Approximation of the functional equation I ( x,I ( x,y )) = I ( x,y )

Abstract

The well-known iterative boolean-like law a →(a → b) = a → b can be generalized to the functional equation I(x, I(x, y)) = I(x, y), where I is a fuzzy implication. In this paper, we discuss an approximation of the equation, I(x, I(x, y)) ≈ I(x, y), i.e., the law is approximately valid. Furthermore, we study the property of approximation preserving with respect to compositions of fuzzy implications. Finally, we give a necessary condition and a sufficient condition for the approximate equation of (S, N)-implications.

Keywords

Functional equation iterative boolean-like law fuzzy implication

1 Introduction

Functional equations involving fuzzy implications play an important role in fuzzy logics [1, 2]. They are generalizations of Boolean laws involving classical implications. They are called Boolean-like laws in fuzzy logics. So it is important to investigate whether Boolean-like laws remain valid. Several Boolean-like laws have been investigated.

Alsina and Trillas [3] studied some standard iterative boolean-like laws derived from the set theoretic laws, such as (A∪ A) ∩(A ∩ A) ^C = ∅, A ∪ B =(A ∩ B) ∪ [(A ∪ B) ∩(A ∩ B) ^C] and (A ∪ A ∪ B) ∪(A ∩ B ∩ B) = A ∪ B.

Shi et al. [4] considered the derived boolean law a →(a → b) = a → b. They gave the conditions of it holds for S-, R- and QL-implications. Xie and Qin [5, 6] investigated under which conditions of it holds for a continuous D-operation and several implications derived from uninorms. Massanet and Torrens [7] also investigated this law for D-operations.

Jayaram [1] studied the law of importation (a ∧ b) → c = a →(b → c) for R-, S-, QL-, g- and f-implications. Mas et al. [10, 11] studied this law for fuzzy implications derived from some discrete t-norms, t-conorms and uninorms. Massanet et al. [12–14] studied this law for some fuzzy implications with a fixed t-norm (or uninorm). Zhou et al. [15, 16] studied this law for fuzzy implications generated by continuous multiplicative generators of t-norms. Li and Qin [17] characterized fuzzy implications satisfying this law with respect to uninorms with continuous underlying operators.

Cruz et al. [18] exposed the conditions under which the generalized Frege’s Law (a ∧ b) → c =(a → b) →(a → c) holds for the (S,N)-implications (R-, QL-, D-, (T,N)-, H-, respectively). Peng et al. [19] gave the conditions under which this law holds for the (U,N)-implications (f-, g-, T-Power based implications, respectively).

Cruz et al. [20, 21] exposed the conditions under which the Boolean-like law a →(b → a) =1 holds for the (S,N)-implications (R-, QL-, D-, (N, T), respectively).

The exchange principle a →(b → c) = b →(a → c) of fuzzy implications as a particular generalization of boolean law has been investigated in [9 , 30]. The relationships of the exchange principle and the law of importation are studied in [31].

Trillas and Alsina [23] studied the law p ∧ q → r =(p → r) ∨(q → r). This law and three laws p ∨ q → r =(p → r) ∧(q → r), p →(q ∧ r) =(p → q) ∧(p → r) and p →(q ∨ r) =(p → q) ∨(p → r) are four basic distributivity equations of fuzzy implications. Balasubramaniam and Rao [24] studied the distributivity of fuzzy implications over t-norms and t-conorms. Qin et al. [25] studied the distributive equations of fuzzy implications based on continuous t-norms. Baczynski and Jayaram [26] studied the distributivity of fuzzy implications over nilpotent or strict t-conorms. Xie et al. [27] studied the distributive equations of fuzzy implications based on continuous t-conorms given as ordinal sums. Su et al. [28] studied the distributivity of ordinal sum implications over t-norms and t-conorms. Su and Hu [28] studied the distributivity of fuzzy implications over additively generated overlap and grouping functions.

It is worth noting that the aforementioned boolean-like laws are required to be exactly valid for fuzzy implications. In other words, scholars focused on the conditions under which these laws are tautologies in fuzzy logics.

