Ch.3 Fuzzy Rules and Fuzzy Reasoning Extension principle, Fuzzy relations, Fuzzy if-then rules, Compositional rule of inference, Fuzzy reasoning

Extension Principle Generalizes crisp math concept to fuzzy framework. Extends point to point mapping to mapping of FS’s. Maps points (x1, x2, …,xn) in X to a crisp set Y to be generalized such that it maps fuzzy subsets of X to Y. f : X Y

If more than one point is mapped to a point in Y, Extension Principle If more than one point is mapped to a point in Y, Ex) A = { (-1, 0.5), (0, 0.8), (1, 1), (2, 0.4) }, and y = x2 Find B=[(y, μB(y))]. B = 0.8/0 + 1/1 + 0.4/4 x μA(x) y = x2 μB(y) -1 0.5 1 max(0.5, 1)=1 0.8 2 0.4 4

Extension Principle: Extension Principle Let A1, A2, …, An be FS’s in X1, X2, …, Xn, respectively. X = and f: X Y, y = f(x1, x2, …, xn), then extension principle allows us to map FS’s in X to Y as follows: B = f(A) where B is the fuzzy image (set) in Y:

Ex) f maps ordered pairs from X1 and X2 to Y, where X1 = {-1, 0, 1}, Extension Principle Ex) f maps ordered pairs from X1 and X2 to Y, where X1 = {-1, 0, 1}, X2 = {-2, 2 }, Y = {-2, -1, 2, 3} and y = f(x1, x2) = x12+x2. FS A1 = 0.5/-1 + 0.1/0 + 0.9/1, A2 = 0.4/-2 + 1/2, then B = f(A1, A2) ? μB(y = -1) = max { min[μA1(x1 = -1), μA2(x2 = -2)], min[μA1(x1 = 1), μA2(x2 = -2)] } = 0.4 Similarly, μB(y = 3) = max[ 0.5, 0.9 ] = 0.9 Thus, B = 0.1/-2 + 0.4/-1 + 0.1/2 +0.9 /3 x1 μA1(x1) x2 μA1xA2(x1, x2) y = x12+x2 μB(y) -1 0.5 -2 0.4 2 1 3 0.9 0.1

Extension Principle Ex)

Extension Principle Ex)

Ex) Extension Principle = bell(x; 1.5, 2, 0.5) if X Y y = f(x) -2 2 -3 -1 1 3 Y y = f(x) 0.2 0.4 0.6 0.8 Membership Grades A 0.5 B = bell(x; 1.5, 2, 0.5) if

Fuzzy Relations A fuzzy relation R is a 2D MF: Ex) x is close to y (x and y are numbers). x depends on y (x and y are events). x and y look alike (x and y are persons or objects). If x is large, then y is small (x is an observed reading and y is a corresponding action).

Fuzzy Relations The max-min composition of two fuzzy relations R1 (defined on X and Y) and R2 (defined on Y and Z): Properties: Associativity: Distributivity over union: Weak distributivity over intersection: Monotonicity:

Max-product composition Fuzzy Relations Max-product composition Max-* composition where * is a T-norm operator.

Fuzzy Relations Ex) Max-min composition: Max-product composition: Derive the degree of relevance between 2 in X and a in Z based on R1 and R2 = “ x is relevant to y,” = “ y is relevant to z,” where

Fuzzy if-than rules Linguistic variable: a variable whose values are words or sentences in a natural language. Ex) Speed is a linguistic variable Age is a linguistic variable Defn: Fuzzy variables are characterized by triples: x = Name of the fuzzy variable X = Universe of discourse R(x) = Fuzzy restriction Ex) 1. x = fast, R(x) = fuzzy set of fast 2. x = old, R(x) = 0.1/20 + 0.2/30 + ··· + 1/70 + 1/80

Defn: A linguistic variable is characterized by a quintuple: Fuzzy if-than rules Defn: A linguistic variable is characterized by a quintuple: where x = Name of the linguistic variable T(x) = A term set of x, i.e., the set of names of linguistic variables of x with each value being a fuzzy variable defined in X G = Syntactic rule for generating the names of the value of x M = Semantic rule for associating each value of x with its meaning

Linguistic Hedge (Modifier) With the linguistic hedges, we can define

Membership functions of the term set T(age) Fuzzy if-than rules Ex) Linguistic variable: Age, X = [0, 100] T(age) = {young, not young, very young, …, middle aged, not middle aged, …, old, not old, very old, not very old, …, not very young and not very old, … } 10 20 30 40 50 60 70 80 90 100 0.2 0.4 0.6 0.8 1 X = age Membership Grades Old Very Young Middle Aged Membership functions of the term set T(age)

