#### Abstract

Type-2 fuzzy reasoning relations are the type-2 fuzzy relations obtained from a group of type-2 fuzzy reasonings by using extended t-(co)norm, which are essential for implementing type-2 fuzzy logic systems. In this paper an algorithm is provided for constructing type-2 fuzzy reasoning relations of SISO type-2 fuzzy logic systems. First, we give some properties of extended t-(co)norm and simplify the expression of type-2 fuzzy reasoning relations in accordance with different input subdomains under certain conditions. And then different techniques are discussed to solve the simplified expressions on the input subdomains by using the related methods on solving fuzzy relation equations. Besides, it is pointed out that the computation amount level of the proposed algorithm is the same as that of polynomials and the possibility of applying the proposed algorithm in the construction of type-2 fuzzy reasoning relations is illustrated on several examples. Finally, the calculation of an arbitrary extended continuous t-norm can be obtained as the special case of the proposed algorithm.

#### 1. Introduction

Type-2 fuzzy sets first proposed by Zadeh in 1975 [1] are fuzzy sets equipped with ordinary fuzzy subsets of as membership grades, henceforth called fuzzy truth values. Then Mizumoto and Tanaka [2, 3] used Zadeh's extension principle to extend minimum and maximum both based on minimum for calculating union and intersection on type-2 fuzzy sets, respectively, and showed that the results of the union and intersection keep the convexity and normality. Based on the theory of type-2 fuzzy sets, Karnik et al. [4] proposed a new fuzzy system called type-2 fuzzy system. Up to now, both the theory and application of type-2 fuzzy systems have been widely researched (see, e.g., [5–8]). What is more, type-2 fuzzy neural networks and type-2 fuzzy classification and pattern recognition have been also studied (see, e.g., [9, 10]). However, the computation process of the extended operations on the noninterval type-2 fuzzy sets is more complex than that of ordinary operations on type-1 fuzzy sets, which blocks the wide use of the noninterval type-2 fuzzy logic systems, type-2 fuzzy neural networks, and so on. In recent years, a heated wave of research about the operation on type-2 fuzzy sets has been set off. For example, Karnik and Mendel [11] further generalized these definitions of operations presented by Mizumoto and Tanaka and gave some analytical formulae for extensions of extended maximum and minimum based on minimum or product. Kawaguchi and Miyakoshi [12, 13] showed that extended continuous t-(co)norms based on arbitrary t-norm satisfy the definitions of type-2 t-(co)norms. C. L. Walker and E. A. Walker [14, 15] considered the algebras of fuzzy truth values equipped with extended maximum and minimum based on minimum. Coupland and John [16, 17] presented geometric methods for performing the operations of extended minimum and maximum based on minimum on type-2 fuzzy sets. Starczewski [18] provided analytical expressions for membership functions of five kinds of extended t-norms. Ling and Zhang [19] reconstructed the framework of set-theoretic operations on triangle type-2 fuzzy sets by presenting polygon type-2 fuzzy sets and gave manageable and simplified formulas for operations on triangle type-2 fuzzy sets. Hu and Kwong [20] discussed extended t-norm on a linearly ordered set with a unit interval and a real number set as special cases.

From the above it can be seen that these research works have well contributed to the properties of extended t-(co)norms and gave many useful results for the calculations of some kinds of extended t-(co)norms. All of these promote the structure of noninterval type-2 fuzzy logic systems since extended t-(co)norms are the important tools in the construction of type-2 fuzzy reasoning relations. Nevertheless, there are still many other extended t-norms whose membership functions lack analytical expressions or feasible algorithms. It hampers the attempt of the construction of type-2 fuzzy reasoning relations by using these extended t-norms. Besides, the work [18] leaves a key problem to us that, except for extended minimum and maximum both based on minimum, no theory guarantees that the results of general extended t-(co)norms on two type-2 fuzzy sets still satisfy the calculation conditions (e.g., convexity and normality). Moreover, there are always more than two fuzzy truth values in the calculation process of the construction of type-2 fuzzy reasoning relations; it may be time-consuming and laborious to proceed the calculation just on two fuzzy truth values each time. It is a natural idea that we can solve the computation in an integral and faster way. This paper is devoted to deal with these problems we have mentioned above. The following rows present our results: we show that the results of extended continuous t-(co)norms based on arbitrary t-norm keep the convexity and normality and simplify the expression of type-2 fuzzy reasoning relations of type-2 fuzzy logic systems with single input and single output (SISO) in accordance with different input subdomains under the condition that all the fuzzy truth values of type-2 fuzzy sets participated in the calculation are required to be convex and normal (Theorem 2). After that, we solve the simplified expressions on three input subdomains (from Theorem 3 to Theorem 9), which demonstrate an algorithm to construct type-2 fuzzy reasoning relations. The complexity of the algorithm is analyzed and it is pointed out that the computation amount level of the proposed algorithm is the same as that of polynomials. And then the possibility of applying the proposed algorithm in the construction of type-2 fuzzy reasoning relations is illustrated on several examples. Besides, the calculation of a class of extended t-norms being broader than those in [18] can be obtained as the special case of the proposed algorithm.

