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Artificial Intelligence Techniques

This session aims to delve into the principles of Fuzzy Logic and Defuzzification within the realm of Artificial Intelligence. Fuzzy Logic, proposed by Lofti Zadeh in 1965, allows for the combination of weighting factors with propositions, enabling a transition from crisp Boolean values to fuzzy interpretations. Through examples, such as temperature categorization, we explore the concept of fuzzy sets and membership functions. We also examine the process of defuzzification, emphasizing the importance of the center of gravity in obtaining meaningful outcomes from fuzzy data.

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Artificial Intelligence Techniques

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  1. Artificial Intelligence Techniques Knowledge Processing 2-MSc

  2. Aims of session To understand • Fuzzy Logic • Defuzzification

  3. Introduction • Lofti Zadeh (1965) proposed Possibilistic Logic which became Fuzzy-Logic. • Allows us to combine weighting factors with propositions. • 0<=T(X)<=1

  4. Boolean v Fuzzy • Boolean Fuzzy T(X^Y) MIN(T(X),T(Y)) T(XvY) MAX(T(X),T(Y)) T(¬X) (1-T(X)) T(XY) MAX((1-T(X),T(Y)) Where X and Y are propositions Any Boolean expression can be converted to a fuzzy expression.

  5. t X Y Min(X,Y) MAX(X,Y) (1-X) MAX((1-X),Y) 0 0 0 0 1 1 0 1 0 1 1 1 1 0 0 1 0 0 1 1 1 1 0 1 So when truth values are 0 and 1 Fuzzy=Boolean Other rules such as De-Morgan’s Laws still apply

  6. Membership functions • A fuzzy set is a set whose membership function takes values between 0 and 1. • Example: Cold, Warm and Hot describe temperature we could define thresholds T1 and T2. • Starting at low temperature as the temperature rises to T1 the temperature becomes Warm. As the temperature rises to T2 the temperature becomes Hot.

  7. What is the problem? • Is there really a crisp change between the definitions?

  8. Answer • Change the shape of the membership function so it not so crisp. • Common one is a triangular functions that have some overlap. • At some temperatures it is possible to be a member of two different sets.

  9. Using the example from Johnson and Picton (1995)

  10. At 8 degrees it is a member of both COLD (0.7) and WARM (0.3) sets. • These are NOT necessarily probabilities, they are not so rigorously defined.

  11. Defuzzication • To calculate final setting need defuzzication rules, this often based around the ‘centre of gravity’ of shaded area. • Why do we need this?

  12. So back to the temperature measures the fuzzy membership can be combined using MIN, MAX and (1-T(X)) operations so IF-THEN can be used. • IF (temperature is COLD) THEN (heating on HIGH) • IF (temperature is WARM) THEN (heating on LOW) • So first rule heating is turned on to HIGH with a membership of 0.7.Second rule heating is turned on to LOW. • So membership can be represented by the heating memebership,

  13. Heater membership • Centre of gravity is point where area to left of the point=area to the right.

  14. Centre of Gravity

  15. Paradoxes and Fuzzy Sets • A useful feature is they can be built on the basis of minimal information and fine-tuned to be more consistent afterwards by observation. • Problem is (Hopgood’s Paradox) is that it is possible that ‘weak’ information can result in strong information on defuzzication.

  16. Not inverse • Defuzzication is not truly the inverse of fuzzification. • If you defuzzify fuzzy data you will often get distortion in the resulting values.

  17. References • Johnson J and Picton P (1995) Mechatronics : designing intelligent machines. - Vol.2 : concepts in artificial intelligence Oxford : Butterworth-Heinemann pg 175-187

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