Post on 02-Jan-2016
description
transcript
1
Granular Computing: A New Problem Solving Paradigm
Tsau Young (T.Y.) Lin
tylin@cs.sjsu.edu dr.tylin@sbcglobal.net
Computer Science Department, San Jose State University, San Jose, CA 95192,
and
Berkeley Initiative in Soft Computing, UC-Berkeley, Berkeley, CA 94720
2
Outline
1. Introduction
2. Intuitive View of Granular Computing
3. A Formal Theory
4. Incremental Development
4.1. Classical Problem Solving Paradigm
4.2. New View of the Universe
4.3. New Problem Solving Paradigm
2
3
Outline
1. Introduction
4
Granular computing The term granular computing is first used by this speaker in 1996-97 to label a subset 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.)
5
Granular computing
Since, then, it has grown into an active research area:
Books, sessions, workshops IEEE task force (send e-mails to join
the task force, please include Full name, affiliation, and E-mail
6
Granular computing
IEEE GrC-conference
http://www.cs.sjsu.edu/~grc/.
7
Granular computing
Historical Notes
1. Zadeh (1979) Fuzzy sets and granularity
2. Pawlak, Tony Lee (1982):Partition Theory(RS)
3. Lin 1988/9: Neighborhood Systems(NS) and Chinese
Wall (a set of binary relations. A non-reflexive. . .)
4. Stefanowski 1989 (Fuzzified partition)
5. Qing Liu &Lin 1990 (Neighborhood system)
8
Granular computing
Historical Notes6. Lin (1992):Topological and Fuzzy Rough Sets
7. Lin & Liu (1993): Operator View of RS and NS
8. Lin & Hadjimichael (1996): Non-classificatory hierarchy
9
Granular computing
Granulation seems to be a natural problem-solving methodology deeply rooted in human thinking.
10
Granular computing
Human body has been granulated into head, neck, and etc. (there are overlapping areas)
The notion is intrinsically fuzzy, vague, and imprecise.
11
Partition theory
Mathematicians have idealized the granulation into
Partition (at least back to Euclid)
12
Partition theory
Mathematicians have developed it into
a fundamental problem solving methodology in mathematics.
13
Partition theory
Rough Set community has applied the idea into Computer Science with reasonable results.
14
Partition theory
But Partition requires Absolutely no overlapping
among granules
(equivalence classes)
15
More General Theory
Partitions is too restrictive for real world applications.
16
More General Theory
Even in natural science, classification does permit a small degree of overlapping;
17
More General Theory
There are beings that are both proper subjects of zoology and botany.
18
More General Theory
A more general theory is needed
19
Outline
2. New Theory - Granular Computing
2
20
Zadeh’s Intuitive notion
Granulation involves
partitioning
a class of objects(points)
into granules,
21
Zadeh’s Intuitive notion
with a granule being
a clump of objects (points)
which are drawn together by indistinguishability, similarity or functionality.
22
Formalization
We will present a formal theory, we believe, that has captures quite an essence of Zadeh’s idea (but not full)
23
Outline
3. A Formal Theory
24
(Single Level) Granulation
Consider two universes(classical sets): 1. V is a universe of objects 2. U is a data/information space 3. To each object p V, we associate
at most one granule U;
The granule is a classical/fuzzy subset.
25
(Single Level) Granulation A granulation is a map: p V B(p) 2U
where B(p) could be an empty set.
26
(Single Universe) Granulation
A (single universe) granulation is a map:
p V B(p) 2V (U=V)
where B(p) is a granule/neighborhood of objects.
27
(Single Level) Granulation
Intuitively B(p) is the collection of objects that are drawn towards p
28
Granulation - Binary Relation
The collection
B={(p, x) | x B(p) p V} VU
is a binary relation
29
(Single Level) Granulation
If B is an equivalence relation the collection
{B(p)} is a partition
30
More General Case
If we consider a set of Bj of binary relations(drawn by various “forces”, such as indistinguishability, similarity or functionality) then
31
More General Case
we have the association
p NS(p)={Nj(p)| Nj(p)={x | (p, x) Bj } j
runs through an index}.
is called multiple level granulation and form a neighborhood system (pre-topological space).
