Research on rule extraction method based on concept lattice of intuitionistic fuzzy language

Hongpeng Yan

Dettagli rivista e numero

Applied Mathematics and Nonlinear Sciences

Dettagli della rivista

Formato: Rivista
eISSN: 2444-8656
Prima pubblicazione: 01 Jan 2016
Frequenza di pubblicazione: 2 volte all'anno
Lingue: Inglese

The rapid progress of artificial intelligence has accelerated the development of society. Nowadays, people punch in and out of work using fingerprint identification, which avoids the need to wait in a queue for a long time; such an arrangement not only saves each worker's time but also ensures that the company does not need to maintain or rely on any manual record of attendance, and that absentees, latecomers and early leavers can be known at a glance. Scanning code payment and scanning code collection are achieved everywhere by a mobile phone [1], which saves people a lot of unnecessary trouble and makes their lives tend to be simplified and convenient. At the same time, ‘talking’ robots can sing, tell jokes and chat with people, which relieves people's mental stress [2]. Therefore, the development of intelligence has promoted the growth of human happiness index and the continuous progress of society.

In the field of artificial intelligence research, to make the research results more beneficial to people's lives, it is necessary that the computer be able to not only handle the calculation of general numerical information but also manage the data processing of intelligent information (that is, non-numerical information) close to the human daily thinking mode [3]. In the natural environment, most things that people come into contact with are uncertain, and in the process of reasoning and decision, more emphasis is placed on the use of natural languages with strong fuzziness, such as ‘a little low’, ‘absolutely true’ and ‘very high’ [4]. Therefore, the development of intelligent information processing is particularly important. It can penetrate into any part of people's daily life, and moreover, computers can be used to imitate the thinking mode of the human brain, which accelerates the development of artificial intelligence [5].

Concept lattice, as an effective mathematical tool, can effectively handle knowledge processing and data mining [6, 7], which are widely used in the research of reasoning decision, machine learning, recommendation system, intelligent evaluation, human factor engineering and other fields [8]. Concept lattice is a formal concept model composed of intension and extension that was first proposed by R Wille in 1982 [9]. Nowadays, it is being widely used in the fields of knowledge representation, data mining and machine learning [10, 11].

Extracting implicit and potential rules from information systems is the key step to achieve knowledge acquisition. In the traditional expert system or fuzzy reasoning system, rules are often given by experts according to their experience, which may lead to some problems [12] owing to the fact that the rules acquired based on classical rough set theory are often clear rules with steep truncation, and compared with fuzzy rules, their generalisation ability and practicability are weak [13]. Intuitionistic fuzzy sets (IFS) constitute an important extension of fuzzy sets; their mathematical description is more in line with the nature of fuzzy objects in the objective world, and IFS have thus become a new research hotspot in recent years [14]. Based on retaining the membership function of fuzzy set, IFS adds a new attribute parameter – non-membership function [15], which can further describe the ‘fuzzy concept of either this or the other’; that is, membership, non-membership and hesitation of IFS can represent the three states of support, opposition and neutrality, respectively [16].

Context and attribute reduction of intuitionistic fuzzy language

Concept lattice is a very effective tool in the field of decision analysis. In real life, people are more accustomed to expressing information with linguistic values rather than numerical values [17], and sometimes consider problems with both pros and cons aspects. In order to better deal with linguistic values expressed in terms of both positive and negative aspects, concept lattice theory is used to deal with IFS and fuzzy linguistic values. Moreover, concept lattice of intuitionistic fuzzy language is established [18], and a method of attribute reduction is proposed in the context of intuitionistic fuzzy language.

2.1

Intuitionistic fuzzy language concept lattice

Definition 1

Define (U,A,S^′) as an formal context of intuitionistic fuzzy language, in which U = {x₁, x₂, ⋯ , x_n} is an object set, A = {a₁, ⋯ , a_m} is attribute set, S^′ is binary relation between U and A, and S^′ ⊆ U × A; regard (1) $S^{'} (x, a) = s_{μ} (x, a), s_{ν} (x, a)$ {S^{'}}(x,a) = {s_\mu}(x,a),{s_\nu}(x,a) as the intuitionistic fuzzy language value of object X under attribute A, where $s_{μ} : U \times A \to {s_{0}, s_{1}, \dots, s_{g}}, s_{v} : U \times A \to {s_{0}, s_{1}, \dots, s_{g}} and 0 \leq μ + v \leq g .$ {s_\mu}:U \times A \to \left\{{{s_0},{s_1}, \cdots,{s_g}} \right\},\quad {s_v}:U \times A \to \left\{{{s_0},{s_1}, \cdots,{s_g}} \right\}{\kern 1pt} {\rm{and}}{\kern 1pt} 0 \le \mu + v \le g.

Further, s_μ(x,a) and s_ν(x,a) represent the linguistic value membership degree and linguistic value non-membership degree between object X and attribute A, respectively [19]. As ∀x,y ∈ U, ∀a,b ∈ A, there are: (2) $\begin{matrix} S^{'} (x, a) \geq S^{'} (y, b) \Leftrightarrow (s_{μ} (x, a) \geq s_{μ} (y, b)) \cap (s_{v} (x, a) \leq s_{v} (y, b)) \\ S^{'} {(x, a)}^{c} = s_{v} (x, a), s_{μ} (x, a) \end{matrix}$ \matrix{{{S^{'}}(x,a) \ge {S^{'}}(y,b) \Leftrightarrow \left({{s_\mu}(x,a) \ge {s_\mu}(y,b)} \right) \cap \left({{s_v}(x,a) \le {s_v}(y,b)} \right)} \cr {{S^{'}}{{(x,a)}^c} = {s_v}(x,a),{s_\mu}(x,a)} \cr}

Definition 2

Define (U,A,S^′) as an intuitionistic fuzzy language formal context, where ∀X ∈ U, $S_{A}^{'}$ S_A^{'} represents a set of object subset X under attribute set A, $B \subseteq S_{A}^{'}$ B \subseteq S_A^{'} represents the attribute intuitionistic fuzzy language values, and at the same time, a pair of derived operators is defined as follows: (3) $X^{<} = \cap {〈 \underset{x \in X}{\land} s_{μ} (x, a), \underset{x \in X}{v} s_{v} (x, a) 〉 a \in A, x \in X}$ {X^ <} = \cap \left\{{\left\langle {\mathop \wedge \limits_{x \in X} {s_\mu}(x,a),\mathop v\limits_{x \in X} {s_v}(x,a)} \right\rangle a \in A,x \in X} \right\} (4) $B^{≺} = \cap {x S^{'} (x, a) \geq B (x, a), x \in U, \forall a \in A}$ {B^ \prec} = \cap \left\{{x{S^{'}}(x,a) \ge B(x,a),x \in U,\forall a \in A} \right\} in which B(x,a) represents the intuitionistic fuzzy language value of object X under attribute A.

Definition 3

Define (U,A,S^′) as an intuitionistic fuzzy language formal context; if a tuple (X,B) satisfies X^≺ = B and X = B^≺, (X,B) is regarded as the concept of intuitionistic fuzzy language form, in which X is the extension of intuitionistic fuzzy language form concept and B is the intension of intuitionistic fuzzy language form concept [20].

Definition 4

Define (U,A,S^′) as an intuitionistic fuzzy language form context, make IFLL (U,A,S^′) represent (U,A,S^′) on the whole intuitionistic fuzzy language concept [21] and IFLL (U,A,S^′) is called intuitionistic fuzzy language concept lattice, for ∀(X₁, B₁), (X₂, B₂) ∈ IFLL (U,A,S^′), defining its partial order relation as (5) $(X_{1}, B_{1}) \leq (X_{2}, B_{2}) \Leftrightarrow X_{1} \subseteq X_{2} (\Leftrightarrow B_{1} \supseteq B_{2})$ \left({{X_1},{B_1}} \right) \le \left({{X_2},{B_2}} \right) \Leftrightarrow {X_1} \subseteq {X_2}\left({\Leftrightarrow {B_1} \supseteq {B_2}} \right) in which (X₁, B₂) is called the child concept of (X₂, B₂) and (X₂, B₂) is called the parent concept of (X₁, B₂).

Define the lower and upper bounds of IFLL (U,A,S^′) as [22] (6) $(X_{1}, B_{1}) \cap (X_{2}, B_{2}) = ({(X_{1} \cup X_{2})}^{≺ ≺}, B_{1} \cap B_{2})$ \left({{X_1},{B_1}} \right) \cap \left({{X_2},{B_2}} \right) = \left({{{\left({{X_1} \cup {X_2}} \right)}^{\prec \prec}},{B_1} \cap {B_2}} \right) (7) $(X_{1}, B_{1}) \cup (X_{2}, B_{2}) = (X_{1} \cap X_{2}, {(B_{1} \cup B_{2})}^{≺ ≺})$ \left({{X_1},{B_1}} \right) \cup \left({{X_2},{B_2}} \right) = \left({{X_1} \cap {X_2},{{\left({{B_1} \cup {B_2}} \right)}^{\prec \prec}}} \right) where IFLL (U,A,S^′) is a complete lattice.

Example 1

Given a formal context of intuitionistic fuzzy language (U,A,S^′) U = (x₁, x₂, x₃), A = (a,b,c) and make the language set S = {s₀ = verylow, s₁ = low, s₂ = medium, s₃ = high, s₄ = veryhigh}. Among them, S^′ (x₁, a) = s₀, s₄ means the possibility that the object x₁ has the attribute A is very low, and the possibility that it does not have the attribute A is very high (Table 1).

Table 1

Formal context of Intuitionistic fuzzy linguistic (U,A,S^′)

U	A	b	c

x₁	〈s₀, s₄〉	〈s₂, s₂〉	〈s₁, s₂〉
x₂	〈s₃, s₀〉	〈s₀, s₂〉	〈s₃, s₁〉
x₃	〈s₁, s₁〉	〈s₃, s₁〉	〈s₂, s₁〉

Table 1 shows the formal context of intuitionistic fuzzy language (U,A,S^′) The generated concept is:

#1 (U, {s₀, s₄s₀, s₂s₁, s₂})

#2 ({x₁x₃}, {s₀, s₄s₂, s₂s₁, s₂})

#3 ({x₂x₃}, {s₁, s₁s₀, s₂s₂, s₁})

#4 ({x₂}, {s₃, s₀s₀, s₂s₃, s₁})

#5 ({x₃}, {s₁, s₁s₃, s₁s₂, s₁})

#6 (∅, {s₃, s₀s₃, s₁s₃, s₁})

The intuitionistic fuzzy language concept lattice is built as shown in Figure 1:

Intuitionistic fuzzy linguistic concept lattice IFLL (U,A,S^′)

According to the above definition, the following properties can be obtained:

Property 1, applicable in the context of intuitionistic fuzzy language form (U,A,S^′): Next, for any X ⊆ U and $B \subseteq S_{A}^{'}$ B \subseteq S_A^{'} , there are:

X ⊆ X^≺≺, B ⊆ B^≺≺

X^≺ = X^≺≺≺, B^≺ = B^≺≺≺

X ⊆ B^≺ ⇔ B ⊆ X^≺

(X^≺≺, X^≺) and (B^≺, B^≺≺) are all concepts.

