Upper and Lower Bounds of the α-Universal Triple I Method for Unified Interval Implications

1  Introduction

As the core of fuzzy logic [1–3], fuzzy reasoning plays an important role in intelligent control, affective computing, machine vision, large models and other fields [4–7]. Fuzzy reasoning itself has two key problems. The first problem is fuzzy modus ponens (FMP):

FMP:FocusingonQ↦PandQ′,calculateP′(1)

The other one is fuzzy modus tollens (FMT):

FMT:Focusing on Q↦P and P′,calculate Q′(2)

Since Zadeh came up with the compositional rule of inference (CRI) [8,9] method in 1973, it has been widely adopted in various fields such as industrial control and artificial intelligence. So far, many scholars have presented various fuzzy reasoning methods. Among them, Wang’s triple I (TI) method [10] is one of the most influential methods.

Due to the strong logical completeness of the TI method, many scholars have carried out more in-depth research in this direction. Many methods are proposed to enhance the TI method, such as the α-triple I constraint degree method [11], the α-universal triple I (α-UTI) method (proposed by us in [12]), the quintuple implicational method [13,14], symmetric implicational methods [15] and similarity-base inference methods [16]. These methods all enhance the performance of the TI method to a certain extent. It is necessary to test the different properties of these methods in order to analyze their performance.

Among them, robustness is a crucial property, which refers to the resistance of the method to perturbations. It can be quantified by calculating the gap between the output of the fuzzy reasoning method with interference and the output without interference. The smaller the gap, the smaller the impact of interference. Distance [17–19] (or dissimilarity) is a common tool, including the Chebyshev distance [20], the Minkowski distance [21] and the Hamming distance [22]. The concepts of maximum ε-disturbance, α-similarity, fuzzy set approximate equality derived from Chebyshev distance are explored in detail in [23–26]. Therefore, the Chebyshev distance is also used in this study to measure the gap between the original output and the disturbed output to evaluate the robustness.

Fuzzy implication plays a key role in fuzzy reasoning strategy. Different implications will also have different effects on the robustness of the method. Reference [26] studied the robustness of the CRI method when adopting R-implications, S-implications and QL-implications, and demonstrates that CRI had strong stability. In [17], the robustness of the BKS (Bandler-Kohout subproduct) method with S-implications and QL-implications was discussed in the context of interval perturbation. Wang et al. [27] discovered the robustness of the TI method.

The main formula of the α-UTI method for FMP is as below [12] (α∈[0,1]):

(Q↦P)↦(Q′↦P′)≥α.(3)

The α-UTI method has been widely recognized in the field of fuzzy reasoning as a generalization of the CRI method and the TI method. The α-UTI method is also a generalization of the universal triple I (UTI) method proposed in [12]. In detail, when α = 1, the α-UTI method degenerates into the UTI method.

The current research gaps in this field include the following two aspects:

•   On the one hand, how to give the solution of chain reasoning, a special form of reasoning, is a tough problem in fuzzy reasoning. In this regard, we can consider starting from the α-UTI method to construct a scheme oriented towards chain reasoning.

•   On the other hand, it is not clear how the interval stability of the α-UTI method is. In particular, can the upper and lower bounds of its reasoning be estimated? This aspect has not yet been discussed. This has aroused our great interest, and also constitutes the research goal of this paper.

In this study, we explore the interval robustness of the α-UTI method, which is embodied by interval stability.

Fig. 1 shows the detailed schematic diagram of the proposed study.

Figure 1: The detailed schematic diagram of the proposed study

The proposed study includes the following aspects:

•   To begin with, we discuss the robustness of the α-UTI method in the case of an individual rule, and give an estimate of the upper and lower bounds. Therein, four kinds of unified interval implications mentioned above are employed in turn. It is important to note that each kind of implication actually contains a lot of specific implications, so this study can incorporate many implications into its framework. The process and flow chart of the corresponding algorithm are given. The results show that the α-UTI method is stable in the individual rule case under the interval perturbation.

•   Moreover, we provide the estimation for upper and lower bounds for the multiple rules case of the α-UTI method. Here, four kinds of unified interval implications are respectively examined in the α-UTI method. The results show that, in the context of the interval perturbation, the α-UTI method is stable for the situation of multiple rules.

