Relationships between knowledge bases and their uncertainty measures

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Relationships between knowledge bases and their uncertainty measures Fei Xia a , Hongxiang Tang a,b,∗ , Sichun Wang a a Guangxi Key Laboratory of Cross-Border E-Commerce Intelligent Information Processing, Guangxi University of Finance and Economics,

Nanning, Guangxi 530003, People’s Republic of China b Panyapiwat Institute of Management, Bangkok, Bangkok 10310, Thailand

Received 8 March 2017; received in revised form 3 October 2017; accepted 27 November 2018

Abstract A knowledge base is a basic concept in rough set theory. Uncertainty measures are critical evaluation tools in machine learning fields. This article investigates relationships between knowledge bases and their uncertainty measures. Firstly, dependence and independence between knowledge bases are proposed, and are characterized by the inclusion degree. Secondly, knowledge distance between knowledge bases is studied. Thirdly, invariant and inverse invariant characteristics of knowledge bases under homomorphisms based on data compression are obtained. Fourthly, measuring uncertainty of knowledge bases is investigated, and an example is provided to illustrate the fact that knowledge granulation, rough entropy, knowledge entropy, and knowledge distance are neither invariant nor inverse invariant under homomorphisms based on data compression. Finally, a numerical experiment is provided to show features of the proposed measures for uncertainty of knowledge bases, and effectiveness analysis is conducted from the two aspects of dispersion and correlation. These results will be helpful for the establishment of a framework of granular computing in knowledge bases. © 2018 Elsevier B.V. All rights reserved. Keywords: Rough set theory; Knowledge base; Dependency; Inclusion degree; Knowledge distance; Homomorphism; Characteristic; Uncertainty; Measure

1. Introduction Rough set theory, presented by Pawlak [12], is a mathematical tool to deal with uncertainty and can be considered as the generalization of classical set theory. It has been successfully applied to intelligent systems, expert systems, * Corresponding author at: Guangxi Key Laboratory of Cross-Border E-Commerce Intelligent Information Processing, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, People’s Republic of China. E-mail addresses: [email protected] (F. Xia), [email protected] (H. Tang), [email protected] (S. Wang).

https://doi.org/10.1016/j.fss.2018.11.016 0165-0114/© 2018 Elsevier B.V. All rights reserved.

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knowledge discovery, pattern recognition, machine learning, signal analysis, image processing, inductive reasoning, and decision analysis, and in many other fields [12–15]. A knowledge base is a basic notion of rough set theory. One of the strengths of rough set theory is that an unknown target concept can be approximately characterized by existing knowledge structures in a knowledge base. For a given knowledge base, one of the tasks in data mining and knowledge discovery is to generate new knowledge through use of known knowledge. From the viewpoint of rough set theory, knowledge means the capacity to classify objects [3,7,12], where objects may be real things, abstract concepts, processes, states, and moments of time. Initially, rough set theory just considers equivalence relations. Every equivalence relation on a given universe determines a classification on this universe and vice versa. One deals with not only a single classification on the universe but also a family of classifications on the universe [3,8]. This leads to the notion of knowledge bases. The study of knowledge bases is an important research issue. Some scholars have done very good work. For example, Qian et al. [17] studied knowledge structure in a tolerance knowledge base by means of set families and proposed knowledge distance between knowledge structures. Li et al. [5] 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. [6] 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 [16] showed that ∗-reductions in a knowledge base are invariant and inverse invariant under homomorphisms. The concept of entropy originates from energetics. It can be used to measure the degree of disorder of a system. The entropy of a system, proposed by Shannon [18], gives a measure of the uncertainty of a system. It has been applied in diverse fields as a useful mechanism for evaluating uncertainty in various modes. Some scholars have applied the extension of entropy and its variants to rough sets. For example, Düntsch and Gediga [2] proposed information entropy and three kinds of conditional entropy in rough sets for prediction of a decision attribute. Beaubouef et al. [1] presented a method measuring uncertainty of rough sets and rough relation databases. Wierman [19] addressed granulation measure to measure uncertainty of information. Yao [23] gave a granularity measure from the angle of granulation. Liang et al. [9,11] studied several measures of knowledge in incomplete and complete information systems. Liang and Shi [10] introduced information entropy, rough entropy, and knowledge granulation in rough set theory; Qian et al. [17] investigated knowledge granulation of knowledge structures in a tolerance knowledge base. The aim of this article is to investigate relationships between knowledge bases and their uncertainty measures. We perform statistical analysis of the proposed measures to determine their effectiveness or merits in a statistical sense. The rest of this article is organized as follows. In Section 2, some basic notions of knowledge bases, relation information systems, and homomorphisms between relation information systems are recalled. In Section 3, dependence and independence between knowledge bases are proposed, and knowledge distance between knowledge bases is introduced. In Section 4, invariant and inverse invariant characteristics of knowledge bases under homomorphisms based on data compression are obtained. In Section 5, some tools for measuring uncertainty of knowledge bases are introduced, and an illustrative example is given. In Section 6, a numerical experiment is given, and effectiveness analysis is conducted from the viewpoint of statistics. Section 7 provides a summary. 2. Preliminaries In this section, we recall some basic notions of knowledge bases, relation information systems, and homomorphisms. Throughout this article, U, V denotes two nonempty finite sets called the universes, 2U denotes the family of all subsets of U , and |X| denotes the cardinality of X ∈ 2U . All mappings are assumed to be surjective. Let U = {x1 , x2 , . . . , xn }, V = {y1 , y2 , . . . , ym }, 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 U = {x1 , x2 , ..., xn }, then R may be represented by the following matrix:

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⎛

R(x1 , x1 ) R(x1 , x2 ) ⎜ R(x2 , x1 ) R(x2 , x2 ) ⎜ M(R) = ⎝ ... ... R(xn , x1 ) R(xn , x2 ) where R(xi , xj ) =



1, 0,

... ... ... ...

3

⎞ R(x1 , xn ) R(x2 , xn ) ⎟ ⎟, ⎠ ... R(xn , xn )

(xi , xj ) ∈ R, / R. (xi , xj ) ∈

Let R be a binary relation on U . Then R is referred to as equivalence if R is reflexive, symmetric, and transitive. We denote the family of all equivalence relations on U by R∗ (U ). Suppose R ∈ R∗ (U ). If R = δ, then R is called a universal relation on U ; if R = , then R is said to be an identity relation on U . Definition 2.1 ([12]). (U, R) is referred to as a knowledge base if R ∈ 2R Example 2.2. Let U ⎛ 1 1 ⎜1 1 ⎜ ⎜0 0 R1 = ⎜ ⎜0 0 ⎜ ⎝1 1 0 0 ⎛ 1 1 ⎜1 1 ⎜ ⎜0 0 R3 = ⎜ ⎜0 0 ⎜ ⎝1 1 1 1

= {x1 , x2 , x3 , x4 , x5 , x6 }. Let ⎛ ⎞ 0 0 1 0 1 ⎜0 0 0 1 0⎟ ⎜ ⎟ ⎜ 1 1 0 1⎟ ⎟ , R2 = ⎜ 0 ⎜0 1 1 0 1⎟ ⎜ ⎟ ⎝0 0 0 1 0⎠ 1 1 0 1 1 ⎛ ⎞ 0 0 1 1 1 ⎜1 0 0 1 1⎟ ⎜ ⎟ ⎜ 1 1 0 0⎟ ⎟ , R4 = ⎜ 0 ⎜0 ⎟ 1 1 0 0⎟ ⎜ ⎝1 0 0 1 1⎠ 0 0 1 1 0

0 1 1 1 1 0

0 1 1 1 1 0

0 1 1 1 1 0

0 1 1 1 1 0

1 1 0 0 1 0

0 0 1 1 0 1

0 0 1 1 0 1

1 1 0 0 1 0

∗ (U )

.

⎞ 1 0⎟ ⎟ 0⎟ ⎟, 0⎟ ⎟ 0⎠ 1 ⎞ 0 0⎟ ⎟ 1⎟ ⎟. 1⎟ ⎟ 0⎠ 1

Then (U, R) is a knowledge base, where R = {R1 , R2 , R3 , R4 }. Because a knowledge base is composed of equivalence relations and an equivalence relation can be expressed as a 0–1 matrix, a knowledge base reflects some of nice properties of∗the data on the universe. ∗ R. Then ind(R) ∈ R (U ). For R ∈ 2R (U ) , denote ind(R) = R∈R

If (x, y) ∈ ind(R), then we denote it by xRy. Suppose we have a knowledge base (U, R). Then (U, ind(R)) is a Pawlak approximation space [12,14]. For x ∈ U , let [x]ind(R) = {y : xRy}. Then [x]ind(R) is said to be the equivalence class of the element x on the equivalence relation ind(R). With the help of the equivalence relation ind(R), the universe U is divided into some subsets. Each of these subsets is an equivalence class. All these subsets form a quotient set induced by ind(R), denoted by U/ind(R). If two elements of the universe U belong to the same equivalence class, we say that these two elements are indistinguishable under the equivalence relation ind(R); that is, they are equal with respect to ind(R). Thus ind(R) represents the classification capability of knowledge base (U, R). ∗ For R ∈ 2R (U ) and x ∈ U , U/ind(R) and [x]ind(R) are denoted by U/R and [x]R , respectively. With the aid of a knowledge base, one can construct a rough set of any subset on the universe by the following definition.