32
Development
4. Incremental Development
2
33
Classical ParadigmWhat do we have?
1. (Divide) Partitioning
2. Quotient Set (Knowledge level)
3. Integration (of subtasks and quotient task)
34
What do we have?
Classical Paradigm
1. Partition of a classical set (Divide)
Absolutely no overlapping among granules
35
Some Mathematics
A partition
Granule A
Granule B
f, g, h i, j, k
Granule Cl, m, n
36
Some Mathematics
Partition Equivalence relation
X Y (Equivalence Relation)
if and only if
both belong to the same class/granule
37
Equivalence Relation Generalized Identity X X (Reflexive)
X Y implies Y X (Symmetric)
X Y, Y Z implies X Z (Transitive)
38
Example
Partition
[0]4 = {. . . , 0, 4, 8, . . .},
[1]4 = {. . . , 1, 5, 9, . . .},
[2]4 = {. . . , 2, 6, 10, . . .},
[3]4 = {. . . , 3, 7, 11, . . .}.
39
Quotient set
{ [0]4 , [1]4 , [2]4, [3]4 }
[0]4 +[1]4 =[1]4 [4]4 +[5]4 =[9]4
[1]4 = [9]4
40
New territories
Granulation (not Partition)
B0 = [0]4 {5, 9},
B1 = [1]4 ={. . . , 1, 5, 9, . . .},
B2 = [2]4 {7},
B3 = [3]4 {6}.
41
New territories
Granulation (not Partition)
B0 B1 = {5, 9},
B2 B3 = {6,7},
Could we define a quotient set ?
42
New territories
If {B0, B1, B2, B3} is a quotient set, then
B0 and B1 are distinct elements, so
B0 B1 (= {5, 9}) should be empty
{B0, B1, B2, B3 } is NOT a set
43
New Paradigm
In general, classical scheme is unavailable for general granulation
We will show that:
classical scheme can be extended to single level granulation
44
New formal theory
New view of the universe
45
Granulated/clustered space
Let V be a set of object with granulation B: V B(p) 2V
V=(V, B) is a granulated/clustered space, called B-space (a pre-topological space).
V is approximation space (called A-space) if B is a
partition.
46
Classical ParadigmWhat do we have?
1. (Divide) Partitioning
2. Quotient Set (Knowledge level)
3. Integration (of subtasks and quotient task)
47
What are in the new paradigm?
1. Partition of B-space (Divide)
2. Quotient B-space (Knowledge)
3. Integration-Approximation (and extension)
48
Integration-Approximations
Some Comments on approximations
49
Lower/Interior approximations
B(p), p V, be a granule
L(X)= {B(p) | B(p) X} (Pawlak)
I(X)= {p | B(p) X} (Lin-topology)
50
Upper/Closure approximations
Let B(p), p V, be an elementary granule
U(X)= {B(p) | B(p) X = } (Pawlak)
C(X)= {p | B(p) X = } (Lin-topology)
51
Upper/Closure approximations
Cl(X)= iCi(X) (Sierpenski-topology)
Where Ci(X)= C(…(C(X))…)
(transfinite steps) Cl(X) is closed.
52
New View
Divide (and Conquer)
Partition of set (generalize) ?
Partition of B-space
(topological partition)
53
New View:B-space
The pair (V, B) is the universe, namely
an object is a pair (p, B(p))
where B: V 2V ; p B(p) is a granulation
54
Derived Partitions
The inverse images of B is a partition (an equivalence relation)
C ={Cp | Cp =B –1 (Bp) p V}
55
Derived Partitions
Cp is called the center class of Bp
A member of Cp is called a center.
