An intuitionistic fuzzy approach to S-approximation spaces

Abstract

In this paper, we will investigate the concept of intuitionistic fuzzy S-approximation spaces. This concept can be employed to manage the uncertainty in the intuitionistic fuzzy problems which are not expressible in terms of intuitionistic fuzzy inclusion relations. Also, we have studied these structures from a three-way decisions approach. The monotone intuitionistic fuzzy S-approximation spaces capture some interesting properties which are discussed as well as complement compatible ones.

Keywords

Intuitionistic fuzzy S-approximation spaces rough set theory intuitionistic fuzzy sets three-way decisions monotonicity complement compatibility

1 Introduction

The concept of information systems, as a representation for knowledge, has many applications in artificial intelligence and data mining. The uncertainty in studying information systems is characterized by a boundary or uncertain region, which for all of its elements, if it is non-empty, we cannot certainly decide whether it belongs or does not belong to a concept. This kind of uncertainty is not avoidable in many applications, so the research community proposed several tools to handle them such as Dempster-Shafer theory of evidence [27], theory of fuzzy sets [42], theory of rough sets [21, 22], and S-approximation spaces [15, 28], to name a few. Among many key differences between these theories, two basic common components can be distinguished, the “knowledge” and the notion of “be a part of” [24]. Knowledge in Pawlak’s rough set theory is represented by a partition over the set of objects [21], in fuzzy set theory by a fuzzy membership function [42], in Dempster-Shafer theory of evidence by a probability distribution over the set of objects [9] and in S-approximation spaces by an arbitrary mapping from a set to a subset of a set of objects (possibly different) [15]. Also, the notion of be a part of in Pawlak’s rough set theory and most of its generalizations [8 , 40], in fuzzy set theory [42] and in Dempster-Shafer theory of evidence [9], is the inclusion relation or some restrictions of it. However, in S-approximation spaces the notion of “be a part of” is independent of the inclusion relation and can be almost any mapping of the form $S : P (W) \times P (W) \to {0, 1}$ [15, 28], so it is called a decider. The concept of S-approximation spaces is a novel tool that can be used for analysis of information systems, which are independent from the inclusion relation. Moreover, S-approximation spaces are over two universes and cast generalized approximation spaces discussed in [8, 23]. The concept of S-approximation spaces is studied in [28] from a three-way decision perspective and also as a unifying theory for rough set models. Moreover, studying S-approximation spaces can lead to a more systematic way of constructing new rough set models or analyzing hybrid models. When we are uncertain about a concept, knowing its similarity and its difference to other concepts at the same time is useful. This idea can be seen as generalizing a mixture of similarity and dissimilarity functions. Note that the application of dissimilarity functions to approximation spaces was first considered in [12], which used a dissimilarity mapping to construct both lower and upper approximations. In general, using both measures of similarity and dissimilarity, especially in the case of uncertainty, is not a new idea in the field. The concept of intuitionistic fuzzy sets is introduced in [2] with a similar idea. Employing this idea in S-approximation spaces leads to a model with capability of approximating crisp, fuzzy and intuitionistic fuzzy sets in an efficient way. The combination of intuitionistic fuzzy set theory with rough set theory was first discussed in [4] and was reintroduced in [16] as is cited in [6]. In their construction, the lower and the upper approximations are intuitionistic fuzzy sets. This notion was also introduced in [26] from a different points of view and was called rough intuitionistic fuzzy set. In their view, the rough intuitionistic fuzzy sets are no longer fuzzy sets, but fuzzy rough sets as in [11]. However, all of these extensions are not constructed by an approximation space, since they are all generalizations of fuzzy rough sets in the sense of [20] as it is known that they have chosen to construct fuzzy rough sets of a rough set [6]. Along the way, in [5], it was shown that fuzzy rough sets are intuitionistic L-fuzzy sets. Moreover, a pair of lower and upper approximation operators were defined in an axiomatic way in [6] and was further studied by [5, 7]. Some works on intuitionistic fuzzy rough sets include [43, 44] where an axiomatic approach is employed to construct the approximation operators. An intuitionistic fuzzy rough set was constructed in [34] based on intuitionistic fuzzy residual implication and relations. Generalizations of rough set theory to two universal sets like [8, 23] are studied from an intuitionistic fuzzy viewpoint like in [19 , 37]. In this case, our approach to S-approximation spaces can be seen as a continuation of the study on S-approximation spaces, where the intuitionistic fuzzy inclusion measure can be any arbitrary mapping of the form $S : IF (W) \times IF (W) \to [0, 1] \times [0, 1]$ . In comparison with rough set theory and Dempster-Shafer theory of evidence, which are defined based on the inclusion relation, this structure can also be applied to problems which cannot be expressed in terms of the inclusion relation, cf. Examples 9 and 12 in [15]. The organization of this paper is as follows. In Section 2, some necessary concepts from intuitionistic fuzzy set theory as well as S-approximation spaces are reminded. The concept of intuitionistic fuzzy S-approximation spaces is proposed in Section 3. In Section 4, these new constructions are considered from a three-way decision theory. Monotonicity property is then introduced in intuitionistic fuzzy S-approximation spaces and its properties are considered in Section 5. Complement compatible intuitionistic fuzzy S-approximation spaces are investigated in Section 6. Some of the results obtained in this paper are illustrated by giving a detailed example in Section 7. Finally, the paper is concluded in Section 8.

2 Preliminaries

2.1 Intuitionistic fuzzy set theory

In this section, we will review some necessary facts of intuitionistic fuzzy sets from [3]. Note that we have made minor changes in the notations for ease of use in this paper. Let U be a finite non-empty set. An intuitionistic fuzzy or IF subset of U like A is defined as A = {〈 x, μ_A (x) , ν_A (x) 〈 | x ∈ U}, where μ_A : U → [0, 1] and ν_A : U → [0, 1] are degree of membership and degree of non-membership of an element x ∈ U to the intuitionistic fuzzy set A, respectively, which satisfy 0 ≤ μ_A (x) + ν_A (x) ≤1, for every x ∈ U. The degree of non-determinacy of or hesitation of the membership of an element x ∈ U to an IF set A is defined as π_A (x) =1 - μ_A (x) - ν_A (x). The set of all intuitionistic subsets of a set U is denoted by $IF (U)$ . For an IF subset A, its necessity and possibility are defined as ◊A = {〈 x, μ_A (x) , 1 - μ_A (x) 〈 | x ∈ U} and □A = {〈 x, 1 - ν_A (x) , ν_A (x) 〈 | x ∈ U}, respectively. The IF complement of an IF set A is defined as A^c = {〈 x, ν_A (x) , μ_A (x) 〈 | x ∈ U}.

