Efficient decomposition of large fuzzy functions and relations
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Efficient Decomposition of Large Fuzzy Functions and Relations. Minimization of Fuzzy Functions. Fuzzy functions are realized in: analog hardware software Why to minimize fuzzy logic functions? Smaller area Lower Power Simpler and faster program

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Efficient Decomposition of Large Fuzzy Functions and Relations

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Efficient decomposition of large fuzzy functions and relations

Efficient Decomposition of Large Fuzzy Functions and Relations


Minimization of fuzzy functions

Minimization of Fuzzy Functions

  • Fuzzy functions are realized in:

    • analog hardware

    • software

  • Why to minimize fuzzy logic functions?

    • Smaller area

    • Lower Power

    • Simpler and faster program

    • Better learning, Occam Razor - not covered here


Minimization approaches to fuzzy functions

Minimization Approaches to Fuzzy Functions

  • Two level minimization (Siy,Kandel, Mukaidono, Lee, Rovatti et al)

  • Algebraic factorization (Wielgus)

  • Genetic algorithms (Thrift, Bonarini, many authors)

  • Fuzzy decision diagrams (Moraga, Perkowski)

  • Functional decomposition (Kandel, Kandel and Francioni)


Graphical representations

Graphical Representations

  • Fuzzy Maps

  • Lattice of variables

  • The Subsumption rule

  • Kandel’s methods to decompose Fuzzy Functions


Identities

Identities

The identities for fuzzy algebra are:

Idempotency: X + X = X, X * X = X

Commutativity: X + Y = Y + X, X * Y = Y * X

Associativity: (X + Y) + Z = X + (Y + Z),

(X * Y) * Z = X * (Y* Z)

Absorption: X + (X * Y) = X, X * (X + Y) = X

Distributivity: X + (Y * Z) = (X + Y) * (X + Z),

X * (Y + Z) = (X * Y) + (X * Z)

Complement: X’’ = X

DeMorgan's Laws: (X + Y)’ = X’ * Y’, (X * Y)’ = X’ + Y’


Transformations

Transformations

Some transformations of fuzzy sets with examples follow:

x’b + xb = (x + x’)b  b

xb + xx’b = xb(1 + x’) = xb

x’b + xx’b = x’b(1 + x) = x’b

a + xa = a(1 + x) = a

a + x’a = a(1 + x’) = a

a + xx’a = a

a + 0 = a

x + 0 = x

x * 0 = 0

x + 1 = 1

x * 1 = x

Examples:

a + xa + x’b + xx’b = a(1 + x) + x’b(1 + x) = a + x’b

a + xa + x’a + xx’a = a(1 + x + x’ + xx’) = a


Differences between boolean logic and fuzzy logic

Differences Between Boolean Logic and Fuzzy Logic

Boolean logic the value of a variable and its inverse are always disjoint (X * X’ = 0) and (X + X’ = 1) because the values are either zero or one.

Fuzzy logic membership functions can be either disjoint or non-disjoint.

Example of a fuzzy non-linear and linear membership function X is shown (a) with its inverse membership function shown in (b).

We first discuss a simplified logic with few literals


Fuzzy intersection and union

Fuzzy Intersection and Union

  • From the membership functions shown in the top in (a), and complement X’ (b) the intersection of fuzzy variable X and its complement X’ is shown bottom in (a).

  • From the membership functions shown in the top in (a), and complement X’ (b) the union of fuzzy variable X and its complement X’ is shown bottom in (b).


Approaches to fuzzy logic decomposition

Our new approach

APPROACHES TO FUZZY LOGIC DECOMPOSITION

  • Kandel's and Francioni's Approach based on graphical representations:

    • Requires subsumption-based reduction to canonical form

    • Graphical: uses S-Maps and Fuzzy Maps

    • Decomposition Implicant Pattern (DIP)

    • Variable Matching DIP’s Table

    • non-algorithmic

    • not scalable to larger functions

    • no software

      Fuzzy to Multiple-valued Function Conversion Approach and use of Generalized Ashenhurst-Curtis Decomposition


New approach fuzzy to multiple valued function conversion and a c decomposition

New Approach: Fuzzy to Multiple-valued Function Conversion and A/C Decomposition

  • Fuzzy Function Ternary Map

  • Fuzzy Function to Three-valued Function Conversion:

    • The MAX operation forms the result

    • The result from the canonical form is the same as from the non-canonical form

      • Thus time consuming reduction to canonical form is not necessary


Fuzzy function ternary map

Fuzzy Function Ternary Map

This shows the mapping between the fuzzy terms and terms in the ternary map.


