Fuzzy Reasoning Symmetric Quintuple-Implication Method for Mixed Information and Its Application

Abstract

Rule-based reasoning with different kinds of uncertain information has been identified in numerous applications within the real world. Any reasoning method must be able to coherently obtain the inference result by composing the given if–then rule with the assertion of the given input. The symmetric quintuple-implication principle was established by introducing symmetry into the five implication operators included. For example, the first, third and fifth implication operators exhibit symmetric properties, i.e., the three implication operators are taken as the same kind of operator and the second and fourth implication operators satisfy symmetry, that is, the two implication operators take the same kind of operator. So, the reasoning method induced by this principle possesses significant advantages in terms of its logical foundation and reductivity. This paper derives and studies reasoning methods for the mixture of fuzzy information and intuitionistic fuzzy information based on the symmetric quintuple-implication principle where all five implication operators satisfy symmetry (also called the quintuple-implication principle). Such reasoning methods are based on the ideas that the input and the given if–then rule can be combined for calculation only when their information representations exhibit consistency. An inconsistent given if–then rule with two different representations should be regarded as the composition of two different consistent given if–then rules with their own unique representations. This paper then elaborates on the methods by employing the possibility and necessity operators and the quintuple-implication principle from the perspective of whether the representation of rule antecedent and rule consequent is consistent or not. The reductivity of all the proposed reasoning methods is also analyzed in detail. This paper mainly contributes to the development of a novel mixed information reasoning framework, along with the introduction of the quintuple-implication principle into reasoning with mixed information. The proposed methods have also been applied to pattern recognition, and several experiments demonstrate that the mixed information reasoning methods based on the quintuple-implication principle are superior to the corresponding methods based on the triple I principle.

1. Introduction

The main pattern of reasoning is to draw conclusions from given if–then rules and uncertain information. In actuality, the types of information are complex and diverse. For example, Zadeh [1] proposed the concept of a fuzzy set (FS) to describe the ambiguity and uncertainty of information. As a generalization of the fuzzy set concept, Atanassov [2] proposed the concept of an intuitionistic fuzzy set (IFS), which utilizes both the membership function and the non-membership function to elaborate on the uncertainty of information more precisely. When it comes to reasoning based on if–then rules and the input information, there could exist circumstances where the information representation in the input is inconsistent with the information representation in the rules. However, different information representations correspond to different reasoning mechanisms. Fuzzy set-based reasoning methods and intuitionistic fuzzy set-based reasoning methods will not be able to solve the inference problem with mixed information. It is hoped that in a hybrid information representation environment, the obtained conclusions will be still valid and interpretable. Therefore, the core of this paper is to study reasoning methods when the problem of interest is filled with mixed information, specifically information represented by a mixture of fuzzy sets and intuitionistic fuzzy sets. This inference problem can also be interpreted as an extended version of modus ponens (MP) with a mixture of fuzzy information and intuitionistic fuzzy information. Here, the rule antecedent and rule consequent can be expressed by fuzzy sets or by intuitionistic fuzzy sets or by the mixture of fuzzy sets and intuitionistic fuzzy sets; the input can be expressed by fuzzy sets or by intuitionistic fuzzy sets. This leads to the following inference problem: Given a nonempty universe of discourse X as the input space, a nonempty universe of discourse Y as the output space, and for all $x\in X$ and $y\in Y$,

$$\begin{array}{c}\mathrm{The}\mathrm{given}\mathrm{rule}:\hspace{1em}\mathrm{If}x\mathrm{is}\mathbb{A},\mathrm{then}y\mathrm{is}\mathbb{B}\hfill \\ \underline{\mathrm{Input}:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\mathrm{is}\mathbb{A}*\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} }\hfill \\ \mathrm{Output}:y\mathrm{is}?\hfill \end{array}$$

where $\mathbb{A}$ and $\mathbb{A}*$ may be fuzzy sets or intuitionistic fuzzy sets of universe X, and $\mathbb{B}$ may be a fuzzy set or an intuitionistic fuzzy set of universe Y.

1.1. The Inference Method Within the Fuzzy Environment

On the basis of fuzzy sets, Wang [3] gave the inference problem, called fuzzy modus ponens (FMP), that is, the inference problem related to the given rule “if x is A, then y is B” and the input “x is $A*$”, where A and $A*$ are fuzzy sets of the universe X, B is a fuzzy set of the universe Y and the output $B*$ is also a fuzzy set of Y. For solving the FMP, Zadeh [4] introduced a compositional rule of inference (CRI). Although it is widely employed in engineering and other fields, the CRI lacks a logical foundation and it fails to satisfy reductivity. Therefore, the results derived from the CRI are not always consistent with human intuition. To overcome this deficiency, Wang [3,5] proposed the triple I principle, which interprets the inference result $B*$ as the solution most supported by the given if–then rule, providing a rigorous logical basis for fuzzy reasoning. Pei [6] put forward a triple I method based on all residuated implications induced by left continuous t-norms. In the reasoning processes of the CRI and the triple I method, the degree to which $A*$ is similar to A is not taken into account. This may give rise to circumstances in which calculations are either meaningless or yield misleading outcomes. Hence, Zhou et al. [7] proposed the quintuple-implication principle for fuzzy reasoning. So far, the quintuple-implication principle has been studied by many scholars. Luo et al. [8] laid down a rigorous logical foundation for the quintuple-implication principle of fuzzy reasoning. Li et al. [9] analyzed the performance of a fuzzy system that satisfies the quintuple-implication principle and studied the robustness and general approximation ability of quintuple-implication reasoning methods under different implications, such as R-implication, S-implication and QL-implication.

1.2. The Inference Method Within the Intuitionistic Fuzzy Environment

An intuitionistic fuzzy modus ponens (IFMP) is a generalization of the FMP from fuzzy sets to intuitionistic fuzzy sets, namely, combining the given rule “if x is $\mathcal{A}$, then y is $\mathcal{B}$” with an input “x is $\mathcal{A}*$” in order to obtain a new output “y is $\mathcal{B}*$”, where $\mathcal{A}$ and $\mathcal{A}*$ are intuitionistic fuzzy sets of the universe X, and $\mathcal{B}$ and $\mathcal{B}*$ are intuitionistic fuzzy sets of the universe Y. Cornelis et al. [10] proffered the residual intuitionistic-implication operator and undertook an in-depth exploration of the CRI for intuitionistic fuzzy reasoning. Zheng et al. [11] generalized the triple I method to intuitionistic fuzzy systems and investigated the reductivity of the triple I method for the IFMP. Li et al. [12] investigated the intuitionistic residual implications and the intuitionistic given if–then rule. Moreover, the approximation properties of the intuitionistic Mamdani, the intuitionistic Larsen and the intuitionistic triple I fuzzy systems are obtained. Duan et al. [13] explored four kinds of intuitionistic similarity and utilized them to analyze the robustness for the triple I method of intuitionistic fuzzy reasoning. Zeng et al. [14] discussed the quintuple-implication principle of intuitionistic fuzzy reasoning by using the intuitionistic t-norm generated by the left continuous t-norm.

1.3. Motivation

The research goal of this paper is to propose a new perspective on solving mixed information reasoning problems to make up for the shortcomings of the existing mixed information reasoning methods and to enrich the research on mixed information reasoning. Specifically, this paper attempts to build a novel framework for reasoning with a mixture of fuzzy information and intuitionistic fuzzy information and then develops the corresponding mixed information reasoning methods under this framework. Concerning reasoning with if–then rules under mixed information types of fuzzy sets and intuitionistic fuzzy sets, there are several reasons supporting this study.

The first comes from the construction of reasoning methods that are closely related to the given rule as well as the way information is represented. Obviously, the FMP and IFMP belong to the case in which the representations of rule antecedent $\mathbb{A}$ and rule consequent $\mathbb{B}$ in the given if–then rule are consistent, either by fuzzy sets or by intuitionistic fuzzy sets. Since the input $\mathbb{A}*$ is expressed in the same way with the representations of $\mathbb{A}$ and $\mathbb{B}$, the induced inference methods are constructed in fuzzy systems with a unique representation according to the existing reasoning principles, such as the triple-implication principle and the quintuple-implication principle. However, the representation of $\mathbb{A}*$ may be different from those of $\mathbb{A}$ and $\mathbb{B}$, or the representations of $\mathbb{A}$ and $\mathbb{B}$ may not be the same. In these cases, reasoning methods based on fuzzy sets or on intuitionistic fuzzy sets cannot directly reach reasonable and effective conclusions.

The second reason comes from the given if–then rules. It has been found that there are eight types of mixture scenarios that can arise, summarized in

Table 1. The mixed information inference problems.

Type	The Given Rule	Rule Antecedent	Rule Consequent	Input	Output
1	Consistent	FS	FS	FS	FS
2		IFS	IFS	IFS	IFS
3		FS	FS	IFS	?
4		IFS	IFS	FS	?
5	Inconsistent	FS	IFS	FS	?
6		IFS	FS	IFS	?
7		FS	IFS	IFS	?
8		IFS	FS	FS	?

Note: “?” indicates the unknown information type.

with regard to the issue whether the representation of information in the given rules is consistent. Here, it is assumed that the information types of $\mathbb{A}$, $\mathbb{A}*$ and $\mathbb{B}$ can be arbitrarily set to be fuzzy or intuitionistic fuzzy. Accordingly, the quality of the inference result depends largely on how to use the given if–then rules to design the inference method that will inevitably involve the conversion between different information types. In designing an inference procedure, it is natural to let the given rules speak for themselves. In other words, the information conversion of the given rule will be carried out only when the given rule itself has a mixed representation, in which case a given if–then rule with mixed information types will consist of two given if–then rules with different single information types.

The third reason comes from the current research status on mixed information reasoning problems and the conversion between different information types. In terms of converting intuitionistic fuzzy sets to fuzzy sets, Atanassov suggested in [2] the possibility operator and the necessity operator to characterize this conversion. Ref. [15] has used this conversion operation from intuitionistic fuzzy sets to fuzzy sets, as well as a simple conversion operation from fuzzy sets to intuitionistic fuzzy sets by setting the non-membership degree as 1-membership degree, to explore the reasoning methods for a fuzzy and intuitionistic fuzzy information mixture on the basis of the triple-implication principle. But in some cases, the inference results given by this reasoning principle are often meaningless, without considering the connection between rule antecedent and input. The application examples described at the end of this paper also illustrate this point. Moreover, the work on mixed information reasoning is currently limited. It merits further investigation to tackle mixed information inference problems based on the quintuple-implication principle for the sake of overcoming the defects of existing methods.

On the basis of the aforementioned analysis, a new class of mixed information reasoning methods is proposed in this paper based on the quintuple-implication principle for the FMP in [7] and for the IFMP in [14]. The proposed methods are grounded in the conversion operators between fuzzy sets and intuitionistic fuzzy sets as presented in [2], starting from whether the given rules are consistent, so as to address the issues listed in Table 1. Moreover, in view of the limited available datasets on mixed information reasoning, this paper collected the datasets represented by fuzzy sets or intuitionistic fuzzy sets from the relevant literature in the field of fuzzy reasoning and constructed the corresponding datasets suitable for the mixed information inference problems studied. These constructed datasets were used to test the effectiveness of the reasoning methods built in this paper. Considering that the pattern recognition problem can be easily solved by transforming it into an inference problem, the datasets constructed in this paper belong to the application scope of pattern recognition, with each dataset containing both fuzzy set and intuitionistic fuzzy set representations. The proposed reasoning methods were compared with the methods in [15] on these constructed datasets.

To end this section, the contribution and structure of this paper are exhibited. The present paper is devoted to the study of solving the inference problems using information that is a hybrid between two different representation forms in a more reasonable way. The main contribution of this work is summarized as below:

1.

A new perspective for addressing the mixed information reasoning problem is offered, as well as the corresponding inference framework designed based on whether the representation of the given if–then rules is consistent;

2.

Using the quintuple-implication principle in fuzzy reasoning and the conversion operation between fuzzy sets and intuitionistic fuzzy sets, the details of the specific methods for all types of mixed information inference models are developed under the established inference framework, accompanied by the discussion of the reductivity of all the methods;

3.

Several pattern recognition tasks with a mixture of fuzzy information and intuitionistic fuzzy information are built to test the proposed reasoning methods, which indicates that these new methods overcome problems with the triple-implication principle and possess a significant advantage in addressing mixed information reasoning tasks.

The structure of this paper is as follows. Section 2 gives some necessary definitions and conclusions. In Section 3, the quintuple-implication reasoning methods to reason with mixed information are constructed from two directions of consistency and inconsistency for the expression of the given if–then rules. Section 4 discusses the reductivity of the proposed methods. In Section 5, four pattern recognition experiments are given to verify that the work presented in this paper is reasonable and effective. The last part is the conclusion.

2. Preliminaries

This section provides some important definitions and conclusions that will be utilized in the sequel.

Definition 1 

([1]). A fuzzy set A on a universe of discourse X is characterized by a membership function $A:X⟶[0,1]$ with the form $A=\{\langle x,A(x)\rangle |x\in X\}$, where for each element x of X, a number $A\left(x\right)$ in the interval $[0,1]$ is interpreted as the degree of membership of x in A. If A satisfies the condition that $\exists x_0\in X,A\left(x_0\right)=1$, then A is called normal. The set of all fuzzy sets on universe X is denoted as $FS\left(X\right)$.