2.2 S-approximation spaces

In this section we remind some preliminary facts and definitions for S-approximation spaces from [15, 28]. An S-approximation space is a quadruple G = (U, W, T, S), where U and W are finite non-empty sets, T is a mapping of the form $T : U \to P (W)$ and S is a mapping of the form $S : P (W) \times P (W) \to {0, 1}$ . In S-approximation spaces, mapping T is called a knowledge component and mapping S is called a decider. The lower and upper approximations of a set X ⊆ W are defined as $\underline{G} (X) = {x \in U | S (T (x), X) = 1}$ and $\bar{G} (X) = {x \in U | S (T (x), X^{c}) = 0}$ , respectively, where X^c is the complement of the set X with respect to theset W. Similarly, for any set X ⊆ W, three regions of positive, negative and boundary are defined as $\begin{matrix} {POS}_{G} X = {x \in U | S (T (x), X) \\ = 1 \land S (T (x), X^{c}) = 0}, \end{matrix}$ $\begin{matrix} {NEG}_{G} X = {x \in U | S (T (x), X) \\ = 0 \land S (T (x), X^{c}) = 1}, \end{matrix}$ ${BND}_{G} X = {x \in U | S (T (x), X) = S (T (x), X^{c})} .$

Using these three regions, measures of G-definability and G-undefinability of a set X ⊆ W can be defined as $α_{G} (X) = 1 - \frac{| {BND}_{G} X |}{| U |}$ and $β_{G} (X) = \frac{| {BND}_{G} X |}{| {POS}_{G} X | + | {NEG}_{G} X |}$ , respectively. For the sake of convenience in notation, we might drop the index G wherever it is clear from the context. As expected, the α and β measures are related as it is stated in the following lemma.

Lemma 2.1.Let G = (U, W, T, S) be an S-approxim-ation space and X ⊆ W. If α_G (X) ≠0, then $β_{G} (X) = \frac{1 - α_{G} (X)}{α_{G} (X)} .$

Otherwise, β_G (X) =∞.

Proof. By the definition of G-definability and G-undefinability of X, we can obtain $\begin{matrix} α_{G} (X) \times β_{G} (X) & = (1 - \frac{| {BND}_{G} X |}{| U |}) \times \\ \frac{| {BND}_{G} X |}{| {POS}_{G} X | + | {NEG}_{G} X |} . \end{matrix}$

By applying the fact that $U = {POS}_{G} X \cup {NEG}_{G} X \cup {BND}_{G} X,$ and the assumption of α_G (X) ≠0, we can obtain $\begin{matrix} α_{G} (X) \times β_{G} (X) = \frac{| {POS}_{G} X | + | {NEG}_{G} X |}{| U |} \\ \times \frac{| {BND}_{G} X |}{| {POS}_{G} X | + | {NEG}_{G} X |} = 1 - α_{G} (X) . \end{matrix}$

Thus, we have $β_{G} (X) = \frac{1 - α_{G} (X)}{α_{G} (X)},$ whenever α_G (X) ≠0. However, if α_G (X) =0 then BND_GX = U and hence, POS_GX = NEG_GX =∅. So, by the definition, β_G (X) =∞ in this case. By investigating different deciders, it can be seen that there are many different deciders than the inclusion relation. Some of them are treated individually in the literature like [13, 25]. In [15], a class of deciders called S-min deciders is investigated. The lower and upper approximations within S-approximation spaces with deciders from this class satisfy certain properties. In [28], an extended class of these deciders is identified with the same set of properties. This class of deciders is called partial monotone deciders. Let S be a relation of the form $S : P (W) \times P (W) \to {0, 1}$ and A, B ⊆ W. We say A ≼ _SB if S (A, B) =1. A relation S is called partial monotone if X ⊆ Y and A ≼ _SX imply that A ≼ _SY for any A, X, Y ⊆ W. An S-approximation space G = (U, W, T, S) is a partial monotone S-approximation space if S is partial monotone.

Proposition 2.2. [28] Let G = (U, W, T, S) be a partial monotone S-approximation space. For all sets X, Y ⊆ W, the followings hold:

X ⊆ Y implies $\bar{G} (X) \subseteq \bar{G} (Y)$ ,

X ⊆ Y implies $\underline{G} (X) \subseteq \underline{G} (Y)$ ,

$\bar{G} (X \cup Y) \supseteq \bar{G} (X) \cup \bar{G} (Y)$ ,

$\bar{G} (X \cap Y) \subseteq \bar{G} (X) \cap \bar{G} (Y)$ ,

$\underline{G} (X \cup Y) \supseteq \underline{G} (X) \cup \underline{G} (Y)$ ,

$\underline{G} (X \cap Y) \subseteq \underline{G} (X) \cap \underline{G} (Y)$ ,

$\bar{G} (X) = (\underline{G} (X^{c}))^{c}$ ,

$\underline{G} (X) = (\bar{G} (X^{c}))^{c}$ .

Proposition 2.3. [28] Let G = (U, W, T, S) be a partial monotone S-approximation space. Then for any X, Y ⊆ W, the followings hold:

X ⊆ Y implies POS_GX ⊆ POS_GY,

X ⊆ Y implies NEG_GY ⊆ NEG_GX,

POS_GX ∪ Y ⊇ POS_GX ∪ POS_GY,

NEG_GX ∪ Y ⊆ NEG_GX ∪ NEG_GY,

POS_GX ∩ Y ⊆ POS_GX ∩ POS_GY,

NEG_GX ∩ Y ⊇ NEG_GX ∩ NEG_GY,

$\begin{matrix} [t] {POS}_{G} X \cap & {NEG}_{G} Y \subseteq & {POS}_{G} X \cap {NEG}_{G} X \cap Y . \end{matrix}$

An S-approximation space G = (U, W, T, S) is called complement compatible if for all x ∈ U and X ⊆ W, S (T (x) , X) =0 or S (T (x) , X^c) =0.

Proposition 2.4.[28] Let G = (U, W, T, S) be a complement compatible S-approximation space. Then for any X ⊆ W, we have $\underline{G} (X) \subseteq \bar{G} (X)$ . Finally, in some proofs of the current paper, the following lemma is used

Lemma 2.5. Suppose thatG = (U, W, T, S) is anS-approximation space and X ⊆ W. Then

POS_GX = NEG_GX^c,

BND_GX = BND_GX^c.

Proof. The proof for the first part is clear from the definitions. The second part is easily shown considering the fact that U is partitioned by the three regions of any X ⊆ W with respect to G and the first part.

3 Intuitionistic fuzzy S-approximation spaces

Extending the ideas of the intuitionistic fuzzy theory into S-approximation spaces can be done in at least two ways:

Approximating an intuitionistic fuzzy set like X ∈ IFS (U) by computing its lower and upper approximations as intuitionistic fuzzy sets with respect to a modified S-approximation space where T : U → IF (W) and S : IF (W) × IF (W) → [0, 1] × [0, 1],

Approximating an intuitionistic fuzzy set like X ∈ IFS (U) by computing its lower and upper approximations as crisp (or fuzzy) sets with respect to a modified S-approximation space.

