Invariant characteristics of knowledge structures in a knowledge base under homomorphisms and their uncertainty measures 1

Abstract

A knowledge base is one basic notion in rough set theory. In a knowledge base, one can approximately describe target notions by using existing knowledge structures. This paper investigates invariant characteristics of knowledge structures in a knowledge base under homomorphisms and their uncertainty measures. First, partial dependence knowledge structures in the same knowledge base is proposed and the concept of knowledge structure bases is presented. Then, invariant and inverse invariant characteristics of knowledge structures in a knowledge base under homomorphisms are obtained. Next, measuring uncertainty of knowledge structures in the same knowledge base is investigated. Finally, two examples are employed to illustrate that knowledge granulation, rough entropy, knowledge entropy and knowledge amount of knowledge structures, and knowledge distance between knowledge structures are neither invariant nor inverse invariant under homomorphisms, and third example shows the feasibility of the proposed measures for uncertainty of knowledge structures in the same knowledge base. These results will be helpful for building a skeleton of granular computing in knowledge bases.

Keywords

Rough set theory knowledge base knowledge granule knowledge structure dependence homomorphism characteristic uncertainty

1 Introduction

Rough set theory, presented by Pawlak [21], is an effective tool to copy with uncertainty. It has been applied to some fields such as intelligent systems, expert systems, knowledge discovery, machine learning, pattern recognition, inductive reasoning, image processing, decision analysis and signal analysis [21 –24].

From the angle of rough set theory, knowledge implies the capacity to classify objects [4 , 21]. Initially, rough set theory just addresses equivalence relations. Every equivalence relation on a given universe ascertains a classification on this universe and vice versa. One handles not only a single classification, but also a collection of classifications [4, 13]. This causes the notion of knowledge bases.

In a knowledge base, a given equivalence relation divides the universe into some disjoint classes, these classes are said to be equivalence classes with respect to this equivalence relation. An equivalence relation may be viewed a special kind of similarity between objects from a data set. If two elements of the universe belong to the same equivalence class, then we may say that they cannot be distinguished under this equivalence relation. Thus, every equivalence class can be viewed as an information granule consisting of indistinguishable objects [14 , 26]. All these equivalence classes or information granules constitutes a vector, this vector is referred to as a knowledge structure of this equivalence relation. Obviously, a knowledge structure is a granular structure in the meaning of granular computing [36 –39]. In a knowledge base, one can approximately describe target notions by using existing knowledge structures.

The study of a knowledge base is an important issue. Some scholars have done some research contributions. For example, Qian et al. [26] studied knowledge structure in a tolerance knowledge base by means of set families and proposed knowledge distance between knowledge structures; Li et al. [8] discussed relationships between knowledge bases and proved that knowledge reductions, coordinate families and necessary relations in a knowledge base are invariant and inverse invariant under homomorphisms; Li et al. [9] studied properties of knowledge structures in a knowledge base by means of condition information amounts, inclusion degrees and lower approximation operators, and gave group, lattice, mapping, and soft characterizations of knowledge structures in a knowledge base; Qin [25] showed that *-reductions in a knowledge base are invariant and inverse invariant under homomorphisms.

To evaluate uncertainty of a system, Shannon [29] introduced the concept of entropy. Some scholars have applied the extension of entropy and its variants to rough sets. For example, D $\overset{˙˙}{u}$ ntsch et al. [2] proposed information entropy and three kinds of conditional entropies in rough sets for predicting a decision attribute; Beaubouef et al. [1] presented a method measuring uncertainty of rough sets and rough relation databases; Wierman et al. [30] addressed granulation measure to measure uncertainty of information; Yao et al. [35] gave a granularity measure from the angle of granulation; Liang et al. [16, 18] studied several measures on knowledge in incomplete and complete information systems; Liang et al. [17] proposed the information entropy, rough entropy and knowledge granulation in rough set theory; Qian et al. [26] introduced knowledge granulation of knowledge structures in a tolerance knowledge base.

On invariant characteristics of knowledge structures in a knowledge base, the facts that the equality and dependence between knowledge structures in a knowledge base are invariant and inverse invariant under homomorphisms have proved (see [9]). In this paper, we will show that the partial dependence and independence between knowledge structures in a knowledge base are invariant and inverse invariant under homomorphisms. Moreover, we will propose some tools for measuring uncertainty of knowledge structures and employ examples to illustrate that the proposed tools and knowledge distance between knowledge structures are neither invariant nor inverse invariant under homomorphisms.

The remaining part of this paper is organized as follows. In Section 2, some basic notions on knowledge bases, relation information systems and homomorphisms are recalled. In Section 3, some of dependence between knowledge structures in the same knowledge base are introduced. In Sections 4, invariant characteristics of knowledge structures in a knowledge base under homomorphisms are obtained. In Sections 5, we introduce some tools for measuring uncertainty of knowledge structures in the same knowledge base. In Sections 6, some examples are given. Section 7 summarizes this paper.

2 Preliminaries

In this section, we recall some basic notions about knowledge bases, relation information systems and homomorphisms.

Throughout this paper, U, V denotes two non-empty finite sets, 2^U denotes the family of all subsets of U, |X| denotes the cardinality of X ∈ 2^U, 2^U×U denotes the family of all binary relations on U and $R^{*} (U)$ denotes the family of all equivalence relations on U. All mappings are assumed to be surjective.

In this paper, put

U = {x ₁, x ₂, ⋯, x _n}, V = {y ₁, y ₂, ⋯, y _m},

δ = U × U, ▵ = {(x, x) : x ∈ U}.

2.1 Knowledge bases

Recall that R is a binary relation on U if R ∈ U × U. If (x, y) ∈ R, then we denote it by xRy.

Let R be a binary relation on U. Then R is referred to as

(1) reflexive, if xRx for any x ∈ U;

(2) symmetric, if xRy implies yRx for any x, y ∈ U;

(3) transitive, if xRy and yRz implies xRz for any x, y, z ∈ U.

Let R be a binary relation on U. Then R is referred to as equivalence if R is reflexive, symmetric and transitive.

Given $R \in R^{*} (U)$ . If R = δ, then R is referred to as a universal relation on U; if R =▵, then R is referred to as an identity relation on U.

Given $R \in R^{*} (U)$ . Then (U, R) is referred to as a Pawlak approximation space [21, 23]. For x ∈ U, put [x] _R = {y : xRy}. Then [x] _R is said to be the equivalence class of the element x with respect to the equivalence relation R. With the help of the equivalence relation R, the universe U is divided into some disjoint subsets. Every of these subsets is an equivalence class. All these subsets forms a quotient set induced by R, denoted by U/R.

Definition 2.1. ([21]). (U, R) is said to be a knowledge base on U, if $ℛ \subseteq R^{*} (U)$ .

By the aid of a given element of a knowledge base, one can construct a rough set of any subset on the universe in the following definition.

Definition 2.2. ([5]). Suppose that (U, R) is a knowledge base. For X ∈ 2^U and R ∈ R, put $\underline{R} (X) = {x \in U : [x]_{R} \subseteq X},$ $\bar{R} (X) = {x \in U : [x]_{R} \cap X \neq \emptyset} .$ Then $\underline{R} (X)$ and $\bar{R} (X)$ are said to be the R-lower and R-upper approximations of X, respectively.

2.2 Relation information systems

Definition 2.3. ([21]). Let U be a set of objects and A a set of attributes. Suppose that U and A are finite sets. Then the pair (U, A) is said to be an information system, if each attribute a ∈ A determines an information function a : U → V _a, where V _a is the set of function values of attribute a.

Given B ⊆ A. Define (x, y) ∈ R _B ⇔ ∀ a ∈ B, a (x) = a (y). Then R _B is an equivalence relation on U.

Wang et al. [31] introduced the following notion of relation information systems as a simplified model of information systems.

Definition 2.4. ([31]). If R ⊆ 2^U×U, then the pair (U, R) is said to be a relation informationsystem.

Clearly, a relation information system is not an information system, and an information system is also not a relation information system. But, the following definition illustrates the fact that there are certain relationships between information systems and relation information systems.

Definition 2.5. ([8]) Given that (U, A) is an information system. Denote R = {R _{a} : a ∈ A}. Then the pair (U, R) is said to be the relation information system induced by (U, A).

In fact, (U, R) in Definition 2.5 is a knowledge base.

Clearly,

(U, R) is a knowledge base ⇒ (U, R) is a relation information system.

2.3 Homomorphisms between relation information systems

Large database often appears in production practice. Homomorphism between relation information systems, proposed by Wang et al. [31], are a kind of tools to study data compression in relation information systems.

Given R ∈ 2^U×U. Put R _s (x) = {y ∈ U : xRy}. Then R _s (x) is referred to as the successor neighborhood of x ∈ U with respect to R ([33]).

Definition 2.6. ([31, 32]) Suppose that f: U → V is a mapping. For R ∈ 2^U×U, put

[x] _f = {u ∈ U : f (u) = f (x)}, (x) _R = {u ∈ U : R _s (u) = R _s (x)}.

If [x] _f ⊆ R _s (u) or [x] _f∩ R _s (u) ≠ ∅ for any x, u ∈ U, then f is referred to as type-1 consistent on R. If [x] _f ⊆ (x) _R for any x ∈ U, then f is referred to as type-2 consistent on R.

Given R ⊆ 2^U×U. If f is type-1 (resp. type-2) consistent on R for every R ∈ R, then f is called type-1 (resp. type-2) consistent on R.