This paper is organized in five sections. The following section contains some preliminary knowledge and the concrete expression of type-2 fuzzy reasoning relations of SISO type-2 fuzzy logic systems. In Section 3 the method for the construction of type-2 fuzzy reasoning relations is investigated under certain conditions on the basis of the properties of extended t-(co)norm and the related methods on solving method of fuzzy relation equations. Section 4 gives several examples by using the presented method. Conclusions are given in Section 5.

#### 2. Preliminaries

A type-2 fuzzy set on the domain is characterized by a membership function , , where , and is called a fuzzy truth value. Convenience to the following writing, we denote by . Moreover, is normal if there exists an such that and convex if, for any and each , . Let be the set of both convex and normal fuzzy truth values. Assume that . Let and be t-norm and t-conorm. Union and intersection on type-2 fuzzy sets are given as follows. For ,
where and are called* extended t-conorm* and* extended t-norm*, respectively. Let be a new domain constructed by two domains , . A type-2 fuzzy set is called a type-2 fuzzy relation between and , where
In the following, we will give the expression of type-2 fuzzy relation from a group of type-2 fuzzy reasoning. This type-2 fuzzy relation is called a* type-2 fuzzy reasoning relation*. Let and be, respectively, type-2 fuzzy sets on input domain and output domain . For a group of type-2 fuzzy reasonings in a SISO type-2 fuzzy logic system
which can be rewritten as and induce the total type-2 fuzzy reasoning relation as follows:
By choosing the suitable and we can obtain that
where and indicate the same t-norm. It is clear that the difficulty on the calculation of type-2 fuzzy reasoning relation is to solve the expression (5). For convenience, we first fix and and denote
Then the expression (5) can be rewritten as

In what follows, we mainly pay attention to working out the expression (7). When changes, and change with it. Then in order to solve , we should reduce the range of as much as possible and then obtain (i.e., the maximum of in ) according to the characteristic of elements in . Next, we will focus on analyzing the condition , which is a fuzzy relation equation if is regarded as a coefficient vector and is regarded as an unknown vector. It is known that fuzzy relation equation was first presented by Sanchez in 1976 [21]. Following it, a lot of work has focused on the solvability conditions and the solution sets. For example, these works [22–24] have systematically introduced some theories of fuzzy relational equations. Bourke and Fisher [25] gave solution algorithms for fuzzy relational equations with max-product composition. Stamou and Tzafestas [26] discussed the resolution of composite fuzzy relation equations based on Archimedean triangular norms. Wang and Xiong [27] investigated the solution sets of a fuzzy relation equation with sup-conjunctor composition in a complete lattice. Next some conceptions and conclusions on fuzzy relation equations will be given.

Let , . Define the partial order There exists no partial order relation between and if and only if .

Define The single fuzzy relation equation constituted by composite relation is as follows: where is the coefficient vector, is known, and is unknown. Let be the solution set of (10). The greatest and minimal elements in are, respectively, called the greatest and minimal solutions of (10). Denote Define . Moreover, some necessary interpretations about the two operations are presented in the following.(1) since .(2)If then ; if , then .(3)Both and are monotone decreasing about the first variable, that is,

since and .

Let In this work it is assumed that is continues and the following results presented in [27] are fitted for (10) on .

