CS621: Artificial Intelligence Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 11– Fuzzy Logic: Inferencing

Representation of Fuzzy sets Let U = {x 1,x 2,…..,x n } |U| = n The various sets composed of elements from U are presented as points on and inside the n-dimensional hypercube. The crisp sets are the corners of the hypercube. (1,0) (0,0) (0,1) (1,1) x1x1 x2x2 x1x1 x2x2 (x 1,x 2 ) A(0.3,0.4) μ A (x 1 )=0.3 μ A (x 2 )=0.4 Φ U={x 1,x 2 } A fuzzy set A is represented by a point in the n-dimensional space as the point {μ A (x 1 ), μ A (x 2 ),……μ A (x n )}

Degree of fuzziness The centre of the hypercube is the “most fuzzy” set. Fuzziness decreases as one nears the corners Measure of fuzziness Called the entropy of a fuzzy set Entropy Fuzzy setFarthest corner Nearest corner

(1,0) (0,0) (0,1) (1,1) x1x1 x2x2 d(A, nearest) d(A, farthest) (0.5,0.5) A

Definition Distance between two fuzzy sets L 1 - norm Let C = fuzzy set represented by the centre point d(c,nearest) = | | + |0.5 – 0.0| = 1 = d(C,farthest) => E(C) = 1

Definition Cardinality of a fuzzy set [generalization of cardinality of classical sets] Union, Intersection, complementation, subset hood

Note on definition by extension and intension S 1 = {x i |x i mod 2 = 0 } – Intension S 2 = {0,2,4,6,8,10,………..} – extension How to define subset hood?

Meaning of fuzzy subset Suppose, following classical set theory we say if Consider the n-hyperspace representation of A and B (1,1) (1,0)(0,0) (0,1) x1x1 x2x2 A. B 1.B 2.B 3 Region where

This effectively means CRISPLY P(A) = Power set of A Eg: Suppose A = {0,1,0,1,0,1,…………….,0,1} – 10 4 elements B = {0,0,0,1,0,1,……………….,0,1} – 10 4 elements Isn’t with a degree? (only differs in the 2 nd element)

Fuzzy definition of subset Measured in terms of “fit violation”, i.e. violating the condition Degree of subset hood = 1- degree of superset hood = m(A) = cardinality of A =

We can show that Exercise 1: Show the relationship between entropy and subset hood Exercise 2: Prove that Subset hood of B in A

Fuzzy sets to fuzzy logic Forms the foundation of fuzzy rule based system or fuzzy expert system Expert System Rules are of the form If then A i Where C i s are conditions Eg: C 1 =Colour of the eye yellow C 2 = has fever C 3 =high bilurubin A = hepatitis

In fuzzy logic we have fuzzy predicates Classical logic P(x 1,x 2,x 3 …..x n ) = 0/1 Fuzzy Logic P(x 1,x 2,x 3 …..x n ) = [0,1] Fuzzy OR Fuzzy AND Fuzzy NOT

Fuzzy Implication Many theories have been advanced and many expressions exist The most used is Lukasiewitz formula t(P) = truth value of a proposition/predicate. In fuzzy logic t(P) = [0,1] t( ) = min[1,1 -t(P)+t(Q)] Lukasiewitz definition of implication

Eg: If pressure is high then Volume is low Pressure High Pressure

Fuzzy Inferencing

Fuzzy Inferencing: illustration through inverted pendulum control problem Core The Lukasiewitz rule t( ) = min[1,1 + t(P) – t(Q)] An example Controlling an inverted pendulum θ = angular velocity Motor i=current

The goal: To keep the pendulum in vertical position (θ=0) in dynamic equilibrium. Whenever the pendulum departs from vertical, a torque is produced by sending a current ‘i’ Controlling factors for appropriate current Angle θ, Angular velocity θ. Some intuitive rules If θ is +ve small and θ. is –ve small then current is zero If θ is +ve small and θ. is +ve small then current is –ve medium

-ve med -ve small Zero +ve small +ve med -ve med -ve small Zero +ve small +ve med +ve small -ve small -ve med -ve small +ve small Zero Region of interest Control Matrix θ.θ. θ

Each cell is a rule of the form If θ is <> and θ. is <> then i is <> 4 “Centre rules” 1.if θ = = Zero and θ. = = Zero then i = Zero 2.if θ is +ve small and θ. = = Zero then i is –ve small 3.if θ is –ve small and θ. = = Zero then i is +ve small 4.if θ = = Zero and θ. is +ve small then i is –ve small 5. if θ = = Zero and θ. is –ve small then i is +ve small

Linguistic variables 1.Zero 2.+ve small 3.-ve small Profiles -ε-ε+ε+ε ε2 ε2 -ε2-ε2 -ε3-ε3 ε3ε3 +ve small -ve small 1 Quantity (θ, θ., i)

Inference procedure 1. Read actual numerical values of θ and θ. 2. Get the corresponding μ values μ Zero, μ (+ve small), μ (-ve small). This is called FUZZIFICATION 3. For different rules, get the fuzzy I-values from the R.H.S of the rules. 4. “Collate” by some method and get ONE current value. This is called DEFUZZIFICATION 5. Result is one numerical value of ‘i’.