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Definition 2.3. Suppose that (U, R) is a knowledge base. For X ∈ 2U , let R(X) = {x ∈ U : [x]R ⊆ X}, R(X) = {x ∈ U : [x]R ∩ X = ∅}. Then R(X) and R(X) are said to be the R-lower approximation and the R-upper approximation with respect to R, respectively. The order pair R(X), R(X) is referred to as the rough set of X with respect to R. For P, Q ∈ 2R

∗ (U )

P osP Q =



, the P-positive region of Q is defined as follows: P(X).

X∈U/Q

Definition 2.4 ([25]). Let (U, P) and (U, Q) be two knowledge bases. (1) (U, P) and (U, Q) are called equivalent if ind(P) = ind(Q). We write (U, P)  (U, Q). (2) (U, P) is said to be finer than (U, Q) or (U, Q) is said to depend on (U, P) if ind(P) ⊆ ind(Q). We write (U, P)  (U, Q). Definition 2.5. Let (U, P) and (U, Q) be two knowledge bases. Then (U, P) and (U, Q) are said to be the same if P = Q. We write (U, P) = (U, Q). Obviously, (U, P) = (U, Q) ⇒ (U, P)  (U, Q) ⇐⇒ (U, P)  (U, Q), (U, Q)  (U, P). Theorem 2.6. Let (U, P) and (U, Q) be two knowledge bases. Then the following are equivalent: (1) (U, P)  (U, Q). (2) For each x ∈ U , [x]P = [x]Q . (3) U/P = U/Q. Proof. This is obvious.

2

Theorem 2.7. Let (U, P) and (U, Q) be two knowledge bases. Then (U, P)  (U, Q) ⇔ for each x ∈ U, [x]P ⊆ [x]Q . Proof. “⇒.” Suppose x ∈ U . For any y ∈ [x]P , (x, y) ∈ ind(P). Since (U, P)  (U, Q), we have ind(P) ⊆ ind(Q). Then (x, y) ∈ ind(Q). So y ∈ [x]Q . This shows that [x]P ⊆ [x]Q . “⇐.” For any (x, y) ∈ ind(P), y ∈ [x]P . Since [x]P ⊆ [x]Q , we have y ∈ [x]Q . Then (x, y) ∈ ind(Q). Thus (U, P)  (U, Q). 2 Theorem 2.8 ([25]). Let (U, P) and (U, Q) be two knowledge bases. Then the following are equivalent: (1) (U, P)  (U, Q). (2) ind(P ∪ Q) = ind(Q). (3) P osP Q = U . Definition 2.9 ([24]). Suppose that A = {X1 , X2 , . . . , Xk } and B = {Y1 , Y2 , . . . , Yl } are two partitions on U : (1) H (A) =

k  i=1

p(Xi )p(U − Xi )

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is referred to as the information amount of A, where p(Xi ) = belongs to Xi .

|Xi | |U |

5

means the probability that the element of U

(2) H (B/A) =

k  l 

p(Xi ∩ Yj )p(Xi − Yj )

i=1 j =1

is said to be the condition information amount of B with respect to A. Theorem 2.10 ([5]). Suppose that (U, P) and (U, Q) are two knowledge bases. Let U/Q = {D1 , D2 , . . . , Dr }. Then the following are equivalent: (1) (U, P)  (U, Q). (2) For each j ≤ r, P(Dj ) = Dj . r (3) P(Dj ) = U . j =1

(4) U/P refines U/Q; that is, for all A ∈ U/P, there exists B ∈ U/Q, A ⊆ B. (5) H ((U/Q)/(U/P)) = 0. 2.2. Relation information systems Definition 2.11 ([12]). 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 referred to as an information system if each attribute a ∈ A determines an information function a : U → Va , where Va is the set of function values of attribute a. If (U, A) is an information system and B ⊆ A, then an equivalence relation (or indiscernibility relation) RB can be defined as follows: (x, y) ∈ RB ⇐⇒ ∀ a ∈ B, a(x) = a(y). Wang et al. [20] introduced the following notion of relation information systems as a simplified model of information systems. Definition 2.12 ([20]). If R ⊆ 2U ×U , then the pair (U, R) is said to be a relation information system. Clearly, a relation information system is not an information system, and an information system is also not a relation information system. The following definition illustrates, however, that there are certain relationships between information systems and relation information systems. Definition 2.13 ([5]). Suppose that (U, A) is an information system. Let R = {R{a} : a ∈ A}. Then the pair (U, R) is said to be the relation information system induced by the information system (U, A). In fact, (U, R) in Definition 2.13 is a knowledge base. Clearly, (U, R) is a knowledge base ⇒ (U, R) is a relation information system. Because an information system reflects some properties on data, the knowledge base induced by this information system also reflects some properties on data.

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2.3. Homomorphisms between relation information systems The data compression in a relation information system includes two aspects of operations on data: one is to reduce the data dimension and the other is to reduce the stored and transferred data volume. Data dimension reduction can be seen as knowledge reduction in relation information systems. Data volume reduction, in mathematical linguistics, can be viewed as a many-to-one mapping between relation information systems. Homomorphisms between relation information systems, introduced by Wang et al. [20], are such mappings. Suppose R ∈ 2U ×U and x ∈ U . Denote Rs (x) = {y ∈ U : xRy}. Then Rs (x) = {y ∈ U : xRy} is called the successor neighborhood of x with respect to R [22]. Definition 2.14 ([20,21]). Suppose f : U → V is a mapping. For R ∈ 2U ×U , let [x]f = {u ∈ U : f (u) = f (x)}, (x)R = {u ∈ U : Rs (u) = Rs (x)}. If [x]f ⊆ Rs (u) or [x]f ∩ Rs (u) = ∅ for any x, u ∈ U , then f is referred to as type 1 consistent with respect to R. If [x]f ⊆ (x)R for any x ∈ U , then f is referred to as type 2 consistent with respect to R. Suppose R ⊆ 2U ×U . If f is type 1 (or type 2) consistent with respect to R for every R ∈ R, then f is called type 1 consistent (or type 2 consistent) with respect to R. Definition 2.15 ([20,21]). Suppose f : U → V is a mapping. Define fˆ : 2U ×U → 2V ×V ,

R → fˆ(R) = ({f (x)} × f (Rs (x))); x∈U

fˆ−1 : 2V ×V → 2U ×U ,

({f −1 (y)} × f −1 (Ts (y))]. T → fˆ−1 (T ) = y∈V

Then fˆ and fˆ−1 are said to be the relation mapping and the inverse relation mapping induced by f , respectively. Clearly, y1 fˆ(R)y2 ⇐⇒ ∃ x1 , x2 ∈ U, y1 = f (x1 ), y2 = f (x2 ), and x1 Rx2 , f (x1 )fˆ(R)f (x2 ) ⇐= x1 Rx2 , x1 fˆ−1 (T )x2 ⇐⇒ f (x1 )Tf (x2 ). For R ⊆ 2U ×U , let fˆ(R) = {fˆ(R) : R ∈ R}. Definition 2.16 ([20]). Suppose that (U, R) is a relation information system. Suppose f : U → V is a mapping. Then the pair (V , fˆ(R)) is referred to as an f -induced relation information system of (U, R). Definition 2.17 ([20]). Suppose that (U, R) is a relation information system and (V , fˆ(R)) is an f -induced relation information system of (U, R). If f is both type 1 and type 2 consistent with respect to R on U , then f is referred to as a homomorphism from (U, R) to (V , fˆ(R)). We write (U, R) ∼ (V , fˆ(R)). Proposition 2.18 ([20]). If f : U → V is both type 1 and type 2 consistent with respect to R ∈ 2U ×U , then fˆ−1 (fˆ(R)) = R.

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Fig. 1. Dependence between (U, P) and (U, Q).

Theorem 2.19 ([5]). If (U, R) ∼f (V , fˆ(R)), then (U, R) is a knowledge base ⇐⇒ (V , fˆ(R)) is a knowledge base. Theorem 2.20 ([5]). Let (U, R) be a relation information system and (V , fˆ(R)) an f -induced relation information system of (U, R). Then (1) (V , fˆ(R)) is a knowledge base on V ⇐⇒ (U, fˆ−1 (fˆ(R))) is a knowledge base on U . (2) If (U, R) ∼f (V , fˆ(R)), then fˆ−1 (fˆ(R)) = R. Theorem 2.21 ([20]). Let (U, R) ∼f (V , fˆ(R)). Suppose P ⊆ R. Then fˆ(ind(P)) = ind(fˆ(P)). Theorem 2.22 ([20]). Let (U, R) ∼f (V , fˆ(R)). Suppose S ⊆ fˆ(R). Then fˆ−1 (ind(S)) = ind(fˆ−1 (S)). 3. Relationships between knowledge bases In this section, we investigate relationships between knowledge bases from the two aspects of dependence and separation. 3.1. Dependence and independence between knowledge bases From observation of Theorem 2.7, dependence and independence between knowledge bases are introduced in the following definition. Definition 3.1. Let (U, P) and (U, Q) be two knowledge bases. (1) (U, P) is said to be finer strictly than (U, Q) or (U, Q) is said to depend strictly on (U, P) if (U, P)  (U, Q) and (U, P)  (U, Q). We write (U, P) ≺ (U, Q). (2) (a) (U, P) is said to be partially finer than (U, Q) or (U, Q) is said to partially dependent on (U, P) if there exists i, [xi ]P ⊆ [xi ]Q . We write (U, P)  (U, Q). (b) (U, P) is said to be partially finer strictly than (U, Q) or (U, Q) is said to partially depend strictly on (U, P) if (U, P)  (U, Q) and (U, P)  (U, Q). We write (U, P) < (U, Q). (3) (U, Q) is said to be independent of (U, P) if for each i, [xi ]P  [xi ]Q . We write (U, P)  (U, Q). Fig. 1a represents (U, P) ≺ (U, Q), Fig. 1b represents (U, P) < (U, Q), Fig. 1c represents (U, P) < (U, Q) and (U, Q) < (U, P), and Fig. 1d represents (U, P)  (U, Q) and (U, Q)  (U, P). Clearly, (U, P)  (U, Q) =⇒ (U, P)  (U, Q), (U, P) ≺ (U, Q) =⇒ (U, P) < (U, Q); (U, P)  (U, Q) ⇐⇒ (U, P)  (U, Q).

