Hyers-Ulam Stability of Quantum Logic Fuzzy Implication
IQBAL H. JEBRIL, NAJAT M. ABDELQADER
Department of Mathematics,
Al-Zaytoonah University of Jordan,
Queen Alia Airport St. 594 11942, Amman,
JORDAN
Abstract: - There are four different types of fuzzy implications in fuzzy logic, referred to as (S, N)-
Implications, R-Implications, QL-Implications, and D-Implications. Only one research work on the
implications of two functional equations for the Hyers-Ulam stability for (S, N) has been published recently.
The Hyers-Ulam stability of two functional equations for QL-Implication is being examined in this study.
Keywords: - Fuzzy implication, QL-Implication, Stability.
Received: May 21, 2022. Revised: February 15, 2023. Accepted: March 23, 2023. Published: April 28, 2023.
1 Introduction
In 1964, the problem of the stability of the
functional equations has been presented to Ulam. He
asked if it is potential to find near a function that
satisfies the equation precisely (S. M. Ulam, 1964)).
In 1941, Gerhard Hyers presented a partial solution,
also in 1978, Song and Rassias established the
Ulam-Hyers-Rassias stability problem of the
functional equation. This problem was named as
Ulam-Hyers-Rassias stability problem. Many
studies have looked into the problems of stability of
several equations. The stability of classical
functional equations including the additive
mappings, quadratic equations, cubic equations in
the fuzzy normed space, and various differential
fuzzy equations was also investigated. In these
equations, however, there is no fuzzy implication,
[5], [8], [10], [11]
The first study on the stability of two functional
equations for fuzzy implication, [13], was finished
in the year 2020, although this study only looks at
the first type of fuzzy implication (S, N)-
implication. Studying this equation's stability for the
QL-Implication problem that this research is trying
to solve is fascinating, [7].
This study examines fuzzy implications that come
near to, but don't completely, satisfy these
equations. The Hyers-Ulam stability of the QL-
implication iterative functional equations is then
established. But, if there is a fuzzy negation N, t-
norm, and t-conorm, we declare that the implication
is QL-Implication as follows:
, [3], [4].
We would examine the Hyers-Ulam stability of the
following two functional equations for QL-
Implication in our work:
1. Study the stability of Hyers-Ulam of the
derived Boolean law which formulated in fuzzy
logic for -
implications.
2. Study the stability of Hyers-Ulam of the
importation law which is formulated in fuzzy
logic for-
implications.
2 Preliminaries
Definition 2.1 [2], a function
which forms a triangular norm (in short, t-norm), if
T is commutative, increasing, associative, and has 1
as identity.
Definition 2.2 [2], a function
which forms a triangular conorm (in short, t-
conorm), if S is commutative, increasing,
associative, and 0 is its identity.
Definition 2.3. [12], a function
which is defined as a negation function, if:
1. , ;
2. , if . .
Proposition 2.1. [12], for: t-norm, t-conorm
and strong negation so, is -dual of when:
, ,
And is -dual of when
Definition 2.4. [1], [6], a function
is a fuzzy implication if , the
following conditions were fulfilled:
WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS
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(I1) and .
(I2) .
(I3) .
In fuzzy logic, where quantum mechanics is
employed as a detriment to conventional reasoning,
QL-implications are presented by analogy. The
extension of the quantum logic implication is now
referred to as a QL operation.
The abbreviation for a quantum logic fuzzy
implication is QL-implication. Using inspiration
from the classical logic equivalence, fuzzy negation,
a t-norm, and a t-conorm provide a QL-implication.
, .
Definition 2.5. [6], let is a t-norm, is a t-conorm
and is fuzzy negation. The QL-implication can be
defined as:
, .
QL-implication created by t-norm, a t-conorm
and fuzzy negation can be denoted by.
3 Stability
The Boolean law and the law of importation are
tautologies in classical logic, but they are two
fundamental features of fuzzy logic. Answers to
their solution have surfaced in recent years, [9]. The
stability of the fuzzy functional equation with fuzzy
implications has not yet been discovered, though.
