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Granular and Rough Computing:Incremental Development

Tsau Young (T.Y.) Lin

tylin@cs.sjsu.edu Computer Science Department, San Jose State University,

San Jose, CA 95192,

and

Berkeley Initiative in Soft Computing, UC-Berkeley, Berkeley, CA 94720

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Introduction The term granular computing is first usedby this speaker in 1996-97 to label asubset of Zadeh’s

granular mathematics

as his research topic in BISC. (Zadeh, L.A. (1998) Some reflections on soft computing, granular

computing and their roles in the conception, design and utilization of information/intelligent systems, Soft Computing, 2, 23-25.)

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Granular computing

IEEE GrC-conference

http://www.cs.sjsu.edu/~grc/.

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Historical Notes

1. Zadeh (1979) Fuzzy Sets and Information granularity(about Dempster-Shaffer Theory(DST))

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Notes

Dempster-Shaffer Theory(DST)

Note: In general, basic probability assignment (bpa) classical probability(cp)

but . . .

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Notes

Dempster-Shaffer Theory(DST)2. Note, but if the given focal elements are

mutually disjoints, then bpa=cp. In this case

2.1. Bel = inner probability(lower bound) 2.2. Pl=Outer probability (upper bound)

(This is NOT general cases—Common errors)

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Historical Notes

2. Pawlak (1982 Dec)

3. Tony Lee (1983 Jan)

Study of relations via partitions

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Historical Notes

Pawlak: Rough Sets, Information systems, Approximations

Lee: Algebraic Theory of Relational Databases

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Historical Notes

4a T. Y. Lin 1988-89: Neighborhood Systems(NS)

( a set of general binary relations)

4b T. Y. Lin (1989) Chinese Wall Security Model

(A study of non-reflexive, symmetric, non-transitive binary relation)

5. Stefanowski (1989) about Fuzzified Partitions

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Historical Notes

6. Lin, Qing Liu & James Huang (1990): Neighborhood system &RS)

7. Lin (1992):Topological and Fuzzy Rough Sets

Lin said, Intuitively, closure is smallest closed set—incorrect.

This is true for topological spaces, not for NS-space

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Historical Notes

8. Lin & Liu (1993): Operator View of RS and NS

Important Results:

8.1. Axiomatic View of RS

8.2. clopen space defines a partition:

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Historical Notes8.3. Simple Proof of item 2

Consider connected components(CC) of clopen space. From general topology theory,

8.3.1. CC is closed.

8.3.2. CCs intersect each other only on boundary.

Since CC is also open, and open set has no boundary.

So CC is disjoint and hence is a partition.

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Historical Notes

9. Lin & Hadjimichael (1996): Non-classificatory hierarchy

This paper builds a tree for nested binary relations;

This result is obvious for equivalence relations

but is very hard for general binary relations

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Granular computing

Granulation seems to be a natural problem-solving methodology deeply rooted in human thinking.

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Granular computing

Human body has been granulated into head, neck, and etc. (there are overlapping areas)

The notion is intrinsically fuzzy, vague, and imprecise.

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Partition theoryMathematicians have idealized the granulation into

Partition (at least as far back to Euclid)

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Partition theory

Mathematicians have developed it into

a fundamental problem solving methodology in mathematics.

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Partition theory

Rough Set community has applied the idea into Computer Science with reasonable results, called

Rough Set Theory(RST)

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Key Views in RST 1. Granulation=Partition: E

2. (V, E): Approximation Space

3. Representations(Information Systems/Tables)

4. Table Processing(Knowledge Processing)

4.1. Reducts and Core

4.2. Value Reducts(Data Mining)

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Key Views in RST

5. Granular/Rough Logic Theory.

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1. Granulation=Partition A Partition of V:

Class A

Class B

f, g, h i, j, k

Class Cl, m, n

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RS Approximations

Upper

approximation

Lower approximation

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Lower/Interior Approximations

L(X)= {B(p) | B(p) X} (Pawlak)

I(X)= {p | B(p) X} (Lin – topology based)

L = I in RS theory

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Upper/Closure Approximations

U(X)= {B(p) | B(p) X } (Pawlak)

C(X)= {p | B(p) X } (Lin - topology based) (A Closure,C(X), may not be closed;error in Lin 1992)

U=C if B is a partition

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Research IssuesIn general case, the coverings(({B(p)) are not

unique, so Pawlak’s notion of Upper/Lower approximations need “MAX” or “MIN”

U(X)= MIN ({B(p) | B(p) X } )L(X)= MAX ({B(p) | B(p) X})

Please do some research (see measure theory)

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Research Issues

From topological space point of view,

Pawlak Style approximations should be replaced by Lin’s style in Topological View.

