Fuzzy Logic Systems

Soft Computing

Soft computing is a set of algorithms, including neural networks, fuzzy logic, and evolutionary algorithms. These algorithms are tolerant of imprecision, uncertainty, partial truth and approximation. It is contrasted with hard computing: algorithms which find provably correct and optimal solutions to problems.

Soft Computing includes:

• Fuzzy Logic
  • use fuzzy rules and approximate reasoning
  • a popular analogy: human knowledge
• Neural Networks
  • network of massively connected nodes
  • a popular analogy: neuron structure in brain
• Genetic Algorithms
  • Derivative-free optimization
  • a popular analogy: biological evolution
• Probabilistic Reasoning
  • Incorporates uncertainty in predicting future events
  • a popular analogy: random action of a human

Fuzzy Logic Systems

Fuzzy logic was first developed by L.A. Zadeh in 1960’s to extend conventional (binary) crisp logic to make it suitable to incorporate knowledge and mimic human-like approximate reasoning.

Two-state (bivalent) crisp logic uses two quantities: true (T) and false (F) as truth values.

However:

• Real-life situations are usually characterized by ambiguity and partial truth.
  • cannot always assume two crisp states T and F with a crisp line dividing them.

Terminology:

• Crisp: only T or F
• Fuzzy: Partial membership is possible (very true? not very true?)

Membership Function

A fuzzy set may be represented by a membership function, which gives the grade (or degree) of membership within the set, of every element of the universe of discourse.

Think of universe as a as set that contains every set in the context.

• A fuzzy set A may be represented as a set of ordered pairs:
  • $A = \{(x, \mu_A(x)); x \in X, \mu_A(x) \in [0,1]\}$
  • The membership function $\mu_A(x)$ maps the elements of the universe of discourse onto numerical values in the interval [0, 1].
• The membership function $\mu_A(x)$ represents the grade of possibility (not probability) that an element $x$ belongs to the fuzzy set $A$.
  • $\mu_A(x) = 1$ => x is definitely an element of A.
  • $\mu_A(x) = 0$ => x is definitely not an element of A.
  • $\mu_A(x) = 0.2$ => the possibility that x is an element of A is 0.2.

Symbolic Representation

If the universe of discourse is discrete with elements $x_i$, then a fuzzy set $A$ may be symbolically represented with a “formal series”

$$A=\mu_A\left(x_1\right) / x_1+\mu_A\left(x_2\right) / x_2+\cdots+\mu_A\left(x_n\right) / x_n=\sum_{x_i \in X} \mu_A\left(x_i\right) / x_i$$

If the universe of discourse is continuous, an equivalent form notation is given in terms of a “symbolic” integration

$$A=\int_{x_i \in X} \mu_A\left(x_i\right) / x_i$$

Note that $\sum$ and $\int$ do not represent summation or integration.

but rather the collection of members in discrete or continuous domain.

Discrete Universe

• E.g. a fuzzy representation of the integer 5:
  • Given by the Zadeh notation: $A = 0.2/3 + 0.3/4 + 1.0/5 + 0.2/6 + 0.1/7$

Triangular Membership Function

• +ve: Linear ⇒ Computationally inexpensive.

• -ve: Non-differentiable ⇒ May not be suitable to be used with gradient-descent optimization algorithms.

• A triangular membership function is characterized by 3 parameters {a, b, c}

  • a < b < c

Trapezoidal Membership Function

• +ve: Linear ⇒ Computationally inexpensive.

• -ve: Non-differentiable ⇒ May not be suitable to be used with gradient-descent optimization algorithms.

• A trapezoidal membership function is characterized by 4 parameters {a, b, c, d},

  • a < b < c < d

Gaussian Membership Function

• +ve: Differentiable ⇒ Suitable to be used with gradient-descent optimization algorithms.
• -ve: Nonlinear ⇒ Computationally expensive.
• A Gaussian membership function is characterized by 2 parameters {σ,c},
  • σ indicating the width
  • c being the center of the membership function.