Property 2, applicable in the context of intuitionistic fuzzy language form (U,A,S^′): For any X₁, X₂ ⊆ U and $B_{1}, B_{2} \subseteq S_{A}^{'}$ {B_1},{B_2} \subseteq S_A^{'} , there are:

$X_{1} \subseteq X_{2} \Rightarrow X_{2}^{≺} \subseteq X_{1}^{≺}$ {X_1} \subseteq {X_2} \Rightarrow X_2^ \prec \subseteq X_1^ \prec , $B_{1} \subseteq B_{2} \Rightarrow B_{2}^{≺} \subseteq B_{1}^{≺}$ {B_1} \subseteq {B_2} \Rightarrow B_2^ \prec \subseteq B_1^ \prec ;

${(X_{1} \cup X_{2})}^{≺} = X_{1}^{≺} \cap X_{2}^{≺}$ {\left({{X_1} \cup {X_2}} \right)^ \prec} = X_1^ \prec \cap X_2^ \prec , ${(B_{1} \cup B_{2})}^{≺} = B_{1}^{≺} \cap B_{2}^{≺}$ {\left({{B_1} \cup {B_2}} \right)^ \prec} = B_1^ \prec \cap B_2^ \prec ;

${(X_{1} \cap X_{2})}^{≺} \supseteq X_{1}^{≺} \cup X_{2}^{≺}$ {\left({{X_1} \cap {X_2}} \right)^ \prec} \supseteq X_1^ \prec \cup X_2^ \prec , ${(B_{1} \cap B_{2})}^{≺} \supseteq B_{1}^{≺} \cup B_{2}^{≺}$ {\left({{B_1} \cap {B_2}} \right)^ \prec} \supseteq B_1^ \prec \cup B_2^ \prec

Theorem 1

If (X₁, B₁) and (X₂, B₂) are two concepts of the formal context of intuitionistic fuzzy language (U,A,S^′), then X₁ ∩ X₂,(B₁ ∪ B₂)^≺≺) and ((X₁ ∪ X₂)^≺≺, B₁ ∩ B₂) are all concepts.

Prove

It is evidently proved by the above nature and definition.

2.2

Attribute reduction in the formal context of intuitionistic fuzzy language

In the era of big data, the amount of information is huge. Under this context, the construction of concept lattice is very complicated, and it is thus necessary to reduce the attributes of concept lattice [23]. In this section, the related definitions and properties of attribute reduction in the context of intuitionistic fuzzy language form are obtained [24].

Definition 5

Define $IFLL (U, A_{1}, S_{1}^{'})$ IFLL\left({U,{A_1},S_1^{'}} \right) and $IFLL (U, A_{2}, S_{2}^{'})$ IFLL\left({U,{A_2},S_2^{'}} \right) as two conceptual lattices of intuitionistic fuzzy language; if $\forall (X^{'}, B^{'}) \in IFLL (U, A_{2}, S_{2}^{'})$ \forall \left({{X^{'}},{B^{'}}} \right) \in IFLL\left({U,{A_2},S_2^{'}} \right) , $\exists (X, B) \in IFLL (U, A_{1}, S_{1}^{'})$ \exists (X,B) \in IFLL\left({U,{A_1},S_1^{'}} \right) so that X = X^′, which is called $IFLL (U, A_{1}, S_{1}^{'})$ IFLL\left({U,{A_1},S_1^{'}} \right) , $IFLL (U, A_{2}, S_{2}^{'})$ IFLL\left({U,{A_2},S_2^{'}} \right) as: (8) $IFLL (U, A_{1}, S_{1}^{'}) \leq IFLL (U, A_{2}, S_{2}^{'})$ IFLL\left({U,{A_1},S_1^{'}} \right) \le IFLL\left({U,{A_2},S_2^{'}} \right)

If $IFLL (U, A_{1}, S_{1}^{'}) \leq IFLL (U, A_{2}, S_{2}^{'})$ IFLL\left({U,{A_1},S_1^{'}} \right) \le IFLL\left({U,{A_2},S_2^{'}} \right) and $IFLL (U, A_{1}, S_{1}^{'}) \geq IFLL (U, A_{2}, S_{2}^{'})$ IFLL\left({U,{A_1},S_1^{'}} \right) \ge IFLL\left({U,{A_2},S_2^{'}} \right) are established, it is called $IFLL (U, A_{1}, S_{1}^{'})$ IFLL\left({U,{A_1},S_1^{'}} \right) , and $IFLL (U, A_{2}, S_{2}^{'})$ IFLL\left({U,{A_2},S_2^{'}} \right) is isomorphic and regarded as $IFLL (U, A_{1}, S_{1}^{'}) ≅ IFLL (U, A_{2}, S_{2}^{'})$ IFLL\left({U,{A_1},S_1^{'}} \right) \cong IFLL\left({U,{A_2},S_2^{'}} \right) .

Under the formal context of intuitionistic fuzzy language (U,A,S^′), ∀D ⊆ A and $S_{D}^{'} = S^{'} \cap (U \times D)$ S_D^{'} = {S^{'}} \cap (U \times D) ; then, $(U, D, S_{D}^{'})$ \left({U,D,S_D^{'}} \right) is also a formal context of intuitionistic fuzzy language [25]. $IFLL (U, D, S_{D}^{'})$ IFLL\left({U,D,S_D^{'}} \right) refers to the formal context of expressing intuitionistic fuzzy language $(U, D, S_{D}^{'})$ \left({U,D,S_D^{'}} \right) Under the concept lattice, for operators X^≺(X ⊆ U), $\forall (X, B) \in IFLL (U, D, S_{D}^{'})$ \forall (X,B) \in IFLL\left({U,D,S_D^{'}} \right) , satisfy: if a ∈ D, X^≺D (a) = X^≺(a).

Theorem 2

Assuming (U,A,S^′) is the formal context of intuitionistic fuzzy language [26], $\forall (X, B) \in IFLL (U, D, S_{D}^{'})$ \forall (X,B) \in IFLL(U,D,S_D^{'}) , D ⊆ A yes B^≺ ⊆ B^≺D.

Proof

It can be easily proved by property 1 and definition 5.

Theorem 3

Assuming (U,A,S^′) is the formal context of intuitionistic fuzzy language, for ∀D ⊆ A, D ≠ ∅ there is always $IFLL (U, A, S^{'}) \leq IFLL (U, D, S_{D}^{'})$ IFLL\left({U,A,{S^{'}}} \right) \le IFLL\left({U,D,S_D^{'}} \right) .

Prove

$\forall (X, B) \in IFLL (U, D, S_{D}^{'})$ \forall (X,B) \in IFLL\left({U,D,S_D^{'}} \right) , X^≺D = B and B^≺D = X, and as can be seen from property 1, (X^≺≺, X^≺) ∈ IFLL (U,A,S^′), X ⊆ X^≺<. According to theorem 2, B = X^≺D ⊆ X^≺ ⇒ X^≺≺ ⊆ B^≺ ⊆ B^≺D = X. Therefore, IFLL(U,A,S^′) ≤ IFLL(U,D,Sb).

Definition 6

(U,A,S^′) is the formal context of intuitionistic fuzzy language; if ∃D ⊆ A, make $IFLL (U, D, S_{D}^{'}) ≅ IFLL (U, A, S^{'})$ IFLL\left({U,D,S_D^{'}} \right) \cong IFLL\left({U,A,{S^{'}}} \right) , and then d is a coordination set. If D is a coordination set and $IFLL (U, D - d, S_{D - d}^{'})$ IFLL\left({U,D - d,S_{D - d}^{'}} \right) , is different from IFLL (U,A,S^′), and then D is called (U,A,S^′). About Jane. All the intersection of (U,A,S^′)reduction is called the core of (U,A,S^′).

The following theorems can be obtained through definition 6:

Theorem 4

Assuming (U,A,S^′) is formal context of intuitionistic fuzzy language, D ⊆ A, D ≠ ∅ and then D is the coordination set. $\Leftrightarrow IFLL (U, D, S_{D}^{'}) \leq IFLL (U, A, S^{'})$ \Leftrightarrow IFLL\left({U,D,S_D^{'}} \right) \le IFLL\left({U,A,{S^{'}}} \right) .

Definition 7

Assumes that all reductions of the formal context of intuitionistic fuzzy language (U,A,S^′) are {D_iD_i isareduction, i ∈ τ} (τ is an index set) [27]. Attribute set a can be divided into the following four parts:

Absolutely necessary attribute (core attribute) b: $b \in \underset{i \in τ}{\cap} D_{i}$ b \in \mathop \cap \limits_{i \in \tau} {D_i}

Relative essential attribute C: $c \in \underset{i \in τ}{\cup} D_{i} - \underset{i \in τ}{\cup} D_{i}$ c \in \mathop \cup \limits_{i \in \tau} {D_i} - \mathop \cup \limits_{i \in \tau} {D_i}

Absolutely unnecessary attribute d: $d \in A - \underset{i \in τ}{\cup} D_{i}$ d \in A - \mathop \cup \limits_{i \in \tau} {D_i}

Unnecessary attribute E: $e \in A - \underset{i \in τ}{\cap} D_{i}$ e \in A - \mathop \cap \limits_{i \in \tau} {D_i}

Theorem 5

Under the formal context of any intuitionistic fuzzy language (U,A,S^′), there must be at least one reduction.

Prove

If for any a ∈ A, there are $IFLL (U, A - {a}, S_{A - {a}}^{'})$ IFLL\left({U,A - \{a\},S_{A - \{a\}}^{'}} \right) different from IFLL (U,A,S^′), then A itself is a reduction.

If it exists a ∈ A, make $IFLL (U, A - {a}, S_{A - {a}}^{'}) = IFLL (U, A, S^{'})$ IFLL\left({U,A - \{a\},S_{A - \{a\}}^{'}} \right) = IFLL\left({U,A,{S^{'}}} \right) ; if for any a₁ ∈ A − {a}, there are $IFLL (U, A - {a, a_{1}}, S_{A - {a, a_{1}}}^{'})$ IFLL\left({U,A - \left\{{a,{a_1}} \right\},S_{A - \left\{{a,{a_1}} \right\}}^{'}} \right) different from IFLL (U,A,S^′), then A−{a₁} is about Jane; if $IFLL (U, A - {a, a_{1}}, S_{A - {a, a_{1}}}^{'}) ≅ IFLL (U, A, S^{'})$ IFLL\left({U,A - \left\{{a,{a_1}} \right\},S_{A - \left\{{a,{a_1}} \right\}}^{'}} \right) \cong IFLL\left({U,A,{S^{'}}} \right) , A − {a,a₁} needs to be discussed. Since A is a finite set, at least one reduction will be found.

Generally, the reduction intuition blurs the formal context of (U,A,S^′) is not unique.

Example 2

For the formal context of intuitionistic fuzzy language shown in Table 2 (U,A,S^′), U = {x₁, x₂, x₃, x₄}, A = {a,b,c,d,e} and the concept lattice of intuitionistic fuzzy language and its reduction are obtained.

