Overview

  • Classical set theory and logic
    • Rigorous
    • Desirable for math proofs
  • Fuzzy logic
    • Allow fuzziness to generalize classical set theory and logic
    • Rule-based approach to model some complex, non-linear, or ill-defined systems
    • Capture human knowledge or heuristic reasoning that cannot be adequately described by two-valued classical logic
    • Defuzzification: convert the combined function into a single numeric output
Classical Two-Valued LogicFuzzy Logic

Classical Set Theory

Binary belonging property.

Characteristic Function

$$\begin{aligned} \mu_A(x) = &1 \text{ if } x \in A\ &0 \text{ if } x \notin A \end{aligned}$$

Relations

$$R = {(x, y) | relation} \subset X \times Y$$

  • Representaiton
    • Relational matrix
    • Set-to-set relation graph
Composition of Relations

$$R \circ S = {(x, z) \in X \times Z | (x, y) \in R \text{ and } (y, z) \in S \text{ for some } y \in Y }$$

Fuzzy Sets

Grade of membership belonging property.

$$A = {(x, \mu_A(x)) | x \in U } = { \frac{\mu_A(x)}{x} | x \in U }$$

Membership Function

$$\mu_A: U \rightarrow [0, 1]$$

If $$\mu_A$$ only takes 0 or 1, $$A$$ is called a crisp (nonfuzzy) set.

Support

Membership values > 0.

$$supp(A) = {x \in U | \mu_A(x) > 0}$$

Height

Supremum/maximum of the membership function.

$$hgt(A) = sup_{x \in U}(\mu_A(x))$$

If $$hgt(A) = 1$$, then $$A$$ is normal.

$$\alpha$$-Level Cut

$$A_{\alpha} = {x \in U | \mu_A(x) \ge \alpha}$$

Convex

If the $$\alpha$$-cut $$A_{\alpha}$$is a convex set for any $$\alpha$$, $$A$$ is convex.

$$\forall x1, x2 \in U, 0 \le \lambda \le 1, \mu_A(\lambda x_1 + (1 - \lambda) x_2) \ge min(\mu_A(x_1), \mu_A(x_2))$$

A normal & convex fuzzy set is a fuzzy number.

Crossover Point

Point $$x \in U$$ where $$\mu_A(x) = \frac{1}{2}$$.

Equality

$$A = B \text{ iff } \mu_A(x) = \mu_B(x) \forall x \in U$$

Subset

$$A \subset B \text{ iff } \mu_A(x) \le \mu_B(x) \forall x \in U$$

Center

Let $$m$$ be the mean value of all points in a set $$M$$ that are the max value in a fuzzy set $$A$$.

$$\begin{aligned} \text{center of A} = &m &\text{if m is finite}\ &\text{min of M} &\text{if m = } \infty\ &\text{max of M} &\text{if m = } -\infty\ \end{aligned}$$

Fuzzy Set Operations

Intersection

$$\mu_{A \cap B}(x) = min(\mu_A(x), \mu_B(x)) \forall x \in U$$

  1. If $$A$$ and $$B$$ are crisp sets, then $$A \cap B$$ defined using fuzzy intersection is the same as using classical set intersection
  2. If $$A$$ and $$B$$ are fuzzy sets, the properties of classical set intersection hold for fuzzy set intersection
t-norms

$$t: [0, 1] \times [0, 1] \rightarrow [0, 1]$$

For any $$a, b, c \in [0, 1]$$,

  • Axiom t1 (boundary condition): $$t(0, 0) = 0$$, $$t(a, 1) = t(1, a) = a$$
  • Axiom t2 (commutativity): $$t(a, b) = t(b, a)$$
  • Axiom t3 (non-decreasing): $$a \le a', b \le b' \Rightarrow t(a, b) \le t(a', b')$$
  • Axiom t4 (associativity): $$t(t(a, b), c) = t(a, t(b, c))$$
Common Classes of t-norms
  • Minimum
    • $$t(a, b) = min(a, b)$$
  • Algebraic product
    • $$t_{ap}(a, b) = ab$$
  • Dubois-Prade
    • $$t_{\alpha}(a, b) = \frac{ab}{max(a, b, \alpha)}$$
  • Drastic product
    • $$\begin{aligned}t_{dp}(a, b) = &a &\text{if } b = 1\&b &\text{if } a = 1\&0 &\text{otherwise}\end{aligned}$$
  • Einstein product
    • $$t_{ep}(a, b) = \frac{ab}{2 - (a+b-ab)}$$