Definition 2.7. ([31, 32]) Given that f : U → V is a mapping. Define $\hat{f} : 2^{U \times U} \to 2^{V \times V},$ $R \to \hat{f} (R) = ⋃_{x \in U} ({f (x)} \times f (R_{s} (x)));$ ${\hat{f}}^{- 1} : 2^{V \times V} \to 2^{U \times U},$ $T \to {\hat{f}}^{- 1} (T) = ⋃_{y \in V} ({f^{- 1} (y)} \times f^{- 1} (T_{s} (y))) .$ Then $\hat{f}$ and ${\hat{f}}^{- 1}$ are said to be the relation mapping and inverse relation mapping induced by f, respectively.

Clearly, $y_{1} \hat{f} (R) y_{2} \Leftrightarrow \exists x_{1}, x_{2} \in U, y_{1} = f (x_{1}), y_{2} = f (x_{2}), x_{1} {Rx}_{2},$ $f (x_{1}) \hat{f} (R) f (x_{2}) \Leftarrow x_{1} {Rx}_{2};$

$x_{1} {\hat{f}}^{- 1} (T) x_{2} \Leftrightarrow \exists y_{1}, y_{2} \in V, y_{1} = f (x_{1}), y_{2} = f (x_{2}), y_{1} {Ty}_{2} \Leftrightarrow f (x_{1}) Tf (x_{2}) .$ For R ⊆ 2^U×U, denote $\hat{f} (ℛ) = {\hat{f} (R) : R \in ℛ} .$

Definition 2.8. ([31]) Suppose that (U, R) is a relation information system. Given f : U → V is a mapping. Then the pair $(V, \hat{f} (ℛ))$ is referred to as an f-induced relation information system of (U, R).

Definition 2.9. ([31]) Let (U, R) be a relation information system. Suppose that $(V, \hat{f} (ℛ))$ is an f-induced relation information system of (U, R). If f is both type-1 and type-2 consistent on R, then f is referred to as a homomorphism from (U, R) to $(V, \hat{f} (ℛ))$ . We write $(U, ℛ) \sim (V, \hat{f} (ℛ)) .$

Proposition 2.10. ([31]). if f : U → V is both type-1 and type-2 consistent with respect to R ∈ 2^U×U, then

${\hat{f}}^{- 1} (\hat{f} (R)) = R .$

Theorem 2.11. ([8]). Given that f : U → V is both type-1 and type-2 consistent on R ∈ 2^U×U. then

$R \in R^{*} (U) \Leftrightarrow \hat{f} (R) \in R^{*} (V) .$

Theorem 2.12. ([8]). Given that f : U → V is a mapping. then

$T \in R^{*} (V) \Leftrightarrow {\hat{f}}^{- 1} (T) \in R^{*} (U) .$

3 Some notions of knowledge structures in a knowledge base

In this section, we recall some notions of knowledge structures in a knowledge base.

Definition 3.1. ([9]) Given R ∈ R^* (U). Then K (R) = ([x ₁] _R, [x ₂] _R, ⋯, [x _n] _R) is said to be the knowledge structure of R.

Example 3.2. $K (δ) = \tilde{U}$ ,

K (▵) = ({x ₁}, {x ₂}, ⋯, {x _n}).

Example 3.3. ([9]) Let P, Q ∈ R^* (U). Suppose that K (P) and K (Q) are the knowledge structures of P and Q, respectively. If for each i, [x _i] _P = [x _i] _Q, then K (P) and K (Q) are called to be the same. We write K (P) = K (Q).

One can investigate relationships between information structures in a knowledge based from the two aspects of separation and dependence.

To characterize the difference between knowledge structures in the same knowledge based, Qian et al. [26] propose the following concept of knowledge distance between knowledge structures.

Definition 3.4. ([26]). Given P, Q ∈ R^* (U). Suppose that K (P) and K (Q) are the knowledge structures of P and Q, respectively. Knowledge distance between K (P) and K (Q) is defined as $ρ (K (P), K (Q)) = \frac{1}{n^{2}} \sum_{i = 1}^{n} ∣ [x_{i}]_{P} \oplus [x_{i}]_{Q} ∣,$ where ∣ [x _i] _P⊕ [x _i] _Q ∣ = ∣ [x _i] _P ∪ [x _i] _Q ∣ - ∣ [x _i] _P ∩ [x _i] _Q ∣.

Definition 3.5 depicts the dependence, partial dependence and independence between knowledge structures in the same knowledge base.

Definition 3.5. ([9]) Let (U, R) be a knowledge base. Suppose that K (P) and K (Q) are the knowledge structures of P ∈ R and Q ∈ R, respectively.

(1) K (Q) is called to depend on K (P), if for each i, [x _i] _P ⊆ [x _i] _Q. We write K (P) ⊏ K (Q).

(2) K (Q) is called to depend partially on K (P), if there exists i, [x _i] _P ⊆ [x _i] _Q. We write K (P) ⊑ K (Q).

(3) K (Q) is said to be independent of K (P), if for each i, [x _i] _P ⊈ [x _i] _Q. We write K (P) ⋈ K (Q).

The following definition makes some changes to Definition 3.5 above.

Definition 3.6. Given P, Q ∈ R^* (U). Suppose that K (P) and K (Q) are the knowledge structures of P and Q, respectively.

(1) K (Q) is called to depend on K (P), if for each i, [x _i] _P ⊆ [x _i] _Q, we write K (P) ⪯ K (Q); K (Q) is called to depend strictly on K (P), if K (P) ⪯ K (Q) and K (P) ≠ K (Q), we write K (P) ≺ K (Q).

(2) K (Q) is called to depend partially on K (P), if there exists i, [x _i] _P ⊆ [x _i] _Q, we write K (P) ⊑ K (Q); K (Q) is referred to as to depend partially strictly on K (P), if K (P) ⊑ K (Q) and K (P) ≠ K (Q), we write K (P) ⊏ K (Q).

(3) K (Q) is said to be independent of K (P), if K (P) ⋈ K (Q).

In Fig. 1, (a) means K (P) ≺ K (Q), (b) means K (P) ⊏ K (Q), (c) means K (P) ⊏ K (Q) and K (Q) ⊏ K (P), (d) means K (P) ⋈ K (Q) and K (Q) ⋈ K (P).

Fig.1
Clearly,K (P) = K (Q ⇔ K (P) ⪯ K (Q), K (Q) ⪯ K (P) ;K (P)⪯ K (Q) ⇒ K (P) ⊑ K (Q), K (P) ≺ K (Q) ⇒ K (P) ⊏ K (Q) ;K (P) ⋈ K (Q) ⇔ K (P) ⋢K (Q).

Definition 3.7. Let (U, R) be a knowledge base. Then K (U, R) = {K (P) : P ∈ R} is called the knowledge structure base of (U, R).

Obviously, K (U, R^* (U)) is the knowledge structure base of the knowledge base (U, R^* (U)).

Example 3.8. Let U = {x ₁, x ₂, x ₃}. Given R = {R _i : 0 ≤ i ≤ 4} where

R ₀ =▵,

R ₁ = ▵ ∪ {(x ₁, x ₂), (x ₂, x ₁)},

R ₂ = ▵ ∪ {(x ₁, x ₃), (x ₃, x ₁)},

R ₃ = ▵ ∪ {(x ₂, x ₃), (x ₃, x ₂)},

R ₄ = δ. Then (U, R) is a knowledge base. So

K (U, R) = {K (▵), K (R ₁), K (R ₂), K (R ₃), K (δ)}, where K (▵) = ({x ₁}, {x ₂}, {x ₃}),

K (R ₁) = ({x ₁, x ₂}, {x ₁, x ₂}, {x ₃}),

K (R ₂) = ({x ₁, x ₃}, {x ₂}, {x ₁, x ₃}),

K (R ₃) = ({x ₁}, {x ₂, x ₃}, {x ₂, x ₃}),

$K (δ) = \tilde{U}$ .

We have

K (▵) ≺ K (R ₁) ⊏ K (▵), K (R ₁) ⊏ K (R ₂) ⊏ K (R ₁), K (R ₁) ⊏ K (R ₃) ⊏ K (R ₁), K (R ₁) ≺ K (δ) ⋈ K (R ₁);

K (▵) ≺ K (R ₂) ⊏ K (▵), K (R ₂) ⊏ K (R ₃) ⊏ K (R ₂), K (R ₂) ≺ K (δ) ⋈ K (R ₂);

K (▵) ≺ K (R ₃) ⊏ K (▵), K (R ₃) ≺ K (δ) bowtieK (R ₃) (see Fig. 2).
Fig.2
Theorem 3.9. ([9]) Suppose P, Q ∈ R^ (U). Then the following are equivalent:

(1) K (P) = K (Q);

(2) U/P = U/Q;

(3) P = Q.

Theorem 3.10. ([9]) Given P, Q ∈ R^* (U). Then the following are equivalent*:

(1) K (P) ⊏ K (Q);

(2) U/P refines U/Q, i.e., for each A ∈ U/P, there exists B ∈ U/Q such that A ⊆ B;

(3) P ⪯ Q.
4 Invariant characteristics of knowledge structures in a knowledge base under homomorphisms

In this section, invariant characteristics of knowledge structures in a knowledge base under homomorphisms are obtained.

