Triple I fuzzy modus tollens method with inconsistent bipolarity information

Abstract

Triple I method was proven to be the effective reasoning method with solid logic foundation, however, the existing works about triple I method all neglect inconsistent bipolarity information in reasoning process, which exists in the way that our brain handles information. Considering that bipolar-valued fuzzy set theory is a powerful tool to describe inconsistent fuzzy bipolarity information, this paper introduces inconsistent bipolarity to triple I method for the first time, and systematically studies the triple I fuzzy modus tollens method with inconsistent bipolarity information based on the regular implication. First, some new concepts about inconsistent fuzzy bipolarity information are introduced. And by introducing the concept of YinYang bipolar-valued induced function, we can induce bipolar-valued operations under the YinYang order relation from the corresponding unipolar ones. Then the triple I bipolar-valued fuzzy modus tollens methods under the YinYang order relation are proposed, together with the solutions. It is proved that the solutions can degenerate into two corresponding unipolar triple I fuzzy modus tollens solutions. Lastly, reductive property, approximation property and robustness of the proposed methods are discussed.

Keywords

Fuzzy set fuzzy reasoning inconsistent bipolarity YinYang order relation

1 Introduction

The famous compositional rule of inference method presented by Zadeh [30] opened the new era of the fuzzy reasoning. However, composition in the Zadeh’s method lacks clear logical sense, which leads to the introduction of triple I method to provide a logic foundation for fuzzy reasoning [22, 24]. Some systematical researches were devoted to triple I methods [13 , 36]. The past century witnessed dramatical improvement in the artificial intelligent reasoning, while there still exists a gap between machine reasoning and human reasoning, since artificial intelligent reasoning methods usually neglect inconsistent bipolarity information. To our best knowledge, the existing works about triple I reasoning all fail to consider inconsistent bipolarity information in reasoning process.

Generally speaking, understanding concepts by means of bipolarity implies that we can capture the tension between both two poles. And such simultaneous bipolar (opposite) views are unavoidable to start understanding the world [14]. Recent fruits in psychology and neurology have proven that the human brain manages positive information in different way than negative information, along with the fact that some kind of inconsistent bipolarity in the way that human brains make reasoning [5]. In some real cases, inconsistent bipolarity is the norm rather than exception. A typical example is the psychology disease-bipolar disorder. Only the patient suffering the opposite episodes of bipolar disorder, mania and depression both to certain extent, i.e., overlapping of two poles appears, can be finally diagnosed as suffering bipolar disorder. See another motion control example about the smart mobile robot moving to its target position in an obstacle space. There are potential conflicts between the positive information concerns possible locations and the negative information concerns obstacle places [3]. Bloch pointed out that the case differs from uncertainty in the standard meaning, since it is not just an uncertainty at each position about the possible location of the robot at this position, and there exists inconsistency.

In 1957, Osgood et al. stressed the importance of bipolar reasoning in human activity and the semantic differential scale was presented to evaluate an object is positive, neural or negative [17]. However, linear logic behind the semantic differential scale prevents it from representing inconsistent bipolarity and it can not model information endowed with fuzziness. Bipolarity and fuzziness both are the inherent parts of human thinking [31], resulting that a proper model is needed to accommodate inconsistent fuzzy bipolarity in reasoning. It has been proven that fuzzy set [29] and its extensions, such as interval-valued fuzzy set [30], Atanassov’s intuitionistic fuzzy set [1 , 26] all fail to handle inconsistent fuzzy bipolarity information [2 , 14]. In view of that, Zhang et al. introduced bipolar-valued fuzzy set to stress the inconsistent bipolarity in fuzzy set theory [31] as follows:

Denote I^P = [0, 1] and I^N = [-1, 0]. Unless otherwise stated, n denotes the natural number. X = {x₁, ⋯ , x_n} and Y = {y₁, ⋯ , y_n} always represent finite discourses. Bipolar-valued fuzzy set in X is defined by the mapping $\tilde{A} : X \to I^{P} \times I^{N}$ , $x \mapsto ({\tilde{A}}^{P} (x), {\tilde{A}}^{N} (x)), for any x \in X,$ where the functions ${\tilde{A}}^{P} : X \to I^{P}, x \mapsto {\tilde{A}}^{P} (x) \in I^{P}$ and ${\tilde{A}}^{N} : X \to I^{N}, x \mapsto {\tilde{A}}^{N} (x) \in I^{N}$ define the satisfaction degree of the element x ∈ X to the property corresponding and the implicit counter-property to the bipolar-valued fuzzy set $\tilde{A}$ in X, respectively.

The necessity to introduce inconsistent bipolarity to fuzzy set and the relationships between bipolar-valued fuzzy set with other fuzzy set models are the hot topics in the recent research about fuzzy set theory [2 , 14]. In the definition of bipolar-valued fuzzy set, if ${\tilde{A}}^{P} (x) \neq 0$ and ${\tilde{A}}^{N} (x) = 0$ , x is regarded as having only positive satisfaction for $\tilde{A}$ ; if ${\tilde{A}}^{P} (x) = 0$ and ${\tilde{A}}^{N} (x) \neq 0$ , x does not satisfy the property of $\tilde{A}$ , but somewhat satisfies the counter property of $\tilde{A}$ . In the above cases, a bipolar-valued fuzzy set obviously degenerates into a fuzzy set [9]. Furthermore, considering the linear transformation φ : I^P → I^N given by φ (t) = t - 1, we see that Atanassov’s intuitionistic pair (μ, ν) ∈ I^P × I^P becomes a new pair (p, q) ∈ I^P × I^N with p + q = μ + ν - 1 ≤0 [2]. That is to say that, Atanassov’s intuitionistic fuzzy set is also the particular case of bipolar-valued fuzzy set [2 , 14] pointed that bipolar-valued fuzzy set is equivalent to the neutrosophic set proposed by Smarandache [21, 35]. The same paper also proposed that the selected denominations within each model might suggest different underlying structures: while the bipolar-valued fuzzy set suggests inconsistence between categories, neutrosophic set suggests a general neutrality that should perhaps jointly cover some of the specific types of neutrality considered in paired approach. Furthermore, a mathematical equivalence is one thing, and what a particular mathematical concept or property may model (its semantics) is another thing, and clearly the latter is what matters more for applications [8]. Although the above mentioned fuzzy models hold similar syntax structures. From the aspect of bipolarity semantics, each model has its own characteristic and holds research value in practice. We see that inconsistent bipolarity has attracted researchers’ attentions, and some instructive results have been obtained [15 , 33].

From the above analysis, we draw the conclusions that:

inconsistent bipolarity is the essence of human brains reasoning and is inevitable in the practical reasoning problems;

although triple I methods have been extended to interval-valued fuzzy set [13] and Atanassov’s intuitionistic fuzzy set [34], et al., inconsistent bipolarity in reasoning process are all neglected;

bipolar-valued fuzzy set is a powerful tool to represent inconsistent fuzzy bipolarity.

This motives us to combine bipolar-valued fuzzy set with triple I method, and triple I YinYang bipolar-valued fuzzy modus tollens (YBFMT) reasoning methods are presented in this paper. The main originality of this paper lies in providing a new inconsistent bipolarity perspective to triple I method, which will bridge the gap between machine reasoning and human reasoning. Furthermore, the existing triple I methods related to fuzzy set, interval-valued fuzzy set and Atanassov’s intuitionistic fuzzy set are proven that are the special cases of our methods.

