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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

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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

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Outline

1. Introduction

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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.)

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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

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Granular computing

IEEE GrC-conference

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

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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)

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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

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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 theory

Mathematicians have idealized the granulation into

Partition (at least 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.

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Partition theory

But Partition requires Absolutely no overlapping

among granules

(equivalence classes)

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More General Theory

Partitions is too restrictive for real world applications.

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More General Theory

Even in natural science, classification does permit a small degree of overlapping;

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More General Theory

There are beings that are both proper subjects of zoology and botany.

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More General Theory

A more general theory is needed

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Outline

2. New Theory - Granular Computing

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Zadeh’s Intuitive notion

Granulation involves

partitioning

a class of objects(points)

into granules,

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Zadeh’s Intuitive notion

with a granule being

a clump of objects (points)

which are drawn together by indistinguishability, similarity or functionality.

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Formalization

We will present a formal theory, we believe, that has captures quite an essence of Zadeh’s idea (but not full)

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Outline

3. A Formal Theory

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(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.

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(Single Level) Granulation A granulation is a map: p V B(p) 2U

where B(p) could be an empty set.

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(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.

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(Single Level) Granulation

Intuitively B(p) is the collection of objects that are drawn towards p

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Granulation - Binary Relation

The collection

B={(p, x) | x B(p) p V} VU

is a binary relation

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(Single Level) Granulation

If B is an equivalence relation the collection

{B(p)} is a partition

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More General Case

If we consider a set of Bj of binary relations(drawn by various “forces”, such as indistinguishability, similarity or functionality) then

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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).

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Development

4. Incremental Development

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Classical ParadigmWhat do we have?

1. (Divide) Partitioning

2. Quotient Set (Knowledge level)

3. Integration (of subtasks and quotient task)

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What do we have?

Classical Paradigm

1. Partition of a classical set (Divide)

Absolutely no overlapping among granules

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Some Mathematics

A partition

Granule A

Granule B

f, g, h i, j, k

Granule Cl, m, n

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Some Mathematics

Partition Equivalence relation

X Y (Equivalence Relation)

if and only if

both belong to the same class/granule

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Equivalence Relation Generalized Identity X X (Reflexive)

X Y implies Y X (Symmetric)

X Y, Y Z implies X Z (Transitive)

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Example

Partition

[0]4 = {. . . , 0, 4, 8, . . .},

[1]4 = {. . . , 1, 5, 9, . . .},

[2]4 = {. . . , 2, 6, 10, . . .},

[3]4 = {. . . , 3, 7, 11, . . .}.

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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

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New territories

Granulation (not Partition)

B0 = [0]4 {5, 9},

B1 = [1]4 ={. . . , 1, 5, 9, . . .},

B2 = [2]4 {7},

B3 = [3]4 {6}.

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New territories

Granulation (not Partition)

B0 B1 = {5, 9},

B2 B3 = {6,7},

Could we define a quotient set ?

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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

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New Paradigm

In general, classical scheme is unavailable for general granulation

We will show that:

classical scheme can be extended to single level granulation

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New formal theory

New view of the universe

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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.

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Classical ParadigmWhat do we have?

1. (Divide) Partitioning

2. Quotient Set (Knowledge level)

3. Integration (of subtasks and quotient task)

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What are in the new paradigm?

1. Partition of B-space (Divide)

2. Quotient B-space (Knowledge)

3. Integration-Approximation (and extension)

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Integration-Approximations

Some Comments on approximations

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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)

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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)

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Upper/Closure approximations

Cl(X)= iCi(X) (Sierpenski-topology)

Where Ci(X)= C(…(C(X))…)

(transfinite steps) Cl(X) is closed.

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New View

Divide (and Conquer)

Partition of set (generalize) ?

Partition of B-space

(topological partition)

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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

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Derived Partitions

The inverse images of B is a partition (an equivalence relation)

C ={Cp | Cp =B –1 (Bp) p V}

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Derived Partitions

Cp is called the center class of Bp

A member of Cp is called a center.

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Derived Partitions

The center class Cp consists of all the points that have the same granule

Center class Cp = {q | Bq= Bp}

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C-quotient set

The set of center classes Cp is a quotient

set

Iran, Iraq. . US, UK, . . .

Russia, Korea

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New Problem Solving Paradigm

(Divide and) Conquer

Quotient set

Topological Quotient space

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Neighborhood of center class C (in the case B is not reflexive)

B-granule/neighborhood C-classes

C-classes

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Neighborhood of center class

B-granule C-classes

C-classes

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Topological partition

Cp -classes

Cp -classes

B-granule/neighborhood

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New Problem Solving Paradigm

(Divide and) Conquer

Quotient set

Topological Quotient space

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Topological partition

Cp -classes

Cp -classes

B-granule/neighborhood

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Topological partition

Cp -classes

Cp -classes

B-granule/neighborhood

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Topological partition

Cp -classes

Cp -classes

B-granule/neighborhood

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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)

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Future Direction

Topological Reduct

Topological Table processing

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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.

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Background

Brewer and Nash (BN) proposed Chinese Wall Security Policy Model (CWSP) 1989 for this purpose

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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."

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A little Fomral

Simple CWSP(SCWSP):

No single agent can read data X and Y

that are in CONFLICT

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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

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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

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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?

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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

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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

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Composition Information Flow

Aggressive CWSP says that a system is secure, if

“(X, Y) CIR X NCIF Y “

“(X, Y) CIR X CIF Y “

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The Problem

Simple CWSP ? Aggressive CWSP

This is a malicious Trojan Horse problem

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Need ACWSP Theorem

Theorem If CIR is anti-reflexive, symmetric and anti-transitive, then

Simple CWSP Aggressive CWSP

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C and CIR classes CIR: Anti-reflexive, symmetric, anti-transitive

CIR-class Cp -classes

Cp -classes

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Application 2

Association mining by Granular/Bitmap computing

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Fundamental Theorem

Theorem 1:

All isomorphic relations have isomorphic patterns

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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)

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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)

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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)

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Isomorphic 2-Associations

K Count K’

(TWENTY, MAR)

3 (20, 3rd)

(MAR, SJ) 3 (3rd, San Jose)

(TWENTY, SJ) 3 (20, San Jose)

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Canonical Model Bitmaps in Granular Forms

Patterns in Granular Forms

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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

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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}

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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)

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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}

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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

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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

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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.

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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)

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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)

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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}=

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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

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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

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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

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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

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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

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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)