Decomposing a MISO fuzzy rules system into its equivalent collection of SISO fuzzy rules systems

Abstract

In this paper, a multiple-input single-output (MISO) fuzzy rules system is decomposed into its equivalent collection of single-input single-output (SISO) fuzzy rules systems. First, the constructive and destructive linguistic models are reviewed. Under certain conditions, the final consequence of the constructive linguistic model working on a MISO fuzzy rules system is equivalent to the final consequence of the constructive linguistic model working on a collection of SISO fuzzy rules systems. Meanwhile, under another conditions, the final consequence of the destructive linguistic model working on a MISO fuzzy rules system is equivalent to the final consequence of the destructive linguistic model working on a collection of the SISO fuzzy rules systems. The decomposition of the MISO fuzzy rules system with fuzzy singleton consequent parts is also investigated. This paper shows the MISO fuzzy rules system requires many more rules than the SISO fuzzy rules system. Through the decomposing process, we can realize the inference result with fewer rules.

Keywords

Fuzzy modeling fuzzy rules system decomposing multiple-input single-output (MISO)single-input single-output (SISO)

1 Introduction

Since the notion of fuzziness was introduced into systems theory by Zadeh [1], various linguistic models based on collections of if-then rules with vague predicates and using fuzzy reasoning have been constructed [2]. A fuzzy rules system with one input variable and one output variable is called a single-input single-output (SISO) fuzzy rules system. A fuzzy rules system with more than one input variable and only one output variable is called a multiple-input single-output (MISO) fuzzy rules system [2, 3]. Mamdani [4] first applied the fuzzy if-then rules system to a steam engine experimental device. There were also many applications of fuzzy if-then rules system in the field of control. An introduction of fuzzy control systems design can be found in [5]. Recently, many researchers achieved some great works in fuzzy control and successfully handled nonsmooth input such as dead-zone, backlash. For example, an adaptive fuzzy control scheme was designed in [6] for a class of MIMO nonlinear systems with nonsymmetrical nonlinear dead-zone input described by unknown function. The fuzzy rules systems are employed to approximate the unknown functions. In [7], the authors solved the problem of adaptive fuzzy inverse compensation control for uncertain nonlinear system whose actuator is subjected to generalized dead-zone nonlinearity. The fuzzy rules systems are embedded into compensation structure to handle uncertain input dynamics. Meanwhile, there were some works focused on discrete-time systems. For example, an adaptive fuzzy controller was designed in [8] for a class of nonlinear discrete-time systems with backlash, and the fuzzy rules systems are used to approximate the unknown nonlinear functions and unknown backlash. In [9], an adaptive fuzzy controller was constructed for a class of nonlinear discrete-time systems with the unknown functions, the dead-zone input, and the external disturbance. The fuzzy rules systems are used to approximate the unknown functions. A fuzzy state observer is designed to estimate the immeasurable states. Fuzzy tracking adaptive control of discrete-time switched nonlinear systems was constructed in [10] based on the common Lyapunov function method and by utilizing the fuzzy rules systems to approximate the unknown nonlinear functions. From the above, fuzzy rules systems play a significant role in the application of control systems. The problem is that in some cases, the number of fuzzy rules becomes increasingly huge. Therefore, the setup and adjustment of fuzzy rules become difficult.

The proposed paper is motivated by Seki’ works [11 –16]. In [11 –15], Seki et al. studied the equivalence conditions between different fuzzy models. In [16], Seki, Ishii and Mizumoto proposed a method for decomposing a MISO fuzzy rules system with n^m rules into its equivalent collection of SISO fuzzy rules systems with n × m rules. Through the decomposition, a huge fuzzy rules system with n^m rules can be realized by studying an equivalent collection of small fuzzy rules systems with n × m rules. However, in [16], the limitation is that the decomposition is only suitable for T-S MISO fuzzy rules system [17], i.e., the consequent parts must be functional type. In the proposed paper, the restriction of functional type consequent parts is removed. The proposed paper investigates the decomposing problem for both the antecedent and consequent parts being represented by fuzzy sets. Compared with the previous works, the main contributions of the proposed paper are as follows.

First, this paper proposes a method to decompose a MISO fuzzy rules system into its equivalent collection of SISO fuzzy rules systems. The restriction in [16] for the functional type consequent parts is removed and a general form is considered. Through the decomposing process, we can realize the inference result of a fuzzy rules system with fewer rules.

Second, this paper also investigates the decomposition of a special type of MISO fuzzy rules system, that is, the MISO fuzzy rules system with fuzzy singleton consequent parts. The general procedure for decomposing is obtained.

The remainder of this paper is organized as follows. Section 2 presents a brief review of the t-norm, t-conorm, constructive linguistic model and destructive linguistic model. Section 3 presents two propositions for decomposing a MISO fuzzy rules system into its equivalent collection of SISO rules systems. Section 4 investigates the decomposition of the MISO fuzzy rules system with fuzzy singleton consequent parts. Section 5 offers some concluding remarks.

2 Preliminaries

This section presents a review of the t-norm, t-conorm, constructive linguistic model and destructive linguistic model.

2.1 Review of t-norm and t-conorm

First, we review t-norm and t-conorm from the reference [18, Ch.5, 10, 11].

Definition 1. A binary operation ⊗: [0,1] × [0,1] ⟶ [0,1] is a t-norm if it satisfies the following:

Identity: 1 ⊗ x = x, for all x ∈ [0, 1].

Commutativity: x ⊗ y = y ⊗ x, for all x, y ∈ [0, 1].

Associativity: x ⊗ (y ⊗ z) = (x ⊗ y) ⊗ z, for all x, y, z ∈ [0, 1].

Monotonicity: w ⊗ y ≦ x ⊗ z, if for all w, x, y, z ∈ [0, 1] and w ≦ x, y ≦ z.

Definition 2. A binary operation ⊕ : [0, 1] × [0, 1] → [0, 1] is a t-conorm if it satisfies the following:

Identity: 0 ⊕ x = x, for all x ∈ [0, 1].

Commutativity: x ⊕ y = y ⊕ x, for all x, y ∈ [0, 1].

Associativity: x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z, for all x, y, z ∈ [0, 1].

Monotonicity: w ⊕ y ≦ x ⊕ z, if for all w, x, y, z ∈ [0, 1] and w ≦ x, y ≦ z.

Corollary 1.Min operator ∧ is the largest of all the t-norms.

Corollary 2.Max operator ∨ is the smallest of all the t-conorms.

Next, we provide an example of t-norm and t-conorm operations, as follows.

Example 1. Minimum t-norm; x ⊗ y = min {x, y}, and 0.5 ⊗ 0.3 = min {0.5, 0.3} =0.3.

Product t-norm; x ⊗ y = xy, and 0.5 ⊗ 0.3 = 0.5 × 0.3 = 0.15.

Lukasiewicz t-norm; x ⊗ y = max {0, x + y - 1}, and 0.5 ⊗ 0.3 = max {0, 0.5 + 0.3 - 1} =0.

Maximum t-conorm; x ⊕ y = max {x, y}, and 0.5 ⊕ 0.3 = max {0.5, 0.3} =0.5.

Lukasiewicz t-conorm; x ⊕ y = min {1, x + y}, and 0.5 ⊕ 0.3 = min {1, 0.5 + 0.3} =0.8.

Definition 3. Let ⊗ be a t-norm and ⊕ be a t-conorm. Then we say ⊗ is distributive over ⊕ if for all x, y, z ∈ [0, 1], x ⊗ (y ⊕ z) = (x ⊗ y) ⊕ (x ⊗ z), and that ⊕ is distributive over ⊗ if for all x, y, z ∈ [0, 1], x ⊕ (y ⊗ z) = (x ⊕ y) ⊗ (x ⊕ z).

Definition 4. Let ⊗ be a t-norm and ⊕ be a t-conorm. Then we say ⊗ is a dual t-norm of ⊕ if for all x, y ∈ [0, 1], $\bar{x \otimes y} = \bar{x} \oplus \bar{y}$ , where $\bar{x} = 1 - x$ and $\bar{y} = 1 - y$ .

Example 2. Let ⊗ be a minimum t-norm, ⊕ a maximum t-conorm, x = 0.3, y = 0.5 and z = 0.7. $\begin{matrix} x \otimes (y \oplus z) = 0.3 \otimes (0.5 \oplus 0.7) \\ = 0.3 \otimes 0.7 = 0.3 \\ (x \otimes y) \oplus (x \otimes z) = (0.3 \otimes 0.5) \oplus (0.3 \otimes 0.7) \\ = 0.3 \oplus 0.3 = 0.3 \end{matrix}$

Because x ⊗ (y ⊕ z) = (x ⊗ y) ⊕ (x ⊗ z), we say a minimum t-norm is distributive over a maximum t-conorm. $\begin{matrix} x \oplus (y \otimes z) = 0.3 \oplus (0.5 \otimes 0.7) = 0.3 \oplus 0.5 = 0.5 \\ (x \oplus y) \otimes (x \otimes z) = (0.3 \oplus 0.5) \otimes (0.3 \oplus 0.7) \\ = 0.5 \otimes 0.7 = 0.5 \end{matrix}$

Because x ⊕ (y ⊗ z) = (x ⊕ y) ⊗ (x ⊕ z), we say a maximum t-conorm is distributive over a minimum t-norm. $\begin{matrix} \bar{x \otimes y} = \bar{0.3 \otimes 0.5} = \bar{0.3} = 0.7 \\ \bar{x} \oplus \bar{y} = \bar{0.3} \oplus \bar{0.5} = 0.7 \oplus 0.5 = 0.7 \end{matrix}$

Because $\bar{x \otimes y} = \bar{x} \oplus \bar{y}$ , we say a minimum t-norm is a dual t-norm of a maximum t-conorm.

Example 3. Let ⊗ be a product t-norm, ⊕ a Lukasiewicz t-conorm, x = 0.7, y = 0.5 and z = 0.3. $\begin{matrix} x \otimes (y \oplus z) = 0.7 \otimes (0.5 \oplus 0.3) \\ = 0.7 \otimes 0.8 = 0.56 \\ (x \otimes y) \oplus (x \otimes z) = (0.7 \otimes 0.5) \oplus (0.7 \otimes 0.3) \\ = 0.35 \oplus 0.21 = 0.56 \end{matrix}$

Because x ⊗ (y ⊕ z) = (x ⊗ y) ⊕ (x ⊗ z), we say a product t-norm is distributive over a Lukasiewicz t-conorm in case the summation of values does not exceed 1.