Definition 2 

([2]). An intuitionistic fuzzy set $\mathcal{A}$ on a universe of discourse X is defined by a membership function ${\mathcal{A}}_t:X⟶[0,1]$ and a non-membership function ${\mathcal{A}}_f:X⟶[0,1]$ under the condition that ${\mathcal{A}}_t\left(x\right)+{\mathcal{A}}_f\left(x\right)\in [0,1]$ for all $x\in X$. Here, write $\mathcal{A}=\left\{\langle x,{\mathcal{A}}_t\left(x\right),{\mathcal{A}}_f\left(x\right)\rangle \right|x\in X\}$ and $\mathcal{A}\left(x\right)=\langle {\mathcal{A}}_t\left(x\right),{\mathcal{A}}_f\left(x\right)\rangle$. If $\mathcal{A}$ satisfies the condition that $\exists x_0\in X,\mathcal{A}\left(x_0\right)=\langle 1,0\rangle$, then $\mathcal{A}$ is called normal. The set of all intuitionistic fuzzy sets on X is denoted as $IFS\left(X\right)$. The intuitionistic fuzzy set $\mathcal{A}$ degenerates into fuzzy sets if and only if ${\mathcal{A}}_t\left(x\right)+{\mathcal{A}}_f\left(x\right)=1$ for all $x\in X$.

Moreover, an intuitionistic fuzzy number α is denoted by $\alpha =\langle {\alpha }_t,{\alpha }_f\rangle$, where ${\alpha }_t,{\alpha }_f\in [0,1]$ and ${\alpha }_t+{\alpha }_f\le 1$. The set of all intuitionistic fuzzy numbers is denoted as $\mathcal{I}*$. A partial order ⪯ on $\mathcal{I}*$ can be defined as ${\alpha }_1⪯{\alpha }_2$ if and only if ${\alpha }_{1t}\le {\alpha }_{2t}$ and ${\alpha }_{1f}\ge {\alpha }_{2f}$ for all ${\alpha }_1,{\alpha }_2\in \mathcal{I}*$. Intuitionistic fuzzy sets ${\mathcal{A}}_1={\mathcal{A}}_2$ if and only if ${{\mathcal{A}}_1}_t\left(x\right)={{\mathcal{A}}_2}_t\left(x\right)$ and ${{\mathcal{A}}_1}_f\left(x\right)={{\mathcal{A}}_2}_f\left(x\right)$ for all ${\mathcal{A}}_1,{\mathcal{A}}_2\subseteq IFS\left(X\right)$.

Definition 3 

([2]). Let $\mathcal{A}=\left\{\langle x,{\mathcal{A}}_t\left(x\right),{\mathcal{A}}_f\left(x\right)\rangle \right|x\in X\}$ be an intuitionistic fuzzy set. Then, the necessity operator is a mapping ♮: $IFS\left(X\right)⟶FS\left(X\right)$ such that $♮\mathcal{A}=\left\{\langle x,{\mathcal{A}}_t\left(x\right)\rangle \right|x\in X\}$; the possibility operator is a mapping ♯: $IFS\left(X\right)⟶FS\left(X\right)$ such that $♯\mathcal{A}=\left\{\langle x,{\mathcal{A}}_{-f}\left(x\right)\rangle \right|x\in X\}$, where ${\mathcal{A}}_{-f}\left(x\right)=1-{\mathcal{A}}_f\left(x\right)$.

Proposition 1 

([2]). Let ¶: $FS\left(X\right)⟶IFS\left(X\right)$ be a mapping and $A=\{\langle x,A(x)\rangle |x\in X\}$ be a fuzzy set. Then, the set $\P A=\{\langle x,A(x),1-A(x)\rangle |x\in X\}$ is an intuitionistic fuzzy set.

Definition 4 

([16]). A t-norm is a binary function $\otimes :[0,1]\times [0,1]⟶[0,1]$, which is commutative, associative, increasing in each variable and contains neutral element 1 such that $x\otimes 1=x$ for all $x\in [0,1]$. A t-conorm is a binary function $\oplus :[0,1]\times [0,1]⟶[0,1]$, which is commutative, associative, increasing in each variable and contains neutral element 0 such that $x\oplus 0=x$ for all $x\in [0,1]$. The t-conorm ⊕ given by $a\oplus b=1-(1-a)\otimes (1-b)$ is said to be the dual t-conorm of ⊗, with the pair $(\otimes ,\oplus )$ as a dual pair. Analogously, the t-norm ⊗ defined by $a\otimes b=1-(1-a)\oplus (1-b)$ is called the dual t-norm of ⊕, with the pair $(\otimes ,\oplus )$ as a dual pair.

Definition 5 

([16]). A binary function $\to :[0,1]\times [0,1]⟶[0,1]$ is called a residual implication (R-implication) if there exists a left-continuous t-norm ⊗ such that $a\to b=\bigvee \{x\in [0,1\left]\right|x\otimes a\le b\}$ for all $a,b\in [0,1]$. Further, ⊗ and → form an adjoint pair $(\otimes ,\to )$ and satisfy the residuation property, i.e., $c\otimes a\le b$ if and only if $c\le a\to b$ for all $a,b,c\in [0,1]$.

Example 1 

([16]). The Łukasiewicz t-norm ${\otimes }_{\mathsf{Ł}u}$ and the corresponding Łukasiewicz implication ${\to }_{\mathsf{Ł}u}$ are expressed as follows: $a{\otimes }_{\mathsf{Ł}u}b=(a+b-1)\vee 0$, $a{\to }_{\mathsf{Ł}u}b=(1-a+b)\wedge 1$ for all $a,b\in [0,1]$.

Definition 6 

([11]). A binary function ${\otimes }_{\mathcal{I}*}:\mathcal{I}*\times \mathcal{I}*⟶\mathcal{I}*$ is called an intuitionistic t-norm induced from t-norm ⊗, if it satisfies $\alpha {\otimes }_{\mathcal{I}*}\beta =\langle {\alpha }_t\otimes {\beta }_t,{\alpha }_f\oplus {\beta }_f\rangle$ for all $\alpha =\langle {\alpha }_t,{\alpha }_f\rangle ,\beta =\langle {\beta }_t,{\beta }_f\rangle \in \mathcal{I}*$.

A binary function ${\oplus }_{\mathcal{I}*}:\mathcal{I}*\times \mathcal{I}*⟶\mathcal{I}*$ is called an intuitionistic t-conorm induced from t-conorm ⊕, if it satisfies $\alpha {\oplus }_{\mathcal{I}*}\beta =\langle {\alpha }_t\oplus {\beta }_t,{\alpha }_f\otimes {\beta }_f\rangle$ for all $\alpha =\langle {\alpha }_t,{\alpha }_f\rangle ,\beta =\langle {\beta }_t,{\beta }_f\rangle \in \mathcal{I}*$, where ⊕ is the dual t-conorm of the t-norm ⊗.

Definition 7 

([11]). A binary operator ${\to }_{\mathcal{I}*}:\mathcal{I}*\times \mathcal{I}*⟶\mathcal{I}*$ is called an intuitionistic residual implication (intuitionistic R-implication) if there exists a left-continuous intuitionistic t-norm ${\otimes }_{\mathcal{I}*}$ such that $\alpha {\to }_{\mathcal{I}*}\beta =\bigvee \{\theta \in \mathcal{I}*|\theta {\otimes }_{\mathcal{I}*}\alpha ⪯\beta \}$, $\forall \alpha ,\beta \in \mathcal{I}*$. Further, ${\otimes }_{\mathcal{I}*}$ and ${\to }_{\mathcal{I}*}$ form an adjoint pair $({\otimes }_{\mathcal{I}*},{\to }_{\mathcal{I}*})$ and satisfy the residuation property, i.e., $\forall \alpha ,\beta ,\gamma \in \mathcal{I}*$, $\gamma {\otimes }_{\mathcal{I}*}\alpha ⪯\beta$ if and only if $\gamma ⪯\alpha {\to }_{\mathcal{I}*}\beta$.

Example 2 

([11]). The intuitionistic Łukasiewicz t-norm ${\otimes }_{{\mathcal{I}}_{\mathsf{Ł}u}^{*}}$ and the corresponding intuitionistic R-implication ${\to }_{{\mathcal{I}}_{\mathsf{Ł}u}^{*}}$ are expressed as follows: $\alpha {\otimes }_{{\mathcal{I}}_{\mathsf{Ł}u}^{*}}\beta =\langle ({\alpha }_t+{\beta }_t-1)\vee 0,({\alpha }_f+{\beta }_f)\wedge 1\rangle$, $\alpha {\to }_{{\mathcal{I}}_{\mathsf{Ł}u}^{*}}\beta =\langle (1-{\alpha }_t+{\beta }_t)\wedge (1-{\alpha }_f+{\beta }_f)\wedge 1,({\beta }_f-{\alpha }_f)\vee 0\rangle$ for all $\alpha ,\beta \in {\mathcal{I}}^{*}$.

Quintuple-Implication Principle for FMP 

([7]). Let → be an R-implication derived from a left-continuous t-norm ⊗ and then $B*$ is the smallest fuzzy subset on Y such that there is the following:

$$(A\left(x\right)\to B\left(y\right))\to ((A*\left(x\right)\to A\left(x\right))\to (A*\left(x\right)\to B*\left(y\right)))=1forallx\in X,y\in Y,$$

where $A,A*$ are fuzzy sets on X, and B is fuzzy sets on Y.

Theorem 1 

([7]). Let → be an R-implication derived from a left-continuous t-norm ⊗ and then the quintuple-implication solution $B*=\left\{\langle y,B*\left(y\right)\rangle \right|y\in Y\}$ for the FMP problem is presented as follows:

$$\begin{array}{c}\hfill B*\left(y\right)=\underset{x\in X}{\vee }\{A*\left(x\right)\otimes ((A*\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to B\left(y\right)))\},y\in Y.\end{array}$$

(1)

Quintuple-Implication Principle for FMP 

([14]). Let ${\to }_{\mathcal{I}*}$ be an intuitionistic R-implication derived from a left-continuous intuitionistic t-norm ${\otimes }_{\mathcal{I}*}$ and then $\mathcal{B}*$ is the smallest intuitionistic fuzzy subset on Y such that there is the following:

$$\left(\mathcal{A}\left(x\right){\to }_{\mathcal{I}*}\mathcal{B}\left(y\right)\right){\to }_{\mathcal{I}*}\left((\mathcal{A}*\left(x\right){\to }_{\mathcal{I}*}\mathcal{A}\left(x\right)){\to }_{\mathcal{I}*}(\mathcal{A}*\left(x\right){\to }_{\mathcal{I}*}\mathcal{B}*\left(y\right))\right)=\langle 1,0\rangle forallx\in X,y\in Y,$$

where $\mathcal{A}$, $\mathcal{A}*$ are intuitionistic fuzzy sets on X, and $\mathcal{B}$ is intuitionistic fuzzy sets on Y.

Theorem 2 

([14]). Let ${\to }_{\mathcal{I}*}$ be an intuitionistic R-implication derived from a left-continuous intuitionistic t-norm ${\otimes }_{\mathcal{I}*}$ and then the quintuple-implication solution $\mathcal{B}*=\left\{\langle y,{\mathcal{B}}_t^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)\rangle \right|y\in Y\}$ for the IFMP problem is presented as follows:

$$\begin{array}{c}\hfill \mathcal{B}*\left(y\right)=\underset{x\in X}{\vee }\{\mathcal{A}*\left(x\right){\otimes }_{\mathcal{I}*}\left((\mathcal{A}*\left(x\right){\to }_{\mathcal{I}*}\mathcal{A}\left(x\right)){\otimes }_{\mathcal{I}*}\left(\mathcal{A}\left(x\right){\to }_{\mathcal{I}*}\mathcal{B}\left(y\right)\right)\right)\},y\in Y,\end{array}$$

(2)

and

$$\begin{array}{cc}\hfill {\mathcal{B}}_t^{*}\left(y\right)=& \underset{x\in X}{\vee }\{{\mathcal{A}}_t^{*}\left(x\right)\otimes ((({\mathcal{A}}_t^{*}\left(x\right)\to {\mathcal{A}}_t\left(x\right))\wedge ({\mathcal{A}}_{-f}^{*}\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right)))\hfill \\ \hfill & \otimes (({\mathcal{A}}_t\left(x\right)\to {\mathcal{B}}_t\left(y\right))\wedge ({\mathcal{A}}_{-f}\left(x\right)\to {\mathcal{B}}_{-f}\left(y\right)))\left)\right\},y\in Y,\hfill \\ \hfill {\mathcal{B}}_f^{*}\left(y\right)=& \underset{x\in X}{\wedge }\{{\mathcal{A}}_f^{*}\left(x\right)\oplus ((1-{\mathcal{A}}_{-f}^{*}\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right))\oplus (1-{\mathcal{A}}_{-f}\left(x\right)\to {\mathcal{B}}_{-f}\left(y\right)))\},y\in Y,\hfill \end{array}$$

where ${\mathcal{A}}_{-f}=1-{\mathcal{A}}_f$, ${\mathcal{A}}_{-f}^{*}=1-{\mathcal{A}}_f^{*}$ and ${\mathcal{B}}_{-f}=1-{\mathcal{B}}_f$, and ⊕ is the dual t-conorm of t-norm ⊗.

