Efficient Decomposition of Large Fuzzy Functions and Relations

Efficient Efficient Decomposition of Decomposition of

Large Fuzzy Large Fuzzy Functions and Functions and

RelationsRelations

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

• 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

• Fuzzy Maps• Lattice of variables• The Subsumption rule• Kandel’s methods to

decompose Fuzzy Functions

IdentitiesThe identities for fuzzy algebra are:

Idempotency: X + X = X, X * X = XCommutativity: X + Y = Y + X, X * Y = Y * XAssociativity: (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’’ = XDeMorgan's Laws: (X + Y)’ = X’ * Y’, (X * Y)’ = X’ + Y’

TransformationsSome transformations of fuzzy sets with examples follow:x’b + xb = (x + x’)b bxb + xx’b = xb(1 + x’) = xbx’b + xx’b = x’b(1 + x) = x’ba + xa = a(1 + x) = aa + x’a = a(1 + x’) = aa + xx’a = aa + 0 = ax + 0 = xx * 0 = 0x + 1 = 1x * 1 = x Examples: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

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• 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

• 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

Our new approach

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

DecompositionDecomposition• 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

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

Fuzzy Function to Three-valued Function ConversionConversion Example

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 forf = x2x’2 +x’1x2

+x1x’2+ x1x’1x’2

x2x’2

x’1x2

x1x’2

x1x’1x’2

Non-canonical

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

canonical

canonicalNon-canonical

F(x,y,z) = xz + x’y’zz’ + yzEntire flow of our methodInitial non-canonical expression

Decomposed Function

Generalized Generalized Ashenhurst-Ashenhurst-Curtis Curtis DecompositionDecomposition

Fuzzy to Fuzzy to Ternary Ternary ConversionConversion

decomposed expression

H(x,y) = Gz + zz’.G(x,y) = x+y

F(x,y,z) = xz + x’y’zz’ + yzEntire flow of our methodInitial non-canonical expression

Decomposition is Decomposition is based on finding based on finding patterns in this tablepatterns in this table

0 1 0

0 1 1

0 1 2

Only three Only three patternspatterns

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

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

G(x,y) = x+y, H(x,y) = Gz+zz’

Fuzzy terms Gz, G’zz’ and zz’ of Hare shown.

G(x,y) = x+y, H(x,y) = Gz+G’zz’

Generalization of Generalization of the Ashenhurst-the Ashenhurst-

Curtis Curtis decomposition decomposition

modelmodel

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

Decomposition is hierarchical At every step many

decompositions exist

Ashenhurst Functional DecompositionAshenhurst Functional DecompositionEvaluates the data function and attempts to

decompose into simpler functions.

if A B = , it is disjoint decompositionif A B , it is non-disjoint decomposition

X B - bound set

A - free set

F(X) = H( G(B), A ), X = AF(X) = H( G(B), A ), X = A BB

A Standard Map of A Standard Map of function ‘z’function ‘z’

Bound Set

Free

Set

a b \ c

z

Columns 0 and 1and

columns 0 and 2are compatible

column compatibility = 2

Explain the concept of Explain the concept of generalized don’t caresgeneralized don’t cares

NEW Decomposition of Multi-Valued NEW Decomposition of Multi-Valued RelationsRelations

if A B = , it is disjoint decompositionif A B , it is non-disjoint decomposition

F(X) = H( G(B), A ), X = A B

Relation Rel

atio

n

Rel

atio

n

A

B

X

Forming a CCG from a K-MapForming a CCG from a K-Map

z

Bound Set

Free

Set

a b \ c Columns 0 and 1 and columns 0 and 2 are compatiblecolumn compatibility index = 2

C1

C2

C0

Column Compatibility

Graph

Forming a CIG from a K-MapForming a CIG from a K-MapColumns 1 and 2 are incompatiblechromatic number = 2

z

a b \ c

C1

C2

C0

Column Incompatibility

Graph

CCG and CIG are CCG and CIG are complementarycomplementary

C1

C2

C0

Column Incompatibility

Graph

C1

C2

C0

Column Compatibility

Graph

Maximal Maximal clique clique coveringcovering

clique clique partitioningpartitioning

Graph Graph coloringcoloring

graph multi-graph multi-coloringcoloring

clique partitioning clique partitioning example.example.

