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1 OPPA European Social Fund Prague & EU: We invest in your future.
2 AE4M33RZN, Fuzzy logic: Introduction, Fuzzy operators Radomír Černoch 19/11/2012 Faculty of Electrical Engineering, CTU in Prague 1 / 44 Basic fuzzy
3 Plan of the lecture Introduction AMKR course so far Criticism of both approaches Basic definitions Crisp sets Vertical representation Horizontal representation Special cases of fuzzy sets Operations on fuzzy sets Negation Conjunction Disjunction Conjunction - disjunction duality Criteria for selecting operators 2 / 44 Basic fuzzy
4 Description logics A description logic is a decideable fragment of first order logic (FOL). 3 / 44 Basic fuzzy
5 Description logics A description logic is a decideable fragment of first order logic (FOL). + Uses concepts, roles and individuals to capture structured knowledge. 3 / 44 Basic fuzzy
6 Description logics A description logic is a decideable fragment of first order logic (FOL). + Uses concepts, roles and individuals to capture structured knowledge. An unexpected fact in the A Box might lead to a contradiction, which is a pain. (See an example in a minute.) 3 / 44 Basic fuzzy
7 Graphical probabilistic models GPM is an efficient representation of large probability distributions. (Photo by ICMA Photos under the CC-BY-SA 2.0.) 4 / 44 Basic fuzzy
8 Graphical probabilistic models GPM is an efficient representation of large probability distributions. + Captures uncertainty well. (Photo by ICMA Photos under the CC-BY-SA 2.0.) 4 / 44 Basic fuzzy
9 Graphical probabilistic models GPM is an efficient representation of large probability distributions. + Captures uncertainty well. + Even unlikely events (tossing head 100 times in a row) can be processed. (Photo by ICMA Photos under the CC-BY-SA 2.0.) 4 / 44 Basic fuzzy
10 Graphical probabilistic models (Photo by ICMA Photos under the CC-BY-SA 2.0.) GPM is an efficient representation of large probability distributions. + Captures uncertainty well. + Even unlikely events (tossing head 100 times in a row) can be processed. Cannot formulate complex statements explicitly, such as Every object in the database has at least one... 4 / 44 Basic fuzzy
11 Example: Smoking friends (1) To illustrate the limitations of DL and GPM, consider an example from [Domingos and Lowd, 2009]. (Image: Matthew Romack under the CC-BY-SA 2.0.) 5 / 44 Basic fuzzy
12 Example: Smoking friends (1) To illustrate the limitations of DL and GPM, consider an example from [Domingos and Lowd, 2009]. Obervation 1: High-school experience. People start or stop smoking in groups of friends. (Image: Matthew Romack under the CC-BY-SA 2.0.) 5 / 44 Basic fuzzy
13 Example: Smoking friends (1) To illustrate the limitations of DL and GPM, consider an example from [Domingos and Lowd, 2009]. (Image: Matthew Romack under the CC-BY-SA 2.0.) Obervation 1: High-school experience. People start or stop smoking in groups of friends. Obervation 2: Six degrees of separation. Everyone is on average approximately six steps away, by way of introduction, from any other person in the world, so that a chain of a friend of a friend statements can be made, on average, to connect any two people in six steps. [Wikipedia, 2012] 5 / 44 Basic fuzzy
14 Example: Smoking friends (2) To formalize the example, let's use description logic ALC: 6 / 44 Basic fuzzy
15 Example: Smoking friends (2) To formalize the example, let's use description logic ALC: Obervation 1: High-school experience. If you have a friend, who is a smoker, you are a smoker as well: friendof Smoker Smoker 6 / 44 Basic fuzzy
16 Example: Smoking friends (2) To formalize the example, let's use description logic ALC: Obervation 1: High-school experience. If you have a friend, who is a smoker, you are a smoker as well: friendof Smoker Smoker Obervation 2: Six degrees of separation. Joining the friendof relation 6 times gives the top relation. friendof... friendof 6 / 44 Basic fuzzy