Theorem 4.1. ([8]) Suppose $(U, ℛ) \sim_{f} (V, \hat{f} (ℛ))$ . Then $(U, ℛ) is a knowledge base \Leftrightarrow (V, \hat{f} (ℛ)) is a knowledge base .$

Theorem 4.2. ([9]) Let (U, R) be a knowledge base. If $(U, ℛ) \sim_{f} (V, \hat{f} (ℛ))$ , then for P, Q ∈ R,

$K (P) = K (Q) \Leftrightarrow K (\hat{f} (P)) = K (\hat{f} (Q)) .$

Theorem 4.3. ([9]) Given that (U, R) is a knowledge base. If $(U, ℛ) \sim_{f} (V, \hat{f} (ℛ))$ , then for P, Q ∈ R,

$K (P) ⪯ K (Q) \Leftrightarrow K (\hat{f} (P)) ⪯ K (\hat{f} (Q)) .$

Corollary 4.4. Suppose that (U, R) is a knowledge base. If $(U, ℛ) \sim_{f} (V, \hat{f} (ℛ))$ , then for P, Q ∈ R,

$K (P) ≺ K (Q) \Leftrightarrow K (\hat{f} (P)) ≺ K (\hat{f} (Q)) .$

Proof. This holds by Theorems 4.2 and 4.3.

Theorem 4.5. Given that (U, R) is a knowledge base. If $(U, ℛ) \sim_{f} (V, \hat{f} (ℛ))$ , then for P, Q ∈ R,

$K (P) ⊑ K (Q) \Leftrightarrow K (\hat{f} (P)) ⊑ K (\hat{f} (Q)) .$

Proof. “⇒". Suppose K (P) ⊑ K (Q). Then, ∃ i, [x _i] _P ⊆ [x _i] _Q. Let y _j = f (x _i).

$\forall v \in [y_{j}]_{\hat{f} (P)}$ , we have $y_{j} \hat{f} (P) v$ . Then, ∃ x′, x″ ∈ U, y _j = f (x′), v = f (x″) and x′Px″.

f (x _i) = f (x′) implies x′ ∈ [x _i] _f. Since f is type-2 consistent on P and Q, we have [x _i] _f ⊆ (x _i) _P, [x _i] _f ⊆ (x _i) _Q. Then x′ ∈ (x _i) _P, x′ ∈ (x _i) _Q. So P _s (x′) = P _s (x _i) = [x _i] _P, [x′] _Q = Q _s (x _i) = [x _i] _Q.

It should be noted that [x _i] _P ⊆ [x _i] _Q. Then P _s (x′) ⊆ [x′] _Q.

x′Px″ implies x″ ∈ P _s (x′). So x″ ∈ [x′] _Q, i.e., x′Qx″.

It should be noted that y _j = f (x′), v = f (x″) and x′Qx″. Then $y_{j} \hat{f} (Q) v$ . This implies $v \in [y_{j}]_{\hat{f} (Q)}$ .

Hence $[y_{j}]_{\hat{f} (P)} \subseteq [y_{j}]_{\hat{f} (Q)}$ . This shows $K (\hat{f} (P)) ⊑ K (\hat{f} (Q))$ .

“⇐". Suppose $K (\hat{f} (P)) ⊑ K (\hat{f} (Q))$ . Then ∃ j, $[y_{j}]_{\hat{f} (P)} \subseteq [y_{j}]_{\hat{f} (Q)}$ . Let f (x _i) = y _j.

∀ u ∈ [x _i] _P, we have x _i Pu. Then $f (x_{i}) \hat{f} (P) f (u),$ i.e., $y_{j} \hat{f} (P) f (u) .$ This implies $f (u) \in [y_{j}]_{\hat{f} (P)}$ .

Since $[y_{j}]_{\hat{f} (P)} \subseteq [y_{j}]_{\hat{f} (Q)}$ , we have $f (u) \in [y_{j}]_{\hat{f} (Q)}$ . Then $y_{j} \hat{f} (Q) f (u)$ , i.e., $f (x_{i}) \hat{f} (Q) f (u)$ . So

∃ x′, x″ ∈ U, f (x _i) = f (x′), f (u) = f (x″) and x′Qx″.

f (x _i) = f (x′) implies x′ ∈ [x _i] _f. Since f is type-2 consistent on Q, we have [x _i] _f ⊆ (x _i) _Q. Then x′ ∈ (x _i) _Q. So x′ ∈ Q _s (x′) = Q _s (x _i). This implies x _i Qx′.

f (u) = f (x″) implies x″ ∈ [u] _f. Since f also is type-2 consistent on Q, we have [u] _f ⊆ (u) _Q. Then x″ ∈ (u) _Q. So x″ ∈ Q _s (x″) = Q _s (u). This implies uQx″.

It should be noted that x′Qx″, x _i Qx′ and u Q x″. Then x _i Qu. This implies u ∈ [x _i] _Q.

Thus [x _i] _P ⊆ [x _i] _Q. This shows K (P) ⊑ K (Q). □

Corollary 4.6. Given that (U, R) is a knowledge base. If $(U, ℛ) \sim_{f} (V, \hat{f} (ℛ))$ , then for P, Q ∈ R,

(1) $K (P) ⊏ K (Q) \Leftrightarrow K (\hat{f} (P)) ⊏ K (\hat{f} (Q)) .$

(2) $K (P) ⋈ K (Q) \Leftrightarrow K (\hat{f} (P)) ⋈ K (\hat{f} (Q)) .$

Proof. (1) This holds by Theorems 4.2 and 4.5.

(2) This follows from Theorem 4.5. □

Theorem 4.2, Theorem 4.3, Corollary 4.4, Theorem 4.5 and Corollary 4.6 mean that the equality, dependence, partial dependence and independence between knowledge structures in a knowledge base are invariant under homomorphisms. But the image knowledge base $(V, \hat{f} (ℛ))$ is more simpler than the original knowledge base (U, R). Therefore, we can research the equality, dependence, partial dependence and independence between knowledge structures in the original knowledge base by means of the results of the image knowledge base.

Example 4.7. (Continued from Example 4.25 in [8])

Let U = {x _i : 1 ≤ i ≤ 15}. Put

U/R ₁ = {{x ₁, x ₂, x ₄, x ₇, x ₈, x ₉, x ₁₀, x ₁₁}, {x ₃, x ₅, x ₆, x ₁₂, x ₁₃, x ₁₄, x ₁₅}},

U/R ₂ = {{x ₁, x ₄, x ₁₁, x ₁₂, x ₁₃, x ₁₄, x ₁₅}, {x ₂, x ₃, x ₅, x ₆, x ₇, x ₈, x ₉, x ₁₀}},

U/R ₃ = {{x ₁, x ₂, x ₄, x ₇, x ₈, x ₉, x ₁₀, x ₁₁, x ₁₂, x ₁₃, x ₁₄, x ₁₅}, {x ₃, x ₅, x ₆}},

U/R ₄ = {{x ₁, x ₂, x ₄, x ₇, x ₈, x ₉, x ₁₀, x ₁₁}, {x ₃, x ₅, x ₆, x ₁₂, x ₁₃, x ₁₄, x ₁₅}} ; R = {R ₁, R ₂, R ₃, R ₄}. Then (U, R) is a knowledge base.

Pick V = {y ₁, y ₂, y ₃, y ₄, y ₅, y ₆}. A mapping f : U → V is defined as follows: $\begin{array}{l} \frac{x_{1}, x_{4}, x_{11} x_{2}, x_{8} x_{3}, x_{6} x_{5} x_{7}, x_{9}, x_{10}}{y_{1} y_{2} y_{3} y_{4} y_{5}} \\ \frac{x_{12}, x_{13}, x_{14}, x_{15}}{y_{6}} \end{array}$

It is easy to check that f is both type-1 and type-2 consistent on R. Thus

$(U, ℛ) \sim_{f} (V, \hat{f} (ℛ))$ , $\hat{f} (ℛ) = {\hat{f} (R_{1}), \hat{f} (R_{2}), \hat{f} (R_{3}), \hat{f} (R_{4})}$ .

By Theorem 4.1, $(V, \hat{f} (ℛ))$ is a knowledge base.

It should be noted that K (R ₁) = K (R ₄);

K (R ₁) ⊏ K (R ₃) ⊏ K (R ₁), K (R ₂) ⊏ K (R ₃) ⊏ K (R ₂), K (R ₃) ⊏ K (R ₄) ⊏ K (R ₃). Then, by Theorem 4.6 and Corollary 4.10(1), $K (\hat{f} (R_{1})) = K (\hat{f} (R_{4}))$ ; $K (\hat{f} (R_{1})) ⊏ K (\hat{f} (R_{3})) ⊏ K (\hat{f} (R_{1}))$ , $K (\hat{f} (R_{2})) ⊏ K (\hat{f} (R_{3})) ⊏ K (\hat{f} (R_{2}))$ , $K (\hat{f} (R_{3})) ⊏ K (\hat{f} (R_{4})) ⊏ K (\hat{f} (R_{3}))$ .

Theorem 4.8. ([8]). Let (U, R) be a relation information system. Given that $(V, \hat{f} (ℛ))$ is an f-induced relation information system of (U, R). Then the following properties hold:

(1) $(V, \hat{f} (ℛ))$ is a knowledge base ⇔ $(U, {\hat{f}}^{- 1} (\hat{f} (ℛ)))$ is a knowledge base;

(2) If $(U, ℛ) \sim_{f} (V, \hat{f} (ℛ))$ , then ${\hat{f}}^{- 1} (\hat{f} (ℛ)) = ℛ$ .

Theorem 4.9. ([9]). Suppose that (U, R) is a knowledge base. If $(U, ℛ) \sim_{f} (V, \hat{f} (ℛ))$ , then for any $S, T \in \hat{f} (ℛ)$ ,

$K (S) = K (T) \Leftrightarrow K ({\hat{f}}^{- 1} (S)) = K ({\hat{f}}^{- 1} (T)) .$

Theorem 4.10. ([9]). Given that (U, R) is a knowledge base. If $(U, ℛ) \sim_{f} (V, \hat{f} (ℛ))$ , then for any $S, T \in \hat{f} (ℛ)$ ,

$K (S) ⪯ K (T) \Leftrightarrow K ({\hat{f}}^{- 1} (S)) ⪯ K ({\hat{f}}^{- 1} (T)) .$

Corollary 4.11. Suppose that (U, R) is a knowledge base. If $(U, ℛ) \sim_{f} (V, \hat{f} (ℛ))$ , then for any $S, T \in \hat{f} (ℛ)$ ,

$K (S) ≺ K (T) \Leftrightarrow K ({\hat{f}}^{- 1} (S)) ≺ K ({\hat{f}}^{- 1} (T)) .$

Proof. This follows from Theorems 4.9 and 4.10.