Generalized Bell Membership Function

• +ve: Differentiable ⇒ Suitable to be used with gradient-descent optimization algorithms.
• -ve: Nonlinear ⇒ Computationally expensive.
• A Generalized Bell (or simply Bell) membership function is characterized by 3 parameters {a, b, c},
  • a and b defining the width and the slope,
  • and c being the center of the membership function,

Fuzzy Logic Operations

Assume two fuzzy sets $A$ and $B$ are given in universe $X$.

• the fuzzy complement $A'$
  • $\mu_{A'}(x) = 1 - \mu_A(x), x \in X$
• the fuzzy union $A\cup B$
  • $\mu_{A\cup B}(x) = \max(\mu_A(x), \mu_B(x)), \forall x \in X$
  • OR => max
• the fuzzy intersection $A\cap B$
  • $\mu_{A\cap B}(x) = \min(\mu_A(x), \mu_B(x)), \forall x \in X$
  • AND => min
• Generalized Fuzzy Complement

Fuzzy Complement

$\mu_{A'}(x) = 1 - \mu_A(x), x \in X$

Example: Discrete Universe: Suppose that the universe of discourse X is defined as X = {0, 7}.

Consider the fuzzy set A in this discrete universe given by:

$A = 0.2/3 + 0.3/4 + 1.0/5 + 0.2/6 + 0.1/7$

Then the complement of fuzzy set A is as follows:

$A' = 1.0/0 + 1.0/1 + 1.0/2 + 0.8/3+0.7/4 + 0.0/5 + 0.8/6 + 0.9/7$

Fuzzy Union

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

Example: Discrete Universe: Suppose that the universe of discourse X denotes the driving speeds in a highway. Consider the fuzzy sets A and B are given by:

Fuzzy state Slow (A): $A = 1.0/40 + 1.0/50 + 0.8/60 + 0.4/80 + 0.1/100$

Fuzzy state Moderate (B): $B = 0.6/50 + 0.8/60 + 1.0/70 + 1.0/80 + 1.0/90 + 1.0/100$

The fuzzy union of A and B are as follows:

$A \cup B = 1.0/40 + 1.0/50 + 0.8/60 + 1.0/70 + 1.0/80 + 1.0/90 + 1.0/100$

Fuzzy Intersection

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

Example: Discrete Universe: Suppose that the universe of discourse X denotes the driving speeds in a highway. Consider the fuzzy sets A and B are given by:

Fuzzy state Slow (A): $A = 1.0/40 + 1.0/50 + 0.8/60 + 0.4/80 + 0.1/100$

Fuzzy state Moderate (B): $B = 0.6/50 + 0.8/60 + 1.0/70 + 1.0/80 + 1.0/90 + 1.0/100$

##The fuzzy intersection of A and B are as follows:

$A \cap B = 0.6/50 + 0.8/60 + 0.4/80 + 0.1/100$

Generalized Fuzzy Complement

A generalized complement operation, denoted by C : [0, 1] → [0, 1], should satisfy the following axioms:

• Boundary conditions
  • $C(\emptyset) = X$ and $C(X) = \emptyset$, where $X$ is the universe of discourse
• Non-increasing:
  • If $\mu_A(x) < \mu_B(X)$ then $C(\mu_A (x)) \geq C(\mu_B (x))$ , and vice versa
• Involutive
  • $C(C(A)) = A$

T-norm (Generalized Intersection)

The intersection of two fuzzy sets A and B is given by an operation T which maps two membership functions to:

$\mu_{A\cap B} (x) = T(\mu_A(x), \mu_B(x))$

Let $a = \mu_A(x)$, $b = \mu_B(x)$.

The t-norm operation or generalized intersection may be represented by $T(a,b)$ or more commonly $aTb$.

