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11_Intro_Fuzzy_Logic and Expert Systems
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1 .Soft Computing Fuzzy logic is part of soft computing
2 .Congress of Computational Intelligence Neural Nets Evolutionary Algorithms Computational Intelligence Fuzzy Logic
3 .Fuzzy Logic and Functions Constructive Induction Decision Trees and Evolutionary Algorithms Fuzzy logic other Learning Neural Nets
4 . The Definition of Fuzzy Logic Membership Function • A person's height membership function graph is shown next with linguistic values of the degree of membership as very tall, tall, average, short and very short being replaced by 0.85, 0.65, 0.50, 0.45 and 0.15.
5 .• In traditional logic, statements can be either true or false, and sets can either contain an element or not. • These logic values and set memberships are typically represented with number 1 and 0. • Fuzzy logic generalizes traditional logic by allowing statements to be somewhat true, partially true, etc. • Likewise, sets can have full members, tall partial members, and so on. • For example, a person whose height is 5’ 9” might be assigned a medium membership of 0.6 in the fuzzy set “tall people”. • The statement “Joe is tall” is 60% true of Joe is 5’9”. • Fuzzy logic is a set of “if--then” statements based on combining fuzzy sets. (Beale & Demuth..Fuzzy Systems Toolbox.)
6 . Fuzzy Sets, Statements, and Rules • A crisp set is simply a collection of objects taken from the universe of objects. • Fuzzy refers to linguistic uncertainty, like the word “tall”. • Fuzzy sets allow objects to have membership in more than one set: – e.g. 6’ 0” has grade 70% in the set “tall” and grade 40% in the set “medium”. • A fuzzy statement describes the grade of a fuzzy variable with an expression: – e.g. Pick a real number greater than 3 and less than 8.
7 . The Definition of Fuzzy Logic Rules • A fuzzy logic system uses fuzzy logic rules, as in an expert system where there are many if-then rules. – A fuzzy logic rule uses membership functions as variables. • A fuzzy logic rule is defined as an if variable(s) and then output fuzzy variable(s). • Fuzzy logic variables are connected together like binary equations with the variables separated with operators of AND, OR, and NOT.
8 . Contents • Review of classical logic and reasoning systems • Fuzzy sets • Fuzzy logic • Fuzzy logic systems applications • Fuzzy Logic Minimization and Synthesis • Practical Examples • Approaches to fuzzy logic decomposition • Our approach to decomposition • Combining methods and future research
9 . Outline •be introduced to the topics of: – fuzzy sets, – fuzzy operators, – fuzzy logic – and come to terms with the technology •learn how to represent concepts using fuzzy logic •understand how fuzzy logic is used to make deductions •familiarise yourself with the `fuzzy' terminology
10 . Review of Traditional Propositional Logic and why it is not
11 .Traditional Logic • One of the main aims of logic is to provide rules which can be employed to determine whether a particular argument is correct or not. • The language of logic is based on mathematics and the reasoning process is precise and unambiguous.
12 . Logical arguments • Any logical argument consists of statements. • A statement is a sentence which unambiguously either holds true or holds false. – Example:Today Example: is Sunday
13 . Predicates • Example: Seven is an even number – This example can be written in a mathematical form as follows: • 7 {x| x is an even number}x| x is an even number} – or in a more concise way: • 7 {x| x is an even number}x|P(x)} } – where | is read as such that and P(x)} stands for `x has property P' and it is known as the predicate. – Note that a predicate is not a statement until some particular x-value is specified. – Once a x value is specified then the predicate becomes a statement whose truth or falsity can be worked out.
14 . For All Quantifier • For all x and y, x2-y2 is the same as (x+y)} *(x-y)} – This example can be written in a mathematical form as well: x,y ((x,yR)} (x2-y2)} =(x+y)} *(x-y)} )} • where the is interpreted as 'for all', is the logical operator AND, and R represents what is termed as the universe of discourse.
15 . Universe of Discourse • Using the universe of discourse one assures that a statement is evaluated for relevant values. – The above predicate is then true only for real numbers. • Similarly for the first example the universe of discourse is most likely to be the set of natural numbers rather than buildings, rivers, or anything else. – Hence, using the concept of the universe of discourse any logical paradoxes can not arise.