The first case captures the second case as a special case. So, in this paper, we will first investigate the first approach and then, the second approach is studied with respect to the results of the first. This approach leads to a modified S-approximation space which we call it intuitionistic fuzzy S-approximation space and is defined formally as Definition 3.1.

Definition 3.1. (Intuitionistic Fuzzy S-approximation Space) An intuitionistic fuzzy S-approximation space, or IF S-approximation space for short, is a quadruple IFG = (U, W, T, S) where U and W are finite non-empty sets, T is mapping of the form $T : U \to IF (W)$ and S is a mapping of the form $S : IF (W) \times IF (W) \to [0, 1] \times [0, 1]$ which satisfies 0 ≤ Proj₁2S (T (x) , X) + Proj₂2S (T (x) , X) ≤ 1, for all x ∈ U and $X \in IF (W)$ where ${Proj}_{i}^{n} (x_{1}, \dots, x_{n}) = x_{i} .$

Remark 1. From now on, for simplicity of notation and clarity, we will use S_μ (A, B) and S_ν (A, B) to denote Proj₁2S (A, B) and Proj₂2S (A, B), respectively.

Definition 3.2. [The intuitionistic fuzzy lower and upper approximations of X] Let IFG = (U, W, T, S) be an IF S-approximation space and $X \in IF (W)$ . Then, the intuitionistic fuzzy lower and upper approximations of X with respect to IFG are defined as $\underline{IFG} (X) = {〈 x, S_{μ} (T (x), X), S_{ν} (T (x), X) 〈 | x \in U} .$ and $\bar{IFG} (X) = {〈 x, S_{ν} (T (x), X^{c}), S_{μ} (T (x), X^{c}) 〈 | x \in U} .$ The intuition behind defining the IF upper approximation is the fact that we expect the degree of non-membership of an element to an IF set like X has some kind of relation with its membership degree to X^c and vice versa. Note that this expectation is not necessary and does not imply any properties on the structures; it is just the intuition of defining the operator in this way. As an example where this expectation might be false is as follows. An element x ∈ U might have the following membership degrees with respect to an IF set X ⊆ U, μ_X (x) =0, ν_X (x) =0, μ_{X
^c} (x) =0, ν_{X
^c} (x) =0. This is quite legal in IF theory.

Remark 2. Note that the Definition 3.2 captures the definitions in [6 , 29], as is shown next. Assume an intuitionistic fuzzy S-approximation space as IFG = (U, W, T, S). For the approximation pair in [19], we assume T (u) = R_s (u) for u ∈ U, S_μ (A, B) = ⋀ _y∈Aμ_B (y) and S_ν (A, B) = ⋁ _y∈Aν_B (y), so the lower and upper approximation operators coincide.

To obtain approximation pair in [6], assume T (u) = R (u), $S_{μ} (A, B) = inf_{x \in B} I (μ_{A} (x), μ_{B} (x))$ and $S_{ν} (A, B) = sup_{x \in B} T (ν_{A} (x), ν_{B} (x))$ . To obtain the pair in [29], assume that T (u) = R (u), S_μ (A, B) = ∧ _w∈W (μ_B (w) ∨ (1 - μ_A (w))) and S_ν (A, B) = ∨ _w∈W (ν_B (w) ∧ ν_A (w)). Note that in [29], R is considered as a fuzzy relation which implies that μ_R(x) (y) + ν_R(x) (y) =1.

It is clear that both $\underline{IFG} (X)$ and $\bar{IFG} (X)$ are intuitionistic fuzzy subsets of set U. So, we can apply the necessity and possibility operators of intuitionistic fuzzy theory to these two sets, as well as other operators. It is notable that by applying these operations on the lower and upper IF approximation sets, we arrive at a dual operator set. This is stated more formally in the following proposition.

Proposition 3.3. Let IFG = (U, W, T, S) be an IF S-approximation space and $X \in IF (W)$ . Then, the followings hold:

$□ \bar{IFG} (X) = {(◊ \underline{IFG} (X^{c}))}^{c}$ ,

$□ \underline{IFG} (X) = {(◊ \bar{IFG} (X^{c}))}^{c}$ ,

$◊ \bar{IFG} (X) = {(□ \underline{IFG} (X^{c}))}^{c}$ ,

$◊ \underline{IFG} (X) = {(□ \bar{IFG} (X^{c}))}^{c}$ .

Remark 3. Note that the inner complement operator, i.e. X^c, is the intuitionistic complement operator and the outer one is the fuzzy complement operator.

Proof. We will just show item (1), because the rest of them can be similarly obtained. For (1), by the definition we have $\begin{matrix} \bar{IFG} (X) = & {〈 x, S_{ν} (T (x), X^{c}), S_{μ} (T (x), X^{c}) 〈 \\ | x \in U}, \end{matrix}$ so, by computing its necessity, we obtain $\begin{matrix} □ \bar{IFG} (X) = & {〈 x, S_{ν} (T (x), X^{c}), 1 - S_{ν} (T (x), X^{c}) 〈 \\ | x \in U} . \end{matrix}$

Similarly, $◊ \underline{IFG} (X^{c})$ equals $\begin{matrix} ◊ \underline{IFG} (X^{c}) = & {〈 x, 1 - S_{ν} (T (x), X^{c}), S_{ν} (T (x), X^{c}) 〈 \\ | x \in U}, \end{matrix}$ so, by computing its complement, we obtain $\begin{matrix} {(◊ \underline{IFG} (X^{c}))}^{c} = \\ {〈 x, S_{ν} (T (x), X^{c}), 1 - S_{ν} (T (x), X^{c}) 〈 | x \in U}, \end{matrix}$ which concludes the proof, that is $□ \bar{IFG} (X) = {(◊ \underline{IFG} (X^{c}))}^{c}$ . The interpretations of S_μ (· , ·) and S_ν (· , ·) are quite natural as in intuitionistic fuzzy set theory. More formally, S_μ (A, B) denotes the degree of acceptance of A as B whereas S_ν (A, B) denotes the degree of rejecting A as B. Let a denotes the acceptance accuracy and r denotes the rejection accuracy in a threshold pair (a, r) where a, r ∈ [0, 1]. This pair can be interpreted as in Table 1. Using these interpretations, we can modify the lower and upper approximation operations in Definition 3.2. using these thresholds. This is considered next.