17 Example: Smoking friends (2) To formalize the example, let's use description logic ALC: Obervation 1: High-school experience. If you have a friend, who is a smoker, you are a smoker as well: friendof Smoker Smoker Obervation 2: Six degrees of separation. Joining the friendof relation 6 times gives the top relation. friendof... friendof What is wrong with this model? 6 / 44 Basic fuzzy
18 Example: Smoking friends (3) If there is one smoker, the whole world starts smoking. (Formally, an interpretation I must satisfy Smoker I = or Smoker I = Δ.) 7 / 44 Basic fuzzy
19 Example: Smoking friends (3) If there is one smoker, the whole world starts smoking. (Formally, an interpretation I must satisfy Smoker I = or Smoker I = Δ.) We start from reasonable assumptions and arrive at counter-intuitive conclusion. What's wrong with our reasoning? 7 / 44 Basic fuzzy
20 Example: Smoking friends (3) If there is one smoker, the whole world starts smoking. (Formally, an interpretation I must satisfy Smoker I = or Smoker I = Δ.) We start from reasonable assumptions and arrive at counter-intuitive conclusion. What's wrong with our reasoning? We would like to express something like ( friendof Smoker Smoker) is mostly true. Fuzzy logic can do that! 7 / 44 Basic fuzzy
21 Conclusion All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life but only to an imagined celestial existence. Bertrand Russel [Russell, 1923] 8 / 44 Basic fuzzy
22 Crisp sets: Definition (Informally:) A crisp set ( ostrá množina ) X is a collection of objects x X that can be finite, countable or overcountable. 9 / 44 Basic fuzzy
23 Crisp sets: Definition (Informally:) A crisp set ( ostrá množina ) X is a collection of objects x X that can be finite, countable or overcountable. We will speak about sets with in relation to a universe set ( univerzum ). 9 / 44 Basic fuzzy
24 Crisp sets: Definition (Informally:) A crisp set ( ostrá množina ) X is a collection of objects x X that can be finite, countable or overcountable. We will speak about sets with in relation to a universe set ( univerzum ). Let P(Δ) be the powerset (a set of all subsets) of Δ (the universe). Then any crisp set is an element in the powerset of its universe: A P(Δ). 9 / 44 Basic fuzzy
25 Crisp sets: Example Equivalent ways of describing a crisp set in IN: A = {1, 3, 5} (1) A = {x IN x 5 and x is odd} (2) 10 / 44 Basic fuzzy
26 Crisp sets: Example Equivalent ways of describing a crisp set in IN: A = {1, 3, 5} (1) A = {x IN x 5 and x is odd} (2) 0 x > 5 μ A (x) = 0 x is even 1 otherwise μ A is called the membership function ( charakteristická funkce, funkce příslušnosti ). 10 / 44 Basic fuzzy (3)
27 Membership function If μ A is a function Δ {0, 1}, the inverse membership function μ -1 A returns objects with the given membership degree: μ -1 A(M) = {x X μ A (x) M} (4) 11 / 44 Basic fuzzy
28 Membership function If μ A is a function Δ {0, 1}, the inverse membership function μ -1 A returns objects with the given membership degree: μ -1 A(M) = {x X μ A (x) M} (4) Example μ -1 A({1}) = {1, 3, 5} (5) 11 / 44 Basic fuzzy
29 Membership function If μ A is a function Δ {0, 1}, the inverse membership function μ -1 A returns objects with the given membership degree: μ -1 A(M) = {x X μ A (x) M} (4) Example μ -1 A({1}) = {1, 3, 5} (5) Note μ -1 A is not an inverse in a strict mathematical sense. The inverse of Δ {0, 1} should be {0, 1} Δ, but μ -1 A P({0, 1}) P(Δ). 11 / 44 Basic fuzzy
30 Check your knowledge: μ =? μ Δ =? μ -1 ({0, 1}) =? 12 / 44 Basic fuzzy
31 Check your knowledge: μ = 0 μ Δ = 1 μ -1 ({0, 1}) = Δ 12 / 44 Basic fuzzy
32 Fuzzy set Definition Fuzzy set ( fuzzy množina ) is an object A described by a generalized membership function μ A Δ [0, 1]. 13 / 44 Basic fuzzy