•   In addition, for the problem of chain reasoning, we put forward a strategy based upon the α-UTI method, which is called the α-UTI reasoning chain method. We provide an intelligent algorithm to estimate its upper and lower bounds, and draw the corresponding flow chart. In this case, the four types of interval implications are adopted in turn. It is found the α-UTI reasoning chain method is stable for the problem of chain reasoning.

•   Lastly, the α-UTI method is analyzed through two application examples in affective computing. Here we propose the scheme of emotional deduction based on the α-UTI method. These examples verify the stability of the α-UTI method.

To sum up, the results show that in three cases of individual rule, multiple rules and reasoning chain, the α-UTI method has good interval stability when it respectively adopts four kinds of unified interval implications mentioned above.

The structure of this paper is as follows. Section 2 gives the fundamental knowledge and related work. Sections 3 and 4 prove the robustness of the α-UTI method from the perspectives of individual rule and multiple rules, respectively. Section 5 probes into the robustness of the α-UTI reasoning chain method and its robustness. Section 6 gives two examples. Section 7 summarizes all the work and provides the prospect.

2  Preliminaries

This section mainly introduces the basic knowledge of fuzzy reasoning, and provides some interval implications and interval connectives.

Definition 2.1 [26–28]. Let Q denote the set of intervals in the range [0,1], that is, Q={[p,q]|0≤p≤q≤1}.

There are two functions L:Q→R and U:Q→R that extract the lower and upper endpoints of an interval, respectively.

L([m,n])=m,U([m,n])=n.(4)

For any Q∈Q, using interval notation, the lower and upper endpoints can also be expressed as Q_ and Q¯. The following definition gives some basic operations of intervals.

Definition 2.2 [28–30]. For any Q1,Q2∈Q, one has

a)   Q1≥Q2 if and only if Q1_≥Q2_,Q1¯≥Q2¯;

b)   Q1⊆Q2 if and only if Q1_≥Q2_,Q1¯≤Q2¯;

c)   Q1+Q2=[Q1_+Q2_,Q1¯+Q2¯];

d)   Q1−Q2=max(|Q1_−Q2_|,|Q1¯−Q2¯|).

In order to facilitate the subsequent operation, we give two tokens, namely 0^=[0,0] and 1^=[1,1]. The definition of interval fuzzy implication is given below.

Definition 2.3 [30,31]. The operation ↦:Q2→Q is an interval fuzzy implication if it satisfies the underlying (5) and (6) :

0^↦1^=1^↦1^=0^↦0^=1^,(5)

1^↦0^=0^.(6)

For the definition of interval fuzzy implication, there are other definitions, such as the need for incrementality with regard to the second variable. But here conditions (5) and (6) are often basic conditions, and generally need to be met. In summary, Definition 2.3 here is a classical and fundamental definition of interval fuzzy implication.

Two important operations are further defined here, in which these two concepts are mutually symmetric.

Definition 2.4 [32,33]. An operation ⊗:Q2→Q is referred to as an interval t-norm, when the operation is commutative, associative, monotonically increasing and satisfies 1^⊗Q=Q(Q∈Q).

Definition 2.5 [28,34]. An operation ⊕:Q2→Q is referred to as an interval t-conorm, when the operation is commutative, associative, monotonically increasing and satisfies 0^⊕Q=Q(Q∈Q).

The interval t-norm and the interval t-conorm are symmetric concepts. As the opposite structure of conventional operations, interval fuzzy negation is defined as below.

Definition 2.6 [28,30,34,35]. An operation N:Q→Q is under the title of an interval fuzzy negation, if it satisfies the subsequent conditions (where Q1,Q2∈Q):

a)   (N1):N(0^)=1^whenN(1^)=0^;

b)   (N2):IfQ1≥Q2thenN(Q1)≤N(Q2);

c)   (N3):IfQ1⊆Q2thenN(Q1)⊇N(Q2).

On the basis of interval fuzzy negation, if it still has:

d)   (N4):N(N(Q))=Q,∀Q∈Q

then this is a strong interval fuzzy negation.

Subsequently, three important classes of interval fuzzy implication are defined as below.