Definition 3.4. (The lower and upper approximations of X) Let IFG = (U, W, T, S) be an IF S-approximation space, (a, r) ∈ [0, 1] ² a pair of thresholds and $X \in IF (W)$ . Then, the lower and the upper approximations of X with respect to IFG and (a, r) are defined as $\begin{matrix} {\underline{IFG}}_{(a, r)} X = {x \in U | S_{μ} (T (x), X) \geq a \\ and S_{ν} (T (x), X) < r}, \end{matrix}$ and $\begin{matrix} {\bar{IFG}}_{(a, r)} X = {x \in U | S_{μ} (T (x), X^{c}) < a \\ and S_{ν} (T (x), X^{c}) \geq r} . \end{matrix}$ In other words, the lower approximation of $X \in IF (W)$ in Definition 3.4. consists of elements x ∈ U which can be accepted as X with accuracy of at least a and cannot be rejected with accuracy of at least r. The upper approximation of $X \in IF (W)$ contains elements x ∈ U that cannot be accepted as X^c with accuracy of at least a and can be rejected as X^c with accuracy of at least r. Note that we could define the lower and upper approximations of a set X in another form. That is ${\underline{IFG}}_{a} X = {x \in U | S_{μ} (T (x), X) \geq a}$ and ${\bar{IFG}}_{r} X = {x \in U | S_{ν} (T (x), X^{c}) \geq r}$ . Then, we have ${\underline{IFG}}_{(a, r)} X = {\underline{IFG}}_{a} X \cap ({\bar{IFG}}_{r} X^{c})^{c}$ and ${\bar{IFG}}_{(a, r)} X = {({\underline{IFG}}_{a} X^{c})}^{c} \cap {\bar{IFG}}_{r} X$ . Thesetwo equalities can be easily obtained from the definitions. Similarly, it is easy to see that ${\underline{IFG}}_{a} X = {\underline{IFG}}_{(a, 1)} X$ and ${\bar{IFG}}_{r} X = {\bar{IFG}}_{(1, r)} X$ for 0 < a, r < 1. Hence, these two operators can be obtained from each other whenever (a, r) ∈ (0, 1) ². However, we stick to the one in the Definition 3.4, because it makes the notation much simpler in the sequel.

Remark 4. Note that the Definition 3.4 captures the definitions in [14]. It suffices to consider T (u) = R (u) and S_μ (A, B) = a if $A_{a \leq}^{μ} \cap A_{r \geq}^{ν} \subseteq B$ and $A_{a \leq}^{μ} \cap A_{r \geq}^{ν} \neq \emptyset$ , otherwise S_μ (A, B) = a - ɛ for some ɛ > 0 where $A_{a \leq}^{μ} = {x \in A | μ_{A} (x) \geq a}$ and $A_{r \geq}^{ν} = {x \in A | ν_{A} (x) \leq r}$ . Also S_ν (A, B) = r - ɛ for some ɛ > 0 if $A_{a \leq}^{μ} \cap A_{r \geq}^{ν} \cap B \neq \emptyset$ or $A_{a \leq}^{μ} \cap A_{r \geq}^{ν} = \emptyset$ ,otherwise S_μ (A, B) = r.

4 Three-way decisions in IF S-approximation spaces

In problem solving, we usually encounter three-way decisions, i.e. decisions of acceptance, rejection and non-commitment. This notion of three regions was embedded in the theory of rough sets [21] and then was studied further in [35 , 39]. Also, there are some recent works like [17] which interprets three-way decisions from an intuitionistic viewpoint. Using the interpretations of Table 1, there are 2⁴ = 16 different situations to consider. Among them, the positive and negative regions of X with respect to an IF S-approximation space and threshold pair (a, r) ∈ [0, 1] ² is defined as $\begin{matrix} \begin{matrix} {POS}_{{IFG}_{(a, r)}} X = {x \in U | S_{μ} (T (x), X) \geq a, \\ S_{μ} (T (x), X^{c}) < a, S_{ν} (T (x), X) < r \\ and S_{ν} (T (x), X^{c}) \geq r}, \end{matrix} \end{matrix}$ and $\begin{matrix} \begin{matrix} {NEG}_{{IFG}_{(a, r)}} X = {x \in U | S_{μ} (T (x), X) < a, \\ S_{μ} (T (x), X^{c}) \geq a, S_{ν} (T (x), X) \geq r \\ and S_{ν} (T (x), X^{c}) < r} . \end{matrix} \end{matrix}$

Then, the boundary regions of X with respect to IFG and threshold pair (a, r) is defined as ${BND}_{{IFG}_{(a, r)}} (X) = U \ ({POS}_{{IFG}_{(a, r)}} (X) \cup {NEG}_{{IFG}_{(a, r)}} X) .$

So, there are 14 possible scenarios for the boundary region of an intuitionistic concept $X \in IF (W)$ with respect to an IF S-approximation space and threshold pair (a, r) ∈ [0, 1] ². These 14 scenarios are divisible into three categories of inconsistency, uncertainty and non-commitment. This is illustrated in Table 2. In other words, (1) Non-Commitment (NC): it happens when we cannot accept x as X and reject as X^c as well as accept x as X^c and reject as X with accuracies of at least a, r, a and r, respectively. (2) InConsistency (IC): this happens whenever we can accept x as X and reject as X^c as well as accept x as X^c and reject as X with accuracies of at least a, r, a and r, respectively. (3) UnCertainty (UC): this last situation contains all the other cases in the boundary region where we lack something, e.g. we cannot accept x as X and reject it as X with accuracies of at least a and r, although we can accept it as X^c and reject it as X^c with accuracies of at least a and r. From Table 2, it is easy to see that IC_{IFG
_(a,r)} (X), UC_{IFG
_(a,r)} (X) and NC_{IFG
_(a,r)} (X) partition the set BND_{IFG
_(a,r)}X. Using these sets, we can define the following set of measures for a set X ∈ IFS (W) with respect to IFG = (U, W, T, S) and a pair of thresholds (a, r) ∈ [0, 1] ². The first measure is the inconsistency measure which is defined as $d_{IC} = \frac{| {IC}_{{IFG}_{(a, r)}} (X) |}{| U |} .$

The measure of generality, based on [38], is defined as ${Generality}_{{IFG}_{(a, r)}} (X) = 1 - \frac{| {NC}_{{IFG}_{(a, r)}} (X) |}{| U |} .$

It is worth to note the difference in the definition of generality measure as in here compared to the one in [38]. Here, we have just considered the non-commitment region, a region whose elements has no decision with them whereas in [38], the boundary region is considered. In addition to these measures, we can define the IFG-definability and IFG-undefinability of a set $X \in IF (W)$ similar to the ones for S-approximation spaces as follows. The IFG-definability of a set $X \in IF (W)$ with respect to a threshold pair (a, r) is defined as $α_{{IFG}_{(a, r)}} (X) = 1 - \frac{| {BND}_{{IFG}_{(a, r)}} X |}{| U |},$ and the IFG-definability of a set $X \in IF (W)$ with respect to a threshold pair (a, r) is defined as $\begin{matrix} β_{{IFG}_{(a, r)}} (X) = \\ \frac{| {BND}_{{IFG}_{(a, r)}} X |}{| {POS}_{{IFG}_{(a, r)}} X | + | {NEG}_{{IFG}_{(a, r)}} X |}, \end{matrix}$ whenever ${POS}_{{IFG}_{(a, r)}} X \neq \emptyset,$ or ${NEG}_{{IFG}_{(a, r)}} X \neq \emptyset,$ otherwise it is considered as β_{IFG
_(a,r)} (X) =∞. As it is shown next, the IFG-definability and IFG-undefinability are connected.