3. The Quintuple-Implication Fuzzy Reasoning Method with Mixed Information

For reasoning with given if–then rules of mixed information, since there may exist the mismatch between the representation of the input and that of the given rules of the knowledge base in real life, the ability of a quintuple-implication reasoning method based on single information is limited. From Table 1, it is easy to find that the information types in both rule antecedent and rule consequent are consistent in Types 1–4, whereas they exhibit inconsistency in Types 5–8. Considering this analysis, in this section, we will delve into the quintuple-implication principle for reasoning with mixed information from these two distinct perspectives of the given if–then rules.

The reasoning framework to solve inference problems Type 1–Type 8

As for the inference models with consistent given if–then rules (Types 1–4), the input information first needs to be consistent with the information representation in the given rules. For example, as for Type 3, the input $\mathcal{A}*$ has the priority to be converted into the same information type $A*$ as the given rule “if x is A, then y is B”; as for Type 4, the input $A*$ has the priority to be converted into an intuitionistic fuzzy set $\mathcal{A}*$, same as the representation of the given rule “if x is $\mathcal{A}$, then y is $\mathcal{B}$”. Then, the subsequent inference steps can be performed to obtain the output with the same representation as the rule, together with the inference result consistent with the representation of the input by employing the reciprocal conversion formula between fuzzy sets and intuitionistic fuzzy sets. Figure 1 shows the processing framework of the first four types of inference models.

Regarding the inference model Types 5–8 with an inconsistency in the representation of rule antecedent and rule consequent, a given rule containing mixed information needs to be initially converted into a set of two distinct given rules, each having different single information representation through the corresponding conversion operations. For example, as for Types 5 and 7, the inconsistent given rule “if x is A, then y is $\mathcal{B}$” is considered to consist of two consistent given rules “if x is A, then y is B” with B converted from $\mathcal{B}$ and “if x is $\mathcal{A}$, then y is $\mathcal{B}$” with $\mathcal{A}$ converted from A; as for Types 6 and 8, the inconsistent given rule “if x is $\mathcal{A}$, then y is B” is viewed as being composed of two consistent given rules “if x is A, then y is B” with A converted from $\mathcal{A}$ and “if x is $\mathcal{A}$, then y is $\mathcal{B}$” with $\mathcal{B}$ converted from B. Next, the input is combined with each of these two given rules, respectively, to carry out the subsequent inference process. At this stage, the two directions will yield their own outputs and the processing in both directions will be found to be partially similar to the reasoning scheme based on consistent given if–then rules. Ultimately, these outputs are aggregated to produce a final inference result that can be expressed in a form consistent with the rule consequent or in alignment with the input. The processing scheme for reasoning under inconsistent given if–then rules is exhibited in Figure 2.

3.1. The Quintuple-Implication Fuzzy Reasoning Method with Mixed Information of Consistent Rules

To address the problems in reasoning under consistent rules, we first discuss the mixed information inference problem of Type 4, described as follows:

$$\begin{array}{c}\mathrm{The}\mathrm{given}\mathrm{rule}:\hspace{1em}\mathrm{If}x\mathrm{is}\mathcal{A},\mathrm{then}y\mathrm{is}\mathcal{B}\hfill \\ \underline{\mathrm{Input}:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\mathrm{is}A*\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} }\hfill \\ \mathrm{Output}:y\mathrm{is}?\hfill \end{array}$$

where $\mathcal{A}$ is an intuitionistic fuzzy set on the universe X, $\mathcal{B}$ is an intuitionistic fuzzy set on the universe Y and $A*$ is a fuzzy set on the universe X. We aim to derive an output based on the referential rule “if x is $\mathcal{A}$, then y is $\mathcal{B}$” and the given input “x is $A*$” by utilizing the quintuple-implication principle.

The method to reasoning with Type 4 mixed information

Step 1: Convert the fuzzy set $A*$ into the intuitionistic fuzzy set $\mathcal{A}*=\left\{\langle x,A*\left(x\right),1-A*\left(x\right)\rangle \right|x\in X\}$ via Proposition 1 in [2], which hence leads to the following problem:

$$\begin{array}{c}\mathrm{The}\mathrm{given}\mathrm{rule}:\hspace{1em}\mathrm{If}x\mathrm{is}\mathcal{A},\mathrm{then}y\mathrm{is}\mathcal{B}\hfill \\ \underline{\mathrm{Input}:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\mathrm{is}\mathcal{A}*\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} }\hfill \\ \mathrm{Output}:y\mathrm{is}\overline{\mathcal{B}}*\hfill \end{array}$$

where $\mathcal{A}$ and $\mathcal{A}*$ are intuitionistic fuzzy sets defined on the universe X, and $\mathcal{B}$ and $\overline{\mathcal{B}}*$ are intuitionistic fuzzy sets on the universe Y.

Step 2: Solve the inference problem in Step 1 and obtain the solution ${\overline{\mathcal{B}}}^{*}=\{\langle y,$${\overline{\mathcal{B}}}_t^{*}\left(y\right),$${\overline{\mathcal{B}}}_f^{*}\left(y\right)\rangle |y\in Y\}$ by applying Theorem 2 in [14] (i.e., quintuple-implication method for Type 2):

$${\overline{\mathcal{B}}}^{*}\left(y\right)=\underset{x\in X}{\vee }\left\{{\mathcal{A}}^{*}\left(x\right){\otimes }_{\mathcal{I}*}\left((\mathcal{A}*\left(x\right){\to }_{\mathcal{I}*}\mathcal{A}\left(x\right)){\otimes }_{\mathcal{I}*}\left(\mathcal{A}\left(x\right){\to }_{\mathcal{I}*}\mathcal{B}\left(y\right)\right)\right)\right\},y\in Y,$$

(3)

$$\begin{array}{cc}\hfill {\overline{\mathcal{B}}}_t^{*}\left(y\right)& =\underset{x\in X}{\vee }\{A*\left(x\right)\otimes \left(\right((A*\left(x\right)\to {\mathcal{A}}_t\left(x\right))\wedge (A*\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right))\hfill \\ \hfill & \hspace{1em}\otimes (({\mathcal{A}}_t\left(x\right)\to {\mathcal{B}}_t\left(y\right))\wedge ({\mathcal{A}}_{-f}\left(x\right)\to {\mathcal{B}}_{-f}\left(y\right)))\left)\right\}\hfill \\ \hfill & =\underset{x\in X}{\vee }\{A*\left(x\right)\otimes ((A*\left(x\right)\to {\mathcal{A}}_t\left(x\right))\otimes (({\mathcal{A}}_t\left(x\right)\to {\mathcal{B}}_t\left(y\right))\hfill \\ \hfill & \hspace{1em}\wedge ({\mathcal{A}}_{-f}\left(x\right)\to {\mathcal{B}}_{-f}\left(y\right))\left)\right)\},y\in Y,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\overline{\mathcal{B}}}_f^{*}\left(y\right)& =\underset{x\in X}{\wedge }\{(1-A*\left(x\right))\oplus ((1-A*\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right))\oplus (1-{\mathcal{A}}_{-f}\left(x\right)\to {\mathcal{B}}_{-f}\left(y\right)))\}\hfill \\ \hfill & =1-\underset{x\in X}{\vee }\{A*\left(x\right)\otimes ((A*\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right))\otimes ({\mathcal{A}}_{-f}\left(x\right)\to {\mathcal{B}}_{-f}\left(y\right)))\},y\in Y,\hfill \end{array}$$

(5)

where ${\to }_{\mathcal{I}*}$ is an intuitionistic R-implication derived from a left-continuous intuitionistic t-norm ${\otimes }_{\mathcal{I}*}$, → is an R-implication derived from a left-continuous t-norm ⊗ and ⊕ is the dual of ⊗.

Step 3: The solution of the Type 4 inference problem is given, as presented in

Table 2. The solutions for the mixed information inference problem of Type 4.

The Given Rule	Input	Output
$\begin{array}{c}\mathrm{If}x\mathrm{is}\mathcal{A},\mathrm{then}y\mathrm{is}\mathcal{B}\end{array}$	$x\mathrm{is}A*$	FS	$B*=\left\{\langle y,B*\left(y\right)\rangle \right|B*\left(y\right)\in [{\overline{\mathcal{B}}}_t^{*}\left(y\right),1-{\overline{\mathcal{B}}}_f^{*}\left(y\right)],y\in Y\}$
		IFS	$\mathcal{B}*=\left\{\langle y,{\mathcal{B}}_t^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)\rangle \right|{\mathcal{B}}_t^{*}\left(y\right)={\overline{\mathcal{B}}}_t^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)={\overline{\mathcal{B}}}_f^{*}\left(y\right),y\in Y\}$

Note: ${\overline{\mathcal{B}}}_t^{*}\left(y\right)=\underset{x\in X}{\vee }\left\{A*\left(x\right)\otimes ((A*\left(x\right)\to {\mathcal{A}}_t\left(x\right))\otimes (({\mathcal{A}}_t\left(x\right)\to {\mathcal{B}}_t\left(y\right))\wedge ({\mathcal{A}}_{-f}\left(x\right)\to {\mathcal{B}}_{-f}\left(y\right))))\right\},y\in Y$, ${\overline{\mathcal{B}}}_f^{*}\left(y\right)=1-\underset{x\in X}{\vee }\left\{A*\left(x\right)\otimes ((A*\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right))\otimes ({\mathcal{A}}_{-f}\left(x\right)\to {\mathcal{B}}_{-f}\left(y\right)))\right\},y\in Y$.

.

According to the rule “if x is $\mathcal{A}$, then y is $\mathcal{B}$”, the output can be obtained in the form of intuitionistic fuzzy sets, denoted as $\mathcal{B}*=\left\{\langle y,{\mathcal{B}}_t^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)\rangle \right|{\mathcal{B}}_t^{*}\left(y\right)={\overline{\mathcal{B}}}_t^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)={\overline{\mathcal{B}}}_f^{*}\left(y\right),y\in Y\}$. Furthermore, to ensure consistency with the information type of the input “x is $A*$”, we also present via Definition 3 in [2] the solution in the form of fuzzy sets, i.e., $B*=\left\{\langle y,B*\left(y\right)\rangle \right|B*\left(y\right)\in [{\overline{\mathcal{B}}}_t^{*}\left(y\right),1-{\overline{\mathcal{B}}}_f^{*}\left(y\right)],y\in Y\}$.

Analogously, the inference problem of Type 3 can also be addressed:

$$\begin{array}{c}\mathrm{The}\mathrm{given}\mathrm{rule}:\hspace{1em}\mathrm{If}x\mathrm{is}A,\mathrm{then}y\mathrm{is}B\hfill \\ \underline{\mathrm{Input}:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\mathrm{is}\mathcal{A}*\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} }\hfill \\ \mathrm{Output}:y\mathrm{is}?\hfill \end{array}$$

where A and B represent two fuzzy sets on X and Y, respectively, and $\mathcal{A}*$ is an intuitionistic fuzzy set on X.

The method to reasoning with Type 3 mixed information

Step 1: Convert the intuitionistic fuzzy sets $\mathcal{A}*$ into fuzzy set $A*=\left\{\langle x,A*\left(x\right)\rangle \right|x\in X\}$ by using the possibility and necessity operators stated in Definition 3 in [2], where $A*\left(x\right)\in [{\mathcal{A}}_t^{*}\left(x\right),{\mathcal{A}}_{-f}^{*}\left(x\right)]$. Since the fuzzy set $A*$ converted from the input $\mathcal{A}*$ is determined by two fuzzy sets ${\mathcal{A}}_t^{*}=♮\mathcal{A}*$ and ${\mathcal{A}}_{-f}^{*}=♯\mathcal{A}*$, to solve the Type 3 inference problem, the following two FMP problems need to be addressed in advance:

$$\begin{array}{c}\mathrm{The}\mathrm{given}\mathrm{rule}:\hspace{1em}\mathrm{If}x\mathrm{is}A,\mathrm{then}y\mathrm{is}B\hfill \\ \underline{\mathrm{Input}:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\mathrm{is}{\mathcal{A}}_t^{*}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} }\hfill \\ \mathrm{Output}:y\mathrm{is}{\tilde{B}}_1^{*}\hfill \end{array}$$

$$\begin{array}{c}\mathrm{The}\mathrm{given}\mathrm{rule}:\hspace{1em}\mathrm{If}x\mathrm{is}A,\mathrm{then}y\mathrm{is}B\hfill \\ \underline{\mathrm{Input}:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\mathrm{is}{\mathcal{A}}_{-f}^{*}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} }\hfill \\ \mathrm{Output}:y\mathrm{is}{\tilde{B}}_2^{*}\hfill \end{array}$$

where ${\mathcal{A}}_t^{*},{\mathcal{A}}_{-f}^{*}$ are fuzzy sets transformed from intuitionistic fuzzy set $\mathcal{A}*$ on X, and A and B represent two fuzzy sets on X and Y, respectively.

Step 2: Obtain the solutions to the above two inference problems. According to Theorem 1 in [7] (i.e., quintuple-implication method for Type 1),

$${\tilde{B}}_1^{*}\left(y\right)=\underset{x\in X}{\vee }\{{\mathcal{A}}_t^{*}\left(x\right)\otimes (({\mathcal{A}}_t^{*}\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to B\left(y\right)))\},y\in Y,$$

(6)

$${\tilde{B}}_2^{*}\left(y\right)=\underset{x\in X}{\vee }\{{\mathcal{A}}_{-f}^{*}\left(x\right)\otimes (({\mathcal{A}}_{-f}^{*}\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to B\left(y\right)))\},y\in Y.$$

(7)

where → is an R-implication derived from a left-continuous t-norm ⊗.