2.2 Review of linguistic models

This subsection reviews the constructive and destructive linguistic models [2, ch. 5]. First, the following fuzzy rules system is considered:

Premise:

Rule R_j: $u_{1} = B_{j}^{1}, u_{2} = B_{j}^{2}, \dots, u_{n} = B_{j}^{n} \to v = D_{j} (z)$ .

Fact: $u_{1} = u_{1}^{0}, u_{2} = u_{2}^{0}, \dots, u_{n} = u_{n}^{0}$ . $Consequence : v = F (z)$ (1) where u_i, i = 1, 2, …, n are the input variables and v is the output variable; $B_{j}^{i}$ and D_j, j = 1, 2, …, m, i = 1, 2, …, n are fuzzy sets; The fact $u_{1}^{0}, u_{2}^{0}, \dots, u_{n}^{0}$ are real numbers and F is the inference result for the fact. Note that the total number of rules is m. The constructive and destructive linguistic models are presented as follows.

The constructive linguistic model:

The relevance degree h_j of the fact “ $u_{1}^{0}, u_{2}^{0}, \dots, u_{n}^{0}$ ” to the antecedent parts “ $B_{j}^{1}, B_{j}^{2}, \dots, B_{j}^{n}$ ” is given as $h_{j} = B_{j}^{1} (u_{1}^{0}) \otimes B_{j}^{2} (u_{2}^{0}) \otimes \dots \otimes B_{j}^{n} (u_{n}^{0})$ (2) where ⊕ stands for any t-norm. The inference result F_j (z) for rule R_j is given as

$\begin{matrix} F_{j} (z) & = & B_{j}^{1} (u_{1}^{0}) \otimes B_{j}^{2} (u_{2}^{0}) \otimes \dots \otimes B_{j}^{n} (u_{n}^{0}) \otimes D_{j} (z) \\ = & h_{j} \otimes D_{j} (z) . \end{matrix}$ (3)

The final consequence F (z) of (1) is aggregated from F₁, F₂, …, F_m by $F (z) = F_{1} (z) \oplus F_{2} (z) \oplus \dots \oplus F_{m} (z)$ (4) where ⊕ stands for any t-conorm. Note that if there is only one input variable, the MISO rules system becomes the SISO rules system.

The destructive linguistic model:

The relevance degree h_j of is the same as (2), i. e., $h_{j} = B_{j}^{1} (u_{1}^{0}) \otimes B_{j}^{2} (u_{2}^{0}) \otimes \dots \otimes B_{j}^{n} (u_{n}^{0})$ where ⊗ stands for any t-norm. The inference result F_j (z) for rule R_j is given as $F_{j} (z) = \bar{h_{j}} \oplus D_{j} (z),$ (5) where $\bar{h_{j}} = 1 - h_{j}$ and ⊕ stands for any t-conorm. The final consequence F (z) of (1) is aggregated from F₁, F₂, …, F_m by $F (z) = F_{1} (z) \otimes F_{2} (z) \otimes \dots \otimes F_{m} (z),$ (6) where ⊗ stands for any t-norm.

3 Decomposing a MISO fuzzy rules system into a collection of SISO fuzzy rules systems

This section presents two propositions for decomposing a MISO fuzzy rules system into its equivalent collection of SISO rules systems. First, the following MISO rules system is considered:

Premise:

Rule R_{j₁j₂…j_n}: $u_{1} = B_{j_{1}}^{1}, u_{2} = B_{j_{2}}^{2}, \dots, u_{n} = B_{j_{n}}^{n} \to v = D_{j_{1} j_{2} \dots j_{n}} (z) .$

Fact: $u_{1} = u_{1}^{0}, u_{2} = u_{2}^{0}, \dots, u_{n} = u_{n}^{0}$ . $Consequence : v = F (z)$ (7) where u_i, i = 1, 2, …, n are the input variables and v is the output variable; $B_{ji}^{i}$ and D_{j₁j₂…j_n}, j_i = 1, 2, …, m_i, i = 1, 2, …, n are fuzzy sets; The fact $u_{1}^{0}, u_{2}^{0}, \dots, u_{n}^{0}$ are real numbers and F is the inference result for the fact. Note that the total number of rules is m₁m₂ … m_n.

For the MISO rules system (7), we associate it with the following collection of SISO rules systems. For each i = 1, 2, …, n, a SISO rules system is presented as follows.

Premise:

Rule $R_{ji}^{i}$ : $u_{i} = B_{ji}^{i} \to v_{i} = D_{ji}^{i} (z)$ .

Fact: $u_{i} = u_{i}^{0}$ . $Consequence : v_{i} = F_{i} (z)$ (8-i) where D_jiⁱ, j_i = 1, 2, …, m_i, i = 1, 2, …, n are fuzzy sets; v_i is the output variable; F_i is the inference result for the fact. The other notations are defined in the same ways as in (7). Note that the total number of rules for the rules system (8-i) is m_i.

Proposition 1.LetF (z) be the final consequence of the constructive linguistic model working on the MISO fuzzy rules system (7). LetF_i (z) be the final consequence of the constructive linguistic model working on the SISO fuzzy rules system (8-i) fori = 1, 2, …, n. Suppose the consequent partD_{j₁j₂…j_n} (z) of (7) and the consequent part $D_{ji}^{i} (z)$ of (8-i) satisfy the following relationship $D_{j_{1} j_{2} \dots j_{n}} (z) = D_{j_{1}}^{1} (z) \otimes D_{j_{2}}^{2} (z) \otimes \dots \otimes D_{jn}^{n} (z),$ (9) where j_i = 1, 2, …, m_i, i = 1, 2, …, n, and ⊗ is a t-norm. Suppose the t-norm ⊗ is distributive over the t-conorm ⊕ in the aggregating process (3) and (4) of the constructive linguistic model. Then, the MISO rules system can be decomposed into a collection of SISO rules systems, and the final consequences of the MISO and SISO rules systems are equivalent as follows $F (z) = F_{1} (z) \otimes F_{2} (z) \otimes \dots \otimes F_{n} (z) = \otimes_{j = 1}^{n} F_{i} (z) .$ (10)

Proof. First, the MISO rules system (7) is considered. From (2), the relevance degree h_{j₁j₂…j_n} of the constructive linguistic model for rules system (7) is given as $h_{j_{1} j_{2} \dots j_{n}} = B_{j_{1}}^{1} (u_{1}^{0}) \otimes B_{j_{2}}^{2} (u_{2}^{0}) \otimes \dots \otimes B_{jn}^{n} (u_{n}^{0}) .$

From (3), the inference result F_{j₁j₂…j_n} (z) of rule R_{j₁j₂…j_n} is given as $\begin{matrix} F_{j_{1} j_{2} \dots j_{n}} (z) & = & h_{j_{1} j_{2} \dots j_{n}} \otimes D_{j_{1} j_{2} \dots j_{n}} (z) \\ = & B_{j_{1}}^{1} (u_{1}^{0}) \otimes B_{j_{2}}^{2} (u_{2}^{0}) \otimes \dots \\ \otimes B_{j_{n}}^{n} (u_{n}^{0}) \otimes D_{j_{1} j_{2} \dots j_{n}} (z) . \end{matrix}$

From (4), the final consequence F (z) of rules system (7) is given as

$\begin{matrix} F (z) & = & \oplus_{j_{1} = 1, j_{2} = 1, \dots, j_{n} = 1}^{m_{1}, m_{2}, \dots, m_{n}} F_{j_{1} j_{2} \dots j_{n}} (z) \\ = & \oplus_{j_{1} = 1, j_{2} = 1, \dots, j_{n} = 1}^{m_{1}, m_{2}, \dots, m_{n}} B_{j_{1}}^{1} (u_{1}^{0}) \otimes B_{j_{2}}^{2} (u_{2}^{0}) \\ \otimes \dots \otimes B_{j_{n}}^{n} (u_{n}^{0}) \otimes D_{j_{1} j_{2} \dots j_{n}} (z) . \end{matrix}$ (11)

Then, the SISO rules system (8-i) is considered. From (2), (3) and (4), the final consequence of the constructive linguistic model working on the SISO rules system (8-i) is given as $F_{i} (z) = \oplus_{j_{1} = 1}^{m_{i}} (B_{j_{i}}^{i} (u_{i}^{0}) \otimes D_{j_{i}}^{i} (z))$ where i = 1, 2, …, n. Then, the aggregation of F_i (z) by the t-norm is given as $\begin{matrix} \otimes_{j = 1}^{n} F_{i} (z) = F_{1} (z) \otimes F_{2} (z) \otimes \dots \otimes F_{m} (z) \\ = [\oplus_{j_{1} = 1}^{m_{1}} (B_{j_{1}}^{1} (u_{1}^{0}) \otimes D_{j_{1}}^{1} (z))] \\ \otimes [\oplus_{j_{2} = 1}^{m_{2}} (B_{j_{2}}^{1} (u_{2}^{0}) \otimes D_{j_{2}}^{1} (z))] \\ \otimes \dots \otimes [\oplus_{j_{n} = 1}^{m_{n}} (B_{j_{n}}^{n} (u_{n}^{0}) \otimes D_{j_{n}}^{n} (z))] \\ = [B_{1}^{1} (u_{1}^{0}) \otimes D_{1}^{1} (z) \otimes B_{1}^{2} (u_{2}^{0}) \otimes D_{1}^{2} (z) \otimes \dots \otimes \\ B_{1}^{n} (u_{n}^{0}) \otimes D_{1}^{n} (z)] \oplus \dots \oplus \\ [B_{m_{1}}^{1} (u_{1}^{0}) \otimes D_{m_{1}}^{1} (z) \otimes B_{m_{2}}^{2} (u_{2}^{0}) \otimes D_{m_{2}}^{2} (z) \\ \otimes \dots \otimes B_{m_{n}}^{n} (u_{n}^{0}) \otimes D_{m_{n}}^{n} (z)] \\ (since \otimes is distributive over \oplus) \\ = [B_{1}^{1} (u_{1}^{0}) \otimes \dots \otimes B_{1}^{n} (u_{n}^{0}) \otimes D_{1}^{1} (z) \otimes \dots \\ \otimes D_{1}^{n} (z)] \oplus \dots \oplus [B_{m_{1}}^{1} (u_{1}^{0}) \otimes \dots \\ \otimes B_{m_{n}}^{n} (u_{n}^{0}) \otimes D_{m_{1}}^{1} (z) \otimes \dots \otimes D_{m_{n}}^{n} (z)] \\ (since \otimes is commutative) \\ = \oplus_{j_{1} = 1, j_{2} = 1, \dots j_{n} = 1}^{m_{1}, m_{2}, \dots, m_{n}} \\ [B_{j_{1}}^{1} (u_{1}^{0}) \otimes B_{j_{2}}^{2} (u_{2}^{0}) \otimes \dots \otimes B_{j_{n}}^{n} (u_{n}^{0}) \\ \otimes D_{j_{1} j_{2} \dots j_{n}} (z)] . \\ (since D_{j_{1} j_{2} \dots j_{n}} (z) = D_{j_{1}}^{1} (z) \otimes \dots \otimes D_{j_{n}}^{n} (z)) \end{matrix}$

The latter is also the final consequence F (z) for the MISO rules system (7), as shown in (11). Thus, the proof is complete.