Lemma 4.1.Let IFG = (U, W, T, S) be an IFS-approximation space, (a, r) a pair of thresholds and $X \in IF (W)$ . If $α_{{IFG}_{(a, r)}}^{X} \neq 0$ , then $β_{{IFG}_{(a, r)}}^{X} = \frac{1 - α_{{IFG}_{(a, r)}}^{X}}{α_{{IFG}_{(a, r)}}^{X}} .$

Otherwise, $β_{{IFG}_{(a, r)}}^{X} = \infty$ .

Proof. The proof is similar to the proof of Lemma 2.1.

The relation between the three regions and the lower and upper approximations of an $X \in IF (W)$ with respect to a pair of thresholds (a, r) ∈ [0, 1] ² is stated in the following lemma.

Lemma 4.2. Let IFG = (U, W, T, S) be an IF S-approximation space, (a, r) ∈ [0, 1] ² a pair of thresholds and $X \in IF (W)$ . Then, ${POS}_{{IFG}_{(a, r)}} X = {\underline{IFG}}_{(a, r)} X \cap {\bar{IFG}}_{(a, r)} X,$ and ${NEG}_{{IFG}_{(a, r)}} X = {\underline{IFG}}_{(a, r)} X^{c} \cap {\bar{IFG}}_{(a, r)} X^{c} .$

Proof. The proof easily follows from the definition. Given the facts that the sets POS_{IFG
_(a,r)}X, NEG_{IFG
_(a,r)}X and BND_{IFG
_(a,r)}X partition U as well as POS_{IFG
_(a,r)}X = NEG_{IFG
_(a,r)}X^c, we can obtain ${BND}_{{IFG}_{(a, r)}} X = {BND}_{{IFG}_{(a, r)}} X^{c} .$ Corollary 4.3. Let IFG = (U, W, T, S) be an IF S-approximation space, (a, r) ∈ [0, 1] ² a pair of thresholds and $X \in IF (W)$ . Then, α_{IFG
_(a,r)} (X) = α_{IFG
_(a,r)} (X^c) and β_{IFG
_(a,r)} (X) = β_{IFG
_(a,r)} (X^c).

5 Monotone IF S-approximation spaces

In searching for certain decision mappings in S-approximation spaces which ascertains some necessary properties (Proposition 2.2) for the lower and upper approximation operators, the notion of partial monotonicity was proposed in [28]. A modified version of it for IF S-approximation spaces can be stated as follows.

Definition 5.1. (Monotone IF S-approximation Space) Let IFG = (U, W, T, S) be an IF S-approximation space. Then, IFG is called monotone if A ⊆ B implies S_μ (T (x) , A) ≤ S_μ (T (x) , B) and S_ν (T (x) , A) ≥ S_ν (T (x) , B) for every x ∈ U and $A, B \in IF (W)$ . In what follows, we will show that choosing a monotone decider mapping preserves most of the properties of Section Proposition 2.2. Note that these properties (or some variants of them) are preserved by the lower and upper approximations defined in [6 , 29], too.

Proposition 5.2. Let IFG = (U, W, T, S) be a monotone IF S-approximation space and $X, Y \in IF (W)$ . Then,

[(M1)] X ⊆ Y implies $\bar{IFG} (X) \subseteq \bar{IFG} (Y)$ ,

[(M2)] X ⊆ Y implies $\underline{IFG} (X) \subseteq \underline{IFG} (Y)$ ,

[(M3)] $\bar{IFG} (X \cup Y) \supseteq \bar{IFG} (X) \cup \bar{IFG} (Y)$ ,

[(M4)] $\bar{IFG} (X \cap Y) \subseteq \bar{IFG} (X) \cap \bar{IFG} (Y)$ ,

[(M5)] $\underline{IFG} (X \cup Y) \supseteq \underline{IFG} (X) \cup \underline{IFG} (Y)$ ,

[(M6)] $\underline{IFG} (X \cap Y) \subseteq \underline{IFG} (X) \cap \underline{IFG} (Y)$ .

Proof. The proof is as follows:

Given X ⊆ Y, then Y^c ⊆ X^c. Hence, bymonotonicity of IFG we have S_μ (T (x) , Y^c) ≤ S_μ (T (x) , X^c) and S_ν (T (x) , Y^c) ≥ S_ν (T (x) , X^c). So, $\bar{IFG} (X) \subseteq \bar{IFG} (Y)$ .

Since IFG is monotone, then X ⊆ Y implies S_μ (T (x) , X) ≤ S_μ (T (x) , Y) and S_ν (T (x) , X) ≥ S_ν (T (x) , Y). So $\underline{IFG} (X) \subseteq \underline{IFG} (Y)$ .

By X ⊆ X ∪ Y and (M1), we have $\bar{IFG} (X) \subseteq \bar{IFG} (X \cup Y)$ . Similarly, we can obtain $\bar{IFG} (Y) \subseteq \bar{IFG} (X \cup Y)$ which concludes the proof.

X ∩ Y ⊆ X and (M1) imply $\bar{IFG} (X \cap Y) \subseteq \bar{IFG} (X)$ . Similarly, we can obtain $\bar{IFG} (X \cap Y) \subseteq \bar{IFG} (Y)$ which concludes the proof.

X ⊆ X ∪ Y and (M2) imply $\underline{IFG} (X) \subseteq \underline{IFG} (X \cup Y)$ . Similarly, we can obtain $\underline{IFG} (Y) \subseteq \underline{IFG} (X \cup Y)$ which concludes the proof.

X ∩ Y ⊆ X and (M2) imply $\underline{IFG} (X \cap Y) \subseteq \underline{IFG} (X)$ . Similarly, we can obtain $\underline{IFG} (X \cap Y) \subseteq \underline{IFG} (Y)$ which concludes the proof.

Proposition 5.3. Let IFG = (U, W, T, S) be a monotone IF S-approximation space, (a, r) a pair of thresholds and $X, Y \in IF (W)$ . Then,

X ⊆ Y implies ${\bar{IFG}}_{(a, r)} X \subseteq {\bar{IFG}}_{(a, r)} Y$ ,

X ⊆ Y implies ${\underline{IFG}}_{(a, r)} X \subseteq {\underline{IFG}}_{(a, r)} Y$ ,

$\begin{matrix} [t] & {\bar{IFG}}_{(a, r)} (X \cup Y) \supseteq {\bar{IFG}}_{(a, r)} (X) \\ \cup {\bar{IFG}}_{(a, r)} (Y) \end{matrix}$ ,

${\bar{IFG}}_{(a, r)} (X \cap Y) \subseteq {\bar{IFG}}_{(a, r)} (X) \cap {\bar{IFG}}_{(a, r)} (Y)$ ,

${\underline{IFG}}_{(a, r)} (X \cup Y) \supseteq {\underline{IFG}}_{(a, r)} (X) \cup {\underline{IFG}}_{(a, r)} (Y)$ ,

${\underline{IFG}}_{(a, r)} (X \cap Y) \subseteq {\underline{IFG}}_{(a, r)} (X) \cap {\underline{IFG}}_{(a, r)} (Y)$ .