In fuzzy logics, the truth degree of a proposition is belongs to [0,1]. One expects to infer from some approximately true premises to obtain other approximately true conclusions. Then various generalized tautologies have emerged in the literature [32–34]. In a similar direction, we should look at the case that these laws are generalized tautologies in fuzzy logics. In other words, we should consider the case that these functional equations are approximately valid in fuzzy logics.

In this paper, we focus on the approximation of the iterative boolean-like law a →(a → b) = a → b. In classical logic, it is a tautology. This law has been generalized as functional equation

$I (x, I (x, y)) = I (x, y), \forall x, y \in [0, 1],$ (1) in fuzzy logic, where I is a fuzzy implication. Researchers [4–7] have characterized the solutions of Equation (1) for several classes of fuzzy implications. Dai [8] have studied the stability of Equation (1) for (S, N)-implications.

The Equation (1) requests that I(x, I(x, y)) is strictly equal to I(x, y). This requirement restricts the potential theoretical and applied value of functional equations in fuzzy logics. In this paper, we relax the requirement. We consider the case that I(x, I(x, y)) is approximately equal to I(x, y). In other words, we consider the case the functional equation I(x, I(x, y)) = I(x, y) is approximately valid in fuzzy logics.

In fuzzy logics, fuzzy implication is employed in fuzzy inference schemas. The fuzzy conditional statement A → B is called a fuzzy IF-THEN rule where A is call the fuzzy antecedent (or input) and B is called the fuzzy consequent (or output). A fuzzy IF-THEN rule A → B is of form $\begin{matrix} Rule (1) : IF x is A THEN y is B . \end{matrix}$

The statement A₁ & A₂ & . . . & A_n → B is a Multi-Input Single-Output (MISO) rule of the form $\begin{matrix} Rule (1^{'}) : IF x_{1} is A_{1} \\ AND x_{2} is A_{2} \\ AND \dots \\ AND x_{n} is A_{n} \\ THEN y is B . \end{matrix}$

Then a question: when A₁ = A₁ = ⋯ = A_n, could Rule (1’) simplify to Rule (1) (or approximately)?

We replace A₁ & A₂ & . . . & A_n → B with A₁ →(A₂ →(. . . →(A_n → B))) so the question is: whether A →(A →(. . . →(A → B))) is approximately equal to A → B?

In the simple case of MISO rule with 2 inputs, the above question is concerned with the equation I(x, I(x, y)) ≈ I(x, y).

This paper is organized as follows. In Section 2, we briefly review the fuzzy implications and their compositions. In Section 3, we give a definition of the approximation of I(x, I(x, y)) = I(x, y). In Section 4, we study the property of approximation preserving with respect to several compositions of fuzzy implications. In Section 5, we consider the necessary condition and the sufficient condition of Equation (2) for (S, N)-implication. In Section 6, we summarize our main results.

2 Preliminaries

2.1 Definitions

Definition 1. [30]. A binary operation I: [0, 1] ² → [0, 1] is called a fuzzy implication if I is nonincreasing on the first variable and nondecreasing on the second one, and I(1, 0) =0, I(0, 0) = I(0, 1) = I(1, 1) =1.

Definition 2. [35]. A unary operation N : [0, 1] → [0, 1] is called a fuzzy negation if it is decreasing and N(1) =0, N(0) =1.

Definition 3. [35] A binary operation T: [0, 1] ² → [0, 1] is called a t-norm if T is commutative, associative, monotone and has 1 as its neutral element.

Definition 4. [35] A binary operation S: [0, 1] ² → [0, 1] is called a t-conorm if S is commutative, associative, monotone and has 0 as its neutral element.

Definition 5. [30]. A fuzzy implication I is called an (S, N)-implication if it is defined as

$I (x, y) = S (N (x), y), \forall x, y \in [0, 1],$ (2) where S is a t-conorm and N is a fuzzy negation.

