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 –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

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 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 = xExamples: 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

Short Introduction: multiple-valued logic {0,1} - binary logic (a special case) {0,1,2} - a ternary logic {0,1,2,3} - a quaternary logic, etc Signals can have values from some set, for instance {0,1,2}, or {0,1,2,3} Minimal value MINMIN MAXMAX 2 Maximal value

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

Conversion Fuzzy Function to Three-valued Function Conversion Example Conversion of the Fuzzy function terms: x 2 x’ 2 x’ 1 x 2 x 1 x’ 2 x 1 x’ 1 x’ 2 In non- canonical form using the MIN operation as shown for f = x 2 x’ 2 +x’ 1 x 2 +x 1 x’ 2 + x 1 x’ 1 x’ 2 x 2 x’ 2 x’ 1 x 2 x 1 x’ 2 x 1 x’ 1 x’ 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 = x 2 x’ 2 +x’ 1 x 2 +x 1 x’ 2 + x 1 x’ 1 x’ 2 conversion is equal to F(x 1 x 2 ) =x’ 1 x 2 +x 1 x’ 2 canonical Non-canonical

F(x,y,z) = xz + x’y’zz’ + yz Entire flow of our method Initial non-canonical expression Decomposed Function Generalized Ashenhurst- Curtis Decomposition Fuzzy to Ternary Conversion decomposed expression H(x,y) = Gz + zz’. G(x,y) = x+y

F(x,y,z) = xz + x’y’zz’ + yz Entire flow of our method Initial non-canonical expression Decomposition is based on finding patterns in this table Only three patterns This way, the table is rewritten to the table from the next page

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 H are shown. G(x,y) = x+y, H(x,y) = Gz+G’zz’

Generalization of the Ashenhurst- Curtis decomposition model

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

Decomposition is hierarchical At every step many decompositions exist

Ashenhurst Functional Decomposition Evaluates 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 = A  B

A Standard Map of function ‘z’ Bound Set Free Set a b \ c z Columns 0 and 1 and columns 0 and 2 are compatible column compatibility = 2 Explain the concept of generalized don’t cares

NEW Decomposition of Multi-Valued Relations 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 A B X

Forming a CCG from a K-Map z Bound Set Free Set a b \ c Columns 0 and 1 and columns 0 and 2 are compatible column compatibility index = 2 C1C1 C2C2 C0C0 Column Compatibility Graph

Forming a CIG from a K-Map Columns 1 and 2 are incompatible chromatic number = 2 z a b \ c C1C1 C2C2 C0C0 Column Incompatibility Graph

CCG and CIG are complementary C1C1 C2C2 C0C0 Column Incompatibility Graph C1C1 C2C2 C0C0 Column Compatibility Graph Maximal clique covering clique partitioning Graph coloring graph multi- coloring

clique partitioning example.

Maximal clique covering example.

G \ c g = a high pass filter whose acceptance threshold begins at c > 1 Map of relation G G \ c From CIG After induction

Cost Function Decomposed Function Cardinality is 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 * 2 n

Example of DFC calculation B1 B2 B3 Cost(B3) =2 2 *1=4 Cost(B1) =2 4 *1=16 Cost(B2) =2 3 *2=16 Total DFC = = 36 Other cost functions

Decomposition Algorithm Find a set of partitions (A i, B i ) of input variables (X) into free variables (A) and bound variables (B) For each partitioning, find decomposition F(X) = H i (G i (B i ), A i ) 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 Since the process is iterative, it is of high importance that minimization of the column multiplicity index is done as fast as possible. absolute minimumAt 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 Free Set

Column Multiplicity-other example Bound Set Free Set AB CD D C X=G(C,D) X=C in this case But how to calculate function H?

relation Decomposition of multiple-valued relation Karnaugh Map Compatibility Graph for columns compatible Kmap of block G Kmap of block H One level of decomposition

Compatibility of columns for 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 ordering But how to select good partitions of variables?

Vacuous variables removing Variables b and d reduce uncertainty o f 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 purchasing a carDiscovering concepts useful for purchasing a car

OTHER APPROACHES TO FUZZY LOGIC MINIMIZATION Graphical 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(x 1, x 2 ) = x’ 1 x 2 + x 1 x’ 1 x’ 2 = x 1 x’ 1 Fuzzy Maps (b) f(x 1, x 2 ) = x 1 x’ 1 (a) f(x 1, x 2 ) =x 1 x’ 1 x 2 + x 1 x’ 1 x’ 2

(b) f(x 1, x 2 ) = x 1 x’ 1 (a) f(x 1, x 2 ) =x 1 x’ 1 x 2 + x 1 x’ 1 x’ 2 x 2 =0.3x 2 ’ =0.7 (a) f(x 1, x 2 ) =min(x 1, x’ 1, 0.3) max min(x 1, x’ 1, 07) = min(0.5, 0.3) max min (0.5, 0.7) = 0.3 max 0.5 = 0.5 = x 1 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.

 x i x’ I  +  ’ x i x’ I  = x i x’ I   x i x’ I  +  ’ x i x’ I  = x i x’ I  The Subsumption Rule 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:  x i x’ I  +  ’ x i x’ I  = x i x’ I  Operations on two variable map are shown with I subsuming i. The Subsumption Rule

Form Needed to Decompose Fuzzy Functions Form requirements: 1.Sum-of-products 2.Canonical Figures show the function x 2 x’ 2 + x’ 1 x 2 + x 1 x’ 2 + x 1 x’ 1 x’ 2 before using the subsuming rules in (a) and after in (d) x’ 1 x 2 + x 1 x’ 2. x1

x 1 x’ 2 + x 1 x’ 1 x’ 2 x 2 x’ 2 + x’ 1 x 2 + x 1 x’ 2 + x 1 x’ 1 x’ 2 x 1 x’ 2 = x 2 x’ 2 + x’ 1 x 2 + x 1 x’ 2 (1+ x’ 1 ) x 1 x’ 2 = x 2 x’ 2 + x’ 1 x 2 + x 1 x’ 2 x 1 x’ 2 = x’ 1 x 2 + x 1 x’ 2.  x i x’ I  +  ’ x i x’ I  = x i x’ I   x i x’ I  +  ’ x i x’ I  = x i x’ 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 d 3 xx’

Fuzzy Logic Circuit Implemented using Multiplexers