Lemma 1. *Let be a continuous t-norm. Then the following items are equivalent. *(1)* if and only if ; that is, there exists , such that if and only if (10) has the greatest solution .*(2)*If , then (10) has the minimum solutions where the th minimum solution is
**
Furthermore, the solution set of (10) can be written as
*

#### 3. The Construction of Type-2 Fuzzy Reasoning Relations

In this section, we will demonstrate the solving process for the expression (7) gradually. First, we will simplify the expression (7) in accordance with three subdomains of . Importantly, for two of these subdomains we will, respectively, reduce into its subdomains and but keeping the values of without change (Theorem 2). Then all the elements in and will be found out (Theorem 3). Following it, and will be further reduced into smaller subsets and still keeping the values of without change, respectively (Theorem 5). Finally, some theorems about how to get the exact value of will be presented on the basis of the characteristics of the on and (Theorems 7 and 9).

It needs to be stated that the proposed method to solve differs from the native algorithm which is just finding the maximal number of from all the elements in (or and ). The native algorithm is impractical due to its huge computation. But what form of the elements in is the key to solving the problem (7). Let . Denote .

Theorem 2. *Let , where = , . Denote
**
Then the following items hold.*(1)*If , then .*(2)*If , then .*(3)*If , then .*

Before the proof of Theorem 2, several conclusions and their proofs will be given in the following and the conclusion (a) is from [18].

(a) Let , where and . Assume that is continuous. Denote Then the following items hold. (1)If , then .(2)If , then .(3)If , then .

(b) Suppose that the conditions is the same as that of (a). Denote Then the following items hold. (1)If , then .(2)If , then .(3)If , then .

*Proof. *This proof is similar as that of (a) in [18] since is also monotone increasing in the first and second variables.

(c) Let , where . Assume that is continuous. Denote Then for every [resp. ], there exists [resp. ] such that and .

*Proof. *Let and . Since , by the monotonicity of , we have or . Assume that . If , then the conclusion is obvious. For the case of , there is
that is,
By the continuity of , it can be inferred that there exists such that . Let . Then there are , , and . Clearly . Similarly, we can prove that if and , there exists such that . Let . Then there are , , and . To sum up, we can conclude that for every , there exists such that , . In a similar way, we can prove that for every , there exists such that and .

(d) Let , where and . Assume that and are continuous. Then . Furthermore, and .

*Proof. *Let , , , , , where . By the continuity of and conclusion (a), we obtain that if , then . For the converse, let . If there exists such that , that is, there exists such that and , then there is . By the monotonicity of , we have or . Without loss of generality, we can assume that . Thus , which leads to a contradiction. Therefore, . In a similar way, we can prove that if , then . To sum up, we have .

Now we will give the proof of convexity. It is obvious that
If , then . From conclusion (a), it can be inferred that the values of and can be obtained on . Denote
Let satisfy
By conclusion (c), there exists such that and . Because and by the convexity of and , we have
From (24) and (25), we get
which implies that (22) holds. If , then . In a similar way, we can prove that . Thus (22) holds. To sum up, there is . It is easy to prove that the conclusion (c) is valid if is replaced with since is also monotone increasing in the first and second variables. Therefore, in a similar way, we can give the proof of .

Next we will give the proof of Theorem 2.

*Proof. *It is known that . From conclusion (d) it can be obtained that and , . Moreover, denote
From conclusion (a), we obtain that if , then = ; if , then = . From conclusion (d) we have and = . Denote and
From the above discussion and conclusion (b), we have that if , then
Similarly, if , then .

From Theorem 2, it can be seen that when , we can omit the calculation process of since , and for other situations can be obtained from or independently. From now on, we will focus on analyzing the cases of and and assume that , where , . Denote The idea about how to find the elements in [, resp.] is to solve the fuzzy relation equation by taking [resp. ] and then obtain the solution in [, resp.]. Thus [, resp.]. In this way, all the elements in [, resp.] can be found. Denote Now we will provide all the elements of and .

Theorem 3. *Assume that is continuous. For every or denote the greatest solution of (31) in as and minimal solution of (31) in as (if any). The following items hold. *(1)*Suppose that . Then for every the solution set of (31) in is denoted by and
*(2)*Suppose that . Then for every the solution set of (31) in is denoted by and
*

*Proof. * From Lemma 1 it is obvious that is the solution set of (31) in . For every , we have and . Then from Lemma 1 it can be inferred that . Thus ; that is, . For the converse case, let . Then . Obviously and is a solution of (31) with the coefficient vector . Denote the solution set of (31) in as . Clearly there exists such that . Thereby ; that is, ; that is, . To sum up, the conclusion holds. In a similar way, we can prove the case .