56
Derived Partitions
The center class Cp consists of all the points that have the same granule
Center class Cp = {q | Bq= Bp}
57
C-quotient set
The set of center classes Cp is a quotient
set
Iran, Iraq. . US, UK, . . .
Russia, Korea
58
New Problem Solving Paradigm
(Divide and) Conquer
Quotient set
Topological Quotient space
59
Neighborhood of center class C (in the case B is not reflexive)
B-granule/neighborhood C-classes
C-classes
60
Neighborhood of center class
B-granule C-classes
C-classes
61
Topological partition
Cp -classes
Cp -classes
B-granule/neighborhood
62
New Problem Solving Paradigm
(Divide and) Conquer
Quotient set
Topological Quotient space
63
Topological partition
Cp -classes
Cp -classes
B-granule/neighborhood
64
Topological partition
Cp -classes
Cp -classes
B-granule/neighborhood
65
Topological partition
Cp -classes
Cp -classes
B-granule/neighborhood
66
Topological Table (2-column)2-columns Binary relation for Column I
US CXWest CX CY ( BX)
UK CXWest CX CZ ( BX)
Iran CYM-east CY CX ( BY)
Iraq CYM-east CY CZ ( BY)
Russia CzEast CZ CX ( Bz)
Korea CzEast CZ Cy ( Bz)
67
Future Direction
Topological Reduct
Topological Table processing
68
Application 1: CWSP
In UK, a financial service company may consulted by competing companies. Therefore it is vital to have a lawfully enforceable security policy.
3
69
Background
Brewer and Nash (BN) proposed Chinese Wall Security Policy Model (CWSP) 1989 for this purpose
70
Policy: Simple CWSP (SCWSP)
"Simple Security", BN asserted that
"people (agents) are only allowed access to information which is notheld to conflict with any other information that they (agents) already possess."
71
A little Fomral
Simple CWSP(SCWSP):
No single agent can read data X and Y
that are in CONFLICT
72
Formal SCWSP
SCWSP says that a system is secure, if
“(X, Y) CIR X NDIF Y “
CIR=Conflict of Interests Binary Relation
NDIF=No direct information flow
73
Formal Simple CWSP
SCWSP says that a system is secure, if
“(X, Y) CIR X NDIF Y “
“(X, Y) CIR X DIF Y “
CIR=Conflict of Interests Binary Relation
74
More Analysis
SCWSP requires no single agent can read X and Y,
but do not exclude the possibility a sequence of agents may read them
Is it secure?
75
Aggressive CWSP (ACWSP)
The Intuitive Wall Model implicitly requires: No sequence of agents can read X and Y:
A0 reads X=X0 and X1,
A1 reads X1 and X1,
. . .An reads Xn=Y
76
Composite Information flow
Composite Information flow(CIF) is
a sequence of DIFs , denoted by such that
X=X0 X1 . . . Xn=Y
And we write X CIF Y
NCIF: No CIF
77
Composition Information Flow
Aggressive CWSP says that a system is secure, if