Primary terms (young, middle aged, old) Fuzzy if-than rules Primary terms (young, middle aged, old) Hedges (very, more or less, quite, extremely, etc) Connectives (and, or, either, neither) Operations on linguistic values: Concentration (very) and dilation (more or less) Contrast intensification Concentration Dilation

Ex) Fuzzy if-than rules NOT(A) = A AND B = A OR B = more or less = DIL(old) not young and not old

extremely old = CON(CON(CON(old)) Fuzzy if-than rules young but not too young extremely old = CON(CON(CON(old)) 10 20 30 40 50 60 70 80 90 100 0.2 0.4 0.6 0.8 1 X = age Membership Grades (a) Primary Linguistic Values Old Young (b) Composite Linguistic Values Not Young and Not Old More or Less Old Extremely Old Young but Not Too Young

Ex) Fuzzy if-than rules Solid line : A Dotted line : INT(A) 1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 X Membership Grades Effects of Contrast Intensifier Solid line : A Dotted line : INT(A) Dashed line : INT(INT(A)) Dashed-dot line : INT(INT(INT(A)))

Orthogonality: Fuzzy if-than rules A term set of a linguistic variable x on X is orthogonal if where the ’s are convex and normal fuzzy sets.

Fuzzy if-than rules Fuzzy if-then rules (fuzzy rule, fuzzy implication, fuzzy conditional statement) assumes the form: If x is A, then y is B. (antecedent, premise), (consequence, conclusion) Two ways to interpret a fuzzy rule: “If pressure is high, then volume is small” A coupled with B: (A and B) A entails B: (Not A or B) Y X A B { A entails B A coupled with B

Fuzzy relation: R = A B Fuzzy if-than rules A coupled with B: (A and B) A entails B: (Not A or B) Material implication: Propositional calculus: Extended propositional calculus: Generalization of modus ponens: where is a T-norm operator where and is a T-norm operator.

Fuzzy implication function f: Fuzzy if-than rules Fuzzy implication function f: Transforming the membership grades of x in A and y in B into those of (x, y) in A B A coupled with B Min Algebraic-product Bounded product Drastic product A entails B Zadeh’s arithmetic rule Zadeh’s max-min rule Boolean fuzzy implication Goguen’s fuzzy implication

1st row: 2nd row: Fuzzy if-than rules fuzzy implication ftns corresponding R

Fuzzy if-than rules

Fuzzy Reasoning Logic Logic is a basis for reasoning. How to combine two propositions:

Reasoning/Inference Rules Modus Ponens (MP) Modus Tollens (MT) Hypothetical Syllogism (HS) 3단 논법 Ex: (1) Gildong is bald. (2) Bald men are rich. Gildong is rich.

p: This tomato is very red. Generalized MP (GMP) Tuned a little bit. p: This tomato is very red. Implication: If a tomato is red, then the tomato is ripe. Conclusion: This tomato is very ripe.

n-valued Logic - 2-valued logic is not enough. Logic operations: indeterminant. Logic operations: - n-valued logic: Note: For

crisp (Koo is a student of EE). Fuzzy Logic Proposition: Subject crisp (Koo is a student of EE).  Fuzzy Logic: Proposition → Fuzzy proposition Predicate → Fuzzy predicate Truth value → A point in the unit interval [0,1]

Fuzzy Logic (Gildong is young.) is very true or more or less true (one point in [0,1]).

Connections for truth value: Fuzzy Logic Defn: The truth value of the proposition “ X is A ” or simply truth value of A is denoted by which is defined to be a point in [0,1] (called numerical truth value) or a possible fuzzy set in [0,1] (called linguistic truth value, e.g., very true) Connections for truth value:

Fuzzy Logic Defn:

Fuzzy Logic Ex)

Fuzzy (or Approximate) Reasoning Fuzzy proposition - conclusion Fuzzy proposition Rules of inference govern the deduction of a proposition from a set of premises. For fuzzy reasoning or approximate reasoning, q = approximate rather than an exact consequence of

4 Types of Rules for Approximate Reasoning Categorical reasoning Qualitative reasoning Syllogistic reasoning Dispositional reasoning (1) Categorical reasoning Premise: X is A Implication: If X is C, then Y is B. where X, Y, Z taking values in U, V, W, and A, B, C are fuzzy predicates.