2.2 Compositions of fuzzy implications

Let I, J be two fuzzy implications, their meet and join are defined as, ∀x, y ∈ [0, 1], $(I \lor J) (x, y) = max (I (x, y), J (x, y)),$ (3) $(I \land J) (x, y) = min (I (x, y), J (x, y)),$ (4) their convex combination is defined as, ∀x, y ∈ [0, 1] , λ ∈ [0, 1],

$K_{λ} (x, y) = λ I (x, y) + (1 - λ) J (x, y),$ (5) their ⊛-composition is defined as ∀x, y ∈ [0, 1],

$(I ⊛ J) (x, y) = I (x, J (x, y)) .$ (6)

The n-th power of I is defined as: For n = 1, I¹(x, y) = I(x, y), and for n ≥ 2,

$\begin{matrix} I^{n} (x, y) = I (x, I^{n - 1} (x, y)) = I^{n - 1} (x, I (x, y)), \\ \forall x, y \in [0, 1] . \end{matrix}$

The following lemma will be used in the proofs of our main conclusions

Lemma 1. [36]. Suppose u, v are two bounded, real valued functions in X, then $| sup_{x \in X} u (x) - sup_{x \in X} v (x) | \leq sup_{x \in X} | u (x) - v (x) |,$ (7) $| inf_{x \in X} u (x) - inf_{x \in X} v (x) | \leq sup_{x \in X} | u (x) - v (x) | .$ (8)

3 Definition of I² ≈ _ɛI

This section defines the approximate extension of I(x, I(x, y)) = I(x, y).

Definition 6. Let I be a fuzzy implication and ɛ ∈ [0, 1]. If

$| I (x, I (x, y)) - I (x, y) | \leq ɛ, \forall x, y \in [0, 1],$ (9) then I² and I are said to be ɛ-approximately equal, denoted by I² ≈ _ɛI.

Remark 1. Equation (1) is a special case (ɛ = 0) of Equation (2). For simplicity, Equation (1) could be denoted by I² = I.

Remark 2. I² ≈ _ɛI also means that I² is (1 - ɛ) equal to I. ɛ can be viewed as the maximum difference between I² and I, and 1 - ɛ represents the equality degree between I² and I.

For a given fuzzy implication I, we can use $max_{x, y \in [0, 1]} | I^{2} (x, y) - I (x, y) |$ to get the parameter ɛ. The parameter ɛ according to the choices of fuzzy implication is given in Table 1.

Table 1

Values of the difference parameter ɛ

Name	Formula	2-power	ɛ
Gödel	$I_{GD} (x, y) = {\begin{matrix} 1, & if x \leq y, \\ y, & if x > y . \end{matrix}$	$I_{GD}^{2} = I_{GD}$	0
Reichenbach	I_RC(x, y) =1 - x + xy	$I_{RC}^{2} (x, y) = 1 - x^{2} + x^{2} y$	$\frac{1}{4}$
Kleene-Dienes	I_KD(x, y) = max(1 - x, y)	$I_{KD}^{2} = I_{KD}$	0
Rescher	$I_{RS} (x, y) = {\begin{matrix} 1, & if x \leq y, \\ 0, & if x > y . \end{matrix}$	$I_{RS}^{2} = I_{RS}$	0
Weber	$I_{WB} (x, y) = {\begin{matrix} 1, & if x < 1, \\ y, & if x = 1 . \end{matrix}$	$I_{WB}^{2} = I_{WB}$	0
Fodor	$I_{FD} (x, y) = {\begin{matrix} 1, & if x \leq y, \\ max (1 - x, y), & if x > y . \end{matrix}$	$I_{FD}^{2} (x, y) = {\begin{matrix} 1, & if x \leq y or x \in [0, 0.5], \\ max (1 - x, y), & otherwise . \end{matrix}$	$\frac{1}{2}$
Least FI	$I_{LE} (x, y) = {\begin{matrix} 1, & if x = 0 or y = 1, \\ 0, & if x > 0 and y < 1 . \end{matrix}$	$I_{LE}^{2} = I_{LE}$	0
Greatest FI	$I_{GR} (x, y) = {\begin{matrix} 1, & if x < 1 or y > 0, \\ 0, & if x = 1 and y = 0 . \end{matrix}$	$I_{GR}^{2} = I_{GR}$	0
Most Strict	$I_{D} (x, y) = {\begin{matrix} 1, & if x = 0, \\ y, & if x > 0 . \end{matrix}$	$I_{D}^{2} = I_{D}$	0

Now we give two examples that for any small nonnegative number ϵ > 0 there exists a fuzzy implication I such that I² ≈ _ϵI, but I² ≠ I.