Lemma: $$t_{dp}(a, b) \le t(a, b) \le min(a, b)$$

Union

$$\mu_{A \cup B}(x) = max(\mu_A(x), \mu_B(x)) \forall x \in U$$

s-norms

$$s: [0, 1] \times [0, 1] \rightarrow [0, 1]$$

For any $$a, b, c \in [0, 1]$$,

  • Axiom s1 (boundary condition): $$s(1, 1) = 1$$, $$s(a, 0) = s(0, a) = a$$
  • Axiom s2 (commutativity): $$s(a, b) = s(b, a)$$
  • Axiom s3 (non-decreasing): $$a \le a', b \le b' \Rightarrow s(a, b) \le s(a', b')$$
  • Axiom s4 (associativity): $$s(s(a, b), c) = s(a, s(b, c))$$
Common Classes of s-norms
  • Maximum
    • $$s(a, b) = max(a, b)$$
  • Algebraic sum
    • $$s_{as}(a, b) = a + b - ab$$
  • Dubois-Prade
    • $$s_{\alpha}(a, b) = \frac{a + b - ab - min(a, b, 1-\alpha)}{max(1-a, 1-b, \alpha)}$$
  • Drastic sum
    • $$\begin{aligned}s_{ds}(a, b) = &a &\text{if } b = 0\&b &\text{if } a = 0\&1 &\text{otherwise}\end{aligned}$$
  • Einstein sum
    • $$t_{ep}(a, b) = \frac{a+b}{1+ab}$$

Lemma: $$max(a, b) \le s(a, b) \le s_{ds}(a, b)$$

Complement

$$\mu_{\bar{A}}(x) = 1 - \mu_A(x) \forall x \in U$$

$$c: [0, 1] \rightarrow [0, 1]$$

For any $$a, b \in [0, 1]$$,

  • Axiom c1 (boundary condition): $$c(0) = 1$$, $$c(1) = 0$$
  • Axiom c2 (non-decreasing): $$a < b \Rightarrow c(a) \ge c(b)$$

Fuzzy Relations

$$P = {((x, y), \mu_P(x, y)) | (x, y) \in X \times Y}$$

$$P \circ Q = {((x, z), \mu_{P \circ Q}(x, z))|(x, z)\in X \times Z}$$

Take t-norm along a path from $$x$$ to $$z$$, then max over all possible paths:

$$\mu{P \circ Q}(x, z) = max{y \in Y} { t(\mu_P(x, y), \mu_Q(y, z)) }$$

  • Max-min composition
    • $$t$$ = min
  • Max-product composition
    • $$t$$ = algebraic product

Linguistic Variables and Fuzzy Propositions

Linguistic Variables

A linguistic variable $$x$$ defined on domain$$U$$ can take fuzzy sets defined on $$U$$ & the numbers of $$U$$ as its value.

Linguistic Hedges/Fuzzy Modifiers
  1. very A is a fuzzy set with membership function $$\mu_{veryA}(x) = [\mu_A(x)]^2$$
  2. more or less A is a fuzzy set with membership function $$\mu_{moreorlessA}(x) = [\mu_A(x)]^{1/2}$$

Fuzzy Propositions

Atomic Fuzzy Proposition

$$x$$ is $$A$$

  • $$x$$: linguistic variable
  • $$A$$: linguistic value (fuzzy set)
  • $$\mu_A(a) \text{ for any } x = a$$: truth value of the proposition
  • $$\mu_A(x)$$: truth function, the degree of truth of the proposition
Compound Fuzzy Proposition

A composition of atomic fuzzy propositions using connectives and, or, not.