Proposition 2.LetF (z) be the final consequence of the destructive linguistic model working on the MISO fuzzy rules system (7). LetF_i (z) be the final consequence of the destructive linguistic model working on the SISO fuzzy rules system (8-i) fori = 1, 2, …, n. Suppose the consequent partD_{j₁j₂…j_n} (z) of (7) and the consequent part $D_{j_{i}}^{i} (z)$ of (8-i) satisfy the following relationship $D_{j_{1} j_{2} \dots j_{n}} (z) = D_{j_{1}}^{1} (z) \oplus D_{j_{2}}^{2} (z) \oplus \dots \oplus D_{j_{n}}^{n} (z)$ (12) where j_i = 1, 2, …, m_i, i = 1, 2, …, n, and ⊕ is a t-conorm. Suppose the t-conorm ⊕ is distributive over the t-norm ⊗ in the aggregating process (5) and (6) of the destructive linguistic model, and ⊗ is the dual t-norm of ⊕. Then, the MISO rules system can be decomposed into a collection of SISO rules systems, and the final consequences of the MISO and SISO rules systems are equivalent as follows $F (z) = F_{1} (z) \oplus F_{2} (z) \oplus \dots \oplus F_{n} (z) = \oplus_{j = 1}^{n} F_{i} (z) .$ (13)

Proof. First, the MISO rules system (7) is considered. The relevance degree h_{j₁j₂…j_n} of the destructive linguistic model for rules system (7) is given as $h_{j_{1} j_{2} \dots j_{n}} = B_{j_{1}}^{1} (u_{1}^{0}) \otimes B_{j_{2}}^{2} (u_{2}^{0}) \otimes \dots \otimes B_{j_{n}}^{n} (u_{n}^{0}) .$

From (5), the inference result F_{j₁j₂…j_n} (z) of rule R_{j₁j₂…j_n} is given as $\begin{matrix} F_{j_{1} j_{2} \dots j_{n}} (z) & = & \bar{h_{j_{1} j_{2} \dots j_{n}}} \oplus D_{j_{1} j_{2} \dots j_{n}} (z) \\ = & \bar{B_{j_{1}}^{1} (u_{1}^{0}) \otimes B_{j_{2}}^{2} (u_{2}^{0}) \otimes . . . \otimes B_{j_{n}}^{n} (u_{n}^{0})} \\ \oplus D_{j_{1} j_{2} \dots j_{n}} (z) . \end{matrix}$

From (6), the final consequence F (z) of rules system (7) is given as

$\begin{matrix} F (z) & = & \otimes_{j_{1} = 1, j_{2} = 1, \dots, j_{n} = 1}^{m_{1}, m_{2}, \dots, m_{n}} F_{j_{1} j_{2} \dots j_{n}} (z) \\ = & \otimes_{j_{1} = 1, j_{2} = 1, \dots, j_{n} = 1}^{m_{1}, m_{2}, \dots, m_{n}} \\ \bar{B_{j_{1}}^{1} (u_{1}^{0}) \otimes B_{j_{2}}^{2} (u_{2}^{0}) \otimes \dots \otimes B_{j_{n}}^{n} (u_{n}^{0})} \\ \oplus D_{j_{1} j_{2} \dots j_{n}} (z) . \end{matrix}$ (14)

Then, the SISO rules system (8-i) is considered. From (5) and (6), the final consequence of the destructive linguistic model working on the SISO rules system (8-i) is given as $F_{i} (z) = \otimes_{j_{i} = 1}^{m_{i}} (\bar{B_{j_{i}}^{i} (u_{i}^{0})} \oplus D_{j_{i}}^{i} (z))$ where i = 1, 2, …, n. Then, the aggregation of F_i (z) by t-conorm ⊕ is given as $\begin{matrix} \oplus_{j = 1}^{n} F_{i} (z) \\ = F_{1} (z) \oplus F_{2} (z) \oplus \dots \oplus F_{m} (z) \\ = [\otimes_{j_{1} = 1}^{m_{1}} (\bar{B_{j_{1}}^{1} (u_{1}^{0})} \oplus D_{j_{1}}^{1} (z))] \\ \oplus [\otimes_{j_{2} = 1}^{m_{2}} (\bar{B_{j_{2}}^{2} (u_{2}^{0})} \oplus D_{j_{2}}^{2} (z))] \\ \oplus \dots \oplus [\otimes_{j_{n} = 1}^{m_{n}} (\bar{B_{j_{n}}^{n} (u_{n}^{0})} \oplus D_{j_{n}}^{n} (z))] \\ = [\bar{B_{1}^{1} (u_{1}^{0})} \oplus D_{1}^{1} (z) \oplus B_{1}^{2} (u_{2}^{0}) \oplus D_{1}^{2} (z) \\ \oplus \dots \oplus \bar{B_{1}^{n} (u_{n}^{0})} \oplus D_{1}^{n} (z)] \otimes \dots \otimes \\ [\bar{B_{m_{1}}^{1} (u_{1}^{0})} \oplus D_{m_{1}}^{1} (z) \oplus \bar{B_{m_{2}}^{2} (u_{2}^{0})} \\ \oplus D_{m_{2}}^{1} (z) \oplus \dots \oplus \bar{B_{m_{n}}^{n} (u_{n}^{0})} \oplus D_{m_{n}}^{n} (z)] \\ (since \oplus is distributive over \otimes) \\ = [\bar{B_{1}^{1} (u_{1}^{0})} \oplus \dots \oplus \bar{B_{1}^{n} (u_{n}^{0})} \oplus D_{1}^{1} (z) \\ \oplus \dots \oplus D_{1}^{n} (z)] \otimes \dots \otimes [B_{m_{1}}^{1} (u_{1}^{0}) \\ \oplus \dots \oplus \bar{B_{m_{n}}^{n} (u_{n}^{0})} \oplus D_{m_{1}}^{1} (z)] \\ \oplus \dots \oplus D_{m_{n}}^{n} (z)] (since \oplus is commutative) \\ = [\bar{B_{1}^{1} (u_{1}^{0}) \otimes \dots \otimes B_{1}^{n} (u_{n}^{0})} \oplus D_{1}^{1} (z) \\ \oplus \dots \oplus D_{1}^{n} (z)] \otimes \dots \\ \otimes [\bar{B_{m_{1}}^{1} (u_{1}^{0}) \otimes \dots \otimes B_{m_{n}}^{n} (u_{n}^{0})} \oplus D_{m_{1}}^{1} (z) \\ \oplus \dots \oplus D_{m_{n}}^{n} (z)] \\ (since \otimes is the dual t - norm of \oplus) \\ = \otimes_{j_{1} = 1, j_{2} = 1, \dots, j_{n} = 1}^{m_{1}, m_{2}, \dots, m_{n}} \\ [\bar{B_{j_{1}}^{1} (u_{1}^{0}) \otimes B_{j_{2}}^{2} (u_{2}^{0}) \otimes \dots \otimes B_{j_{n}}^{n} (u_{n}^{0})} \\ \oplus D_{j_{1} j_{2} \dots j_{n}} (z)] . (since D_{j_{1} j_{2} \dots j_{n}} (z) \\ = D_{j_{1}}^{1} (z) \oplus \dots \oplus D_{j_{n}}^{n} (z)) \end{matrix}$

The latter is also the final consequence F (z) for the MISO rules system (7), as shown in (14). Thus, the proof is complete.

Next, the conditions in Propositions 1 and 2 are discussed. There are two conditions to be satisfied in Proposition 1. First condition, the t-norm ⊗ is distributive over the t-conorm ⊕. If we let ⊗ be a minimum t-norm, ⊕ a maximum t-conorm, then ⊗ is distributive over ⊕ as shown in Example 2. The other t-norms and t-conorms satisfying first condition can be also found. Let ⊗ be a product t-norm, ⊕ a Lukasiewicz t-conorm. If the summation of values does not exceed 1, the Lukasiewicz t-conorm will operate as the sum operation. Then the product t-norm is distributive over the Lukasiewicz t-conorm as shown in Example 3. Besides, any t-norm is distributive over the maximum t-conorm and any t-conorm is distributive over the minimum t-norm. In Proposition 2, one more condition to be satisfied is that the t-norm ⊗ must be a dual t-norm of the t-conorm ⊕. Let ⊗ be a minimum t-norm, ⊕ a maximum t-conorm. Then, the above condition is satisfied as shown in Example 2. There are other t-norms and t-conorms are dual such as product t-norm and probabilistic sum t-conorm, Lukasiewicz t-norm and bounded sum t-conorm, etc. The other condition is that Equations (9 and 12) must be satisfied. This condition is discussed in Section 5. To summarize, for Proposition 1, two conditions must be satisfied. First, the t-norm ⊗ must be distributive over the t-conorm ⊕. Second, Equation (9) must be satisfied. For Proposition 2, three conditions must be satisfied. First, the t-conorm ⊕ must be distributive over the t-norm ⊗. Second, the t-norm ⊗ must be a dual t-norm of the t-conorm ⊕. Third, Equation (12) must be satisfied.

From Propositions 1 and 2, this paper proposes the conditions for decomposing a MISO fuzzy rules system into its equivalent collection of SISO fuzzy rules systems. The following example is provided to illustrate the decomposing process.

