Fuzzy Sets and Systems. Lecture 4 (Fuzzy Logic) Bu- Ali Sina University Computer Engineering Dep. Spring 2010

Transcription

1 Fuzzy Sets and Systems Lecture 4 (Fuzzy Logic) Bu- Ali Sina University Computer Engineering Dep. Spring 2010

2 Outline Fuzzy Logic Classical logic- an overview Multi-valued logic Fuzzy logic Fuzzy proposition Unconditional Unqualified Qualified Conditional Unqualified Qualified Fuzzy Quantifier Linguistic hedges

3 Classical Logic Logic is the study of methods and principles of reasoning in all its possible forms. Propositions - statements that are required to be true or false. Instead of propositions, we use logic variables. Logic variable may asses one of the two truth values, if it is substituted by a particular proposition. Propositional logic studies the rules by which new logic variables can be produced from some given logic variables. The internal structure of the propositions behind the variables does not matter! Logic function assigns a truth value to a combination of truth values of its variables:

4 Classical Logic

5 Classical Logic Denfinition If v is a logic variable, then v and are logic formulae; If v1 and v2 are logic formulae, then v1 v2 and v1 v2 are also logic formulae; v Tautology is (any) logic formula that corresponds to a logic function one. Contradiction is (any) logic formula that corresponds to a logic function zero.

6 Inference rules are tautologies used for making deductive inferences. Examples: modus ponens modus tollens hypothetical syllogism Propositions are, in general, of the form x is P where x is a symbol of a subject and P is a predicate that characterizes a property. x is any element of universal set X, while P is a function on X, which for each value of x forms a proposition. P(x) is called predicate; it becomes true or false for any particular value of x.

7 Multivalued logic Third truth value is allowed: truth: 1, false: 0, intermediate: 1/2 quasi-tautology is a logic formula that never assumes truth value 0; quasi-contradiction is a logic formula that never assumes truth value 1.

8 Multivalued logics n-valued logics. The set of truth values: Truth values are interpreted as degrees of truth. Primitives in n-valued logics of Lukasiewicz, denoted by Ln, are:

9 Possible operators for AND in fuzzy logic µ ( x) µ ( x) A max{ 0, µ ( x) + µ ( x) 1} A µ ( x) µ ( x) A 2 [ µ ( x) + µ ( x) µ ( x) µ ( x)] B µ ( x) µ ( x) A µ ( x) + µ ( x) µ ( x) µ ( x) A B A B A B A B B B B

10 µ ( x) max( µ ( x), µ A + B = A B ( x)) Let A and B be fuzzy subsets of the universe X={-3, -2, -1, 0, 1, 2, 3, 4} A= 0.6/ / / / / / / /4 B= 0.2/ / / / / / / /4 µa B = 0.6/ / / / / / / /4

11 Possible operators for OR in fuzzy logic µ ( x) + µ ( x) µ ( x) µ ( x) A B A B µ A ( x) + µ B( x) 2µ A ( x) µ B( x) 1 µ ( x) µ ( x) A B µ ( x) + µ ( x) A B 1 + µ ( x) µ ( x)] A min{ 1, µ ( x) + µ ( x)} A B B

12 µ ( x) = 1 ( x) µ A A Let A be fuzzy subset of the universe X={-3, -2, -1, 0, 1, 2, 3, 4} A = 0.6/ / / / / / / /4 A = 0.4/ / / / / / / /4

13 Approximate reasoning Types of fuzzy linguistic terms Fuzzy predicates: tall, young, small, median Fuzzy truth values: true, false, very true Fuzzy probabilities: likely, unlikely, very likely Fuzzy quantifiers: many, few, most

14 Fuzzy logic - Fuzzy proposition The range of truth values of fuzzy propositions is not only {0; 1}, but [0; 1]. The truth of a fuzzy proposition is a matter of degree. Classification of fuzzy propositions: Unconditional and unqualified propositions The temperature is high Unconditional and qualified propositions The temperature is high is very true Conditional and unqualified propositions If the temperature is high, then it is hot Conditional and qualified propositions If the temperature is high, then it is hot is true

15 Unconditional and unqualified propositions Propositional form p: χ is A The temperature (35ºC) is (high). χ is a variable A is some property or predicate p: χ is A is true T(p x ) = the degree of truth of p x p x : χ=x is A T( p x ) = A(x) The degree of x belong to χ

16 Example p 65 : Humidity of 65% is high The degree of p 65 is T(p 65 ) = H(65) =0.25 T( p x ) = H(x) The degree of x belong to χ

17 Unconditional and qualified propositions Propositional form p: χ is A is S Humidity of 65% is high is very false S is a fuzzy truth qualifier The degree of truth, T s (p x ) of the qualified proposition p x : χ=x is A is S is T s (p x ) = S(A(x))

18 example p 65 : Humidity of 65% is high is very true The degree of truth of p 65 is T s (p 65 ) = S(A(x)) = S(0.25) =

19 example p 65 : Humidity of 65% is high is very false The degree of truth of p 65 is T s (p 65 ) = S(A(x)) = S(0.25) = 0.5