Step 3: The solution of the Type 3 inference problem is derived, as presented in

Table 3. The solutions for the mixed information inference problem of Type 3.

The Given Rule	Input	Output
$\begin{array}{c}\mathrm{If}x\mathrm{is}A,\mathrm{then}y\mathrm{is}B\end{array}$	$x\mathrm{is}\mathcal{A}*$	FS	$B*=\left\{\langle y,B^{*}\left(y\right)\rangle \right|B^{*}\left(y\right)\in [{\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right),{\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right)],y\in Y\}$
		IFS	$\mathcal{B}*=\left\{\langle y,{\mathcal{B}}_t^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)\rangle \right|{\mathcal{B}}_t^{*}\left(y\right)={\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)=1-{\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right),y\in Y\}$

Note: ${\tilde{B}}_1^{*}\left(y\right)=\underset{x\in X}{\vee }\{{\mathcal{A}}_t^{*}\left(x\right)\otimes (({\mathcal{A}}_t^{*}\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to B\left(y\right)))\},y\in Y$, ${\tilde{B}}_2^{*}\left(y\right)=\underset{x\in X}{\vee }\{{\mathcal{A}}_{-f}^{*}\left(x\right)\otimes (({\mathcal{A}}_{-f}^{*}\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to B\left(y\right)))\},y\in Y$.

.

The fuzzy output $B*=\left\{\langle y,B*\left(y\right)\rangle \right|y\in Y\}$ will be acquired via Equations (6) and (7) with the same information representation as the rule “if x is A, then y is B”, where $B*\left(y\right)\in [{\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right),{\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right)]$. By combining Equations (6) and (7) with Definition 3 in [2], the intuitionistic fuzzy output $\mathcal{B}*=\left\{\langle y,{\mathcal{B}}_t^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)\rangle \right|y\in Y\}$ of the same type as the input ”x is $\mathcal{A}$ ” can likewise be provided, where ${\mathcal{B}}_t^{*}\left(y\right)={\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)=1-{\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right)$.

According to Equations (4)–(7), the following proposition is presented when the t-norm is Łukasiewicz t-norm ${\otimes }_{Ł\mathrm{u}}$.

Proposition 2. 

(1) For the Type 4 inference problem,

$$\begin{array}{cc}\hfill {\overline{\mathcal{B}}}_t^{*}\left(y\right)=& \underset{x\in X}{\vee }\left\{\right(A*\left(x\right)+((1-A*\left(x\right)+{\mathcal{A}}_t\left(x\right))\wedge 1+(1-{\mathcal{A}}_t\left(x\right)+{\mathcal{B}}_t\left(x\right))\hfill \\ \hfill & \wedge (1-{\mathcal{A}}_{-f}\left(x\right)+{\mathcal{B}}_{-f}\left(x\right))\wedge 1-1)\vee 0)-1)\vee 0\},y\in Y,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\overline{\mathcal{B}}}_f^{*}\left(y\right)=& 1-\underset{x\in X}{\vee }\left\{\right(A*\left(x\right)+((1-A*\left(x\right)+{\mathcal{A}}_{-f}\left(x\right))\wedge 1+(1-{\mathcal{A}}_{-f}\left(x\right)\hfill \\ \hfill & +{\mathcal{B}}_{-f}\left(y\right))\wedge 1-1)\vee 0-1)\vee 0\},y\in Y.\hfill \end{array}$$

(9)

(2) For the Type 3 inference problem,

$$\begin{array}{cc}\hfill {\tilde{B}}_1^{*}\left(y\right)=& \underset{x\in X}{\vee }\left\{\right({\mathcal{A}}_t^{*}\left(x\right)+((1-{\mathcal{A}}_t^{*}\left(x\right)+A\left(x\right))\wedge 1+(1-A\left(x\right)+B\left(y\right))\wedge 1-1)\hfill \\ \hfill & \vee 0-1)\vee 0\},y\in Y,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\tilde{B}}_2^{*}\left(y\right)=& \underset{x\in X}{\vee }\left\{\right({\mathcal{A}}_{-f}^{*}\left(x\right)+((1-{\mathcal{A}}_{-f}^{*}\left(x\right)+A\left(x\right))\wedge 1+(1-A\left(x\right)+B\left(y\right))\wedge 1-1)\hfill \\ \hfill & \vee 0-1)\vee 0\},y\in Y.\hfill \end{array}$$

(11)

3.2. The Quintuple-Implication Fuzzy Reasoning Method with Mixed Information of Inconsistent Rules

To reason with inconsistent if–then rules (Types 5–8 mixed information inference problems), it is essential to first gain a precise and comprehensive understanding of the inconsistent rules, conceptualizing one inconsistent rule as comprising two distinct consistent rules, each containing unique information representation.

By this analysis, for each of the Types 5–8 inference problems, the input will be combined with the two consistent rules to conduct the inference process in their respective directions. Further, the conclusions in these two directions will be aggregated to obtain the final inference result for each type, as depicted in Figure 2. The final inference result will be expressed according to the representation of the input and rule consequent. When the representation of the input and rule consequent is the same, the final inference result will also be written with full consideration in the form that is different from the representation of the two, in addition to giving the final result with the same representation as the two.

Consider the Type 6 inference problem:

$$\begin{array}{c}\mathrm{The}\mathrm{given}\mathrm{rule}:\hspace{1em}\mathrm{If}x\mathrm{is}\mathcal{A},\mathrm{then}y\mathrm{is}B\hfill \\ \underline{\mathrm{Input}:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\mathrm{is}\mathcal{A}*\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} }\hfill \\ \mathrm{Output}:y\mathrm{is}?\hfill \end{array}$$

where $\mathcal{A}$ and $\mathcal{A}*$ represent two intuitionistic fuzzy sets, respectively, defined on the universe X, and B is a fuzzy set on the universe Y.

The method to reasoning with Type 6 mixed information

Step 1: Determine the output given the intuitionistic fuzzy if–then rule and the input.

The following IFMP problem (i.e., Type 2 inference problem) needs to be addressed:

$$\begin{array}{c}\mathrm{The}\mathrm{given}\mathrm{rule}:\hspace{1em}\mathrm{If}x\mathrm{is}\mathcal{A},\mathrm{then}y\mathrm{is}\mathcal{B}\hfill \\ \underline{\mathrm{Input}:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\mathrm{is}\mathcal{A}*\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} }\hfill \\ \mathrm{Output}:y\mathrm{is}\overline{\mathcal{B}}*\hfill \end{array}$$

where $\mathcal{B}=\{\langle y,B(y),1-B(y)\rangle |y\in Y\}$ is an intuitionistic fuzzy set on the universe Y converted from fuzzy set $B=\{\langle y,B(y)\rangle |y\in Y\}$, and A and $\mathcal{A}*$ represent two intuitionistic fuzzy sets, respectively, defined on the universe X.

Theorem 2 in [14] is utilized to obtain the intuitionistic fuzzy output ${\overline{\mathcal{B}}}^{*}=\{\langle y,$${\overline{\mathcal{B}}}_t^{*}\left(y\right),$${\overline{\mathcal{B}}}_f^{*}\left(y\right)\rangle |$$y\in Y\}$:

$$\begin{array}{c}\hfill \overline{\mathcal{B}}*\left(y\right)=\underset{x\in X}{\vee }\{\mathcal{A}*\left(x\right){\otimes }_{\mathcal{I}*}\left((\mathcal{A}*\left(x\right){\to }_{\mathcal{I}*}\mathcal{A}\left(x\right)){\otimes }_{\mathcal{I}*}\left(\mathcal{A}\left(x\right){\to }_{\mathcal{I}*}\mathcal{B}\left(y\right)\right)\right)\},y\in Y.\end{array}$$

(12)

where ${\to }_{\mathcal{I}*}$ is an intuitionistic R-implication derived from a left-continuous intuitionistic t-norm ${\otimes }_{\mathcal{I}*}$, and further,

$$\begin{array}{cc}\hfill {\overline{\mathcal{B}}}_t^{*}\left(y\right)=& \underset{x\in X}{\vee }\{{\mathcal{A}}_t^{*}\left(x\right)\otimes ((({\mathcal{A}}_t^{*}\left(x\right)\to {\mathcal{A}}_t\left(x\right))\wedge ({\mathcal{A}}_{-f}^{*}\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right)))\hfill \\ \hfill & \otimes ({\mathcal{A}}_{-f}\left(x\right)\to B\left(y\right))\left)\right\},y\in Y\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\overline{\mathcal{B}}}_f^{*}\left(y\right)=& \underset{x\in X}{\wedge }\{(1-{\mathcal{A}}_{-f}^{*}\left(x\right))\oplus ((1-{\mathcal{A}}_{-f}^{*}\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right))\oplus (1-{\mathcal{A}}_{-f}\left(x\right)\to B\left(y\right)))\}\hfill \\ \hfill =& 1-\underset{x\in X}{\vee }\{{\mathcal{A}}_{-f}^{*}\left(x\right)\otimes (({\mathcal{A}}_{-f}^{*}\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right))\otimes ({\mathcal{A}}_{-f}\left(x\right)\to B\left(y\right)))\},y\in Y\hfill \end{array}$$

(14)

where → is an R-implication derived from a left-continuous t-norm ⊗ and ⊕ is the dual t-conorm of ⊗. In order to obtain the final inference result (Step 3), here, the fuzzy output ${\overline{B}}^{*}=\left\{\langle y,{\overline{B}}^{*}\left(y\right)\rangle \right|{\overline{B}}^{*}\left(y\right)\in [{\overline{\mathcal{B}}}_t^{*}\left(y\right),{\overline{\mathcal{B}}}_{-f}^{*}\left(y\right)],y\in Y\}$ is also given, where ${\overline{\mathcal{B}}}_{-f}^{*}\left(y\right)=1-{\overline{\mathcal{B}}}_f^{*}\left(y\right)$.

Step 2: Determine the output given the fuzzy if–then rule and the input $\mathcal{A}*$.

Having considered that this inference issue overlaps with the Type 3 inference problem, the following two FMP problems will be addressed:

$$\begin{array}{c}\mathrm{The}\mathrm{given}\mathrm{rule}:\hspace{1em}\mathrm{If}x\mathrm{is}{\mathcal{A}}_t,\mathrm{then}y\mathrm{is}B\hfill \\ \underline{\mathrm{Input}:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\mathrm{is}{\mathcal{A}}_t^{*}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} }\hfill \\ \mathrm{Output}:y\mathrm{is}{\tilde{B}}_1^{*}\hfill \end{array}$$

$$\begin{array}{c}\mathrm{The}\mathrm{given}\mathrm{rule}:\hspace{1em}\mathrm{If}x\mathrm{is}{\mathcal{A}}_{-f},\mathrm{then}y\mathrm{is}B\hfill \\ \underline{\mathrm{Input}:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\mathrm{is}{\mathcal{A}}_{-f}^{*}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} }\hfill \\ \mathrm{Output}:y\mathrm{is}{\tilde{B}}_2^{*}\hfill \end{array}$$

where ${\mathcal{A}}_t$ and ${\mathcal{A}}_{-f}$ are fuzzy sets on X converted from intuitionistic fuzzy set $\mathcal{A}$; ${\mathcal{A}}_t^{*}$ and ${\mathcal{A}}_{-f}^{*}$ are fuzzy sets on X converted from intuitionistic fuzzy set $\mathcal{A}*$; and B represents a fuzzy set on Y. The above reasoning problems are solved according to Theorem 1 in [7]:

$$\begin{array}{cc}\hfill {\tilde{B}}_1^{*}\left(y\right)=& \underset{x\in X}{\vee }\{{\mathcal{A}}_t^{*}\left(x\right)\otimes (({\mathcal{A}}_t^{*}\left(x\right)\to {\mathcal{A}}_t\left(x\right))\otimes ({\mathcal{A}}_t\left(x\right)\to B\left(y\right)))\},y\in Y,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\tilde{B}}_2^{*}\left(y\right)=& \underset{x\in X}{\vee }\{{\mathcal{A}}_{-f}^{*}\left(x\right)\otimes (({\mathcal{A}}_{-f}^{*}\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right))\otimes ({\mathcal{A}}_{-f}\left(x\right)\to B\left(y\right)))\},y\in Y,\hfill \end{array}$$

(16)

where → is an R-implication derived from a left-continuous t-norm ⊗.

Thus, from Equations (15) and (16), we can obtain the fuzzy output ${\tilde{B}}^{*}=\left\{\langle y,{\tilde{B}}^{*}\left(y\right)\rangle \right|y\in Y\}$, ${\tilde{B}}^{*}\left(y\right)\in [{\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right),{\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right)]$ and the intuitionistic fuzzy output ${\tilde{\mathcal{B}}}^{*}=\left\{\langle y,{\tilde{\mathcal{B}}}_t^{*}\left(y\right),{\tilde{\mathcal{B}}}_f^{*}\left(y\right)\rangle \right|y\in Y\}$, ${\tilde{\mathcal{B}}}_t^{*}\left(y\right)={\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right)$, ${\tilde{\mathcal{B}}}_f^{*}\left(y\right)=1-{\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right)$.