Definition 2.7 [30]. An interval fuzzy implication is called an R-interval implication denoted by ↦⊗ whenever there exists an interval t-norm ⊗ and ↦⊗ is designed by

Q1↦⊗Q2=sup{Q3∈Q:Q1⊗Q3≤Q2}(7)

Definition 2.8 [30]. An interval fuzzy implication is under the title of an S-interval implication denoted by ↦⊕,N whenever there exist an interval t-conorm and an interval fuzzy negation such that ↦⊕,N is given by

Q1↦⊕,NQ2=N(Q1)⊕Q2(8)

Definition 2.9 [30]. An interval fuzzy implication is called a QL-interval implication represented by ↦⊗,⊕,N whenever there exist an interval t-norm, an interval t-conorm and an interval fuzzy negation letting ↦⊗,⊕,N be provided by

Q1↦⊗,⊕,NQ2=N(Q1)⊕(Q1⊗Q2)(9)

The interval t-norm can also be regarded as an interval fuzzy implication, called the interval t-norm implication.

Definition 2.10 [26]. Suppose μl,μu∈F(Θ) and μl≤μr(θ∈Θ), namely

μl_(θ)≤μu_(θ),μl¯(θ)≤μu¯(θ)(10)

then [μl,μu] is called the fuzzy interval on Θ.

In what follows, the idea of interval disturbance is provided.

Definition 2.11 [26]. Suppose S∈F(Θ) and [μl,μu] is the fuzzy interval on Θ. If μl≤S≤μu(θ∈Θ), that means

μl_(θ)≤S_(θ)≤μu_(θ),μl¯(θ)≤S¯(θ)≤μu¯(θ)(11)

holds, then [μl,μu] is called the interval disturbance of S and we denote S∈[μl,μu].

Definition 2.12 [26]. Suppose X,Y∈F(Θ), Z∈F(Ω) and X∈[xl,xu], Y∈[yl,yu], Z∈[zl,zu]. A fuzzy reasoning strategy f is robust, if for any ε>0, a fuzzy interval [λl,λu] on Ω and δ>0 such that λu−λl<ε(ω∈Ω) (namely λu_(ω)−λl_(ω)<ε and λu¯(ω)−λl¯(ω)<ε are established) and f(X,Y,Z)∈[λl,λu] whenever xu−xl<δ(θ∈Θ), yu−yl<δ(θ∈Θ) and zu−zl<δ(ω∈Ω) (namely xu_(θ)−xl_(θ)<δ, xu¯(θ)−xl¯(θ)<δ, yu_(θ)−yl_(θ)<δ, yu¯(θ)−yl¯(θ)<δ, zu_(ω)−zl_(ω)<δ and zu¯(ω)−zl¯(ω)<δ are established).

In this work, ∨ and ∧ stand for supremum and infimum, respectively.

Lemma 2.1 [26]. Let σ1 and σ2 be real-valued, bounded maps on Θ, and S,R∈F(Θ). The following a)~j) can be obtained:

a)   σ1∨σ2=max{σ1,σ2}=(σ1+σ2)/2+|σ1−σ2|/2;

b)   σ1∧σ2=min{σ1,σ2}=(σ1+σ2)/2−|σ1−σ2|/2;

c)   −|σ1−σ2|≤|σ1|−|σ2|≤|σ1−σ2|;

d)   ∧θ∈ΘS(θ)+∧θ∈ΘR(θ)≤∧θ∈Θ(S(θ)+R(θ));

e)   ∨θ∈Θ(S(θ)+R(θ))≤∨θ∈ΘS(θ)+∨θ∈ΘR(θ);

f)   ∨θ∈ΘS(θ)=−∧θ∈Θ(−S(θ));

g)   ∧θ∈Θ(S(θ)∧R(θ))=(∧θ∈ΘS(θ))∧(∧θ∈ΘR(θ));

h)   ∧θ∈Θ(S(θ)∨R(θ))≥(∧θ∈ΘS(θ))∨(∧θ∈ΘR(θ));

i)   ∨θ∈Θ(S(θ)∧R(θ))≤(∨θ∈ΘS(θ))∧(∨θ∈ΘR(θ));

j)   ∨θ∈Θ(S(θ)∨R(θ))=(∨θ∈ΘS(θ))∨(∨θ∈ΘR(θ)).