33 Fuzzy set Definition Fuzzy set ( fuzzy množina ) is an object A described by a generalized membership function μ A Δ [0, 1]. For better readability, A(x) μ A (x). 13 / 44 Basic fuzzy
34 Fuzzy set Definition Fuzzy set ( fuzzy množina ) is an object A described by a generalized membership function μ A Δ [0, 1]. For better readability, A(x) μ A (x). The set of all fuzzy subsets of a crisp universe Δ will be denoted as F(Δ). 13 / 44 Basic fuzzy
35 Source: [Zimmermann, 2001] 14 / 44 Basic fuzzy
36 Fuzzy set: Properties (1) Cardinality is the size of a fuzzy set. A = A(x) (6) x 15 / 44 Basic fuzzy
37 Fuzzy set: Properties (1) Cardinality is the size of a fuzzy set. A = A(x) (6) x Height of a fuzzy set is the highest value of the membership function. Height(A) = sup {α x Δ, A(x) = α} (7) 15 / 44 Basic fuzzy
38 Fuzzy set: Properties (2) Support ( nosič ) is the set of objects contained in the fuzzy set at least a bit. Supp(A) = {x X A(x) > 0} = μ -1 A((0, 1]) (8) 16 / 44 Basic fuzzy
39 Fuzzy set: Properties (2) Support ( nosič ) is the set of objects contained in the fuzzy set at least a bit. Supp(A) = {x X A(x) > 0} = μ -1 A((0, 1]) (8) Core ( jádro ) is the set of objects fully contained in the fuzzy set. Core(A) = {x X A(x) = 1} = μ -1 A({1}) (9) 16 / 44 Basic fuzzy
40 Horizontal representation The α-level ( α-hladina ) of a fuzzy set A is a crisp set μ -1 A(M) = {x X A(x) M} (10) 17 / 44 Basic fuzzy
41 Horizontal representation The α-level ( α-hladina ) of a fuzzy set A is a crisp set μ -1 A(M) = {x X A(x) M} (10) The α-cut ( α-řez ) of a fuzzy set A is a crisp set R A (α) = {x X A(x) α} = β [α,1] μ -1 A(β) (11) Sometimes we speak about a strong α-cut ( ostrý α-řez ), where in the definition is replaced by >. 17 / 44 Basic fuzzy
42 Horizontal representation The α-level ( α-hladina ) of a fuzzy set A is a crisp set μ -1 A(M) = {x X A(x) M} (10) The α-cut ( α-řez ) of a fuzzy set A is a crisp set R A (α) = {x X A(x) α} = β [α,1] μ -1 A(β) (11) Sometimes we speak about a strong α-cut ( ostrý α-řez ), where in the definition is replaced by >. For better readability A -1 (x) μ -1 A(x). 17 / 44 Basic fuzzy
43 Core = [0,28] 0.6-level = {40} The set Age of Young Men with its properties. 0.6-cut = [0,40] Support = [0,60] 18 / 44 Basic fuzzy
44 Check your knowledge: R A (0) =? Core(A) =? Height(A) = sup {? R A?} 19 / 44 Basic fuzzy
45 Check your knowledge: R A (0) = Δ Core(A) = R A (1) Height(A) = sup {α [0, 1] R A (α) } 19 / 44 Basic fuzzy
46 Converting vertical and horizontal representation Horizontal representation the α-cuts R. Vertical representation the characteristic function μ. 1 2 : From the definition on the previous slide. 2 1 : By taking the highest α-level containing x: A(x) = max{α [0, 1] x R A (α)} (12) 20 / 44 Basic fuzzy
47 Special cases of fuzzy sets Definition Fuzzy interval A is a fuzzy set on Δ = IR s.t. R A (α) is a closed interval for all α [0, 1] R A (1) is not empty. Supp(A) is finite. Special cases of fuzzy intervals Fuzzy number A is a fuzzy interval s.t. Core(A) = 1 Trapezoidal interval will be denoted by a, b, c, d. Triangular number will be denoted by a, b, c = a, b, b, c. A crisp interval [a, b] is also a, a, b, b. 21 / 44 Basic fuzzy
48 Operations on fuzzy sets set operation propositional operation P(Δ) P(Δ) {0, 1} {0, 1} P(Δ) P(Δ) P(Δ) {0, 1} 2 {0, 1} P(Δ) P(Δ) P(Δ) {0, 1} 2 {0, 1} 22 / 44 Basic fuzzy
49 Operations on fuzzy sets set operation propositional operation P(Δ) P(Δ) {0, 1} {0, 1} P(Δ) P(Δ) P(Δ) {0, 1} 2 {0, 1} P(Δ) P(Δ) P(Δ) {0, 1} 2 {0, 1} We can use the logical operators to define the set operators: A = {x Δ (x A)} A B = {x Δ (x A) (x B)} A B = {x Δ (x A) (x B)} (LS1) (LS2) (LS3) Therefore we will cover the logical negation, conjunction and disjunction. We get the set operations for free. 22 / 44 Basic fuzzy