Fuzzy function to three valued function conversion example

Fuzzy Function to Three-valued Function Conversion Example

Non-canonical

Conversion of the Fuzzy function terms:

x2x’2

x’1x2

x1x’2

x1x’1x’2

In non-canonical form using the MIN operation as shown for

f = x2x’2 +x’1x2 +x1x’2+ x1x’1x’2

x2x’2

x’1x2

x1x’2

x1x’1x’2


The max operation forms the result

The MAX operation forms the result

  • Combining the three-valued term functions into a single three-valued function is performed using the MAX Operation


The result from the canonical form is the same as from the non canonical form

F = x2x’2 +x’1x2 +x1x’2+ x1x’1x’2

conversion is equal to

F(x1x2) =x’1x2+x1x’2

The result from the canonical form is the same as from the non-canonical form

canonical

canonical

Non-canonical


Entire flow of our method

Initial non-canonical expression

Entire flow of our method

F(x,y,z) = xz + x’y’zz’ + yz

decomposed expression

G(x,y) = x+y

H(x,y) = Gz + zz’.

Fuzzy to Ternary Conversion

Generalized Ashenhurst-Curtis Decomposition

Decomposed Function


Entire flow of our method1

Initial non-canonical expression

Entire flow of our method

F(x,y,z) = xz + x’y’zz’ + yz

Only three patterns

0 1 0

Decomposition is based on finding patterns in this table

0 1 1

0 1 2

This way, the table is rewritten to the table from the next page


Efficient decomposition of large fuzzy functions and relations

0 1 0

0 1 1

0 1 2


Multiple valued function minimized and converted to fuzzy circuit

Two solutions are obtained in this case

Multiple-valued function minimized and converted to fuzzy circuit

Fuzzy terms Gz, G’zz’ and zz’ of H

are shown.

G(x,y) = x+y,

H(x,y) = Gz+zz’

G(x,y) = x+y,

H(x,y) = Gz+G’zz’


Generalization of the ashenhurst curtis decomposition model

Generalization of the Ashenhurst-Curtis decomposition model


Efficient decomposition of large fuzzy functions and relations

This kind of tables known from Rough Sets, Decision Trees, etc Data Mining


Efficient decomposition of large fuzzy functions and relations

Decomposition is hierarchical

At every step many decompositions exist


Ashenhurst functional decomposition

Ashenhurst Functional Decomposition

X

A - free set

Evaluates the data function and attempts to decompose into simpler functions.

F(X) = H( G(B), A ), X = A B

B - bound set

if A B = , it is disjoint decomposition

if A B  , it is non-disjoint decomposition


A standard map of function z

A Standard Map of function ‘z’

Explain the concept of generalized don’t cares

Bound Set

a b \ c

Columns 0 and 1

and

columns 0 and 2

are compatible

column

compatibility = 2

Free Set

z


New decomposition of multi valued relations

Relation

NEW Decomposition of Multi-Valued Relations

F(X) = H( G(B), A ), X = A B

A

X

Relation

Relation

B

if A B = , it is disjoint decomposition

if A B  , it is non-disjoint decomposition


Forming a ccg from a k map

Bound Set

a b \ c

C0

C1

Free Set

C2

Forming a CCG from a K-Map

Columns 0 and 1 and columns 0 and 2 are compatible

column compatibility index = 2

Column Compatibility Graph

z


Efficient decomposition of large fuzzy functions and relations

Forming a CIG from a K-Map

a b \ c

C0

C1

C2

z

Columns 1 and 2 are incompatible

chromatic number = 2

Column Incompatibility Graph


Efficient decomposition of large fuzzy functions and relations

C0

C0

C1

C1

C2

C2

Column Compatibility Graph

Column Incompatibility Graph

CCG and CIG are complementary

Maximal clique covering

clique partitioning

Graph coloring

graph multi-coloring


Clique partitioning example

clique partitioning example.


Maximal clique covering example

Maximal clique covering example.


Efficient decomposition of large fuzzy functions and relations

\ c

\ c

G

G

Map of relation G

After induction

From CIG

g = a high pass filter whose acceptance threshold begins at

c > 1


Cost function

Cost Function

Decomposed Function Cardinalityis the total cost of all blocks.

Cost is defined for a single block in terms of the block’s n inputs and m outputs

Cost := m * 2n


Example of dfc calculation

B2

B1

B3

Cost(B2) =23*2=16

Cost(B3) =22*1=4

Cost(B1) =24*1=16

Example of DFC calculation

Total DFC = 16 + 16 + 4 = 36

Other cost functions


Decomposition algorithm

Decomposition Algorithm

  • Find a set of partitions (Ai, Bi) of input variables (X) into free variables (A) and bound variables (B)

  • For each partitioning, find decompositionF(X) = Hi(Gi(Bi), Ai) such that column multiplicity is minimal, and calculate DFC

  • Repeat the process for all partitioning until the decomposition with minimum DFC is found.


Algorithm requirements

Algorithm Requirements

  • Since the process is iterative, it is of high importance that minimization of the column multiplicity index is done as fast as possible.

  • At the same time, for a given partitioning, it is important that the value of the column multiplicity is as close to the absolute minimum value


Column multiplicity

Bound Set

1

3

Free Set

4

2

Column Multiplicity


Column multiplicity other example

D

1

0

AB

C

0

0

Free Set

0

1

1

1

X

Column Multiplicity-other example

Bound Set

CD

1

3

4

2

1

3

2

4

X=G(C,D)

X=C in this case

But how to calculate function H?


Decomposition of multiple valued relation

Decomposition of multiple-valued relation

compatible

Compatibility Graph for columns

Karnaugh Map

Kmap of block G

One level of decomposition

Kmap of block H


Efficient decomposition of large fuzzy functions and relations

Compatibility of columns for Relations is not transitive !

This is an important difference between decomposing functions and relations


Decomposition of relations

Decomposition of Relations

Now H is a relation

which can be either decomposed or minimized directly in a sum-of-products fashion


Variable ordering

Variable ordering

But how to select good partitions of variables?


Vacuous variables removing

Vacuous variables removing

  • Variables b and d reduce uncertainty of y to 0 which means they provide all the information necessary for determination of the output y

  • Variables a and c are vacuous


Efficient decomposition of large fuzzy functions and relations

Example of removing inessential variables (a) original function (b)variable a removed (c) variable b removed, variable c is no longer inessential.


Discovering new concepts

Discovering new concepts

  • Discovering concepts useful for purchasing a car


Other approaches to fuzzy logic minimization

OTHER APPROACHES TO FUZZY LOGIC MINIMIZATION

Graphical Representations

  • Fuzzy to Multiple-valued Function Conversion Approach

  • Fuzzy Logic Decision Diagrams Approach

  • Fuzzy Logic Multiplexer


Fuzzy maps

Fuzzy Maps

Fuzzy map may be regarded as an extension of the Veitch diagram, which forms also the basis for the Karnauph map.

The functions shown in (a) and (b) are equivalent to f(x1, x2) =x’1 x2+x1 x’1 x’2 =x1 x’1

(b) f(x1, x2) =x1 x’1

(a) f(x1, x2) =x1 x’1 x2+x1 x’1 x’2


Efficient decomposition of large fuzzy functions and relations

(a) f(x1, x2) =x1 x’1 x2+x1 x’1 x’2

x2=0.3

x2’=0.7

(a) f(x1, x2) =min(x1 , x’1, 0.3) max min(x1 ,x’1 , 07) = min(0.5, 0.3) max min (0.5, 0.7) = 0.3 max 0.5 = 0.5 = x1 x’1

Assuming max value of x1 x’1

Please check other values

(b) f(x1, x2) =x1 x’1


Lattice of two variables

Lattice of Two Variables

  • Shows the relationship of all the possible terms.

  • Shows which two terms can be reduced to a single term.


The subsumption rule

The Subsumption Rule

Used to reduce a fuzzy logic function.

 xi x’I +’ xi x’I  = xi x’I 

Operations on two variable map are shown with I subsuming i.


The subsumption rule1

The Subsumption Rule

  • Used to reduce a fuzzy logic function.

  • The subsumption rule is:

     xi x’I +’ xi x’I  = xi x’I 

  • Operations on two variable map are shown with I subsuming i.


Form needed to decompose fuzzy functions

Form Needed to Decompose Fuzzy Functions

  • Form requirements:

  • Sum-of-products

  • Canonical

  • Figures show the function x2 x’2 +x’1 x2 +x1 x’2 +x1 x’1 x’2

  • before using the subsuming rules in (a) and after in (d) x’1 x2 +x1 x’2 .

x1


Efficient decomposition of large fuzzy functions and relations

 xi x’I +’ xi x’I  = xi x’I 

subsumption

Let us use subsumption to verify:

x2 x’2 +x’1 x2 +x1 x’2 +x1 x’1 x’2

= x2 x’2 +x’1 x2 +x1 x’2(1+x’1)

= x2 x’2 +x’1 x2 +x1 x’2

= x’1 x2 +x1 x’2 .


Graphical representations1

Graphical Representations

  • Fuzzy Maps

  • Lattice of two variables

  • The Subsumption rule

  • Form to Decompose a Fuzzy Functions


Approaches to fuzzy logic decomposition1

APPROACHES TO FUZZY LOGIC DECOMPOSITION

  • Graphical Representations

    Fuzzy Logic Decision Diagrams Approach

  • Fuzzy Logic Multiplexer


Fuzzy logic decision diagrams approach

Fuzzy Logic Decision Diagrams Approach

w w’(z + x’ z z’ + xz) + w’ (x’z z’ + xz) + w(z+xz)+ xz


Result of example using fldd

Result of Example using (FLDD)

(w+x)z + (w+x)’ z’z = wz + xz + w’x’z’z + w w’z + w w’ x’ z z’


Fuzzy logic multiplexer

Fuzzy Logic Multiplexer

d3xx’


Fuzzy logic circuit implemented using multiplexers

Fuzzy Logic Circuit Implemented using Multiplexers


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