Theorem 4.12. Suppose that (U, R) is a knowledge base. If $(U, ℛ) \sim_{f} (V, \hat{f} (ℛ))$ , then for any $S, T \in \hat{f} (ℛ)$ , $K (S) ⊑ K (T) \Leftrightarrow K ({\hat{f}}^{- 1} (S)) ⊑ K ({\hat{f}}^{- 1} (T)) .$

Proof. “⇒". Suppose K (S) ⊑ K (T). Then, ∃ j, [y _j] _S ⊆ [y _j] _T. Let f (x _i) = y _j.

$\forall u \in [x_{i}]_{{\hat{f}}^{- 1} (S)}$ , we have $x_{i} {\hat{f}}^{- 1} (S) u$ . Then f (x _i) Sf (u), i.e., y _j Sf (u). This implies f (u) ∈ [y _j] _S. Since [y _j] _S ⊆ [y _j] _T, we have f (u) ∈ [y _j] _T. This implies y _j Tf (u), i.e., f (x _i) Tf (u). Then $x_{i} {\hat{f}}^{- 1} (T) u$ . So $u \in T_{{\hat{f}}^{- 1} (T)} (x_{i})$ .

Thus $[x_{i}]_{{\hat{f}}^{- 1} (S)} \subseteq [x_{i}]_{{\hat{f}}^{- 1} (T)}$ . This shows $K ({\hat{f}}^{- 1} (S)) ⊑ K ({\hat{f}}^{- 1} (T)) .$

"⇐". Suppose $K ({\hat{f}}^{- 1} (S)) ⊑ K ({\hat{f}}^{- 1} (T))$ . Then, ∃ i, $[x_{i}]_{{\hat{f}}^{- 1} (S)} \subseteq [x_{i}]_{{\hat{f}}^{- 1} (T)}$ . Let y _j = f (x _i).

∀ v ∈ [y _j] _S, we have y _j Sv. Let f (u) = v. It should be noted that f (x _i) Sf (u). Then $x_{i} {\hat{f}}^{- 1} (S) u$ . This implies $u \in [x_{i}]_{{\hat{f}}^{- 1} (S)}$ . Since $[x_{i}]_{{\hat{f}}^{- 1} (S)} \subseteq [x_{i}]_{{\hat{f}}^{- 1} (T)}$ , we have $u \in [x_{i}]_{{\hat{f}}^{- 1} (T)}$ . Then $x_{i} {\hat{f}}^{- 1} (T) u$ . Then f (x _i) Tf (u), i.e., y _j Tv. This implies v ∈ [y _j] _T. So [y _j] _S ⊆ [y _j] _T.

Hence K (S) ⊑ K (T). □

Corollary 4.13. Given that (U, R) is a knowledge base. If $(U, ℛ) \sim_{f} (V, \hat{f} (ℛ))$ , then for any $S, T \in \hat{f} (ℛ)$ ,

(1) K (S) ⊏ K (T) ⇔ $K ({\hat{f}}^{- 1} (S)) ⊏ K ({\hat{f}}^{- 1} (T))$ .

(2) K (S) ⋈ K (T) ⇔ $K ({\hat{f}}^{- 1} (S)) ⋈ K ({\hat{f}}^{- 1} (T))$ .

Proof. (1) This follows from Theorem 4.9 and Theorem 4.12.

(2) This holds by Theorem 4.12.

Theorem 4.9, Theorem 4.10, Corollary 4.11, Theorem 4.12 and Corollary 4.13 mean that the equality, dependence, partial dependence and independence between knowledge structures in a knowledge base are inverse invariant under homomorphisms.

5 Measuring uncertainty of knowledge structures in a knowledge base

In this section, we propose some tools for measuring uncertainty of knowledge structures in the same knowledge base and give their properties.

5.1 Granularity measures for knowledge structures in a knowledge base

In a knowledge base, knowledge granulation is the average measure of knowledge granules of a given knowledge structure. It may be used to characterize the classification ability of a given knowledge structure [17 , 33].

We first propose axiom definition of knowledge granulation of knowledge structures in the same knowledge base, which makes a little adjustment to Definition 5 in [26].

Definition 5.1. Let K (U, R^* (U)) be the knowledge structure base of the knowledge base (U, R^* (U)). Suppose that G : K (U, R^* (U)) → (- ∞, + ∞) is a function. Then G is called a knowledge granulation function in K (U, R^* (U)), if G satisfies the following conditions:

(1) Non-negativity: ∀ P ∈ R^* (U), G (K (P)) ≥0;

(2) Invariability: ∀ P, Q ∈ R^* (U), if K (P) = K (Q), then G (K (P)) = G (K (Q));

(3) Monotonicity: ∀ P, Q ∈ R^* (U), if K (P) ≺ K (Q), then G (K (P)) < G (K (Q)).

Here, ∀ P ∈ R^* (U), G (K (P)) is called knowledge granulation of the knowledge structure K (P).

Similar to Definition 2 in [16], knowledge granulation of a given knowledge structure in the same knowledge base is given in the following definition.

Definition 5.2. Suppose P ∈ R^* (U). Knowledge granulation of the knowledge structure K (P) is defined as $G (K (P)) = \frac{1}{n^{2}} \sum_{i = 1}^{m} | X_{i} |^{2},$ where U/P = {X ₁, X ₂, ⋯, X _m}.

Proposition 5.3. Given P ∈ R^* (U). Then $G (K (P)) = \frac{1}{n^{2}} \sum_{i = 1}^{n} | [x_{i}]_{P} | .$ where K (P) = ([x ₁] _P, [x ₂] _P, ⋯, [x _n] _P).

Proof. Denote U/P = {X ₁, X ₂, ⋯, X _m}.

Suppose X _i = {x _i1, x _i2, ⋯, x _{is
_i}}, |X _i| = s _i. Then

$\sum_{i = 1}^{m} s_{i} = n, X_{i} = [x_{i 1}]_{P} = [x_{i 2}]_{P} = \dots = [x_{{is}_{i}}]_{P} .$

So | [x _i] _P| = | [x _i1] _P| = | [x _i2] _P| = ⋯ = | [x _{is
_i}] _P| = s _i.

Thus, ∀ i,

$| [x_{i}]_{P} |^{2} = s_{i} | [x_{i}]_{P} | = \sum_{k = 1}^{s_{i}} | [x_{ik}]_{P} | .$

Hence

$G (K (P)) = \frac{1}{n^{2}} \sum_{i = 1}^{m} | [x_{i}]_{P} |^{2} = \frac{1}{n^{2}} \sum_{i = 1}^{m} \sum_{k = 1}^{s_{i}} | [x_{ik}]_{P} |$

$= \frac{1}{n^{2}} \sum_{i = 1}^{n} | [x_{i}]_{P} | .$

□

Propositions 5.3 shows the fact that knowledge granulation of a give knowledge structure is fully based on knowledge granules of this knowledge structure.

Proposition 5.4. Suppose P ∈ R^* (U). Then $\frac{1}{n} \leq G (K (P)) \leq 1 .$ Moreover, if P is an identity relation, then E _r (K (P)) achieves the minimum value $\frac{1}{n}$ ; if P is a universal relation, then E _r (K (P)) achieves the maximum value 1, specially, G (K (∅)) =1.

Proof. Since ∀ i, 1 ≤ | [x _i] _P| ≤ n, $n \leq \sum_{i = 1}^{n} | [x_{i}]_{P} | \leq n^{2}$ .

By Proposition 5.3, $\frac{1}{n} \leq G (K (P)) \leq 1 .$

If P is an identity relation, then ∀ i, | [x _i] _P|=1. So $G (K (P)) = \frac{1}{n}$ .

If P is a universal relation, then ∀ i, | [x _i] _P| = n. So G (K (P)) =1.

Theorem 5.5. Given P, Q ∈ R^* (U). If K (P) ≺ K (Q), then G (K (P)) < G (K (Q)).

Proof. By Proposition 5.3,

$G (K (P)) = \frac{1}{n^{2}} \sum_{i = 1}^{n} | [x_{i}]_{P} |, G (K (Q)) = \frac{1}{n^{2}} \sum_{i = 1}^{n} | [x_{i}]_{Q} | .$

Since K (P) ≺ K (Q), we have ∀ i, [x _i] _P ⊆ [x _i] _Q and ∃ j, [x _j] _P ⊊ [x _j] _Q. Then, ∀ i, | [x _i] _P| ≤ | [x _i] _Q| and ∃ j, | [x _j] _P| < | [x _j] _Q|.

Hence G (K (P)) < G (K (Q)). □

This theorem illustrates the fact that the knowledge granulation measure increases when the available knowledge becomes coarser, and it decreases when the available knowledge becomes finer. That is to say, the more uncertain the available knowledge is, the bigger the knowledge granulation value becomes. Thus, we can come to the conclusion that the knowledge granulation introduced in Definition 5.2 may be used to measure the certainty degree of a given knowledge structure in the same knowledge base.

Theorem 5.6. GK in Definition 5.2 is a knowledge granulation function under Definition 5.1.

Proof. (1) Obviously, “non-negativity" holds.