Example 1. For any ϵ > 0, we consider the following function $I_{ɛ} (x, y) = {\begin{matrix} 1, & if x \leq y, \\ max (y - ɛ, 0), & if x > y . \end{matrix}$ (10) It easy to verify that I_ɛ is a fuzzy implication. And we can obtain that $I_{ɛ}^{2} (x, y) = {\begin{matrix} 1, & if x \leq y, \\ max (y - 2 ɛ, 0), & if x > y . \end{matrix}$ (11) For any x, y ∈ [0, 1], if x ≤ y, then I_ɛ(x, (I_ɛ(x, y)) = I_ɛ(x, y); if x > y, then $\begin{matrix} | I_{ɛ} (x, y) - I_{ɛ}^{2} (x, y) | \\ = & | max (y - ɛ, 0) - max (y - 2 ɛ, 0) | \\ \leq & max (| (y - ɛ) - (y - 2 ɛ) |, | 0 - 0 |) (by Lemma 1) \\ = & | (y - ɛ) - (y - 2 ɛ) | \\ = & ɛ . \end{matrix}$ Thus we have I² ≈ _ɛI. Moreover, if there exist x, y (by taking x > y > ɛ > 0) such that $I_{ɛ} (x, y) - I_{ɛ}^{2} (x, y) = ɛ \neq 0$ , then we have I² ≠ I.

Example 2. For any ϵ > 0, there exists N₀ such that $\frac{1}{N_{0}} \leq ϵ$ . Then we consider the following function $I_{N_{0}} (x, y) = {\begin{matrix} 1, & if x \leq y, \\ \frac{y}{N_{0}}, & if x > y . \end{matrix}$ (12) It easy to verify that I_{N
₀} is a fuzzy implication. For any x, y ∈ [0, 1], if x ≤ y, then I_{N
₀}(x, (I_{N
₀}(x, y)) = I_{N
₀}(x, y); if x > y, then $\begin{matrix} | I_{N_{0}} (x, y) - I_{N_{0}} (x, (I_{N_{0}} (x, y)) | \\ = & \frac{y}{N_{0}} - I_{N_{0}} (x, \frac{y}{N_{0}}) \\ = & \frac{y}{N_{0}} - \frac{y}{N_{0}^{2}} \\ = & \frac{(N_{0} - 1) y}{N_{0}^{2}} \\ \leq & \frac{(N_{0} - 1)}{N_{0}^{2}} \\ \leq & \frac{N_{0}}{N_{0}^{2}} \\ \leq & \frac{1}{N_{0}} \\ \leq & ϵ . \end{matrix}$ Thus we have I² ≈ _ϵI. Moreover, if there exist y such that $I_{N_{0}} (x, y) - I_{N_{0}} (x, (I_{N_{0}} (x, y)) = \frac{(N_{0} - 1) y}{N_{0}^{2}} \neq 0$ , then we have I² ≠ I.

4 Properties of I² ≈ _ɛI

This section gives some properties of I² ≈ _ɛI.

Lemma 2. If ɛ₁ ≤ ɛ₂ then I² ≈ _ɛ₁I ⇒ I² ≈ _ɛ₂I.

Theorem 1. If I² ≈ _ɛI, then Iⁿ⁺¹ ≈ _ɛIⁿ.

Proof. It can be proved by mathematical induction. The formula holds for n = 1, assume it holds for n = k - 1, we will prove it for n = k. If I^k ≈ _ɛI^k-1, i.e.,

$I^{k - 1} (x, y) - ɛ \leq I (x, I^{k - 1} (x, y)) \leq I^{k - 1} (x, y) + ɛ$ (13) By taking y = I(x, y), we have $\begin{matrix} \begin{matrix} I^{k - 1} (x, I (x, y)) - ɛ \\ \leq I (x, I^{k - 1} (x, I (x, y))) \\ \leq I^{k - 1} (x, I (x, y)) + ɛ \\ \Rightarrow & I^{k} (x, y) - ɛ \\ \leq I (x, I^{k} (x, I (x, y))) \\ \leq I^{k} (x, y) + ɛ \\ \Rightarrow & I^{k} (x, y) - ɛ \leq I^{k + 1} (x, y) \leq I^{k} (x, y) + ɛ \\ \Rightarrow & I^{k + 1} \approx_{ɛ} I^{k} . \end{matrix} \end{matrix}$

We thus get Iⁿ⁺¹ ≈ _ɛIⁿ.□

In the above theorem, we show that I² ≈ _ɛI ⇒ Iⁿ⁺¹ ≈ _ɛIⁿ. However, the converse is not true, see the following example.