“(X, Y) CIR X NCIF Y “
“(X, Y) CIR X CIF Y “
78
The Problem
Simple CWSP ? Aggressive CWSP
This is a malicious Trojan Horse problem
79
Need ACWSP Theorem
Theorem If CIR is anti-reflexive, symmetric and anti-transitive, then
Simple CWSP Aggressive CWSP
80
C and CIR classes CIR: Anti-reflexive, symmetric, anti-transitive
CIR-class Cp -classes
Cp -classes
81
Application 2
Association mining by Granular/Bitmap computing
82
Fundamental Theorem
Theorem 1:
All isomorphic relations have isomorphic patterns
83
Illustrations:Table Kv1 TWENTY MAR NY)
v2 TEN MAR SJ)
v3 TEN FEB NY)
v4 TEN FEB LA)
v5 TWENTY MAR SJ)
v6 TWENTY MAR SJ)
v7 TWENTY APR SJ)
v8 THIRTY JAN LA)
v9 THIRTY JAN LA)
84
Illustrations: Table K’ v1 20 3rd New York)
v2 10 3rd San Jose)
v3 10 2nd New York)
v4 10 2nd Los Angels)
v5 20 3rd San Jose)
v6 20 3rd San Jose)
v7 20 4th San Jose)
v8 30 1st Los Angels)
v9 30 1st Los Angels)
85
Illustrations: Patterns in Kv1 TWENTY MAR NY)
v2 TEN MAR SJ)
v3 TEN FEB NY)
v4 TEN FEB LA)
v5 TWENTY MAR SJ)
v6 TWENTY MAR SJ)
v7 TWENTY APR SJ)
v8 THIRTY JAN LA)
v9 THIRTY JAN LA)
86
Isomorphic 2-Associations
K Count K’
(TWENTY, MAR)
3 (20, 3rd)
(MAR, SJ) 3 (3rd, San Jose)
(TWENTY, SJ) 3 (20, San Jose)
87
Canonical Model Bitmaps in Granular Forms
Patterns in Granular Forms
88
Table K’ v1 20 3rd
v2 10 3rd
v3 10 2nd
v4 10 2nd
v5 20 3rd
v6 20 3rd
v7 20 4th
v8 30 1st
v9 30 1st
89
Illustration: KGDM K GDM
v1 20 3rd {v1 v5 v6 v7 } {v1 v2 v5 v6}
v2 10 3rd {v2 v3 v4} {v1 v2 v5 v6}
v3 10 2nd {v2 v3 v4} {v3 v4}
v4 10 2nd {v2 v3 v4} {v3 v4}
v5 20 3rd {v1 v5 v6 v7 } {v1 v2 v5 v6}
v6 20 3rd {v1 v5 v6 v7 } {v1 v2 v5 v6}
v7 20 4th {v1 v5 v6 v7 } {v7}
v8 30 1st {v8 v9} {v8 v9}
v9 30 1st {v8 v9} {v8 v9}
90
Illustration: KGDM K GDM v1 20 3rd (100011100
)(110011000)
v2 10 3rd (011100000)
(110011000)
v3 10 2nd (011100000)
(001100000)
v4 10 2nd (011100000)
(001100000)
v5 20 3rd (100011100)
(110011000)
v6 20 3rd (100011100)
(110011000)
v7 20 4th (100011100)
(110011000)
v8 30 1st (000000011)
(000000011)
v9 30 1st (000000011)
(000000011)
91
Granular Data Model (of K’ )
NAME Elementary Granules10 (011100000)={v2 v3 v4}
20 (100011100) ={v1 v5 v6 v7 }
30 (000000011)={v8 v9}
1st (000000011)={v8 v9}
2nd (001100000)={v3 v4}
3rd (110011000)={v1 v2 v5 v6}
4th (110011000)={v7}
92
Associations in Granular Forms
K Cardinality of Granules
(20, 3rd)
|{v1 v5 v6 v7 } {v1 v2 v5 v6 }|=
|{v1 v5 v6 }|=3
(10, 2nd) |{v2 v3 v4 } {v3 v4 }|=
|{v3 v4 }|=2
(30, 1st) |{v8 v9 } {v8 v9 }|=
|{v8 v9 }|=2
93
Associations in Granular Forms
K Cardinality of Granules
(20, 3rd)
|{v1 v5 v6 v7 } {v1 v2 v5 v6 }|=
|{v1 v5 v6 }|=3
(3rd, SJ)