Example 4. Considering two inputs u₁ and u₂, the rules system is given as

$\begin{matrix} Rule R_{11} : u_{1} = B_{1}^{1}, u_{2} = B_{1}^{2} \to v = D_{11} \\ Rule R_{12} : u_{1} = B_{1}^{1}, u_{2} = B_{2}^{2} \to v = D_{12} \\ Rule R_{21} : u_{1} = B_{2}^{1}, u_{2} = B_{1}^{2} \to v = D_{21} \\ Rule R_{22} : u_{1} = B_{2}^{1}, u_{2} = B_{2}^{2} \to v = D_{22}, \end{matrix}$ (15) where the fuzzy sets $B_{j}^{i}$ , i = 1, 2 and j = 1, 2, are shown in Fig. 1. The fuzzy set D₁₁ is represented by trapezoid a-b-c-h, D₁₂ by trapezoid i-c-e-f, D₂₁ by triangle i-c-h, and D₂₂ by trapezoid i-c-d-g, as shown in Fig. 2. The x coordinates of a, b, c, d, e, f, g, h and i are 0, 1, 2, 2.5, 3, 6, 4, 3 and 1. First, we solve the problem by the MISO rules system (15). Choose the constructive linguistic model, and choose t-norm ⊗ = min operation, t-conorm ⊕ = max operation. The defuzzification is by the center-of-gravity method. The inference result by the constructive linguistic model working on the MISO rules system (15) is shown in Fig. 3.

Fig.1

Fuzzy sets for $B_{1}^{i}$ and $B_{2}^{i}$ , i = 1, 2.

Fig.2

Fuzzy sets for D₁₁, D₁₂, D₂₁ and D₂₂.

Fig.3

Inference result by the constructive linguistic model working on the MISO rules system (15).

Then, we solve the problem by decomposing the MISO rules system into a collection of SISO rules systems. We associate the MISO rules system (15) with the following SISO rules systems. $\begin{matrix} Rule R_{11} : u_{1} = B_{1}^{1} \to v_{1} = D_{1}^{1} \\ Rule R_{12} : u_{1} = B_{2}^{1} \to v_{1} = D_{1} \end{matrix}$ (16-1) $\begin{matrix} Rule R_{21} : u_{2} = B_{1}^{2} \to v_{2} = D_{1}^{2} \\ Rule R_{22} : u_{2} = B_{2}^{2} \to v_{2} = D_{2}^{2}, \end{matrix}$ (16-2) where the fuzzy sets $B_{j}^{i}$ , i = 1, 2 and j = 1, 2, are the same as in (15). The fuzzy sets $D_{1}^{1}$ is represented by trapezoid a-b-e-f, $D_{2}^{1}$ by trapezoid i-c-d-g, $D_{1}^{2}$ by trapezoid a-b-c-h, and $D_{2}^{2}$ by trapezoid i-c-e-f, as shown in Fig. 2. Choose the constructive linguistic model, and choose t-norm ⊗ = min operation, t-conorm ⊕ = max operation. The defuzzification is by the center-of-gravity method. From Fig. 2, Equation (9) of Proposition 1 is satisfied as follows. $\begin{matrix} D_{1}^{1} \otimes D_{1}^{2} = trapezoid a - b - e - f \otimes trapezoid \\ a - b - c - h = trapezoid a - b - c - h = D_{11} \\ D_{1}^{1} \otimes D_{2}^{2} = trapezoid a - b - e - f \otimes trapezoid \\ i - c - e - f = trapezoid i - c - e - f = D_{12} \\ D_{2}^{1} \otimes D_{1}^{2} = trapezoid i - c - d - g \otimes trapezoid \\ a - b - c - h = triangle i - c - h = D_{21} \\ D_{2}^{1} \otimes D_{2}^{2} = trapezoid i - c - d - g \otimes trapezoid \\ i - c - e - f = trapezoid i - c - d - g = D_{22} \end{matrix}$

The other condition in Proposition 1 is also satisfied such that t-norm ⊗ is distributive over t-conorm ⊕. From Proposition 1, the defuzzification result of the constructive linguistic model working on the collection of the SISO rules systems (16-1) and (16-2) is equal to the defuzzification result of the constructive linguistic model working on the MISO rules system (15). Thus, we decompose the MISO rules system (15) into an equivalent collection of the SISO rules systems (16-1) and (16-2). The inference result is the same as in Fig. 3.

4 Decomposition of the MISO rules system with fuzzy singleton consequent parts

This section investigates the decomposition of the MISO rules system with fuzzy singleton consequent parts. Decomposition for the constructive and destructive linguistic models is both investigated. For clarity, first considering two inputs u₁ and u₂, the rules system is given as

$\begin{matrix} Rule R_{11} : u_{1} & = & B_{1}^{1}, u_{2} = B_{1}^{2} \to v = D_{11} \\ = & w_{1} / g_{1}^{1} (u_{1}) + w_{2} / g_{1}^{2} (u_{2}) \\ Rule R_{12} : u_{1} & = & B_{1}^{1}, u_{2} = B_{2}^{2} \to v = D_{12} \\ = & w_{1} / g_{1}^{1} (u_{1}) + w_{2} / g_{2}^{2} (u_{2}) \\ Rule R_{21} : u_{1} & = & B_{2}^{1}, u_{2} = B_{1}^{2} \to v = D_{21} \\ = & w_{1} / g_{2}^{1} (u_{1}) + w_{2} / g_{1}^{2} (u_{2}) \\ Rule R_{22} : u_{1} & = & B_{2}^{1}, u_{2} = B_{2}^{2} \to v = D_{22} \\ = & w_{1} / g_{2}^{1} (u_{1}) + w_{2} / g_{2}^{2} (u_{2}) \end{matrix}$ (17) where $B_{1}^{1}$ , $B_{1}^{2}$ , $B_{1}^{1}$ , $B_{2}^{2}$ , D₁₁, D₁₂, D₂₁, and D₂₂ are fuzzy sets; v is the output variable; $g_{1}^{1}$ , $g_{1}^{2}$ , $g_{2}^{1}$ , and $g_{2}^{2}$ are functions; w₁ and w₂ are the weights and w₁ ≦ 1, w₂ ≦ 1; The notation $w_{i} / g_{j}^{i} (u_{i})$ means it is a fuzzy singleton with membership value w_i at $g_{j}^{i} (u_{i})$ for i = 1, 2 and j = 1, 2.

Next, the MISO rules system (17) is decomposed into a collection of SISO rules systems. We associate (17) with the following rules systems. $\begin{matrix} Rule R_{11} : u_{1} \\ = B_{1}^{1} \to v_{1} = D_{1}^{1} = 1 / g_{1}^{1} (u_{1}) + 0 / g_{2}^{1} (u_{1}) \\ + w_{2} / g_{1}^{2} (u_{2}) + w_{2} / g_{2}^{2} (u_{2}) \\ Rule R_{12} : u_{1} \\ = B_{2}^{1} \to v_{1} = D_{2}^{1} = 1 / g_{1}^{1} (u_{1}) + 1 / g_{2}^{1} (u_{1}) \\ + w_{2} / g_{1}^{2} (u_{2}) + w_{2} / g_{2}^{2} (u_{2}) \end{matrix}$ (18-1) $\begin{matrix} Rule R_{21} : u_{2} \\ = B_{1}^{2} \to v_{2} = D_{1}^{2} = w_{1} / g_{1}^{1} (u_{1}) + w_{1} / g_{2}^{1} (u_{1}) \\ + 1 / g_{1}^{2} (u_{2}) + 0 / g_{2}^{2} (u_{2}) \\ Rule R_{22} : u_{2} \\ = B_{2}^{2} \to v_{2} = D_{2}^{2} = w_{1} / g_{1}^{1} (u_{1}) + w_{1} / g_{2}^{1} (u_{1}) \\ + 0 / g_{1}^{2} (u_{2}) + 1 / g_{2}^{2} (u_{2}) \end{matrix}$ (18-2)

Equation (9) of Proposition 1 is satisfied as follows. $\begin{matrix} D_{1}^{1} \otimes D_{1}^{2} \\ = (1 / g_{1}^{1} + 0 / g_{2}^{1} + w_{2} / g_{1}^{2} + w_{2} / g_{2}^{2}) \\ \otimes (w_{1} / g_{1}^{1} + w_{1} / g_{2}^{1} + 1 / g_{1}^{2} + 0 / g_{2}^{2}) \\ = (1 \otimes w_{1}) / g_{1}^{1} + (0 \otimes w_{1}) / g_{2}^{1} + (w_{2} \otimes 1) / g_{1}^{2} \\ + (w_{2} \otimes 0) / g_{2}^{2} = w_{1} / g_{1}^{1} + w_{2} / g_{1}^{2} = D_{11} \\ D_{1}^{1} \otimes D_{2}^{2} \\ = (1 / g_{1}^{1} + 0 / g_{2}^{1} + w_{2} / g_{1}^{2} + w_{2} / g_{2}^{2}) \\ \otimes (w_{1} / g_{1}^{1} + w_{1} / g_{2}^{1} + 0 / g_{1}^{2} + 1 / g_{2}^{2}) \\ = (1 \otimes w_{1}) / g_{1}^{1} + (0 \otimes w_{1}) / g_{2}^{1} + (w_{2} \otimes 0) / g_{1}^{2} \\ + (w_{2} \otimes 1) / g_{2}^{2} = w_{1} / g_{1}^{1} + w_{2} / g_{2}^{2} = D_{12} \\ D_{2}^{1} \otimes D_{1}^{2} \\ = (0 / g_{1}^{1} + 1 / g_{2}^{1} + w_{2} / g_{1}^{2} + w_{2} / g_{2}^{2}) \\ \otimes (w_{1} / g_{1}^{1} + w_{1} / g_{2}^{1} + 1 / g_{1}^{2} + 0 / g_{2}^{2}) \\ = (0 \otimes w_{1}) / g_{1}^{1} + (1 \otimes w_{1}) / g_{2}^{1} + (w_{2} \otimes 1) / g_{1}^{2} \\ + (w_{2} \otimes 0) / g_{2}^{2} = w_{1} / g_{2}^{1} + w_{2} / g_{1}^{2} = D_{21} \\ D_{2}^{1} \otimes D_{2}^{2} \\ = (0 / g_{1}^{1} + 1 / g_{2}^{1} + w_{2} / g_{1}^{2} + w_{2} / g_{2}^{2}) \\ \otimes (w_{1} / g_{1}^{1} + w_{1} / g_{2}^{1} + 0 / g_{1}^{2} + 1 / g_{2}^{2}) \\ = (0 \otimes w_{1}) / g_{1}^{1} + (1 \otimes w_{1}) / g_{2}^{1} + (w_{2} \otimes 0) / g_{1}^{2} \\ + (w_{2} \otimes 1) / g_{2}^{2} = w_{1} / g_{2}^{1} + w_{2} / g_{2}^{2} = D_{22} \end{matrix}$

Thus, Equation (9) of Proposition 1 is satisfied. Choose the t-norm and the t-conorm such that the t-norm is distributive over the t-conorm. For example, the minimum t-norm and the maximum t-conorm satisfy the above condition. From Proposition 1, the constructive linguistic model for the SISO rules systems (18-1) and (18-2) is equivalent to the constructive linguistic model for the MISO rules system (17).