T-norm Properties:

• non-decreasing
  • if $a \leq b$ and $c \leq d$ then $aTc \leq bTd$
• commutativity
  • $aTb = bTa$
• associativity
  • $(aTb)Tc = aT(bTc)$
• Boundary conditions
  • $aT1 = a$
  • $aT0 = 0$

Two general forms of T-norms are:

$$\begin{aligned} & 1-\min \left[1,\left((1-a)^p+(1-b)^p\right)^{1 / p}\right], p \geq 1 \\ & \max [0,(\lambda+1)(a+b-1)-\lambda a b], \lambda \geq-1 \end{aligned}$$

Four of the well known T-norm operators are:

$$\begin{array}{ll} \text { Min: } & T(a, b)=\min (a, b)=a \wedge b \\ \text { Algebraic product: } & T(a, b)=a \times b \\ \text { Bounded product: } & T(a, b)=0 \vee(a+b-1) \\ \text { Basic product: } & T(a, b)= \begin{cases}a & \text { if } b=1 \\ b & \text { if } a=1 \\ 0 & \text { if } a, b<1\end{cases} \end{array}$$

S-norm (Generalized Union)

$S(\cdot)$ is known as S-norm operator, or T-conorm operator

The union of two fuzzy sets A and B is given by an operation S which maps two membership functions to:

$\mu_{A\cup B} (x) = S(\mu_A(x), \mu_B(x))$

Let $a = \mu_A(x)$, $b = \mu_B(x)$.

According to DeMorgan’s Law, there exists a complementary S-norm associated to every T-norm, and vice versa.

$aSb = 1 − (1 − a)T (1 − b)$

• $aSb = 1 − \min[(1 − a), (1 − b)] = 1 − (1 − a) = a \quad \text{ if } a \geq b$
• $aSb = 1 − \min[(1 − a), (1 − b)] = 1 − (1 − b) = b \quad \text{ if } a < b$

$aTb = 1 − (1 − a)S(1 − b)$

S-norm Properties:

• non-decreasing
• commutativity
• associativity
• Boundary conditions

Two general forms of S-norms are:

$$\begin{aligned} & \min \left[1,\left(a^p+b^p\right)^{1 / p}\right], \quad p \geq 1 \\ & \min [1, a+b+\lambda a b], \quad \lambda \geq-1 \end{aligned}$$

Four of the well known S-norm operators are:

$$\begin{array}{ll} \text { Max: } & S(a, b)=\max (a, b)=a \vee b \\ \text { Algebraic sum: } & T(a, b)=a+b-a b \\ \text { Bounded sum: } & T(a, b)=1 \wedge(a+b) \\ \text { Basic sum: } & S(a, b)= \begin{cases}a & \text { if } b=0 \\ b & \text { if } a=0 \\ 1 & \text { if } a, b<1\end{cases} \end{array}$$

T-norm and S-norm Convertion

Example:

If the S-norm of two fuzzy variables $a$ and $b$ is given by

$$aSb = \min(1, a+b+\lambda ab) \quad \text{for } \lambda \geq -1$$

Then, using de Morgan law, determine its dual T-norm $(aTb)$.

Recall deMorgan’s law:

$$aSb = 1 − (1 − a)T(1 − b)\\ aTb = 1 − (1 − a)S(1 − b)$$

and note that

$$-max(a,b) = +min(-a,-b)\\ -min(a,b) = +max(-a,-b)$$

Therefore:

$$\begin{align} (1 − a)S(1 − b) & = \min[1, (1-a)+(1-b) + \lambda(1-a)(1-b)]\\ & = \min[1, 2-a-b + \lambda(1-a-b+ab)]\\ & = \min[1, 2-a-b + \lambda-\lambda a- \lambda b+ \lambda ab)]\\ & = \min[1, (2+\lambda) - (\lambda+1)a - (\lambda+1) b + \lambda ab] \end{align}$$

$$\begin{align} aTb & = 1 - (1 − a)S(1 − b)\\ & = 1 - \min[1, (2+\lambda) - (\lambda+1)a - (\lambda+1) b + \lambda ab]\\ & = 1 + \max[-1, -(2+\lambda) + (\lambda+1)a + (\lambda+1) b - \lambda ab]\\ & = \max[0, 1-(2+\lambda) + (\lambda+1)a + (\lambda+1) b - \lambda ab]\\ & = \max[0, -(\lambda+1) + (\lambda+1)a + (\lambda+1) b - \lambda ab]\\ & = \max[0, (\lambda+1)(a+b-1) - \lambda ab] \end{align}$$

