Efficient Decomposition of Large Fuzzy Functions and Relations

Marek Perkowski

+ Portland State University, Dept. Electrical Engineering, Portland, Oregon 97207, Tel. 503-725-5411, Fax (503) 725-

4882, USA

FUZZY LOGIC IN FUZZY LOGIC IN ROBOTICSROBOTICS

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

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 two variables

• The Subsumption rule

• Kandel’s Form to Decompose Fuzzy Functions

IdentitiesThe 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’

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).

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).

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

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.

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:

Kandel's and Kandel's and Francioni's Approach Francioni's Approach

• Decomposition Implicant Pattern (DIP)

• Variable Matching DIP’s Table

• S-Maps

• Example using Kandel and Francioni approach

Decomposition Implicant Pattern (DIP)

Variable Matching DIPs Table

h(X) h’(X) DIP

xi

xixj

x’ix’j

x’ixj

xix’j

xixj + x’ix’j

x’i

x’i x’j

xi xj

xi + x’j

x’i + xj

xix’j + x’i xj

-

1

2

3

4

5

Tabular form of Decomposition Implicant Pattern (DIP) used in Kandel’s and Francioni’s approach

S-Maps

• Arrange two-variable fuzzy maps for n variables.• This method is just done by iteration to form an n variable

S-map.• This shows X1 is made up of repeated X2 and X3 two

variable maps.

Example using Kandel and Francioni approach

f = x’y’zz’ + xz + w’x’zz’ + wyz

From DIP 1 implies:g(w,y) =wy, G’ (w,y) = w’ + y’f = (wy)z + (w’ + y’) x’zz’ + xz

By substituting: G(w,y) = G(Y) and G’(w,y) = G’(Y)

f = F[(G(w,y), x,z)] = G(Y)z + G’(Y) x’zz’ + xz

APPROACHES TO FUZZY LOGIC DECOMPOSITION

• Kandel's and Francioni's Approach based on graphical representations:– 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 Example– The MAX operation forms the result– The result from canonical form are 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 form 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

Fuzzy to Multiple-valued Function Conversion Approach

Example– Fuzzy to Multiple-valued Function Conversion

Example• Fuzzy function to Multiple-valued function

• Input and results of decomposition

• Multiple-valued function to fuzzy function with circuit

– Method of Doing More Examples• Using Mathcad to do the MIN, MAX Operations

– Fuzzy Function Decomposition Results

F(x,y,z) = xz + x’y’zz’ + yzEntire flow of our method

Initial 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 method

Initial 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

Multiple-valued function to fuzzy function with circuitTwo solutions are obtained,

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’

Contents

• Fuzzy logic

• Fuzzy logic systems applications

• Approaches to fuzzy logic decomposition

Decomposition program

• Conclusion

Generalization of Generalization of the Ashenhurst-the Ashenhurst-

Curtis Curtis decomposition decomposition

modelmodel

Decomposition is hierarchical At every step many

decompositions exist

Functional DecompositionFunctional DecompositionEvaluates the data function and attempts to

decompose into simpler functions.

if A B = , it is disjoint decomposition

if 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 function ‘z’A Standard Map of function ‘z’Bound Set

Fre

e S

et

a b \ c

z

Columns 0 and 1and

columns 0 and 2are compatible

column compatibility = 2

Decomposition of Multi-Valued Decomposition of Multi-Valued RelationsRelations

if A B = , it is disjoint decomposition

if 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

Fre

e S

et

a b \ cColumns 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

C1

C2

C0

Column Compatibility

Graph

Column Incompatibility

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 10

00 0 0 – 101 – 1 0 011 1 – 1 010 1 1 0 0

Bound Set

Fre

e S

et

1 2 3 4

Column Multiplicity-other Column Multiplicity-other exampleexample

3

2

1

4

00 01 11 10

00 0 0 – 101 – 1 0 011 1 – 1 -10 1 1 0 0

Bound Set

1 2 3 4

Fre

e S

et

ABCD D

C

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 Checking and Correction for Relations Example

• Function that needs checked and corrected shown in a decomposition-map.

Compatibility Graph Show Cliques

• Cliques before checking and correction:

clique 0 = 0 1 2

clique 1 = 0 3

• Cliques after:

clique 0 = 0

clique 1 = 0 3

clique 2 = 1 2

• Compatibility graph and corrected cliques shown left

Discovering new concepts

• Discovering concepts useful for purchasing a carpurchasing a car

Variable orderingVariable ordering

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.

Need to Decompose Multiple-valued Functions and Relations

• Multiple-valued and Inconsistent Data

• Ways to Create Relations– Decomposition Process to Create Relations– Program to Change Inconsistency data into

Relations

Decomposition Structure

• General flow chart of RMVGUD Program.

Results of Relation Decomposition

3

21

18

68

157

244

132

323

1066

4

10

10

hayes

flare 1

flare 2

No. of

Blocks

No. of

cubes

No. of

Cubes

No. of

Inputs

Output FileInput FileFile

CONCLUSION CONCLUSION • Fuzzy functions are efficiently realized as multi-level

networks of fuzzy operators

• The approach has been extended also to relations. Relation is decomposed to network of relations and functions, non-decomposable relations are realized using other methods as the simplest functions

• Advantages of the new approach to fuzzy logic decomposition

– Eliminates the need for time-consuming conversion to canonical form

– Eliminates the use of S-maps

– Entirely algorithmic

– Enables decomposition of large fuzzy expressions

– Software tool exists