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Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory 2. Introduction to fuzzy logic Philippe De Wilde Department of Computer Science Heriot-Watt University Edinburgh Vloeberghs Leerstoel 2010 Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Outline 1 Fuzzy Sets and Fuzzy Logic 2 Fuzzy Graphs 3 Fuzzy Associative Memory Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Six Lectures 1. History of decision making under uncertainty. Game: repeated prisoner’s dilemma. 2. Introduction to fuzzy logic. Game: rock, scissors, paper. 3. Financial markets and fuzzy games. Game: fuzzy chess. 4. Coupled networks in the brain. Game: Monty Hall’s dilemma. 5. Neuroeconomics. Game: ultimatum game. 6. The AI of deviations of rationality. Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Linguistic Variables Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Membership Function membership 1 fast 0 30 60 90 120 speed µfast (speed), µA (x). Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Membership Function Deﬁnition Deﬁnition (Universe of discourse) The universe of discourse is a set X , discrete ({x1 , . . . , xn }), or continuous (union of intervals on the real line). Deﬁnition (Membership Function) A membership function is a function µA : X → [0, 1]. Deﬁnition (Fuzzy Set) A fuzzy set is deﬁned by a membership function, it consists of some elements x of a universe of discourse X together with their membership values (or degrees) µa (x). Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Basic Logic Operations Deﬁnition (AND) µA∩B (x) = min(µA (x), µB (x)), ∀x ∈ X . Deﬁnition (OR) µA∪B (x) = max(µA (x), µB (x)), ∀x ∈ X . Deﬁnition (NOT, optional) µ¬A (x) = 1 − µA (x), ∀x ∈ X . Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Extension Principle A function transforming a set into another set will transform a membership function into another membership function, using the extension principle. Deﬁnition (Extension Principle) If f : X → Y is a function transforming universe of discourse X into Y , then fuzzy set µA (x) is transformed into µB (y ): maxy =f (x) µa (x) if f −1 (y ) = ∅, µB (y ) = 0 otherwise. Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Example of Extension Principle y = x2 The extension principle is powerful, and can be used to create fuzzy arithmetic. Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Fuzzy Numbers µ10(x) x 0 10 Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Decision Making under Fuzziness Deﬁnition Let X be a set of options. A fuzzy goal is a fuzzy set µG (x), x ∈ X . A fuzzy constraint is a fuzzy set µC (x), x ∈ X . A fuzzy decision is a fuzzy set µD (x), x ∈ X , with µD (x) = min(µG (x), µC (x)). A crisp decision x ∗ can be derived from a fuzzy decision by defuzziﬁcation: x ∗ = arg max µD (x). x∈X There are several ways to defuzzify, for example the centre of gravity of the area under the curve. Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Example of Decision Making µG(x) µC(x) x X* x ∗ is the decision, subject to constraints C and goal G. Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Uncertainty of a Functional Dependency Uncertainty can be represented by additive noise, e.g. y = x 2 + ξ, with ξ a random variable. The noise can also be on the parameters of the functional relationship, e.g. y = x (2+ξ) , or y = ξx 2 . Zadeh proposed a radically different way of looking at this, where the curve of a function becomes a union of squares, and each point in the union belongs to the function to a certain degree. This is the fuzzy graph. Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Fuzzy Graph Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Union of Cartesian Products The fuzzy graph is a union: (A1 × B1 ) ∪ (A2 × B2 ) ∪ . . . (An × Bn ). Deﬁnition (Fuzzy graph) If X and Y are universes of discourse, f ∗ : X → Y is a fuzzy graph iff n f∗ = i=1 Ai × Bi , µf ∗ (u, v ) = maxi min(µAi (u), µBi (v )), u ∈ X, v ∈ Y. max and min come from (u is in A1 AND v is in B1 ), OR (u is in A2 AND v is in B2 ), OR ... Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Fuzzy Graph as a Set of Rules e.g. f ∗ represents: If u is small then v is large else (or) if u is medium then v is medium else if u is large then v is small. If u is A1 then v is B1 else if u is A2 then v is B2 else if u is A3 then v is B3 . Bi can coincide with Bj , but the Ai need to be different, just as f(x) needs to be unique for a function. Fuzzy probability distributions are possible! Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Intersection of Two Fuzzy Graphs Equivalent of solving two simultaneous equations. Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Inversion of a Fuzzy Graph If a fuzzy set B is given, ﬁnd the fuzzy set A corresponding with it according to f ∗ . Minimum and maximum of a fuzzy graph can also be deﬁned. Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Fuzzy Control: Rules for Stopping a Car 1 input, 1 rule, 1 output If you go too fast, brake hard. 1 input, 2 rules, 1 output If you go too fast, brake hard, or, if you go fast, brake. 2 inputs, 4 rules, 1 output If you go too fast and the wall is very close, brake hard, or If you go fast and the wall is very close, brake, or If you go too fast and the wall is close, brake, or If you go fast and the wall is close, slow down. Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Fuzzy Associative Memory too fast fast very close brake hard brake close brake slow down µ2 µ2 µ 2 . . . 1 2 3 µ1 1 µ11 µ12 µ13 . . . µ1 2 µ21 µ22 µ23 . . . µ1 3 µ31 µ32 µ33 . . . . . . . . . . . .. . . . . . That’s how an expert lists the rules she uses. Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory 2 Inputs Fire Rules 2 inputs x 1 and x 2 x 1 belongs to the input membership functions µ1 , µ1 , µ1 , . . . 1 2 3 to degrees µ1 (x 1 ), µ1 (x 1 ), µ1 (x 1 ), . . .. 1 2 3 x 2 belongs to the input membership functions µ2 , µ2 , µ2 , . . . 1 2 3 to degrees µ2 (x 2 ), µ2 (x 2 ), µ2 (x 2 ), . . .. 1 2 3 output membership function µij ﬁres at degree min[µ1 (x 1 ), µ2 (x 2 )], using min because of the ’and’ in the i j rules. output membership function µij is truncated at min[µ1 (x 1 ), µ2 (x 2 )]. i j Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Truncated Output Membership Function µij(z) min[µi1(x1),µj2(x2)] z truncation = min µij (z), min[µ1 (x 1 ), µ2 (x 2 )] i j = min µij (z), µ1 (x 1 ), µ2 (x 2 ) i j Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Combination of all Output Membership Functions Max, because a collection of rules is combined with ’or’. maxi,j min µij (z), µ1 (x 1 ), µ2 (x 2 ) i j Sum can be used instead of max. z Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Defuzziﬁcation using Centre of Gravity f (z) = maxi,j min µij (z), µ1 (x 1 ), µ2 (x 2 ) i j ∞ −∞ zf (z)dz Centre of gravity y = ∞ −∞ f (z)dz y is the defuzziﬁed output, the control Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Applications in automotive, process control, consumer electronics, defence Abe Mamdani Philippe De Wilde 2. Introduction to fuzzy logic Fuzzy Sets and Fuzzy Logic Fuzzy Graphs Fuzzy Associative Memory Summary Membership functions deﬁne fuzzy sets. Fuzzy graphs express relationships between linguistic variables, a collection of fuzzy if-then-else rules. Fuzzy inference uses fuzzy associative memory, and defuzziﬁcation. Next game theory ﬁnancial markets fuzzy games Philippe De Wilde 2. Introduction to fuzzy logic Appendix For Further Reading Further Reading and Picture Credits I Lotﬁ Zadeh Fuzzy Logic, Neural Networks, and Soft Computing. Communications of the ACM, 37(3):77–84, 1994. Philippe De Wilde 2. Introduction to fuzzy logic

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fuzzy logic, fuzzy sets, membership functions, membership function, fuzzy set, linguistic variables, fuzzy systems, control system, fuzzy inference, fuzzy system, fuzzy relations, fuzzy control, fuzzy logic systems, truth value, rule base

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