine the law in information system encounter difficulty, because Pawlak Z rough sets and S-rough sets have no law characteristic. What is a law? A function is a law. In or-der to solve the difficulty, Shi[1,2] put forward function S-rough sets and function rough sets in 2005, gave their structure, and indicated that function rough sets are the static form of function S-rough sets. Refs. [3―9] gave further discussion of function S-rough sets. Function S-rough sets are defined by R-function equivalence class [u] which proposes dynamic characteristic and func-tion rough sets is defined by R-function equivalence class [u] which proposes static characteristic. Function S-rough sets generalized S-rough sets and developed the application domain of S-rough sets. In the sense of mathematics structure and static-dynamic characteristic, Pawlak Z rough sets are the special case of S-rough sets, and S-rough sets are the special case of function S-rough sets. On the contrary, function S-rough sets are the general form of S-rough sets, and S-rough sets are the general form of Pawlak Z rough sets. Especially, S-rough sets improved Pawlak Z rough sets, and function S-rough sets improved S-rough sets; both improvements do not change the equiva-lence relation R[9] in Pawlak Z rough sets.
The output law of information system is a function in the time interval [t1, t k] (the multiple outputs are many functions on time interval [t1, t k]). The partial difference between output law and given law leads the output law of information system to drift and make the output law of in-formation system apart from the law point of law, so it comes into being the study of the output law identification of system information. Function S-rough sets give a new study idea and tool for law identification study of information system.
The crossing, amalgamate, and pervasion of function S-rough sets and law identification in information system will be a new research direction of law identification of information system.
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