Grade of Inclusion

In the context of fuzzy logic, a set may be partially included in another set.

$$\begin{equation} \begin{aligned} & \mu_{A \subset B}(x)=\left\{\begin{array}{cl} 1 & , \text { if } \mu_A(x)<\mu_B(x) \\ \mu_A(x) T \mu_B(x) & , \text { otherwise } \end{array}\right. \\ & \mu_{A \subseteq B}(x)=\left\{\begin{array}{cl} 1 & , \text { if } \mu_A(x) \leq \mu_B(x) \\ \mu_A(x) T \mu_B(x) & , \text { otherwise } \end{array}\right. \end{aligned} \end{equation}$$

Grade of Equality

fuzzy relation $A=B$

$$\begin{equation} \begin{aligned} & \mu_{A = B}(x)=\left\{\begin{array}{cl} 1 & , \text { if } \mu_A(x)=\mu_B(x) \\ \mu_A(x) T \mu_B(x) & , \text { otherwise } \end{array}\right. \\ \end{aligned} \end{equation}$$

Dilation and Contraction

Let A be a fuzzy set in the universe X with membership function $\mu_A$.

Dilation

• K-th dilation = $\mu_A^{1/K}(x)$

Example : Consider the following fuzzy set:

$A = 0.0/1 + 0.5/2 + 1.0/3 + 0.5/4 + 0.0/5$

The second dilation of fuzzy set A is given as follows:

$A^{1/2} = 0.0^{1/2}/1 + 0.5^{1/2}/2 + 1.0^{1/2}/3 + 0.5^{1/2}/4 + 0.0^{1/2}/5$

$= 0.0/1 + 0.7071/2 + 1.0/3 + 0.7071/4 + 0.0/5$

Contraction:

• K-th contraction = $\mu_A^{K}(x)$

Example : Consider the following fuzzy set:

$A = 0.0/1 + 0.5/2 + 1.0/3 + 0.5/4 + 0.0/5$

Contraction of fuzzy set A is given as follows:

$A^2 = 0.0^2/1 + 0.5^2/2 + 1.0^2/3 + 0.5^2/4 + 0.0^2/5$

$= 0.0/1 + 0.25/2 + 1.0/3 + 0.25/4 + 0.0/5$

Implication (if-then)

Consider two fuzzy sets: A in X and B in Y .

The fuzzy implication A → B is a fuzzy relation in the Cartesian product X × Y defined by:

Larsen implication:

$$\underset{\forall(x, y) \in(X \times Y)}{\mu_{A \rightarrow B}(x, y)}=\mu_A(x) \mu_B(y)$$

Mamdai implication:

$$\underset{\forall(x, y) \in(X \times Y)}{\mu_{A \rightarrow B}(x, y)}=\min \left[\mu_A(x), \mu_B(y)\right]$$

Zadeh implication:

$$\underset{\forall(x, y) \in(X \times Y)}{\mu_{A \rightarrow B}(x, y)}=\max \left[\min \left\{\mu_A(x), \mu_B(y)\right\}, 1-\mu_A(x)\right]$$

Dienes-Rascher implication:

$$\underset{\forall(x, y) \in(X \times Y)}{\mu_{A \rightarrow B}(x, y)}=\max \left[1-\mu_A(x), \mu_B(y)\right]$$

Lukasiewicz implication:

$$\underset{\forall(x, y) \in(X \times Y)}{\mu_{A \rightarrow B}(x, y)}=\min \left[1,1-\mu_A(x)+\mu_B(y)\right]$$

Height of a Fuzzy Set

The height (also called modal grade) of a fuzzy set is the maximum value of its membership function.