Step 3: Obtain the final result of the Type 6 inference problem by consolidating the outcomes of Steps 1 and 2, as presented in

Table 4. The solutions for the mixed information inference problem of Type 6.

The Given Rule	Input	Output
$\mathrm{if}x\mathrm{is}\mathcal{A},\mathrm{then}y\mathrm{is}B$	$x\mathrm{is}\mathcal{A}*$	FS	$B*=\left\{\langle y,B*\left(y\right)\rangle \right|B*\left(y\right)\in [{\overline{\mathcal{B}}}_t^{*}\left(y\right)\wedge {\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right),{\overline{\mathcal{B}}}_{-f}^{*}\left(y\right)\vee {\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right)],y\in Y\}$
		IFS	$\mathcal{B}*=\left\{\langle y,{\mathcal{B}}_t^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)\rangle \right|{\mathcal{B}}_t^{*}={\overline{\mathcal{B}}}_t^{*}\left(y\right)\wedge {\tilde{B}}_1^{*}\left(y\right)\wedge {\overline{B}}_2^{*}\left(y\right),{\mathcal{B}}_f^{*}=1-{\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right),y\in Y\}$

Note: ${\overline{\mathcal{B}}}_t^{*}\left(y\right)=\underset{x\in X}{\vee }\{{\mathcal{A}}_t^{*}\left(x\right)\otimes ((({\mathcal{A}}_t^{*}\left(x\right)\to {\mathcal{A}}_t\left(x\right))\wedge ({\mathcal{A}}_{-f}^{*}\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right)))\otimes ({\mathcal{A}}_{-f}\left(x\right)\to B\left(y\right)))\},y\in Y$, ${\overline{\mathcal{B}}}_f^{*}\left(y\right)=1-\underset{x\in X}{\vee }\{{\mathcal{A}}_{-f}^{*}\left(x\right)\otimes (({\mathcal{A}}_{-f}^{*}\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right))\otimes ({\mathcal{A}}_{-f}\left(x\right)\to B\left(y\right)))\},y\in Y$, ${\tilde{B}}_1^{*}\left(y\right)=\underset{x\in X}{\vee }\{{\mathcal{A}}_t^{*}\left(x\right)\otimes (({\mathcal{A}}_t^{*}\left(x\right)\to {\mathcal{A}}_t\left(x\right))\otimes ({\mathcal{A}}_t\left(x\right)\to B\left(y\right)))\},y\in Y$, ${\tilde{B}}_2^{*}\left(y\right)=\underset{x\in X}{\vee }\{{\mathcal{A}}_{-f}^{*}\left(x\right)\otimes (({\mathcal{A}}_{-f}^{*}\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right))\otimes ({\mathcal{A}}_{-f}\left(x\right)\to B\left(y\right)))\},y\in Y$.

.

It is easy to prove the following: by combining ${\overline{B}}^{*}$ and $\tilde{B}*$, we obtain the fuzzy $B*=\left\{\langle y,B*\left(y\right)\rangle \right|B*\left(y\right)\in [{\overline{\mathcal{B}}}_t^{*}\left(y\right)\wedge {\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right),{\overline{\mathcal{B}}}_{-f}^{*}\left(y\right))\vee {\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right)],y\in Y\}$; from $\overline{\mathcal{B}}*$ and $\tilde{\mathcal{B}}*$, we obtain the intuitionistic fuzzy $\mathcal{B}*=\left\{\langle y,{\mathcal{B}}_t^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)\rangle \right|{\mathcal{B}}_t^{*}\left(y\right)={\overline{\mathcal{B}}}_t^{*}\left(y\right)\wedge {\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)=1-{\overline{\mathcal{B}}}_{-f}^{*}\left(y\right)\vee {\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right),y\in Y\}$; moreover, there exists ${\overline{\mathcal{B}}}_{-f}^{*}\left(y\right)=1-{\overline{\mathcal{B}}}_f^{*}\left(y\right)={\tilde{B}}_2^{*}\left(y\right)$ by referring to Equations (14) and (16).

Similar to the Type 6 inference problem, the solutions of Types 5, 7 and 8 are shown in

Table 5. The solutions for the mixed information inference problem of Types 5, 7 and 8.

Type	The Given Rule	Input	Output
Type 5	$\begin{array}{c}\mathrm{if}x\mathrm{is}A,\mathrm{then}y\mathrm{is}\mathcal{B}\end{array}$	$x\mathrm{is}A*$	FS	$B*=\left\{\langle y,B*\left(y\right)\rangle \right|B*\left(y\right)\in [{\tilde{B}}_1^{*}\left(y\right),{\tilde{B}}_2^{*}\left(y\right)],y\in Y\}$
			IFS	$\mathcal{B}*=\left\{\langle y,{\mathcal{B}}_t^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)\rangle \right|{\mathcal{B}}_t^{*}={\tilde{B}}_1^{*}\left(y\right),{\mathcal{B}}_f^{*}=1-{\tilde{B}}_2^{*}\left(y\right),y\in Y\}$
Type 7	$\begin{array}{c}\mathrm{if}x\mathrm{is}A,\mathrm{then}y\mathrm{is}\mathcal{B}\end{array}$	$x\mathrm{is}\mathcal{A}*$	FS	$B*=\left\{\langle y,B*\left(y\right)\rangle \right|B*\left(y\right)\in [{\overline{\mathcal{B}}}_t^{*}\left(y\right)\wedge {\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right),{\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right)],y\in Y\}$
			IFS	$\mathcal{B}*=\left\{\langle y,{\mathcal{B}}_t^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)\rangle \right|{\mathcal{B}}_t^{*}={\overline{\mathcal{B}}}_t^{*}\left(y\right)\wedge {\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right),{\mathcal{B}}_f^{*}=1-{\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right),y\in Y\}$
Type 8	$\begin{array}{c}\mathrm{if}x\mathrm{is}\mathcal{A},\mathrm{then}y\mathrm{is}B\end{array}$	$x\mathrm{is}A*$	FS	$B*=\left\{\langle y,B*\left(y\right)\rangle \right|B*\left(y\right)\in [{\overline{\mathcal{B}}}_t^{*}\left(y\right)\wedge {\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right),{\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right)],y\in Y\}$
			IFS	$\mathcal{B}*=\left\{\langle y,{\mathcal{B}}_t^{*}\left(y\right),{\mathcal{B}}_f^{*}\left(y\right)\rangle \right|{\mathcal{B}}_t^{*}={\overline{\mathcal{B}}}_t^{*}\left(y\right)\wedge {\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right),{\mathcal{B}}_f^{*}=1-{\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right),y\in Y\}$

Note: In Type 5, ${\overline{\mathcal{B}}}_t^{*}\left(y\right)=\underset{x\in X}{\vee }\{A*\left(x\right)\otimes ((A*\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to {\mathcal{B}}_t\left(y\right)))\},y\in Y$, ${\overline{\mathcal{B}}}_f^{*}\left(y\right)=1-\underset{x\in X}{\vee }\{A*\left(x\right)\otimes ((A*\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to {\mathcal{B}}_{-f}\left(y\right)))\},y\in Y$, ${\tilde{B}}_1^{*}\left(y\right)=\underset{x\in X}{\vee }\{A*\left(x\right)\otimes ((A*\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to {\mathcal{B}}_t\left(y\right)))\},y\in Y$, ${\tilde{B}}_2^{*}\left(y\right)=\underset{x\in X}{\vee }\{A*\left(x\right)\otimes ((A*\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to {\mathcal{B}}_{-f}\left(y\right)))\},y\in Y$. Note: In Type 7, ${\overline{\mathcal{B}}}_t^{*}\left(y\right)=\underset{x\in X}{\vee }\{{\mathcal{A}}_t^{*}\left(x\right)\otimes (({\mathcal{A}}_{-f}^{*}\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to {\mathcal{B}}_t\left(y\right)))\},y\in Y$, ${\overline{\mathcal{B}}}_f^{*}\left(y\right)=1-\underset{x\in X}{\vee }\{{\mathcal{A}}_{-f}^{*}\left(x\right)\otimes (({\mathcal{A}}_{-f}^{*}\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to {\mathcal{B}}_{-f}\left(y\right)))\},y\in Y$, ${\tilde{B}}_1^{*}\left(y\right)=\underset{x\in X}{\vee }\{{\mathcal{A}}_t^{*}\left(x\right)\otimes (({\mathcal{A}}_t^{*}\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to {\mathcal{B}}_t\left(y\right)))\},y\in Y$, ${\tilde{B}}_2^{*}\left(y\right)=\underset{x\in X}{\vee }\{{\mathcal{A}}_{-f}^{*}\left(x\right)\otimes (({\mathcal{A}}_{-f}^{*}\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to {\mathcal{B}}_{-f}\left(y\right)))\},y\in Y$. Note: In Type 8, ${\overline{\mathcal{B}}}_t^{*}\left(y\right)=\underset{x\in X}{\vee }\{A*\left(x\right)\otimes ((A*\left(x\right)\to {\mathcal{A}}_t\left(x\right))\otimes ({\mathcal{A}}_{-f}\left(x\right)\to B\left(y\right)))\},y\in Y$, ${\overline{\mathcal{B}}}_f^{*}\left(y\right)=1-\underset{x\in X}{\vee }\{A*\left(x\right)\otimes ((A*\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right))\otimes ({\mathcal{A}}_{-f}\left(x\right)\to B\left(y\right)))\},y\in Y$, ${\tilde{B}}_1^{*}\left(y\right)=\underset{x\in X}{\vee }\{A*\left(x\right)\otimes ((A*\left(x\right)\to {\mathcal{A}}_t\left(x\right))\otimes ({\mathcal{A}}_t\left(x\right)\to B\left(y\right)))\},y\in Y$, ${\tilde{B}}_2^{*}\left(y\right)=\underset{x\in X}{\vee }\{A*\left(x\right)\otimes ((A*\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right))\otimes ({\mathcal{A}}_{-f}\left(x\right)\to B\left(y\right)))\},y\in Y$.

.

Proposition 3. 

For the Type 6 inference problem, if the t-norm is Łukasiewicz t-norm ${\otimes }_{{\mathcal{I}}_{\mathsf{Ł}u}^{*}}$, Equations (13)–(16) can be rewritten as

$$\begin{array}{cc}\hfill {\overline{\mathcal{B}}}_t^{*}\left(y\right)=& \underset{x\in X}{\vee }\{({\mathcal{A}}_t^{*}\left(x\right)+((1-{\mathcal{A}}_t^{*}\left(x\right)+{\mathcal{A}}_t\left(x\right))\wedge 1+(1-{\mathcal{A}}_t\left(x\right)+B\left(y\right))\wedge 1-1)\vee 0)-1)\vee 0\},y\in Y,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\overline{\mathcal{B}}}_f^{*}\left(y\right)=& 1-\underset{x\in X}{\vee }\{({\mathcal{A}}_{-f}^{*}\left(x\right)+((1-{\mathcal{A}}_{-f}^{*}\left(x\right)+{\mathcal{A}}_{-f}\left(x\right))\wedge 1+(1-{\mathcal{A}}_{-f}\left(x\right)+B\left(y\right))\wedge 1-1)\vee 0-1)\vee 0\},y\in Y,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\tilde{B}}_1^{*}\left(y\right)=& \underset{x\in X}{\vee }\{({\mathcal{A}}_t^{*}\left(x\right)+((1-{\mathcal{A}}_t^{*}\left(x\right)+{\mathcal{A}}_t\left(x\right))\wedge 1+(1-{\mathcal{A}}_t\left(x\right)+B\left(y\right))\wedge 1-1)\vee 0-1)\vee 0\},y\in Y,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\tilde{B}}_2^{*}\left(y\right)=& \underset{x\in X}{\vee }\{({\mathcal{A}}_{-f}^{*}\left(x\right)+((1-{\mathcal{A}}_{-f}^{*}\left(x\right)+{\mathcal{A}}_{-f}\left(x\right))\wedge 1+(1-{\mathcal{A}}_{-f}\left(x\right)+B\left(y\right))\wedge 1-1)\vee 0-1)\vee 0\},y\in Y.\hfill \end{array}$$

(20)

4. Reductivity Analysis

Reductivity is an important property frequently employed as a criterion for evaluating the efficacy of reasoning methods. A reasoning method for the FMP or IFMP (with a unique information type) is considered to be reductive if the equality of the input to the rule antecedent implies that the output is equal to the rule consequent. When discussing the reductivity of methods for resolving inference problems in Table 1 that involve a mixture of information types, how to define the reductivity is a critical issue.

As to methods developed specifically for addressing the Types 3-8 inference problems, there are two representation options for the output, either in the form of fuzzy sets or in the form of intuitionistic fuzzy sets. Having considered the inconsistency of the representations among the input, rule antecedent and rule consequent, several findings are mentioned below:

1.

It is possible to make the input and the antecedent of the rule equal when the representation types of the input and the antecedent of the rule are consistent;

2.

It is possible to make the input and the antecedent of the rule equal by using the conversion formula from intuitionistic fuzzy sets to fuzzy sets when the input $\mathcal{A}*$ is an intuitionistic fuzzy set and the antecedent A of the rule is a fuzzy set;

3.