|{v1 v2 v5 v6} {v2 v5 v6 v7}|=
|{v2 v5 v6 }|=3
(20, SJ) |{v1 v5 v6 v7 }{v2 v5 v6 v7}|=
|{v5 v6 v7}|= 3
94
Fundamental Theorems
1. All isomorphic relations are isomorphic to the canonical model (GDM)
2. A granule of GDM is a high frequency pattern if it has high support.
95
Relation Lattice Theorems
1. The granules of GDM generate a lattice of granules with join = and meet=.
This lattice is called Relational Lattice by Tony Lee (1983)
2. All elements of lattice can be written as join of prime (join-irreducible elements)
(Birkoff & MacLane, 1977, Chapter 11)
96
Find Association by Linear Inequalities
Theorem. Let P1, P2, are primes (join-irreducible) in the Canonical Model. then
G=x1* P1 x2* P2 is a High Frequency Pattern, If
|G|= x1* |P1| +x2* |P2| + th,
(xj is binary number)
97
Join-irreducible elements
101st {v2 v3 v4}{v8 v9}=
20 1st {v1 v5 v6 v7 } {v8 v9}=
30 1s {v8 v9} {v8 v9}= {v8 v9}
10 2nd {v2 v3 v4} {v3 v4}= {v3 v4}
20 2nd {v1 v5 v6 v7 } {v3 v4}=
30 2nd {v8 v9} {v3 v4}=
10 3rd {v2 v3 v4}{v1 v2 v5 v6}= {v2}
20 3rd {v1v5v6v7}{v1 v2 v5 v6}= {v1 v5 v6}
30 3rd {v8 v9} {v1 v2 v5 v6}=
10 4th {v2 v3 v4} {v7}=
20 4th {v1 v5 v6 v7 }{v7}= {v7}
30 4th {v8 v9}{v7}=
98
AM by Linear Inequalities
|x1*{v1v5v6}=(20, 3rd)
+x2*{v2} =(10, 3rd)
+x3*{v3v4}=(10, 2nd)
+x4*{v7} =(20, 4th)
+x5*{v8v9} =(30, 1st)|
= x1*3+x2*1+x3*2+x4*1+ x5*2
99
AM by Linear Inequalities|x1*{v1v5v6}+x2*{v2}+x3*{v3v4}+x4*{v7}+x5*{v8v9}|
= x1*3+x2*1+x3*2+x4*1+ x5*2
1. x1=1
2. x2 =1, x3 =1, or x2 =1, x5 =1
3. x3 =1, x4 =1 or x3 =1, x5 =1
4. x4 =1, x5 =1
100
AM by Linear Inequalities|x1*{v1v5v6}+x2*{v2}+x3*{v3v4}+x4*{v7}+x5*{v8v9}|
= x1*3+x2*1+x3*2+x4*1+ x5*2
1. x1=1
|1*{v1v5v6} | = 1*3=3
(20, 3rd) |{v1 v5 v6 v7 } {v1 v2 v5 v6 }|=
|{v1 v5 v6 }|=3
101
AM by Linear Inequalities|x1*{v1v5v6}+x2*{v2}+x3*{v3v4}+x4*{v7}+x5*{v8v9}|
= x1*3+x2*1+x3*2+x4*1+ x5*2
x2 =1, x3 =1, or x2 =1, x5 =1
|x2*{v2}+x3*{v3v4}| =(1020, 3rd)
|x2*{v2}+x5*{v8v9}| =(10, 2nd) (10, 3rd)
x3 =1, x4 =1 or x3 =1, x5 =1
x4 =1, x5 =1
102
AM by Linear Inequalities
|x1*{v1v5v6}+x2*{v2}+x3*{v3v4}+x4*{v7}+x5*{v8v9}|
= x1*3+x2*1+x3*2+x4*1+ x5*2x3 =1, x4 =1 or x3 =1, x5 =1
| x3*{v3v4}+x4*{v7}| =(10, 2nd 3rd)
| x3*{v3v4}+x5*{v8v9}| =(10, 2nd) (30, 1st)
x4 =1, x5 =1
103
AM by Linear Inequalities
|x1*{v1v5v6}+x2*{v2}+x3*{v3v4}+x4*{v7}+x5*{v8v9}|
= x1*3+x2*1+x3*2+x4*1+ x5*2x4 =1, x5 =1
| x3*{v3v4}+x5*{v8v9}| =(20, 4st) (30, 1st)