16 .Existential Quantifier • Another type of quantifier is the existential quantifier ()} . • The existential quantifier is interpreted as 'there exists' or 'for some' and describes a statement as being true for at least one element of the set. • For example, (x)} ( river(x)} name(x)} =Amazon )}
17 .Connectives and their truth tables • A number of connectives exist. – Their sole purpose is to allow us to join together predicates or statements in order to form more complicated ones. • Such connectives are NOT (~)} , AND ()} , OR ()} . – Strictly speaking NOT is not a connective since it only applies to a single predicate or statement. • In traditional logic the main tools of reasoning are tautologies, such as the modus ponens (A(AB)))} )} B)) ( means implies)} .
18 . Truth Tables A B And AÙB |AÚB Or | ~ A Not True True True True False True False False True False False True False True True False False False False True This everything will hold true, we will just do a small modification to the material on logic from the last quarter
19 . Identities of Fuzzy Logic or how fuzzy logic differs from classical
20 . Identities of Fuzzy Logic • The form of identities used in fuzzy variables are the same as elements in fuzzy sets. • The definition of an element in a fuzzy set, {(x,u a(x))}, is the same as a fuzzy variable x and this form will be used in the remainder of the paper. • Fuzzy functions are made up of fuzzy variables. The identities for fuzzy algebra are: Idempotency: X + X = X, X * X = X Commutativity: X + Y = Y + X, X * Y = Y * X Associativity: (X + Y) + Z = X + (Y + Z), (X * Y) * Z = X * (Y* Z) Absorption: X + (X * Y) = X, X * (X + Y) = X Distributivity: X + (Y * Z) = (X + Y) * (X + Z), X * (Y + Z) = (X * Y) + (X * Z) Complement: X’’ = X DeMorgan's Laws: (X + Y)’ = X’ * Y’, (X * Y)’ = X’ + Y’
21 .Transformations of Fuzzy Logic Formulas Some transformations of fuzzy sets with examples follow: x’b + xb = (x + x’)b b xb + xx’b = xb(1 + x’) = xb x’b + xx’b = x’b(1 + x) = x’b a + xa = a(1 + x) = a a + x’a = a(1 + x’) = a a + xx’a = a a+0=a x+0=x x*0=0 x+1=1 x*1=x Examples: a + xa + x’b + xx’b = a(1 + x) + x’b(1 + x) = a + x’b a + xa + x’a + xx’a = a(1 + x + x’ + xx’) = a
22 . Differences Between Boolean Logic and Fuzzy Logic • In Boolean logic the value of a variable and its inverse are always disjoint (X * X’ = 0) and (X + X’ = 1) because the values are either zero or one. • Fuzzy logic membership functions can be either disjoint or non-disjoint. • Example of a fuzzy non- linear and linear membership function X is shown (a) with its inverse membership function shown in (b).
23 . Fuzzy Intersection and Union • From the membership functions shown in the top in (a), and complement X’ (b) the intersection of fuzzy variable X and its complement X’ is shown bottom in (a). • From the membership functions shown in the top in (a), and complement X’ (b) the union of fuzzy variable X and its complement X’ is shown bottom in (b). Fuzzy Fuzzy union intersection
24 .Validation of Fuzzy Functions valid inconsistent • Two fuzzy functions are valid iff the function outputs are 0.5 under all possible assignments. • This is like doing EXOR of two binary functions shown in (b) which is the same as union. • Two fuzzy functions are inconsistent iff the function output is 0.5 under all possible assignments. Thus, if the output of the two fuzzy functions is < 0.5 then the two fuzzy functions are inconsistent. • This is like exnor of two binary functions of shown in (a) which is the same as intersection.
25 . Fuzzy Logic as an answer to problems with traditional logic
26 . Fuzzy Logic • The concept of fuzzy logic was introduced by L.A Zadeh in a 1965 paper. • Aristotelian concepts have been useful and applicable for many years. • B))ut these traditional approaches present us with certain problems: – Cannot express ambiguity – Lack of quantifiers – Cannot handle exceptions
27 .Traditional Logic Problems – Cannot express ambiguity: • Consider the predicate `X is tall'. • Providing a specific person we can turn the predicate into a statement. • But what is the exact meaning of the word `tall'? • What is `tall' to some people is not tall to others. – Lack of quantifiers: • Another problem is the lack of being able to express statements such as `Most of the goals came in the first half '. • The `most' quantifier cannot be expressed in terms of the universal and/or existential quantifiers.
28 .Traditional Logic Problems – Cannot handle exceptions: • Another limitation of traditional predicate logic is expressing things that are sometimes, but not always true.
29 .Traditional sets