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

Prof. 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.

Prof. 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.

Prof. 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)

Prof. 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

Prof. 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.

Prof. 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

Prof. 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

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

Prof. 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”.

Prof. 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.

Prof. 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) height, h 1.0 very tall somewhat tall

Prof. 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

Prof. 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”,