The rest of the paper is structured as follows. In Section 2, firstly, some new concepts for inconsistent fuzzy bipolarity information under the YinYang order relation are introduced. Then, based on introducing the concept of YinYang bipolar-valued induced function, a bridge connecting bipolar-valued operations under the YinYang order relation with the corresponding unipolar ones is constructed. Lastly, triple I methods about the YBFMT problems are obtained and it is proved that solutions of the presented methods based on the regular implication can degenerate into two corresponding triple I fuzzy modus tollens (FMT) solutions. In Section 3, firstly, with the help of the new proposed concept of YinYang bipolar-valued fuzzy biresiduum, it is proved that the proposed methods hold robustness for all the regular implications. Secondly, by introducing the new concept of YinYang bipolar-valued fuzzy metric, it is proved that the proposed methods holds the approximation property only for a special class of regular implications. Thirdly, the proposed methods are proved to possess reductive property only for a special class of regular implications. The paper is concluded in Section 4.

2 Triple I fuzzy modus tollens method with inconsistent bipolarity information

In this section, some new operations for inconsistent fuzzy bipolarity information under the YinYang order relation will be introduced. Based on which, FMT problems will be extended to YBFMT problems and the related triple I methods based on the regular implication will be discussed.

2.1 New concepts for inconsistent fuzzy bipolarity information under the YinYang order relation

First, some related notions are reviewed before introducing the new concepts.

Denote L = {a = (a^P, a^N) ∣ a^P ∈ I^P, a^N ∈ I^N}. And for any a, b ∈ L, the YinYang order relation ≤^L in L was defined in [32] as a ≤ ^Lb if and only if a^P ≤ b^P, a^N ≥ b^N. It is obvious that 1^L = (1, - 1) , 0^L = (0, 0) are the maximum element and minimum element in L about the order relation ≤^L, respectively. For any a, b, c ∈ L, some operations under the YinYang order relation were proposed in [32]: a′ = (1 - a^P, - 1 - a^N) , a ∧ ^Lb = (a^P ∧ b^P, a^N ∨ b^N) , a ∨ ^Lb = (a^P ∨ b^P, a^N ∧ b^N), where ∨ means “max” and ∧ means “min”.

Definition 1. For any a ∈ L, if the mapping ⊗^L : L ⊗ ^LL → L satisfies the commutativity, associativity, monotonicity, and boundary condition that, a ⊗ ^L1^L = a, then ⊗^L is called the YinYang bipolar-valued triangular norm in L.

Definition 2. Let ⊗^L be the YinYang bipolar-valued triangular norm in L, for any a, b, c ∈ L, if ⊗^L satisfies $(\lor_{i}^{L} a_{i}) \otimes^{L} b = \lor_{i}^{L} (a_{i} \otimes^{L} b)$ , then ⊗^L is called a left-continuous YinYang bipolar-valued triangular norm in L. And there exists a binary operation →^L ∈ L such that (⊗ ^L, → ^L) satisfies the YinYang bipolar-valued residual principle: a ⊗ ^Lb ≤ ^Lc iff a ≤ ^Lb → ^Lc, where →^L is given by a → ^Lb = ∨ ^L {c ∈ L ∣ a ⊗ ^Lc ≤ ^Lb}, then (⊗ ^L, → ^L) is called YinYang bipolar-valued residual pair and →^L is called the YinYang bipolar-valued regular implication. Specifically, when →^L satisfies the condition: a → ^Lb = b′ → ^La′, then →^L is called YinYang bipolar-valued contrapositive symmetric regular implication.

Definition 3. Let f^P : I^P × I^P ⋯ × I^P → I^P be the n operation in I^P, a₁, ⋯ , a_n ∈ L, then YinYang bipolar-valued induced function $f_{P}^{L} : L \times L \dots \times L \to L$ in L is defined as follows: $\begin{matrix} f_{P}^{L} (a_{1}, \dots, a_{n}) \\ = (f^{P} (a_{1}^{P}, \dots, a_{n}^{P}), - f^{P} (- a_{1}^{N}, \dots, - a_{n}^{N})) \end{matrix}$

Remark 1. On one hand, for any $a_{i} = (a_{i}^{P}, a_{i}^{N}) \in L (i = 1, \dots, n)$ , we have $- a_{i}^{N} \in I^{P} (i = 1, \dots, n)$ , then $f^{P} (- a_{1}^{N}, \dots, - a_{n}^{N}) \in I^{P} (i = 1, \dots, n)$ , it follows that $- f^{P} (- a_{1}^{N}, \dots, - a_{n}^{N}) \in I^{N} (i = 1, \dots, n)$ , that is to say $f_{P}^{L} (a_{1}, \dots, a_{n}) \in L$ .

On the other hand, for any $a_{i} = (a_{i}^{P}, a_{i}^{N}), b_{i} = (b_{i}^{P}, b_{i}^{N}) \in L (i = 1, \dots, n)$ , if the n operation in I^P satisfies the condition that when $a_{i}^{P} \leq b_{i}^{P} (i = 1, \dots, n)$ , $f^{P} (a_{1}^{P}, \dots, a_{n}^{P}) \leq f^{P} (b_{1}^{P}, \dots, b_{n}^{P})$ , then for any a_i ≤ ^Lb_i (i = 1, ⋯ , n), we get $a_{i}^{P} \leq b_{i}^{P} (i = 1, \dots, n)$ and $- a_{i}^{N} \leq - b_{i}^{N} (i = 1, \dots, n)$ . We have $f^{P} (a_{1}^{P}, \dots, a_{n}^{P}) \leq$ $f^{P} (b_{1}^{P}, \dots, b_{n}^{P})$ and $f^{P} (- a_{1}^{N}, \dots, - a_{n}^{N}) \leq f^{P} (- b_{1}^{N}, \dots, - b_{n}^{N})$ , it follows that $f^{P} (a_{1}^{P}, \dots, a_{n}^{P}) \leq f^{P} (b_{1}^{P}, \dots, b_{n}^{P})$ and $- f^{P} (- a_{1}^{N},$ $\dots, - a_{n}^{N})$ $\geq - f^{P} (- b_{1}^{N}, \dots, - b_{n}^{N})$ . We prove that, for any a_i ≤ ^Lb_i (i = 1, ⋯ , n), $f_{P}^{L} (a_{1}, \dots, a_{n}) \leq^{L} f_{P}^{L} (b_{1}, \dots, b_{n})$ . That is to say the operations in L induced by YinYang bipolar-valued induced function keep the YinYang order relation. Thus, Definition 3 is reasonable.

It is obvious that the YinYang bipolar-valued induced function constructs a bridge connecting operations in L under the YinYang order relation with the corresponding ones in I^P. Based on which, we can induce operations in L under the YinYang order relation from the corresponding ones in I^P. In the following paper, we call the YinYang bipolar-valued regular implication induced from the regular implication in I^P by the YinYang bipolar-valued induced function the induced YinYang bipolar-valued regular implication. And some properties about the induced YinYang bipolar-valued regular implication in L are given as follows:

Proposition 1. Let →^L be the induced YinYang bipolar-valued regular implication in L. For any a, b, c ∈ L, the following properties hold:

a → ^Lb = 1^L if and only if a ≤ ^Lb;

c ≤ ^La → ^Lb if and only if a ≤ ^Lc → ^Lb;

c → ^L (a → ^Lb) = a → ^L (c → ^Lb);

1^L → ^La = a;

$b \to^{L} \land_{i}^{L} a_{i} = \land_{i}^{L} (b \to^{L} a_{i})$ ;

$b \to^{L} \lor_{i}^{L} a_{i} = \lor_{i}^{L} (b \to^{L} a_{i})$ .