When the input $A*$ is a fuzzy set and the antecedent of the rule $\mathcal{A}$ is an intuitionistic fuzzy set (in which ${\mathcal{A}}_t\ne 1-{\mathcal{A}}_f$), the equality between the input and the antecedent of the rule cannot be realized at this time according to the conversion formula from fuzzy sets to intuitionistic fuzzy sets.

Thus, the reductivity, to our knowledge, can be interpreted in the way that for each of Types 3, 5, 6 and 7, when the equality between the input and the antecedent of the rule is achieved, the method is called reductive if the output of this method with the same representation as the rule consequent is proved to be equal to the rule consequent. As for Types 4 and 8, the discussion of reductivity is not applicable to the methods for these two inference problems because it is not feasible to make the input and the antecedent of the rule equal.

Definition 8. 

(1)

For Type 3, the method is called reductive if the assumption that the input $A*$ converted from $\mathcal{A}*$ is equal to the rule antecedent A implies that the fuzzy output $B*$ is equal to the rule consequent B.

(2)

For Type 5, the method is called reductive if the assumption that the input $A*$ is equal to the rule antecedent A implies that the intuitionistic fuzzy output $\mathcal{B}*$ is equal to the rule consequent $\mathcal{B}$.

(3)

For Type 6, the method is called reductive if the assumption that the input $\mathcal{A}*$ is equal to the rule antecedent A implies that the fuzzy output $B*$ is equal to the rule consequent B.

(4)

For Type 7, the method is called reductive if the assumption that the input $A*$ converted from $\mathcal{A}*$ is equal to the rule antecedent A implies that the intuitionistic fuzzy output $\mathcal{B}*$ is equal to the rule consequent $\mathcal{B}$.

Theorem 3. 

If A is normal, then the proposed method for the Type 3 mixed information inference problem is reductive.

Proof. 

If $A*=A$ and A is normal, i.e., $A\left(x_0\right)=1,x_0\in X$, then there are

$$\begin{array}{cc}\hfill B\left(y\right)\ge & {\tilde{B}}_1^{*}\left(y\right)\hfill \\ \hfill =& \underset{x\in X}{\vee }\{{\mathcal{A}}_t^{*}\left(x\right)\otimes (({\mathcal{A}}_t^{*}\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to B\left(y\right)))\}\hfill \\ \hfill \ge & A\left(x_0\right)\otimes ((A\left(x_0\right)\to A\left(x_0\right))\otimes (A\left(x_0\right)\to B\left(y\right)))=B\left(y\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill B\left(y\right)\ge & {\tilde{B}}_2^{*}\left(y\right)\hfill \\ \hfill =& \underset{x\in X}{\vee }\{{\mathcal{A}}_{-f}^{*}\left(x\right)\otimes (({\mathcal{A}}_{-f}^{*}\left(x\right)\to A\left(x\right))\otimes (A\left(x\right)\to B\left(y\right)))\}\hfill \\ \hfill \ge & A\left(x_0\right)\otimes ((A\left(x_0\right)\to A\left(x_0\right))\otimes (A\left(x_0\right)\to B\left(y\right)))=B\left(y\right).\hfill \end{array}$$

Then, ${\tilde{B}}_1^{*}={\tilde{B}}_2^{*}=B$ can be obtained, and there exists the fuzzy output $B*=\left\{\langle y,B*\left(y\right)\in [{\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right),{\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right)]\rangle \right|y\in Y\}$$=\left\{\langle y,B*\left(y\right)\in [B\left(y\right),B\left(y\right)]\rangle \right|y\in Y\}=\{\langle y,$$B\left(y\right)\rangle |y\in Y\}=B.$ Thus, the reductivity of the method proposed for Type 3 is proved. □

Theorem 4. 

If $\mathcal{A}$ or A is normal, then the proposed methods for the Types 5–7 mixed information inference problems are reductive.

Proof. 

In Type 6, if $\mathcal{A}*=\mathcal{A}$ and $\mathcal{A}$ is normal, i.e., $\mathcal{A}\left(x_0\right)=\langle 1,0\rangle ,x_0\in X$, then

$$\begin{array}{cc}\hfill \mathcal{B}\left(y\right)=& \mathcal{A}\left(x_0\right){\otimes }_{\mathcal{I}*}\left(\left(\mathcal{A}\left(x_0\right){\to }_{\mathcal{I}*}\mathcal{A}\left(x_0\right)\right){\otimes }_{\mathcal{I}*}\left(\mathcal{A}\left(x_0\right){\to }_{\mathcal{I}*}\mathcal{B}\left(y\right)\right)\right)\hfill \\ \hfill =& \mathcal{A}*\left(x_0\right){\otimes }_{\mathcal{I}*}\left(\left(\mathcal{A}*\left(x_0\right){\to }_{\mathcal{I}*}\mathcal{A}\left(x_0\right)\right){\otimes }_{\mathcal{I}*}\left(\mathcal{A}\left(x_0\right){\to }_{\mathcal{I}*}\mathcal{B}\left(y\right)\right)\right)\hfill \\ \hfill ⪯& \underset{x\in X}{\vee }\left\{\mathcal{A}*\left(x\right){\otimes }_{\mathcal{I}*}\left(\left(\mathcal{A}*\left(x\right){\to }_{\mathcal{I}*}\mathcal{A}\left(x\right)\right){\otimes }_{\mathcal{I}*}\left(\mathcal{A}\left(x\right){\to }_{\mathcal{I}*}\mathcal{B}\left(y\right)\right)\right)\right\}\hfill \\ \hfill =& \overline{\mathcal{B}}*\left(y\right)⪯\mathcal{B}\left(y\right),\hfill \end{array}$$

where $\mathcal{B}$ is an intuitionistic fuzzy set converted from fuzzy set B by Proposition 1. Thus, $\overline{\mathcal{B}}*=\left\{\langle y,{\overline{\mathcal{B}}}_t^{*}\left(y\right),{\overline{\mathcal{B}}}_f^{*}\left(y\right)\rangle \right|y\in Y\}$ is equivalent to $\mathcal{B}=\{\langle y,B(y),1-B(y)\rangle |y\in Y\}$. In addition, there are

$$\begin{array}{cc}\hfill B\left(y\right)\ge & {\tilde{B}}_1^{*}\left(y\right)\hfill \\ \hfill =& \underset{x\in X}{\vee }\{{\mathcal{A}}_t^{*}\left(x\right)\otimes (({\mathcal{A}}_t^{*}\left(x\right)\to {\mathcal{A}}_t\left(x\right))\otimes ({\mathcal{A}}_t\left(x\right)\to B\left(y\right)))\}\hfill \\ \hfill \ge & {\mathcal{A}}_t\left(x_0\right)\otimes (({\mathcal{A}}_t\left(x_0\right)\to {\mathcal{A}}_t\left(x_0\right))\otimes ({\mathcal{A}}_t\left(x_0\right)\to B\left(y\right)))=B\left(y\right),\hfill \\ \hfill B\left(y\right)\ge & {\tilde{B}}_2^{*}\left(y\right)\hfill \\ \hfill =& \underset{x\in X}{\vee }\{{\mathcal{A}}_{-f}^{*}\left(x\right)\otimes (({\mathcal{A}}_{-f}^{*}\left(x\right)\to {\mathcal{A}}_{-f}\left(x\right))\otimes ({\mathcal{A}}_{-f}\left(x\right)\to B\left(y\right)))\}\hfill \\ \hfill \ge & {\mathcal{A}}_{-f}\left(x_0\right)\otimes (({\mathcal{A}}_{-f}\left(x_0\right)\to {\mathcal{A}}_{-f}\left(x_0\right))\otimes ({\mathcal{A}}_{-f}\left(x_0\right)\to B\left(y\right)))=B\left(y\right).\hfill \end{array}$$

Through the above analysis, it can be obtained that ${\tilde{B}}_1^{*}={\tilde{B}}_2^{*}=B$. Therefore, the fuzzy output $B*=\{\langle y,B*\left(y\right)\in [{\overline{\mathcal{B}}}_t^{*}\left(y\right)\wedge {\tilde{B}}_1^{*}\left(y\right)\wedge {\tilde{B}}_2^{*}\left(y\right),{\overline{\mathcal{B}}}_{-f}^{*}\left(y\right))\vee {\tilde{B}}_1^{*}\left(y\right)\vee {\tilde{B}}_2^{*}\left(y\right)]\rangle |y\in Y\}=\left\{\langle y,B*\left(y\right)\in [B\left(y\right),B\left(y\right)]\rangle \right|y\in Y\}=\left\{\langle y,B\left(y\right)\rangle \right|y\in Y\}=B$. It is proved that the method proposed for Type 6 is reductive.

Likewise, it can be proven that the methods proposed for Types 5 and 7 are reductive. □

5. Application in Pattern Recognition

Observe that pattern recognition problems involving a mixture of fuzzy and intuitionistic fuzzy information in real life can be solved by first transforming these practical problems into corresponding mixed information inference problems and then applying reasoning methods to yield the recognition results. The proposed reasoning methods were evaluated on four pattern recognition tasks with a mixture of fuzzy information and intuitionistic fuzzy information: the pattern recognition task was accompanied by a comparison with other existing reasoning methods for fuzzy and intuitionistic fuzzy-mixed information. Figure 3 depicts how to solve practical problems using the methods established in Section 3 for reasoning with mixed information.

Since the partial order ⪯ is unable to order all intuitionistic fuzzy numbers, a total order ${⪯}_{Xu}$ based on the score function $SF$ and the accuracy function $AF$ is provided.

Definition 9 

([17]). Let $\alpha =\langle {\alpha }_t,{\alpha }_f\rangle \in \mathcal{I}*$ be an intuitionistic fuzzy number. $SF\left(\alpha \right)={\alpha }_t-{\alpha }_f$ is called a score function and $AF\left(\alpha \right)={\alpha }_t+{\alpha }_f$ is called an accuracy function.

$\alpha {⪯}_{Xu}\beta$ holds for the two intuitionistic fuzzy numbers $\alpha =\langle {\alpha }_t,{\alpha }_f\rangle$ and $\beta =\langle {\beta }_t,{\beta }_f\rangle$ if and only if (i) $SF\left(\alpha \right)<SF\left(\beta \right)$ and (ii) $SF\left(\alpha \right)=SF\left(\beta \right)$ and $AF\left(\alpha \right)\le AF\left(\beta \right)$.

5.1. Method for Pattern Recognition

Let $K=\{K_1,K_2,\cdots ,K_n\}$ be a set of patterns and $X=\{x_1,x_2,\cdots ,x_m\}$ be a set of attributes. ${\mathbb{A}}_i\left(x_j\right)$$(i=1,2,\cdots ,n$ and $j=1,2,\cdots ,m)$ is the value of the attributes $x_j$, and ${\mathbb{B}}_i\left(y\right)$ denote the evaluation value of ith known patterns. Both ${\mathbb{A}}_i\left(x_j\right)$ and ${\mathbb{B}}_i\left(y\right)$ are depicted by a fuzzy number or an intuitionistic fuzzy number, as indicated in

Table 6. The relationship between the patterns and their attributes.

	${\mathit{x}}_1$	${\mathit{x}}_2$	⋯	${\mathit{x}}_{\mathit{m}}$
${\mathbb{B}}_1\left(y\right)$	${\mathbb{A}}_1\left(x_1\right)$	${\mathbb{A}}_1\left(x_2\right)$	⋯	${\mathbb{A}}_1\left(x_m\right)$
${\mathbb{B}}_2\left(y\right)$	${\mathbb{A}}_2\left(x_1\right)$	${\mathbb{A}}_2\left(x_2\right)$	⋯	${\mathbb{A}}_2\left(x_m\right)$
⋮	⋮	⋮	⋮	⋮
${\mathbb{B}}_n\left(y\right)$	${\mathbb{A}}_n\left(x_1\right)$	${\mathbb{A}}_n\left(x_2\right)$	⋯	${\mathbb{A}}_n\left(x_m\right)$

. $\mathbb{A}*\left(x_j\right)$ represents the value of the test sample G with respect to attribute $x_j$, and it is also depicted by a fuzzy number or an intuitionistic fuzzy number, as shown in

Table 7. The relationship between the test sample and its attributes.

	${\mathit{x}}_1$	${\mathit{x}}_2$	⋯	${\mathit{x}}_{\mathit{m}}$
G	$\mathbb{A}*\left(x_1\right)$	$\mathbb{A}*\left(x_2\right)$	⋯	$\mathbb{A}*\left(x_m\right)$

. The goal of this study is to recognize the sample G into these n distinct patterns.

Step 1: Determine which type of mixed information inference problem this pattern recognition problem belongs to, as in Table 1. The data presented in Table 6 can be interpreted as the if–then rules within the inference problem, and the data in Table 7 can be viewed as the input within the inference problem. Then, the following n inference problems will be resolved:

$$\begin{array}{c}\mathrm{The}\mathrm{given}\mathrm{rule}:\mathrm{If}{x}_1\mathrm{is}{\mathbb{A}}_i,x_2\mathrm{is}{\mathbb{A}}_i,\cdots ,x_m\mathrm{is}{\mathbb{A}}_i,\mathrm{then}y\mathrm{is}{\mathbb{B}}_i\hfill \\ \underline{\mathrm{Input}:x_1\mathrm{is}\mathbb{A}*,x_2\mathrm{is}\mathbb{A}*,\cdots ,x_m\mathrm{is}{\mathbb{A}}^{*}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} }\hfill \\ \mathrm{Output}:y\mathrm{is}{\mathbb{B}}_i^{*}\hfill \end{array}(i=1,2,\cdots ,n)$$

Step 2: Calculate the outputs ${\mathbb{B}}_i^{*}\left(y\right)$ using the proposed reasoning methods in Table 2, Table 3, Table 4 and Table 5.