  • and
    • $$\mu_{A \cap B}(x, y) = t(\mu_A(x), \mu_B(y))$$
  • or
    • $$\mu_{A \cup B}(x, y) = s(\mu_A(x), \mu_B(y))$$
  • not
    • $$\mu_{\bar{A}}(x) = c(\mu_A(x))$$

Fuzzy IF-THEN Rules

$$\text{IF } \text{ THEN } $$

$$P1 \rightarrow Q1 = \overline{P1} \lor Q1 = (P1 \land Q1) \lor \overline{P1}$$

  • $$P1$$: antecedent; fuzzy proposition containing linguistic variable $$x$$ defined on domain $$U$$
  • $$Q1$$: consequent; fuzzy proposition containing linguistic variable $$y$$ defined on domain $$V$$
  • Truth function : $$\mu_{P1 \rightarrow P2}(x, y)$$

Dienes-Rescher Implication

$$\mu{P1 \rightarrow P2}(x, y) = max{1-\mu{P1}(x), \mu_{P2}(y)}$$

Zadeh Implication

$$\mu{P1 \rightarrow P2}(x, y) = max{min[\mu{P1}(x), \mu{P2}(y)], 1-\mu{P1}(x)}$$

Godel Implication

$$\begin{aligned} \mu{P1 \rightarrow P2}(x, y) = &1 &\text{if }\mu{P1}(x) \le \mu{P2}(y)\ &\mu{P2}(x)&\text{otherwise}\end{aligned}$$

Everyday IF-THEN Rule

$$\text{IF } \text{ THEN } \text{ ELSE } $$

$$P1 \rightarrow Q1 = P1 \land P2$$

  • Truth function $$\mu_{P1 \rightarrow P2}(x, y)$$ = Relational matrix $$U \circ V$$

Mamdani Minimum Implication

$$\mu{P1 \rightarrow P2}(x, y) = min{\mu{P1}(x), \mu_{P2}(y)}$$

Mamdani Product Implication

$$\mu{P1 \rightarrow P2}(x, y) = \mu{P1}\mu_{P2}(y)$$

Fuzzy Inference for Approximate Reasoning

Tautology

$$p \rightarrow q$$ is identically true is denoted as $$p \Rightarrow q$$

  • Modus Ponens
    • $$[p \land (p \rightarrow q)] \Rightarrow q$$
  • Modus Tollens
    • $$[\overline{q} \land (p \rightarrow q)] \Rightarrow \overline{p}$$
  • Hypothetical Syllogism
    • $$[(p \rightarrow q) \land (q \rightarrow r)] \Rightarrow (p \rightarrow r)$$

Modus Ponens Inference Rule

Generalized Modus Ponens Inference Rule

Compositional Rule of Inference

$$\mu{B'}(y) = sup_x t[\mu{A'}(x), \mu{A \rightarrow B}(x, y)]$$ or $$\mu{B'}(y) = supx [\mu{A'}(x) * \mu_{A \rightarrow B}(x, y)]$$

Also called sup-star composition.

Crisp Singleton Input

A fuzzy set $$F$$ is called a crisp singleton if it contains a single element $$a$$ with a grade of membership 1, i.e. $$F = \frac{1}{a}$$.

For example, $$A' = \frac{1}{21}$$:

  • Minimum implication: $$\mu_M(y)$$ clipped at $$\mu_L(21)$$
  • Product implication: $$\mu_M(y)$$ scaled by $$\mu_L(21)$$
Multiple Non-Interacting Inputs

$$\mu_{A \times C}(x, z) = t(\mu_A(x), \mu_C(z))$$

$$\begin{aligned} \mu{B'}(y) &= sup{x, z} t[\mu{A' \times C'}(x, z), \mu{(A \times C) \rightarrow B}((x, z), y)]\ &= sup{x, z} t[t(\mu{A'}(x), \mu{C'}(z)), \mu{(A \times C) \rightarrow B}((x, z), y)]\ &= sup{x, z} t[\mu{A'}(x), \mu{C'}(z), \mu{A}(x), \mu{B}(y), \mu{C}(z)] \end{aligned}$$

Minimum Implication

Product Implication

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