Example 3. Consider the Fodor implication

$I_{FD} (x, y) = {\begin{matrix} 1, & if x \leq y, \\ max (1 - x, y), & otherwise . \end{matrix}$ (14) From Table 8 in [9], its n-power is, for n ≥ 2, $I_{KD}^{n} (x, y) = {\begin{matrix} 1, & if x \leq y or x \in [0, 0.5], \\ max (1 - x, y), & otherwise . \end{matrix}$ (15) Thus we have Iⁿ⁺¹ = Iⁿ for n ≥ 2. But I² ≠ I, for example, I_KD(0.5, 0.4) =0.5 and $I_{KD}^{2} (0.5, 0.4) = 1$ .

Theorem 2. If I² ≈ _ɛI, then Iⁿ ≈ _(n-1)ɛI.

Proof. From Theorem 4, we have Iⁿ ≈ _ɛI^n-1 for all n ≥ 2. Then $\begin{matrix} \begin{matrix} | I^{n} (x, y) - I (x, y) | \\ = & | (I^{n} (x, y) - I^{n - 1} (x, y)) \\ + \dots + (I^{2} (x, y) - I (x, y)) | \\ \leq & | I^{n} (x, y) - I^{n - 1} (x, y) | \\ + \dots + | I^{2} (x, y) - I (x, y) | \\ \leq & (n - 1) ɛ . \end{matrix} \end{matrix}$ We thus get Iⁿ ≈ _(n-1)ɛI.□

Theorem 3. Let K_λ(x, y) be the convex combination of I(x, y) and J(x, y). If I² ≈ _ɛ₁I and J² ≈ _ɛ₂J, then $K_{λ}^{2} \approx_{ɛ_{1} \lor ɛ_{2}} K_{λ}$ .

Proof. From |I²(x, y) - I(x, y) | ≤ ɛ₁ and |J²(x, y) - J(x, y) | ≤ ɛ₂, we have $\begin{matrix} \begin{matrix} | K_{λ}^{2} (x, y) - K_{λ} (x, y) | \\ = & | λ I (x, I (x, y)) + (1 - λ) J (x, J (x, y)) \\ - λ I (x, y) - (1 - λ) J (x, y) | \\ = & | λ (I (x, I (x, y)) - I (x, y)) \\ + (1 - λ) (J (x, J (x, y)) - J (x, y)) | \\ \leq & λ | I (x, I (x, y)) - I (x, y) | \\ + (1 - λ) | J (x, J (x, y)) - J (x, y) | \\ \leq & λ ɛ_{1} + (1 - λ) ɛ_{2} \\ \leq & λ (ɛ_{1} \lor ɛ_{2}) + (1 - λ) (ɛ_{1} \lor ɛ_{2}) \\ = & ɛ_{1} \lor ɛ_{2} . \end{matrix} \end{matrix}$

We thus get $K_{λ}^{2} \approx_{ɛ_{1} \lor ɛ_{2}} K_{λ}$ .□

Theorem 4. Let I and J be two fuzzy implications, if I² ≈ _ϵI, then $I^{2} \lor J \approx_{ɛ} I \lor J,$ (16) $I^{2} \land J \approx_{ɛ} I \land J .$ (17)

Proof. For any x, y ∈ [0, 1], we have $\begin{matrix} (I^{2} \lor J) (x, y) - (I \lor J) (x, y) \\ = & max (I^{2} (x, y), J (x, y)) - max (I (x, y), J (x, y)) \\ \leq & max (| I^{2} (x, y) - I (x, y) | \\ , | J (x, y) - J (x, y) |) (by Lemma 1) \\ = & | I^{2} (x, y) - I (x, y) | \\ \leq & ɛ . \end{matrix}$ Thus we have I² ∨ J ≈ _ɛI ∨ J. Similarly, we can get I² ∧ J ≈ _ɛI ∧ J.□

Theorem 5. Let I and J be two fuzzy implications, if I² ≈ _ɛI, then

$I^{2} ⊛ J \approx_{ɛ} I ⊛ J .$ (18)