Maximal clique covering Maximal clique covering example.example.

G

\ c

g = a high pass filter whose acceptance threshold begins at

c > 1

Map of relation GMap of relation G

G

\ c

From CIG After induction

Cost FunctionCost 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 calculationExample of DFC calculation

B1

B2

B3

Cost(B3) =22*1=4Cost(B1) =24*1=16

Cost(B2) =23*2=16

Total DFC = 16 + 16 + 4 = 36

Other cost functionsOther cost functions

Decomposition AlgorithmDecomposition 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 RequirementsAlgorithm 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 minimumabsolute minimum value

Column MultiplicityColumn Multiplicity

3

2

1

4

00 01 11 1000 0 0 – 101 – 1 0 011 1 – 1 010 1 1 0 0

Bound Set

Free

Set

1 2 3 4

Column Multiplicity-other Column Multiplicity-other exampleexample

3

2

1

4

00 01 11 1000 0 0 – 101 – 1 0 011 1 – 1 -10 1 1 0 0

Bound Set

1 2 3 4

Free

Set

AB CD DC

0

1

0 1

00 00

11 11

X=G(C,D)X=C in this case

But how to calculate function H?

Decomposition of multiple-valued relationrelation

Karnaugh Map Compatibility Graph for columns

compatible

Kmap of block G

Kmap of block H

One level of decomposition

Compatibility of columns for Compatibility of columns for Relations is not transitive Relations is not transitive !!

This is an important difference between decomposing functions and relations

Decomposition of Relations

Now H is a relation

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

Variable orderingVariable ordering

But how to select good But how to select good partitions of variables?partitions of variables?

Vacuous variables removingVacuous 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

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 concepts useful for purchasing a carpurchasing a car

OTHER APPROACHES TO FUZZY LOGIC

MINIMIZATIONGraphical Representations• Fuzzy to Multiple-valued Function Conversion

Approach• Fuzzy Logic Decision Diagrams Approach• Fuzzy Logic Multiplexer

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

Fuzzy MapsFuzzy Maps

(b) f(x1, x2) = x1 x’1(a) f(x1, x2) =x1 x’1 x2+ x1 x’1 x’2

(b) f(x1, x2) = x1 x’1

(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

Lattice of Two Variables• Shows the

relationship of all the possible terms.

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

xxi i x’x’I I ++ ’ x’ xi i x’x’I I = x = xi i x’x’I I

The The Subsumption Subsumption

RuleRule

Used to reduce a fuzzy logic function.

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

• 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.

The Subsumption RuleThe Subsumption Rule

Form Needed to Decompose Fuzzy Functions

Form requirements:

1. Sum-of-products2. 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

x2 x’2 + x’1 x2 + xx1 1 x’x’2 2 ++ xx1 1 x’x’1 1 x’x’22

= x2 x’2 + x’1 x2 + xx1 1 x’x’22 (1+ x’1) = x2 x’2 + x’1 x2 + xx1 1 x’x’22

= x’1 x2 + xx1 1 x’x’22 .

xxi i x’x’I I ++ ’ x’ xi i x’x’I I = x = xi i x’x’I I subsumption

Let us use subsumption to verify:

Graphical Representations• Fuzzy Maps• Lattice of two variables• The Subsumption rule• Form to Decompose a Fuzzy

Functions

APPROACHES TO FUZZY LOGIC DECOMPOSITION

• Graphical Representations

Fuzzy Logic Decision Diagrams Approach• Fuzzy Logic Multiplexer

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)

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

Fuzzy Logic Multiplexer

d3xx’

Fuzzy Logic Circuit Implemented using Multiplexers