Proof. The proof is as follows:

(M1) Given X ⊆ Y, then we have Y^c ⊆ X^c. Hence, monotonicity of IFG implies $S_{μ} (T (x), Y^{c}) \leq S_{μ} (T (x), X^{c}),$ and S_ν (T (x) , Y^c) ≥ S_ν (T (x) , X^c). Assume that $x \in \bar{{IFG}_{(a, r)}} (X)$ , then we have S_μ (T (x) , X^c) < a and S_ν (T (x) , X^c) ≥ r which implies S_μ (T (x) , Y^c) ≤ S_μ (T (x) , X^c) < a and $S_{ν} (T (x), Y^{c}) \geq S_{ν} (T (x), X^{c}) \geq r .$

So, $x \in \bar{{IFG}_{(a, r)}} (Y)$ .

(M2) Since IFG is monotone, then X ⊆ Y implies S_μ (T (x) , X) ≤ S_μ (T (x) , Y) and S_ν (T (x) , X) ≥ S_ν (T (x) , Y). Assume that $x \in \underline{{IFG}_{(a, r)}} (X)$ , henceS_μ (T (x) , X) ≥ a and S_ν (T (x) , X) < r. So, a ≤ S_μ (T (x) , X) ≤ S_μ (T (x) , Y), and r > S_ν (T (x) , X) ≥ S_ν (T (x) , Y), which implies that $x \in \underline{{IFG}_{(a, r)}} (Y)$ .

(M3) By X ⊆ X ∪ Y and (M1), we have ${\bar{IFG}}_{(a, r)} (X) \subseteq {\bar{IFG}}_{(a, r)} (X \cup Y) .$

Similarly, we can obtain ${\bar{IFG}}_{(a, r)} (Y) \subseteq {\bar{IFG}}_{(a, r)} (X \cup Y),$ which concludes the proof.

(M4) By X ∩ Y ⊆ X and (M1), we have ${\bar{IFG}}_{(a, r)} (X \cap Y) \subseteq {\bar{IFG}}_{(a, r)} (X) .$

Similarly, we can obtain ${\bar{IFG}}_{(a, r)} (X \cap Y) \subseteq {\bar{IFG}}_{(a, r)} (Y)$ which concludes the proof.

(M5) By X ⊆ X ∪ Y and (M2), we have ${\underline{IFG}}_{(a, r)} (X) \subseteq {\underline{IFG}}_{(a, r)} (X \cup Y) .$

Similarly, we can obtain ${\underline{IFG}}_{(a, r)} (Y) \subseteq {\underline{IFG}}_{(a, r)} (X \cup Y),$ which concludes the proof.

(M6) By X ∩ Y ⊆ X and (M2), we have ${\underline{IFG}}_{(a, r)} (X \cap Y) \subseteq {\underline{IFG}}_{(a, r)} (X) .$

Similarly, we can obtain ${\underline{IFG}}_{(a, r)} (X \cap Y) \subseteq {\underline{IFG}}_{(a, r)} (Y),$ which concludes the proof.

Similar to the S-approximation spaces in [28], the three-way decision regions in the intuitionistic fuzzy ones preserve certain properties, as is shown next.

Proposition 5.4.Let IFG = (U, W, T, S) be a monotone IF S-approximation space, (a, r) a pair of thresholds and $X, Y \in IF (W)$ . Then,

X ⊆ Y implies

${POS}_{{IFG}_{(a, r)}} X \subseteq {POS}_{{IFG}_{(a, r)}} Y,$

X ⊆ Y implies

${NEG}_{{IFG}_{(a, r)}} Y \subseteq {NEG}_{{IFG}_{(a, r)}} X,$

POS_{IFG
_(a,r)}X ∪ Y ⊇ POS_{IFG
_(a,r)}X ∪ POS_{IFG
_(a,r)}Y,

POS_{IFG
_(a,r)}X ∩ Y ⊆ POS_{IFG
_(a,r)}X ∩ POS_{IFG
_(a,r)}Y,

NEG_{IFG
_(a,r)}X ∪ Y ⊆ NEG_{IFG
_(a,r)}X ∪ NEG_{IFG
_(a,r)}X,

NEG_{IFG
_(a,r)}X ∩ Y ⊇ NEG_{IFG
_(a,r)}X ∩ NEG_{IFG
_(a,r)}X.

Proof. The proof is as follows:

Given X ⊆ Y, then Y^c ⊆ X^c. Hence, bymonotonicity of IFG we have S_μ (T (x) , Y^c) ≤ S_μ (T (x) , X^c) and S_ν (T (x) , Y^c) ≥ S_ν (T (x) , X^c). Similarly, S_μ (T (x) , X) ≤ S_μ (T (x) , Y) and S_ν (T (x) , X) ≥ S_ν (T (x) , Y). Assume that x ∈ POS_{IFG
_(a,r)}X, then $\begin{matrix} S_{μ} (T (x), X) \geq a, S_{μ} (T (x), X^{c}) < a, \\ S_{ν} (T (x), X) < r, S_{ν} (T (x), X^{c}) \geq r . \end{matrix}$

By combining these inequalities, we can derive $\begin{matrix} S_{μ} (T (x), Y) \geq S_{μ} (T (x), X) \geq a, \\ S_{μ} (T (x), Y^{c}) \leq S_{μ} (T (x), X^{c}) < a, \\ S_{ν} (T (x), Y) \leq S_{ν} (T (x), X) < r, \\ S_{ν} (T (x), Y^{c}) \geq S_{ν} (T (x), X^{c}) \geq r, \end{matrix}$ which implies that x ∈ POS_{IFG
_(a,r)}Y.

Given X ⊆ Y, then Y^c ⊆ X^c. By (M1), POS_{IFG
_(a,r)}Y^c ⊆ POS_{IFG
_(a,r)}X^c. So, using POS_{IFG
_(a,r)}X = NEG_{IFG
_(a,r)}X^c, we have NEG_{IFG
_(a,r)}Y ⊆ NEG_{IFG
_(a,r)}X.

By X ⊆ X ∪ Y and (M1), we have ${POS}_{{IFG}_{(a, r)}} X \subseteq {POS}_{{IFG}_{(a, r)}} X \cup Y .$ Similarly, we can obtain POS_{IFG
_(a,r)}Y ⊆ POS_{IFG
_(a,r)}X ∪ Y which concludes the proof.