$hgt(A) = \underset{x\in X}{\sup}\mu_A((x))$

also called the modal element value, or simply modal point.

Support Set of a Fuzzy Set

A crisp set containing all the elements (in the universe) whose membership grades are greater than zero.

$S = \{x \in X | \mu_A(x) > 0\}$

Alpha-cut of a Fuzzy Set

The α-cut of a fuzzy set A is the crisp set denoted by $A_α$ formed by the elements of A whose membership function grades are greater than or equal to a specified threshold value α ∈ [0, 1].

$A_a = \{x \in X | \mu_A(x) \geq \alpha\}, \alpha \in [0,1]$

Three Measures of Fuzziness

A measure of fuzziness of a fuzzy set A in the universe X defines the closeness of its membership function $μ_A$ to the most fuzzy grade (0.5).

• Closeness to grade 0.5 ($A_1$),
  • $A_1 = \int_{x\in S} f(x)dx$
    • where
      • $S$ denotes the support set
      • $f(x) = \mu_A(x)$ for $\mu_A(x) \leq 0.5$
      • $f(x) = 1 - \mu_A(x)$ otherwise
• Distance from 1/2−cut ($A_2$),
  • $A_2 = \int_{x \in X} |\mu_A(x) - \mu_{A_{1/2}}(x)|dx$
• Inverse of distance from the complement ($A_3$).
  • $A_3 = 2 \int_{x\in X} |\mu_A(x) - 0.5|dx$
    • can be derived to $\int_{x\in X} | \mu_A(x) - \mu_{\bar{A}}(x)|dx$

Relationship Between Different Measures of Fuzziness

$$A_1 = A_2 = \frac{1}{2}(S *1 - A_3)$$

Relations

For boolean relation in crisp sets and binary logic:

• the relation is equivalent to the logical expression (proposition)
• Can be proved using truth table

E.g. Consider a fuzzy logic proposition $A$ with membership function $μ_A$.

• $\mu_A$ may be taken to represent the degree of validity of the relation $A$ is True.
• $1 - \mu_A$ represents the degree of validity of the fuzzy relation $A$ is NOT True.

Cartesian Product of Fuzzy Sets

Consider a fuzzy set $A_1$ defined on the universe $X_1$ and a second fuzzy set $A_2$ defined on a different (independent, orthogonal) universe $X_2$.

• The Cartesian product $A_1 × A_2$ is the fuzzy subset of the Cartesian space $X_1 × X_2$. Its membership function is given by:
  • $\mu_{(A_1 \times A_2)}(x_1, x_2) = \min[\mu_{A_1}(x_1), \mu_{A_2}(x_2)], \forall(x_1, x_2) \in (X_1 \times X_2)$
  • note that it is a “AND” operation

Extension Principle

Consider a mapping function $f: X \rightarrow Y$
Let $A$ be a fuzzy set on $X$

$$A=\frac{\mu_A\left(x_1\right)}{x_1}+\frac{\mu_A\left(x_2\right)}{x_2}+\cdots+\frac{\mu_A\left(x_n\right)}{x_n}$$

The Extension Principle states that the image of $A$ under mapping $f(\cdot)$ is expressed on $Y$

$$\begin{aligned} B=f(A) & =\frac{\mu_A\left(x_1\right)}{y_1}+\frac{\mu_A\left(x_2\right)}{y_2}+\cdots+\frac{\mu_A\left(x_n\right)}{y_n}, \text { where } \\ y_1 & =f\left(x_1\right), y_2=f\left(x_2\right), \ldots, y_n=f\left(x_n\right) \end{aligned}$$

presuming that the function $f$ is a one-to-one mapping.
If $f$ is a many-to-one mapping, i.e., $\exists x_1 \neq x_2$ such that $y_1=f\left(x_1\right)=f\left(x_2\right)=y_2$, Then $\mu_B(y)=\max \left[\mu_A\left(x_1\right), \mu_A\left(x_2\right)\right]$.