50 Fuzzy negation Fuzzy negation is a non-increasing, involutive, unary function [0, 1] [0, 1] s.t. if α β then β α (N1) α = α (N2) 23 / 44 Basic fuzzy
51 Fuzzy negation Fuzzy negation is a non-increasing, involutive, unary function [0, 1] [0, 1] s.t. if α β then β α (N1) α = α (N2) Example Standard ( standardní ), Łukasiewicz negation α = 1 α (13) 23 / 44 Basic fuzzy
52 The fuzzy set complement is a defined using (LS1). 23 / 44 Basic fuzzy Fuzzy negation Fuzzy negation is a non-increasing, involutive, unary function [0, 1] [0, 1] s.t. if α β then β α (N1) α = α (N2) Example Standard ( standardní ), Łukasiewicz negation α = 1 α (13)
53 Fuzzy negation: More examples Cosine negation Sugeno negation Yager negation α = (cos(πα) + 1)/2 (14) cos 1 α α = Sλ 1 + λa (15) Yλ α = (1 aλ ) 1/λ (16) 24 / 44 Basic fuzzy
54 Fuzzy negation: Properties The axioms (N1) and (N2) imply more properties of fuzzy negations: Theorem 53 Every fuzzy negation is a continuous decreasing bijective generalization of the propositional negation 25 / 44 Basic fuzzy
55 Fuzzy negation: Proof of 53 Injective (f(a) = f(b) a = b): Take 2 values, whose negations are equal: α = β. By (N2) α = α. The can be substituted using the assumption: α = β. Using (N1) gives β = β. Therefore α = β. Every non-increasing function (N1) which is injective, must be decreasing. If α < β WLOG, then α β. Then either α > β and is decreasing, or α = β, which contradicts the injectivity. 26 / 44 Basic fuzzy
56 Fuzzy negation: Proof of 53 Surjective y x.f(x) = y: We seek a value of β for each α s.t. α = β. Using injectivity, the condition is equivalent to α = β. Using (N2), we find the value of β for any α: β = α. Bijection is an injective and surjective function (by definition). Continuous: Every decreasing bijection is continuous. Boundary values: Let 0 = α and suppose that α < 1. Then from surjectivity, there must be some other β > 0 s.t. β = 1. This contracits monotonicity, because 0 < β. The other boundary value is proven similarly. 27 / 44 Basic fuzzy
57 Fuzzy conjunctions (t-norms) Fuzzy t-norm (triangluar norm, conjunction) is a binary, comutative, operation s.t. α β = β α α (β γ) = (α β) γ if β γ then (α β) (α γ) (α 1) = α (T1) (T2) (T3) (T4) 28 / 44 Basic fuzzy
58 Fuzzy conjunctions (t-norms) Fuzzy t-norm (triangluar norm, conjunction) is a binary, comutative, operation s.t. α β = β α α (β γ) = (α β) γ if β γ then (α β) (α γ) (α 1) = α (T1) (T2) (T3) (T4) The fuzzy set intersection is a defined using (LS2). 28 / 44 Basic fuzzy
59 Fuzzy conjunctions: Examples Standard (Gödel, Zadeh) α β = min (α, β) (17) Łukasiewicz α β = max (α + β 1, 0) (18) Algebraic product ( součinová ) α β = α β (19) Weak ( drastická ) α if β = 1 α β = β if α = 1 29 / 44 Basic fuzzy 0 otherwise (20)
60 Fuzzy conjunctions: Visualization [Wikipedia] Standard Algebraic Łukasiewicz 30 / 44 Basic fuzzy Drastic
61 Fuzzy conjunctions: Properties (1) Theorem 60 The weak and standard conjunctions provide a lower and upper bound on all possible conjunctions: Proof Assume WLOG α β. (α β) (α β) (α β) (21) β = 1 The condition (T4) gives the same result for all conjunctions. β < 1 α β = 0, which gives the lower bound. The upper bound is rewritten using the definition of standard conjunction (17): α β = α. From (T4) follows that α = α 1 α β. 31 / 44 Basic fuzzy
62 Therefore α β = α. There is only one such conjunction:. 32 / 44 Basic fuzzy Fuzzy conjunctions: Properties (2) Theorem 61 The standard conjunction is the only idempotent conjunction: α α = α (22) Proof Assume WLOG α β. α = α α (T3) = α β (T3) = α 1 (T4) = α (23)
63 Fuzzy disjunctions (s-norm) Fuzzy s-norm (t-conorm, disjunction) is a binary operation s.t. α β = β α α (β γ) = (α β) γ if β γ then (α β) (α γ) (α 0) = α (S1) (S2) (S3) (S4) 33 / 44 Basic fuzzy