Step 3: Select the largest ${\mathbb{B}}_i^{*}\left(y\right)$ as the final result, and the test sample G is recognized as the pattern $K_i$.

Note that the choice of representation form (fuzzy sets or intuitionistic fuzzy sets) for ${\mathbb{B}}_i^{*}$ depends on the user. When ${\mathbb{B}}_i^{*}$ is expressed as fuzzy sets $B_i^{*}=\left\{\langle y,B_i^{*}\left(y\right)\rangle \right|y\in Y\}$, there exist $B_i^{*}\left(y\right)\in [a,b]$, $a\le b$ and $a,b\in [0,1]$ according to the solution representations provided in Table 2, Table 3, Table 4 and Table 5. (The left endpoint a and the right endpoint b represent the minimum and maximum numbers of $B_i^{*}\left(y\right)$, respectively). For the convenience of comparison, $B_i^{*}\left(y\right)$ is taken as a, and the $B_i^{*}\left(y\right)$ can be sorted by the order ≤. Then, the pattern $K_i$ corresponding to the largest value of $B_i^{*}\left(y\right)$ is the pattern to which the test sample G belongs. When ${\mathbb{B}}_i^{*}$ is expressed as intuitionistic fuzzy sets ${\mathcal{B}}_i^{*}$, the order ${⪯}_{Xu}$ is used for comparison and the pattern $K_i$ corresponding to the largest ${\mathcal{B}}_i^{*}\left(y\right)$ is identified as the pattern to which the test sample G belongs.

5.2. Application in Pattern Recognition with Mixed Information

For the following recognition problems, the R-implication involved takes the Łukasiewicz implication ${\to }_{Ł\mathrm{u}}$ induced by the Łukasiewicz t-norm ${\otimes }_{Ł\mathrm{u}}$, and the intuitionistic R-implication takes the intuitionistic Łukasiewicz implication ${\to }_{{\mathcal{I}}_{Ł\mathrm{u}}^{*}}$ induced by the intuitionistic Łukasiewicz t-norm ${\otimes }_{{\mathcal{I}}_{Ł\mathrm{u}}^{*}}$. The Łukasiewicz t-norm and the Łukasiewicz implication are typical t-norms and R-implications, so this paper uses them for pattern recognition applications. The proposed methods are also applicable to other t-norms and their corresponding induced R-implications. Since research on mixed information reasoning methods is relatively limited, and few methods can directly address applications in pattern recognition involving mixed information of fuzzy and intuitionistic fuzzy, only the methods discussed in [15] are considered for the comparison analysis. In the future, we will try to broaden the range of methods that can be used for comparative analysis.

5.2.1. Application in Type 4 Mixed Information Pattern Recognition

Example 3. 

There is a set of patterns $K=\{K_1,K_2,$$K_3\}$ and a set of attributes $X=\{x_1,x_2,\cdots x_6\}$. The relationships ${\mathcal{A}}_i\left(x_j\right)$ ($i=1,2,3$, $j=1,2,\cdots ,6$) between the three known patterns and their attributes from [18,19], as well as the evaluation values $B_i\left(y\right)$ of known patterns, are presented by intuitionistic fuzzy numbers in

Table 8. The relationships of known patterns $K_1,K_2,K_3$ and their attributes in Example 3.

	${\mathcal{B}}_{\mathit{i}}\left(\mathit{y}\right)$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$
$K_1$	〈0.87,0.05〉	〈0.94,0.00〉	〈0.88,0.00〉	〈0.82,0.00〉	〈0.78,0.02〉	〈0.75,0.05〉	〈0.72,0.08〉
$K_2$	〈0.81,0.18〉	〈0.86,0.07〉	〈0.92,0.04〉	〈0.98,0.01〉	〈0.98,0.00〉	〈0.95,0.00〉	〈0.92,0.00〉
$K_3$	〈0.85,0.12〉	〈0.66,0.14〉	〈0.72,0.08〉	〈0.78,0.02〉	〈0.84,0.00〉	〈0.90,0.00〉	〈0.96,0.00〉

. Now, determine which the unknown pattern G described by fuzzy numbers in

Table 9. The attributes of the unknown pattern G in Example 3.

	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$
G	0.83	0.76	0.79	0.82	0.85	0.78

belongs to.

It can be determined that this pattern recognition belongs to the Type 4 mixed information inference problem according to the rules provided in Table 8 and the input given in Table 9. Using the reasoning method to the Type 4 mixed information inference problem in Table 2, the fuzzy outputs $B_i^{*}\left(y\right)$ associated with the unknown G are shown in

Table 10. The fuzzy outputs $B_i^{*}\left(y\right)$ and recognition result in Example 3.

Method	${\mathit{K}}_1$	${\mathit{K}}_2$	${\mathit{K}}_3$	Recognition Result
ETIM [15]	0.85	0.72	0.83	$K_1$
DTIM [15]	0.85	0.72	0.83	$K_1$
The proposed method	0.76	0.72	0.73	$K_1$

Note: Bold indicates the maximum present in that group of data.

and the intuitionistic fuzzy outputs ${\mathcal{B}}_i^{*}\left(y\right)$ associated with G are given in

Table 11. The intuitionistic fuzzy outputs ${\mathcal{B}}_i^{*}\left(y\right)$ and recognition result in Example 3.

Method	${\mathit{K}}_1$	${\mathit{K}}_2$	${\mathit{K}}_3$	Recognition Result
ETIM [15]	〈0.85,0.15〉	〈0.72,0.28〉	〈0.83,0.17〉	$K_1$
DTIM [15]	〈0.85,0.15〉	〈0.72,0.28〉	〈0.83,0.17〉	$K_1$
The proposed method	〈0.76,0.15〉	〈0.72,0.28〉	〈0.73,0.17〉	$K_1$

Note: Bold indicates the maximum present in that group of data.

. By borrowing the order ≤ for $B_i^{*}\left(y\right)$ in Table 10 and the order ${⪯}_{Xu}$ for ${\mathcal{B}}_i^{*}\left(y\right)$ in Table 11, there exist the maximum $B_1^{*}\left(y\right)=0.76$ and the maximum ${\mathcal{B}}_1^{*}\left(y\right)=\langle 0.76,0.15\rangle$, which implies that the unknown pattern G belongs to the known pattern $K_1$.

Example 4. 

There is a patient G, and the symptoms of the disease are described over the feature space $X=\{x_1,x_2,\cdots ,x_5\}$, including Temperature, Headache, Stomachache, Cough and Chest Pain. $K=\{K_1,K_2,\cdots ,K_5\}$ describes five possible diseases, Viral Fever, Malaria, Typhoid, Stomach Problem and Heart Problem, from which the patient may suffer.

Table 12. Symptoms characteristic for the diagnoses in Example 4.

	${\mathcal{B}}_{\mathit{i}}\left(\mathit{y}\right)$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$
Viral Fever	〈0.75,0.05〉	〈0.40,0.00〉	〈0.30,0.50〉	〈0.10,0.70〉	〈0.40,0.30〉	〈0.10,0.70〉
Malaria	〈0.75,0.05〉	〈0.70,0.00〉	〈0.20,0.60〉	〈0.00,0.90〉	〈0.70,0.00〉	〈0.10,0.80〉
Typhoid	〈0.75,0.05〉	〈0.30,0.30〉	〈0.60,0.10〉	〈0.20,0.70〉	〈0.20,0.60〉	〈0.10,0.90〉
Stomach Problem	〈0.75,0.05〉	〈0.10,0.70〉	〈0.20,0.40〉	〈0.80,0.00〉	〈0.20,0.70〉	〈0.20,0.70〉
Heart Problem	〈0.75,0.05〉	〈0.10,0.80〉	〈0.00,0.80〉	〈0.20,0.80〉	〈0.20,0.80〉	〈0.80,0.10〉

displays the symptoms ${\mathcal{A}}_i\left(x_j\right)$$(i=1,2,\cdots ,5$, $j=1,2,\cdots ,5)$ for each disease as in [20,21,22], and the assessment value ${\mathcal{B}}_i$, all represented by intuitionistic fuzzy numbers.

Table 13. Symptoms characteristic for the patient G in Example 4.

	Temperature	Headache	Stomachache	Cough	Chest Pain
G	0.80	0.80	0.10	0.20	0.10

exhibits the symptoms of the patient G using fuzzy numbers. The goal is to determine which of the known five diseases the patient has.

Solving this pattern recognition problem is similar to Example 3. The fuzzy results $B_i^{*}\left(y\right)$ and the intuitionistic fuzzy results ${\mathcal{B}}_i^{*}\left(y\right)$ are presented in

Table 14. The fuzzy outputs $B_i^{*}\left(y\right)$ and the disease of the patient G in Example 4.

Method	Viral Fever	Malaria	Typhoid	Stomach Problem	Heart Problem	Recognition Result
ETIM [15]	0.80	0.80	0.80	0.80	0.80	−
DTIM [15]	0.80	0.80	0.80	0.80	0.80	−
The proposed method	0.35	0.65	0.60	0.20	0.20	Malaria

Note: “−” indicates that the result was not identified, and bold indicates the maximum present in that group of data.

and

Table 15. The intuitionistic fuzzy outputs ${\mathcal{B}}_i^{*}\left(y\right)$ and the disease of the patient G in Example 4.

Method	Viral Fever	Malaria	Typhoid	Stomach Problem	Heart Problem	Recognition Result
ETIM [15]	〈0.80,0.20〉	〈0.80,0.20〉	〈0.80,0.20〉	〈0.80,0.20〉	〈0.80,0.20〉	−
DTIM [15]	〈0.80,0.20〉	〈0.80,0.20〉	〈0.80,0.20〉	〈0.80,0.20〉	〈0.80,0.20〉	−
The proposed method	〈0.35,0.25〉	〈0.65,0.25〉	〈0.60,0.20〉	〈0.20,0.40〉	〈0.20,0.80〉	Malaria

Note: “−” indicates that the result was not identified, and bold indicates the maximum present in that group of data.

, respectively. It can be concluded that patient G most likely suffers from Malaria.

Table 10 and Table 11 respectively and Table 14 and Table 15 also, respectively, display the fuzzy outcomes and the intuitionistic fuzzy outcomes of Examples 3 and 4 derived from ETIM and DTIM in [15].

The proposed method obtained the same results in Example 3 as ETIM and DTIM, demonstrating that the proposed method is feasible and effective. But ETIM and DTIM obtained the same solutions in Example 4, whereas the proposed method can provide effective results, indicating that the proposed method is superior to ETIM and DTIM.

5.2.2. Application in Type 6 Mixed Information Pattern Recognition

Example 5. 

Consider a mineral identification task. $X=\{x_1,x_2,$$\cdots ,x_6\}$ represents the six characteristics that a mineral possesses, and $K_1,K_2,\cdots ,$$K_5$, respectively, represent five typical mineral-producing areas. The data in

Table 16. The characteristics of five minerals from the corresponding known areas in Example 5.

	${\mathit{B}}_{\mathit{i}}\left(\mathit{y}\right)$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$
$K_1$	0.75	〈0.7390,0.1250〉	〈0.0330,0.8180〉	〈0.1880,0.6260〉	〈0.4920,0.3580〉	〈0.0200,0.6280〉	〈0.7390,0.1250〉
$K_2$	0.75	〈0.1240,0.6650〉	〈0.0300,0.8250〉	〈0.0480,0.8000〉	〈0.1360.0.6480〉	〈0.0190,0.8230〉	〈0.3930,0.6530〉
$K_3$	0.75	〈0.4490,0.3870〉	〈0.6620,0.2980〉	〈1.0000,0.0000〉	〈1.0000,0.0000〉	〈1.0000,0.0000〉	〈1.0000,0.0000〉
$K_4$	0.75	〈0.2800,0.7150〉	〈0.5210,0.3680〉	〈0.4700,0.4230〉	〈0.2950,0.6580〉	〈0.1880,0.8060〉	〈0.7350,0.1180〉
$K_5$	0.75	〈0.3260,0.4520〉	〈1.0000,0.0000〉	〈0.1820,0.7250〉	〈0.1560,0.7650〉	〈0.0490,0.8960〉	〈0.6750,0.2630〉

and

Table 17. The characteristic of a mineral G to be identified in Example 5.

	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$
G	〈0.6290,0.3030〉	〈0.5240,0.3560〉	〈0.2100,0.6890〉	〈0.2180,0.7530〉	〈0.0690,0.8760〉	〈0.6580,0.2560〉

from [23] present the characteristic values ${\mathcal{A}}_i\left(x_j\right)$$(i=1,2,\cdots ,5$, $j=1,2,\cdots ,6)$ for the five known minerals and the characteristic values $\mathcal{A}*\left(x_j\right)$ for the unknown mineral G, written in the form of intuitionistic fuzzy sets. The evaluated value $B_i\left(y\right)$ of the corresponding production area for each mineral in Table 16 is given in the form of fuzzy sets. The goal is to ascertain the specific mineral area among the five from which the mineral G originates.