Table 2

Intuitionistic fuzzy linguistic for mal context (U,A,S^′)

U	a	b	c	d	e

x₁	〈s₄, s₀〉	〈s₃, s₁〉	〈s₁, s₂〉	〈s₃, s₀〉	〈s₃, s₀〉
x₂	〈s₃, s₀〉	〈s₃, s₁〉	〈s₃, s₀〉	〈s₁, s₂〉	〈s₁, s₃〉
x₃	〈s₀, s₃〉	〈s₁, s₂〉	〈s₀, s₄〉	〈s₃, s₁〉	〈s₁, s₂〉
x₄	〈s₃, s₁〉	〈s₃, s₀〉	〈s₂, s₀〉	〈s₁, s₂〉	〈s₁, s₃〉

The formal context of intuitionistic fuzzy language shown in Table 2 has nine concepts:

#1 ({U}, {s₀, s₃s₁, s₂s₀, s₄s₁, s₂s₁, s₃})

#2 ({x₁x₂x₄}, {s₃, s₁s₃, s₁s₁, s₂s₁, s₂s₁, s₃})

#3 ({x₁x₂}, {s₃, s₀s₃, s₁s₁, s₂s₁, s₂s₁, s₃})

#4 ({x₁x₃}, {s₀, s₃s₁, s₂s₀, s₄s₃, s₁s₁, s₂})

#5 ({x₂x₄}, {s₃, s₁s₃, s₁s₂, s₀s₁, s₂s₁, s₃})

#6 ({x₁}, {s₄, s₀s₃, s₁s₁, s₂s₄, s₀s₃, s₀})

#7 ({x₂}, {s₃, s₀s₃, s₁s₃, s₀s₁, s₂s₁, s₃})

#8 ({x₄}, {s₃, s₁s₃, s₀s₂, s₀s₁, s₂s₁, s₃})

#9 (∅, {s₃, s₁s₃, s₁s₁, s₂s₁, s₂s₁, s₃})

Its intuitionistic fuzzy language concept lattice IFLL (U,A,S^′) is shown in Figure 2.

Define D₁ = {a,b,c,d}, in the context of intuitionistic fuzzy language form $(U, D_{1}, S_{D_{1}}^{'})$ \left({U,{D_1},S_{{D_1}}^{'}} \right) ; the generated concept is

#1 (U, {s₀, s₃s₁, s₂s₀, s₄s₁, s₂})

#2 ({x₁x₂x₄}, {s₃, s₁s₃, s₁s₁, s₂s₁, s₂})

#3 ({x₁x₂}, {s₃, s₀s₃, s₁s₁, s₂s₁, s₂})

#4 ({x₁x₃}, {s₀, s₃s₁, s₂s₀, s₄s₃, s₁})

#5 ({x₂x₄}, {s₃, s₁s₃, s₁s₂, s₀s₁, s₂})

#6 ({x₁}, {s₄, s₀s₃, s₁s₁, s₂s₄, s₀})

#7 ({x₂}, {s₃, s₀s₃, s₁s₃, s₀s₁, s₂})

#8 ({x₄}, {s₃, s₁s₃, s₀s₂, s₀s₁, s₂})

#9 (∅, {s₃, s₁s₃, s₁s₁, s₂s₁, s₂})

Its intuitionistic fuzzy language concept lattice $IFLL (U, D_{1}, S_{D_{1}}^{'})$ IFLL\left({U,{D_1},S_{{D_1}}^{'}} \right) is shown in Figure 3.

Intuitionistic fuzzy linguistic concept lattice $IFLL (U, D_{1}, S_{D_{1}}^{'})$ IFLL\left({U,{D_1},S_{{D_1}}^{'}} \right)

As can be seen from Figures 2 and 3, $IFLL (U, D_{1}, S_{D_{1}}^{'}) ≅ IFLL (U, A, S^{'})$ IFLL\left({U,{D_1},S_{{D_1}}^{'}} \right) \cong IFLL\left({U,A,{S^{'}}} \right) but does not exist. d₁ ∈ D₁ to make $IFLL (U, D_{1} - {d_{1}}, S_{D_{1} - (d_{1})}^{'}) ≅ IFLL (U, A, S^{'})$ IFLL\left({U,{D_1} - \left\{{{d_1}} \right\},S_{{D_1} - \left({{d_1}} \right)}^{'}} \right) \cong IFLL\left({U,A,{S^{'}}} \right) , and thus D₁ = {a,b,c,d} is a reduction of (U,A,S^′). Similarly, D₂ = {a,b,c,e} is a reduction of (U,A,S^′).

To sum up, intuitionistic fuzzy language formal context of (U,A,S^′) has two reductions: D₁ = {a,b,c,d} and D₂ = {a,b,c,e}. That is, the core attributes are a, b and c, the relative necessary attributes are d and e, and there are no unnecessary attributes [28].

Corollary 1

Assuming that (U,A,S^′) is the formal context of intuitionistic fuzzy language, then the core attribute is reduction. ⇔ The reduction is the only one.

Corollary 2

Assuming that (U,A,S^′) is the formal context of intuitionistic fuzzy language, then a ∈ A is an unnecessary attribute. ⇔ A − {a} is a coordination set [29].

Corollary 3

Assuming that (U,A,S^′) is the formal context of intuitionistic fuzzy language, then a ∈ A is a core attribute. ⇔ A − {a} is not a coordination set.

The reduction D on the concept lattice of intuitionistic fuzzy language needs to satisfy the following two conditions:

D is a coordination set;

∀d ∈ D, D − {d} is not a coordination set.

Therefore, for attribute reduction, it is necessary to determine whether the attribute subset is a coordination set, and the judgment theorem of the coordination set is given as follows:

Theorem 6

It is assumed (U,A,S^′) that formal context of intuitionistic fuzzy language, D ⊆ A, D ≠ ∅. So, D is the coordination set ⇔ B^≺≺D≺D = B^≺, $\forall B \subseteq S_{A}^{'}$ \forall B \subseteq S_A^{'} .

Prove

Necessity: Assuming that D is a coordinated set, according to theorem 4, there is $IFLL (U, D, S_{D}^{'}) \leq IFLL (U, A, S^{'})$ IFLL\left({U,D,S_D^{'}} \right) \le IFLL(U,A,{S^{'}}) , and for $\forall B \subseteq S_{A}^{'}$ \forall B \subseteq S_A^{'} , (B^≺, B^≺≺) ∈ IFLL (U,A,S^′). According to definition 5, $\exists B^{'} \subseteq S_{A}^{'}$ \exists {B^{'}} \subseteq S_A^{'} satisfies $(B^{≺}, B^{'}) \in IFLL (U, D, S_{D}^{'})$ \left({{B^ \prec},{B^{'}}} \right) \in IFLL(U,D,S_D^{'}) . Therefore, B^′ = B^≺≺D and □^≺ = □^′≺□; therefore B^≺≺D≺D = B^≺.

Sufficiency: $\forall B \subseteq S_{A}^{'}$ \forall B \subseteq S_A^{'} , B^≺≺D≺D = B^≺. Asuming that (X,B) ∈ IFLL(U,A,S^′), that is, there is X^≺ = B, B^≺ = X is established. Take B^′ = B^≺≺D, B^′≺D = B^≺≺D≺D = B^≺ = X. Therefore, $(X, B^{'}) \in IFLL (U, D, S_{D}^{'})$ (X,{B^{'}}) \in IFLL(U,D,S_D^{'}) , that is, for ∀(X,B) ∈ IFLL(U,A,S^′), $\exists B^{'} \subseteq S_{A}^{'}$ \exists B' \subseteq S_A^{'} satisfies $(X, B^{'}) \in IFLL (U, D, S_{D}^{'})$ (X,{B^{'}}) \in IFLL(U,D,S_D^{'}) , and thus D is the coordination set.

Corollary 4

Assuming that (U,A,S^′) is the formal context of intuitionistic fuzzy language, D ⊆ C, D ≠ ∅. So D is the coordination set ⇔ $\forallB\subseteqS_{A}^{'}$ \forall B \subseteq S_A^{'} , $\exists B^{'} \subseteq S_{A}^{'}$ \exists {B^{'}} \subseteq S_A^{'} satisfy B^′≺D = B^≺.

Theorem 7

Assuming that (U,A,S^′) is the formal context of intuitionistic fuzzy language, D ⊆ A, D ≠ ∅, E = A − D. So, D is the coordination set ⇔ $\forall B \subseteq S_{E}^{'}$ \forall B \subseteq S_E^{'} , $\exists B^{'} \subseteq S_{D}^{'}$ \exists {B^{'}} \subseteq S_D^{'} satisfy B^′≺D = B^≺E.

Proof

Necessity: D is the coordination set of (U,A,S^′); then, for $\forall B_{1} \subseteq S_{A}^{'}$ \forall {B_1} \subseteq S_A^{'} , $B_{1}^{≺} = B_{1}^{≺_{D}}$ B_1^ \prec = B_1^{{\prec _D}} . With regard to $\forall B \subseteq S_{E}^{'}$ \forall B \subseteq S_E^{'} , set B₂ = B ∪ B₁, in which $B_{1} \subseteq S_{D}^{'}$ {B_1} \subseteq S_D^{'} , $B_{1}^{≺} = B^{≺_{B}}$ B_1^ \prec = {B^{{\prec _B}}} . According to corollary 4, $\exists B \subseteq S_{D}^{'}$ \exists B \subseteq S_D^{'} satisfies B^′≺D = B^≺E.

Sufficiency: For $\forall B \subseteq S_{E}^{'}$ \forall B \subseteq S_E^{'} , $\exists B \subseteq S_{D}^{'}$ \exists B \subseteq S_D^{'} satisfies B^′<D = B^≺E. Suppose (X,B₂) ∈ IFLL (U,A,S^′), then X^≺ = B₂ and $X = B_{2}^{≺}$ X = B_2^ \prec . Assume that B₂ = B ∪ B₁, in which $B \subseteq S_{E}^{'}$ B \subseteq S_E^{'} , $B_{1} \subseteq S_{D}^{'}$ {B_1} \subseteq S_D^{'} ; then B₂^≺ = (B ∪ B₁)^≺ = B^≺ ∩ B₁^≺ = B^≺D ∩ B₁^≺E. Therefore $\exists B^{'} \subseteq S_{D}^{'}$ \exists {B^{'}} \subseteq S_D^{'} satisfies $B^{' ≺_{D}} = B_{1}^{≺ E}$ {B^{'{\prec _D}}} = B_1^{\prec E} , and therefore B^≺D ∩ B₁^≺E = B₁^≺D ∩ B^′≺D = (B1 ∪ B^′)^≺D = X. $({(B_{1} \cup B^{'})}^{<_{D}}, {(B_{1} \cup B^{'})}^{<_{D} <_{D}}) \in IFLL (U, D, S_{D}^{'})$ \left({{{\left({{B_1} \cup {B^{'}}} \right)}^{{< _D}}},{{\left({{B_1} \cup {B^{'}}} \right)}^{{< _D}{< _D}}}} \right) \in IFLL\left({U,D,S_D^{'}} \right) , and thus $IFLL (U, D, S_{D}^{'}) \leq IFLL (U, A, S^{'})$ IFLL\left({U,D,S_D^{'}} \right) \le IFLL\left({U,A,{S^{'}}} \right) .

To sum up, D is the coordination set of (U,A,S^′).

2.3

Attribute reduction of intuitionistic fuzzy language concept lattice based on discernibility matrix

Discernibility matrix is a common tool for attribute reduction. The identification matrix and identification function are used to reduce the concept lattice of intuitionistic fuzzy language [30].