By X ∩ Y ⊆ X and (M1), we have ${POS}_{{IFG}_{(a, r)}} X \cap Y \subseteq {POS}_{{IFG}_{(a, r)}} X .$ Similarly, we can obtain POS_{IFG
_(a,r)}X ∩ Y ⊆ POS_{IFG
_(a,r)}Y which concludes the proof.

By X ⊆ X ∪ Y and (M2), we have ${NEG}_{{IFG}_{(a, r)}} X \supseteq {NEG}_{{IFG}_{(a, r)}} X \cup Y .$ Similarly, we can obtain NEG_{IFG
_(a,r)}Y ⊇ NEG_{IFG
_(a,r)}X ∪ Y which concludes the proof.

By X ∩ Y ⊆ X and (M2), we have ${NEG}_{{IFG}_{(a, r)}} X \cap Y \supseteq {NEG}_{{IFG}_{(a, r)}} X .$ Similarly, we can obtain NEG_{IFG
_(a,r)}X ∩ Y ⊇ NEG_{IFG
_(a,r)}Y which concludes the proof.

6 Complement compatible IF S-approximation spaces

The notion of complement compatibility was introduced in [28] in order to make the lower approximation of a set included into the upper approximation of that set. For the same reason, we will introduce the notion of complement compatibility for intuitionistic fuzzy S-approximation spaces as follows.

Definition 6.1. (Complement Compatible IF S-approximation Space) Let IFG = (U, W, T, S) be an IF S-approximation space. Then, IFG is called complement compatible if S_μ (T (x) , A) ≤ S_ν (T (x) , A^c) for every x ∈ U and $A \in IF (W)$ .

Proposition 6.2. Let IFG = (U, W, T, S) be a complement compatible IF S-approximation space and $X \in IF (W)$ . Then, $\underline{IFG} (X) \subseteq \bar{IFG} (X)$ .

Proof. Given that IFG is a complement compatible IF S-approximation space, then S_μ (T (x) , X) ≤ S_ν (T (x) , X^c) and S_ν (T (x) , X) ≥ S_μ (T (x) , X^c). By Definition 3.2., these inequalities imply that $\underline{IFG} (X) \subseteq \bar{IFG} (X)$ .

7 Illustrative example

The concept of intuitionistic S-approximation spaces can be used in case-based reasoning applications, e.g. there are two sets of diseases and symptoms with a mapping among them. With these sets and relation, one would like to diagnose which disease(s) a patient suffers, with respect to a set of obsevered symptoms. Let U = {u₁, …, u₅} be a set of diseases and W = {w₁, …, w₅} be a set of symptoms. For each disease u_i, there is an intuitionistic fuzzy set of symptoms, $T (u_{i}) \in IF (W)$ given by an expert. This knowledge is given in Table 3 (it is taken from [31]).

We will use two intuitionistic fuzzy decider mappings which are listed as follows. For $A, B \in IF (W)$ , the first one is the default intuitionistic fuzzy inclusion function, i.e. $S_{1} (A, B) = {\begin{matrix} (1, 1) & if \forall w \in W, μ_{A} (w) = μ_{B} (w) and ν_{A} (w) = ν_{B} (w) \\ (1, 0) & if \forall w \in W, μ_{A} (w) \leq μ_{B} (w) and ν_{A} (w) \geq ν_{B} (w) \\ and \exists w^{'} \in W, (μ_{A} (w^{'}) \neq μ_{B} (w^{'}) or ν_{A} (w^{'}) \neq ν_{B} (w^{'})) \\ (0, 1) & if \forall w \in W, ν_{A} (w) \leq ν_{B} (w) and μ_{A} (w) \geq μ_{B} (w) \\ and \exists w^{'} \in W, (μ_{A} (w^{'}) \neq μ_{B} (w^{'}) or ν_{A} (w^{'}) \neq ν_{B} (w^{'})) \\ (0, 0) & otherwise \end{matrix} .$

The second one is a hybrid of similarity measures taken from [10, 18] and is defined as $\begin{matrix} \begin{matrix} S_{2}^{p} (A, B) = & (1 - \frac{1}{\sqrt[p]{| W |}} \sqrt[p]{\sum_{i = 1}^{| W |} {| m_{A} (i) - m_{B} (i) |}^{p}}, \\ min {\frac{1}{\sqrt[p]{| W |}} \sqrt[p]{\sum_{i = 1}^{| W |} {(φ_{t (A, B)} (i) + φ_{f (A, B)} (i))}^{p}}, \\ \frac{1}{\sqrt[p]{| W |}} \sqrt[p]{\sum_{i = 1}^{| W |} {| m_{A} (i) - m_{B} (i) |}^{p}}}), \end{matrix} \end{matrix}$ where $\begin{matrix} m_{A} (i) = \frac{μ_{A} (w_{i}) + 1 - ν_{A} (w_{i})}{2}, \\ φ_{t (A, B)} (i) = | \frac{μ_{A} (w_{i}) - μ_{B} (w_{i})}{2} |, \end{matrix}$ and $φ_{f (A, B)} (i) = | \frac{1 - ν_{A} (w_{i})}{2} - \frac{1 - ν_{B} (w_{i})}{2} | .$

It is easy to see that S_is are intuitionistic fuzzy deciders for i = 1, 2. Assume that there is a patient with observations $X \in IF (W)$ as $\begin{matrix} X = \frac{w_{1}}{0.8, 0.1} + \frac{w_{2}}{0.6, 0.1} + \frac{w_{3}}{0.2, 0.8} + \frac{w_{4}}{0.6, 0.1} \\ + \frac{w_{5}}{0.1, 0.6}, \end{matrix}$

where $\frac{w_{i}}{α, β}$ in $A \in IF (W)$ represents μ_A (w_i) = α and ν_A (w_i) = β for w_i ∈ W.

Given intuitionistic fuzzy S-approximation spaces IFG_i = (U, W, T, S_i) for i = 1, 2, for X we can compute

$\begin{matrix} \underline{{IFG}_{1}} (X) = & \frac{u_{1}}{0.0, 0.0} + \frac{u_{2}}{0.0, 0.0} + \frac{u_{3}}{0.0, 0.0} \\ + \frac{u_{4}}{0.0, 0.0} + \frac{u_{5}}{0.0, 0.0}, \end{matrix}$ $\begin{matrix} \bar{{IFG}_{1}} (X) = & \frac{u_{1}}{0.0, 0.0} + \frac{u_{2}}{0.0, 0.0} + \frac{u_{3}}{0.0, 0.0} \\ + \frac{u_{4}}{0.0, 0.0} + \frac{u_{5}}{0.0, 0.0}, \\ \underline{{IFG}_{2}} (X) = & \frac{u_{1}}{0.70504, 0.29496} + \frac{u_{2}}{0.58708, 0.41292} \\ + \frac{u_{3}}{0.56641, 0.43359} + \\ \frac{u_{4}}{0.73074, 0.26926} + \frac{u_{5}}{0.47038, 0.52962}, \\ \bar{{IFG}_{2}} (X) = & \frac{u_{1}}{0.47223, 0.52777} + \frac{u_{2}}{0.32016, 0.67984} \\ + \frac{u_{3}}{0.4626, 0.5374} + \\ \frac{u_{4}}{0.46314, 0.53686} + \frac{u_{5}}{0.32016, 0.67984} . \end{matrix}$