Proof. The proof is trivial according to the properties of regular implication in I^P. □

Some examples about the induced YinYang bipolar-valued regular implication in L are given as follows:

Example 1. For any a, b ∈ L, induced Lukasiewicz YinYang bipolar-valued regular implication $\to_{Lu}^{L}$ , induced Gödel YinYang bipolar-valued regular implication $\to_{G}^{L}$ , induced Product YinYang bipolar-valued regular implication $\to_{π}^{L}$ , and induced R₀-YinYang bipolar-valued regular implication $\to_{0}^{L}$ are given as follows: $\begin{matrix} a^{P} \to_{Lu}^{L} b^{P} \\ = ((1 - a^{P} + b^{P}) \land 1, (- 1 - a^{N} + b^{N}) \lor - 1), \\ a \to_{G}^{L} b = {\begin{matrix} 1^{L}, & a^{P} \leq b^{P}, a^{N} \geq b^{N}, \\ (1, b^{N}), & a^{P} \leq b^{P}, a^{N} < b^{N}, \\ (b^{P}, - 1), & a^{P} > b^{P}, a^{N} \geq b^{N}, \\ (b^{P}, b^{N}), & a^{P} > b^{P}, a^{N} < b^{N}, \end{matrix} \\ a \to_{π}^{L} b = {\begin{matrix} 1^{L}, & a^{P} \leq b^{P}, a^{N} \geq b^{N}, \\ (1, - \frac{b^{N}}{a^{N}}), & a^{P} \leq b^{P}, a^{N} < b^{N}, \\ (\frac{b^{P}}{a^{P}}, - 1), & a^{P} > b^{P}, a^{N} \geq b^{N}, \\ (\frac{b^{P}}{a^{P}}, - \frac{b^{N}}{a^{N}}), & a^{P} > b^{P}, a^{N} < b^{N}, \end{matrix} \end{matrix}$

Specifically, $\to_{Lu}^{L}$ and $\to_{0}^{L}$ are the YinYang bipolar-valued contrapositive symmetric regular implications.

Remark 2. Example 1 demonstrates that when the inconsistent bipolarity considered, the induced bipolar-valued regular implications are different from the corresponding interval-valued regular implications given in [13] and Atanassov’s intuitionistic regular implications given in [34]. In fact, the latter two both are the special cases of the corresponding bipolar-valued regular implications. Considering triple I methods strongly depend on implication operations, although there have existed results about triple I methods related to Atanassov’s intuitionistic fuzzy set and interval-valued fuzzy set, difference among the implication operations demonstrates that necessity to study triple I methods based on bipolar-valued fuzzy set.

2.2 Methods

In this subsection, the triple I YBFMT problems with solutions will be discussed. Furthermore, we will prove that triple I YBFMT solution can degenerate into two corresponding triple I FMT solutions.

In the following paper, all of the fuzzy sets in X are denoted by F (X) and all of the bipolar-valued fuzzy sets in X are denoted by $\tilde{F} (X)$ . Next, we will extend the FMT problem to the YBFMT problem, and present the corresponding principles. $\begin{matrix} Suppose that \tilde{A} \to^{L} \tilde{B}, and given that \\ {\tilde{B}}_{*}, calculate {\tilde{A}}_{*} . \end{matrix}$

Where, $\tilde{A}$ , ${\tilde{A}}_{*} \in \tilde{F} (X)$ , $\tilde{B}$ , ${\tilde{B}}_{*} \in \tilde{F} (Y)$ , and →^L is the YinYang bipolar-valued fuzzy implication in L. The corresponding problem is denoted by YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ .

Definition 4. Triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ solution ${\tilde{A}}_{*}$ should be the biggest bipolar-valued fuzzy set under the order relation ≤^L in X satisfying $(\tilde{A} (x) \to^{L} \tilde{B} (y)) \to^{L} ({\tilde{A}}_{*} (x) \to^{L} {\tilde{B}}_{*} (y)) = 1^{L} .$

Furthermore, for any a ∈ L, a-triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ solution ${\tilde{A}}_{*}^{a}$ should be the biggest bipolar-valued fuzzy set under the order relation ≤^L in X satisfying $(\tilde{A} (x) \to^{L} \tilde{B} (y)) \to^{L} ({\tilde{A}}_{*}^{a} (x) \to^{L} {\tilde{B}}_{*} (y)) \geq^{L} a$

Next theorem will present the method to get the solution of a-triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ problem.

Theorem 1. The a-triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ solution ${\tilde{A}}_{*}^{a} (x)$ based on the induced YinYang bipolar-valued regular implication $\to_{P}^{L}$ is given by (1):

Specifically, when a = 1^L, the triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ solution ${\tilde{A}}_{*}$ based on the induced YinYang bipolar-valued regular implication $\to_{P}^{L}$ is expressed by (2):

Proof. By (1), for any y ∈ Y, ${\tilde{A}}_{*}^{a} (x) \leq^{L} a \to_{P}^{L} ((\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \to_{P}^{L} {\tilde{B}}_{*} (y))$ obviously holds.

Since $\to_{P}^{L}$ is the induced YinYang bipolar-valued regular implication, for any y ∈ Y, by property 2° in Proposition 1, we have $a \leq^{L} {\tilde{A}}_{*}^{a} (x) \to_{P}^{L} ((\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \to_{P}^{L} {\tilde{B}}_{*} (y)), x \in X$

From property 3° in Proposition 1, for any y ∈ Y, $(\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \to_{P}^{L} ({\tilde{A}}_{*}^{a} (x) \to_{P}^{L} {\tilde{B}}_{*} (y)) \geq^{L} a, x \in X .$

We prove that ${\tilde{A}}_{*}^{a} (x) = a \to_{P}^{L} ((\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \to_{P}^{L} {\tilde{B}}_{*} (y))$ .

On the other hand, let $\tilde{φ}$ be the bipolar-valued fuzzy set in X that satisfies for any y ∈ Y, $(\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \to_{P}^{L} (\tilde{φ} (x) \to_{P}^{L} {\tilde{B}}_{*} (y)) \geq^{L} ax \in X .$

Considering $\to_{P}^{L}$ is the induced YinYang bipolar-valued regular implication, by property 2° in Proposition 1, then we have

$a \to_{P}^{L} ((\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \to_{P}^{L} {\tilde{B}}_{*} (y)) \geq^{L} \tilde{φ} (x) . x \in X .$

It follows that ${\tilde{A}}_{*}^{a} (x) \geq^{L} \tilde{φ} (x)$ . That is to say ${\tilde{A}}_{*}^{a} (X)$ is the a-triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ solution based on the induced YinYang bipolar-valued regular implication.