Lemma 2.2 [17] Let σ1 and σ2 be real-valued, bounded maps on Θ, then it can also be obtained:

a)   ∨θ∈Θσ1(θ)−∨θ∈Θσ2(θ)≤∨θ∈Θ(σ1(θ)−σ2(θ));

b)   ∧θ∈Θσ1(θ)−∧θ∈Θσ2(θ)≤∨θ∈Θ(σ1(θ)−σ2(θ));

c)   ∧θ∈Θσ1(θ)−∧θ∈Θσ2(θ)≥∧θ∈Θ(σ1(θ)−σ2(θ));

d)   ∨θ∈Θσ1(θ)−∨θ∈Θσ2(θ)≥∨θ∈Θ(σ1(θ)−σ2(θ)).

Example 2.1 Let [i,j],[h,k]∈Q. Here are some common interval fuzzy implications:

a)   [i,j]↦RL[h,k]=[(1−i+j)∧(1−j+k)∧1,(1−j+k)∧1](Lukasiewicz (R-) interval implication);

b)   [i,j]↦SL[h,k]=[(1−j+h)∧1,(1−i+k)∧1] (Lukasiewicz (S-) interval implication);

c)   [i,j]↦G[h,k]=[((1−i)∨h)∧((1−j)∨k),(1−j)∨k] (G interval implication);

d)   [i,j]↦L[h,k]=[((1−i)∨h)∧((1−j)∨k),(1−i)∨k] (L interval implication);

e)   [i,j]↦KD[h,k]=[(1−j)∨h,(1−i)∨k] (Kleene-Dienes interval implication);

f)   [i,j]↦R[h,k]=[1−j+jh,1−i+ik] (Reichenbach interval implication);

g)   [i,j]↦M[h,k]=[[(1−j)∧(1−k)]∨h,[(1−i)∧(1−h)]∨k] (M interval implication).

Example 2.2 Here are three representative interval t-norms, which are the Gödel norm ⊗G, the Goguen norm ⊗Go and the Lukasiewicz norm ⊗L. The residual interval fuzzy implications are ↦G, ↦Go and ↦L.

a)   ↦G and ⊗G are characterized as follows:

[i,j]↦G[h,k]={[h,k],i>h,j>k[h,1],i>h,j≤k[1,1],i≤h,j≤k[k,k],i≤h,j>k[i,j]⊗G[h,k]=[i∧h,j∧k](12)

b)   ↦Go and ⊗Go are given as below:

[i,j]↦Go[h,k]={[hi∧kj,kj],i>h,j>k[hi,1],i>h,j≤k[1,1],i≤h,j≤k[kj,kj],i≤h,j>k[i,j]⊗Go[h,k]=[ih,jk](13)

c)   ↦L and ⊗L are expressed as follows:

[i,j]↦L[h,k]={[(1−i+h)∧(1−j+k),1−j+k],i>h,j>k[1−i+h,1],i>h,j≤k[1,1],i≤h,j≤k[1−j+n,1−j+n],i≤h,j>k[i,j]⊗L[h,k]=[0∨(i+h−1),0∨(j+k−1)](14)

3  Interval Perturbation of the α-UTI Method for an Individual Rule

For an individual rule, assume that [ξl,ξu] and [φl,φu] are fuzzy intervals on Θ and [ψl,ψu] is a fuzzy interval on Ω. Denote UTI1(ω)=UTI1(Ξ,Φ,Ψ)(ω) (ω∈Ω). For simplicity of calculation, we give the following notations:

∧θ(f)=infθ∈Θ{f(θ)},∨θ(f)=supθ∈Θ{f(θ)}(15)

Theorem 3.1 Let Ξ,Φ∈F(Θ), Ψ∈F(Ω) and Ξ∈[ξl,ξu], Φ∈[φl,φu], Ψ∈[ψl,ψu]. If ↦ is an R-interval implication, then the result of the α-UTI method is as follows:

∨θ{[∧θ(φl_),∧θ(φl¯)]⊗[([∨θ(ξu_),∨θ(ξu¯)]↦[ψl_(ω),ψl¯(ω)])⊗[α_,α¯]]}≤UTI1(ω)≤∨θ{[∨θ(φu_),∨u(φu¯)]⊗[([∧θ(ξl_),∧θ(ξl¯)]↦[ψu_(ω),ψu¯(ω)])⊗[α_,α¯]]}(16)