Note that each of these approximation spaces represents a mathemtical model of a specialist, e.g. a doctor. Let assume (a, r) = (0.5, 0.5). Then, $\begin{matrix} {\underline{{IFG}_{1}}}_{(a, r)} X = & \emptyset, {\bar{{IFG}_{1}}}_{(a, r)} X = \emptyset, \\ {\underline{{IFG}_{2}}}_{(a, r)} X = & {u_{1}, u_{2}, u_{3}, u_{4}}, {\bar{{IFG}_{2}}}_{(a, r)} X = \emptyset . \end{matrix}$

We can compute the three-way decision regions for this patient whose symptoms are represented by set X. For thresholds (a, r) = (0.5, 0.5), we have $\begin{matrix} {POS}_{{IFG}_{1}_{(a, r)}} X = & \emptyset, {NEG}_{{IFG}_{1}_{(a, r)}} X = \emptyset, \\ {BND}_{{IFG}_{1}_{(a, r)}} X = U, \\ {POS}_{{IFG}_{2}_{(a, r)}} X = & \emptyset, {NEG}_{{IFG}_{2}_{(a, r)}} X = {u_{5}}, \\ {BND}_{{IFG}_{2}_{(a, r)}} X = {u_{1}, u_{2}, u_{3}, u_{4}} . \end{matrix}$

Given these three regions for set X, we can compute the IFG_i-definability and -undefinability measures as $\begin{matrix} α_{{IFG}_{1}_{(a, r)}} (X) & = 0 & β_{{IFG}_{1}_{(a, r)}} (X) & = \infty, \\ α_{{IFG}_{2}_{(a, r)}} (X) & = 0.2 & β_{{IFG}_{2}_{(a, r)}} (X) & = 4 . \end{matrix}$

Also, the inconsistency and generality measures are obtained as follows $\begin{matrix} d_{{IC}_{{IFG}_{1}_{(a, r)}}} (X) = 0, {Generality}_{{IFG}_{1}_{(a, r)}} (X) = 0, \\ d_{{IC}_{{IFG}_{2}_{(a, r)}}} (X) = 0.8, {Generality}_{{IFG}_{2}_{(a, r)}} (X) = 1 . \end{matrix}$

These results can be interpreted as “by certainty of at least %50 and error of at most %50, the patient with symptoms X is not suffering from disease u₅ with respect to the second doctor”. Similarly, we can interpret that the first doctor cannot decide which disease(s) the patient suffers and at the same time, is suspected on all of the disease. In this case, each doctor might need to do further tests and/or tolerate the thresholds. Similarly, for (a′, r′) = (0.7, 0.3) we have $\begin{matrix} {\underline{{IFG}_{1}}}_{(a^{'}, r^{'})} X = & \emptyset, & {\bar{{IFG}_{1}}}_{(a^{'}, r^{'})} X = & \emptyset, \\ {\underline{{IFG}_{2}}}_{(a^{'}, r^{'})} X = & {u_{1}, u_{4}}, & {\bar{{IFG}_{2}}}_{(a^{'}, r^{'})} X = & U . \end{matrix}$

Then, the three regions of decision for this patient are computed as follows: $\begin{matrix} {POS}_{{IFG}_{1}_{(a^{'}, r^{'})}} X = & \emptyset, {NEG}_{{IFG}_{1}_{(a^{'}, r^{'})}} X = \emptyset, \\ {BND}_{{IFG}_{1}_{(a^{'}, r^{'})}} X = U, \\ {POS}_{{IFG}_{2}_{(a^{'}, r^{'})}} X = & {u_{1}, u_{4}}, {NEG}_{{IFG}_{2}_{(a^{'}, r^{'})}} X = \emptyset, \\ {BND}_{{IFG}_{2}_{(a^{'}, r^{'})}} X = {u_{2}, u_{3}, u_{5}} . \end{matrix}$

Given these three regions for set X, we can compute the IFG_i-definability and -undefinability measures as $\begin{matrix} α_{{IFG}_{1}_{(a^{'}, r^{'})}} (X) & = 0 & β_{{IFG}_{1}_{(a^{'}, r^{'})}} (X) & = \infty, \\ α_{{IFG}_{2}_{(a^{'}, r^{'})}} (X) & = 0.4 & β_{{IFG}_{2}_{(a^{'}, r^{'})}} (X) & = 1.5 . \end{matrix}$

Also, the inconsistency and generality measures are obtained as follows $\begin{matrix} d_{{IC}_{{IFG}_{1}_{(a^{'}, r^{'})}}} (X) & = 0 & {Generality}_{{IFG}_{1}_{(a^{'}, r^{'})}} (X) & = 0, \\ d_{{IC}_{{IFG}_{2}_{(a^{'}, r^{'})}}} (X) & = 0.6 & {Generality}_{{IFG}_{2}_{(a^{'}, r^{'})}} (X) & = 1 . \end{matrix}$

These results can be interpreted in the same way as before, i.e. changing the thresholds did not work for the first doctor, however the second one decides that patient suffers from diseases u₁ and u₄ with acceptance accuracy of at least %70 and rejection error of at most %20. However, the patient is suspicious of being infected by the disease U \ {u₁, u₄}. Note that with both pairs of thresholds, the second doctor comes up with contradictory decisions for boundary elements. This is happened since S₂ (A, B) does not satisfy complement compatibility condition (Definition 6.1).

8 Conclusion and future research directions

The main contributions of this paper are as follows:

Introducing and studying the notion of intuitionistic fuzzy S-approximation operators and three pairs of lower and upper approximation operators,

Introducing and studying a pair of thresholds and giving its interpretations along defining five situations regarding these thresholds, that is (a, r)-acceptance, (a, r)-rejection, non-commitment, inconsistent and uncertain,

Defining and studying three-way regions for an intuitionistic fuzzy S-approximation space using a pair of thresholds,

Introducing measures of inconsistency and generality regarding three regions,

Introducing and investigating two new subclasses of intuitionistic fuzzy S-approximation spaces, monotone and complement compatible ones,

Future works shall study possible applications of (intuitionistic fuzzy) S-approximation spaces in information systems, decision systems, multi-granular computing in addition to the problem of communicating among (intuitionistic fuzzy) S-approximation spaces with different degrees of granularity. Also, defining neighborhood systems [41] for (intuitionistic fuzzy) S-approximation spaces like the ones in [30, 32], operations among these construction like [15] and probabilistic interpretations of these spaces similar to [1] as well as introducing uncertainty measures like the ones in [33] are some other fruitful and open research directions.

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