When a = 1^L, according to property 4° in Proposition 1, it is easy to prove that $\begin{matrix} {\tilde{A}}_{*} (x) = {⋀_{y \in Y}}^{L} {(\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \to_{P}^{L} {\tilde{B}}_{*} (y)} \\ x \in X □ \end{matrix}$ According to Proposition 1 and Theorem 1, we can derive the following corollary easily:

Corollary 1. Let a ∈ L. The a-triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ solution ${\tilde{A}}_{*}^{a} (x)$ and the triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ solution ${\tilde{A}}_{*}$ based on the induced YinYang bipolar-valued regular implication have the following relationship: ${\tilde{A}}_{*}^{a} (x) = a \to_{P}^{L} {\tilde{A}}_{*} (x) .$

Based on Theorem 1, to any a-triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ based on the induced YinYang bipolar-valued regular implication, we can get its solution. And the Theorem 2 will discuss relationships between the a-triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ solution based on the induced YinYang bipolar-valued regular implication with the corresponding a^P-triple I FMT (A, B, B_∗) solution. $\begin{matrix} a \to_{0}^{L} b & = & {\begin{matrix} 1^{L}, & a^{P} \leq b^{P}, a^{N} \geq b^{N}, \\ (1, (- 1 - a^{N}) \land b^{N}), & a^{P} \leq b^{P}, a^{N} < b^{N}, \\ ((1 - a^{P}) \lor b^{P}, - 1), & a^{P} > b^{P}, a^{N} \geq b^{N}, \\ ((1 - a^{P}) \lor b^{P}, (- 1 - a^{N}) \land b^{N}), & a^{P} > b^{P}, a^{N} < b^{N}, \end{matrix} \\ {\tilde{A}}_{*}^{a} (x) & = & {⋀_{y \in Y}}^{L} {a \to_{P}^{L} ((\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \to_{P}^{L} {\tilde{B}}_{*} (y))} x \in X \end{matrix}$ (1) ${\tilde{A}}_{*} (x) = {⋀_{y \in Y}}^{L} {(\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \to_{P}^{L} {\tilde{B}}_{*} (y)} x \in X$ (2)

Theorem 2. Let a ∈ L. The a-triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ solution ${\tilde{A}}_{*}^{a}$ based on the induced YinYang bipolar-valued regular implication can be expressed as follows: $\begin{matrix} {\tilde{A}}_{*}^{a} (x) & = & (⋀_{y \in Y} {a^{P} \to^{P} (({\tilde{A}}^{P} (x) \to^{P} {\tilde{B}}^{P} (y)) \to^{P} {\tilde{B}}_{*}^{P} (y))}, - ⋀_{y \in Y} {(- a^{N}) \to^{P} (((- {\tilde{A}}^{N} (x)) \to^{P} (- {\tilde{B}}^{N} (y))) \to^{P} (- {\tilde{B}}_{*}^{N} (y)))}) \\ = & ({\tilde{A}}_{*}^{a^{P}} (x), - {\tilde{A}}_{*}^{- a^{N}} (x)) x \in X \end{matrix}$

where $\begin{matrix} {\tilde{A}}_{*}^{a^{P}} (x) & = & ⋀_{y \in Y} {a^{P} \to^{P} (({\tilde{A}}^{P} (x) \to^{P} {\tilde{B}}^{P} (y)) \to^{P} \\ {\tilde{B}}_{*}^{P} (y))} x \in X \end{matrix}$ is the a^P-triple I FMT $({\tilde{A}}^{P}, {\tilde{B}}^{P}, {\tilde{B}}_{*}^{P})$ solution given in [18], and $\begin{matrix} {\tilde{A}}_{*}^{- a^{N}} (x) & = & ⋀_{y \in Y} {(- a^{N}) \to^{P} ((- {\tilde{A}}^{N} (x)) \to^{P} \\ (- {\tilde{B}}^{N} (y)) \to^{P} (- {\tilde{B}}_{*}^{N} (y)))} x \in X \end{matrix}$ is the (- a^N)-triple I FMT $(- {\tilde{A}}^{N}, - {\tilde{B}}^{N}, - {\tilde{B}}_{*}^{N})$ solution given in [18].

Proof. For any a ∈ L,

$\begin{matrix} {\tilde{A}}_{*}^{a} (x) & = & {⋀_{y \in Y}}^{L} {a \to_{P}^{L} ((\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \to_{P}^{L} {\tilde{B}}_{*} (y))} x \in X \\ = & {⋀_{y \in Y}}^{L} {a \to_{P}^{L} (({\tilde{A}}^{P} (x) \to^{P} {\tilde{B}}^{P} (y), - ((- {\tilde{A}}^{N} (x)) \to^{P} (- {\tilde{B}}^{N} (y))) \to_{P}^{L} {\tilde{B}}_{*} (y))} x \in X \\ = & {⋀_{y \in Y}}^{L} {a \to_{P}^{L} (({\tilde{A}}^{P} (x) \to^{P} {\tilde{B}}^{P} (y)) \to^{P} {\tilde{B}}_{*}^{P} (y), - ((- {\tilde{A}}^{N} (x)) \to^{P} (- {\tilde{B}}^{N} (y))) \to^{P} (- {\tilde{B}}_{*}^{N} (y)))} x \in X \\ = & {⋀_{y \in Y}}^{L} ({a^{P} \to^{P} ({\tilde{A}}^{P} (x) \to^{P} {\tilde{B}}^{P} (y) \to^{P} {\tilde{B}}_{*}^{P} (y)), - {(- a^{N}) \to^{P} ((- {\tilde{A}}^{N} (x)) \to^{P} (- {\tilde{B}}^{N} (y)) \to^{P} (- {\tilde{B}}_{*}^{N} (y)))}) x \in X \\ = & (⋀_{y \in Y} {a^{P} \to^{P} ({\tilde{A}}^{P} (x) \to^{P} {\tilde{B}}^{P} (y) \to^{P} {\tilde{B}}_{*}^{P} (y))}, - ⋀_{y \in Y} {(- a^{N}) \to^{P} ((- {\tilde{A}}^{N} (x)) \to^{P} (- {\tilde{B}}^{N} (y)) \to^{P} (- {\tilde{B}}_{*}^{N} (y)))}) x \in X \end{matrix}$

According to the a^P-triple I principle, it is trivial to prove that ${\tilde{A}}_{*}^{a^{P}} (x)$ is the a^P-triple I FMT $({\tilde{A}}^{P}, {\tilde{B}}^{P}, {\tilde{B}}_{*}^{P})$ solution, and ${\tilde{A}}_{*}^{- a^{N}} (x)$ is the (- a^N)-triple I FMT $(- {\tilde{A}}^{N},$ $- {\tilde{B}}^{N}, - {\tilde{B}}_{*}^{N})$ solution. □

Theorem 2 proves that the a-triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ solution based on the induced YinYang bipolar-valued regular implication given in (1) can degenerate into two corresponding triple I FMT (A, B, B_∗) solutions, which reflects the relation between triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ problem and triple I FMT (A, B, B_∗) problem.

3 Performance analysis

In this section, we will prove that the new proposed reasoning methods hold robustness, approximation property, and reductive property, which will guarantee the feasibility, validity, and practical potential of the methods used in the practical control problems.

3.1 Robustness analysis

If small disturbances of input just cause small changes of output, then the corresponding reasoning method is robust. Some concepts have been proposed to discuss robustness of the fuzzy reasoning method [4 , 23]. Among which, the fuzzy biresiduum s^P : I^P × I^P → I^P was given in [10] as follows: $\begin{matrix} s^{P} (a^{P}, b^{P}) = a^{P} \leftrightarrow^{P} \\ b^{P} = (a^{P} \to^{P} b^{P}) \land (b^{P} \to^{P} a^{P}) \end{matrix}$

Where, →^P is the regular implication in I^P.