Definition 8

Assume that (U,A,S^′) is the formal context of intuitionistic fuzzy language, (X_i,B_i) and (X_j,B_j), and (X_i,B_i),(X_j,B_j) ∈ IFLL (U,A,S^′). Then, (9) $\tilde{D} ((X_{i}, B_{i}), (X_{j}, B_{j})) = {\begin{array}{l} {a \in A μ_{X_{1}} (a) > μ_{X_{j}} (a) {orv}_{X_{i}} (a) < v_{X_{j}} (a)}, X_{i} \subset X_{j}, X_{i}, X_{j} \neq U, \emptyset \\ \emptyset else \end{array}$ \tilde D\left({\left({{X_i},{B_i}} \right),\left({{X_j},{B_j}} \right)} \right) = \left\{{\matrix{{\left\{{a \in A{\mu _{{X_1}}}(a) > {\mu _{{X_j}}}(a)or{v_{{X_i}}}(a) < {v_{{X_j}}}(a)} \right\},{X_i} \not\subset {X_j},{X_i},{X_j} \ne U,\emptyset} \hfill \cr {\emptyset \quad {\rm{else}}} \hfill \cr}} \right.

${\tilde{D}}^{R} = (\tilde{D} ((X_{i}, B_{i}), (X_{j}, B_{j})) (X_{j}, B_{j})), (X_{i}, B_{i}), (X_{j}, B_{j}) \in IFLL (U, A, S^{'}))$ {\tilde D^R} = \left. {\left({\tilde D\left({\left({{X_i},{B_i}} \right),\left({{X_j},{B_j}} \right)} \right)\left({{X_j},{B_j}} \right)} \right),\left({{X_i},{B_i}} \right),\left({{X_j},{B_j}} \right) \in IFLL\left({U,A,{S^{'}}} \right)} \right) is regarded as an identifiable matrix on the formal context of intuitionistic fuzzy language.

Theorem 8

Assume that (U,A,S^′) is the formal context of intuitionistic fuzzy language, D ⊆ A, D ≠ ∅, ∀ (X_i,B_i), (X_j,B_j) ∈ IFLL (U,A,S^′), (X_i,B_i) ≠ (X_j,B_j); then the following two propositions are equivalent: (1)

D is the coordination set.

(2)

If $\tilde{D} ((X_{i}, B_{i}), (X_{j}, B_{j})) \neq \emptyset$ \tilde D\left({\left({{X_i},{B_i}} \right),\left({{X_j},{B_j}} \right)} \right) \ne \emptyset , then $D \cap \tilde{D} ((X_{i}, B_{i}), (X_{j}, B_{j})) \neq \emptyset$ D\bigcap\nolimits_ \tilde D\left({\left({{X_i},{B_i}} \right),\left({{X_j},{B_j}} \right)} \right) \ne \emptyset , $\tilde{D} ((X_{i}, B_{i}), (X_{j}, B_{j})) \in {\tilde{D}}^{R}$ \tilde D\left({\left({{X_i},{B_i}} \right),\left({{X_j},{B_j}} \right)} \right) \in {\tilde D^R}

Proof

(1)⇒ (2): Assuming that (2) does not hold true, $\exists \tilde{D} ((X_{i}, B_{i}), (X_{j}, B_{j})) \neq \emptyset \in {\tilde{D}}^{R}$ \exists \tilde D\left({\left({{X_i},{B_i}} \right),\left({{X_j},{B_j}} \right)} \right) \ne \emptyset \in {\tilde D^R} , X_i ⊄ X_j satisfy $D \cap \tilde{D} ((X_{i}, B_{i}), (X_{j}, B_{j})) = \emptyset$ D\bigcap\nolimits_ \tilde D\left({\left({{X_i},{B_i}} \right),\left({{X_j},{B_j}} \right)} \right) = \emptyset . That is, ∀a ∈ D satisfies μ _{X_i}(a) > μ _{X_j}(a).

Definition 9

Assume that (U,A,S^′) is the intuitionistic fuzzy linguistic form context; D^*R is the identification matrix on (U,A,S^′); define (10) $\begin{matrix} f (\tilde{D} R) = \overset{\land}{\tilde{D} ((X_{i}, B_{i}), (X_{j}, B_{j})) \in {\tilde{D}}^{R}} {\overset{\lor}{a_{k} \in \tilde{D} ((X_{i}, B_{i}), (X_{j}, B_{j}))} a_{k}} \\ \tilde{D} ((X_{i}, B_{i}), (X_{j}, B_{j}) \neq \emptyset \end{matrix}$ \matrix{{f(\tilde DR) = \mathop {\tilde D(({X_i},{B_i}),({X_j},{B_j})) \in {{\tilde D}^R}}\limits^ \wedge \left\{{\mathop {{a_k} \in \tilde D(({X_i},{B_i}),({X_j},{B_j}))}\limits^ \vee {a_k}} \right\}} \cr {\tilde D(({X_i},{B_i}),({X_j},{B_j}) \ne \emptyset} \cr}

Then, $f ({\tilde{D}}^{R})$ f\left({{{\tilde D}^R}} \right) is the identification function of (U,A,S^′).

Theorem 9

Assume that (U,A,S^′) is the formal context of intuitionistic fuzzy language, and the minimal disjunctive normal form of the discernibility function is defined as (11) $Λ ({\tilde{D}}^{R}) = \begin{matrix} p \\ \lor \\ k = 1 \end{matrix} [\begin{matrix} q_{k} \\ \land a_{r} \\ r = 1 \end{matrix}]$ \Lambda \left({{{\tilde D}^R}} \right) = \matrix{p \cr \vee \cr {k = 1} \cr} \left[ {\matrix{{{q_k}} \cr {\wedge {a_r}} \cr {r = 1} \cr}} \right]

Make B_k = {a_rr = 1,2, ⋯ , q_k}; then, {B_kk = 1,2, ⋯ , p} are all reductions of intuitionistic fuzzy language form context (U,A,S^′).

It can be seen from the above that to find the attribute reduction of the conceptual lattice of intuitionistic fuzzy language, the following conditions need to be obtained:

The smallest coordination set D of $D \cap \tilde{D} ((X_{i}, B_{i}), (X_{j}, B_{j})) = \emptyset$ D\bigcap\nolimits_ \tilde D\left({\left({{X_i},{B_i}} \right),\left({{X_j},{B_j}} \right)} \right) = \emptyset , and $\tilde{D} ((X_{i}, B_{i}), (X_{j}, B_{j})) = \emptyset$ \tilde D\left({\left({{X_i},{B_i}} \right),\left({{X_j},{B_j}} \right)} \right) = \emptyset .

Corollary 5

D is (U,A,S^′) coordination set on ⇔ X^≺₀D ⊆ X^α<.

Table 3 gives the intuitionistic fuzzy language form context of (U,A,S^′) identifiable matrix.

Table 3

Discernible matrix of Intuitionistic fuzzy linguistic formal context (U,A,S^′)

□_□/□_□	□₁	□₂	□₄	□₁□₂	□₁□₃	□₂□₄	□₁□₂□₄

□₁		ade	ade			ade
□₂	c		ac		abc
□₄	bc	b		bc	abc
□₁□₂			a		abc	a
□₁□₃		de	de	de		de	de
□₂□₄	c			c	abc
□₁□₂□₄					abc

Note. Since ∅ is not considered in the process of attribute reduction, the ∅ is omitted in the identification matrix. $\begin{array}{l} Λ ({\tilde{D}}^{*}) & = (a \lor d \lor e) \land (b \lor c) \land (a \lor c) \land (a \lor b \lor c) \land (d \lor e) \land a \land b \land c = a \land b \land c \land (d \lor e) \\ = (a \land b \land c \land d) \lor (a \land b \land c \land e) \end{array}$ \matrix{{\Lambda \left({{{\tilde D}^*}} \right)} \hfill & {= (a \vee d \vee e) \wedge (b \vee c) \wedge (a \vee c) \wedge (a \vee b \vee c) \wedge (d \vee e) \wedge a \wedge b \wedge c = a \wedge b \wedge c \wedge (d \vee e)} \hfill \cr {} \hfill & {= (a \wedge b \wedge c \wedge d) \vee (a \wedge b \wedge c \wedge e)} \hfill \cr}

The formal context of intuitionistic fuzzy language shown in Table 3 has two reductions, namely: and. Among them, a, b and c, are core essential attributes; d and e are relatively necessary attributes. These observations are consistent with the conclusion of example 2.

Rule extraction method based on intuitionistic fuzzy language concept lattice

Rule extraction is an important branch in the field of concept lattice. In the era of big data, knowledge acquired from complex data is largely given in the form of rules [31]. Based on the formal context of intuitionistic fuzzy language, this paper puts forward the formal context of intuitionistic fuzzy language decision, studies the acquisition method of rules, discusses the similarity between intuitionistic fuzzy language sets and constructs a rule extraction method based on concept lattice of intuitionistic fuzzy language.

3.1

Rule extraction under the decision formal context of intuitionistic fuzzy language

Definition 10

Assume that (U,A,S^′, T,R^′) is a decision formal context of intuitionistic fuzzy language, in which (U,A,S^′) and (U,T,R^′) are the context of intuitionistic fuzzy language, A ∩ T = ∅, S^′ is Intuitionistic fuzzy language binary relation between U ×A, R^′ is intuitionistic fuzzy language binary relation between U × T, A is called conditional attribute set and T is called decision attribute set.

Under the decision formal context of intuitionistic fuzzy language (U,A,S^′, T,R^′), $R_{T}^{'}$ R_T^{'} A set of intuitionistic fuzzy language values representing the object set U under the decision attribute set T [32].

Definition 11

Assuming that (U,A,S^′, T,R^′) is an intuitionistic fuzzy language decision formal context, if IFLL (U,A,S^′) ≤ IFLL (U,T,R^′), the context is regarded to be coordinated.

IFLL (U,A,S^′) is a conditional concept lattice of intuitionistic fuzzy language, whereas IFLL (U,T,R^′) is a decision concept lattice for intuitionistic fuzzy language.

Definition 12

Assuming that (U,A,S^′, T,R^′) is a decision formal context of intuitionistic fuzzy language, IFLL (U,A,S^′) ≤ IFLL (U,T,R^′); if (X,B) ∈ IFLL (U,A,S^′), (Y,C) ∈ IFLL (U,T,R^′), satisfy X = Y (X ≠ U, ∅), it is called B→ C as the rule derived under the context of decision, which is recorded as if B, then C, and all the rules are recorded in Φ in (a).

3.2

Similarity of language values in the context of intuitionistic fuzzy language decision formal

Definition 13

Assuming that (U,A,S^′) is a formal context of intuitionistic fuzzy language, for ∀X ⊆ U $B, G \subseteq S_{A}^{'}$ B,G \subseteq S_A^{'} , (X,B) ∈ IFLL (U,A,S^′) then the distance between two intuitionistic fuzzy language sets B = {s_α₁, s_β₁ s_α₂, s_β₂ ⋯ s_{α_m}, s_{β_m}} and G = {s_γ₁, s_λ₁ s_γ₂, s_λ₂ ⋯ s_{γ_m}, s_{λ_m}} is: (12) $d (B, G) = \frac{1}{m} \sum_{l = 1}^{m} \frac{| α_{l} - γ_{l} | + | β_{l} - λ_{l} |}{2 g}$ d(B,G) = {1 \over m}\mathop {\sum\limits_{l = 1}^m}{{\left| {{\alpha _l} - {\gamma _l}} \right| + \left| {{\beta _l} - {\lambda _l}} \right|} \over {2g}}

Among them, l = 1, ⋯ , m.