Example:

$$\begin{equation} \begin{aligned} A & =\frac{0.1}{-2}+\frac{0.4}{-1}+\frac{0.8}{0}+\frac{0.9}{1}+\frac{0.3}{2} \\ f & : x \mapsto x^2-3 \\ B & =\frac{0.1}{1}+\frac{0.4}{-2}+\frac{0.8}{-3}+\frac{0.9}{-2}+\frac{0.3}{1} \\ & =\frac{(0.1 \vee 0.3)}{1}+\frac{(0.4 \vee 0.9)}{-2}+\frac{(0.8)}{-3} \\ & =\frac{0.3}{1}+\frac{0.9}{-2}+\frac{0.8}{-3} \end{aligned} \end{equation}$$

Projection

Given the relation $R$ defined by $R=\int_{X \times Y} \frac{\mu_R(x, y)}{(x, y)}$
The first projection is a fuzzy set that results by eliminating the second set $Y$ of $X \times Y$ by projecting the relation on $X$.

Projection 1:

$$R_1=\int_X \frac{\mu_{R_{\mathbf{1}}}(x)}{x}, \quad \quad \mu_{R_{\mathbf{1}}}(x)=\bigvee_y \max \left[\mu_R(x, y)\right]$$

The second projection is a fuzzy set that results by eliminating the first set $X$ of $X \times Y$ by projecting the relation on $Y$.

Projection 2:

$$R_2=\int_Y \frac{\mu_{R_2}(y)}{y}, \quad \quad \mu_{R_2}(y)=\bigvee_x \max \left[\mu_R(x, y)\right]$$

The total projection is a combined projection over the spaces $X$ and $Y$ and is defined by

$$\mu_{R_T}(x, y)=\bigvee_x \bigvee_y \max \left[\mu_R(x, y)\right]=\bigvee_y \bigvee_x \max \left[\mu_R(x, y)\right]$$

Cylindrical Extension

• Basically extend 1D projection into 2D projection by repeating the axis along another axis

Projection Example

Example for Projection in discrete cases:

Given:

$$\underset{y}{R=\left[\begin{array}{cccccc} 0.1 & 0.2 & 0.4 & 0.8 & 1.0 & 0.6 \\ 0.2 & 0.4 & 0.8 & 0.9 & 0.8 & 0.6 \\ 0.5 & 0.9 & 1.0 & 0.8 & 0.4 & 0.2 \end{array}\right]x}\\$$

Then the first projection $R_1$, second projection $R_2$ and total projection $R_T$

$$\begin{equation} \begin{aligned} & R = \frac{0.1}{(x_1, y_1)} + \cdots + \frac{0.6}{(x_1, y_6)} + \frac{0.2}{(x_2, y_1)}+ \cdots + \frac{0.6}{(x_2, y_6)}+ \frac{0.5}{(x_3, y_1)}+ \cdots + \frac{0.2}{(x_3, y_6)} \\ & R^1=\sum_i \frac{\mu_{R^1}\left(x_i\right)}{x_i}=\frac{1.0}{x_1}+\frac{0.9}{x_2}+\frac{1.0}{x_3}=\left[\begin{array}{l} 1.0 \\ 0.9 \\ 1.0 \end{array}\right] \\ & R^2=\sum_j \frac{\mu_{R^2}\left(y_j\right)}{y_j}=\frac{0.5}{y_1}+\frac{0.9}{y_2}+\frac{1.0}{y_3}+\frac{0.9}{y_4}+\frac{1.0}{y_5}+\frac{0.6}{y_6} =\left[\begin{array}{llllll} 0.5 & 0.9 & 1.0 & 0.9 & 1.0 & 0.6 \end{array}\right]^T \\ & R^T=1 \end{aligned} \end{equation}$$