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Fig. 2. Relationships among knowledge bases.

Example 3.2. Let U = {x1 , x2 , x3 , x4 }. Let R1 =  ∪ {(x1 , x2 ), (x2 , x1 )}, R2 =  ∪ {(x1 , x3 ), (x3 , x1 ), (x2 , x4 ), (x4 , x2 )}, R3 =  ∪ {(x1 , x2 ), (x2 , x1 ), (x1 , x3 ), (x3 , x1 ), (x2 , x3 ), (x3 , x2 )}, R4 =  ∪ {(x2 , x3 ), (x3 , x2 )}, R5 =  ∪ {(x2 , x3 ), (x3 , x2 ), (x2 , x4 ), (x4 , x2 ), (x3 , x4 ), (x4 , x3 )}. Then Ri ∈ R∗ (U ) (i = 1, 2, 3, 4, 5). Pick O = {R3 }, P = {R1 , R3 }, Q = {R1 , R2 , R4 }, and R = {R4 , R5 }. Then (U, O), (U, P), (U, Q), and (U, R) are four knowledge bases. So (U, Q) ≺ (U, P) ≺ (U, O), (U, Q) ≺ (U, R) ≺ (U, O); (U, P) < (U, R), (U, R) < (U, P) (see Fig. 2). Denote  = {(U, R) : R ∈ 2R

∗ (U )

}.

Definition 3.3. A mapping D :  ×  → [0, 1] is referred to as the inclusion degree on  if (1) 0 ≤ D((U, Q)/(U, P)) ≤ 1; (2) (U, P)  (U, Q) implies D((U, Q)/(U, P)) = 1; (3) (U, P)  (U, Q)  (U, R) implies D((U, P)/(U, R)) ≤ D((U, P)/(U, Q)). Definition 3.4. For any P, Q ∈ 2R D((U, Q)/(U, P)) =

n  l=1

where χ[xl ]Q ([xl ]P ) =



1, 0,

∗ (U )

, define

|[xl ]Q |

n χ[xl ]Q ([xl ]P ), i=1 |[xi ]Q |

if [xl ]P ⊆ [xl ]Q , if [xl ]P  [xl ]Q .

Proposition 3.5. D in Definition 3.4 is the inclusion degree under Definition 3.3. Proof. Let P, Q, R ∈ 2R

∗ (U )

.

(1) Obviously, 0 ≤ D((U, Q)/(U, P)) ≤ 1.

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(2) Suppose (U, P)  (U, Q). Then ind(P) ⊆ ind(Q). So for each l, [xl ]P ⊆ [xl ]Q . This implies for each l, χ[xl ]Q ([xl ]P ) = 1. Thus D((U, Q)/(U, P)) = 1. (3) Suppose (U, P)  (U, Q)  (U, R). Then ind(P) ⊆ ind(Q) ⊆ ind(R). So for each l, [xl ]P ⊆ [xl ]Q ⊆ [xl ]R . By Definition 3.4, D((U, P)/(U, R)) =

n  l=1

D((U, P)/(U, Q)) =

n  l=1

|[xl ]P | χ[xl ]P ([xl ]R ), i=1 |RP (xi )|

n

|[xl ]P | χ[xl ]P ([xl ]Q ). i=1 |RP (xi )|

n

If [xl ]Q  [xl ]P , then [xl ]R  [xl ]P . This illustrates that χ[xl ]P ([xl ]Q ) = 0 implies χ[xl ]P ([xl ]R ) = 0. Thus D((U, P)/(U, R)) ≤ D((U, P)/(U, Q)). Hence D in Definition 3.4 is the inclusion degree under Definition 3.3.

2

Example 3.6 (Continued from Example 3.2). We have U/P = {{x1 , x2 }, {x3 }, {x4 }), U/Q = {{x1 }, {x2 }, {x3 }, {x4 }}. Then [x1 ]P = [x2 ]P = {x1 , x2 }, [x3 ]P = {x3 }, [x4 ]P = {x4 }; [x1 ]Q = {x1 }, [x2 ]Q = {x2 }, [x3 ]Q = {x3 }, [x4 ]Q = {x4 }. We have D((U, P)/(U, Q)) =

|[x1 ]P | |[x2 ]P | χ[x1 ]P ([x1 ]Q ) + χ[x2 ]P ([x2 ]Q ) 4 4

|[xi ]P | |[xi ]P |

i=1

i=1

|[x4 ]P | |[x3 ]P | χ[x3 ]P ([x3 ]Q ) + χ[x4 ]P ([x4 ]Q ) + 4 4

|[xi ]P | |[xi ]P | i=1

i=1

|[x2 ]P | |[x3 ]P | |[x4 ]P | |[x1 ]P | + + + = 4 4 4 4



|[xi ]P | |[xi ]P | |[xi ]P | |[xi ]P | i=1

i=1

i=1

i=1

2 2 1 1 = + + + = 1, 6 6 6 6 |[x1 ]Q | |[x2 ]Q | D((U, Q)/(U, P)) = χ[x1 ]Q ([x1 ]P ) + χ[x2 ]Q ([x2 ]P ) 4 4

|[xi ]Q | |[xi ]Q | i=1

i=1

|[x4 ]Q | |[x3 ]Q | χ[x3 ]Q ([x3 ]P ) + χ[x4 ]Q ([x4 ]P ) + 4 4

|[xi ]Q | |[xi ]Q | i=1

i=1

|[x3 ]Q | |[x4 ]Q | 1 1 1 = + = + = . 4 4 4 4 2

|[xi ]Q | |[xi ]Q | i=1

i=1

9

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This example illustrates that D((U, P)/(U, Q)) + D((U, Q)/(U, P)) = 1. The following theorem shows that dependence and independence between knowledge bases can be quantitatively described by the inclusion degree. Theorem 3.7. Suppose P, Q ∈ 2R

∗ (U )

. Then

(1) (U, P)  (U, Q) ⇐⇒ D((U, Q)/(U, P)) = 1. (2) (U, P)  (U, Q) ⇐⇒ D((U, Q)/(U, P)) = 0. (3) (U, P)  (U, Q) ⇐⇒ 0 < D((U, Q)/(U, P)) ≤ 1. Proof. (1) “=⇒.” Obviously. “⇐=.” Let |[xl ]Q | = ql ,

n 

|[xl ]Q | = q.

l=1

Then q =

n

ql . Since D((U, Q)/(U, P)) = 1, we have

l=1 n 

ql χ[xl ]Q ([xl ]P ) = q =

l=1

n 

ql .

l=1

Then n 

ql (1 − χ[xl ]Q ([xl ]P )) = 0.

l=1

Thus for each l, 1 − χ[xl ]Q ([xl ]P ) = 0. It follows that for each l, [xl ]P ⊆ [xl ]Q . Hence (U, P)  (U, Q). (2) “=⇒.” Since (U, P)  (U, Q), we have for all l, [xl ]P  [xl ]Q . Then for each l, χ[xl ]Q ([xl ]P ) = 0. Thus D((U, Q)/(U, P)) = 0. “⇐=.” Since D((U, Q)/(U, P)) = 0, we obtain that for each l, χ[xl ]Q ([xl ]P ) = 0. Then for all l, [xl ]P  [xl ]Q . Thus (U, P)  (U, Q). (3) This holds by (1) and (2). 2 3.2. Knowledge distance between knowledge bases Considering the separation between knowledge bases, in this subsection, we propose the concept of knowledge distance between knowledge bases and give some of its properties. For A, B ∈ 2U , denote A ⊕ B = A ∪ B − A ∩ B. Then A ⊕ B is called the symmetric difference between A and B. It is well known that | A ⊕ B |=| A ∪ B | − | A ∩ B |.

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11

Fig. 3. A, B and C.