(2) Given $P, Q \in R^{*} (U)$ . If K (P) = K (Q), then ∀ i, [x _i] _P = [x _i] _Q.

By Proposition 5.3, G (K (P)) = G (K (Q)).

(3) “Monotonicity" follows from Theorem 5.5.

5.2 Entropy measures for knowledge structures in a knowledge base

In physics, entropy is often used to measure out-of-order degree of a system. The bigger the entropy value is, the higher out-of-order of a system will be. Shannon [29] applied the concept of entropy in physics to information theory for measuring uncertainty of the structure of a system.

Definition 5.7. ([2]). Let A = {X ₁, X ₂, ⋯⋯, X _k} be one partition on U. Then $H (A) = - \sum_{i = 1}^{m} \frac{| X_{i} |}{n} {log}_{2} \frac{| X_{i} |}{n}$ is referred to as the information entropy of A, where $p (X_{i}) = \frac{∣ X_{i} ∣}{n}$ means the probability that the element of U belongs to X _i.

Similarly, knowledge entropy of a knowledge structure in the same knowledge base is given in the following definition.

Definition 5.8. Suppose P ∈ R^* (U). Knowledge entropy of the knowledge structure K (P) is defined as $H (K (P)) = - \sum_{i = 1}^{m} \frac{| X_{i} |}{n} {log}_{2} \frac{| X_{i} |}{n},$ where U/P = {X ₁, X ₂, ⋯, X _m}. When some $\frac{| X_{i} |}{n}$ equals 0, then 0 · log ₂0 = 0.

Proposition 5.9. Given P ∈ R^* (U). Then

$H (K (P)) = - \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{| [x_{i}]_{P} |}{n} .$ where K (P) = ([x ₁] _P, [x ₂] _P, ⋯, [x _n] _P).

Proof. Denote $U / P = {X_{1}, X_{2}, \dots, X_{m}} .$

Suppose X _i = {x _i1, x _i2, ⋯, x _{is
_i}}, |X _i| = s _i. Then $\sum_{i = 1}^{m} s_{i} = n, X_{i} = [x_{i 1}]_{P} = [x_{i 2}]_{P} = \dots = [x_{{is}_{i}}]_{P} .$

So | [x _i] _P| = | [x _i1] _P| = | [x _i2] _P| = ⋯ = | [x _{is
_i}] _P| = s _i.

Thus, ∀ i,

$\frac{| [x_{i}]_{P} |}{n} {log}_{2} \frac{| [x_{i}]_{P} |}{n} = s_{i} \frac{1}{n} {log}_{2} \frac{| [x_{i}]_{P} |}{n}$ $= \sum_{k = 1}^{s_{i}} \frac{1}{n} {log}_{2} \frac{| [x_{ik}]_{P} |}{n} .$

Hence $\begin{matrix} H (K (P)) & = - \sum_{i = 1}^{m} \frac{| [x_{i}]_{P} |}{n} {log}_{2} \frac{| [x_{i}]_{P} |}{n} \\ = - \sum_{i = 1}^{m} \sum_{k = 1}^{s_{i}} \frac{1}{n} {log}_{2} \frac{| [x_{ik}]_{P} |}{n} \\ = - \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{| [x_{i}]_{P} |}{n} . \end{matrix}$

□

Propositions 5.9 shows the fact that knowledge entropy of a give knowledge structure is fully based on knowledge granules of this knowledge structure.

Theorem 5.10. Suppose P, Q ∈ R^* (U). If K (P) ≺ K (Q), then H (K (Q)) < H (K (P)).

Proof. By Proposition 5.9,

$H (K (P)) = - \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{| [x_{i}]_{P} |}{n}, H (K (Q))$

$= - \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{| [x_{i}]_{Q} |}{n} .$

Since K (P) ≺ K (Q), similar to the proof of Theorem 5.5, we have ∀ i, 1 ≤ | [x _i] _P| ≤ | [x _i] _Q| and ∃ j, 1 ≤ | [x _j] _P| < | [x _j] _Q|.

Then, ∀ i,

$- {log}_{2} \frac{| [x_{i}]_{P} |}{n} = {log}_{2} \frac{n}{| [x_{i}]_{P} |}$ $\geq {log}_{2} \frac{n}{| [x_{i}]_{Q} |} = - {log}_{2} \frac{| [x_{i}]_{Q} |}{n}$

and ∃ j,

$- {log}_{2} \frac{| [x_{j}]_{P} |}{n} = {log}_{2} \frac{n}{| [x_{j}]_{P} |}$ $> {log}_{2} \frac{n}{| [x_{j}]_{Q} |} = - {log}_{2} \frac{| [x_{j}]_{Q} |}{n} .$ Hence H (K (Q)) < H (K (P)). □

From Theorem 5.10, we can come to the conclusion that the knowledge entropy proposed in Definition 5.8 may be used to measure the certainty degree of a given knowledge structure in the same knowledge base. That is to say, the more uncertain the available knowledge is, the smaller the knowledge entropy value becomes.

Rough entropy, introduced by Yao [35], is used to measure granularity of a given partition. It is also called co-entropy by some scholars [1]. Similarly, rough entropy of a given knowledge structure in a knowledge base is proposed in the following definition.

Similar to Definition 6 in [16], rough entropy of of a knowledge structure in the same knowledge base is given in the following definition.

Definition 5.11. Given P ∈ R^* (U). Rough entropy of the knowledge structure K (P) is defined as $E_{r} (K (P)) = - \sum_{i = 1}^{m} \frac{| X_{i} |}{n} {log}_{2} \frac{1}{| X_{i} |},$ where U/P = {X ₁, X ₂, ⋯, X _m}.

Proposition 5.12. Suppose P ∈ R^* (U). Then $E_{r} (K (P)) = - \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{1}{| [x_{i}]_{P} |} .$ where K (P) = ([x ₁] _P, [x ₂] _P, ⋯, [x _n] _P).

Proof. Denote $U / P = {X_{1}, X_{2}, \dots, X_{m}} .$

Suppose X _i = {x _i1, x _i2, ⋯, x _{is
_i}}, |X _i| = s _i. Then $\sum_{i = 1}^{m} s_{i} = n, X_{i} = [x_{i 1}]_{P} = [x_{i 2}]_{P} = \dots = [x_{{is}_{i}}]_{P} .$ So $| [x_{i}]_{P} | = | [x_{i 1}]_{P} | = | [x_{i 2}]_{P} | = \dots = | [x_{{is}_{i}}]_{P} | = s_{i} .$ Thus, ∀ i,

$| [x_{i}]_{P} | {log}_{2} \frac{1}{| [x_{i}]_{P} |} = s_{i} {log}_{2} \frac{1}{| [x_{i}]_{P} |} = \sum_{k = 1}^{s_{i}} {log}_{2} \frac{1}{| [x_{ik}]_{P} |} .$

Hence $\begin{matrix} E_{r} (K (P)) & = & - \sum_{i = 1}^{m} \frac{| [x_{i}]_{P} |}{n} {log}_{2} \frac{1}{| [x_{i}]_{P} |} \\ = & - \frac{1}{n} \sum_{i = 1}^{m} | [x_{i}]_{P} | {log}_{2} \frac{1}{| [x_{i}]_{P} |} \\ = & - \frac{1}{n} \sum_{i = 1}^{m} \sum_{k = 1}^{s_{i}} {log}_{2} \frac{1}{| [x_{ik}]_{P} |} \\ = & - \frac{1}{n} \sum_{i = 1}^{n} {log}_{2} \frac{1}{| [x_{i}]_{P} |} \\ = & - \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{1}{| [x_{i}]_{P} |} . \end{matrix}$ □

Propositions 5.12 shows the fact that rough entropy of a give knowledge structure is fully based on knowledge granules of this knowledge structure.

Proposition 5.13. Given P ∈ R^* (U). Then 0 ≤ E _r (K (P)) ≤ log ₂ n.

Moreover, if P is an identity relation, then E _r (K (P)) achieves the minimum value 0; if P is a universal relation, then E _r (K (P)) achieves the maximum value log ₂ n.

Proof. Since ∀ i, 1 ≤ | [x _i] _P| ≤ n, $0 \leq - {log}_{2} \frac{1}{| [x_{i}]_{P} |} = {log}_{2} | [x_{i}]_{P} | \leq {log}_{2} n$ . Then $0 \leq - \sum_{i = 1}^{n} {log}_{2} \frac{1}{| [x_{i}]_{P} |} \leq n {log}_{2} n$ .

By Proposition 5.12, $0 \leq E_{r} (K (P)) \leq {log}_{2} n .$

If P is an identity relation, then ∀ i, | [x _i] _P|=1. So E _r (K (P)) =0.

If P is a universal relation, then ∀ i, | [x _i] _P| = n. So E _r (K (P)) = log ₂ n.

Theorem 5.14. Suppose P, Q ∈ R^* (U). If K (P) ≺ K (Q), then E _r (K (P)) < E _r (K (Q)).

Proof. By Proposition 5.12,

$E_{r} (K (P)) = - \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{1}{| [x_{i}]_{P} |},$

$E_{r} (K (Q)) = - \sum_{i = 1}^{n} \frac{1}{n} {log}_{2} \frac{1}{| [x_{i}]_{Q} |} .$

Since K (P) ≺ K (Q), similar to the proof of Theorem 5.5, we have ∀ i, 1 ≤ | [x _i] _P| ≤ | [x _i] _Q| and ∃ j, 1 ≤ | [x _j] _P| < | [x _j] _Q|.