To discuss robustness of the proposed methods, we will extend the concept of fuzzy biresiduum in I^P to L and propose a similar degree to measure the disturbances of the proposed methods.

Definition 5. Let s^P be the fuzzy biresiduum in I^P. And $s_{P}^{L} = (s^{P}, s_{P}^{N})$ is the operation in L induced from s^P by the YinYang bipolar-valued induced function: $\begin{matrix} s_{P}^{L} (a, b) & = & (s^{P} (a^{P}, b^{P}), - s^{P} (- a^{N}, - b^{N})) \\ = & (a \to_{P}^{L} b) \land^{L} (b \to_{P}^{L} a) \end{matrix}$ then $s_{P}^{L}$ is called YinYang bipolar-valued fuzzy biresiduum. Furthermore, for any $\tilde{A}, \tilde{B} \in \tilde{F} (X)$ , define $\begin{matrix} | s_{P}^{L} | = min {s^{P} (a^{P}, b^{P}), s^{P} (- a^{N}, - b^{N})} \\ | S_{P}^{L} (\tilde{A}, \tilde{B}) | = | ⋀_{x \in X}^{L} s_{P}^{L} (\tilde{A} (x), \tilde{B} (x)) | \end{matrix}$ then $| S_{P}^{L} (\tilde{A}, \tilde{B}) |$ is called similar degree of $\tilde{A}$ and $\tilde{B}$ .

According to the properties about s^P in I^P given in Lemmas 2.2 and 2.3 in [7], and Proposition 1, we can easily prove the following propositions: Proposition 2. Let $s_{P}^{L}$ be YinYang bipolar-valued fuzzy biresiduum. For any a, b, c, d ∈ L, the following properties hold:

$s_{P}^{L} (a, 1^{L}) = a$ ;

a = b if and only if $s_{P}^{L} (a, b) = 1^{L}$ if and only if $| s_{P}^{L} (a, b) | = 1$ ;

$s_{P}^{L} (a, b) = s_{P}^{L} (b, a)$ , $| s_{P}^{L} (a, b) | = | s_{P}^{L} (b, a) |$ ;

$s_{P}^{L} (a, b) \otimes_{P}^{L} s_{P}^{L} (b, c) \leq^{L} s_{P}^{L} (a, c)$ ;

$s_{P}^{L} (a, b) \land^{L} s_{P}^{L} (c, d) \leq^{L} s_{P}^{L} (a \lor^{L} c, b \lor^{L} d)$ ;

$s_{P}^{L} (a, b) \otimes_{P}^{L} s_{P}^{L} (c, d) \leq^{L} s_{P}^{L} (a \otimes_{P}^{L} c, b \otimes_{P}^{L} d)$ ;

$s_{P}^{L} (a, b) \otimes_{P}^{L} s_{P}^{L} (c, d) \leq^{L} s_{P}^{L} (a \to_{P}^{L} c, b \to_{P}^{L} d)$ .

If $s_{P}^{L} (a, b) \leq^{L} s_{P}^{L} (c, d)$ , then $| s_{P}^{L} (a, b) | \leq | s_{P}^{L} (c, d) |$ .

Proposition 3. Let $\tilde{A}, \tilde{B} \in \tilde{F} (X)$ and $s_{P}^{L}$ be the YinYang bipolar-valued fuzzy biresiduum. Then

$\land_{x \in X}^{L} s_{P}^{L} (\tilde{A} (x), \tilde{B} (x)) \leq^{L} s_{P}^{L} (\land_{x \in X}^{L} \tilde{A} (x), \land_{x \in X}^{L} \tilde{B} (x))$ ;

$\land_{x \in X}^{L} s_{P}^{L} (\tilde{A} (x), \tilde{B} (x)) \leq^{L} s_{P}^{L} (\lor_{x \in X}^{L} \tilde{A} (x), \lor_{x \in X}^{L} \tilde{B} (x))$ .

Next theorem will discuss robustness of the proposed method.

Theorem 3. Suppose that $\tilde{A}, {\tilde{A}}^{\circ}, {\tilde{A}}_{*}, {\tilde{A}}_{*}^{\circ} \in \tilde{F} (X)$ , $\tilde{B}, {\tilde{B}}^{\circ}, {\tilde{B}}_{*}, {\tilde{B}}_{*}^{\circ} \in \tilde{F} (Y)$ . Let ${\tilde{A}}_{*}^{a}$ be the a-triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ solution, and $({\tilde{A}}_{*}^{\circ})^{a}$ be the a-triple I YBFMT $({\tilde{A}}^{\circ}, {\tilde{B}}^{\circ}, {\tilde{B}}_{*}^{\circ})$ solution given by (1). Given that δ₁, δ₂, δ₃ > 0, if $\begin{matrix} | S_{P}^{L} (\tilde{A}, {\tilde{A}}^{\circ}) | \geq δ_{1}, | S_{P}^{L} (\tilde{B}, {\tilde{B}}^{\circ}) | \geq δ_{2}, \\ | S_{P}^{L} ({\tilde{B}}_{*}, {\tilde{B}}_{*}^{\circ}) | \geq δ_{3} \end{matrix}$ then $| S_{P}^{L} ({\tilde{A}}_{*}^{a}, ({\tilde{A}}_{*}^{\circ})^{a}) | \geq δ_{1} \otimes^{P} δ_{2} \otimes^{P} δ_{3}$

Specifically, when $\tilde{A} = {\tilde{A}}^{\circ}$ and $\tilde{B} = {\tilde{B}}^{\circ}$ , if $| S_{P}^{L} ({\tilde{B}}_{*}, {\tilde{B}}_{*}^{\circ}) | \geq δ_{3}$ , then we have $| S_{P}^{L} ({\tilde{A}}_{*}^{a}, ({\tilde{A}}_{*}^{\circ})^{a}) | \geq δ_{3}$ .

Proof. Firstly, suppose that a = 1^L. To (2), according to Theorem 2, Propositions 2 and 3, we have