Definition 3.8 ([4]). Suppose we have a nonempty set X. Let ρ : X × X → R be a mapping. Then (X, ρ) is called a pseudometric space if (1) for any x, y ∈ X, ρ(x, y) ≥ 0, and for any x ∈ X, ρ(x, x) = 0; (2) for any x, y ∈ X, ρ(x, y) = ρ(y, x); (3) for any x, y, z ∈ X, ρ(x, z) ≤ ρ(x, y) + ρ(y, z). In this case, ρ is called a pseudometric on X. Definition 3.9. For P, Q ∈ 2R ρ((U, P), (U, Q)) =

∗ (U )

, the knowledge distance between (U, P) and (U, Q) is defined as

n 1  | [xi ]P ⊕ [xi ]Q | . n2 i=1

Lemma 3.10. Let A, B ∈ 2U . Then A = B ⇐⇒| A ⊕ B |= 0. Proof. “=⇒” is obvious. “⇐=.” Suppose | A ⊕ B |= 0. Then | A ∪ B − A ∩ B |= 0. It follows that A ∪ B − A ∩ B = ∅. So A ∪ B ⊆ A ∩ B. Since A ∩ B ⊆ A ∪ B, we have A ∩ B = A ∪ B. Thus A = B. 2 Lemma 3.11. Let A, B, C ∈ 2U . Then | A ⊕ B | + | B ⊕ C |≥| A ⊕ C | . Proof. See Fig. 3, denote A1 = A − (A ∩ B + A ∩ C − A ∩ B ∩ C), A2 = A ∩ B − A ∩ B ∩ C, A3 = B − (A ∩ B + B ∩ C − A ∩ B ∩ C), A4 = A ∩ C − A ∩ B ∩ C, A5 = A ∩ B ∩ C, A6 = B ∩ C − A ∩ B ∩ C, A7 = C − (A ∩ C + B ∩ C − A ∩ B ∩ C). Then | A ⊕ B |=| A1 | + | A4 | + | A3 | + | A6 |, | B ⊕ C |=| A2 | + | A3 | + | A4 | + | A7 |, | A ⊕ C |=| A1 | + | A2 | + | A6 | + | A7 | .

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So | A ⊕ B | + | B ⊕ C | − | A ⊕ C |= 2(| A3 | + | A4 |) > 0. Thus | A ⊕ B | + | B ⊕ C |≥| A ⊕ C |. 2 Lemma 3.12. Let A, B, C ∈ 2U . If A ⊆ B ⊆ C or C ⊆ B ⊆ A, then | A ⊕ B | + | B ⊕ C |=| A ⊕ C | . Proof. Suppose A ⊆ B ⊆ C. Then | A ⊕ B |=| A ∪ B | − | A ∩ B |=| B | − | A |, | B ⊕ C |=| B ∪ C | − | B ∩ C |=| C | − | B |, | A ⊕ C |=| A ∪ C | − | A ∩ C |=| C | − | A | . Thus | A ⊕ B | + | B ⊕ C |=| A ⊕ C | . Suppose C ⊆ B ⊆ A. Similarly, we can prove that | A ⊕ B | + | B ⊕ C |=| A ⊕ C | .

2

Theorem 3.13. Let (U, P) and (U, Q) be two knowledge bases. Then ρ((U, P), (U, Q)) = 0 ⇐⇒ (U, P)  (U, Q). Proof. Obviously, ρ((U, P), (U, Q)) = 0 ⇔ ∀ i, | [xi ]P ⊕ [xi ]Q |= 0. By Lemma 3.10, ∀ i, | [xi ]P ⊕ [xi ]Q |= 0 ⇔ ∀ i, [xi ]P = [xi ]Q . By Theorem 2.6, (U, P)  (U, Q) ⇐⇒ ∀ i, [xi ]P = [xi ]Q . Thus ρ((U, P), (U, Q)) = 0 ⇐⇒ (U, P)  (U, Q).

2

Corollary 3.14. Let (U, P) and (U, Q) be two knowledge bases. Then D((U, Q)/(U, P)) = 1, ρ((U, P), (U, Q)) = 0 ⇐⇒ (U, P) ≺ (U, Q). Proof. This holds by Theorems 3.7 and 3.13.

2

Theorem 3.15. (, ρ) is a pseudometric space. Proof. Obviously, the pseudometric axioms (1) and (2) hold. ∗ Suppose P, Q, R ∈ 2R (U ) . By Lemma 3.11, | [xi ]P ⊕ [xi ]Q | + | [xi ]Q ⊕ [xi ]R |≥| [xi ]P ⊕ [xi ]R | .

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Then ρ((U, P), (U, Q)) + ρ((U, Q), (U, R)) n n 1  1  = 2 | [xi ]P ⊕ [xi ]Q | + 2 | [xi ]Q ⊕ [xi ]R | n n i=1

i=1

n 1  (| [xi ]P ⊕ [xi ]Q | + | [xi ]Q ⊕ [xi ]R |) = 2 n i=1

n 1  | [xi ]P ⊕ [xi ]R | ≥ 2 n i=1

= ρ((U, P), (U, R)). Thus (, ρ) is a pseudometric space. Proposition 3.16. Suppose P, Q ∈ 2R (1)

∗ (U )

2 . Then

1 0 ≤ ρ((U, P), (U, Q)) ≤ 1 − ; n ∗ (2) If (U, P)  (U, Q) and ind(P ) is an identity relation on U , then ρ((U, P), (U, P ∗ )) ≤ ρ((U, Q), (U, P ∗ )); (3) If (U, P)  (U, Q), then ρ((U, P), (U, ∅)) ≥ ρ((U, Q), (U, ∅)). Proof. (1) Obviously, 1 ≤| [xi ]P ∪ [xi ]Q |≤ n and 1 ≤| [xi ]P ∩ [xi ]Q |≤ n (i = 1, 2, . . . , n). Then 0 ≤| [xi ]P ⊕ [xi ]Q |≤ n − 1 (i = 1, 2, . . . , n). Thus 0 ≤ ρ((U, P), (U, Q)) ≤

n 1  n2 − n 1 (n − 1) = =1− . 2 2 n n n i=1

(2) Since (U, P)  (U, Q), for any i, we have [xi ]P ⊆ [xi ]Q . Then for any i, | [xi ]P |≤| [xi ]Q |. Thus ρ((U, P), (U, P ∗ )) =

n n 1  1  | [x ] ⊕ {x } |= (| [xi ]P ∪ {xi } | − | [xi ]P ∩ {xi } |) i P i n2 n2 i=1

i=1

n n 1  1  (| [xi ]P | −1) ≤ 2 (| [xi ]Q | −1) = ρ((U, Q), (U, P ∗ )). = 2 n n i=1

i=1

(3) Note that (U, P)  (U, Q). Then for any i, | [xi ]P |≤| [xi ]Q |. Thus ρ((U, P), (U, ∅)) =

n n 1  1  | [x ] ⊕ U |= (| [xi ]P ∪ U | − | [xi ]P ∩ U |) i P n2 n2 i=1

i=1

n n 1  1  (n− | [xi ]P |) ≥ 2 (n− | [xi ]Q |) = ρ((U, Q), (U, ∅)). = 2 n n i=1

i=1

2

13

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Proposition 3.17. Suppose P, P ∗ ∈ 2R

∗ (U )

. If ind(P ∗ ) is an identity relation on U , then

1 ρ((U, P), (U, P ∗ )) + ρ((U, P), (U, ∅)) = 1 − . n Proof. ρ((U, P), (U, P ∗ )) + ρ((U, P), (U, ∅)) n n 1  1  = 2 | [xi ]P ⊕ {xi } | + 2 | [xi ]P ⊕ U | n n = =

1 n2 1 n2

i=1 n 

(| [xi ]P | −1) +

i=1 n  i=1

1 n2

i=1 n 

(n− | [xi ]P |)

i=1

1 (n − 1) = 1 − . 2 n

Proposition 3.18. Suppose P, Q, R ∈ 2R

∗ (U )

. If (U, P)  (U, Q)  (U, R) or (U, R)  (U, Q)  (U, P), then

ρ((U, P), (U, Q)) + ρ((U, Q), (U, R)) = ρ((U, P), (U, R)). Proof. Since (U, P)  (U, Q)  (U, R) or (U, R)  (U, Q)  (U, P), we have [xi ]P ⊆ [xi ]Q ⊆ [xi ]R or [xi ]R ⊆ [xi ]Q ⊆ [xi ]P (i = 1, 2, . . . , n). By Lemma 3.12, | [xi ]P ⊕ [xi ]Q | + | [xi ]Q ⊕ [xi ]R |=| [xi ]P ⊕ [xi ]R | (i = 1, 2, . . . , n), ρ((U, P), (U, Q)) + ρ((U, Q), (U, R)) n n 1  1  = 2 | [xi ]P ⊕ [xi ]Q | + 2 | [xi ]Q ⊕ [xi ]R | n n i=1

i=1

n 1  = 2 (| [xi ]P ⊕ [xi ]Q | + | [xi ]Q ⊕ [xi ]R |) n

=

1 n2

i=1 n 

| [xi ]P ⊕ [xi ]R |

i=1

= ρ((U, P), (U, R)).