Theorem 10

Assuming that B = {s_α₁, s_β₁ s_α₂, s_β₂ ⋯ s_{α_m}, s_{β_m}}, G = {s_γ₁, s_λ₁ s_γ₂, s_λ₂ ⋯ s_{γ_m}, s_{λ_m}} for any two intuitionistic fuzzy language sets, the distance d(B,G) of language sets in the context of intuitionistic fuzzy language forms satisfies the following properties:

0 ≤ d(B,G) ≤ 1;

d(B,G) = 0 if and only if B = G B=G;

d(B,G) = d(G,B);

Assume Q = {〈s_κ₁, s_ρ₁〉, 〈s_κ₂, _ρ₂〉 ⋯ 〈s_{κ_m}, s_{ρ_m}〉} as an intuitionistic fuzzy language set, B ⊆ G ⊆ Q, then d(B,G) ≤ d(B,Q), d(G,Q) ≤ d(B,Q).

Proof

α_l,β_l,κ_l,ρ_l ∈ [0, g]

|α_l − γ_l| ∈ [0, g],|β_l − λ_l| ∈ [0, g],|α_l − γ_l| + |β_l − λ_l| ∈ [0,2g],

then $0 \leq \frac{| α_{l} - γ_{l} | + | β_{l} - λ_{l} |}{2 g} \leq 1 \Rightarrow 0 \leq \frac{1}{n} \sum_{l = 1}^{m} \frac{| α_{l} - γ_{l} | + | β_{l} - λ_{l} |}{2 g} \leq 1$ 0 \le {{\left| {{\alpha _l} - {\gamma _l}} \right| + \left| {{\beta _l} - {\lambda _l}} \right|} \over {2g}} \le 1 \Rightarrow 0 \le {1 \over n}\sum\limits_{l = 1}^m {{\left| {{\alpha _l} - {\gamma _l}} \right| + \left| {{\beta _l} - {\lambda _l}} \right|} \over {2g}} \le 1

$d (B, G) = 0 \Leftrightarrow \frac{1}{n} \sum_{l = 1}^{m} \frac{| α_{l} - γ_{l} | + | β_{l} - λ_{l} |}{2 g} = 0$ d(B,G) = 0 \Leftrightarrow {1 \over n}\sum\limits_{l = 1}^m {{\left| {{\alpha _l} - {\gamma _l}} \right| + \left| {{\beta _l} - {\lambda _l}} \right|} \over {2g}} = 0

$\Leftrightarrow \frac{| α_{l} - γ_{l} | + | β_{l} - λ_{l} |}{2 g} = 0$ \Leftrightarrow {{\left| {{\alpha _l} - {\gamma _l}} \right| + \left| {{\beta _l} - {\lambda _l}} \right|} \over {2g}} = 0

⇔ |α_l − γ_l| = 0 and |β_l − λ_l| = 0

⇔α_l = γ_l and β_l = λ_l

⇔B = G

$d (B, G) = \frac{1}{n} \sum_{l = 1}^{m} \frac{| α_{l} - γ_{l} | + | β_{l} - λ_{l} |}{2 g} = \frac{1}{n} \sum_{l = 1}^{m} \frac{| γ_{l} - α_{l} | + | λ_{l} - β_{l} |}{2 g} = d (G, B)$ d(B,G) = {1 \over n}\sum\limits_{l = 1}^m {{\left| {{\alpha _l} - {\gamma _l}} \right| + \left| {{\beta _l} - {\lambda _l}} \right|} \over {2g}} = {1 \over n}\sum\limits_{l = 1}^m {{\left| {{\gamma _l} - {\alpha _l}} \right| + \left| {{\lambda _l} - {\beta _l}} \right|} \over {2g}} = d(G,B)

If B ⊆ G ⊆ Q, yes α_l ≤ γ_l ≤ κ₁, β_l ≥ λ_l ≥ ρ_l, |α_l − γ_l| ≤ |α_l − κ_l|, |β_l − λ_l| ≤ |β_l − ρ_l|, then $d (B, G) = \frac{1}{n} \sum_{l = 1}^{m} \frac{| α_{l} - γ_{l} | + | β_{l} - λ_{l} |}{2 g} \leq \frac{1}{n} \sum_{l = 1}^{m} \frac{| α_{l} - κ_{l} | + | β_{l} - ρ_{l} |}{2 g} = d (B, Q)$ d(B,G) = {1 \over n}\sum\limits_{l = 1}^m {{\left| {{\alpha _l} - {\gamma _l}} \right| + \left| {{\beta _l} - {\lambda _l}} \right|} \over {2g}} \le {1 \over n}\sum\limits_{l = 1}^m {{\left| {{\alpha _l} - {\kappa _l}} \right| + \left| {{\beta _l} - {\rho _l}} \right|} \over {2g}} = d(B,Q)

Similarly, d(GQ)≤ d(B,Q) can be proved.

To sum up, theorem 10 is realised.

Definition 14

Assuming that (U,A,S^′) is a formal context of intuitionistic fuzzy language, for ∀X ⊆ U, $B, G \subseteq S_{A}^{'}$ B,G \subseteq S_A^{'} , (X,B) ∈ IFLL (U,A,S^′), then the similarity between two intuitionistic fuzzy languages is B = {〈s_α₁, s_β₁〉〈 s_α₂, s_β₂〉 ⋯ 〈 s_{α_m}, s_{β_m}〉} and G = {〈s_γ₁, s_λ₁〉〈s_γ₂, s_λ₂〉 ⋯ 〈 s_{γ_m}, s_{λ_m}〉}: (13) ${sim}_{II} (B, G) = 1 - d (B, G) = 1 - \frac{1}{m} \sum_{l = 1}^{m} \frac{| α_{l} - γ_{l} | + | β_{l} - λ_{l} |}{2 g}$ {\rm{si}}{{\rm{m}}_{{\rm{II}}}}(B,G) = 1 - d(B,G) = 1 - {1 \over m}\sum\limits_{l = 1}^m {{\left| {{\alpha _l} - {\gamma _l}} \right| + \left| {{\beta _l} - {\lambda _l}} \right|} \over {2g}}

Theorem 11

Assuming that B = {〈s_α₁, s_β₁〉〈s_α₂, s_β₂〉 ⋯ 〈s_{α_m}, s_{β_m}〉}, G = {〈s_γ₁, s_λ₁〉〈s_γ₂, s_λ₂〉 ⋯ 〈s_{γ_m}, s_{λ_m}〉}: are any two intuitionistic fuzzy language sets, then the similarity of language sets in the context of intuitionistic fuzzy language forms. sim_II(B,G) satisfying the following properties:

0 ≤ sim_II(B,G) ≤ 1;

sim_II(B,G) = 1 if and only if B=G;

sim_II(B,G) = sim_II(G,B);

Assume Q = {〈s_κ₁, s_ρ₁〉, 〈s_κ₂, _ρ₂〉 ⋯ 〈s_{κ_m}, s_{ρ_m}〉} as any intuitionistic fuzzy language set, B ⊆ G ⊆ Q, then sim_II(B,G) ≥ sim_II(B,Q), sim_II(G,Q) ≥ sim_II(B,Q).

Definition 15

Assuming that (U,A,S^′) is a formal context of intuitionistic fuzzy language, for ∀X ⊆ U, $B, G \subseteq S_{A}^{'}$ B,G \subseteq S_A^{'} , (X,B) ∈ IFLL (U,A,S^′), w_II = (w₁, w₂, …,w_m)^T is the weight vector then the weighted distance between two intuitionistic fuzzy language sets B = {〈s_α₁, s_β₁〉〈s_α₂, s_β₂〉 ⋯ 〈s_{α_m}, s_{β_m}〉} and G = {〈s_γ₁, s_λ₁〉〈s_γ₂, s_λ₂〉 ⋯ 〈s_{γ_m}, s_{λ_m}〉} is: (14) $d_{w} (B, G) = \sum_{l = 1}^{m} \frac{w_{l} (| α_{l} - γ_{l} | + | β_{l} - λ_{l} |)}{2 g}$ {d_w}(B,G) = \sum\limits_{l = 1}^m {{{w_l}\left({\left| {{\alpha _l} - {\gamma _l}} \right| + \left| {{\beta _l} - {\lambda _l}} \right|} \right)} \over {2g}} where w_l ∈ [0,1] and $\sum_{l = 1}^{m} w_{l} = 1 (l = 1, \dots, m)$ \sum\limits_{l = 1}^m {w_l} = 1(l = 1, \cdots,m) .

Theorem 12

Assuming B = {〈s_α₁, s_β₁〉〈s_α₂, s_β₂〉 ⋯ 〈s_{α_m}, s_{β_m}〉}, G = {〈s_γ₁, s_λ₁〉〈s_γ₂, s_λ₂〉 ⋯ 〈s_{γ_m}, s_{λ_m}〉} as any two intuitionistic fuzzy language sets, w_II = (w₁, w₂, …,w_m) is the weight vector, where w_l ∈ [0,1] and $\sum_{l = 1}^{m} w_{l} = 1 (l = 1, \dots, m)$ \sum\limits_{l = 1}^m {w_l} = 1(l = 1, \cdots,m) , and the weighted distance of language set in the context of language form is intuitively fuzzy. d_w(B,G) satisfies the following properties:

0 ≤ d_w(B,G) ≤ 1;

dw(B,G)=0 if and only if B=G;

d_w(B,G)=d_w(GB);

Assume Q = {〈s_κ₁, s_ρ₁〉, 〈s_κ₂, _ρ₂〉 ⋯ 〈s_{κ_m}, s_{ρ_m}〉} as any intuitionistic fuzzy language set, B ⊆ G ⊆ Q, then: d_w(B,G) ≤ d_w(B,Q), d_w(G,Q) ≤ d_w(B,Q).

Definition 16

Assuming that (U,A,S^′) is a formal context of intuitionistic fuzzy language, for ∀X ⊆ U, $B, G \subseteq S_{A}^{'}$ B,G \subseteq S_A^{'} , (X,B) ∈ IFLL (U,A,S^′), w_II = (w₁, w₂, …,w_m)^T is the weight vectorthen, the weighted similarity between two intuitionistic fuzzy language sets B = {〈s_α₁, s_β₁〉〈s_α₂, s_β₂〉 ⋯ 〈s_{α_m}, s_{β_m}〉} and G = {〈s_γ₁, s_λ₁〉〈s_γ₂, s_λ₂〉 ⋯ 〈s_{γ_m}, s_{λ_m}〉} is: (15) ${sim}_{w_{I}} (B, G) = 1 - d_{w} (B, G) = 1 - \sum_{l = 1}^{n} \frac{ω_{l} (| α_{l} - γ_{l} | + | β_{l} - λ_{l} |)}{2 g}$ {\rm{si}}{{\rm{m}}_{{w_{\rm{I}}}}}(B,G) = 1 - {d_w}(B,G) = 1 - \sum\limits_{l = 1}^n {{{\omega _l}\left({\left| {{\alpha _l} - {\gamma _l}} \right| + \left| {{\beta _l} - {\lambda _l}} \right|} \right)} \over {2g}} where w_l ∈ [0,1] and $\sum_{l = 1}^{m} w_{l} = 1 (l = 1, \dots, m)$ \sum\limits_{l = 1}^m {w_l} = 1(l = 1, \cdots,m) .