In response to Ulam's query from 1940, researchers
are looking at stability issues as they pertain to
functional equations (S.M.Ulam, 1964), he
suggested the stability question mentioned below:
Let are 2 groups, D (·, ·) is a metric on .
A number ɛ > 0 is given, and there is a δ > 0 as if a
mapping fulfilled
For all α, β
K1,
So, there is a group homomorphism
and:
For all α
K1 .
When the reply is in the affirmative, the equation h
(αβ) = h (α) h (β) of the homomorphism is then
named stable.
In another formulation, the homomorphism equation
is said to be stable if and only if all the
approximations can be made using this equation's
solution.
Hyers originally proposed a solution to the Ulams
puzzle in 1941, and he established the following
theorem:
Theorem 3.1 [13], let is a function
between the 2 Banach spaces K1 and K2 as
for some ɛ ≥ 0,
for every α, β K1. There is only a unique function
satisfying , and
for any α, β
.
Because of the Ulam question and answer of Hyers,
that type of stability is named Hyers-Ulam stability.
In 2020, [13], explored the law of importation for
(S, N)-an implication that is the first kind of fuzzy
implication and Hyers Ulam stability for Boolean
law, [13]. They look at hazy implications that,
although not quite fitting these equations, come
near.
3.1 The Study of Hyers-Ulam Stability for
Quantum Logic
There are four different kinds of fuzzy implication
(S, N), as well as R, QL, D-implication, and
functional equations as attributes. Recently, the
stability of these functional equations for (S, N)
implication was explored. We attempt to prove the
Hyers-Ulam stability of two functional equations for
QL-implication, a different kind of fuzzy
implication, in the present section, [13].
Fuzzy implication properties come in many different
forms, including identification, importation law,
exchange principle, and others. In fuzzy thinking,
these qualities are crucial, [4]. Earlier works on
functional equations have always been solutions-
oriented.
(1)
is referred to as a fuzzy implication in what is
known as derived Boolean law. There have been
several types of research regarding the solution of
this functional equation for various implications
since Shi and his associates identified the solution
for equation (1) for various types of fuzzy
implications, [4].
(2)
In equation (2), which is frequently referred to as
the importation law, stands for a fuzzy implication
and stands for the t-norm. Jayaram clarified the
importation law’s resolution for several murky
consequences, [3]. Many investigations that
followed concentrated on solving equation (2) with
various implications.
Theorem 3.2 [13], let 1 2 is a function
between the two Banach spaces as:
(3)
For some of , all , there is only a
unique function satisfying
(4)
(5)
There are numerous requirements for the stability of
several functional equations, even though
difficulties with the stability of the functional
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equations with fuzzy implications have received
little attention.
The current portion introduces the study of Hyers-
Ulam stability for equations (1) and (2) for QL-
implication.
Many studies on stability exist, including one that
examines the stability of conventional functional
equations, [9].
Yet, these equations do not have any fuzzy
implications. 2020 saw the study of Hyers-Ulam
stability for two functional equations, [13].
Finding a stable QL-implication is what we're
aiming for, so in other expressions when we take a
fuzzy implication satisfying the inequality.
(6)
We try finding a mapping
fulfilling
(1) is known as a QL-implication.
(2)
(3)
(4) Is the unique QL-implication satisfying (2),
(3), is very small.
In this work, we use the minimum t-
norm , t-conorm
Minimum TM is a t-norm that is the strongest and
is the weakest t-conorm
Definition 3.1 [12], for a t-norm T, t-conorm and
strong negation N then S is N-dual of T when
and T is N-dual of,
If (
Definition 3.2 [1], let T be a t-norm, is a t-
conorm and N is a fuzzy negation. QL-implication
can be defined by:
Lemma 3.1 When N is defined as a continuous
fuzzy negation. [13], for all
as If any function fulfilling the functional
equation is very close to a real solution for the
functional equations, it is defined as stable. For QL-
implication, we will discuss two functional
equations and their Hyers-Ulam stability.
3.2 Stability of Boolean Law
We provide our latest finding about the stability of
two functional equations for QL-implication
throughout the following section.