2

To illustrate the importance and rationality of the above results with regard to knowledge distance between knowledge bases, the following example is given. Example 3.19 (Continued from Example 3.2). By Definition 3.9, one can obtain that 4 6 4 , ρ((U, O), (U, Q)) = , ρ((U, O), (U, R)) = , 16 16 16 2 2 2 ρ((U, P), (U, Q)) = , ρ((U, P), (U, R)) = , ρ((U, Q), (U, R)) = . 16 16 16 This example illustrates the following facts (see Fig. 4): ρ((U, O), (U, P)) =

(1) We have (U, P) ≺ (U, O), (U, R) ≺ (U, O), and (U, Q) = (U, ∅). It is clear that 2 6 ρ((U, P), (U, Q)) = ≤ = ρ((U, O), (U, Q)), 16 16 6 2 ≤ = ρ((U, O), (U, Q)). ρ((U, R), (U, Q)) = 16 16

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Fig. 4. Knowledge distance between knowledge bases.

(2) We have (U, Q) ≺ (U, P) ≺ (U, O) and (U, Q) ≺ (U, R) ≺ (U, O). It is clear that 2 4 6 + = = ρ((U, Q), (U, O)), 16 16 16 4 6 2 + = = ρ((U, Q), (U, O)). ρ((U, Q), (U, R)) + ρ((U, R), (U, O)) = 16 16 16 ρ((U, Q), (U, P)) + ρ((U, P), (U, O)) =

4. Invariant and inverse invariant characteristics of knowledge bases under homomorphisms In this section, invariant and inverse invariant characteristics of knowledge bases under homomorphisms are obtained. Theorem 4.1. Let (U, R) ∼f (V , fˆ(R)). Suppose P, Q ⊆ R∗ (U ). If ind(P) ∈ R, ind(Q) ∈ R, then (U, P)  (U, Q) ⇐⇒ (V , fˆ(P))  (V , fˆ(Q)). Proof. “=⇒.” Suppose (U, P)  (U, Q). Then for all i, [xi ]P ⊆ [xi ]Q . Let yj = f (xi ). For all v ∈ [yj ]fˆ(P ) , we have yj fˆ(P)v; that is, (yj , v) ∈ ind(fˆ(P)). By Theorem 2.21, ind(fˆ(P)) = fˆ(ind(P)). Then (yj , v) ∈ fˆ(ind(P)); that is, yj fˆ(ind(P))v. Thus there exists x  , x  ∈ U , yj = f (x  ), v = f (x  ), and x  Px  . f (xi ) = f (x  ) implies x  ∈ [xi ]f . Since (U, R) ∼f (V , fˆ(R)) and ind(P), ind(Q) ∈ R, we have f are both type 2 consistent with respect to ind(P) and ind(Q). This implies [xi ]f ⊆ (xi )ind(P ) , [xi ]f ⊆ (xi )ind(Q) . Then x  ∈ (xi )ind(P ) , x  ∈ (xi )ind(Q) . So [x  ]P = (ind(P))s (x  ) = (ind(P))s (xi ) = [xi ]P , [x  ]Q = (ind(Q))s (x  ) = (ind(Q))s (xi ) = [xi ]Q . Note that [xi ]P ⊆ [xi ]Q . Then [x  ]P ⊆ [x  ]Q . x  Px  implies x  ∈ [x  ]P . So x  ∈ [x  ]Q ; that is, x  Qx  . Note that yj = f (x  ), v = f (x  ), and x  Qx  . Then yj fˆ(Q)v; that is, (yj , v) ∈ ind(fˆ(Q)). By Theorem 2.21, ind(fˆ(Q)) = fˆ(ind(Q)). Then (yj , v) ∈ fˆ(ind(Q)); that is, yj fˆ(ind(Q))v. This implies v ∈ [yj ]fˆ(Q) . Hence [yj ] ˆ ⊆ [yj ] ˆ . This shows (V , fˆ(P))  (V , fˆ(Q)). f (P )

f (Q)

“⇐=.” Suppose (V , fˆ(P))  (V , fˆ(Q)). Then for all j , [yj ]fˆ(P ) ⊆ [yj ]fˆ(Q) . Let f (xi ) = yj . For all u ∈ [xi ]P , we have xi Pu. Then f (xi )fˆ(ind(P))f (u); that is, yj fˆ(ind(P))f (u). So (yj , f (u)) ∈ ˆ f (ind(P)). By Theorem 2.21,

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ind(fˆ(P)) = fˆ(ind(P)). Then (yj , f (u)) ∈ ind(fˆ(P)); that is, yj fˆ(P)v. This implies f (u) ∈ [yj ]fˆ(P ) . Since [yj ]fˆ(P ) ⊆ [yj ]fˆ(Q) , we have f (u) ∈ [yj ]fˆ(Q) . Then yj fˆ(Q)f (u); that is, f (xi )fˆ(Q)f (u). So (f (xi ), f (u)) ∈ ind(fˆ(Q)). By Theorem 2.21, ind(fˆ(Q)) = fˆ(ind(Q)). Then (f (xi ), f (u)) ∈ fˆ(ind(Q)); that is, f (xi )fˆ(ind(Q))f (u). So there exists x  , x  ∈ U , f (xi ) = f (x  ), f (u) = f (x  ), and x  Qx  . f (xi ) = f (x  ) implies x  ∈ [xi ]f . Since (U, R) ∼f (V , fˆ(R)) and ind(Q) ∈ R, we have f is type 2 consistent with respect to ind(Q). This implies [xi ]f ⊆ (xi )ind(Q) . Then x  ∈ (xi )ind(Q) . So x  ∈ (ind(Q))s (x  ) = (ind(Q))s (xi ). This implies xi Qx  . f (u) = f (x  ) implies x  ∈ [u]f . Since f is also type 2 consistent with respect to ind(Q), we have [u]f ⊆ (u)ind(Q) . Then x  ∈ (u)ind(Q) . So x  ∈ (ind(Q))s (x  ) = (ind(Q))s (u). This implies uQx  . Note that x  Qx  , xi Qx  , and uQx  . Then xi Qu. This implies u ∈ [xi ]Q . Thus [xi ]P ⊆ [xi ]Q . This shows (U, P)  (U, Q). 2 Corollary 4.2. Let (U, R) ∼f (V , fˆ(R)). Suppose P, Q ⊆ R∗ (U ). If ind(P) ∈ R, ind(Q) ∈ R, then (U, P) = (U, Q) ⇐⇒ (V , fˆ(P)) = (V , fˆ(Q)). Proof. This holds by Theorem 4.1.

2

Corollary 4.3. Let (U, R) ∼f (V , fˆ(R)). Suppose P, Q ⊆ R∗ (U ). If ind(P) ∈ R, ind(Q) ∈ R, then (U, P) ≺ (U, Q) ⇐⇒ (V , fˆ(P)) ≺ (V , fˆ(Q)). Proof. This holds by Theorem 4.1 and Corollary 4.2.

2

Theorem 4.4. Let (U, R) ∼f (V , fˆ(R)). Suppose P, Q ⊆ R∗ (U ). If ind(P) ∈ R, ind(Q) ∈ R, then (U, P)  (U, Q) ⇐⇒ (V , fˆ(P))  (V , fˆ(Q)). Proof. “=⇒.” Suppose (U, P)  (U, Q). Then there exists i, [xi ]P ⊆ [xi ]Q . Let yj = f (xi ). For all v ∈ [yj ]fˆ(P ) , we have yj fˆ(P)v; that is, (yj , v) ∈ ind(fˆ(P)). By Theorem 2.21, ind(fˆ(P)) = fˆ(ind(P)). Then (yj , v) ∈ fˆ(ind(P)); that is, yj fˆ(ind(P))v. Thus there exists x  , x  ∈ U , yj = f (x  ), v = f (x  ), and x  Px  . f (xi ) = f (x  ) implies x  ∈ [xi ]f . Since (U, R) ∼f (V , fˆ(R)) and ind(P), ind(Q) ∈ R, we have f are both type 2 consistent with respect to ind(P) and ind(Q). This implies [xi ]f ⊆ (xi )ind(P ) , [xi ]f ⊆ (xi )ind(Q) . Then x  ∈ (xi )ind(P ) , x  ∈ (xi )ind(Q) . So [x  ]P = (ind(P))s (x  ) = (ind(P))s (xi ) = [xi ]P , [x  ]Q = (ind(Q))s (x  ) = (ind(Q))s (xi ) = [xi ]Q . Note that [xi ]P ⊆ [xi ]Q . Then [x  ]P ⊆ [x  ]Q . x  Px  implies x  ∈ [x  ]P . So x  ∈ [x  ]Q ; that is, x  Qx  .

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Note that yj = f (x  ), v = f (x  ), and x  Qx  . Then yj fˆ(Q)v; that is, (yj , v) ∈ ind(fˆ(Q)). By Theorem 2.21, ind(fˆ(Q)) = fˆ(ind(Q)). Then (yj , v) ∈ fˆ(ind(Q)) that is, yj fˆ(ind(Q))v. This implies v ∈ [yj ]fˆ(Q) . Hence [yj ] ˆ ⊆ [yj ] ˆ . This shows (V , fˆ(P))  (V , fˆ(Q)). f (P )

f (Q)

“⇐=.” Suppose (V , fˆ(P))  (V , fˆ(Q)). Then there exists j , [yj ]fˆ(P ) ⊆ [yj ]fˆ(Q) . Let f (xi ) = yj . For all u ∈ [xi ]P , we have xi Pu. Then f (xi )fˆ(ind(P))f (u); that is, yj fˆ(ind(P))f (u). So (yj , f (u)) ∈ ˆ f (ind(P)). By Theorem 2.21, ind(fˆ(P)) = fˆ(ind(P)).