Theorem 13

Assuming that B = {〈s_α₁, s_β₁〉〈s_α₂, s_β₂〉 ⋯ 〈s_{α_m}, s_{β_m}〉}, G = {〈s_γ₁, s_λ₁〉〈s_γ₂, s_λ₂〉 ⋯ 〈s_{γ_m}, s_{λ_m}〉}: are any two intuitionistic fuzzy language sets, w_II = (w₁, w₂, …,w_m) is the weight vector, where w_l ∈ [0,1] and $\sum_{l = 1}^{m} w_{l} = 1 (l = 1, \dots, m)$ \sum\limits_{l = 1}^m {w_l} = 1(l = 1, \cdots,m) . Then the weighted similarity of language sets in the context of intuitionistic fuzzy language forms sim_w_II(B,G) satisfying the following properties:

0 ≤ sim_wI(B,G) ≤ 1;

sim_wII(B,G) = 1 if and only if B=G;

sim_wI(B,G) = sim_wI(G,B);

Assume Q = {〈s_κ₁, s_ρ₁〉〈s_κ₂, _ρ₂〉 ⋯ 〈s_{κ_m}, s_{ρ_m}〉} as any intuitionistic fuzzy language set, B ⊆ G ⊆ Q, then:

{sim}_{W_{I}} (B, G) \geq {sim}_{W_{I}} (B, Q), {sim}_{W_{II}} (G, Q) \geq {sim}_{W_{I}} (B, Q)

{\rm{si}}{{\rm{m}}_{{W_{\rm{I}}}}}(B,G) \ge {\rm{si}}{{\rm{m}}_{{W_{\rm{I}}}}}(B,Q),{\rm{si}}{{\rm{m}}_{{W_{{\rm{II}}}}}}(G,Q) \ge {\rm{si}}{{\rm{m}}_{{W_{\rm{I}}}}}(B,Q)

Note. When each element in the weight vector is evenly distributed, that is $w_{1} = w_{2} = \dots = w_{m} = \frac{1}{m}$ {w_1} = {w_2} = \ldots = {w_m} = {1 \over m} , the weighted similarity degenerates into similarity, which means that similarity is a special case of weighted similarity.

3.3

A rule extraction method based on concept lattice of intuitionistic fuzzy language

Under the context of intuitionistic fuzzy language decision formal, language rules are extracted. First, the conditional concept lattice and decision concept lattice are constructed; then, according to the detailed relationship between concept lattices, the rule set under the context of intuitionistic fuzzy language decision formal is obtained. Finally, the given intuitionistic fuzzy language set is compared with each rule antecedent in the rule set, and according to the formula of weighted similarity, the rules of a given intuitionistic fuzzy language set are judged. The specific steps are as follows:

Step 1: Generate all concepts (X,B) on the formal context (U,A,S^′) of intuitionistic fuzzy language to construct conditional concept lattice. IFLL (U,A,S^′);

Step 2: Generate all concepts (Y,C) on the formal context (U,T,R^′) of intuitionistic fuzzy language to construct decision concept lattice. IFLL (U,T,R^′);

Step 3: Judge the relationship between the two concept lattices IFLL (U,A,S^′) and IFLL (U,T,R^′). According to X = Y, all rules under the context of intuitionistic fuzzy language decision form are obtained, and the satisfied rules are added to the rule set. Φ(A);

Step 4: Construct a given intuitionistic fuzzy language set: $G = {〈 s_{γ_{1}}, s_{λ_{1}} 〉〈 s_{γ_{2}}, s_{λ_{2}} 〉 \dots 〈 s_{γ_{m}}, s_{λ_{m}} 〉}$ G = \left\{{\left\langle {{s_{{\gamma _1}}},{s_{{\lambda _1}}}} \right\rangle \left\langle {{s_{{\gamma _2}}},{s_{{\lambda _2}}}} \right\rangle \ldots \left\langle {{s_{{\gamma _m}}},{s_{{\lambda _m}}}} \right\rangle} \right\}

Combine intuitionistic fuzzy language set G with rule set Φ(A). Compare the antecedent B of each rule in this set; if there is B=G and B→ C, the algorithm ends and G→ C is output; if there is no B=G, proceed to step 5;

Step 5: According to the weighted similarity formula: (16) ${sim}_{w_{I}} (B, G) = 1 - \sum_{l = 1}^{m} \frac{ω_{l} (| α_{l} - γ_{l} | + | β_{l} - λ_{l} |)}{2 g}$ {\rm{si}}{{\rm{m}}_{{w_{\rm{I}}}}}(B,G) = 1 - \sum\limits_{l = 1}^m {{{\omega _l}\left({\left| {{\alpha _l} - {\gamma _l}} \right| + \left| {{\beta _l} - {\lambda _l}} \right|} \right)} \over {2g}}

Calculate the weighted similarity between intuitionistic fuzzy language set G and antecedent B in each rule, so as to obtain the maximum weighted similarity;

Step 6: Compare the maximum weighted similarity with the similarity threshold; if it is greater than or equal to the similarity threshold, then the rule B^′→ C^′ corresponding to the maximum weighted similarity is the rule to be searched, then G→ C^′, the algorithm ends, and G→ C^′ is output. If the maximum weighted similarity is less than the similarity threshold, return to step 1 and add the context until the matching rule is found.

3.4

Examples of disease diagnosis based on rule extraction method of intuitionistic fuzzy language concept lattice

By the symptoms and diseases of some patients, the possibility of a patient with certain symptoms suffering from these diseases can be judged. Assuming that (U,A,S^′, T,R^′) is a formal context of intuitionistic fuzzy language decision, U = {x₁, x₂, x₃, x₄} represents patient set, A = {a,b,c,d,e} represents symptom set and T = {t₁, t₂, t₃, t₄} represents disease set. The specific definitions of each of the symptoms under set A are as follows: a is fever, b is headache, c is stomach-ache, d is cough and e is chest pain. The specific definition of disease T is as follows: t₁ is viral fever, t₂ is typhoid fever, t₃ is gastropathy and t₄ is thoracic lung disease. The language set S = {s₀ = extremely few, s₁ = few, s₂ = medium, s₃ = many, S₄ = extremely many} indicates the frequency of certain symptoms in patients, and the linguistic set R = { r₀ = small, r₁ = small, r₂ = medium, r₃ = large, r₄ = extremely large} indicates that patients suffer from certain diseases. The formal context of intuitionistic fuzzy language decision is shown in Table 4.

Table 4

Intuitionistic fuzzy linguistic formal decision context (U,A,S^′, T,R^′)

	a	b	c	d	e	t₁	t₃	t₃	t₄

x₁	〈s₃, s₁〉	〈s₂, s₂〉	〈s₀, s₄〉	〈s₂, s₂〉	〈s₂, s₁〉	〈r₃, r₀〉	〈r₂, r₂〉	〈r₀, r₄〉	〈r₂, r₁〉
x₂	〈s₄, s₀〉	〈s₂, s₁〉	〈s₄, s₀〉	〈s₃, s₀〉	〈s₃, s₁〉	〈r₄, r₀〉	〈r₃, r₁〉	〈r₄, r₀〉	〈r₃, r₁〉
x₃	〈s₁, s₂〉	〈s₃, s₁〉	〈s₂, s₁〉	〈s₀, s₃〉	〈s₁, s₂〉	〈r₁, r₂〉	〈r₂, r₂〉	〈r₂, r₁〉	〈r₁, r₂〉
x₄	〈s₂, s₂〉	〈s₄, s₀〉	〈s₃, s₁〉	〈s₄, s₀〉	〈s₀, s₃〉	〈r₃, r₁〉	〈r₄, r₀〉	〈r₃, r₀〉	〈r₀, r₃〉

Step 1: Construct a conditional concept lattice of intuitionistic fuzzy language;

According to Table 4, the concepts under the context of (U,A,S^′) form are as follows:

#1 (∅,{〈s₄, s₀〉, 〈s₄, s₀〉〈s₄, s₀〉〈s₄, s₀〉〈s₃, s₁〉}),

#2 ({x₂}, {〈s₄, s₀〉〈s₂, s₁〉〈s₄, s₀〉〈s₃, s₀〉〈s₃, s₁〉}),

#3 ({x₃}, {〈s₁, s₂〉〈s₃, s₁〉〈s₂, s₁〉〈s₀, s₃〉〈s₁, s₂〉}),

#4 ({x₄}, {〈s₂, s₂〉〈s₄, s₀〉〈s₃, s₁〉〈s₄, s₀〉〈s₀, s₃〉}),

#5 ({x₁x₂}, {〈s₃, s₁〉〈s₂, s₂〉〈s₀, s₄〉〈s₂, s₂〉〈s₂, s₁〉}),

#6 ({x₂x₃}, {〈s₁, s₂〉〈s₂, s₁〉〈s₂, s₁〉〈s₀, s₃〉〈s₁, s₂〉}),

#7 ({x₂x₄}, {〈s₂, s₂〉〈s₂, s₁〉〈s₃, s₁〉〈s₃, s₀〉〈s₀, s₃〉}),

#8 ({x₃x₄}, {〈s₁, s₂〉〈s₃, s₁〉〈s₂, s₁〉〈s₀, s₃〉〈s₀, s₃〉}),

#9 ({x₁x₂x₃}, {〈s₁, s₂〉〈s₂, s₂〉〈s₀, s₄〉〈s₀, s₃〉〈s₁, s₂〉}),

#10 ({x₁x₂x₄}, {〈s₂, s₂〉〈s₂, s₂〉〈s₀, s₄〉〈s₂, s₂〉〈s₀, s₃〉}),

#11 ({x₂x₃x₄}, {〈s₁, s₂〉〈s₂, s₁〉〈s₂, s₁〉〈s₀, s₃〉〈s₀, s₃〉}),

#12 (U,{〈s₁, s₂〉〈s₂, s₂〉〈s₀, s₄〉〈s₀, s₃〉〈s₀, s₃〉}).

Construct conditional concept lattice of intuitionistic fuzzy language according to concepts IFLL (U,A,S^′), as indicated in Figure 4.

Intuitionistic fuzzy linguistic conditional concept lattice IFLL (U,A,S^′)

Step 2: Construct the concept lattice of intuitionistic fuzzy language decision;

The concepts under the formal context (U,T,R^′) are as follows:

#1 (∅, {〈r₄, r₀〉〈r₄, r₀〉〈r₄, r₀〉〈r₃, r₁〉}), #2 ({x₂}, {〈r₄, r₀〉〈r₃, r₁〉〈r₄, r₀〉〈r₃, r₁〉})

#3 ({x₄}, {〈r₃, r₁〉〈r₄, r₀〉〈r₃, r₀〉〈r₀, r₃〉}), #4 ({x₁x₂}, {〈r₃, r₀〉〈r₂, r₂〉〈r₀, r₄〉〈r₂, r₁〉}),

#5 ({x₂x₃}, {〈r₁, r₂〉〈r₂, r₂〉〈r₂, r₁〉〈r₁, r₂〉}), #6 ({x₂x₄}, {〈r₃, r₁〉〈r₃, r₁〉〈r₃, r₀〉〈r₀, r₃〉})

#7 ({x₁x₂x₃}, {〈r₁, r₂〉〈r₂, r₂〉〈r₀, r₄〉〈r₁, r₂〉}), #8 ({x₁x₂x₄}, {〈r₃, r₁〉〈r₂, r₂〉〈r₀, r₄〉〈r₀, r₃〉})

#9 ({x₂x₃x₄}, {〈r₁, r₂〉〈r₂, r₂〉〈r₂, r₁〉〈r₀, r₃〉}), #10 (U, {〈r₁, r₂〉〈r₂, r₂〉〈r₀, r₄〉〈r₀, r₃〉})

Construct intuitionistic fuzzy language decision concept lattice according to concepts IFLL (U,T,R^′) as indicated in Figure 5.