Theorem 3.3 [4], let I (the family of fuzzy
implication) be an defined using
a continuous negation N, a t-norm T, and a t-
conorm, then I fulfills derived Boolean law only if
for any [0,1]. Here, we can introduce
the solution to the problem of stability of iterative
functional equations
for QL-implication.
Theorem 3.4 Let I be an
defined using a continuous negation N and at-norm
T and a t-conorm when for some ɛ> 0, I fulfills
the inequality (6), there is a Q
satisfying Equation 1 and
|Q (p, q) − I (p, q)| ≤ ɛ, for all
(7)
Proof
(1) Let then Q is
a QL-implication satisfying Equation 1.
(2) Now we prove the Inequality (7). Let ,
then
and , so
, (8)
p [0, 1].
For any p, q [0, 1], if , then by
Using Eq. (8)
If , then
Thus we have
,
for any p, ɤ [0, 1]. Moreover
(9)
for any [0, 1]. As N is defined as a continuous
negation, the N range becomes [0,1]. So, the
equation mentioned above can be rewritten in the
following form:
p, ɤ [0, 1].
(10)
So, ɛ for any p, γ [0, 1].
(1) However isn’t unique.
Let:
(p) = (1+ɛ). N (p) 1.
For all p [0, 1], N1 (p) remains a continuous
negation by previous lemma. Clearly,
Satisfies Equation (1).
Also, we have
From
and
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≤ 1
So,
for any [0,1].
Combined with
ɛ for any [0,1].
We obtain
for any [0,1].
Thus
| ɛ, for any [0,1].
3.3 Stability of Law of Importation
To find the stability of the law of importation
, we will count the
problem mentioned below for the case of minimum
t-norm, i.e.
(11)
Same to Ulam’s question, here we get the problem
below:
A fuzzy implication is given which fulfills
inequality (11)
(12)
for all , [0, 1], we try finding a mapping Q:
[0, 1]2 → [0, 1] fulfilling:
(1) Q is a ;
(2)
(3)
(4) Q is the unique
satisfying (2) and (3).
The error δ is defined as a real positive number. It
has to be very small.
Theorem 3.5. Let I is a QL-implication which is
defined by the strong negation N and a t-conorm S,
so it fulfills Eq. (11) with t-norm T only when T
=TM
Theorem 3.6 Let I is a QL-implication which is
defined using the strong negation N and a t-norm T
and a t-conorm. when for some ɛ > 0, I satisfy
inequality (12), so there is a QL-implication Q
fulfilling Eq. (11) and
for any p, q[0,1]. (13)
Proof.
(i) Let , then Q
is a QL-implication satisfying Equation (11)
(ii) Now we prove Equation (13). Let σ = 0,
and .
=
=
= = .
Thus,
We have
As N is continuous, the range of N remains [0,1].
After that, the equation mentioned above can be
rewritten as follows:
for
any ].
So, then we have
For any
For all the strong negation N, there is a new strong
negation N1 as:
1 ɛ for all p [0, 1].
So, the (S, N)-implication in the above theorem isn’t
unique.
Let , then we
have, for any p [0, 1].
1
0 1
and we have
from ɛ. Then we obtain
for any p [0,1].
Then is a new
QL-implication fulfilling Eq. (11) and (13) and this
shows uniqueness.
4 Conclusion and Future Work
In this work, we attempted to demonstrate the
Hyers-Ulam stability of two functional equations for
QL-implications with N strong continuous negation,
T the lowest t-norm, and S the maximum t-conorm.
The other QL-implication Q is close to I with a tiny
inaccuracy and satisfies these equations if the two
functional equations hold. This indicates that there
exists a solution for equations (1) and (2) under
sufficient constraints on the functions involved in
"near" any solution of the inequality (6). (2). There
are more functional equations and fuzzy
implications for future investigation. It would be
necessary to provide more information and talk
about the stability issue with those equations.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-Iqbal H. Jebril, carried out the evaluation,
structures and wrote the paper.
-Najat M. Abdelqader, contribute in the results
evaluation.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The authors would like to thank the Al-Zaytoonah
University for providing the necessary scientific
research supplies to implement the research.
Conflict of Interest
The authors have no conflict of interest to declare
that is relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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