Then (yj , f (u)) ∈ ind(fˆ(P)); that is, yj fˆ(P)v. This implies f (u) ∈ [yj ]fˆ(P ) . Since [yj ]fˆ(P ) ⊆ [yj ]fˆ(Q) , we have f (u) ∈ [yj ]fˆ(Q) . Then yj fˆ(Q)f (u); that is, f (xi )fˆ(Q)f (u). So (f (xi ), f (u)) ∈ ind(fˆ(Q)). By Theorem 2.21, ind(fˆ(Q)) = fˆ(ind(Q)). Then (f (xi ), f (u)) ∈ fˆ(ind(Q)); that is, f (xi )fˆ(ind(Q))f (u). So there exists x  , x  ∈ U , f (xi ) = f (x  ), f (u) = f (x  ), and x  Qx  . f (xi ) = f (x  ) implies x  ∈ [xi ]f . Since (U, R) ∼f (V , fˆ(R)) and ind(Q) ∈ R, we have f is type 2 consistent with respect to ind(Q). This implies [xi ]f ⊆ (xi )ind(Q) . Then x  ∈ (xi )ind(Q) . So x  ∈ (ind(Q))s (x  ) = (ind(Q))s (xi ). This implies xi Qx  . f (u) = f (x  ) implies x  ∈ [u]f . Since f is also type 2 consistent with respect to ind(Q), we have [u]f ⊆ (u)ind(Q) . Then x  ∈ (u)ind(Q) . So x  ∈ (ind(Q))s (x  ) = (ind(Q))s (u). This implies uQx  . Note that x  Qx  , xi Qx  , and uQx  . Then xi Qu. This implies u ∈ [xi ]Q . Thus [xi ]P ⊆ [xi ]Q . This shows (U, P)  (U, Q). 2 Corollary 4.5. Let (U, R) ∼f (V , fˆ(R)). Suppose P, Q ⊆ R∗ (U ). If ind(P) ∈ R, ind(Q) ∈ R, then (1) (U, P) < (U, Q) ⇐⇒ (V , fˆ(P)) < (V , fˆ(Q)); (2) (U, P)  (U, Q) ⇐⇒ (V , fˆ(P))  (V , fˆ(Q)). Proof. (1) This holds by Corollary 4.2 and Theorem 4.4. (2) This follows from Theorem 4.4. 2 Theorem 4.1, Corollary 4.2, Corollary 4.3, Theorem 4.4 and Corollary 4.5 illustrate the fact that the equality, dependence, partial dependence, and independence between knowledge bases are invariant under homomorphisms. But the image knowledge bases (V , fˆ(P)) and (V , fˆ(Q)) are simpler than the original knowledge bases (U, P) and (U, Q). Therefore we can study the equality, dependence, partial dependence, and independence between original knowledge bases by means of the image knowledge base. Example 4.6. Let U = {xi : 1 ≤ i ≤ 15}. Let U/R1 = {{x1 , x2 , x4 , x7 , x8 , x9 , x10 , x11 }, {x3 , x5 , x6 , x12 , x13 , x14 , x15 }}, U/R2 = {{x1 , x4 , x11 , x12 , x13 , x14 , x15 }, {x2 , x3 , x5 , x6 , x7 , x8 , x9 , x10 }}, U/R3 = {{x1 , x2 , x4 , x7 , x8 , x9 , x10 , x11 , x12 , x13 , x14 , x15 }, {x3 , x5 , x6 }}, U/R4 = {{x1 , x2 , x4 , x7 , x8 , x9 , x10 , x11 }, {x3 , x5 , x6 , x12 , x13 , x14 , x15 }}. Then (U, Ri ) is a knowledge base (i = 1, 2, 3, 4).

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Let R = {ind(R1 ), ind(R2 ), ind(R3 ), ind(R4 )}. Then (U, R) is also a knowledge base. Let V = {y1 , y2 , y3 , y4 , y5 , y6 }. A mapping f : U → V is defined as follows: x1 , x4 , x11

x2 , x8

x3 , x6

x5

x7 , x9 , x10

x12 , x13 , x14 , x15

y1

y2

y3

y4

y5

y6

.

It is easy to verify that f is both type 1 and type 2 consistent with respect to R on U . Thus (U, R) ∼f (V , fˆ(R)) and fˆ(R) = {fˆ(R1 ), fˆ(R2 ), fˆ(R3 ), fˆ(R4 )}. By Theorem 2.19, (V , fˆ(R)) is a knowledge base. Note that (U, R1 ) = (U, R4 ), (U, R1 ) < (U, R3 ) < (U, R1 ), (U, R2 ) < (U, R3 ) < (U, R2 ), (U, R3 ) < (U, R4 ) < (U, R3 ). Then by Corollaries 4.2 and 4.5(1), (V , fˆ(R1 )) = (U, fˆ(R4 )), (V , fˆ(R1 )) < (U, fˆ(R3 )) < (U, fˆ(R1 )), (V , fˆ(R2 )) < (U, fˆ(R3 )) < (U, fˆ(R2 )), (V , fˆ(R3 )) < (U, fˆ(R4 )) < (U, fˆ(R3 )). Theorem 4.7. Let (U, R) ∼f (V , fˆ(R)). Suppose S, T ⊆ fˆ(R). If ind(fˆ−1 (S)) ∈ R, ind(fˆ−1 (T )) ∈ R, then (V , S)  (V , T ) ⇐⇒ (U, fˆ−1 (S))  (U, fˆ−1 (T )). Proof. “=⇒.” Suppose (V , S)  (V , T ). Then for all j , [yj ]S ⊆ [yj ]T . Let f (xi ) = yj . For all u ∈ [xi ]fˆ−1 (S ) , we have xi fˆ−1 (S)u; that is, (xi , u) ∈ ind(fˆ−1 (S)). By Theorem 2.22, ind(fˆ−1 (S)) = fˆ−1 (ind(S)). Then (xi , u) ∈ fˆ−1 (ind(S)); that is, xi fˆ−1 (ind(S))u. This implies f (xi )Sf (u); that is, yj Sf (u). This implies f (u) ∈ [yj ]S . Since [yj ]S ⊆ [yj ]T , we have f (u) ∈ [yj ]T . This implies yj T f (u); that is, f (xi )T f (u). So (f (xi ), f (u)) ∈ ind(T ). This implies (xi , u) ∈ fˆ−1 (ind(T )). By Theorem 2.22, fˆ−1 (ind(T )) = ind(fˆ−1 (T )). Then (xi , u) ∈ ind(fˆ−1 (T )); that is, xi fˆ−1 (T )u. So u ∈ [xi ]fˆ−1 (T ) . Thus [xi ] ˆ−1 ⊆ [xi ] ˆ−1 . This shows (U, fˆ−1 (S))  (U, fˆ−1 (T )). f

(S )

f

(T )

“⇐=.” Suppose (U, fˆ−1 (S))  (U, fˆ−1 (T )). Then for all i, [xi ]fˆ−1 (S ) ⊆ [xi ]fˆ−1 (T ) . Let yj = f (xi ). For all v ∈ [yj ]S , we have yj Sv. Let f (u) = v. Note that f (xi )Sf (u); that is, (f (xi ), f (u)) ∈ ind(S). So (xi , u) ∈ fˆ−1 (ind(S)). By Theorem 2.22, fˆ−1 (ind(S)) = ind(fˆ−1 (S)).

Then (xi , u) ∈ ind(fˆ−1 (S)); that is, xi fˆ−1 (S)u. This implies u ∈ [xi ]fˆ−1 (S ) . Since [xi ]fˆ−1 (S ) ⊆ [xi ]fˆ−1 (T ) , we have u ∈ [xi ] ˆ−1 . Then xi fˆ−1 (T )u; that is, (xi , u) ∈ ind(fˆ−1 (T )). By Theorem 2.22, f

(T )

ind(fˆ−1 (T )) = fˆ−1 (ind(T )). Then (xi , u) ∈ fˆ−1 (ind(T )); that is, xi fˆ−1 (ind(T ))u. This implies f (xi )T f (u); that is, yj T v. This implies v ∈ [yj ]T . So [yj ]S ⊆ [yj ]T . Hence (V , S)  (V , T ). 2

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Corollary 4.8. Let (U, R) ∼f (V , fˆ(R)). Suppose S, T ⊆ fˆ(R). If ind(fˆ−1 (S)) ∈ R, ind(fˆ−1 (T )) ∈ R, then (V , S) = (V , T ) ⇐⇒ (U, fˆ−1 (S)) = (U, fˆ−1 (T )). Proof. This holds by Theorem 4.7.

2

Corollary 4.9. Let (U, R) ∼f (V , fˆ(R)). Suppose S, T ⊆ fˆ(R). If ind(fˆ−1 (S)) ∈ R, ind(fˆ−1 (T )) ∈ R, then (V , S) ≺ (V , T ) ⇐⇒ (U, fˆ−1 (S)) ≺ (U, fˆ−1 (T )). Proof. This holds by Theorem 4.7 and Corollary 4.8.