Proof. For any x, y ∈ [0, 1], let z = J(x, y), then $\begin{matrix} (I^{2} ⊛ J) (x, y) & = & I^{2} (x, J (x, y)) \\ = & I^{2} (x, z) \\ \approx_{ɛ} & I (x, z) \\ = & I (x, J (x, y)) \\ = & (I ⊛ J) (x, y) \end{matrix}$ Thus we get I² ⊛ J ≈ _ɛI ⊛ J.□ Example 4. Consider the Kleene-Dienes implication I_KD(x, y) = max(1 - x, y), Reichhenbach implication I_RC(x, y) =1 - x + xy and its 2-power $I_{RC}^{2} (x, y) = 1 - x^{2} + x^{2} y$ . Since $I_{KD} (x, y) \leq I_{RC} (x, y) \leq I_{RC}^{2} (x, y), \forall x, y \in [0, 1]$ , then I_RC ∨ I_KD = I_RC, I_RC ∧ I_KD = I_KD, $I_{RC}^{2} \lor I_{KD} = I_{RC}^{2}$ , $I_{RC}^{2} \land I_{KD} = I_{KD}$ . From Table 1, we have $I_{RC}^{2} \lor I_{KD} \approx_{1 / 4} I_{RC} \lor I_{KD}$ and $I_{RC}^{2} \land I_{KD} = I_{RC} \land I_{KD}$ which implies $I_{RC}^{2} \land I_{KD} \approx_{1 / 4} I_{RC} \land I_{KD}$ . Moreover, we have I_RC ⊛ I_KD = max(1 - x², 1 - x + xy) and $I_{RC}^{2} ⊛ I_{KD} = max (1 - x^{3}, 1 - x^{2} + x^{2} y)$ , then $\begin{matrix} I_{RC}^{2} ⊛ I_{KD} - I_{RC} ⊛ I_{KD} \\ = & max (1 - x^{2}, 1 - x + xy) \\ - max (1 - x^{3}, 1 - x^{2} + x^{2} y) \\ \leq & max (| (1 - x^{2}) - (1 - x^{3}) | \\ , | (1 - x + xy) - (1 - x^{2} + x^{2} y) |) (by Lemma 1) \\ = & max (\frac{4}{27}, \frac{1}{4}) \\ = & \frac{1}{4} . \end{matrix}$

Thus we have $I_{RC}^{2} ⊛ I_{KD} \approx_{1 / 4} I_{RC} ⊛ I_{KD}$ .

5 Characterizations of (S, N)-implications satisfying I² ≈ _ɛI

We first give a sufficient condition of Equation (2) for (S, N)-implication.

Theorem 6. Let I be an (S, N)-implication. If S satisfies $max (x, y) \leq S (x, y) \leq max (x, y) + \frac{ɛ}{2},$ (19) then I satisfies Equation (2).

Proof. First, since S(x, y) ≥ max(x, y), we have for any x, y ∈ [0, 1], $\begin{matrix} \begin{matrix} I (x, I (x, y)) \\ = & S (N (x), S (N (x), y)) \\ \geq & max (N (x), max (N (x), y)) \\ = & max (N (x), y) \end{matrix} \end{matrix}$ Then from Equation (11), we have $\begin{matrix} \begin{matrix} I (x, I (x, y)) \\ = & S (N (x), S (N (x), y)) \\ \leq & max (N (x), max (N (x), y) + \frac{ɛ}{2}) + \frac{ɛ}{2} \\ \leq & max (N (x), y) + ɛ . \end{matrix} \end{matrix}$

So we have max(N(x) , y) ≤ I(x, I(x, y)) ≤ max(N(x) , y) + ɛ. By the assumption of S, we have $max (N (x), y) \leq I (x, y) = S (N (x), y) \leq max (N (x), y) + \frac{ɛ}{2}$ . Thus I satisfies Equation (2).□

We then give a necessary condition of Equation (2) for (S, N)-implication.

Lemma 3. Let S be a t-conorm, for any ɛ > 0, the following statements are equivalent,

S(x, x) ≤ x + ɛ: ∀x ∈ [0, 1],

S(x, y) ≤ max(x, y) + ɛ: ∀x, y ∈ [0, 1],

Proof. (ii)⇒(i): It is straightforward.