Further, there are innumerable solutions satisfying Equation (9). Therefore, it turns out the corresponding SISO rules systems exist innumerably. For example, the following SISO rules systems are considered. $\begin{matrix} Rule R_{11} : u_{1} \\ = B_{1}^{1} \to v_{1} = D_{1}^{1} = w_{1} / g_{1}^{1} (u_{1}) \\ + 0 / g_{2}^{1} (u_{1}) + 1 / g_{1}^{2} (u_{2}) + 1 / g_{2}^{2} (u_{2}) \\ Rule R_{12} : u_{1} \\ = B_{2}^{1} \to v_{1} = D_{2}^{1} = 0 / g_{1}^{1} (u_{1}) \\ + w_{1} / g_{2}^{1} (u_{1}) + 1 / g_{1}^{2} (u_{2}) + 1 / g_{2}^{2} (u_{2}) \end{matrix}$ (19-1) $\begin{matrix} Rule R_{21} : u_{2} \\ = B_{1}^{2} \to v_{2} = D_{1}^{2} = 1 / g_{1}^{1} (u_{1}) \\ + 1 / g_{2}^{1} (u_{1}) + w_{2} / g_{1}^{2} (u_{2}) + 0 / g_{2}^{2} (u_{2}) \\ Rule R_{22} : u_{2} \\ = B_{2}^{2} \to v_{2} = D_{2}^{2} = 1 / g_{1}^{1} (u_{1}) \\ + 1 / g_{2}^{1} (u_{1}) + 0 / g_{1}^{2} (u_{2}) + w_{2} / g_{2}^{2} (u_{2}) \end{matrix}$ (19-2)

For (19-1) and (19-2), Equation (9) is also satisfied as follows. $\begin{matrix} D_{1}^{1} \otimes D_{1}^{2} \\ = (w_{1} / g_{1}^{1} + 0 / g_{2}^{1} + 1 / g_{1}^{2} + 1 / g_{2}^{2}) \\ \otimes (1 / g_{1}^{1} + 1 / g_{2}^{1} + w_{2} / g_{1}^{2} + 0 / g_{2}^{2}) \\ = (w_{1} \otimes 1) / g_{1}^{1} + (0 \otimes 1) / g_{2}^{1} + (1 \otimes w_{2}) / g_{1}^{2} \\ + (1 \otimes 0) / g_{2}^{2} = w_{1} / g_{1}^{1} + w_{2} / g_{1}^{2} = D_{11} \\ D_{1}^{1} \otimes D_{2}^{2} \\ = (w_{1} / g_{1}^{1} + 0 / g_{2}^{1} + 1 / g_{1}^{2} + 1 / g_{2}^{2}) \\ \otimes (1 / g_{1}^{1} + 1 / g_{2}^{1} + 0 / g_{1}^{2} + w_{2} / g_{2}^{2}) \\ = (w_{1} \otimes 1) / g_{1}^{1} + (0 \otimes 1) / g_{2}^{1} + (1 \otimes 0) / g_{1}^{2} \\ + (1 \otimes w_{2}) / g_{2}^{2} = w_{1} / g_{1}^{1} + w_{2} / g_{2}^{2} = D_{12} \\ D_{2}^{1} \otimes D_{1}^{2} \\ = (0 / g_{1}^{1} + w_{1} / g_{2}^{1} + 1 / g_{1}^{2} + 1 / g_{2}^{2}) \\ \otimes (1 / g_{1}^{1} + 1 / g_{2}^{1} + w_{2} / g_{1}^{2} + 0 / g_{2}^{2}) \\ = (0 \otimes 1) / g_{1}^{1} + (w_{1} \otimes 1) / g_{2}^{1} + (1 \otimes w_{2}) / g_{1}^{2} \\ + (1 \otimes 0) / g_{2}^{2} = w_{1} / g_{2}^{1} + w_{2} / g_{1}^{2} = D_{21} \\ D_{2}^{1} \otimes D_{2}^{2} \\ = (0 / g_{1}^{1} + w_{1} / g_{2}^{1} + 1 / g_{1}^{2} + 1 / g_{2}^{2}) \\ \otimes (1 / g_{1}^{1} + 1 / g_{2}^{1} + 0 / g_{1}^{2} + w_{2} / g_{2}^{2}) \\ = (0 \otimes 1) / g_{1}^{1} + (w_{1} \otimes 1) / g_{2}^{1} + (1 \otimes 0) / g_{1}^{2} \\ + (1 \otimes w_{2}) / g_{2}^{2} = w_{1} / g_{2}^{1} + w_{2} / g_{2}^{2} = D_{22} \end{matrix}$

Therefore, the constructive linguistic model for the MISO rules system (17) is also equivalent to the constructive linguistic model for the collection of SISO rules systems (19-1) and (19-2).

Meanwhile, if the destructive linguistic model is chosen, the MISO rules system (17) can be decomposed into a collection of SISO rules systems as follows. $\begin{matrix} Rule R_{11} : u_{1} & = & B_{1}^{1} \to v_{1} = D_{1}^{1} = w_{1} / g_{1}^{1} (u_{1}) \\ Rule R_{12} : u_{1} & = & B_{2}^{1} \to v_{1} = D_{2}^{1} = w_{1} / g_{2}^{1} (u_{1}) \end{matrix}$ (20-1) $\begin{matrix} Rule R_{21} : u_{2} & = & B_{1}^{2} \to v_{2} = D_{1}^{2} = w_{2} / g_{1}^{2} (u_{2}) \\ Rule R_{22} : u_{2} & = & B_{2}^{2} \to v_{2} = D_{2}^{2} = w_{2} / g_{2}^{2} (u_{2}) \end{matrix}$ (20-2)

Equation (12) of Proposition 2 is satisfied as follows. $\begin{matrix} D_{1}^{1} \oplus D_{1}^{2} & = & w_{1} / g_{1}^{1} + w_{2} / g_{1}^{2} = D_{11} \\ D_{1}^{1} \oplus D_{2}^{2} & = & w_{1} / g_{1}^{1} + w_{2} / g_{2}^{2} = D_{12} \\ D_{2}^{1} \oplus D_{1}^{2} & = & w_{1} / g_{2}^{1} + w_{2} / g_{1}^{2} = D_{21} \\ D_{2}^{1} \oplus D_{2}^{2} & = & w_{1} / g_{2}^{1} + w_{2} / g_{2}^{2} = D_{22} \end{matrix}$

Choose the t-norm and the t-conorm such that the t-conorm ⊕ is distributive over the t-norm ⊗, and ⊗ is the dual t-norm of ⊕. For example, the minimum t-norm and the maximum t-conorm satisfy the above conditions. From Proposition 2, the destructive linguistic model for MISO rules system (17) is equivalent to the destructive linguistic model for the collection of SISO rules systems (20-1) and (20-2).

Now, the general form is investigated. Consider the following MISO rules system.

$\begin{matrix} Rule R_{j_{1} j_{2} \dots j_{n}} : u_{1} = B_{j_{1}}^{1}, u_{2} \\ = B_{j_{2}}^{2}, \dots, u_{n} = B_{j_{n}}^{n} \to v \\ = D_{j_{1} j_{2} \dots j_{n}} = w_{1} / g_{j_{1}}^{1} (u_{1}) + w_{2} / g_{j_{2}}^{2} (u_{2}) \\ + \dots + w_{n} g_{j_{n}}^{n} (u_{n}) \end{matrix}$ (21) where j_i = 1, 2, …, m_i, i = 1, 2, …, n; u_i, i = 1, 2, …, n are the input variables; $B_{j_{i}}^{i}$ , j_i = 1, 2, …, m_i, i = 1, 2, …, n are the fuzzy sets; v is the output variable; $g_{j_{i}}^{i}$ are functions, j_i = 1, 2, …, m_i, i = 1, 2, …, n; w_i is the weight and w_i ≦ 1 for i = 1, 2, …, n; The total number of rules is M = m₁m₂ … m_n. The notation $w_{i} / g_{j}^{i} (u_{i})$ means it is a fuzzy singleton with membership value w_i at $g_{j_{i}}^{i} (u_{i})$ for i = 1, 2, …, n.

We associate the rules system (21) with the following rules systems: $Rule R_{j_{i}} : u_{i} = B_{j_{i}}^{i} \to v = D_{j_{i}}^{i},$ (22-i) where j_i = 1, 2, …, m_i, i = 1, 2, …, n, and $D_{j_{i}}^{i}$ is as follows.