Then Cylindrical Extension:

$$\begin{equation} \begin{aligned} & C\left(R^1\right)=\left[\begin{array}{llllll} 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\ 0.9 & 0.9 & 0.9 & 0.9 & 0.9 & 0.9 \\ 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \end{array}\right] \\ & C\left(R^2\right)=\left[\begin{array}{llllll} 0.5 & 0.9 & 1.0 & 0.9 & 1.0 & 0.6 \\ 0.5 & 0.9 & 1.0 & 0.9 & 1.0 & 0.6 \\ 0.5 & 0.9 & 1.0 & 0.9 & 1.0 & 0.6 \end{array}\right] \end{aligned} \end{equation}$$

Compositional Rule of Inference

A typical fuzzy logic system’s knowledge base is composed of a series of if-then rules.

• Each rule is a fuzzy relation employing the (if-then) fuzzy implication.
• When all individual rules (relations) are aggregated (executed), they result in one fuzzy relation represented by a fuzzy set, say K, and a multi-variable membership function.
• In a fuzzy decision making process, the rule base K is first collectively matched with the available data.

Suppose that the available data (context) is denoted by the fuzzy set (or relation) $D$ and the inference (action) is denoted by a fuzzy set (or relation) $I$.

• Then, the compositional rule of inference states that

$$I=D \circ K, \quad \quad \circ: \text { ``composed with" }$$

• The membership function of the inference I (also called decision, or action) is determined as

$$\mu_I=\sup _y\left\{\min \left[\mu_D, \mu_K\right]\right\}$$

where the inference $I$ is the output of the knowledge base decision making system.

• This is known as the sup-min (or max-min) composition.
• Another method to compute the inference $I$ is through the sup-product composition.

$$\mu_I=\sup _y\left\{\mu_D \cdot \mu_K\right\}$$

Fuzzy Reasoning

Fuzzy reasoning, also known as approximate reasoning (AR), is an inference procedure that derives conclusions from a set of if-then rules.

• Fuzzy reasoning is basically the extension of the well known composition of elements and functions.

• Let us assume that F is a fuzzy relation on $X \times Y$.
• Let $A$ be a fuzzy set in $X$.
• To obtain the resulting fuzzy set B, we first construct a cylindrical extension of A, C(A).
• The inference of C(A) and F leads to the antecedent of the projection of C(A)∩F.
• This leads to the fuzzy set B.
• C(A)∩F(x,y) is known as the composition operator $R_1 \circ R_2$
  • Where $C(A) = R_1$, $\circ$ is the generalized operator ($\cap$ In this case), $R_2 = F(x,y)$

Modus Ponens (MP)

• The rule of inference in conventional logic is modus ponens.
• Modus Ponens leads to inference of truth of a proposition B from the truth of A and the implementation A → B.

Case 1: Single Rule with Single Antecedent

Rule: if $x$ is $A$ then $y$ is $B$

Fact: x is $A′$

Conclusion: $y$ is $B'$

Relation:

$$R: A \rightarrow B$$

Basically:

$$B' = A' \circ R = A' \circ (A \rightarrow B)$$

In detail (in this case):

$$B' = A' \wedge A \wedge B$$

$$\mu_{B'}(y) = \bigvee_x [\mu_{A'}(x) \wedge \mu_A (x)] \wedge \mu_B(y)$$

where

• $\bigvee_x [\mu_{A'}(x) \wedge \mu_A (x)]$ is the degree of validity $w$

Case 2: Single Rule with Multiple Antecedents

Rule: If $x$ is $A$ and $y$ is $B$, then $z$ is $C$

Fact: $x$ is $A'$ and $y$ is $B'$,

Conclusion: $z$ is $C'$

Relation:

$$R: A \times B \rightarrow C \\ = R: A \times B \times C$$

$$R = \int_{X\times Y \times Z} \frac{\mu_A(x) \wedge \mu_B(y) \wedge \mu_C(z)}{(x,y,z)}$$

