Page 70 - IJOCTA-15-4

B. Shiri et.al. / IJOCTA, Vol.15, No.4, pp.610-624 (2025)
Table 1. Table of mathematical notations 2. What is a fuzzy number?

Notation Description There is no standardized definition of a fuzzy
21,22
α
∇ x(t) Fractional nabla difference number. Several authors identify two types
of definitions relevant to interval analysis: those
of order α
∇ −α x(t) Fractional nabla sum of order α featuring continuous membership functions and
those with semi-continuous membership func-
⃗ α Vector of fractional orders: 21
[α 1 , . . . , α ν ] T tions. According to the critiques in Ref. , a
trapezoidal membership function does not qual-
⃗x(t) Vector of variables:
[x 1 (t), . . . , x ν (t)] T ify as a fuzzy number. It assigns multiple values
without associated uncertainty. Thereby, it rep-
W Weight matrix of size ν × ν
resents a set rather than a number.
⃗ p Bias vector
A key condition for a fuzzy set to qualify as
⃗q Vector of activation functions: a fuzzy number is that its membership function
[q 1 , . . . , q ν ] T
is normal. That is, there must exist a unique el-
⃗ ⊕ Element-wise fuzzy sum
ement for which the membership function value
⃗ ⊖ Element-wise H-difference
is equal to 1. Significantly, this normal element
(d, f 1 , f 2 ) Triple representation
must be unique. This uniqueness specifically pre-
of a fuzzy number
vents the fuzzy number from taking the form
C µ (r) r-cut of the fuzzy number µ
of a trapezoidal membership function. A trape-
Γ(·) Gamma function zoidal function has multiple elements with con-
sgn(·) Sign function stant membership values over an interval, violat-
ing the uniqueness of the normal component. As
a result, the classical definitions of a fuzzy number
Table 2. Table of notations for spaces
are refined in the following manner:
Notation Description
Definition 1. A membership function µ : R →
R Set of real numbers
[0, 1] is a fuzzy number (µ ∈ R F ), if:
Set of natural numbers
N a
Normal:: There exists a unique d ∈ R such that
starting from a
µ(d) = 1.
R F Set of fuzzy numbers
+ Compact support:: Supp(µ) = {w : µ(w) > 0}
U Set of functions f : [0, 1] → R
is compact.
that are m.d.c. with f(1) = 0
Convexity: µ(x) is a convex down function
S Set of shape functions
(concave function) that means
with additional conditions
on f(0) µ(tw 1 + (1 − t)w 2 ) ≥ tµ(w 1 ) + (1 − t)µ(w 2 )
−1
S Set of inverse shape functions for all w 1 , w 2 ∈ R and t ∈ [0, 1].
Upper continuous: µ is right continuous on
(−∞, d) and left continuous on (d, ∞).
Table 3. Table of abbreviations The r-cut set of a fuzzy number is defined as
C µ (r) = {w|µ(w) ≥ r}, r ∈ [0, 1].
Abbreviation Description
IFDSs Incommensurate fuzzy The boundaries of these sets are defined by
∗
fractional nabla C (r) = sup C µ (r),
µ
difference systems
and
RNN Recurrent Neural Network
C µ∗ (r) = inf C µ (r).
NN Neural Network
H-difference Hukuhara Difference Theorem 1. Let µ be a fuzzy number. Then,
GH-difference Generalized Hukuhara
(1) d ∈ C µ (r) and C µ (r) ̸= ∅ for all r ∈ [0, 1].
Difference
(2) ∃ d ∈ R such that C µ (1) = {d}.
m.d.c. Monotonically Decreasing (3) C µ (r 2 ) ⊆ C µ (r 1 ) for r 1 ≤ r 2 .
and Continuous
(4) C µ (r) is a closed and bounded interval
m.i.c. Monotonically Increasing
and Continuous for all r ∈ [0, 1], particularly C µ (r) =
∗
[C µ∗ (r), C (r)].
SGD Stochastic Gradient Descent µ ∗
(5) C µ∗ : [0, 1] → R and C : [0, 1] → R are
µ
well-defined functions.
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