Then ∀ i,

$- {log}_{2} \frac{1}{| [x_{i}]_{P} |} = {log}_{2} | [x_{i}]_{P} |$ $\leq {log}_{2} | [x_{i}]_{Q} | = - {log}_{2} \frac{1}{| [x_{i}]_{Q} |},$

and ∃ j,

$- {log}_{2} \frac{1}{| [x_{j}]_{P} |} = {log}_{2} | [x_{j}]_{P} |$ $< {log}_{2} | [x_{j}]_{Q} | = - {log}_{2} \frac{1}{| [x_{j}]_{Q} |} .$ Hence E _r (K (P)) < E _r (K (Q)). □

This theorem illustrates the fact that the rough entropy measure increases when the available knowledge becomes coarser, and it decreases when the available knowledge becomes finer. That is to say, the more uncertain the available knowledge is, the bigger the rough entropy value becomes. Thus, we can come to the conclusion that the rough entropy proposed in Definition 5.11 may be used to measure the certainty degree of a given knowledge structure in the same knowledge base.

Theorem 5.15. E _r in Definition 5.11 is a knowledge granulation function under Definition 5.1.

Proof. (1) Obviously, “non-negativity" holds.

(2) Given $P, Q \in R^{*} (U)$ . If K (P) = K (Q), then ∀ i, [x _i] _P = [x _i] _Q.

By Proposition 5.12, E _r (K (P)) = E _r (K (Q)).

(3) “Monotonicity" follows from Theorem 5.14.

5.3 Knowledge amounts of knowledge structures in a knowledge base

Definition 5.16. ([40]) Let A = {X ₁, X ₂, ⋯⋯, X _k} be a partition on U. then $H () = \sum_{i = 1}^{k} p (X_{i}) p (U - X_{i})$ is said to be the information amount of A, where $p (X_{i}) = \frac{∣ X_{i} ∣}{n}$ (resp. $p (U - X_{i}) = \frac{| U - X_{i} |}{n}$ ) means the probability that the element of U belongs to X _i (resp.U - X _i).

Similarly, knowledge amount of a knowledge structure in the same knowledge base is given in the following definition.

Definition 5.17. Given P ∈ R^* (U). Knowledge amount of the knowledge structure K (P) is defined as

$E (K (P)) = \sum_{i = 1}^{m} \frac{| X_{i} |}{n} \frac{| U - X_{i} |}{n},$ where U/P = {X ₁, X ₂, ⋯, X _m}.

Proposition 5.18. Given P ∈ R^* (U). Then $E (K (P)) = \sum_{i = 1}^{n} \frac{1}{n} (1 - \frac{| [x_{i}]_{P} |}{n}) .$ where K (P) = ([x ₁] _P, [x ₂] _P, ⋯, [x _n] _P).

Proof. Denote U/P = {X ₁, X ₂, ⋯, X _m}.

Suppose X _i = {x _i1, x _i2, ⋯, x _{is
_i}}, |X _i| = s _i. Then $\sum_{i = 1}^{m} s_{i} = n, X_{i} = [x_{i 1}]_{P} = [x_{i 2}]_{P} = \dots = [x_{{is}_{i}}]_{P} .$

So | [x _i] _P| = | [x _i1] _P| = | [x _i2] _P| = ⋯ = | [x _{is
_i}] _P| = s _i.

Since {X ₁, X ₂, ⋯, X _m} is a partition on U, ∀ i, we have

$U - X_{i} = (⋃_{k = 1}^{i - 1} X_{k}) ⋃ (⋃_{k = i + 1}^{m} X_{k}) .$

Then

$| U - X_{i} | = \sum_{k = 1}^{i - 1} | X_{k} | + \sum_{k = i + 1}^{m} | X_{k} |$ = |U| - | [x _i] _P| = n - | [x _i] _P|.

Thus, ∀ i, $| [x_{i}]_{P} | | U - X_{i} | = s_{i} (n - | [x_{i}]_{P} |) = \sum_{k = 1}^{s_{i}} (n - | [x_{ik}]_{P} |) .$

Hence $\begin{matrix} E (K (P)) & = \sum_{i = 1}^{m} \frac{| [x_{i}]_{P} |}{n} \frac{| U - X_{i} |}{n} \\ = \sum_{i = 1}^{m} \sum_{k = 1}^{s_{i}} \frac{n - | [x_{ik}]_{P} |}{n^{2}} \\ = \sum_{i = 1}^{n} \frac{n - | [x_{i}]_{P} |}{n^{2}} \\ = \sum_{i = 1}^{n} \frac{1}{n} (1 - \frac{| [x_{i}]_{P} |}{n}) . \end{matrix}$ □

Proposition 5.18 shows the fact that knowledge amount of a give knowledge structure is fully based on knowledge granules of this knowledge structure.

Theorem 5.19. Let P, Q ∈ R^* (U). If K (P) ≺ K (Q), then E (K (Q)) < E (K (P)).

Proof. By Proposition 5.18,

$E (K (P)) = \sum_{i = 1}^{n} \frac{1}{n} (1 - \frac{| [x_{i}]_{P} |}{n}),$

$E (K (Q)) = \sum_{i = 1}^{n} \frac{1}{n} (1 - \frac{| [x_{i}]_{Q} |}{n}) .$

Since K (P) ≺ K (Q), similar to the proof of Theorem 5.5, we have ∀ i, 1 ≤ | [x _i] _P| ≤ | [x _i] _Q| and ∃ j, $1 \leq | [x_{j}]_{P} | < | [x_{j}]_{Q} | .$

Hence E (K (Q)) < E (K (P)). □

From Theorem 5.19, we can come to the conclusion that the knowledge amount proposed in Definition 5.17 may be used to measure the certainty degree of a given knowledge structure in the same knowledge base. That is to say, the more uncertain the available knowledge is, the smaller the knowledge amount value becomes.

5.4 Properties

Theorem 5.20. Given P, Q ∈ R^* (U). Then G (K (P)) + E (K (P)) =1.

Proof. Denote $U / P = {X_{1}, X_{2}, \dots, X_{m}} .$

Suppose X _i = {x _i1, x _i2, ⋯, x _{is
_i}}, |X _i| = s _i. Then $\sum_{i = 1}^{m} s_{i} = n$ .

Since {X ₁, X ₂, ⋯, X _m} is a partition on U, ∀ i, we have

$U - X_{i} = (⋃_{k = 1}^{i - 1} X_{k}) ⋃ (⋃_{k = i + 1}^{m} X_{k}) .$

Then

$| U - X_{i} | = \sum_{k = 1}^{i - 1} | X_{k} | + \sum_{k = i + 1}^{m} | X_{k} |$ = |U| - | [x _i] _P| = n - | [x _i] _P|.

Then $\begin{matrix} E (K (P)) & = & \sum_{i = 1}^{m} \frac{| [x_{i}]_{P} |}{n} \frac{| U - X_{i} |}{n} \\ = & \sum_{i = 1}^{m} \frac{s_{i} (n - s_{i})}{n^{2}} \\ = & \sum_{i = 1}^{m} \frac{s_{i}}{n} - \sum_{i = 1}^{m} \frac{s_{i}^{2}}{n^{2}} \\ = & 1 - \sum_{i = 1}^{m} \frac{| [x_{i}]_{P} |^{2}}{n^{2}} \\ = & 1 - G (K (P)) . \end{matrix}$ Thus G (K (P)) + E (K (P)) =1. □

Corollary 5.21. Suppose P ∈ R^* (U). Then $0 \leq E (K (P)) \leq 1 - \frac{1}{n} .$

Proof. By Proposition 5.4, $\frac{1}{n} \leq G (K (P)) \leq 1$ .

By Theorem 5.20, E (K (P)) =1 - G (K (P)).

Thus $0 \leq E (K (P)) \leq 1 - \frac{1}{n}$ .

Theorem 5.22. Given P ∈ R^* (U). Then E _r (K (P)) + H (K (P)) = log ₂ n.

Proof. Denote U/P = {X ₁, X ₂, ⋯, X _m}.

Suppose X _i = {x _i1, x _i2, ⋯, x _{is
_i}}, |X _i| = s _i. Then $\sum_{i = 1}^{m} s_{i} = n$ .

Thus $\begin{matrix} E_{r} (K (P)) + H (K (P)) \\ = - \sum_{i = 1}^{m} \frac{| [x_{i}]_{P} |}{n} {log}_{2} \frac{1}{| [x_{i}]_{P} |} - \sum_{i = 1}^{m} \frac{| [x_{i}]_{P} |}{n} {log}_{2} \\ \frac{| [x_{i}]_{P} |}{n} \\ = - \sum_{i = 1}^{m} \frac{s_{i}}{n} {log}_{2} \frac{1}{s_{i}} - \sum_{i = 1}^{m} \frac{s_{i}}{n} {log}_{2} \frac{s_{i}}{n} \\ = - \sum_{i = 1}^{m} \frac{s_{i}}{n} ({log}_{2} 1 - {log}_{2} s_{i} + {log}_{2} s_{i} - {log}_{2} n) \\ = - \sum_{i = 1}^{m} \frac{s_{i}}{n} ({log}_{2} 1 - {log}_{2} n) = - \sum_{i = 1}^{m} \frac{s_{i}}{n} (- {log}_{2} n) \\ = {log}_{2} n . \end{matrix}$ □

Corollary 5.23. Let P ∈ R^* (U). Then $0 \leq H (K (P)) \leq \log_{2} n .$

Proof. By Proposition 5.13, 0 ≤ E _r (K (P)) ≤ log ₂ n.

By Theorem 5.22, H (K (P)) = log ₂ n - E _r (K (P))

Thus 0 ≤ H (K (P)) ≤ log ₂ n.

