Fuzzy and Decomposition - Embedded Intelligent Robotics - Lecture Slides, Slides of Robotics

Download Fuzzy and Decomposition - Embedded Intelligent Robotics - Lecture Slides and more Slides Robotics in PDF only on Docsity!

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

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’

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).

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.

docsity.com

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 canonical Non-canonical

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 docsity.com

Generalization of the Ashenhurst- Curtis decomposition model

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