1. CS621: Artificial Intelligence Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 10– Uncertainty: Fuzzy Logic

2. Uncertainty Studies Uncertainty Study Probability Based Information Theory based Fuzzy Logic Based Probabilistic Reasoning Markov Processes & Graphical Models Entropy Centric Algos Qualitative Reasoning Bayesian Belief Network

3. Outlook (0) Temp (T) Humidit y (H) Windy (W) Decision to play (D) SunnyHigh FN SunnyHigh TN CloudyHigh FY RainMedHighFY RainColdLowNY RainColdLowTN CloudyColdLowTY To-play-or-not-to-play-tennis data vs. Climatic-Condition from Ross Quinlan’s paper on ID3 (1986), C4.5 (1993)

4. Weather (0) Temp (T) Humidit y (H) Windy (W) Decision (D) SunnyMedHighFN SunnyColdLowFY RainMedLowFY SunnyMedLowTY CloudyMedHighTY CloudyHighLowFY RainHigh TN

5. Outlook Rain CloudySunny Windy YesHumidity High Low Yes No T F Yes No

6. Rule Base R1: If outlook is sunny and if humidity is high then Decision is No. R2: If outlook is sunny and if humidity is low then Decision is Yes. R3: If outlook is cloudy then Decision is Yes.

7. Fuzzy Logic

8. Fuzzy Logic tries to capture the human ability of reasoning with imprecise information Models Human Reasoning Works with imprecise statements such as: In a process control situation, “If the temperature is moderate and the pressure is high, then turn the knob slightly right” The rules have “Linguistic Variables”, typically adjectives qualified by adverbs (adverbs are hedges).

9. Underlying Theory: Theory of Fuzzy Sets Intimate connection between logic and set theory. Given any set ‘S’ and an element ‘e’, there is a very natural predicate, μ s (e) called as the belongingness predicate. The predicate is such that, μ s (e) = 1,iff e ∈ S = 0,otherwise For example, S = {1, 2, 3, 4}, μ s (1) = 1 and μ s (5) = 0 A predicate P(x) also defines a set naturally. S = {x | P(x) is true} For example, even(x) defines S = {x | x is even}

10. Fuzzy Set Theory (contd.) Fuzzy set theory starts by questioning the fundamental assumptions of set theory viz., the belongingness predicate, μ, value is 0 or 1. Instead in Fuzzy theory it is assumed that, μ s (e) = [0, 1] Fuzzy set theory is a generalization of classical set theory also called Crisp Set Theory. In real life belongingness is a fuzzy concept. Example: Let, T = set of “tall” people μ T (Ram) = 1.0 μ T (Shyam) = 0.2 Shyam belongs to T with degree 0.2.

11. Linguistic Variables Fuzzy sets are named by Linguistic Variables (typically adjectives). Underlying the LV is a numerical quantity E.g. For ‘tall’ (LV), ‘height’ is numerical quantity. Profile of a LV is the plot shown in the figure shown alongside. μ tall (h) 1 2 3 4 5 60 height h 1 0.4 4.5

12. Example Profiles μ rich (w) wealth w μ poor (w) wealth w

13. Example Profiles μ A (x) x x Profile representing moderate (e.g. moderately rich) Profile representing extreme

14. Concept of Hedge Hedge is an intensifier Example: LV = tall, LV 1 = very tall, LV 2 = somewhat tall ‘very’ operation: μ very tall (x) = μ 2 tall (x) ‘somewhat’ operation: μ somewhat tall (x) = √(μ tall (x)) 1 0 h μ tall (h) somewhat tall tall very tall

15. 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 )}

16. 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

17. (1,0) (0,0) (0,1) (1,1) x1x1 x2x2 d(A, nearest) d(A, farthest) (0.5,0.5) A

18. Definition Distance between two fuzzy sets L 1 - norm Let C = fuzzy set represented by the centre point d(c,nearest) = |0.5-1.0| + |0.5 – 0.0| = 1 = d(C,farthest) => E(C) = 1

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

20. 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?