Intuitionistic fuzzy linguistic decision concept lattice IFLL (U,T,R^′)

Step 3: Obtain the rule set. Φ(A).

As can be seen from Figures 4 and 5, IFLL (U,A,S^′) ≤ IFLL (U,T,R^′), and the rule set can be made available as follows: $\begin{array}{l} Φ (A) = & {{〈 s_{4}, s_{0} 〉〈 s_{2}, s_{1} 〉〈 s_{4}, s_{0} 〉〈 s_{3}, s_{0} 〉〈 s_{3}, s_{1} 〉} \to {〈 r_{4}, r_{0} 〉〈 r_{3}, r_{1} 〉〈 r_{4}, r_{0} 〉〈 r_{3}, r_{1} 〉}, \\ {〈 s_{2}, s_{2} 〉〈 s_{4}, s_{0} 〉〈 s_{3}, s_{1} 〉〈 s_{4}, s_{0} 〉〈 s_{0}, s_{3} 〉} \to {〈 r_{3}, r_{1} 〉〈 r_{4}, r_{0} 〉〈 r_{3}, r_{0} 〉〈 r_{0}, r_{3} 〉}, \\ {〈 s_{3}, s_{1} 〉〈 s_{2}, s_{2} 〉〈 s_{0}, s_{4} 〉〈 s_{2}, s_{2} 〉〈 s_{2}, s_{1} 〉} \to {〈 r_{3}, r_{0} 〉〈 r_{2}, r_{2} 〉〈 r_{0}, r_{4} 〉〈 r_{2}, r_{1} 〉}, \\ {〈 s_{1}, s_{2} 〉〈 s_{2}, s_{1} 〉〈 s_{2}, s_{1} 〉〈 s_{0}, s_{3} 〉〈 s_{1}, s_{2} 〉} \to {〈 r_{1}, r_{2} 〉〈 r_{2}, r_{2} 〉〈 r_{2}, r_{1} 〉〈 r_{1}, r_{2} 〉}, \\ {〈 s_{2}, s_{2} 〉〈 s_{2}, s_{1} 〉〈 s_{3}, s_{1} 〉〈 s_{3}, s_{0} 〉〈 s_{0}, s_{3} 〉} \to {〈 r_{3}, r_{1} 〉〈 r_{3}, r_{1} 〉〈 r_{3}, r_{0} 〉〈 r_{0}, r_{3} 〉}, \\ {〈 s_{1}, s_{2} 〉〈 s_{2}, s_{2} 〉〈 s_{0}, s_{4} 〉〈 s_{0}, s_{3} 〉〈 s_{1}, s_{2} 〉} \to {〈 r_{1}, r_{2} 〉〈 r_{2}, r_{2} 〉〈 r_{0}, r_{4} 〉〈 r_{1}, r_{2} 〉}, \\ {〈 s_{2}, s_{2} 〉〈 s_{2}, s_{2} 〉〈 s_{0}, s_{4} 〉〈 s_{2}, s_{2} 〉〈 s_{0}, s_{3} 〉} \to {〈 r_{3}, r_{1} 〉〈 r_{2}, r_{2} 〉〈 r_{0}, r_{4} 〉〈 r_{0}, r_{3} 〉}, \\ {〈 s_{1}, s_{2} 〉〈 s_{2}, s_{1} 〉〈 s_{2}, s_{1} 〉〈 s_{0}, s_{3} 〉〈 s_{0}, s_{3} 〉} \to {〈 r_{1}, r_{2} 〉〈 r_{2}, r_{2} 〉〈 r_{2}, r_{1} 〉〈 r_{0}, r_{3} 〉}} . \end{array}$ \matrix{{\Phi (A) =} \hfill & {\left\{{\left\{{\left\langle {{s_4},{s_0}} \right\rangle \left\langle {{s_2},{s_1}} \right\rangle \left\langle {{s_4},{s_0}} \right\rangle \left\langle {{s_3},{s_0}} \right\rangle \left\langle {{s_3},{s_1}} \right\rangle} \right\}} \right. \to \left\{{\left\langle {{r_4},{r_0}} \right\rangle \left\langle {{r_3},{r_1}} \right\rangle \left\langle {{r_4},{r_0}} \right\rangle \left\langle {{r_3},{r_1}} \right\rangle} \right\},} \hfill \cr {} \hfill & {\,\,\,\left\{{\left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_4},{s_0}} \right\rangle \left\langle {{s_3},{s_1}} \right\rangle \left\langle {{s_4},{s_0}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle} \right\} \to \left\{{\left\langle {{r_3},{r_1}} \right\rangle \left\langle {{r_4},{r_0}} \right\rangle \left\langle {{r_3},{r_0}} \right\rangle \left\langle {{r_0},{r_3}} \right\rangle} \right\},} \hfill \cr {} \hfill & {\,\,\,\left\{{\left\langle {{s_3},{s_1}} \right\rangle \left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_0},{s_4}} \right\rangle \left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_2},{s_1}} \right\rangle} \right\} \to \left\{{\left\langle {{r_3},{r_0}} \right\rangle \left\langle {{r_2},{r_2}} \right\rangle \left\langle {{r_0},{r_4}} \right\rangle \left\langle {{r_2},{r_1}} \right\rangle} \right\},} \hfill \cr {} \hfill & {\,\,\,\left\{{\left\langle {{s_1},{s_2}} \right\rangle \left\langle {{s_2},{s_1}} \right\rangle \left\langle {{s_2},{s_1}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle \left\langle {{s_1},{s_2}} \right\rangle} \right\} \to \left\{{\left\langle {{r_1},{r_2}} \right\rangle \left\langle {{r_2},{r_2}} \right\rangle \left\langle {{r_2},{r_1}} \right\rangle \left\langle {{r_1},{r_2}} \right\rangle} \right\},} \hfill \cr {} \hfill & {\,\,\,\left\{{\left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_2},{s_1}} \right\rangle \left\langle {{s_3},{s_1}} \right\rangle \left\langle {{s_3},{s_0}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle} \right\} \to \left\{{\left\langle {{r_3},{r_1}} \right\rangle \left\langle {{r_3},{r_1}} \right\rangle \left\langle {{r_3},{r_0}} \right\rangle \left\langle {{r_0},{r_3}} \right\rangle} \right\},} \hfill \cr {} \hfill & {\,\,\,\left\{{\left\langle {{s_1},{s_2}} \right\rangle \left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_0},{s_4}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle \left\langle {{s_1},{s_2}} \right\rangle} \right\} \to \left\{{\left\langle {{r_1},{r_2}} \right\rangle \left\langle {{r_2},{r_2}} \right\rangle \left\langle {{r_0},{r_4}} \right\rangle \left\langle {{r_1},{r_2}} \right\rangle} \right\},} \hfill \cr {} \hfill & {\,\,\,\left\{{\left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_0},{s_4}} \right\rangle \left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle} \right\} \to \left\{{\left\langle {{r_3},{r_1}} \right\rangle \left\langle {{r_2},{r_2}} \right\rangle \left\langle {{r_0},{r_4}} \right\rangle \left\langle {{r_0},{r_3}} \right\rangle} \right\},} \hfill \cr {} \hfill & {\,\,\,\left\{{\left\langle {{s_1},{s_2}} \right\rangle \left\langle {{s_2},{s_1}} \right\rangle \left\langle {{s_2},{s_1}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle} \right\} \to \left. {\left\{{\left\langle {{r_1},{r_2}} \right\rangle \left\langle {{r_2},{r_2}} \right\rangle \left\langle {{r_2},{r_1}} \right\rangle \left\langle {{r_0},{r_3}} \right\rangle} \right\}} \right\}.} \hfill \cr}

Step 4: Judge whether there is a rule precursor in the rule set (a) that is the same as the known patient symptoms; by comparing the rule sets Φ(A) with symptoms of an existing patient, it can be found that there is no rule antecedent with it.

Step 5: Calculate the weighted similarity between the intuitive fuzzy language set G of the patient's symptoms and each rule antecedent in the rule set, so as to obtain the maximum weighted similarity;

Taking the possibility of the patient suffering from viral fever as an example, the weight corresponding to viral fever is given as $w_{II}^{1} = {0.4, 0.05, 0.2, 0.1, 0.25}$ w_{{\rm{II}}}^1 = \{0.4,0.05,0.2,0.1,0.25\} , assuming that: $\begin{array}{l} B_{1} = {〈 s_{4}, s_{0} 〉〈 s_{2}, s_{1} 〉〈 s_{4}, s_{0} 〉〈 s_{3}, s_{0} 〉〈 s_{3}, s_{1} 〉} \\ B_{2} = {〈 s_{2}, s_{2} 〉〈 s_{4}, s_{0} 〉〈 s_{3}, s_{1} 〉〈 s_{4}, s_{0} 〉〈 s_{0}, s_{3} 〉} \\ B_{3} = {〈 s_{3}, s_{1} 〉〈 s_{2}, s_{2} 〉〈 s_{0}, s_{4} 〉〈 s_{2}, s_{2} 〉〈 s_{2}, s_{1} 〉} \\ B_{4} = {〈 s_{1}, s_{2} 〉〈 s_{2}, s_{1} 〉〈 s_{2}, s_{1} 〉〈 s_{0}, s_{3} 〉〈 s_{1}, s_{2} 〉} \\ B_{6} = {〈 s_{1}, s_{2} 〉〈 s_{2}, s_{2} 〉〈 s_{0}, s_{4} 〉〈 s_{0}, s_{3} 〉〈 s_{1}, s_{2} 〉} \\ B_{7} = {〈 s_{2}, s_{2} 〉〈 s_{2}, s_{2} 〉〈 s_{0}, s_{4} 〉〈 s_{2}, s_{2} 〉〈 s_{0}, s_{3} 〉} \\ B_{8} = {〈 s_{1}, s_{2} 〉〈 s_{2}, s_{1} 〉〈 s_{2}, s_{1} 〉〈 s_{0}, s_{3} 〉〈 s_{0}, s_{3} 〉} \end{array}$ \matrix{{{B_1} = \left\{{\left\langle {{s_4},{s_0}} \right\rangle \left\langle {{s_2},{s_1}} \right\rangle \left\langle {{s_4},{s_0}} \right\rangle \left\langle {{s_3},{s_0}} \right\rangle \left\langle {{s_3},{s_1}} \right\rangle} \right\}} \hfill \cr {{B_2} = \left\{{\left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_4},{s_0}} \right\rangle \left\langle {{s_3},{s_1}} \right\rangle \left\langle {{s_4},{s_0}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle} \right\}} \hfill \cr {{B_3} = \left\{{\left\langle {{s_3},{s_1}} \right\rangle \left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_0},{s_4}} \right\rangle \left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_2},{s_1}} \right\rangle} \right\}} \hfill \cr {{B_4} = \left\{{\left\langle {{s_1},{s_2}} \right\rangle \left\langle {{s_2},{s_1}} \right\rangle \left\langle {{s_2},{s_1}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle \left\langle {{s_1},{s_2}} \right\rangle} \right\}} \hfill \cr {{B_6} = \left\{{\left\langle {{s_1},{s_2}} \right\rangle \left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_0},{s_4}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle \left\langle {{s_1},{s_2}} \right\rangle} \right\}} \hfill \cr {{B_7} = \left\{{\left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_0},{s_4}} \right\rangle \left\langle {{s_2},{s_2}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle} \right\}} \hfill \cr {{B_8} = \left\{{\left\langle {{s_1},{s_2}} \right\rangle \left\langle {{s_2},{s_1}} \right\rangle \left\langle {{s_2},{s_1}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle} \right\}} \hfill \cr}