$\begin{matrix} D_{1}^{1} & = & w_{1} / g_{1}^{1} + 0 / g_{2}^{1} + \dots + 0 / g_{m_{1}}^{1} + 1 / g_{1}^{2} \\ + \dots + 1 / g_{m_{2}}^{2} + \dots + 1 / g_{1}^{n} + \dots + 1 / g_{m_{n}}^{n} \\ D_{2}^{1} & = & 0 / g_{1}^{1} + w_{1} / g_{2}^{1} + 0 / g_{3}^{1} + \dots + 0 / g_{m_{1}}^{1} \\ + 1 / g_{1}^{2} + \dots + 1 / g_{m_{2}}^{2} + \dots + 1 / g_{1}^{n} \\ + \dots + 1 / g_{m_{n}}^{n} \\ ⋮ \\ D_{m_{1}}^{1} & = & 0 / g_{1}^{1} + \dots + 0 / g_{m_{1} - 1}^{1} + w_{1} / g_{m_{1}}^{1} + 1 / g_{1}^{2} \\ + \dots + 1 / g_{m_{2}}^{2} + \dots + 1 / g_{1}^{n} + \dots + 1 / g_{m_{n}}^{n} \\ D_{1}^{2} & = & 1 / g_{1}^{1} + \dots + 1 / g_{m_{1}}^{1} + w_{2} / g_{1}^{2} + 0 / g_{2}^{2} + \dots \\ + 0 / g_{m_{2}}^{2} + 1 / g_{1}^{3} + \dots + 1 / g_{m_{3}}^{3} + \dots \\ + 1 / g_{1}^{n} + \dots + 1 / g_{m_{n}}^{n} \\ D_{2}^{2} & = & 1 / g_{1}^{1} + \dots + 1 / g_{m_{1}}^{1} + 0 / g_{1}^{2} + w_{2} / g_{2}^{2} \\ + 0 / g_{3}^{2} + \dots + 0 / g_{m_{2}}^{2} + 1 / g_{1}^{3} + \dots \\ + 1 / g_{m_{3}}^{3} + \dots + 1 / g_{1}^{n} + \dots + 1 / g_{m_{n}}^{n} \\ ⋮ \\ D_{m_{2}}^{2} & = & 1 / g_{1}^{1} + \dots + 1 / g_{m_{1}}^{1} + 0 / g_{1}^{2} + \dots \\ + 0 / g_{m_{2} - 1}^{2} + w_{2} / g_{m_{2}}^{2} + 1 / g_{1}^{3} + \dots \\ + 1 / g_{m_{3}}^{3} + \dots + 1 / g_{1}^{n} + \dots + 1 / g_{m_{n}}^{n} \\ ⋮ \\ D_{1}^{n} & = & 1 / g_{1}^{1} + \dots + 1 / g_{m_{1}}^{1} + 1 / g_{1}^{2} + \dots + 1 / g_{m_{2}}^{2} \\ + \dots + 1 / g_{1}^{n - 1} + \dots + 1 / g_{m_{n - 1}}^{n - 1} \\ + w_{n} / g_{1}^{n} + 0 / g_{2}^{n} + \dots + 0 / g_{m_{n}}^{n} \\ D_{2}^{n} & = & 1 / g_{1}^{1} + \dots + 1 / g_{m_{1}}^{1} + 1 / g_{1}^{2} + \dots + 1 / g_{m_{2}}^{2} \\ + \dots + 1 / g_{1}^{n - 1} + \dots + 1 / g_{m_{n - 1}}^{n - 1} + 0 / g_{1}^{n} \\ + w_{n} / g_{2}^{n} + 0 / g_{3}^{n} + \dots + 0 / g_{m_{n}}^{n} \\ ⋮ \\ D_{m_{n}}^{n} & = & 1 / g_{1}^{1} + \dots + 1 / g_{m_{1}}^{1} + 1 / g_{1}^{2} + \dots + 1 / g_{m_{2}}^{2} \\ + \dots + 1 / g_{1}^{n - 1} + \dots + 1 / g_{m_{n - 1}}^{n - 1} + 0 / g_{1}^{n} \\ + \dots + 0 / g_{m_{n - 1}}^{n} + w_{n} / g_{m_{n}}^{n} \end{matrix}$ (23)

Note that the total number of rules in (22-i), i = 1, 2 … , n, is m₁ + m₂ + … + m_n. It is easy to prove the following equations. $\begin{matrix} D_{1}^{1} \otimes D_{1}^{2} \otimes \dots \otimes D_{1}^{n} \\ = w_{1} / g_{1}^{1} + w_{2} / g_{1}^{2} + \dots + w_{n} / g_{1}^{n} \\ = D_{11 \dots 1} \\ D_{1}^{1} \otimes D_{2}^{2} \otimes D_{1}^{3} \otimes \dots \otimes D_{1}^{n} \\ = w_{1} / g_{1}^{1} + w_{2} / g_{2}^{2} + w_{3} / g_{1}^{3} + \dots + w_{n} / g_{1}^{n} \\ = D_{121 \dots 1} \\ ⋮ \\ D_{m_{1}}^{1} \otimes D_{m_{2}}^{2} \otimes \dots \otimes D_{m_{n}}^{n} \\ = w_{1} / g_{m_{1}}^{1} + w_{2} / g_{m_{2}}^{2} + \dots + w_{n} / g_{m_{n}}^{n} \\ = D_{m_{1} m_{2} \dots m_{n}} \end{matrix}$

Therefore, Equation (9) is satisfied. Choose the t-norm and the t-conorm such that the t-norm is distributive over the t-conorm. From Proposition 1, the constructive linguistic model for the collection of SISO rules systems (22-i) is equivalent to the constructive linguistic model for the MISO rules system (21).

Moreover, there are innumerable solutions satisfying Equation (9). For example, the following $D_{j_{i}}^{i}$ are considered.

$\begin{matrix} D_{1}^{1} & = & 1 / g_{1}^{1} + 0 / g_{2}^{1} + \dots + 0 / g_{m_{1}}^{1} + w_{2} / g_{1}^{2} \\ + \dots + w_{2} / g_{m_{2}}^{2} + \dots + w_{n} / g_{1}^{n} \\ + \dots + w_{n} / g_{m_{n}}^{n} \\ D_{2}^{1} & = & 0 / g_{1}^{1} + 1 / g_{2}^{1} + 0 / g_{3}^{1} + \dots + 0 / g_{m_{1}}^{1} \\ + w_{2} / g_{1}^{2} + \dots + w_{2} / g_{m_{2}}^{2} + \dots + w_{n} / g_{1}^{n} \\ + \dots + w_{n} / g_{m_{n}}^{n} \\ ⋮ \\ D_{m_{1}}^{1} & = & 0 / g_{1}^{1} + \dots + 0 / g_{m_{1} - 1}^{1} + 1 / g_{m_{1}}^{1} + w_{2} / g_{1}^{2} \\ + \dots + w_{2} / g_{m_{2}}^{2} + \dots + w_{n} / g_{1}^{n} \\ + \dots + w_{n} / g_{m_{n}}^{n} \\ D_{1}^{2} & = & w_{1} / g_{1}^{1} + \dots + w_{1} / g_{m_{1}}^{1} + 1 / g_{1}^{2} + 0 / g_{2}^{2} \\ + \dots + 0 / g_{m_{2}}^{2} + w_{3} / g_{1}^{3} + \dots + w_{3} / g_{m_{3}}^{3} \\ + \dots + w_{n} / g_{1}^{n} + \dots + w_{n} / g_{m_{n}}^{n} \\ D_{2}^{2} & = & w_{1} / g_{1}^{1} + \dots + w_{1} / g_{m_{1}}^{1} + 0 / g_{1}^{2} + 1 / g_{2}^{2} \\ + 0 / g_{3}^{2} + \dots + 0 / g_{m_{2}}^{2} + w_{3} / g_{1}^{3} + \dots \\ + w_{3} / g_{m_{3}}^{3} + \dots + w_{n} / g_{1}^{n} + \dots + w_{n} / g_{m_{n}}^{n} \\ ⋮ \\ D_{m_{2}}^{2} & = & w_{1} / g_{1}^{1} + \dots + w_{1} / g_{m_{1}}^{1} + 0 / g_{1}^{2} + \dots \\ + 0 / g_{m_{2} - 1}^{2} + 1 / g_{m_{2}}^{2} + w_{3} / g_{1}^{3} + \dots \\ + w_{3} / g_{m_{3}}^{3} + \dots + w_{n} / g_{1}^{n} + \dots + w_{n} / g_{m_{n}}^{n} \\ ⋮ \\ D_{1}^{n} & = & w_{1} / g_{1}^{1} + \dots + w_{1} / g_{m_{1}}^{1} + w_{2} / g_{1}^{2} + \dots \\ + w_{2} / g_{m_{2}}^{2} + \dots + w_{n - 1} / g_{1}^{n - 1} + \dots \\ + w_{n - 1} / g_{m_{n - 1}}^{n - 1} + 1 / g_{1}^{n} + 0 / g_{2}^{n} \dots + 0 / g_{m_{n}}^{n} \\ D_{2}^{n} & = & w_{1} / g_{1}^{1} + \dots + w_{1} / g_{m_{1}}^{1} + w_{2} / g_{1}^{2} + \dots \\ + w_{2} / g_{m_{2}}^{2} + \dots + w_{n - 1} / g_{1}^{n - 1} + \dots \\ + w_{n - 1} / g_{m_{n - 1}}^{n - 1} + 0 / g_{1}^{n} + 1 / g_{2}^{n} + 0 / g_{3}^{n} \\ + \dots + 0 / g_{m_{n}}^{n} \\ ⋮ \\ D_{m_{n}}^{n} & = & w_{1} / g_{1}^{1} + \dots + w_{1} / g_{m_{1}}^{1} + w_{2} / g_{1}^{2} + \dots \\ + w_{2} / g_{m_{2}}^{2} + \dots + w_{n - 1} / g_{1}^{n - 1} + \dots \\ + w_{n - 1} / g_{m_{n - 1}}^{n - 1} + 0 / g_{1}^{n} + \dots + 0 / g_{m_{n - 1}}^{n} \\ + 1 / g_{m_{n}}^{n} \end{matrix}$ (24)

It is easy to prove $D_{j_{1} j_{2} \dots j_{n}} (z) = D_{j_{1}}^{1} (z) \otimes D_{j_{2}}^{2} (z) \otimes \dots \otimes D_{j_{n}}^{n} (z)$ for j_i = 1, 2, …, m_i, i = 1, 2, …, n. Therefore, the equivalent SISO rules systems exist innumerably.

From above, it is clear the constructive linguistic model for the SISO rules systems is a special case of the constructive linguistic model for the MISO rules system. However, it follows the number of rules is m₁ + m₂ + … + m_n to realize the result of the constructive linguistic model for the SISO rules systems (22-i), and the number of rules is m₁m₂ … m_n to realize the result of the constructive linguistic model for the MISO rules system (21). Thus, too many rules are required in the MISO rules system.

Then, the general form for the destructive linguistic model is investigated. Consider the MISO rules system (21) and the SISO rules systems (22-i) for i = 1, 2, …, n. $D_{j_{i}}^{i}$ is as follows.

$\begin{matrix} D_{1}^{1} = w_{1} / g_{1}^{1}, D_{2}^{1} = w_{1} / g_{2}^{1}, \dots, D_{m_{1}}^{1} = w_{1} / g_{m_{1}}^{1}, \\ D_{1}^{2} = w_{2} / g_{1}^{2}, D_{2}^{2} = w_{2} / g_{2}^{2}, \dots, D_{m_{1}}^{2} = w_{2} / g_{m_{2}}^{2}, \\ ⋮ \\ D_{1}^{n} = w_{n} / g_{1}^{n}, D_{2}^{n} = w_{n} / g_{2}^{n}, \dots, D_{m_{n}}^{n} = w_{n} / g_{m_{n}}^{n} \end{matrix}$ (25)

It is easy to prove the following equations. $\begin{matrix} D_{1}^{1} \oplus D_{1}^{2} \oplus \dots \oplus D_{1}^{n} \\ = w_{1} / g_{1}^{1} + w_{2} / g_{1}^{2} + \dots + w_{n} / g_{1}^{n} \\ = D_{11 \dots 1} \\ D_{1}^{1} \oplus D_{2}^{2} \oplus D_{1}^{3} \oplus \dots \oplus D_{1}^{n} \\ = w_{1} / g_{1}^{1} + w_{2} / g_{2}^{2} + w_{3} / g_{1}^{3} + \dots + w_{n} / g_{1}^{n} \\ = D_{121 \dots 1} \\ ⋮ \\ D_{m_{1}}^{1} \oplus D_{m_{2}}^{2} \oplus \dots \oplus D_{m_{n}}^{n} \\ = w_{1} / g_{m_{1}}^{1} + w_{2} / g_{m_{2}}^{2} + \dots + w_{n} / g_{m_{n}}^{n} \\ = D_{m_{1} m_{2} \dots m_{n}} \end{matrix}$

Therefore, Equation (12) is satisfied. Choose the t-norm and the t-conorm such that the t-conorm ⊕ is distributive over the t-norm ⊗, and ⊗ is the dual t-norm of ⊕. From Proposition 2, the destructive linguistic model for the collection of SISO rules systems (22-i) is equivalent to the destructive linguistic model for the MISO rules system (21). For clarity, the following example modified from Seki et al. [16] is investigated.

