Utrecht University INFOB2KI 2019-2020 The Netherlands ARTIFICIAL INTELLIGENCE Uncertainty: fuzzy systems Lecturer: Silja Renooij These slides are part of the INFOB2KI Course Notes available from www.cs.uu.nl/docs/vakken/b2ki/schema.html
Outline Boolean logic and crisp sets Fuzzy logic & sets Inferences with fuzzy rules
Reasoning models Classical models (logic, exact) – If Pressure = 10 atm then Volume = 2.5 cc Imprecise models – If Pressure ≥ 5 atm then Volume ≤ 6 cc Probabilistic models – If Pressure ≥ 5 atm then P(Volume = 6 cc) = 0.9 Vague models – If Pressure = HIGH then Volume = LOW
Boolean logic & Crisp sets I Boolean logic uses sharp distinctions. It forces us to draw lines between members of a class and non‐members. tall men For instance, we may say, Tom is tall because his height is 181 cm. If we drew a line at 180 cm, we would find that David, who is 179 cm, is small.
Boolean Logic & Crisp sets II Consider a universe of discourse (interest) X and its elements x . In classical set theory, a crisp subset A of X is defined by the characteristic function f A ( x ) of A : 1, if x A ( ) f A ( x ) : X {0, 1}, where f A x 0, if x A
The trouble with crisp sets Sorites Paradox: the paradox of the heap Consider a heap of sand from which grains are individually removed. One might construct the following argument: 1,000,000 grains of sand is a heap of sand (Premise 1) A heap of sand minus one grain is still a heap. (Premise 2) Repeated application of Premise 2 eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand … …at some point it must stop being a heap!
Fuzzy reasoning Experts rely on common sense when they solve problems. How can we represent expert knowledge that uses vague and ambiguous terms in a computer? Fuzzy logic is based on the idea that all things admit of degrees. Temperature, height, speed, distance, beauty – all come on a sliding scale. – The motor is running really hot . – Tom is a very tall guy.
A bit of History Multi‐valued logic: First introduced in the 1930s by Jan Lukasiewicz (Polish philosopher) work led to an inexact reasoning technique often called possibility theory. extended into a formal system of mathematical logic by Lofti Zadeh in 1965 new multi‐valued logic for representing and manipulating fuzzy terms was called fuzzy logic . Unlike other logical systems, it deals with imprecise or uncertain knowledge.
Fuzzy logic and Fuzzy sets Fuzzy logic is multi‐valued, unlike Boolean logic; it deals with degrees of membership and degrees of truth. uses the continuum between 0 (completely false) and 1 (completely true), accepting that things can be partly true and partly false at the same time. tall men, to a certain degree
Fuzzy Sets vs Crisp sets The set of Tall people Degree of Membership Name Height, cm Crisp Fuzzy Chris 208 1 1.00 Frank 205 1 1.00 John 198 1 0.98 Tom 181 1 0.82 David 179 0 0.78 Mike 172 0 0.24 Bob 167 0 0.15 Steven 158 0 0.06 Bill 155 0 0.01 Peter 152 0 0.00
Fuzzy sets vs Crisp sets Degree of Crisp Sets Membership 1.0 Short Average Short Tall 0.8 Tall Men 0.6 0.4 0.2 0.0 150 160 170 180 190 200 210 Height, cm Degree of Fuzzy Sets Membership 1.0 0.8 Short Average Tall 0.6 0.4 Tall 0.2 0.0 150 160 170 180 190 200 210
Fuzzy sets Consider a universe of discourse (interest) X and its elements x . In fuzzy set theory, a fuzzy subset A of X is defined by the membership function µ A ( x ) of A : µ A ( x ) : X [0,1], where µ A ( x ) = 1 if x is totally in A; µ A ( x ) = 0 if x is not in A; 0 < µ A ( x ) < 1 if x is partly in A.
Fuzzy set representation Typical functions that can be used to represent a fuzzy set are sigmoid, Gaussian and Pi. However, these functions increase the time of computation. in practice, most applications use – linear fit functions, or – just a number of ordered pairs mapping x to µ (discrete) ( x ) Fuzzy Subset A X 1 0 x Crisp Subset A Fuzziness Fuzziness
Fuzzy set notations Consider a discrete universe of discourse (interest) X and its elements x . Different notations are used to describe fuzzy (sub)set A of X : A = {(x 1 , 0.4), (x 2 , 0.3), (x 3 , 1), (x 4 , 0.6)} i.e. A = {(x, µ A (x)) | x in X} Or, using Zadeh’s notation: A = 0.4/x 1 + 0.3/x 2 + 1/x 3 + 0.6/x 4 i.e. A = ∑ x µ A (x)/x (not fractions!)
Fuzzy sets and probability Take care! Fuzzy membership values may look like probabilities, but have a totally different meaning!