Therefore, $\begin{array}{l} {sim}_{w {II}^{2}} (B_{1}, G) = 0.219, {sim}_{w_{Π^{1}}} (B_{2}, G) = 0.606, {sim}_{w_{I}^{1}} (B_{3}, G) = 0.581, \\ {sim}_{w_{II}^{1}} (B_{4}, G) = 0.688, {sim}_{w_{II}^{1}} (B_{5}, G) = 0.625, {sim}_{w_{II}^{1}} (B_{6}, G) = 0.756 \\ {sim}_{w_{I}^{1}} (B_{7}, G) = 0.756, {sim}_{w_{I}^{1}} (B_{8}, G) = 0.775 \end{array}$ \matrix{{{\rm{si}}{{\rm{m}}_{w{\rm{I}}{{\rm{I}}^2}}}\left({{B_1},G} \right) = 0.219,\quad {\rm{si}}{{\rm{m}}_{{w_{{\Pi ^1}}}}}\left({{B_2},G} \right) = 0.606,\quad {\rm{si}}{{\rm{m}}_{w_{\rm{I}}^1}}\left({{B_3},G} \right) = 0.581,} \hfill \cr {{\rm{si}}{{\rm{m}}_{{w_{{\rm{II}}}}^1}}\left({{B_4},G} \right) = 0.688,\quad {\rm{si}}{{\rm{m}}_{{w_{{\rm{II}}}}^1}}\left({{B_5},G} \right) = 0.625,\quad {\rm{si}}{{\rm{m}}_{{w_{{\rm{II}}}}^1}}\left({{B_6},G} \right) = 0.756} \hfill \cr {{\rm{si}}{{\rm{m}}_{w_{\rm{I}}^1}}\left({{B_7},G} \right) = 0.756,\quad {\rm{si}}{{\rm{m}}_{w_{\rm{I}}^1}}\left({{B_8},G} \right) = 0.775} \hfill \cr}

It can be seen from the above that the weighted similarity between G and rule antecedent B_s is the largest.

Step 6: Output the result.

When the similarity threshold is selected as 0.75, the maximum weighted similarity under viral fever is obtained. When sim_{w_I¹} (B_S,G) = 0.775 > 0.75, the following rule is obtained: ${〈 s_{0}, s_{3} 〉〈 s_{2}, s_{0} 〉〈 s_{0}, s_{3} 〉〈 s_{1}, s_{3} 〉〈 s_{0}, s_{4} 〉} \to {〈 r_{1}, r_{2} 〉}$ \left\{{\left\langle {{s_0},{s_3}} \right\rangle \left\langle {{s_2},{s_0}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle \left\langle {{s_1},{s_3}} \right\rangle \left\langle {{s_0},{s_4}} \right\rangle} \right\} \to \left\{{\left\langle {{r_1},{r_2}} \right\rangle} \right\}

That is, the possibility of the patient suffering from viral fever is 〈r₁, r₂〉. Similarly, the given corresponding weight of typhoid fever $w_{II}^{2} = {0.3, 0.4, 0.1, 0.1, 0.1}$ w_{{\rm{II}}}^2 = \{0.3,0.4,0.1,0.1,0.1\} , weight corresponding to stomach diseases w_II³ = {0.1,0.2,0.5,0.1,0.1}, chest and lung diseases corresponding to the right $w_{II}^{4} = {0.05, 0.05, 0.1, 0.1, 0.7}$ w_{{\rm{II}}}^4 = \{0.05,0.05,0.1,0.1,0.7\} . After calculation, ${sim}_{W_{II}^{2}} (B_{8}, G)$ {\rm{si}}{{\rm{m}}_{W_{{\rm{II}}}^2}}\left({{B_{\rm{8}}},G} \right) , sim_{w_II³} (B₆, G) and sim_{w_II⁴} (B₇, G) are, respectively, the m aximum weighted similarity; so: ${〈 s_{0}, s_{3} 〉〈 s_{2}, s_{0} 〉〈 s_{0}, s_{3} 〉〈 s_{1}, s_{3} 〉〈 s_{0}, s_{4} 〉} \to {〈 r_{1}, r_{2} 〉〈 r_{2}, r_{2} 〉〈 r_{0}, r_{4} 〉〈 r_{0}, r_{3} 〉}$ \left\{{\left\langle {{s_0},{s_3}} \right\rangle \left\langle {{s_2},{s_0}} \right\rangle \left\langle {{s_0},{s_3}} \right\rangle \left\langle {{s_1},{s_3}} \right\rangle \left\langle {{s_0},{s_4}} \right\rangle} \right\} \to \left\{{\left\langle {{r_1},{r_2}} \right\rangle \left\langle {{r_2},{r_2}} \right\rangle \left\langle {{r_0},{r_4}} \right\rangle \left\langle {{r_0},{r_3}} \right\rangle} \right\}

That is, the possibility of the patient suffering from viral fever is 〈r₁, r₂〉, the possibility of typhoid fever is 〈r₂, r₂〉, the possibility of gastropathy is 〈r₀, r₄〉 and the possibility of suffering from thoracic lung disease is 〈r₀, r₃〉.

Taking the threshold as 〈r₂, r₂〉, that is, when the probability of suffering from a certain disease is greater than or equal to 〈r₂, r₂〉, it should be noticed. Therefore, the patient should pay attention to typhoid fever.

The above examples show that the results are consistent with the analysis in real life, which indicates that the algorithm is effective and practical. In this paper, intelligent diagnosis with linguistic values in the field of disease diagnosis can better express the description of the disease, reduce data loss and thus get smarter diagnosis results.

Conclusion

Based on the concept lattice of intuitionistic fuzzy language, the rule extraction in the context of intuitionistic fuzzy language decision formal is defined in this paper. In order to handle the rules on the concept lattice, the similarity is defined. In addition, by considering the various influences of each attribute on the object, the weight is added to the similarity, and the weighted similarity is proposed. Finally, using an example connected with disease diagnosis, it is demonstrated that the algorithm proposed in this paper is effective and that the proposed decision method has strong operability and is suitable for practical application.

Figure e tabelle

Formal context of Intuitionistic fuzzy linguistic (U,A,S′)

U	A	b	c

x₁	〈s₀, s₄〉	〈s₂, s₂〉	〈s₁, s₂〉
x₂	〈s₃, s₀〉	〈s₀, s₂〉	〈s₃, s₁〉
x₃	〈s₁, s₁〉	〈s₃, s₁〉	〈s₂, s₁〉

Intuitionistic fuzzy linguistic for mal context (U,A,S′)

U	a	b	c	d	e

x₁	〈s₄, s₀〉	〈s₃, s₁〉	〈s₁, s₂〉	〈s₃, s₀〉	〈s₃, s₀〉
x₂	〈s₃, s₀〉	〈s₃, s₁〉	〈s₃, s₀〉	〈s₁, s₂〉	〈s₁, s₃〉
x₃	〈s₀, s₃〉	〈s₁, s₂〉	〈s₀, s₄〉	〈s₃, s₁〉	〈s₁, s₂〉
x₄	〈s₃, s₁〉	〈s₃, s₀〉	〈s₂, s₀〉	〈s₁, s₂〉	〈s₁, s₃〉

Discernible matrix of Intuitionistic fuzzy linguistic formal context (U,A,S′)

□_□/□_□	□₁	□₂	□₄	□₁□₂	□₁□₃	□₂□₄	□₁□₂□₄

□₁		ade	ade			ade
□₂	c		ac		abc
□₄	bc	b		bc	abc
□₁□₂			a		abc	a
□₁□₃		de	de	de		de	de
□₂□₄	c			c	abc
□₁□₂□₄					abc

Intuitionistic fuzzy linguistic formal decision context (U,A,S′, T,R′)

	a	b	c	d	e	t₁	t₃	t₃	t₄

x₁	〈s₃, s₁〉	〈s₂, s₂〉	〈s₀, s₄〉	〈s₂, s₂〉	〈s₂, s₁〉	〈r₃, r₀〉	〈r₂, r₂〉	〈r₀, r₄〉	〈r₂, r₁〉
x₂	〈s₄, s₀〉	〈s₂, s₁〉	〈s₄, s₀〉	〈s₃, s₀〉	〈s₃, s₁〉	〈r₄, r₀〉	〈r₃, r₁〉	〈r₄, r₀〉	〈r₃, r₁〉
x₃	〈s₁, s₂〉	〈s₃, s₁〉	〈s₂, s₁〉	〈s₀, s₃〉	〈s₁, s₂〉	〈r₁, r₂〉	〈r₂, r₂〉	〈r₂, r₁〉	〈r₁, r₂〉
x₄	〈s₂, s₂〉	〈s₄, s₀〉	〈s₃, s₁〉	〈s₄, s₀〉	〈s₀, s₃〉	〈r₃, r₁〉	〈r₄, r₀〉	〈r₃, r₀〉	〈r₀, r₃〉

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Applied Mathematics and Nonlinear Sciences

Research on rule extraction method based on concept lattice of intuitionistic fuzzy language

Hongpeng Yan
Hongpeng Yan

Pubblicato online: 20 May 2022

Volume & Edizione: AHEAD OF PRINT

Pagine: -

Ricevuto: 04 Mar 2022

Accettato: 10 Apr 2022

Dettagli rivista e numero

Applied Mathematics and Nonlinear Sciences

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Articolo

Fig. 1

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Fig. 5

Figure e tabelle

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Formal context of Intuitionistic fuzzy linguistic (U,A,S′)

Intuitionistic fuzzy linguistic for mal context (U,A,S′)

Discernible matrix of Intuitionistic fuzzy linguistic formal context (U,A,S′)

Intuitionistic fuzzy linguistic formal decision context (U,A,S′, T,R′)

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Applied Mathematics and Nonlinear Sciences

Research on rule extraction method based on concept lattice of intuitionistic fuzzy language

Hongpeng YanHongpeng Yan

Pubblicato online: 20 May 2022

Volume & Edizione: AHEAD OF PRINT

Pagine: -

Ricevuto: 04 Mar 2022

Accettato: 10 Apr 2022

Dettagli rivista e numero

Applied Mathematics and Nonlinear Sciences

Anteprima PDF

Articolo

Fig. 1

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Figure e tabelle

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Formal context of Intuitionistic fuzzy linguistic (U,A,S′)

Intuitionistic fuzzy linguistic for mal context (U,A,S′)

Discernible matrix of Intuitionistic fuzzy linguistic formal context (U,A,S′)

Intuitionistic fuzzy linguistic formal decision context (U,A,S′, T,R′)

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Hongpeng Yan
Hongpeng Yan