Example 5. Considering two inputs u₁ and u₂, the MISO rules system is given as

$\begin{matrix} Rule R_{j_{1} j_{2}} : u_{1} = B_{j_{1}}^{1}, u_{2} = B_{j_{2}}^{2} \to z = D_{j_{1} j_{2}} \\ = w_{1} / (c_{j_{1}}^{1} + d_{j_{1}}^{1} u_{1}) + w_{2} / (c_{j_{2}}^{2} + d_{j_{2}}^{2} u_{2}), \end{matrix}$ (26) where j₁, j₂ = 1, 2, 3, $c_{1}^{1} = - 0.1$ , $c_{2}^{1} = - 0.2$ , $c_{3}^{1} = - 0.5$ , $c_{1}^{2} = 1$ , $c_{2}^{2} = - 0.1$ , $c_{3}^{2} = 0.2$ , $d_{1}^{1} = - 0.4$ , $d_{2}^{1} = - 0.6$ , $d_{3}^{1} = 0.2$ , $d_{1}^{2} = 0.7$ , $d_{2}^{2} = - 0.2$ , $d_{3}^{2} = 0.5$ , w₁ = 0.5, w₂ = 0.5, and the fuzzy sets in Fig. 4 are used. D_{j
₁
j
₂}, j₁, j₂ = 1, 2, 3 are fuzzy singletons with weight w₁ at $c_{j_{1}}^{1} + d_{j_{1}}^{1} u_{1}$ and weight w₂ at $c_{j_{2}}^{2} + d_{j_{2}}^{2} u_{2}$ . First, we solve the problem by the MISO rules system (26). Choose the constructive linguistic model with the selection of t-norm ⊗= product operation, and t-conorm ⊕= the Lukasiewitz t-conorm, as shown in Example 1. The defuzzification is by weighted aggregation with its membership values as weights. The inference result of the MISO rules system (26) is obtained as shown in Fig. 5.

Fig.4

Fuzzy sets for $B_{1}^{i}$ , $B_{2}^{i}$ , and $B_{3}^{i}$ , i = 1, 2.

Fig.5

Inference result of the MISO rules system.

Then, we solve the problem by decomposing the MISO rules system into a collection of SISO rules systems. We associate the MISO rules system (26) with the following SISO rules systems $\begin{matrix} Rule R_{11} : u_{1} = B_{1}^{1} \to v_{1} = D_{1}^{1} \\ Rule R_{12} : u_{1} = B_{2}^{1} \to v_{1} = D_{2}^{1} \\ Rule R_{13} : u_{1} = B_{3}^{1} \to v_{1} = D_{3}^{1} \end{matrix}$ (27-1) $\begin{matrix} Rule R_{21} : u_{2} = B_{1}^{2} \to v_{2} = D_{1}^{2} \\ Rule R_{22} : u_{2} = B_{2}^{2} \to v_{2} = D_{2}^{2} \\ Rule R_{23} : u_{2} = B_{3}^{2} \to v_{2} = D_{3}^{2} . \end{matrix}$ (27-2)

From (23), $D_{1}^{1}$ , $D_{2}^{1}$ , $D_{3}^{1}$ , $D_{1}^{2}$ , $D_{2}^{2}$ and $D_{3}^{2}$ are chosen as follows. $\begin{matrix} D_{1}^{1} & = & w_{1} / (c_{1}^{1} + d_{1}^{1} u_{1}) + 0 / (c_{2}^{1} + d_{2}^{1} u_{1}) \\ + 0 / (c_{3}^{1} + d_{3}^{1} u_{1}) + 1 / (c_{1}^{2} + d_{1}^{2} u_{2}) \\ + 1 / (c_{2}^{2} + d_{2}^{2} u_{2}) + 1 / (c_{3}^{2} + d_{3}^{2} u_{2}) \\ D_{2}^{1} & = & 0 / (c_{1}^{1} + d_{1}^{1} u_{1}) + w_{1} / (c_{2}^{1} + d_{2}^{1} u_{1}) \\ + 0 / (c_{3}^{1} + d_{3}^{1} u_{1}) + 1 / (c_{1}^{2} + d_{1}^{2} u_{2}) \\ + 1 / (c_{2}^{2} + d_{2}^{2} u_{2}) + 1 / (c_{3}^{2} + d_{3}^{2} u_{2}) \\ D_{3}^{1} & = & 0 / (c_{1}^{1} + d_{1}^{1} u_{1}) + 0 / (c_{2}^{1} + d_{2}^{1} u_{1}) \\ + w_{1} / (c_{3}^{1} + d_{3}^{1} u_{1}) + 1 / (c_{1}^{2} + d_{1}^{2} u_{2}) \\ + 1 / (c_{2}^{2} + d_{2}^{2} u_{2}) + 1 / (c_{3}^{2} + d_{3}^{2} u_{2}) \\ D_{1}^{2} & = & 1 / (c_{1}^{1} + d_{1}^{1} u_{1}) + 1 / (c_{2}^{1} + d_{2}^{1} u_{1}) \\ + 1 / (c_{3}^{1} + d_{3}^{1} u_{1}) + w_{2} / (c_{1}^{2} + d_{1}^{2} u_{2}) \\ + 0 / (c_{2}^{2} + d_{2}^{2} u_{2}) + 0 / (c_{3}^{2} + d_{3}^{2} u_{2}) \\ D_{2}^{2} & = & 1 / (c_{1}^{1} + d_{1}^{1} u_{1}) + 1 / (c_{2}^{1} + d_{2}^{1} u_{1}) \\ + 1 / (c_{3}^{1} + d_{3}^{1} u_{1}) + 0 / (c_{1}^{2} + d_{1}^{2} u_{2}) \\ + w_{2} / (c_{2}^{2} + d_{2}^{2} u_{2}) + 0 / (c_{3}^{2} + d_{3}^{2} u_{2}) \\ D_{3}^{2} & = & 1 / (c_{1}^{1} + d_{1}^{1} u_{1}) + 1 / (c_{2}^{1} + d_{2}^{1} u_{1}) \\ + 1 / (c_{3}^{1} + d_{3}^{1} u_{1}) + 0 / (c_{1}^{2} + d_{1}^{2} u_{2}) \\ + 0 / (c_{2}^{2} + d_{2}^{2} u_{2}) + w_{2} / (c_{3}^{2} + d_{3}^{2} u_{2}), \end{matrix}$ where w₁ = w₂ = 0.5, and $c_{j}^{i}$ , i = 1, 2, j = 1, 2, 3, are the same as shown in (26). Equation (9) of Proposition 1 is satisfied as follows. $\begin{matrix} D_{1}^{1} \otimes D_{1}^{2} & = & w_{1} / (c_{1}^{1} + d_{1}^{1} u_{1}) + w_{2} / (c_{1}^{2} + d_{1}^{2} u_{2}) \\ = & D_{11} \\ D_{1}^{1} \otimes D_{2}^{2} & = & w_{1} / (c_{1}^{1} + d_{1}^{1} u_{1}) + w_{2} / (c_{2}^{2} + d_{2}^{2} u_{2}) \\ = & D_{12} \\ D_{1}^{1} \otimes D_{3}^{2} & = & w_{1} / (c_{1}^{1} + d_{1}^{1} u_{1}) + w_{2} / (c_{3}^{2} + d_{3}^{2} u_{2}) \\ = & D_{13} \\ D_{2}^{1} \otimes D_{1}^{2} & = & w_{1} / (c_{2}^{1} + d_{2}^{1} u_{1}) + w_{2} / (c_{1}^{2} + d_{1}^{2} u_{2}) \\ = & D_{21} \\ D_{2}^{1} \otimes D_{2}^{2} & = & w_{1} / (c_{2}^{1} + d_{2}^{1} u_{1}) + w_{2} / (c_{2}^{2} + d_{2}^{2} u_{2}) \\ = & D_{22} \\ D_{2}^{1} \otimes D_{3}^{2} & = & w_{1} / (c_{2}^{1} + d_{2}^{1} u_{1}) + w_{2} / (c_{3}^{2} + d_{3}^{2} u_{2}) \\ = & D_{23} \end{matrix}$

The inference result of the constructive linguistic model for the SISO rules system (27-1) is obtained as follows. $\begin{matrix} v_{1} & = & (B_{1}^{1} (u_{1}) \otimes D_{1}^{1}) \oplus (B_{2}^{1} (u_{1}) \otimes D_{2}^{1}) \\ \oplus (B_{3}^{1} (u_{1}) \otimes D_{3}^{1}) \\ = & w_{1} B_{1}^{1} (u_{1}) / (c_{1}^{1} + d_{1}^{1} u_{1}) + 0 / (c_{2}^{1} + d_{2}^{1} u_{1}) \\ + 0 / (c_{3}^{1} + d_{3}^{1} u_{1}) + B_{1}^{1} (u_{1}) / (c_{1}^{2} + d_{1}^{2} u_{2}) \\ + B_{1}^{1} (u_{1}) / (c_{2}^{2} + d_{2}^{2} u_{2}) + B_{1}^{1} (u_{1}) / (c_{3}^{2} + d_{3}^{2} u_{2}) \\ + 0 / (c_{1}^{1} + d_{1}^{1} u_{1}) + w_{1} B_{2}^{1} (u_{1}) / (c_{1}^{2} + d_{1}^{2} u_{1}) \\ + 0 / (c_{3}^{1} + d_{3}^{1} u_{1}) + B_{2}^{1} (u_{1}) / (c_{1}^{2} + d_{1}^{2} u_{2}) \\ + B_{2}^{1} (u_{1}) / (c_{2}^{2} + d_{2}^{2} u_{2}) + B_{2}^{1} (u_{1}) / (c_{3}^{2} + d_{3}^{2} u_{2}) \\ + 0 / (c_{1}^{1} + d_{1}^{1} u_{1}) + 0 / (c_{2}^{1} + d_{2}^{1} u_{1}) \\ + w_{1} B_{3}^{1} (u_{1}) / (c_{3}^{1} + d_{3}^{1} u_{1}) + B_{3}^{1} (u_{1}) / (c_{1}^{2} + d_{1}^{2} u_{2}) \\ + B_{3}^{1} (u_{1}) / (c_{2}^{2} + d_{2}^{2} u_{2}) + B_{3}^{1} (u_{1}) / (c_{3}^{2} + d_{3}^{2} u_{2}) \end{matrix}$