$\begin{matrix} | S_{P}^{L} ({\tilde{A}}_{*}, {\tilde{A}}_{*}^{\circ}) | = | ⋀_{x \in X}^{L} s_{P}^{L} ({\tilde{A}}_{*} (x), {\tilde{A}}_{*}^{\circ} (x)) | \\ = | ⋀_{x \in X}^{L} s_{P}^{L} (⋀_{y \in Y}^{L} ((\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \to_{P}^{L} {\tilde{B}}_{*} (y)), ⋀_{y \in Y}^{L} ({\tilde{A}}^{\circ} (x) \to_{P}^{L} {\tilde{B}}^{\circ} (y)) \to_{P}^{L} {\tilde{B}}_{*}^{\circ} (y))) | \\ \geq | ⋀_{x \in X}^{L} ⋀_{y \in Y}^{L} s_{P}^{L} ((\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \to_{P}^{L} {\tilde{B}}_{*} (y), ({\tilde{A}}^{\circ} (x) \to_{P}^{L} {\tilde{B}}^{\circ} (y)) \to_{P}^{L} {\tilde{B}}_{*}^{\circ} (y)) | \\ \geq | ⋀_{x \in X}^{L} ⋀_{y \in Y}^{L} (s_{P}^{L} (\tilde{A} (x) \to_{P}^{L} \tilde{B} (y), {\tilde{A}}^{\circ} (x) \to_{P}^{L} {\tilde{B}}^{\circ} (y)) \otimes_{P}^{L} s_{P}^{L} ({\tilde{B}}_{*} (y), {\tilde{B}}_{*}^{\circ} (y))) | \\ \geq | ⋀_{x \in X}^{L} ⋀_{y \in Y}^{L} (s_{P}^{L} (\tilde{A} (x), ({\tilde{A}}^{\circ} (x)) \otimes_{P}^{L} s_{P}^{L} (\tilde{B} (y), {\tilde{B}}^{\circ} (y)) \otimes_{P}^{L} s_{P}^{L} ({\tilde{B}}_{*} (y), {\tilde{B}}_{*}^{\circ} (y))) | \\ = | ⋀_{x \in X} ⋀_{y \in Y} (s^{P} ({\tilde{A}}^{P} (x), ({\tilde{A}}^{\circ})^{P} (x)) \otimes^{P} s^{P} ({\tilde{B}}^{P} (y), ({\tilde{B}}^{\circ})^{P} (y)) \otimes^{P} s^{P} ({\tilde{B}}_{*}^{P} (y), ({\tilde{B}}^{\circ})_{*}^{P} (y))), \\ ⋁_{x \in X} ⋁_{y \in Y} (- s^{P} (- {\tilde{A}}^{N} (x), - ({\tilde{A}}^{\circ})^{N} (x)) \otimes^{P} - s^{P} (- {\tilde{B}}^{N} (y), - ({\tilde{B}}^{\circ})^{N} (y)) \otimes^{P} - s^{P} (- {\tilde{B}}_{*}^{N} (y), - ({\tilde{B}}^{\circ})_{*}^{N} (y))) | \\ = | ⋀_{x \in X} ⋀_{y \in Y} (s^{P} ({\tilde{A}}^{P} (x), ({\tilde{A}}^{\circ})^{P} (x)) \otimes^{P} s^{P} ({\tilde{B}}^{P} (y), ({\tilde{B}}^{\circ})^{P} (y)) \otimes^{P} s^{P} ({\tilde{B}}_{*}^{P} (y), ({\tilde{B}}^{\circ})_{*}^{P} (y))), \\ - ⋀_{x \in X} ⋀_{y \in Y} (s^{P} (- {\tilde{A}}^{N} (x), - ({\tilde{A}}^{\circ})^{N} (x)) \otimes^{P} s^{P} (- {\tilde{B}}^{N} (y), - ({\tilde{B}}^{\circ})^{N} (y)) \otimes^{P} s^{P} (- {\tilde{B}}_{*}^{N} (y), - ({\tilde{B}}^{\circ})_{*}^{N} (y))) | \\ = min {⋀_{x \in X} ⋀_{y \in Y} (s^{P} ({\tilde{A}}^{P} (x), ({\tilde{A}}^{\circ})^{P} (x)) \otimes^{P} s^{P} ({\tilde{B}}^{P} (y), ({\tilde{B}}^{\circ})^{P} (y)) \otimes^{P} s^{P} ({\tilde{B}}_{*}^{P} (y), ({\tilde{B}}^{\circ})_{*}^{P} (y))), \\ ⋀_{x \in X} ⋀_{y \in Y} (s^{P} (- {\tilde{A}}^{N} (x), - ({\tilde{A}}^{\circ})^{N} (x)) \otimes^{P} s^{P} (- {\tilde{B}}^{N} (y), - ({\tilde{B}}^{\circ})^{N} (y)) \otimes^{P} s^{P} (- {\tilde{B}}_{*}^{N} (y), - ({\tilde{B}}^{\circ})_{*}^{N} (y)))} \end{matrix}$

If $| S_{P}^{L} (\tilde{A}, {\tilde{A}}^{\circ}) | \geq δ_{1}, | S_{P}^{L} (\tilde{B}, {\tilde{B}}^{\circ}) | \geq δ_{2},$ and $| S_{P}^{L} ({\tilde{B}}_{*}, {\tilde{B}}_{*}^{\circ}) | \geq δ_{3}$ , then $\begin{matrix} S^{P} ({\tilde{A}}^{P}, ({\tilde{A}}^{\circ})^{P}) \geq δ_{1}, S^{P} ({\tilde{B}}^{P}, ({\tilde{B}}^{\circ})^{P}) \geq δ_{2}, \\ S^{P} ({\tilde{B}}_{*}^{P}, ({\tilde{B}}_{*}^{\circ})^{P}) \geq δ_{3} and \\ S^{P} (- {\tilde{A}}^{N}, (- {\tilde{A}}^{\circ})^{N}) \geq δ_{1}, S^{P} (- {\tilde{B}}^{N}, - ({\tilde{B}}^{\circ})^{N}) \geq δ_{2}, \\ S^{P} (- {\tilde{B}}_{*}^{N}, - ({\tilde{B}}_{*}^{\circ})^{N}) \geq δ_{3} \end{matrix}$

According to Theorem 2, ${\tilde{A}}_{*}^{P}$ is the triple I FMT $({\tilde{A}}^{P}, {\tilde{B}}^{P}, {\tilde{B}}_{*}^{P})$ solution and $({\tilde{A}}_{*}^{\circ})^{P}$ is the triple I FMT $(({\tilde{A}}^{\circ})^{P}, ({\tilde{B}}^{\circ})^{P}, ({\tilde{B}}_{*}^{\circ})^{P})$ solution. Then, according to Theorem 4.2 in [7], we have $⋀_{x \in X} ⋀_{y \in Y} (S^{P} ({\tilde{A}}^{P} (x), ({\tilde{A}}^{\circ})^{P} (x)) \otimes^{P} S^{P} ({\tilde{B}}^{P} (y), ({\tilde{B}}^{\circ})^{P} (y)) \otimes^{P} S^{P} ({\tilde{B}}_{*}^{P} (y), ({\tilde{B}}^{\circ})_{*}^{P} (y))) \geq δ_{1} \otimes^{P} δ_{2} \otimes^{P} δ_{3}$

Similarly, we have $\begin{matrix} ⋀_{x \in X} ⋀_{y \in Y} (S^{P} (- {\tilde{A}}^{N} (x), - ({\tilde{A}}^{\circ})^{N} (x)) \otimes^{P} S^{P} (- {\tilde{B}}^{N} (y), - ({\tilde{B}}^{\circ})^{N} (y)) \otimes^{P} S^{P} (- {\tilde{B}}_{*}^{N} (y), - ({\tilde{B}}^{\circ})_{*}^{N} (y))) \\ \geq δ_{1} \otimes^{P} δ_{2} \otimes^{P} δ_{3}, \end{matrix}$

It follows that $| S_{P}^{L} ({\tilde{A}}_{*}, {\tilde{A}}_{*}^{\circ}) | \geq δ_{1} \otimes^{P} δ_{2} \otimes^{P} δ_{3}$

Furthermore, when a < 1^L, according to Corollary 1, we have $\begin{matrix} | S_{P}^{L} ({\tilde{A}}_{*}^{a}, ({\tilde{A}}_{*}^{\circ})^{a}) | \\ = | S_{P}^{L} ((a \to_{P}^{L} {\tilde{A}}_{*}), (a \to_{P}^{L} {\tilde{A}}_{*}^{\circ})) | \end{matrix}$