20 Conditional and unqualified propositions Propositional form χ is A γ is B p: if χ is A, then γ is B p x,y : if A(x), then B(y) is true Fuzzy implication A(x) B(y) The degree of truth T(p x,y ) = I[A(x), B(y)] = min[1, 1- A(x)+B(y)] If Tina is young, then John is old Lukasiewicz implication

21 example p: if a textbook is large, then it is expensive The degree of truth of p is T (p x,y ) = min[1, 1- L(x)+E(y)] T (p 600, 45 ) = min[1, 1- L(600)+E(45)] = min[1, ] = 0.5 T (p 450, 42 ) = min[1, 1- L(450)+E(42)] = min[1, ] = 0.65

22 Conditional and qualified propositions Propositional form p: if χ is A, then γ is B is S If a textbook is large, then it is expensive is very true The degree of truth T s (p x,y ) = S[T(p x,y )]

23 Fuzzy quantifiers Two quantifiers of predicate logic Universal quantifier: all, Existential quantifier: there exist, Fuzzy quantifiers Absolute quantifiers About a dozen, at most about 10, at least about 100 About 20 hotels are in close proximity to the center of the city Relative quantifiers All snakes are reptiles ( x) ( Sx Rx) Some snakes are poisonous ( x) ( Sx Px) Almost all snakes are poisonous Most, almost all, about half, about 20% Almost all hotels are in close proximity to the center of the city

24 Examples: p: There are about 3 high-fluent students in the group q: Almost all students in the group are high-fluent To determine the truth value of a quantified proposition, we need to know 1-how many students in the group are high-fluent i.e., cardinality of a fuzzy set High-fluent 2- how much is that value about 3 i.e., membership of the obtained value to the fuzzy set About 3 or 1. how many students in the group are high-fluent, relatively to the size of the group i.e., cardinality of a fuzzy set High-fluent divided by the size of the group 2. how much is that value almost all i.e., membership of the obtained value to the fuzzy set Almost all.

25 Group = { Adam, Bob, Cathy, David, Eve }. Fluency is represented by the value from the interval [0, 100]. Fuzzy set F represents High fluency on [0,100]. Fuzzy set Q represents fuzzy quantifier about 3.

26 Linguistic hedges Special linguistic terms by which other linguistic terms are modified. Very, more,less, fairly, extremely In general for fuzzy proposition p: x is F and a linguistic hedge H we can had a modified proposition Hp: x is HF Modifier Any linguistic hedge H may be interpreted as a unary operation h on the unit interval [0,1] thus: HA(x) = h(a(x)) h(a) = a 1/2, a 2, a 3, Weak modifier, fairly strong modifier, very Very strong modifier, very very

27 Example Propositions p 1 : John is young p 2 : John is very young p 3 : John is fairly young Assume John is 26 years old, the degree of truth of the propositions are Young(26) = 0.8 Very young(26) = = 0.64 Fairly young (26) = 0.8 1/2 = 0.89 Strong assertion is less true

28 Fuzzy Inference

29 Fuzzy inference rules Inference rules in classical logic based on the various tautologies. Inference rules can be generalized within the framework of fuzzy logic To facilitate approximate reasoning. Here we describe generalizations for three classical inference rules: modus ponens, modus tollens, hypothetical syllogism.

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31 Fuzzy Inference rule For a given fuzzy relation R on X x Y, and a given fuzzy set A on X, a fuzzy set B on Y can be derived for all y ɛ Y, so that In matrix form, compositional rule of inference is

32 Fuzzy inference rules The fuzzy relation R is given by (one or more) conditional fuzzy propositions. For a given fuzzy proposition: p : If X is A; then Y is B Is determined for all x ɛ X and y ɛ Y by where J stands for a fuzzy implication.

33 Generalized Modus ponens Using relation R obtained from given proposition p (previous slide) and given another proposition q of the form q : x is A' we may conclude that y is B' by the compositional rule of inference. B is calculated by This procedure is called a generalized modus ponens.

34 Example

35 Generalized modus tollens, Another inference rule in fuzzy logic, which is a generalized modus tollens, is expressed by the following schema: the compositional rule of inference has the form

36 Example

37 Generalization of hypothetical syllogism The generalized hypothetical syllogism is expressed by the following schema: In this case, X, v, :Z are variables taking values in sets X, Y, Z, respectively, and A, B, C are fuzzy sets on sets X, Y, Z, respectively. For each conditional fuzzy proposition in (8.43), there is a fuzzy relation

38 Generalization of hypothetical syllogism, Given R1, R2, R3, obtained by these equations, we say that the generalized hypothetical syllogism holds if This equation may also be written in the matrix form

39 Example

40 Inference from conditional and qualified propositions Given a conditional and qualified fuzzy proposition p of the form where S is a fuzzy truth qualifier, and a fact is in the form "X is A'," we want to make an inference in the form y is B'." method of truth-value restrictions, is based on a manipulation of linguistic truth values. The method involves the following four steps.

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42 Example

43 Homework 8-2, 8-4, 8-6, 8-9,