1. 08-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 1 CS 621 Artificial Intelligence Lecture 5 – 08/08/05 Prof. Pushpak Bhattacharyya FUZZY LOGIC & INFERENCING

2. 08-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 2 Fuzzy Logic Fuzzy Logic : Reasoning with qualitative information This is more realistic than predicate calculus, because in real life we need to deal with qualitative statements. Examples: ● In process control: Chemical plant: Rule : If the temperature is moderately high & the pressure is medium, then turn the knob slightly right.

3. 08-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 3 Fuzzy Logic (more) Dealing with precise numerical information is often inconvenient, not suitable for humans. ● Weather is sunny today. ● It is very cold in Himalayas. ● Rich people have a lot of worries.

4. 08-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 4 Fuzzy Set Deep connection between logic & set. Any set comes with a predicate, the predicate is born with the set. Membership / Characteristic / Discriminative predicate Example: S = {2,3,5,a,b,c} X = universe = {1,2,3,...10,a,b,c,....z} 1  S (does not belong) a  S (belongs)

5. 08-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 5 Membership Predicate µ S (x) where x  X, S is a set. In the reverse way, Given any predicate, we can describe a set, trivially. Given P(x 1, x 2, x 3,..... x n ) the set that is born with P is the set of tuples for which P is true. Set Logic

6. 08-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 6 Basics of Fuzzy Set Theory Generalization of crisp set theory. Fundamental observation: µ S (x) is no longer 0/1. Rather µ S (x) is between [0,1], both included. Example: Crisp Set, S 1 = {2,4,6,8,10} µ S1 (x) is a predicate which denotes x to be an even number less than or equal to 10. Given any ‘a’ which is a number, the µ S1 (x) question produces 0/1.

7. 08-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 7 Digression A set is derived in one of the two ways. (a) By extension (b) By intension (a) requires 'listing'. S 1 is {2,4,6,8,10} -- needs finiteness. (b) needs a closed form expression related to properties. S 1 = {x | x mod 2 = 0 and x <= 10} Needed in all set theories. S 2 = set of tall people µ S2 (x) = [0,1] What is x? x is called the underlying numerical quantity. x = height

8. 08-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 8 More Examples S 3 = set of rich people µ S3 (x), x = asset in money unit. S 4 = set of knowledgable people µ S4 (x), x = amount of knowledge.(type-2 fuzzy sets) Why is S 2 / S 3 fuzzy. Given a height h i, it is not fully precisely defined whether µ S2 (h i ) is 0/1. Rather, there is a degree by which h i  S 2 height = 1 ft µ S2 (x) depends on the world of discourse. µ S2 (x = 1 ft) = 0, µ S2 (x = 8 ft) = 1

9. 08-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 9 Profile of 'Tall' Plot on 2-dimension. x axis : underlying numerical quantity y axis : µ µ Tall (h) 0.0 12 3456 789 height, h 1.0

10. 08-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 10 Shapes of Profiles Shapes of profiles are obtained from experiments and expert judgement. Statistically obtained by % count (sometimes). Profile itself is somewhat “vague”.

11. 08-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 11 Notion of Linguistic Variable John is tall (adjective) - set of tall people is fuzzy Americans are mostly (adverb) rich (adjective) Definition – A linguistic variable is the predicate of a sentence and typically is an adjective (often qualified by an adverb). A linguistic variable to be amenable to fuzzy logic, must have an underlying numerical quantity.

12. 08-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 12 Hedges Hedges are entities to deal with adverb. John is tall Jack is very tall Jill is somewhat tall very --> squaring the µ function somewhat --> taking square root of µ function µ Tall (h) 0.0 1 23456 789 height, h 1.0 very tall somewhat tall

13. 08-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 13 Set of Support & Summary Definition: The elements for which the µ value is > 0. Summary : ● Information in qualitative reasoning ● Connection between logic & set ● Membership predicate ● Fuzzy set definition

14. 08-08-05Prof. Pushpak Bhattacharyya, IIT Bombay 14 References ● Klir, G. J. and T. A. Folger [1988], Fuzzy Sets, Uncertainty, and Information. Prentice Hall, Englewood Cliffs, New Jersey. ● Earl Cox [1999], Fuzzy Systems Handbook: A Practioner's Guide to Building, Using and Maintain Fuzzy Systems. Morgan Kaufmann Publishers, 2nd edition ● Bart Kosto [1994], Fuzzy Thinking : The New Science of Fuzzy Logic, Hyperion; Reprint edition ● “Journal of Fuzzy Sets & Systems”, http://www.elsevier.nl/locate/fss