From Propositions 2 and 3, we have $\begin{matrix} | S_{P}^{L} ((a \to_{P}^{L} {\tilde{A}}_{*}), (a \to_{P}^{L} {\tilde{A}}_{*}^{\circ})) | \\ = | ⋀_{x \in X}^{L} s_{P}^{L} (a \to_{P}^{L} {\tilde{A}}_{*} (x), a \to_{P}^{L} {\tilde{A}}_{*}^{\circ} (x)) | \\ \geq | ⋀_{x \in X}^{L} (s_{P}^{L} (a, a) \otimes_{P}^{L} s_{P}^{L} ({\tilde{A}}_{*} (x), {\tilde{A}}_{*}^{\circ} (x)) | \\ = | ⋀_{x \in X}^{L} (1^{L} \otimes_{P}^{L} s_{P}^{L} ({\tilde{A}}_{*} (x), {\tilde{A}}_{*}^{\circ} (x)) | \\ = | ⋀_{x \in X}^{L} (s_{P}^{L} ({\tilde{A}}_{*} (x), {\tilde{A}}_{*}^{\circ} (x)) | \geq δ_{1} \otimes^{P} δ_{2} \otimes^{P} δ_{3} \end{matrix}$

Specifically, when $\tilde{A} = {\tilde{A}}^{\circ}$ and $\tilde{B} = {\tilde{B}}^{\circ}$ , if $| S_{P}^{L} ({\tilde{B}}_{*}, {\tilde{B}}_{*}^{\circ}) | \geq δ_{3}$ , it is easy to prove that $| S_{P}^{L} ({\tilde{A}}_{*}^{a}, ({\tilde{A}}_{*}^{\circ})^{a}) | \geq δ_{3}$ . □

Theorem 3 shows that the proposed a-triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ method is robust. Furthermore, we notice that robustness of a-triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ method is strongly related to the choices of the corresponding YinYang bipolar-valued regular implication, while, it is not related to the parameter a at all.

3.2 Approximation property analysis

A good fuzzy reasoning method should hold the continuity property, i.e., a small input deviation will not result in a great deviation of the conclusion [12]. In this subsection, the approximation property, a special case of continuity of the reasoning methods will be discussed. To discuss approximation property, first, the concepts of YinYang bipolar-valued fuzzy metric is introduced.

Definition 6. For any $\tilde{A}, \tilde{B}, \tilde{C} \in \tilde{F} (X)$ , if the operation $Φ^{L} : \tilde{F} (X) \times \tilde{F} (X) \to I^{P}$ satisfies the following conditions:

$0 \leq Φ^{L} (\tilde{A}, \tilde{B}) \leq 1$ and $Φ^{L} (\tilde{A}, \tilde{B}) = 0$ if and only if $\tilde{A} = \tilde{B}$ ;

$Φ^{L} (\tilde{A}, \tilde{B}) = Φ^{L} (\tilde{B}, \tilde{A})$ ;

$Φ^{L} (\tilde{A}, \tilde{C}) \leq Φ^{L} (\tilde{A}, \tilde{B}) + Φ^{L} (\tilde{B}, \tilde{C})$ ,

then Φ^L is called the YinYang bipolar-valued fuzzy metric in

\tilde{F} (X)

For any $\tilde{A}, \tilde{B} \in \tilde{F} (X)$ , then define $(D_{P}^{L})_{S} (\tilde{A}, \tilde{B}) = 1 - | S_{P}^{L} (\tilde{A}, \tilde{B}) |$ then it is obvious that $(D_{P}^{L})_{S}$ is the YinYang bipolar-valued fuzzy metric in $\tilde{F} (X)$ .

To prove the approximation property of the proposed methods, similar to the concept of continuous symmetric implication defined in [19], the definition of the continuous contrapositive symmetric implication in I^P is given as follows:

Definition 7. Let (⊗ ^P, → ^P) be the residual pair in I^P. For any B ∈ F (Y), g : F (Y) → ^PF (X) is defined as g (B_∗) (x) = ∨ _y∈Y (B_∗ (y) ⊗ ^P (A (x) → ^PB (y)), if the following inequality holds $⋀_{x \in X} (A (x) \leftrightarrow^{P} g (B_{*}) (x)) \geq ⋀_{y \in Y} (B (y) \leftrightarrow^{P} B_{*} (y))$ then →^P is said to be continuous contrapositive symmetric implication.

A method to solve YBFMT problems can be seen as a mapping $\tilde{g} : \tilde{F} (Y) \to \tilde{F} (X)$ , i.e., each input ${\tilde{B}}_{*} \in \tilde{F} (Y)$ corresponds to an output ${\tilde{A}}_{*} = \tilde{g} ({\tilde{B}}_{*})$ . Then, we can propose the following definition.

Definition 8. Let Φ^L be the YinYang bipolar-valued fuzzy metric. Method $\tilde{g}$ is said to hold the approximation property at $\tilde{A} \in \tilde{F} (X)$ in YinYang bipolar-valued fuzzy metric Φ^L, if for any ɛ > 0, ${\tilde{B}}_{*} \in \tilde{F} (Y)$ there exists δ > 0 such that $Φ^{L} (\tilde{g} ({\tilde{B}}_{*}), \tilde{A}) < ɛ$ when $Φ^{L} ({\tilde{B}}_{*}, \tilde{B}) < δ$ .

Next, we well discuss the approximation property of the triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ method given in (2).

Theorem 4. Suppose that $\tilde{A}, {\tilde{A}}_{*} \in \tilde{F} (X)$ , $\tilde{B}, {\tilde{B}}_{*}, \in \tilde{F} (Y)$ . Let $\to_{P}^{L}$ be the YinYang bipolar-valued contrapositive symmetric regular implication induced from the continuous contrapositive symmetric implication →^P in I^P by the YinYang bipolar-valued induced function, then the triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ method given in (2) holds the approximation property at $\tilde{A}$ in YinYang bipolar-valued fuzzy metric $(D_{P}^{L})_{S}$ .

Proof. Note that the YinYang bipolar-valued fuzzy modus ponens (YBFMP) problem is presented as follows: $\begin{matrix} Suppose that \tilde{A} \to^{L} \tilde{B}, and given that \\ {\tilde{A}}_{*}, calculate {\tilde{B}}_{*} \end{matrix}$

Similar to the proof of Theorem 1, the triple I YBFMP $(\tilde{A}, \tilde{B}, {\tilde{A}}_{*})$ solution ${\tilde{B}}_{*}$ based on the induced YinYang bipolar-valued regular implication can be expressed as follows: ${\tilde{B}}_{*} (y) = {⋁_{x \in X}}^{L} {(\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \otimes_{P}^{L} {\tilde{A}}_{*} (x)} y \in Y$

On the other hand, when $\to_{P}^{L}$ is the YinYang bipolar-valued contrapositive symmetric regular implication. The YBFMT problem can be seen as the following special YBFMP problem $Suppose {\tilde{B}}^{'} \to^{L} {\tilde{A}}^{'}, and given {\tilde{B}}_{*}, calculate {\tilde{A}}_{*}$

That is to say when $\to_{P}^{L}$ is the YinYang bipolar-valued contrapositive symmetric regular implication, the triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ solution ${\tilde{A}}_{*} (x)$ given in (2) can also be expressed as follows: ${\tilde{A}}_{*} (x) = {⋁_{y \in Y}}^{L} {(\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \otimes_{P}^{L} {\tilde{B}}_{*} (y)} x \in X$