2

Theorem 4.10. Let (U, R) ∼f (V , fˆ(R)). Suppose S, T ⊆ fˆ(R). If ind(fˆ−1 (S)) ∈ R, ind(fˆ−1 (T )) ∈ R, then (V , S)  (V , T ) ⇐⇒ (U, fˆ−1 (S))  (U, fˆ−1 (T )). Proof. “=⇒.” Suppose (V , S)  (V , T ). Then there exists j , [yj ]S ⊆ [yj ]T . Let f (xi ) = yj . For all u ∈ [xi ]fˆ−1 (S ) , we have xi fˆ−1 (S)u; that is, (xi , u) ∈ ind(fˆ−1 (S)). By Theorem 2.22, ind(fˆ−1 (S)) = fˆ−1 (ind(S)). Then (xi , u) ∈ fˆ−1 (ind(S)); that is, xi fˆ−1 (ind(S))u. This implies f (xi )Sf (u); that is, yj Sf (u). This implies f (u) ∈ [yj ]S . Since [yj ]S ⊆ [yj ]T , we have f (u) ∈ [yj ]T . This implies yj T f (u); that is, f (xi )T f (u). So (f (xi ), f (u)) ∈ ind(T ). This implies (xi , u) ∈ fˆ−1 (ind(T )). By Theorem 2.22, fˆ−1 (ind(T )) = ind(fˆ−1 (T )). Then (xi , u) ∈ ind(fˆ−1 (T )); that is, xi fˆ−1 (T )u. So u ∈ [xi ]fˆ−1 (T ) . Thus [xi ] ˆ−1 ⊆ [xi ] ˆ−1 . This shows (U, fˆ−1 (S))  (U, fˆ−1 (T )). f

(S )

f

(T )

“⇐=.” Suppose (U, fˆ−1 (S))  (U, fˆ−1 (T )). Then there exists i, [xi ]fˆ−1 (S ) ⊆ [xi ]fˆ−1 (T ) . Let yj = f (xi ). For all v ∈ [yj ]S , we have yj Sv. Let f (u) = v. Note that f (xi )Sf (u); that is, (f (xi ), f (u)) ∈ ind(S). So (xi , u) ∈ −1 ˆ f (ind(S)). By Theorem 2.22, fˆ−1 (ind(S)) = ind(fˆ−1 (S)).

Then (xi , u) ∈ ind(fˆ−1 (S)); that is, xi fˆ−1 (S)u. This implies u ∈ [xi ]fˆ−1 (S ) . Since [xi ]fˆ−1 (S ) ⊆ [xi ]fˆ−1 (T ) , we have u ∈ [xi ] ˆ−1 . Then xi fˆ−1 (T )u; that is, (xi , u) ∈ ind(fˆ−1 (T )). By Theorem 2.22, f

(T )

ind(fˆ−1 (T )) = fˆ−1 (ind(T )). Then (xi , u) ∈ fˆ−1 (ind(T )); that is, xi fˆ−1 (ind(T ))u. This implies f (xi )T f (u); that is, yj T v. This implies v ∈ [yj ]T . So [yj ]S ⊆ [yj ]T . Hence (V , S)  (V , T ). 2 Corollary 4.11. Let (U, R) ∼f (V , fˆ(R)). Suppose S, T ⊆ fˆ(R). If ind(fˆ−1 (S)) ∈ R, ind(fˆ−1 (T )) ∈ R, then (1) (V , S) < (V , T ) ⇐⇒ (U, fˆ−1 (S)) < (U, fˆ−1 (T )); (2) (V , S)  (V , T ) ⇐⇒ (U, fˆ−1 (S))  (U, fˆ−1 (T )). Proof. (1) This follows from Corollary 4.8 and Theorem 4.10. (2) This holds by Theorem 4.10. 2 Theorem 4.7, Corollary 4.8, Corollary 4.9, Theorem 4.10, and Corollary 4.11 mean that the equality or dependence or partial dependence or independence between knowledge bases is inverse invariant under homomorphisms.

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Example 4.12 (Continued from Example 4.6). We have (U, R) ∼ (V , fˆ(R)) and ind(R1 ), ind(R2 ) ∈ R. By Definition 3.9, one can obtain that 15 1  108 ρ((U, R1 ), (U, R2 )) = 2 | [xi ]R1 ⊕ [xi ]R2 |= , 15 225 i=1

ρ((V , fˆ(R1 )), (V , fˆ(R2 ))) =

6 1  1 | [xi ]fˆ(R1 ) ⊕ [xi ]fˆ(R2 ) |= . 62 2 i=1

Thus ρ((U, R1 ), (U, R2 )) = ρ((V , fˆ(R1 )), (V , fˆ(R2 ))). Pick S = fˆ(R1 ), T = fˆ(R2 ). By Theorem 2.22, fˆ−1 (S) = R1 , fˆ−1 (T ) = R2 . Thus ρ((V , S), (V , T )) = ρ((U, fˆ−1 (S), (U, fˆ−1 (T ))). This example illustrates the fact that knowledge distance between knowledge bases in a given knowledge base is neither invariant nor inverse invariant under homomorphisms. 5. Some tools for measuring uncertainty of knowledge bases In this section, we investigate measuring uncertainty of knowledge bases. 5.1. Granularity measures of knowledge bases We first propose an axiom definition of knowledge granulation of knowledge bases. ∗

Definition 5.1. Suppose that G : 2R (U ) → (−∞, +∞) is a function. Then G is referred to as a knowledge granulation ∗ function on 2R (U ) if G satisfies the following conditions: ∗

(1) Nonnegativity: For all P ∈ 2R (U ) , G(P) ≥ 0. ∗ (2) Invariability: for all P, Q ∈ 2R (U ) , if (U, P)  (U, Q), then G(P) = G(Q). ∗ R (3) Monotonicity: for all P, Q ∈ 2 (U ) , if (U, P) ≺ (U, Q), then G(P) < G(Q). Moreover, for P ∈ 2R

∗ (U )

, G(P) is referred to as the knowledge granulation of the knowledge base (U, P).

Similar to Definition 2 in [9], the knowledge granulation of a given knowledge base is given in the following definition. Definition 5.2. Suppose that (U, P) is a knowledge base. Knowledge granulation of (U, P) is defined as G(P) =

m 1  |Xi |2 , n2 i=1

where U/P = {X1 , X2 , . . . , Xm }.

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Proposition 5.3. Let (U, P) be a knowledge base. Then G(P) =

n 1  |[xi ]P |. n2 i=1

Proof. Denote U/P = {X1 , X2 , . . . , Xm }. Suppose Xi = {xi1 , xi2 , . . . , xisi }, |Xi | = si . Then m 

si = n, Xi = [xi1 ]P = [xi2 ]P = · · · = [xisi ]P .

i=1

So |Xi | = |[xi1 ]P | = |[xi2 ]P | = · · · = |[xisi ]P | = si . Thus for all i, |Xi |2 = si |Xi | =

si 

|[xik ]P |.

k=1

Hence m m si n 1  1  1  2 G(P) = 2 |Xi | = 2 |[xik ]P | = 2 |[xi ]P |. n n n i=1

i=1 k=1

2

i=1

Proposition 5.4. Let (U, P) be a knowledge base. Then 1 ≤ G(P) ≤ 1. n Moreover, if ind(P) is an identity relation on U , then G(P) achieves the minimum value n1 ; if ind(P) is a universal relation on U , then G(P) achieves the maximum value 1, specially, G(∅) = 1. Proof. Since ∀ i, 1 ≤ |[xi ]P | ≤ n, n ≤

n

i=1

|[xi ]P | ≤ n2 .

By Proposition 5.3, 1 ≤ G(P) ≤ 1. n If ind(P) is an identity relation on U , then for all i, |[xi ]P | = 1. So G(P) = n1 . If ind(P) is a universal relation on U , then for all i, |[xi ]P | = n. So G(P) = 1. 2 Theorem 5.5. Let (U, P) and (U, Q) be two knowledge bases. If (U, P) ≺ (U, Q), then G(P) < G(Q). Proof. By Proposition 5.3, G(P) =

n n 1  1  |[x ] |, G(Q) = |[xi ]Q |. i P n2 n2 i=1

i=1

Since (U, P) ≺ (U, Q), for all i, [xi ]P ⊆ [xi ]Q , and there exists j , [xj ]P  xi ]Q . Then for all i, |[xi ]P | ≤ |[xi ]Q |, and there exists j , 1 ≤ |[xj ]P | < |[xj ]Q |. Hence G(P) < G(Q).

2

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This theorem illustrates the fact that the knowledge granulation increases when the available knowledge becomes coarser, and it decreases when the available knowledge becomes finer. In other words, the more uncertain the available knowledge is, the bigger the knowledge granulation becomes. Thus we can conclude that the knowledge granulation introduced in Definition 5.2 can be used to evaluate the certainty degree of a given knowledge base. Theorem 5.6. G in Definition 5.2 is a knowledge granulation function under Definition 5.1. Proof. (1) Obviously, “nonnegativity” holds. ∗ (2) Suppose P, Q ∈ 2R (U ) . If (U, P)  (U, Q), then for all i, [xi ]P = [xi ]Q . By Proposition 5.3, G(P) = G(Q). (3) “Monotonicity” follows from Theorem 5.5. 2 5.2. Entropy measures of knowledge bases In physics, entropy is often used to measure the degree of disorder of a system. The bigger the entropy is, the higher the degree of disorder of a system will be. Shannon [18] applied the concept of entropy in physics to information theory for measurement of the uncertainty of a system. Similarly to Definition 8 in [9], the knowledge entropy of a given knowledge base is defined as follows. Definition 5.7. Suppose that (U, P) is a knowledge base. Knowledge entropy of (U, P) is defined as H (P) = −

m  |Xi |

n

i=1

log2

|Xi | , n

where U/P = {X1 , X2 , . . . , Xm }. When some

|Xi | n

equals 0, then 0 · log2 0 = 0.