Observe that the rules are represented in the mixed form of fuzzy sets and intuitionistic fuzzy sets, and the input is expressed by intuitionistic fuzzy sets with the same representation type as the rule antecedents. Therefore, this mineral identification problem is a Type 6 mixed information inference problem. Using the reasoning method to the Type 6 mixed information inference problem in Table 4, the fuzzy outputs $B_i^{*}\left(y\right)$ associated with the unknown mineral G are shown in

Table 18. The fuzzy outputs $B_i^{*}\left(y\right)$ and recognition result in Example 5.

Method	${\mathit{K}}_1$	${\mathit{K}}_2$	${\mathit{K}}_3$	${\mathit{K}}_4$	${\mathit{K}}_5$	Recognition Result
ETIM [15]	0.5330	0.6580	0.6290	0.6290	0.6580	−
DTIM [15]	0.6580	0.6580	0.6290	0.6580	0.6580	−
The proposed method	0.5330	0.2610	0.3560	0.5260	0.6510	$K_5$

Note: “−” indicates that the result was not identified, and bold indicates the maximum present in that group of data.

and the intuitionistic fuzzy outputs ${\mathcal{B}}_i^{*}\left(y\right)$ associated with G are given in

Table 19. The intuitionistic fuzzy outputs ${\mathcal{B}}_i^{*}\left(y\right)$ and recognition result in Example 5.

Method	${\mathit{K}}_1$	${\mathit{K}}_2$	${\mathit{K}}_3$	${\mathit{K}}_4$	${\mathit{K}}_5$	Recognition Result
ETIM [15]	〈0.5330,0.3560〉	〈0.6580,0.2560〉	〈0.6290,0.3030〉	〈0.6290.0.3030〉	〈0.6580,0.2560〉	−
DTIM [15]	〈0.6580,0.3560〉	〈0.6580,0.2560〉	〈0.6290,0.3030〉	〈0.6580,0.3030〉	〈0.6580,0.2560〉	−
The proposed method	〈0.5330,0.3420〉	〈0.2610,0.6070〉	〈0.5240,0.3560〉	〈0.5260,0.3420〉	〈0.6510,0.2630〉	$K_5$

Note: “−” indicates that the result was not identified, and bold indicates the maximum present in that group of data.

. By choosing the maximum of the five possible values in Table 18 and Table 19, it can draw a conclusion that the production area of the mineral G is $K_5$.

Example 6. 

The Indian government has released a global tender proposal to strengthen infrastructure construction and now hopes to pick a contractor from six different contractors: Jaihind Road Builders P. L. ($K_1$), J.K. Construction ($K_2$), Build Quick Infrastructure P. L. ($K_3$), Relcon Infra Projects L. ($K_4$), Tata Infrastructure L. ($K_5$) and Birla P. L. ($K_6$). These contractors are assessed based on the following four attributes $x_1,x_2,x_3,x_4$, namely, tender price, completion time, technical capability and background experience. The attribute values ${\mathcal{A}}_i\left(x_j\right)$$(i=1,2,\cdots ,6$, $j=1,2,3,4)$ for each contractor are presented in

Table 20. The relationships of six contractors and their attributes in Example 6.

	${\mathit{B}}_{\mathit{i}}\left(\mathit{y}\right)$	Tender Price	Completion Time	Technical Capability	Background Experience
$K_1$	0.90	〈0.81,0.19〉	〈0.90,0.10〉	〈0.81,0.28〉	〈0.67,0.20〉
$K_2$	0.80	〈0.84,0.10〉	〈0.81,0.11〉	〈0.60,0.20〉	〈0.72.0.19〉
$K_3$	0.92	〈0.67,0.13〉	〈0.78,0.21〉	〈0.92,0.05〉	〈0.81,0.12〉
$K_4$	0.85	〈0.77,0.12〉	〈0.83,0.11〉	〈0.74,0.24〉	〈0.71,0.20〉
$K_5$	0.86	〈0.84,0.15〉	〈0.69,0.20〉	〈0.71,0.20〉	〈0.72,0.27〉
$K_6$	0.77	〈0.79,0.02〉	〈0.81,0.17〉	〈0.66,0.30〉	〈0.78,0.10〉

with the form of intuitionistic fuzzy sets. The evaluated value $B_i$ for each contractor is given in Table 20 with the form of fuzzy sets. The attribute values $\mathcal{A}*$ of the ideal contractor have been given by the government in

Table 21. The characteristic of the ideal standards G in Example 6.

	Tender Price	Completion Time	Technical Capability	Background Experience
Ideal Standards	〈0.72,0.22〉	〈0.84,0.04〉	〈0.92,0.08〉	〈0.85,0.13〉

with the form of intuitionistic fuzzy sets. Now, the task is to select the appropriate contractor to undertake this project.

Solving this pattern recognition problem is similar to Example 5. The fuzzy results $B_i^{*}\left(y\right)$ and the intuitionistic fuzzy results ${\mathcal{B}}_i^{*}\left(y\right)$ are presented in

Table 22. The fuzzy outputs $B_i^{*}\left(y\right)$ and the ideal contractor in Example 6.

Method	${\mathit{K}}_1$	${\mathit{K}}_2$	${\mathit{K}}_3$	${\mathit{K}}_4$	${\mathit{K}}_5$	${\mathit{K}}_6$	Recognition Result
ETIM [15]	0.92	0.92	0.89	0.92	0.92	0.92	−
DTIM [15]	0.92	0.92	0.92	0.92	0.92	0.92	−
The proposed method	0.78	0.71	0.89	0.74	0.72	0.66	$K_3$

Note: “−” indicates that the result was not identified, and bold indicates the maximum present in that group of data.

and

Table 23. The intuitionistic fuzzy outputs ${\mathcal{B}}_i^{*}\left(y\right)$ and the ideal contractor in Example 6.

Method	${\mathit{K}}_1$	${\mathit{K}}_2$	${\mathit{K}}_3$	${\mathit{K}}_4$	${\mathit{K}}_5$	${\mathit{K}}_6$	Recognition Result
ETIM [15]	〈0.92,0.04〉	〈0.92,0.08〉	〈0.85,0.04〉	〈0.92,0.08〉	〈0.92,0.04〉	〈0.92,0.08〉	−
DTIM [15]	〈0.92,0.04〉	〈0.92,0.08〉	〈0.92,0.04〉	〈0.92,0.08〉	〈0.92,0.04〉	〈0.92,0.08〉	−
The proposed method	〈0.78,0.16〉	〈0.71,0.20〉	〈0.89,0.11〉	〈0.74,0.17〉	〈0.72,0.28〉	〈0.66,0.23〉	$K_3$

Note: “−” indicates that the result was not identified, and bold indicates the maximum present in that group of data.

, respectively. It can be concluded that the appropriate contractor is $K_3$.

Table 18, Table 19, Table 22 and Table 23 also, respectively, display the fuzzy outcomes and the intuitionistic fuzzy outcomes of Examples 5 and 6 derived from ETIM and DTIM in [15].

By comparing the results in Examples 5 and 6, it can be observed that ETIM and DTIM did not yield reliable outcomes. This is due to the fact that the triple-implication principle used by ETIM and DTIM does not take into account the connection between the input and rule antecedents, which is what the quintuple-implication principle is good at. Therefore, the results obtained by the proposed method are more reasonable.

5.3. Comparison Analysis

In the following, the pattern recognition problems provided in Examples 3–6 will be utilized to conduct a comprehensive analysis of the existing and proposed inference methods with mixed information.

Table 24. Summary of existing reasoning methods with mixed information for processing pattern recognition examples.

Method	Type 4		Type 6
	Example 3	Example 4	Example 5	Example 6
	Can Solve?
ETIM [15]	✓	✕	✕	✕
DTIM [15]	✓	✕	✕	✕
The proposed method	✓	✓	✓	✓

Note: “✓” indicates that the reasoning method with mixed information can solve the pattern recognition problem, and “✕” indicates that the reasoning method with mixed information cannot solve the pattern recognition problem.

summarizes the recognition outcomes of existing reasoning methods with mixed information in various pattern recognition instances. Examples 3 and 4 belong to Type 4 (inference problems with mixed information of consistent rules), while Examples 5 and 6 belong to Type 6 (inference problems with mixed information of inconsistent rules). According to Example 3, the same results for ETIM, DTIM and the method in this paper are achieved, indicating that all of these methods can be used to some extent to solve pattern recognition problems. In contrast, from the results of Examples 4–6, the patterns to which the test samples belong in these data cannot be provided by ETIM and DTIM. The rationale lies in that (1) the relationship between the input and rule antecedents is taken into account in the quintuple-implication principle as opposed to the triple-implication principle and (2), for mixed information reasoning problem with inconsistent rules, the method proposed herein comprehensively integrates the two information conversion modes, from fuzzy sets to intuitionistic fuzzy sets and from intuitionistic fuzzy sets to fuzzy sets, instead of considering the two conversion modes separately as in [15]. All these results in Examples 3–6 justify the use of the methods in this paper over the methods in [15].

It is evident that the datasets in Examples 3–6 are not derived from the real world. Considering the fact that datasets from the real world can be directly used for mixed information reasoning are relatively few and the construction process of these datasets is complex, the test of the proposed methods on the real-world datasets will be one of future research work.

Regarding the computational complexity of the methods built in this paper, this is mainly related to the specific type of mixed information inference problem. The less information needs to be converted, the lower the computational complexity. When solving the mixed information inference problems of Type 3 and Type 4, only the inputs need to be converted into the same information type as the rule, and the computational complexity is lower. And for Types 5–8 mixed information inference problems, due to the inconsistency between the representations of rule antecedent and rule consequent, it is necessary to consider the directions from fuzzy sets to intuitionistic fuzzy sets and from intuitionistic fuzzy sets to fuzzy sets at the same time, and aggregate the results obtained from the two directions, so the computational complexity is higher. Although the methods proposed in this paper are comparatively more intricate than those presented in [15], they can yield significantly more compelling outcomes in practical applications.

To summarize, the constructed mixed information reasoning methods in this paper are mainly based on the quintuple-implication principle, which has a sound logical foundation. They can be regarded as the extension of the quintuple-implication reasoning method based on fuzzy sets [7] or the extension of the quintuple-implication reasoning method based on intuitionistic fuzzy sets [14]. Furthermore, compared to the existing mixed information reasoning methods based on the triple I principle proposed in [15], the reasoning methods developed in this paper employ the same conversion operators between fuzzy sets and intuitionistic fuzzy sets as those used in [15]. Meanwhile, these two kinds of methods exhibit reductive properties in the inference problems of Types 3, 5, 6 and 7 but are not appropriate to discuss reductivity in the inference problems of Type 4 and Type 8. In addition, when designing the inference process and calculating the inference results, the mixed information reasoning methods in this paper comprehensively consider the inference results in both the direction of converting fuzzy sets into intuitionistic fuzzy sets and the direction of converting intuitionistic fuzzy sets into fuzzy sets for the inference problems with the given inconsistent rules. Compared with the methods those consider only a single conversion direction in [15], the final inference results from our methods are more advantageous. When the inference problem involves the mixture of fuzzy information and intuitionistic fuzzy information, no matter what the mixture type is, the mixed information reasoning methods constructed in this paper can directly give reasonable inference results. These analyses imply that the methods constructed in this paper provide more technical support for solving mixed information reasoning problems and have great potential and broad prospect in practical applications. In view of the fact that the methods proposed in this paper are capable of deriving more insightful conclusions and trends from the data, they can assist managers in making more informed and rational judgments and decisions in complex environments with the mixture of fuzzy information and intuitionistic fuzzy information.

6. Conclusions

A new scheme in this paper has been described to solve inference problems that are infused with various combinations of fuzzy information and intuitionistic fuzzy information. This scheme was based on the idea that the key to the rule-based reasoning is the precise use of the given if–then rules. For Type 3 and Type 4 inference problems characterized by consistent if–then rules, only the representation type of the input was adjusted to match that of the rule for building the reasoning method while maintaining the invariability of the consistent rule. For Types 5–8 inference problems with inconsistent if–then rules, a mixed representation rule was reinterpreted as two distinct single representation rules to construct specific reasoning methods. Therefore, the mixed rule needed to be converted differently twice, where each converted object involves only one of rule antecedent and rule consequent, and the input representation must be consistent with the representation of the converted rule. Subsequently, guided by the possibility and necessity operators as well as the quintuple-implication principle, the methods tailored to each type were developed, alongside the outputs in both fuzzy set representation and intuitionistic fuzzy set representation. The reductivity of the proposed methods for mixed information inference problems of Types 3–8 was also studied in detail. In addition, the proposed methods demonstrated significant effectiveness over the mixed information reasoning methods under the triple-implication principle and the same conversion operators in the multiple recognition tasks studied.

This paper primarily addresses the inference problem associated with mixed fuzzy and intuitionistic fuzzy information. When it comes to other types of hybrid information, the methods constructed in this paper will not work directly. Moreover, the methods in this paper are related to the conversion operators between fuzzy sets and intuitionistic fuzzy sets, and only the relatively simple conversion operations between the two are used, so the methods proposed in this paper have room for improvement. Possible directions for future work would be to study the mixed information reasoning methods induced by other conversion operators between fuzzy sets and intuitionistic fuzzy sets to overcome the limitations of the conversion operators employed in this paper, or to study the reasoning methods under the mixture of fuzzy and picture fuzzy information or the mixture of intuitionistic fuzzy and picture fuzzy information, leading to more interesting results.