The inference result of the constructive linguistic model for the rules system (27-2) is as follows. $\begin{matrix} v_{2} & = & (B_{1}^{2} (u_{2}) \otimes D_{1}^{2}) \oplus (B_{2}^{2} (u_{2}) \otimes D_{2}^{2}) \\ \oplus (B_{3}^{2} (u_{2}) \otimes D_{3}^{2}) \\ = & B_{1}^{2} (u_{2}) / (c_{1}^{1} + d_{1}^{1} u_{1}) + B_{1}^{2} (u_{2}) / (c_{2}^{1} + d_{2}^{1} u_{1}) \\ + B_{1}^{2} (u_{2}) / (c_{3}^{1} + d_{3}^{1} u_{1}) + w_{2} B_{1}^{2} (u_{2}) / \\ (c_{1}^{2} + d_{1}^{2} u_{2}) + 0 / (c_{2}^{2} + d_{2}^{2} u_{2}) + 0 / (c_{3}^{2} + d_{3}^{2} u_{2}) \\ + B_{2}^{2} (u_{2}) / (c_{1}^{1} + d_{1}^{1} u_{1}) + B_{2}^{2} (u_{2}) / (c_{2}^{1} + d_{2}^{1} u_{1}) \\ + B_{2}^{2} (u_{2}) / (c_{3}^{1} + d_{3}^{1} u_{1}) + 0 / (c_{1}^{2} + d_{1}^{2} u_{2}) \\ + w_{2} B_{2}^{2} (u_{2}) / (c_{2}^{2} + d_{2}^{2} u_{2}) + 0 / (c_{3}^{2} + d_{3}^{2} u_{2}) \\ + B_{3}^{2} (u_{2}) / (c_{1}^{1} + d_{1}^{1} u_{1}) + B_{3}^{2} (u_{2}) / (c_{2}^{1} + d_{2}^{1} u_{1}) \\ + B_{3}^{2} (u_{2}) / (c_{3}^{1} + d_{3}^{1} u_{1}) + 0 / (c_{1}^{2} + d_{1}^{2} u_{2}) \\ + 0 / (c_{2}^{2} + d_{2}^{2} u_{2}) + w_{2} B_{3}^{2} (u_{2}) / (c_{3}^{2} + d_{3}^{2} u_{2}) \end{matrix}$

Aggregate v₁ and v₂ by t-norm ⊗ to obtain final result v as $\begin{matrix} v & = & v_{1} \otimes v_{2} \\ = & [(B_{1}^{1} (u_{1}) \otimes D_{1}^{1}) \oplus (B_{2}^{1} (u_{1}) \otimes D_{2}^{1}) \\ \oplus (B_{3}^{1} (u_{1}) \otimes D_{3}^{1})] \otimes [(B_{1}^{2} (u_{2}) \otimes D_{1}^{2}) \\ \oplus (B_{2}^{2} (u_{2}) \otimes D_{2}^{2}) \oplus (B_{3}^{2} (u_{2}) \otimes D_{3}^{2})] \\ = & (B_{1}^{1} (u_{1}) \otimes B_{1}^{2} (u_{2}) \otimes D_{1}^{1} \otimes D_{1}^{2}) \\ \oplus (B_{1}^{1} (u_{1}) \otimes B_{2}^{2} (u_{2}) \otimes D_{1}^{1} \otimes D_{2}^{2}) \\ \oplus (B_{1}^{1} (u_{1}) \otimes B_{3}^{2} (u_{2}) \otimes D_{1}^{1} \otimes D_{3}^{2}) \\ \oplus (B_{2}^{1} (u_{1}) \otimes B_{1}^{2} (u_{2}) \otimes D_{2}^{1} \otimes D_{1}^{2}) \\ \oplus (B_{2}^{1} (u_{1}) \otimes B_{2}^{2} (u_{2}) \otimes D_{2}^{1} \otimes D_{2}^{2}) \\ \oplus (B_{2}^{1} (u_{1}) \otimes B_{3}^{2} (u_{2}) \otimes D_{2}^{1} \otimes D_{3}^{2}) \\ \oplus (B_{3}^{1} (u_{1}) \otimes B_{1}^{2} (u_{2}) \otimes D_{3}^{1} \otimes D_{1}^{2}) \\ \oplus (B_{3}^{1} (u_{1}) \otimes B_{2}^{2} (u_{2}) \otimes D_{3}^{1} \otimes D_{2}^{2}) \\ \oplus (B_{3}^{1} (u_{1}) \otimes B_{3}^{2} (u_{2}) \otimes D_{3}^{1} \otimes D_{3}^{2}) \\ = & w_{1} (B_{1}^{1} (u_{1}) B_{1}^{2} (u_{2}) + B_{1}^{1} (u_{1}) B_{2}^{2} (u_{2}) \\ + B_{1}^{1} (u_{1}) B_{3}^{2} (u_{2})) / (c_{1}^{1} + d_{1}^{1} u_{1}) \\ + w_{1} (B_{2}^{1} (u_{1}) B_{1}^{2} (u_{2}) + B_{2}^{1} (u_{1}) B_{2}^{2} (u_{2}) \\ + B_{2}^{1} (u_{1}) B_{3}^{2} (u_{2})) / (c_{2}^{1} + d_{2}^{1} u_{1}) \\ + w_{1} (B_{3}^{1} (u_{1}) B_{1}^{2} (u_{2}) + B_{3}^{1} (u_{1}) B_{2}^{2} (u_{2}) \\ + B_{3}^{1} (u_{1}) B_{3}^{2} (u_{2})) / (c_{3}^{1} + d_{3}^{1} u_{1}) \\ + w_{2} (B_{1}^{1} (u_{1}) B_{1}^{2} (u_{2}) + B_{2}^{1} (u_{1}) B_{1}^{2} (u_{2}) \\ + B_{3}^{1} (u_{1}) B_{1}^{2} (u_{2})) / (c_{1}^{2} + d_{1}^{2} u_{2}) \\ + w_{2} (B_{1}^{1} (u_{1}) B_{2}^{2} (u_{2}) + B_{2}^{1} (u_{1}) B_{2}^{2} (u_{2}) \\ + B_{3}^{1} (u_{1}) B_{2}^{2} (u_{2})) / (c_{2}^{2} + d_{2}^{2} u_{2}) \\ + w_{2} (B_{1}^{1} (u_{1}) B_{3}^{2} (u_{2}) + B_{2}^{1} (u_{1}) B_{3}^{2} (u_{2}) \\ + B_{3}^{1} (u_{1}) B_{3}^{2} (u_{2})) / (c_{3}^{2} + d_{3}^{2} u_{2}) \end{matrix}$

The defuzzification result v^* of the fuzzy set v is by weighted aggregation with its membership value as the weight. The defuzzification result v^* is obtained as shown in Fig. 6. Note that Figs. 5 and 6 show the same results as expected. In this example, the inference results are obtained from the MISO rules system and the SISO rules systems and they are the same as expected. Note that the rules number of (27-1) and (27-2) is 6 and the rules number of (26) is 9. Using the decomposing process, we can realize the result with fewer rules.

Fig.6

Inference result by the constructive linguistic model working on the SISO rules systems.

5 Conclusion

This paper proposes a method to decompose a MISO fuzzy rules system into its equivalent collection of SISO fuzzy rules systems. From Proposition 1, suppose the consequent parts of rules satisfy (9) and the t-norm ⊗ is distributive over the t-conorm ⊕. Then, the final consequence of the constructive linguistic model working on a MISO fuzzy rules system is equivalent to the final consequence of the constructive linguistic model working on a collection of SISO fuzzy rules systems. From Proposition 2, suppose the consequent parts of rules satisfy (12), and the t-conorm ⊕ is distributive over the t-norm ⊗, and ⊗ is the dual t-norm of ⊕. Then, the final consequence of the destructive linguistic model working on a MISO fuzzy rules system is equivalent to the final consequence of the destructive linguistic model working on a collection of SISO fuzzy rules systems.

In the decomposing process, the main work is to find the consequent parts of the equivalent SISO rules systems as in (9) and (12). Occasionally, it is difficult to find the consequent parts of the equivalent SISO rules systems. This paper shows that it is easy to find the consequent parts of the equivalent SISO rules systems for a special type of MISO fuzzy rules system, that is, the MISO fuzzy rules system with fuzzy singleton consequent parts. The general procedure for the decomposition of the MISO fuzzy rules system with fuzzy singleton consequent parts is investigated. For the constructive linguistic model, the consequent parts of the equivalent SISO rules systems are found as in (23) and (24). For the destructive linguistic model, the consequent parts of the equivalent SISO rules systems are found as in (25). Choose the t-norm and t-conorm satisfying the conditions in Proposition 1 or 2. The MISO fuzzy rules system (21) can be decomposed into its equivalent collection of SISO fuzzy rules systems (22-i), i = 1, 2, …, n. However, to realize the result of the MISO rules system (21) requires m₁m₂ … m_n rules, and to realize the result of equivalent SISO rules systems (22-i), i = 1, 2, …, n, requires m₁ + m₂ + … + m_n rules. Through the decomposing process, we can realize the inference result with fewer rules.

Summarizing the above results, the main advantages of this paper are that two propositions are proposed for decomposing a MISO fuzzy rules system into its equivalent collection of SISO fuzzy rules systems. Through the decomposing process, the number of fuzzy rules can be sharply reduced. For example, a MISO fuzzy rules system with ten inputs, and 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 rules can be decomposed to a collection of SISO rules system with 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 rules. Moreover, the general procedure for decomposing the MISO fuzzy rules system with fuzzy singleton consequent parts is obtained. The main disadvantage of the paper is that the consequent parts of the equivalent SISO rules systems are not easy to find. The future research is to study this problem.

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