(i)⇒(ii): Since S is increasing in each variable, we consider two cases: Case 1, if x ≥ y, then S(x, y) ≤ S(x, x) ≤ x + ɛ = max(x, y) + ɛ. Case 2, if x > y, then S(x, y) ≤ S(y, y) ≤ y + ɛ = max(x, y) + ɛ.□

Theorem 7. Let I be an (S, N)-implication defined by a continuous negation N and a t-conorm S. If I satisfies Equation (2) then S satisfies

$max (x, y) \leq S (x, y) \leq max (x, y) + ɛ .$ (20)

Proof. S(x, y) ≥ max(x, y) is clear. By S(x, 0) = x, ∀ x ∈ [0, 1], we have S(N(x) , 0) = N(x) , ∀ x ∈ [0, 1], then $\begin{matrix} \begin{matrix} I (x, I (x, 0)) \\ = & S (N (x), S (N (x), 0)) \\ = & S (N (x), N (x)) . \end{matrix} \end{matrix}$ Since I satisfies Equation (2), then $\begin{matrix} \begin{matrix} I (x, I (x, 0)) \leq I (x, 0) + ɛ \\ \Rightarrow & S (N (x), S (N (x), 0)) \leq S (N (x), 0) + ɛ \\ \Rightarrow & S (N (x), N (x)) \leq N (x) + ɛ . \end{matrix} \end{matrix}$

Since N is continuous, then the range of N is [0, 1]. So we have S(x, x) ≤ x + ɛ, ∀ x ∈ [0, 1]. By Lemma 5, we get S(x, y) ≤ max(x, y) + ɛ, ∀ x, y ∈ [0, 1]. Thus the proof is complete.□

Example 5. Consider the Reichhenbach implication I_RC(x, y) =1 - x + xy and its 2-power $I_{RC}^{2} (x, y) = 1 - x^{2} + x^{2} y$ . Reichhenbach implication is an (S,N)-implication generated from S_P(x, y) = x + y - xy and N(x) =1 - x. From Table 1, we know that $I_{RC}^{2} \approx_{1 / 4} I_{RC}$ . Clearly S_P(x, y) ≥ max(x, y), moreover $\begin{matrix} \begin{matrix} S_{P} (x, y) - max (x, y) \\ = & x + y - xy - max (x, y) \\ \geq & \frac{1}{4}, \end{matrix} \end{matrix}$ and $S_{P} (x, y) - max (x, y) = \frac{1}{4}$ when $x = y = \frac{1}{2}$ . Thus $max (x, y) \leq S_{P} (x, y) \leq max (x, y) + \frac{1}{4}$ .

Remark 3. Shi et al. [4] showed that an (S,N)-implication I obtained by a t-conorm S and a continuous fuzzy negation N satisfies I² = I iff S is maximum t-conorm. This result could be seen as a corollary of Theorems 6 and 7 by taking ɛ = 0.

6 Conclusion

In this paper, the approximation of the functional equation I(x, I(x, y)) = I(x, y) has been investigated. Obviously, the new approximate equation proposed in the present paper is an extension of the old equation. The new approximate equation is more comprehensive than the old equation because the latter is a special case of the former. A necessary condition and a sufficient condition of I² ≈ _ɛI for (S,N)-implication are given, and properties of I² ≈ _ɛI can be summarized as following:

I² ≈ _ɛI ⇒ Iⁿ⁺¹ ≈ _ɛIⁿ but not vice versa.

I² ≈ _ɛI ⇒ Iⁿ ≈ _(n-1)ɛI.

$I^{2} \approx_{ɛ_{1}} I \land J^{2} \approx_{ɛ_{2}} J \Rightarrow K_{λ}^{2} \approx_{ɛ_{1} \lor ɛ_{2}} K_{λ}$ .

I² ≈ _ɛI ⇒ I² ∨ J ≈ _ɛI ∨ J, I² ∧ J ≈ _ɛI ∧ J .

In the forthcoming work, we will continue to discuss the approximation of other functional equations. Indeed, much research of the approximation of functional equations involving fuzzy implications are still needed to fully comprehend their properties and potential.

Footnotes

Acknowledgments

This project was supported by the National Science Foundation of China under Grant No. 62006168 and Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ21A010001.

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