Followed by

$\begin{matrix} ⋀_{x \in X}^{L} (\tilde{A} (x) \leftrightarrow_{P}^{L} {\tilde{A}}_{*} (x)) \\ = ⋀_{x \in X}^{L} (\tilde{A} (x) \leftrightarrow_{P}^{L} {⋁_{y \in Y}}^{L} {(\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \otimes_{P}^{L} {\tilde{B}}_{*} (y)}) \\ = ⋀_{x \in X}^{L} ({\tilde{A}}^{P} (x) \leftrightarrow^{P} ⋁_{y \in Y} {({\tilde{A}}^{P} (x) \to^{P} {\tilde{B}}^{P} (y)) \otimes^{P} {\tilde{B}}_{*}^{P} (y)}, \\ (- {\tilde{A}}^{N} (x)) \leftrightarrow^{P} ⋁_{y \in Y} {(- {\tilde{A}}^{N} (x) \to^{P} - {\tilde{B}}^{N} (y)) \otimes^{P} - {\tilde{B}}_{*}^{N} (y)}) \\ = (⋀_{x \in X} ({\tilde{A}}^{P} (x) \leftrightarrow^{P} ⋁_{y \in Y} {({\tilde{A}}^{P} (x) \to^{P} {\tilde{B}}^{P} (y)) \otimes^{P} {\tilde{B}}_{*}^{P} (y)}), \\ - (⋀_{x \in X} ({\tilde{A}}^{N} (x) \leftrightarrow^{P} ⋁_{y \in Y} {(- {\tilde{A}}^{N} (x) \to^{P} - {\tilde{B}}^{N} (y)) \otimes^{P} \\ {\tilde{B}}_{*}^{N} (y)})) \\ \geq^{L} (⋀_{y \in Y} ({\tilde{B}}^{P} (y) \leftrightarrow^{P} B_{*}^{P} (y)), - ⋀_{y \in Y} (- {\tilde{B}}^{N} (y) \leftrightarrow^{P} - B_{*}^{N} (y))) \end{matrix}$

Then we have $| ⋀_{x \in X}^{L} (\tilde{A} (x) \leftrightarrow_{P}^{L} {\tilde{A}}_{*} (x)) | \geq | ⋀_{y \in Y}^{L} (\tilde{B} (y) \leftrightarrow_{P}^{L} B_{*} (y)) |$

It follows that $\begin{matrix} 1 - | ⋀_{x \in X}^{L} (\tilde{A} (x) \leftrightarrow_{P}^{L} {\tilde{A}}_{*} (x)) | \\ \leq 1 - | ⋀_{y \in Y}^{L} (\tilde{B} (y) \leftrightarrow_{P}^{L} B_{*} (y)) | \end{matrix}$ that is to say $(D_{P}^{L})_{S} (\tilde{A}, {\tilde{A}}_{*}) \leq (D_{P}^{L})_{S} (\tilde{B}, {\tilde{B}}_{*})$ .

For any ɛ > 0, set δ = ɛ, then $(D_{P}^{L})_{S} (\tilde{A}, {\tilde{A}}_{*}) < ɛ$ when $(D_{P}^{L})_{S} (\tilde{B}, {\tilde{B}}_{*}) < δ$ . Thus, the method given in (2) holds the approximation property at $\tilde{A}$ in YinYang bipolar-valued fuzzy metric $(D_{P}^{L})_{S}$ . □ Theorem 4 proves that the triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ given in (2) holds the approximation property for the YinYang bipolar-valued contrapositive symmetric regular implication.

3.3 Reductive property analysis

This subsection will prove that the triple I BVFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ method based on the regular implication possesses the reductive property. First, we will extend the concept of reductive property from the FMT problem to the YBFMT problem.

Definition 9. For an algorithm to solve the YBFMT problem, if ${\tilde{B}}_{*} = {\tilde{B}}^{'}$ implies ${\tilde{A}}_{*} = {\tilde{A}}^{'}$ whenever condition P is satisfied, then the algorithm is called P-reductive.

Theorem 5. Suppose that $\tilde{A}, {\tilde{A}}_{*} \in \tilde{F} (X)$ , $\tilde{B}, {\tilde{B}}_{*}, \in \tilde{F} (Y)$ . Let $\to_{P}^{L}$ be the YinYang bipolar-valued contrapositive symmetric regular implication. The triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ method given in (2) is P-reductive. P-reductive means that there exists y ∈ Y such that ${\tilde{B}}^{'} (y) = 1^{L}$ .

Proof. Since $\to_{P}^{L}$ is the YinYang bipolar-valued contrapositive symmetric regular implication, we have ${\tilde{B}}_{*} = {\tilde{B}}^{'}$ . As mentioned in Theorem 4, the triple I YBFMT $(\tilde{A}, \tilde{B}, {\tilde{B}}_{*})$ solution ${\tilde{A}}_{*} (x)$ given in (2) can also be expressed as: ${\tilde{A}}_{*} (x) = {⋁_{y \in Y}}^{L} {(\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \otimes_{P}^{L} {\tilde{B}}_{*} (y)} x \in X$

It follows that, if there exists y ∈ Y such that ${\tilde{B}}^{'} (y) = 1^{L}$ , then for any x ∈ X, $\begin{matrix} {\tilde{A}}_{*} (x) & = & {⋁_{y \in Y}}^{L} {(\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \otimes_{P}^{L} {\tilde{B}}_{*} (y)} \\ = & {⋁_{y \in Y}}^{L} {({\tilde{B}}^{'} (x) \to_{P}^{L} {\tilde{A}}^{'} (x)) \otimes_{P}^{L} {\tilde{B}}_{*} (y)} \\ \geq^{L} & (1^{L} \to_{P}^{L} {\tilde{A}}^{'} (x)) \otimes_{P}^{L} 1^{L} = {\tilde{A}}^{'} (x) \end{matrix}$

On the other hand, when ${\tilde{B}}_{*} = {\tilde{B}}^{'}$ , for any x ∈ X and y ∈ Y, we have $\begin{matrix} (\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) = ({\tilde{B}}^{'} (x) \to_{P}^{L} {\tilde{A}}^{'} (x)) \\ = ({\tilde{B}}_{*} (x) \to_{P}^{L} {\tilde{A}}^{'} (x)) \end{matrix}$ it follows that $(\tilde{A} (x) \to_{P}^{L} \tilde{B} (y)) \otimes_{P}^{L} {\tilde{B}}_{*} (x) \leq^{L} {\tilde{A}}^{'} (x)$ that is to say ${\tilde{A}}_{*} (x) \leq^{L} {\tilde{A}}^{'} (x)$ , then we prove that ${\tilde{A}}_{*} (x) = {\tilde{A}}^{'} (x)$ . □

Theorems 3–5 in this section have proven that the proposed methods hold robustness, approximation property and reductive property, which will guarantee the methods can be used in the practical control problems.

4 Conclusion

In this paper, the triple I bipolar-valued fuzzy modus tollens methods under the YinYang order relation were presented and the properties of the new methods were discussed. We have concluded that the proposed methods hold robustness to all the regular implication. While, only to a special class of regular implication, the proposed methods hold approximation property and reductive property. We not only prove that the existing works about triple I methods are the special cases of the new methods, but also provides a new inconsistent bipolarity perspective to artificial reasoning, making it more close to human brain reasoning. In the future work, the application of proposed method to solve practical control problems will be discussed.

Footnotes

Acknowledgments

This work was supported in part by the Joint Key Grant of National Natural Science Foundation of China and Zhejiang Province (U1509217), the National Natural Science Foundation of China (61503191, 61502239), the Natural Science Foundation of Jiangsu Province, China (BK20150924, BK20150933).

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