6 Examples

Example 6.1. (Continued from Example 4.7) We have

R ₁, R ₂ ∈ R and $(U, ℛ) \sim (V, \hat{f} (ℛ)) .$

By Definition 3.4, one can obtain that $ρ (K (R_{1}), K (R_{2})) = \frac{1}{15^{2}} \sum_{i = 1}^{15} ∣ [x_{i}]_{R_{1}} \oplus [x_{i}]_{R_{2}} ∣ = \frac{108}{225},$ $ρ (K (\hat{f} (R_{1})), K (\hat{f} (R_{2}))) = \frac{1}{6^{2}} \sum_{i = 1}^{6} ∣ [x_{i}]_{\hat{f} (R_{1})}$ $\oplus [x_{i}]_{\hat{f} (R_{2})} ∣ = \frac{1}{2} .$

Thus $ρ (K (R_{1}), K (R_{2})) \neq ρ (K (\hat{f} (R_{1})), K (\hat{f} (R_{2}))) .$

Pick $S = \hat{f} (R_{1}), T = \hat{f} (R_{2})$ . By Proposition 2.11, ${\hat{f}}^{- 1} (S) = R_{1}, {\hat{f}}^{- 1} (T) = R_{2} .$ Thus $ρ (K (S), K (T)) \neq ρ (K ({\hat{f}}^{- 1} (S), K ({\hat{f}}^{- 1} (T))) .$

This example illustrates the fact that knowledge distance between knowledge structures in a given knowledge base is neither invariant nor inverse invariant under homomorphisms.

Example 6.2. (Continued from Example 4.7) We have

U/R ₂ = {{x ₁, x ₄, x ₁₁, x ₁₂, x ₁₃, x ₁₄, x ₁₅}, {x ₂, x ₃, x ₅, x ₆, x ₇, x ₈, x ₉, x ₁₀}},

$V / \hat{f} (R_{2}) = {{y_{1}, y_{6}}, {y_{2}, y_{3}, y_{4}, y_{5}}} .$ And R ₂ ∈ R, $(U, ℛ) \sim (V, \hat{f} (ℛ)) .$

By Definition 5.2,

$G (K (R_{2})) = \frac{1}{15^{2}} \sum_{i = 1}^{15} | [x_{i}]_{R_{2}} |^{2} = \frac{171}{45},$

$G (K (\hat{f} (R_{2}))) = \frac{1}{6^{2}} \sum_{j = 1}^{6} | [y_{j}]_{\hat{f} (R_{2})} |^{2} = 2 .$

By Definition 5.8, $\begin{matrix} H (K (R_{2})) & = - \sum_{i = 1}^{15} \frac{| [x_{i}]_{R_{2}} |}{15} {log}_{2} \frac{| [x_{i}]_{R_{2}} |}{15} \\ = \frac{49}{15} {log}_{2} \frac{15}{7} + \frac{64}{15} {log}_{2} \frac{15}{8}, \end{matrix}$ $\begin{matrix} H (\hat{f} (R_{2})) & = - \sum_{j = 1}^{6} \frac{| [y_{j}]_{\hat{f} (R_{2})} |}{6} {log}_{2} \frac{| [y_{j}]_{\hat{f} (R_{2})} |}{6} \\ = \frac{2}{3} {log}_{2} 3 + \frac{8}{3} {log}_{2} \frac{3}{2} . \end{matrix}$

By Definition 5.11,

$E_{r} (R_{2}) = - \sum_{i = 1}^{15} \frac{| [x_{i}]_{R_{2}} |}{15} {log}_{2} \frac{1}{| [x_{i}]_{R_{2}} |} = \frac{64}{5} + \frac{49}{15} {log}_{2} 7,$

$E_{r} (K (\hat{f} (R_{2}))) = - \sum_{j = 1}^{6} \frac{| [y_{j}]_{\hat{f} (R_{2})} |}{6} {log}_{2} \frac{1}{| [y_{j}]_{\hat{f} (R_{2})} |} = 6 .$

By Definition 5.17,

$E (R_{2}) = \sum_{i = 1}^{15} \frac{| [x_{i}]_{R_{2}} |}{15} (1 - \frac{| [x_{i}]_{R_{2}} |}{15}) = \frac{56}{15},$

$E (\hat{f} (R_{2})) = \sum_{j = 1}^{6} \frac{| [y_{j}]_{\hat{f} (R_{2})} |}{6} (1 - \frac{| [y_{j}]_{\hat{f} (R_{2})} |}{6}) = \frac{4}{3} .$

Thus $G (K (R_{2})) \neq G (K (\hat{f} (R_{2}))), E_{r} (R_{2}) \neq$ $E_{r} (K (\hat{f} (R_{2}))),$

$H (K (R_{2})) \neq H (\hat{f} (R_{2})), E (R_{2}) \neq E (\hat{f} (R_{2})) .$

Pick $Q = \hat{f} (R_{2})$ . By Proposition 2.11, ${\hat{f}}^{- 1} (Q) = R_{2}$ .

Thus $GK (Q) \neq GK ({\hat{f}}^{- 1} (Q)), H (Q) \neq$ $H ({\hat{f}}^{- 1} (Q)),$

$E_{r} (Q) \neq E_{r} ({\hat{f}}^{- 1} (Q)), E (Q) \neq E ({\hat{f}}^{- 1} (Q)) .$

This example illustrates the fact that knowledge granulation, knowledge entropy, rough entropy and knowledge amount of knowledge structures in a given knowledge base are neither invariant nor inverse invariant under homomorphisms.

Below, we give a numerical experiment example to show the feasibility of the proposed measures.

Example 6.3. We construct a numerical experiment on the postoperative patient data set that comes from UCI Repository of machine learning databases, which is shown as Table 1. The experiment data in Table 1 is a portion randomly selected from the postoperative patient data set. Table 1 can be expressed as the information system (U, A) of postoperative patients where U = {x ₁, x ₂, x ₃, x ₄, x ₅, x ₆, x ₇, x ₈, x ₉, x ₁₀} is the set of postoperative patients and A = {internal temperature, surface temperature, oxygen saturation, last measurement of blood pressure, stability of patient’s surface temperature, stability of patient’s core temperature, stability of patient’s blood pressure, patient’s perceived comfort at discharge, discharge decision} = {a ₁, a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈, a ₉} is the set of attributes.

Table 1
The information system (U, A) of postoperative patients

U a ₁ a ₂ a ₃ a ₄ a ₅ a ₆ a ₇ a ₈ a ₉

x ₁ mid low excellent mid stable stable stable 15 A

x ₂ mid high excellent high stable stable stable 10 S

x ₃ high low excellent high stable stable mod-stable 10 A

x ₄ mid low good high stable unstable mod-stable 15 A

x ₅ mid mid excellent high stable stable stable 10 A

x ₆ high low good mid stable stable unstable 15 S

x ₇ mid low excellent high stable stable mod-stable 5 S

x ₈ high mid excellent mid unstable unstable stable 10 S

x ₉ mid high good mid stable stable stable 10 S

x ₁₀ mid low excellent mid unstable stable mod-stable 10 S

U	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆	a ₇	a ₈	a ₉
x ₁	mid	low	excellent	mid	stable	stable	stable	15	A
x ₂	mid	high	excellent	high	stable	stable	stable	10	S
x ₃	high	low	excellent	high	stable	stable	mod-stable	10	A
x ₄	mid	low	good	high	stable	unstable	mod-stable	15	A
x ₅	mid	mid	excellent	high	stable	stable	stable	10	A
x ₆	high	low	good	mid	stable	stable	unstable	15	S
x ₇	mid	low	excellent	high	stable	stable	mod-stable	5	S
x ₈	high	mid	excellent	mid	unstable	unstable	stable	10	S
x ₉	mid	high	good	mid	stable	stable	stable	10	S
x ₁₀	mid	low	excellent	mid	unstable	stable	mod-stable	10	S

(1) Denote P _i = ind ({a _i} (i = 1, 2, 3, 4, 5, 6, 7, 8, 9),

PP = {P ₁, P ₂, P ₃, P ₄, P ₅, P ₆, P ₇, P ₈, P ₉}. Then (U, PP) is the knowledge base induced by the information system (U, A).

From Table 1, we have

U/P ₁ = {{x ₁, x ₂, x ₄, x ₅, x ₇, x ₉, x ₁₀}, {x ₃, x ₆, x ₈}},

U/P ₂ = {{x ₁, x ₃, x ₄, x ₆, x ₇, x ₁₀}, {x ₂, x ₉}, {x ₅, x ₈}},

U/P ₃ = {{x ₁, x ₂, x ₄, x ₅, x ₇, x ₉, x ₁₀}, {x ₃, x ₆, x ₈}},

U/P ₄ = {{x ₁, x ₆, x ₈, x ₉, x ₁₀}, {x ₂, x ₃, x ₄, x ₅, x ₇}},

U/P ₅ = {{x ₁, x ₂, x ₃, x ₄, x ₅, x ₆, x ₇, x ₉}, {x ₈, x ₁₀}},

U/P ₆ = {{x ₁, x ₂, x ₃, x ₅, x ₆, x ₇, x ₉, x ₁₀}, {x ₄, x ₈}},

U/P ₇ = {{x ₁, x ₂, x ₅, x ₈, x ₉}, {x ₃, x ₄, x ₇, x ₁₀}, {x ₆}},

U/P ₈ = {{x ₁, x ₄, x ₆}, {x ₂, x ₃, x ₅, x ₈, x ₉, x ₁₀}, {x ₇}},

U/P ₉ = {{x ₁, x ₃, x ₄, x ₅}, {x ₂, x ₆, x ₇, x ₈, x ₉, x ₁₀}}. By Proposition 5.3,

$G (K (P_{1})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{1}} | = 0.58 G (K (P_{2})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{2}} | = 0.44$ , $G (K (P_{3})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{3}} | = 0.58 G (K (P_{4})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{4}} | = 0.5$ , $G (K (P_{5})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{5}} | = 0.68 G (K (P_{6})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{6}} | = 0.68$ , $G (K (P_{7})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{7}} | = 0.42 G (K (P_{8})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{8}} | = 0.46$ , $G (K (P_{9})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{P_{9}} | = 0.52$ .