Proof: The R-interval implication is decreasing with respect to the first parameter and increasing with respect to the second parameter. From Lemma 2.1 and Lemma 2.2, it can be obtained as follows:

∨θ{[∧θ(φl_),∧θ(φl¯)]⊗[([∨θ(ξu_),∨θ(ξu¯)]↦[ψl_(ω),ψl¯(ω)])⊗[α_,α¯]]}=∨θ∈Θ{∧θ∈Θ[φl_(θ),φl¯(θ)]⊗[(∨θ∈Θ[ξu_(θ),ξu¯(θ)]↦[ψl_(ω),ψl¯(ω)])⊗[α_,α¯]]}≤UTI1(ω).(17)

Moreover, one has

UTI1(ω)=∨θ∈Θ{[Φ_(θ),Φ¯(θ)]⊗[([Ξ_(θ),Ξ¯(θ)]↦[Ψ_(ω),Ψ¯(ω)])⊗[α_,α¯]]}≤∨θ∈Θ{∨θ∈Θ[φu_(θ),φu¯(θ)]⊗[(∧θ∈Θ[ξl_(θ),ξl¯(θ)]↦[ψu_(ω),ψu¯(ω)])⊗[α_,α¯]]}=∨θ{[∨θ(φu_),∨θ(φu¯)]⊗[([∧θ(ξl_),∧θ(ξl¯)]↦[ψu_(ω),ψu¯(ω)])⊗[α_,α¯]]}.(18)

In summary, we have proved that (16) holds.

Theorem 3.1 gives a estimation of the corresponding α-UTI solutions for the case of the interval perturbation with an individual rule. In other words, it respectively provides an upper bound and a lower bound of the α-UTI solutions for R-interval implications.

Theorem 3.2, Theorem 3.3 and Theorem 3.4 can be achieved in a similar mode, in which the interval t-norm implication, the S-interval implication, the QL-interval implication are employed, respectively.

Theorem 3.2 Let Ξ,Φ∈F(Θ), Ψ∈F(Ω) and Ξ∈[ξl,ξu], Φ∈[φl,φu], Ψ∈[ψl,ψu]. If ↦ is an interval t-norm implication, then the conclusion of the upper and lower bounds of the α-UTI method can be given as below:

∨θ{[∧θ(φl_),∧θ(φl¯)]⊗[([∧θ(ξl_),∧θ(ξl¯)]↦[ψl_(ω),ψl¯(ω)])⊗[α_,α¯]]}≤UTI1(ω)≤∨θ{[∨θ(φu_),∨θ(φu¯)]⊗[([∨θ(ξu_),∨θ(ξu¯)]↦[ψu_(ω),ψu¯(ω)])⊗[α_,α¯]]}.(19)

Theorem 3.3 Let Ξ,Φ∈F(Θ), Ψ∈F(Ω) and Ξ∈[ξl,ξu], Φ∈[φl,φu], Ψ∈[ψl,ψu]. If ↦ is an S-interval implication, then the conclusion of the upper and lower bounds of the α-UTI method can be given as below:

∨θ{[∧θ(φl_),∧θ(φl¯)]⊗[(N[∨θ(ξu_),∨θ(ξu¯)]⊕1[ψl_(ω),ψl¯(ω)])⊗[α_,α¯]]}≤UTI1(ω)≤∨θ{[∨θ(φu_),∨θ(φu¯)]⊗[(N[∧θ(ξl_),∧θ(ξl¯)]⊕1[ψu_(ω),ψu¯(ω)])⊗[α_,α¯]]}(20)

Theorem 3.4 Let Ξ,Φ∈F(Θ), Ψ∈F(Ω) and Ξ∈[ξl,ξu], Φ∈[φl,φu], Ψ∈[ψl,ψu]. If ↦ is a QL-interval implication, then the conclusion of the upper and lower bounds of the α-UTI method can be found as below:

∨θ{[∧θ(φl_),∧θ(φl¯)]⊗[(N([∨θ(ξu_),∨θ(ξu¯)])⊕1([∧θ(ξl_),∧θ(ξl¯)]⊗1[ψl_(ω),ψl¯(ω)]))⊗[α_,α¯]]}≤UTI1(ω)≤∨θ{[∨θ(φu_),∨θ(φu¯)]⊗[(N([∧θ(ξl_),∧θ(ξl¯)])⊕1([∨θ(ξu_),∨θ(ξu¯)]⊗1[ψu_(ω),ψu¯(ω)]))⊗[α_,α¯]]}(21)

In light of Theorem 3.1 to Theorem 3.4, in the case of an individual rule, if related operators including the interval t-norm, the interval t-conorm and so on, then the α-UTI method is stable with regard to S-, QL- and interval t-norm implication. The α-UTI method is stable with respect to R-interval implication if the R-interval implication and the interval t-norm are continuous. All in all, the α-UTI method is robust for the individual rule.

Suppose that for the α-UTI method, there exists a perturbation array of input (Ξ,Ψ,Φ) as follows:

([ξlm,ξum],[ψlm,ψum],[φlm,φum])(m=1,2,…).(22)

Besides, the consequent properties are satisfied:

limm→∞∨θ∈Θ(ξum−ξlm)=limm→∞∨ω∈Ω(ψum−ψlm)=limm→∞∨θ∈Θ(φum−φlm)=0.(23)

In detail, we have from (23) that

limm→∞∨θ∈Θ(ξum−ξlm)=limm→∞∨θ∈Θ([ξum_,ξum¯]−[ξlm_,ξlm¯])=limm→∞∨θ∈Θ{max(|ξum_−ξlm_|,|ξum¯−ξum¯|)}=0.(24)

Others can be analogously expanded.

Considering that λlm,λum are given as upper and lower bounds on the output of the α-UTI method, it means that for ([ξlm,ξum],[ψlm,ψum],[φlm,φum])(m=1,2,…), one has

UTI1(ω)∈[λlm,λum](25)

When the continuity condition is satisfied, then

limm→∞∨θ∈Θ(λrm−λum)=0(26)

holds. In other words, if the above-mentioned continuity condition is effective, the outcome of the α-UTI method stably converges to a value. As a consequence, in light of the stable definition given in Definition 2.12, the α-UTI method is stable in the individual rule case.

For Theorem 3.1, we give an intelligent algorithm (see the following Algorithm 1) to deal with such operations. For the other theorems, we can get corresponding algorithms.

Fig. 2 shows the flow chart of Algorithm 1.

Figure 2: The flow chart of Algorithm 1

4  Interval Perturbation of the α-UTI Method for Multiple Rules

For multiple rules, the notation for representing fuzzy intervals is similar for the individual rule. It is denoted that (ω∈Ω)

UTIn(ω)=UTI(Ξ1,....,Ξn,Ψ1,....,Ψn,Φ)(ω)(27)

Theorem 4.1 Assume that Ξi,Φ∈F(Θ), Ψi∈F(Ω) where Ξi∈[ξli,ξui], Φ∈[φl,φu], Ψi∈[ψli,ψui] (i=1,⋯,n). When ↦ is an S-interval implication, the result of the α-UTI method is as follows:

∨θ{[∧θ(φl_),∧u(φl¯)]⊗[∧i=1n(N[∨θ(ξui_),∨θ(ξui¯)]⊕1[ψli_(ω),ψli¯(ω)])⊗[α_,α¯]]}≤UTIn(ω)≤∨θ{[∨θ(φu_),∨u(φu¯)]⊗[∧i=1n(N[∧θ(ξli_),∧θ(ξli¯)]⊕1[ψui_(ω),ωui¯(ω)])⊗[α_,α¯]]}.(28)

Proof: When ↦ is an S-interval implication, there are an interval t-conorm ⊕1 and an interval fuzzy negation N letting

Q1↦⊕1,NQ2=N(Q1)⊕1Q2(29)

hold for two fuzzy sets Q1,Q2.