Corollary 4. *Assume that is continuous. Let or . Denote the greatest solution of (31) in as and minimal solution of (31) in as (if any). The following hold. *(1)*Let . Equation (31) has a solution in if and only if there exists such that .*(2)*Let . Equation (31) has a solution in if and only if .*

*Proof. * Equation (31) has a solution in , if and only if , and if and only if there exists such that .

Equation (31) has a solution in , if and only if , and if and only if .

Next, on the basis of Theorem 3 we will further find subsets of and but keeping the values of without change.

Theorem 5. *Assume that is continuous. The following items hold. *(1)*Suppose that and for every the greatest solution of (31) in is (if any). Denote
* *Then
*(2)*Suppose that and for every minimal solutions of (31) in are , (if any). Denote
* *Then
*

*Proof. * Clearly, for every , (31) has a solution in . From Theorem 3, there is ; that is, . Then
For the converse case, let . Then and . We have . Denote . From the convexity of and and the monotonicity of t-norm, there is
That is, for every there exists , such that . Thus
Combined with (39) and (41), it can be shown that the conclusion holds.

Clearly, for every , (31) has a solution in . From Theorem 3, there is , where ; that is, . Thus
For the converse case, let . Then and . There exists , such that . Denote . From the convexity of and and the monotonicity of t-norm, there is
That is, for every , there exists , such that . Thus
Combined with (42) and (44), conclusion holds.

If , from Theorem 5 it can be seen that all of the elements in can be obtained when all of the elements in and the greatest solutions of the corresponding equation (31) in are obtained. The following lemma describes the characteristics of the elements in . Denote

Lemma 6. *Let and . Assume that is continuous. Then (31) has a solution in if and only if there exists such that
*

*Proof. *For the first, we will prove that , . It can be seen that since for every . Therefore, ; that is, . Obviously .

Let satisfy (46). Since , it can be inferred that (31) is solvable in and is a minimal solution in from Lemma 1, where . Then we have since . That is, , which verifies that (31) has a solution in by Corollary 4.

For the converse case, let be a solution of (31). Then . There exists such that . Thereby , and . Thus . It can be seen that equation is solvable and its minimal solution is by Lemma 1. Because is also a solution, we have . So . Thereby, . To sum up, ; that is, satisfies (46).

Now we will solve the formula (5) with the situation of . For every denote the greatest solution of (31) in by (if any), where . From Theorem 5 and Lemma 6 it can be inferred that Denote Obviously can be viewed as a union of subsets, where the th subset is as follows: That is, . Notice that, for any and , it may appear that . However, it will not affect our final results. The following theorem provides a method to obtain when .

Theorem 7. *Let . Assume that is continuous. Denote
**
Then the following items hold: *(1)*, ,*(2)*.*

*Proof. * Without loss of the generality, we prove the case of . Clearly, is bounded in . Then there exists such that reaches the maximum , . Similarly, there exists such that reaches the maximum , . From Lemma 6, it is easy to see that
Let satisfy (46). Then there are
Thus
Therefore, .

Clearly, since and .

If , then by Theorem 5 all elements in can be obtained when all of the elements in and minimal solutions of the corresponding equation (31) in are obtained. The following lemma describes the characteristics of the elements in . Denote

Lemma 8. *Let and . Assume that is continuous. Then (31) has a solution in and if and only if there exists such that
*

*Proof. *For the first, we will give the proof of for every . Note that since . Then . Therefore, since . We obtain that , which indicates that always exists and . Obviously .

Let satisfy (55). Because , from Lemma 1 it can be inferred that (31) has a solution in and the greatest solution is
Furthermore, for every we have since . Then , which indicates that (31) has a solution in by Corollary 4.

For the conversion, assume that (31) has a solution in . There exists such that . Then since . That is to say, . On the other hand, there is by Corollary 4. Thus for every , we have , which indicates that . Therefore, .

Next we will solve the formula (5) with the situation of . For every , denote minimal solutions of (31) in by (if any). From Theorem 5 and Lemma 8, it can be seen that For every satisfying (55) there must exist such that it has the following form: Then Denote Thus can be viewed as a union of subsets, where the th subset is as follows: