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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Minimization of fuzzy functions
Minimization of Fuzzy Functions Relations

  • 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 Relations

  • 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 Relations

  • Fuzzy Maps

  • Lattice of variables

  • The Subsumption rule

  • Kandel’s methods to decompose Fuzzy Functions


Identities
Identities Relations

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 Relations

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 Relations

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 Relations

  • 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 Relations

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 and A/C Decomposition

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 and A/C DecompositionConversion 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 and A/C Decomposition

  • 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 = x and A/C Decomposition2x’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 and A/C Decomposition

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 and A/C Decomposition

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 and A/C Decomposition

0 1 1

0 1 2


Multiple valued function minimized and converted to fuzzy circuit

Two solutions are obtained in this case and A/C Decomposition

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’




Efficient decomposition of large fuzzy functions and relations

Decomposition is hierarchical etc Data Mining

At every step many decompositions exist


Ashenhurst functional decomposition
Ashenhurst Functional Decomposition etc Data Mining

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’ etc Data Mining

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 etc Data Mining

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 etc Data Mining

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 etc Data Mining

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

C etc Data Mining0

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




Efficient decomposition of large fuzzy functions and relations

\ etc Data Mining 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 etc Data Mining

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 etc Data Mining

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 etc Data Mining

  • 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 etc Data Mining

  • 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 etc Data Mining

1

3

Free Set

4

2

Column Multiplicity


Column multiplicity other example

D etc Data Mining

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 etc Data Miningrelation

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 etc Data Mining!

This is an important difference between decomposing functions and relations


Decomposition of relations
Decomposition of Relations etc Data Mining

Now H is a relation

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


Variable ordering
Variable ordering etc Data Mining

But how to select good partitions of variables?


Vacuous variables removing
Vacuous variables removing etc Data Mining

  • 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 function (b)

  • Discovering concepts useful for purchasing a car


Other approaches to fuzzy logic minimization
OTHER APPROACHES TO FUZZY LOGIC MINIMIZATION function (b)

Graphical Representations

  • Fuzzy to Multiple-valued Function Conversion Approach

  • Fuzzy Logic Decision Diagrams Approach

  • Fuzzy Logic Multiplexer


Fuzzy maps
Fuzzy Maps function (b)

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) function (b) 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 function (b)

  • 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 function (b)

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 function (b)

  • 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 function (b)

  • 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

 x function (b)i 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 function (b)

  • 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 function (b)

  • Graphical Representations

    Fuzzy Logic Decision Diagrams Approach

  • Fuzzy Logic Multiplexer


Fuzzy logic decision diagrams approach
Fuzzy Logic Decision Diagrams Approach function (b)

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) function (b)

(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 function (b)

d3xx’