64 Fuzzy disjunctions (s-norm) Fuzzy s-norm (t-conorm, disjunction) is a binary operation s.t. α β = β α α (β γ) = (α β) γ if β γ then (α β) (α γ) (α 0) = α Union The fuzzy set union is a defined using the disjunction: (S1) (S2) (S3) (S4) μ A B (x) = μ A (x) μ B (x) (24) 33 / 44 Basic fuzzy
65 Fuzzy disjunctions: Examples (1) Standard (Gödel, Zadeh) α β = max(α, β) (25) Łukasiewicz α β = min(α + β, 1) (26) Algebraic sum ( součinová ) α β = α + β α β (27) 34 / 44 Basic fuzzy
66 Fuzzy disjunctions: Examples (2) Weak ( drastická ) Einstein α if β = 0 α β = β if α = 0 1 otherwise α α + β β = 1 + αβ (28) (29) 35 / 44 Basic fuzzy
67 Fuzzy disjunctions: Visualization [Wikipedia] Standard Algebraic Łukasiewicz 36 / 44 Basic fuzzy Drastic
68 Fuzzy disjunctions: Properties The standard and weak disjunctions provide a lower and upper bound on all possible conjunctions: (α β) (α β) (α β) (30) 37 / 44 Basic fuzzy
69 Fuzzy disjunctions: Properties The standard and weak disjunctions provide a lower and upper bound on all possible conjunctions: (α β) (α β) (α β) (30) The standard disjunctions is the only idempotent conjunction: α α = α (31) 37 / 44 Basic fuzzy
70 Conjunction - disjunction duality A If is a fuzzy conjunction, then α β = ( α β) is a fuzzy disjunction (dual to w.r.t. ). 38 / 44 Basic fuzzy
71 Conjunction - disjunction duality A If is a fuzzy conjunction, then α β = ( α β) is a fuzzy disjunction (dual to w.r.t. ). B If is a fuzzy disjunction, then α β = ( α β) is a fuzzy conjunction (dual to w.r.t. ). 38 / 44 Basic fuzzy
72 Standard operations, 38 / 44 Basic fuzzy are dual w.r.t. any negation. Conjunction - disjunction duality A If is a fuzzy conjunction, then α β = ( α β) is a fuzzy disjunction (dual to w.r.t. ). B If is a fuzzy disjunction, then α β = ( α β) is a fuzzy conjunction (dual to w.r.t. ). Theorems Łukasiewicz operations, are dual w.r.t. standard negation. Algebraic operations, are dual w.r.t. standard negation.
73 39 / 44 Basic fuzzy Degree of membership for 20 items into the sets make of metal, being a container and being a metalic container. [Zimmermann, 2001]
74 Scatterplot of membership degree for being a metalic container vs. make of metal being a container. (standard conjunction). [Zimmermann, 2001] 40 / 44 Basic fuzzy
75 Scatterplot of membership degree for being a metalic container vs. make of metal being a container (algebraic conjunction). [Zimmermann, 2001] 41 / 44 Basic fuzzy
76 Criteria for selecting operators (1) Axiomatic strength: The set of valid theorems may differ based on the choice of t-norms and s-norms (see tutorials). Empirical fit: Using fuzzy theory for a model of the real world, the chosen operator should match the real behavior of the system. Adaptability: Operators in a generic system should be able to fit several use cases. One way of increasing adaptibility is to use operators with parameters (e.g. Yager and Sugeno negations). 42 / 44 Basic fuzzy
77 Criteria for selecting operators (2) 4. Computational efficiency: Evaluating e.g. the standard negation is usually faster than the Yager negation, which contains the power. 5. Aggregating behavior: When the operators combines a large number of operands, does the value tends to go to 0 (conjunction) or 1 (disjunction). The standard operators behave differently than the algebraic ones. 43 / 44 Basic fuzzy
78 Bibliography Domingos, P. and Lowd, D. (2009). Markov Logic: An Interface Layer for Artificial Intelligence. Morgan & Claypool. Russell, B. (1923). Vagueness. In Australasian Journal of Psychology and Philosophy, page Wikipedia (2012). Six degrees of separation Wikipedia, the free encyclopedia. [Online; accessed 15-October-2012]. Zimmermann, H.-J. (2001). Fuzzy Set Theory and its Applications. Springer. 44 / 44 Basic fuzzy
79 OPPA European Social Fund Prague & EU: We invest in your future.
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