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Other Closure approximations Cl(X)= iCi(X) (Sierpenski defined)

where Ci(X)= C(…C(C(X))…)(transfinite steps) Cl(X) is closed.

This is the closure in classical topology.

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Other Closure approximationsMay consider finite steps Cls:

Clk(X)= i=1k Ci(X)

Chinese Wall Security Policy Model

(Consider all Clk, k =1, 2, . . .)

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Representation Theory Given a partition(named COLOR)

Class A

Class B

f, g, h i, j, k

Class Cl, m, n

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Representation Theory

Review RS approaches:

V: the universe, V, of entities {f,g,h,I,j,k,l,m}

and is partitioned into

A, B, C equivalence classes

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Representation Theory(information table)

All classes A, B, C are named Yellow, Red, Blue. First tuple means (see next page)

“f “ is a member of “A,” so its

COLOR is “Yellow”

Here COLOR is the name of the given Partition (equivalence relation)

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Summarized in Table Format f A Yellowg A Yellowh A Yellowi B Redj B Redk B Redl C Bluem C Bluen C Blue

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One columns Information Table f Yellow The left hand side is ag Yellow table in which the h Yellow Universe V of entities i Red is represented by onej Red column, called COLOR. k Red Each row representl Blue the color of one entitym Bluen Blue

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Representation Theory for granulation

Lat few pages explain how RS has handled Knowledge representations

We will Extend the idea to binary relations

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Representation Theory Given a granulation(has overlapping)

Neighborhood A

Neighborhood B

f, g, h

i, j, k, l

Neighborhood C

m, n

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Neighborhood of x, N(x)x has N(x) Is in N(x) Name of N(x)f A A Ug A A Uh A A Ui B A, B Vj B B Vk B B V l C B,C Wm C C W

n C C W

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The Center set of N(x)x has N(x)f A f, g, h have the neighborhood Ag A Each of f, g, h is the center of Ah A The set of all centers is calledi B the center set of A and is j B denoted by C(A)k B So i,j,k form the center set C(B)l C l, m, n form the center set C(C)m C

n C

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Topological Information Table Ix Name of N(x) The left hand side

f U is a table in which

g U the Universe V of entities

h U is represented by one

i V column, called ?????.

j V Each row represent one

k V entity Syntactically is the

l W same as information table;

m W However U, V, W are

n W Related

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Binary Relation B1 (approach one)Name Name The left hand side is a table that

U U Represents the binary relation of the

U V interactions among attribute values

V V (See Lin 1998)

V U (U, V) is in B1, if UV

V W

W W

W V (W, V) is in B1, if WV

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Topological Information Table IIx Center set Name of C(-) The left hand side

f C(A) U’ is a table in which the first

g C(A) U’ row means: an entity “f” is in

h C(A) U’ a unique center set “C(A)” and

i C(B) V’ “C(A)” has unique name “U’ ”

j C(B) V’k C(B) V’ l C(C) W’m C(C) W’

n C(C) W’

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Topological Information Table IIx Name of C(-) The left hand side is a table in

which

f U’ the Universe V of entities is represented

g U’ by one column

h U’ Each row represents an entity and

i V’ the name of its center set

j V’k V’ l W’m W’

n W’

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Binary Relation B2 (approach Two)Name Name The left hand side is a table that

U’ U’ represents the binary relation of the

U’ V’ interactions among attribute values

V’ V Lin 2004 (IEEE-CIS news letter)

V’ U (U’, V’) is in B2, if AC(B)

V’ W (A is neighborhood of member in U’)

W’ W V’ is name of C(B)

W’ V (V’, U’) is in B2, if BC(A)

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B1 and B2

1.B1 is symmetric, and B1 does not describe the interactions among U, V, W completely.

2.B2 may not be symmetric,moreover

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B1 and B2

If the given granulation under consideration is symmetric

then B2 does describe the interactions among U’, V’, W’ completely.

(In fact B2 is the “quotient” of the granulation)