Approximate Reasoning (a) Projection rules of inference

Approximate Reasoning (b) Conjunction/Particularization Y is “don’t care”

Approximate Reasoning (c) Disjunction/Cartesian product (d) Negation

Approximate Reasoning (e) Entailment rule of inference X is A A B X is B Ex) Mary is very young Very young young Mary is young (f) Compositional rule of inference (X,Y) is R Y is A R R

Approximate Reasoning Ex) p = X is small q = X and Y are approximately equal U = {1, 2, 3, 4} small = 1/1 + 0.6/2 + 0.2/3 approxi. equal = 1/(1,1)+ 1/(2,2)+ 1/(3,3)+ 1/(4,4)+ 0.5/(1,2)+ 0.5/(2,1) +0.5/(2,3) +0.5/(3,2)+ 0.5/(3,4)+ 0.5/(4,3) X is small (X,Y) approximately equal =

Approximate Reasoning A possible interpretation of the inference may be the following: X is small. X and Y are approximately equal. Y is more or less small.

Approximate Reasoning (g) Generalized Modus Ponens (GMP) X is A If X is B, then Y is C Y is where MP: Partial matching by GMP. (h) Extension principle f(X) is f(A) Ex) X is small is small

Approximate Reasoning (2) Qualitative Reasoning I/O relation of a system is expressed as a collection of if-then rules in which the antecedents and consequents involve linguistic variables: If X is and Y is , then Z is . If X is and Y is , then Z is . Ex) R1: If pressure is high, then volume is small. R2: If pressure is low, then volume is large. If pressure is medium, then volume is where

Fuzzy reasoning (approximate reasoning): An inference procedure that derives conclusion from a set of fuzzy rules and known facts Compositional rule of inference y y=f(x) b a x a and b : points y = f(x) : a curve a and b : intervals y = f(x) : an interval-valued function

Compositional rule of inference: Fuzzy Reasoning Compositional rule of inference: Ex) Let F : A B, fuzzy relation, and A: a fuzzy set of X. Find the resulting fuzzy set B: where denotes the composition operator.

Fuzzy Reasoning

Fuzzy reasoning (Generalized Modus Ponens) Rule : “if the tomato is red, then it is ripe” Fact : “the tomato is more or less red” Infer : “the tomato is more or less ripe” fact : x is A, rule : If x is A, then y is B, conclusion : y is B. fact : x is A’, rule : If x is A, then y is B, conclusion : y is B’. where A, B, A’ and B’ are fuzzy sets, and A’ and B’ are close to A and B, respectively.

Def: Approximate reasoning (fuzzy reasoning) Assume the fuzzy implication A B is expressed as a fuzzy relation R on X x Y. Then the fuzzy set induced by “x is ” and the fuzzy rule “If x is A, then y is B” is defined by Or, equivalently,

Single rule with single antecedent Fuzzy Reasoning Single rule with single antecedent Fact : x is A’ Rule : If x is A, then y is B Conclusion : y is B’ A X w A’ B Y x is A’ B’ y is B’ min

Single rule with multiple antecedents Fuzzy Reasoning Single rule with multiple antecedents Fact : x is A’ and y is B’ Rule : If x is A and y is B, then z is C Conclusion : z is C’

Thm 3.1: Decomposition method Fuzzy Reasoning Thm 3.1: Decomposition method The resulting consequence can be expressed as the inferred fuzzy set of a GMP problem for a single fuzzy rule with a single antecedent. A B min X Y w A’ B’ C Z C’ x is A’ y is B’ z is C’ w1 w2

Multiple rules with multiple antecedents Fuzzy Reasoning Multiple rules with multiple antecedents Fact : x is A’ and y is B’ Rule 1 : If x is A1 and y is B1, then z is C1 Rule 2 : If x is A2 and y is B2, then z is C2 Conclusion : z is C’

Graphic representation Fuzzy Reasoning Graphic representation A1 B1 A2 B2 min X Y w1 w2 A’ B’ C1 C2 Z C’ x is A’ y is B’ z is C’

Fuzzy reasoning divided into four steps: Degrees of compatibility (match) Compare the known facts with the antecedents of fuzzy rules Firing strength Degree to which the antecedent part of the rule is satisfied Qualified (induced) consequent MFs Apply the firing strength to the consequent MF of a rule Overall output MF Aggregate all the qualified consequent MFs