It follows from Lemma 2.1 and Lemma 2.2 that one has

∨θ{[∧θ(φl_),∧θ(φl¯)]⊗[∧i=1n(N[∨θ(ξui_),∨θ(ξui¯)]⊕1[ψli_(ω),ψli¯(ω)])⊗[α_,α¯]]}=∨θ∈Θ{∧θ∈Θ[φl_(θ),φl¯(θ)]⊗[∧i=1n(N(∨θ∈Θ[ξui_(θ),ξui¯(θ)])⊕1[ψli_(ω),ψli¯(ω)])⊗[α_,α¯]]}≤UTIn(ω).(30)

In a similar structure, we can also find

UTIn(ω)=∨θ∈Θ{[Φ_(θ),Φ¯(θ)]⊗[∧i=1n(N[Ξi_(θ),Ξi¯(θ)]⊕1[Ψi_(ω),Ψi¯(ω)])⊗[α_,α¯]]}≤∨θ∈Θ{∨θ∈Θ[φu_(θ),φu¯(θ)]⊗[∧i=1n(N(∧θ∈Θ[ξli_(θ),ξli¯(θ)])⊕1[ψui_(ω),ψui¯(ω)])⊗[α_,α¯]]}=∨θ{[∨θ(φu_),∨θ(φu¯)]⊗[∧i=1n(N[i~θ(ξli_), i~θ(ξli¯)]⊕1[ψui_(ω),ψui¯(ω)])⊗[α_,α¯]]}.(31)

To sum up, we can find that (28) is valid. The proof is accomplished.

Theorem 4.2, Theorem 4.3 and Theorem 4.4 can be proved in an analogous mode, in which four kinds of unified interval implications are employed in turn.

Theorem 4.2 Assume that Ξi,Φ∈F(Θ), Ψi∈F(Ω) where Ξi∈[ξli,ξui], Φ∈[φl,φu], Ψi∈[ψli,ψui](i=1,⋯,n). When ↦ is an R-interval implication, the conclusion of the upper and lower bounds of the α-UTI method can be given as below:

∨θ{[∧θ(φl_),∧θ(φl¯)]⊗[∧i=1n([∨θ(ξui_),∨θ(ξui¯)]↦[ψli_(ω),ψli¯(ω)])⊗[α_,α¯]]}≤UTIn(ω)≤∨θ{[∨θ(φu_),∨θ(φu¯)]⊗[∧i=1n([∧θ(ξli_),∧θ(ξli¯)]↦[ψui_(ω),ψui¯(ω)])⊗[α_,α¯]]}.(32)

Theorem 4.3 Assume that Ξi,Φ∈F(Θ), Ψi∈F(Ω) where Ξi∈[ξli,ξui], Φ∈[φl,φu], Ψi∈[ψli,ψui](i=1,⋯,n). When ↦ is an interval t-norm implication, the result of the α-UTI method is as follows:

∨θ{[∧θ(φl_),∧θ(φl¯)]⊗[∧i=1n([∧θ(ξli_),∧θ(ξli¯)]↦[ψli_(ω),ψli¯(ω)])⊗[α_,α¯]]}≤UTIn(ω)≤∨θ{[∨θ(φu_),∨θ(φu¯)]⊗[∧i=1n([∨θ(ξui_),∨θ(ξui¯)]↦[ψui_(ω),ψui¯(ω)])⊗[α_,α¯]]}.(33)

Theorem 4.4 Assume that Ξi,Φ∈F(Θ), Ψi∈F(Ω) where Ξi∈[ξli,ξui], Φ∈[φl,φu], Ψi∈[ψli,ψui](i=1,⋯,n). When ↦ is a QL-interval implication, the conclusion of the upper and lower bounds of the α-UTI method can be given as below:

∨θ{[∧θ(φl_),∧θ(φl¯)]⊗[∧i=1n(N([∨θ(ξui_),∨θ(ξui¯)])⊕1([∧θ(ξli_),∧θ(ξli¯)]⊗1[ψli_(ω),ψli¯(ω)]))⊗[α_,α¯]]}≤UTIn(ω)≤∨θ{[∨θ(φu_),∨θ(φu¯)]⊗[∧i=1n(N([∧θ(ξli_),∧θ(ξli¯)])⊕1([∨θ(ξui_),∨θ(ξui¯)]⊗1[ψui_(ω),ψui¯(ω)]))⊗[α_,α¯]]}.(34)