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Fuzzy Foundations for EEs, CpEs, CSs

Fuzzy Foundations for EEs, CpEs, CSs. By P. D. Olivier, Ph.D., P.E. Fuzzy Foundations. Fuzzy “stuff” can be developed from two (albeit related) classical areas Classical (crisp) logic leads to Fuzzy Logic (path taken here)

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Fuzzy Foundations for EEs, CpEs, CSs

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  1. Fuzzy Foundationsfor EEs, CpEs, CSs By P. D. Olivier, Ph.D., P.E.

  2. Fuzzy Foundations • Fuzzy “stuff” can be developed from two (albeit related) classical areas • Classical (crisp) logic leads to Fuzzy Logic (path taken here) • Classical set theory leads to Fuzzy Set Theory (more common path, see book) • Logic and set theory are related since • AND and INTERSECTION are related • OR and UNION are related • NOT and COMPLEMENT are related

  3. Invented by ancient Greeks, used by “classical scholars”, used by mathematicians Every statement is either TRUEorFALSE Statements can be combined with the logical connections AND and OR A statement can be modified with NOT Truth tables are used to evaluate the truth value (i.e. TRUEness or FALSEness) of a complicated statement Logical IF – THEN statements are very important, used to express “THEOREMS” Classical Logic

  4. Truth Table examples Note: Logical IF premise THEN conclusion not programming IF condition THEN action {IF p THEN q}  {NOT(p)ORq}

  5. Boolean Algebra • Based on Classical Logic • Truth values TRUE=T=1, FALSE=F=0 • Mathematizes classical logic • Formulas evaluate truth values

  6. Truth Table example • TV(p) = truth value of p = 0 or 1 • TV(pANDq) = min(TV(p),TV(q)) = TV(p)*TV(q) = … • TV(pORq) = max(TV(p),TV(q)) • = TV(p)+TV(q)-TV(p)*TV(q) = … • TV(NOT(p))=1-TV(p) • Any logical expression can be expressed in terms of AND, OR, NOT. TV{IF p THEN q} = TV{NOT(p)ORq} = (1-TV(p))+TV(q)- (1-TV(p))*TV(q)

  7. Fuzzy Logic • Truth values are continuous between 0 and 1 • Choose mathematical formulas for AND, OR, NOT • Compute truth values of complicated statements using chosen formulas • Are there other reasonable formulas of AND, OR, NOT? • What kind of vagueness does FL help with? • TV() function related to the characteristic function in classical set theory and classical logic • TV() function related to the membership function in FL

  8. Types of Vagueness • Imprecision: Inaccurate measurement • Statistical: Precise, incomplete, measurements • Classification (membership in a set) • Determining membership in a group based on a measurement(s) • Fuzzy Logic/Set theory helps when set membership is not clear. Consider the set of TALL people. Determine if a given person is tall. Context, subjectivity.

  9. Crisp SET operations • An element x is either in a set or not in the set. • VENN diagrams • Union of A and B • Intersection of A and B • Complement of A

  10. Table 2.2 • Convert set equations to logical equations • CORRECTION: item one should read (A’)’=A

  11. Mathematizing CRISP set theory • Characteristic function • Complement • Intersection Others? • Union

  12. Fuzzy Set theory • Characteristic functions become fuzzy membership functions • Fuzzy membership functions produce continuum of values between 0 and 1 • Not just 0 or 1. • The value of the membership function at a point is the membership value of the point in the set.

  13. Fuzzy Set theory - Logic • Interpretation of Membership functions • truth value of a statement • Level of membership in a set • We will go back and forth between interpretations as convenient. • Fuzzy sets  Fuzzy membership functions

  14. Example 2.7: Expensive Cars • Logic statement: • Car X is an expensive car • Set theory statement • X is an element of the set of expensive cars • Consider Ferraris, Rolls Royce’s, Mercedes, BMWs, Buicks, Toyotas • Produce a membership function

  15. Example 2.8: Natural numbers close to 6 • Logic statement • n is a Natural number close to 6 • Set theory statement • n is an element of the set of Natural numbers close to 6 • Consider the natural numbers 3 … 9 • Produce a membership function

  16. Typical Fuzzy sets • Increasing (s or gamma functions) pg 50 • Decreasing (z or L functions) • Approximating (triangular/lambda, trapezoidal, bell) • Linguistic variables • Age • Old, young, middle aged, very old, very young • Temperature • Hot, cold, tepid, very hot, very cold, comfortable • Generic variable • NB, NM, NS, Z, PS, PM, PB

  17. Mathematical shorthand • For all, or for every • There exists • Such that • With respect to : or s.t. w.r.t.

  18. 2.1.5 Properties of Fuzzy Sets (see pp 52-54) • Support • Width • sup • inf • Nucleus • Height • convexity Largest membership degree

  19. 2.1.6 Operations on Fuzzy Sets (see pp. 55-61) • Equality • Subset and strict subset • Superset and strict superset • Union, intersection and complement • Intersections are described by Triangular-norms (T-norms) • Archimedean • Unions are described by Triangular co-norms (S-norms)

  20. T-norms (generic intersection) • A triangular norm (T-norm) is a binary function (operator) that is • Commutative • Associative • Non dedreasing • T-norm identity is 1

  21. Archimedean T-norms • A T-norm that satisfies T-1 to T-4, together with

  22. S-norms (generic union) • A triangular co-norm (S-norm) is a binary function (operator) that is • Commutative • Associative • Non dedreasing • S-norm identity is 0

  23. Complement • The complement function (operator) is a unary operator that has the following properties • Boundary values • Non-increasing • idempotent

  24. Exercises • Prove that min(a,b) and a*b are T-norms • Prove that max(a,b) and a+b-a*b are S-norms • Prove that min(a,b) and max(a,b) are conjugate T and S norms according to eq. 2.44 • Prove that a*b and a+b-a*b are conjugate T and S norms according to eq. 2.44 • Prove that 1-a is a complement operation PROVE means to demonstrate to a skeptic that the conclusion follows from the basic rules of mathematics.

  25. Classical to Fuzzy Relations • A classical relation is a set of tuples • Binary relation (x,y) • Ternary relation (x,y,z) • N-ary relation (x1,…xn) • Connection with Cross product • Married couples • Nuclear family • Points on the circumference of a circle • Sides of a right triangle that are all integers

  26. Characteristic Function • Any set has a characteristic function. • A relation is a set of points • Review definition of characteristic function • Apply this definition to a set defined by a relation

  27. Properties of some binary relations • Reflexive • Anti-reflexive • Symmetric • Anti-symmetric • Transitive • Equivalence • Partial order • Total order • Assignment: Classify: =,<,>,<=,>=

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