Basically:

$$C' = (A' \times B') \circ (A \times B \rightarrow C)$$

In detail (in this case):

$$C' = [A' \wedge A \wedge B' \wedge B] \wedge C$$

$$\begin{align} \mu_{C'}(z) &= \bigvee_{x,y} [\mu_{A'}(x) \wedge \mu_{B'}(y)] \wedge [\mu_{A}(x) \wedge \mu_{B}(y) \wedge \mu_C(z)]\\ &= \bigvee_{x,y} [\mu_{A'}(x) \wedge \mu_{B'}(y) \wedge \mu_{A}(x) \wedge \mu_{B}(y)] \wedge \mu_C(z)\\ &= \bigvee_{x,y} [\mu_{A'}(x) \wedge \mu_{A}(x)] \wedge \bigvee_{x,y}[\mu_{B'}(y) \wedge \mu_{B}(y)] \wedge \mu_C(z) \end{align}$$

where

• $\bigvee_{x,y} [\mu_{A'}(x) \wedge \mu_{A}(x)]$ is the degree of validity of $x : w_1$
• $\bigvee_{x,y}[\mu_{B'}(y) \wedge \mu_{B}(y)]$ is the degree of validity of $y : w_2$
• $\bigvee_{x,y} [\mu_{A'}(x) \wedge \mu_{A}(x)] \wedge \bigvee_{x,y}[\mu_{B'}(y) \wedge \mu_{B}(y)]$ is the firing strength

Case 3: Multiple Rules with Multiple Antecedents

Rules:

• If $x$ is $A_1$ and $y$ is $B_1$, then $z$ is $C_1$
• Else if $x$ is $A_2$ and $y$ is $B_2$, then $z$ is $C_2$

Fact: $x$ is $A'$ and $y$ is $B'$,

Conclusion: $z$ is $C'$

Relation:

$$R_1 : A_1 \times B_1 \rightarrow C_1 \\ R_2 : A_2 \times B_2 \rightarrow C_2$$

Basically:

$$C' = (A' \times B') \circ (R_1 \cup R_2) = [(A' \times B') \circ R_1] \cup [(A' \times B') \circ R_2] \\ = C'_1 \cup C'_2 \quad \text{where } C'_i \text{ is the inferred consequent of rule i}$$

In detail (in this case):

as shown previously for $C'_i = (A' \times B') \circ R_i$

$$\mu_{C_i'}(z) = \bigvee_x [\mu_{A'}(x) \wedge \mu_{A_i}(x)] \wedge \bigvee_{y}[\mu_{B'}(y) \wedge \mu_{B_i}(y)] \wedge \mu_{C_i}(z)$$

Fuzzy Inference System (FIS)

Fuzzy input => [Rules] => Fuzzy output

Fuzzification

• Fuzzification refers to the representation of a crisp value by a membership function.
• It is needed prior to applying the composition (CRI), when the data (measurements) are crisp values, as common in control applications.
• It may be argued that the process of fuzzification amounts to giving up the accuracy of crisp data.

Discrete case of fuzzification

Continuous case of fuzzification

• Singleton method
• Triangular function method
• Gaussian function method

Defuzzification

• Usually, the decision (control action) of a fuzzy logic controller is a fuzzy value and is represented by a membership function.
• Because low-level control actions are typically crisp, the control inference must be defuzzified for physical purposes such as actuation.

Methods of defuzzification

• Centroid method
• Mean of maxima method
• Threshold methods

Fuzzy Models (Different Inferencing Systems)

Different inferencing procedures have been used in the literature.

The main difference among them is the aggregation and the defuzzification.

Mamdani fuzzy model

• Output is the maximum

Sugeno fuzzy model

• Output is the average or weighted average