Operators • Complement /negation • Intersection / AND • Union / OR • containment / implication • Equality • Cardinality • Empty set
Boolean Logic/Crisp Sets: ops Not A B x ( ) A x A x B A A A Complement Containment x A x B x A x B A B B A B A Intersection Union
Fuzzy ops: complement/negation Boolean/Crisp: Who does not belong to the set? Fuzzy: How much do elements not belong to the set? Standard negation: Complement ¬A of fuzzy set A can be found as follows: ¬A ( x ) = 1 A ( x ) Example: consider set X and fuzzy subset A X = {a, b, c, d, e} A = 1/a + 0.3/b + 0.2/c + 0.8/d + 0/e A = 0/a + 0.7/b + 0.8/c + 0.2/d + 1/e
Negation: alternatives Minimum negation ¬A ( x ) = 1 if A ( x )=0 = 0 otherwise Maximum negation ¬A ( x ) = 0 if A ( x )=1 = 1 otherwise
Fuzzy ops: Intersection/AND Boolean/Crisp: Which element belongs to both sets? Fuzzy: How much of the element is in both sets? Standard intersection: (minimum) Intersection of two fuzzy subsets A and B of X is the lower membership in both sets of each element: A B ( x ) = min [ A ( x ), B ( x )] = A ( x ) B ( x ) Example: consider set X and two fuzzy subsets A,B X = {a, b, c, d, e} A = 1/a + 0.3/b + 0.2/c + 0.8/d + 0/e B = 0.6/a + 0.9/b + 0.1/c + 0.3/d + 0.2/e A B = 0.6/a + 0.3/b + 0.1/c + 0.3/d + 0/e
Intersection: alternatives Product A B ( x ) = A ( x ) ∙ B ( x ) Lukasiewicz A B ( x ) = max[ A ( x )+ B ( x )‐1, 0 ] Drastic A B ( x ) = min[ A ( x ), B ( x ) ], if A ( x )=1 or B ( x )=1 = 0, otherwise
Fuzzy operations: Union/OR Boolean/Crisp: Which element belongs to either set? Fuzzy: How much of the element is in either set? Standard union: (maximum) ‐ reverse of intersection ‐ union of two fuzzy subsets A and B of X is the largest membership value of each element in either set: A B (x) = max [ A (x), B (x)] = A (x) B (x) Example: consider set X and two fuzzy subsets A,B X = {a, b, c, d, e} A = 1/a + 0.3/b + 0.2/c + 0.8/d + 0/e B = 0.6/a + 0.9/b + 0.1/c + 0.3/d + 0.2/e A B = 1/a + 0.9/b + 0.2/c + 0.8/d + 0.2/e
Union: alternatives Probabilistic sum A B (x) = A (x) + B (x) ‐ A (x)∙ B (x) Lukasiewicz A B ( x ) = min[ A ( x )+ B ( x ), 1 ] Drastic A B ( x ) = max[ A ( x ), B ( x ) ], if A ( x )=0 or B ( x )=0 = 1, otherwise
Fuzzy ops: containment(inclusion)/implication Boolean/Crisp: Which sets belong to which other sets? Fuzzy: Which sets belong to other sets? Inclusion: – each element can belong less to the subset than to the larger set. – Fuzzy set A X is included in (is a subset of) another fuzzy set, B X : A ( x ) B ( x ), x X Example: Consider X = {1, 2, 3} and fuzzy subsets A , B A = 0.3/1 + 0.5/2 + 1.0/3 B = 0.5/1 + 0.55/2 + 1.0/3 then A is a subset of B, or A B
Fuzzy Equality Fuzzy set A is considered equal to a fuzzy set B , IF AND ONLY IF (iff): A ( x ) = B ( x ), x X Example: A = 0.3/1 + 0.5/2 + 1.0/3 B = 0.3/1 + 0.5/2 + 1.0/3 therefore A = B
Fuzzy cardinality Crisp Sets: How many elements belong to the set? Fuzzy Sets: What is the total membership value of the set? Cardinality (aka sigma count) for finite sets : card A = A ( x 1 ) + A ( x 2 ) + … A ( x n ) = Σ A ( x i ) for i =1.. n Example: Consider X = {1, 2, 3} and fuzzy subsets A and B A = 0.3/1 + 0.5/2 + 1.0/3 card A = 1.8 B = 0.5/1 + 0.55/2 + 1.0/3 card B = 2.05
Empty Fuzzy Set A fuzzy set A is empty IF AND ONLY IF: A ( x ) = 0, x X Example: Consider X = {1, 2, 3} and fuzzy set A A = 0/1 + 0/2 + 0/3 then A is empty
Fuzzy to Crisp operations ‐cuts (alpha‐cuts) Support core
Fuzzy to crisp: alpha cut An ‐cut or ‐level set of a fuzzy set A X is – a crisp set A X , such that: A ={ x X | A ( x ) } Example: Consider X = {1, 2, 3} and A = 0.3/1 + 0.5/2 + 1.0/3 then A 0.5 = {2, 3}, A 0.1 = {1, 2, 3}, A 1 = {3} For continuous X and A, A will be a subinterval of X (see previous slide)
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