By Proposition 5.9,

$H (K (P_{1})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{1}} |}{10} \approx 0.882, H (K (P_{2})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{2}} |}{10} \approx 1.371$ , $H (K (P_{3})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{3}} |}{10} \approx 0.882, H (K (P_{4})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{4}} |}{10} = 1$ , $H (K (P_{5})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{5}} |}{10} \approx 0.722, H (K (P_{6})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{6}} |}{10} \approx 0.722$ , $H (K (P_{7})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{7}} |}{10} \approx 1.361, H (K (P_{8})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{8}} |}{10} \approx 1.296$ , $H (K (P_{9})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{P_{9}} |}{10} \approx 0.971$ . By Proposition 5.12,

$E_{r} (K (P_{1})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{1}} |} \approx 2.44, E_{r} (K (P_{2})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{2}} |} \approx 1.951$ , $E_{r} (K (P_{3})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{3}} |} \approx 2.44, E_{r} (K (P_{4})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{4}} |} \approx 2.322$ , $E_{r} (K (P_{5})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{5}} |} = 2.6, E_{r} (K (P_{6})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{6}} |} = 2.6$ . $E_{r} (K (P_{7})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{7}} |} \approx 1.961, E_{r} (K (P_{8})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{8}} |} \approx 2.026$ . $E_{r} (K (P_{9})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{P_{9}} |} \approx 2.351$ . By Proposition 5.18, $E (K (P_{1})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{1}} |}{10}) = 0.42, E (K (P_{2})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{2}} |}{10}) = 0.56$ . $E (K (P_{3})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{3}} |}{10}) = 0.42, E (K (P_{4})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{4}} |}{10}) = 0.5$ . $E (K (P_{5})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{5}} |}{10}) = 0.32, E (K (P_{6})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{6}} |}{10}) = 0.32$ . $E (K (P_{7})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{7}} |}{10}) = 0.58, E (K (P_{8})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{8}} |}{10}) = 0.54$ . $E (K (P_{9})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{P_{9}} |}{10}) = 0.48$ .

The knowledge granulation and knowledge amount are shown in Fig. 3. It can be seen that G (K (P)) + E (K (P)) =1. The rough entropy and knowledge entropy are shown in Fig. 4. It can be seen that E _r (K (P)) + H (K (P)) = log ₂ 10.

Fig.3

Fig.4

(2) Put

Q _i = ind ({a ₁, a ₂, ⋯, a _i}) (i = 1, 2, ⋯, 9).

Then

U/Q ₁ = {{x ₁, x ₂, x ₄, x ₅, x ₇, x ₉, x ₁₀}, {x ₃, x ₆, x ₈}},

U/Q ₂ = {{x ₁, x ₄, x ₇, x ₁₀}, {x ₂, x ₉}, {x ₃, x ₆}, {x ₅}, {x ₈}},

U/Q ₃ = {{x ₁, x ₇, x ₁₀}, {x ₂}, {x ₃}, {x ₄}, {x ₅}, {x ₆}, {x ₈}, {x ₉}},

U/Q ₄ = {{x ₁, x ₁₀}, {x ₂}, {x ₃}, {x ₄}, {x ₅}, {x ₆}, {x ₇}, {x ₈}, {x ₉}},

U/Q ₅ = {{x ₁}, {x ₂}, {x ₃}, {x ₄}, {x ₅}, {x ₆}, {x ₇}, {x ₈}, {x ₉}, {x ₁₀}},

U/Q ₆ = ! {{x ₁}, {x ₂}, {x ₃}, {x ₄}, {x ₅}, {x ₆}, {x ₇}, {x ₈}, {x ₉}, {x ₁₀}},

U/Q ₇ = {{x ₁}, {x ₂}, {x ₃}, {x ₄}, {x ₅}, {x ₆}, {x ₇}, {x ₈}, {x ₉}, {x ₁₀}},

U/Q ₈ = {{x ₁}, {x ₂}, {x ₃}, {x ₄}, {x ₅}, {x ₆}, {x ₇}, {x ₈}, {x ₉}, {x ₁₀}},

U/Q ₉ = {{x ₁}, {x ₂}, {x ₃}, {x ₄}, {x ₅}, {x ₆}, {x ₇}, {x ₈}, {x ₉}, {x ₁₀}}.

By Proposition 5.3,

$G (K (Q_{1})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{Q_{1}} | = 0.58 G (K (Q_{2})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{Q_{2}} | = 0.26$ , $G (K (Q_{3})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{Q_{3}} | = 0.16 G (K (Q_{4})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{Q_{4}} | = 0.12$ , $G (K (Q_{5})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{Q_{5}} | = 0.1 G (K (Q_{6})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{Q_{6}} | = 0.1$ , $G (K (Q_{7})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{Q_{7}} | = 0.1 G (K (Q_{8})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{Q_{8}} | = 0.1$ . $G (K (Q_{9})) = \frac{1}{10^{2}} \sum_{i = 1}^{10} | [x_{i}]_{Q_{9}} | = 0.1$

By Proposition 5.9,

$H (K (Q_{1})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{Q_{1}} |}{10} \approx 0.882, H (K (Q_{2})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{Q_{2}} |}{10} \approx 2.122$ , $H (K (Q_{3})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{Q_{3}} |}{10} \approx 2.847, H (K (Q_{4})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{Q_{4}} |}{10} \approx 3.122$ ,

$H (K (Q_{5})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{Q_{5}} |}{10} \approx 3.322, H (K (Q_{6})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{Q_{6}} |}{10} \approx 3.322$ , $H (K (Q_{7})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{Q_{7}} |}{10} \approx 3.322, H (K (Q_{8})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{Q_{8}} |}{10} \approx 3.322$ , $H (K (Q_{9})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{| [x_{i}]_{Q_{9}} |}{10} \approx 3.322$ . By Proposition 5.12,

$E_{r} (K (Q_{1})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{Q_{1}} |} \approx 2.44, E_{r} (K (Q_{2})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{Q_{2}} |} = 1.2$ . $E_{r} (K (Q_{3})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{Q_{3}} |} \approx 0.476, E_{r} (K (Q_{4})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{Q_{4}} |} = 0.2$ . $E_{r} (K (Q_{5})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{Q_{5}} |} = 0, E_{r} (K (Q_{6})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{Q_{6}} |} = 0$ . $E_{r} (K (Q_{7})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{Q_{7}} |} = 0, E_{r} (K (Q_{8})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{Q_{8}} |} = 0$ . $E_{r} (K (Q_{9})) = - \sum_{i = 1}^{10} \frac{1}{10} \log_{2} \frac{1}{| [x_{i}]_{Q_{9}} |} = 0$ . By Proposition 5.18,

$E (K (Q_{1})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{Q_{1}} |}{10}) = 0.42, E (K (Q_{2})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{Q_{2}} |}{10}) = 0.74$ . $E (K (Q_{3})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{Q_{3}} |}{10}) = 0.84, E (K (Q_{4})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{Q_{4}} |}{10}) = 0.88$ . $E (K (Q_{5})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{Q_{5}} |}{10}) = 0.9, E (K (Q_{6})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{Q_{6}} |}{10}) = 0.9$ . $E (K (Q_{7})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{Q_{7}} |}{10}) = 0.9, E (K (Q_{8})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{Q_{8}} |}{10}) = 0.9$ . $E (K (Q_{9})) = \sum_{i = 1}^{10} \frac{1}{10} (1 - \frac{| [x_{i}]_{Q_{9}} |}{10}) = 0.9$ .

The results of uncertainty measures are shown in Fig. 5. It can be seen the fact that with the attribute subset B ⊆ A growth, the knowledge granulation and rough entropy of K (ind (B)) are both monotonically decreasing. Meanwhile, the knowledge amount and knowledge entropy of K (ind (B)) are both monotonically increasing with the attribute subset B ⊆ A growth. This means that knowledge granulation, rough entropy, knowledge amount and knowledge entropy can be applied to measuring uncertainty of a given knowledge structure in the same knowledge base. □

Fig.5

7 Conclusions

In this paper, knowledge structures in the same knowledge base are further studied and invariant characteristics of knowledge structures in a knowledge base under homomorphisms have been obtained (see Table 2). Moreover, measuring uncertainty of knowledge structures in the same knowledge base is investigated. These results will be important in establishing a framework of granular computing in knowledge bases. In the future, we will study the other invariant characteristics of knowledge structures in a knowledge base under homomorphisms and consider applications of the proposed measures in knowledge discovery of knowledge bases.

Table 2

Invariant and inverse invariant characteristics of knowledge structures in a knowledge base under homomorphisms

Characteristics	Invariant	Inverse invariant
Equality between knowledge structures (=)	√ ([9])	√ ([9])
Dependence between knowledge structures (⪯)	√ ([9])	√ ([9])
Dependence strictly between knowledge structures (≺)	√	√
Dependence partially between knowledge structures (⊑)	√	√
Dependence strictly partially between knowledge structures (⊏)	√	√
Indpendence between knowledge structures (⋈)	√	√
Knowledge distance between knowledge structures ρ)	×	×
Knowledge granulation of knowledge structures (G)	×	×
Knowledge entropy of knowledge structures (H)	×	×
Rough entropy of knowledge structures (E_r )	×	×
Knowledge amount of knowledge structures (E)	×	×

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