Proposition 5.8. Let (U, P) be a knowledge base. Then H (P) = −

n  1 i=1

n

log2

|[xi ]P | . n

Proof. Denote U/P = {X1 , X2 , . . . , Xm }. Suppose Xi = {xi1 , xi2 , . . . , xisi }, |Xi | = si . Then m 

si = n, Xi = [xi1 ]P = [xi2 ]P = · · · = [xisi ]P .

i=1

So |Xi | = |[xi1 ]P | = |[xi2 ]P | = · · · = |[xisi ]P | = si . Thus for all i, i |Xi | |Xi | 1 |Xi |  1 |[xik ]P | log2 = si log2 = log2 . n n n n n n

s

k=1

Hence H (P) = − =−

m  |Xi | i=1 n  i=1

n

i  |Xi | 1 |[xik ]P | =− log2 n n n

m

log2

1 |[xi ]P | log2 . 2 n n

s

i=1 k=1

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Theorem 5.9. Let (U, P) and (U, Q) be two knowledge bases. If (U, P) ≺ (U, Q), then H (Q) < H (P). Proof. By Proposition 5.8, H (P) = −

n  1 i=1

n

1 |[xi ]P | |[xi ]Q | , H (Q) = − log2 . n n n n

log2

i=1

Note that (U, P) ≺ (U, Q). Then similarly to the proof of Theorem 5.5, for all i, 1 ≤ |[xi ]P | ≤ |[xi ]Q |, and there exists j , 1 ≤ |[xj ]P | < |[xj ]Q |. Then for all i, − log2

|[xi ]P | n n |[xi ]Q | = log2 ≥ log2 = − log2 , n |[xi ]P | |[xi ]Q | n

and there exists j , − log2

|[xj ]P | |[xj ]Q | n n = log2 > log2 = − log2 . n |[xj ]P | |[xj ]Q | n 2

Hence H (Q) < H (P).

From Theorem 5.9 we can conclude that the knowledge entropy proposed in Definition 5.7 can be used to evaluate the certainty degree of a given knowledge base. In other words, the more uncertain the available knowledge is, the smaller the knowledge entropy becomes. Rough entropy, introduced by Yao [23], is used to measure the granularity of a given partition. It is also called co-entropy by some scholars [1]. Similarly to Definition 6 in [9], the rough entropy of a given knowledge base is proposed in the following definition. Definition 5.10. Let (U, P) be a knowledge base. Rough entropy of (U, P) is defined as Er (P) = −

m  |Xi |

n

i=1

log2

1 , |Xi |

where U/P = {X1 , X2 , . . . , Xm }. Proposition 5.11. Let (U, P) be a knowledge base. Then Er (P) = −

n  1 i=1

n

log2

1 . |[xi ]P |

Proof. Denote U/P = {X1 , X2 , . . . , Xm }. Suppose Xi = {xi1 , xi2 , . . . , xisi }, |Xi | = si . Then m 

si = n, Xi = [xi1 ]P = [xi2 ]P = · · · = [xisi ]P .

i=1

So |Xi | = |[xi1 ]P | = |[xi2 ]P | = · · · = |[xisi ]P | = si .

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Thus for all i, i  1 1 1 |Xi | log2 log2 = si log2 = . |Xi | |Xi | |[xik ]P |

s

k=1

Hence m  |Xi |

Er (P) = −

n

i=1 si m  

1 n

=−

i=1

log2

i=1 k=1 n 

=−

i=1

1 1 1 |Xi | log2 =− |Xi | n |Xi | m

log2

|[xik ]P |

1 1 log2 . n |[xi ]P |

1 1 log2 n |[xi ]P | n

1

=−

i=1

2

Proposition 5.12. Let (U, P) be a knowledge base. Then 0 ≤ Er (P) ≤ log2 n. Moreover, if ind(P) is an identity relation on U , then Er (P) achieves the minimum value 0; if ind(P) is a universal relation on U , then Er (P) achieves the maximum value log2 n, specially, Er (∅) = log2 n. Proof. Since for all i, 1 ≤ |[xi ]P | ≤ n, 0 ≤ − log2

1 |[xi ]P |

= log2 |[xi ]P | ≤ log2 n. Then 0 ≤ −

n

i=1

log2

1 |[xi ]P |

≤

n log2 n. By Proposition 5.11, 0 ≤ Er (P) ≤ log2 n. If ind(P) is an identity relation on U , then for all i, |[xi ]P | = 1. So Er (P) = 0. If ind(P) is a universal relation on U , then for all i, |[xi ]P | = n. So Er (P) = log2 n.

2

Theorem 5.13. Let (U, P) and (U, Q) be two knowledge bases. If (U, P) ≺ (U, Q), then Er (P) < Er (Q). Proof. By Proposition 5.11, Er (P) = −

n  1 i=1

n

1 1 1 , Er (Q) = − log2 . |[xi ]P | n |[xi ]Q | n

log2

i=1

Note that (U, P) ≺ (U, Q). Then, similarly to the proof of Theorem 5.5, for all i, 1 ≤ |[xi ]P | ≤ |[xi ]Q |, and there exists j , 1 ≤ |[xj ]P | < |[xj ]Q |. Then for all i, − log2

1 1 = log2 |[xi ]P | ≤ log2 |[xi ]Q | = − log2 , |[xi ]P | |[xi ]Q |

and there exists j , − log2

1 1 = log2 |[xj ]P | < log2 |[xj ]Q | = − log2 . |[xj ]P | |[xj ]Q |

Hence Er (P) < Er (Q).

2

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From Theorem 5.13, we can conclude that the rough entropy proposed in Definition 5.10 can be used to evaluate the certainty degree of a given knowledge base. In other words, the more uncertain the available knowledge is, the bigger the rough entropy becomes. Theorem 5.14. Er in Definition 5.10 is a knowledge granulation function under Definition 5.1. Proof. (1) Obviously, “nonnegativity” holds. ∗ (2) Suppose P, Q ∈ 2R (U ) . If (U, P)  (U, Q), then for all i, [xi ]P = [xi ]Q . By Proposition 5.11, Er (P) = Er (Q). (3) “Monotonicity” follows from Theorem 5.13. 2 5.3. Knowledge amounts of knowledge bases Similarly to Definition 10 in [9], the knowledge amount of a given knowledge base is proposed in the following definition. Definition 5.15. Let (U, P) be a knowledge base. The knowledge amount of (U, P) is defined as E(P) =

m  |Xi | |U − Xi | , n n i=1

where U/P = {X1 , X2 , . . . , Xm },

|Xi | n

(or

|U −Xi | ) n

represents the probability of Xi (or U − Xi ) within the universe U .

Proposition 5.16. Let (U, P) be a knowledge base. Then E(P) =

n  1 i=1

n

(1 −

|[xi ]P | ). n

Proof. Denote U/P = {X1 , X2 , . . . , Xm }. Suppose Xi = {xi1 , xi2 , . . . , xisi }, |Xi | = si . Then m 

si = n, Xi = [xi1 ]P = [xi2 ]P = · · · = [xisi ]P .

i=1

So |Xi | = |[xi1 ]P | = |[xi2 ]P | = · · · = |[xisi ]P | = si . Since {X1 , X2 , . . . , Xm } is a partition on U , for all i, we have U − Xi = (

i−1

m



Xk ) ( Xk ).

k=1

k=i+1

Then |U − Xi | =

i−1  k=1

|Xk | +

m 

|Xk | = |U | − |Xi | = n − |Xi |.

k=i+1

Thus for all i, |Xi ||U − Xi | = si (n − |Xi |) =

si  (n − |[xik ]P |). k=1

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Hence m m si  |Xi | |U − Xi |   n − |[xik ]P | E(P) = = n n n2 i=1

i=1 k=1

n  n − |[xi ]P |

=

n2

i=1

=

n  1 i=1

n

(1 −

|[xi ]P | ). n

2

Theorem 5.17. Let (U, P) and (U, Q) be two knowledge bases. If (U, P) ≺ (U, Q), then E(Q) < E(P). Proof. By Proposition 5.16, E(P) =

n  1

1 |[xi ]P | |[xi ]Q | (1 − ), E(Q) = (1 − ). n n n n

i=1

n

i=1

Note that (U, P) ≺ (U, Q). Then similarly to the proof of Theorem 5.5, for all i, 1 ≤ |[xi ]P | ≤ |[xi ]Q |, and there exists j , 1 ≤ |[xj ]P | < |[xj ]Q |. 2

Hence E(Q) < E(P).

This theorem illustrates the fact that the knowledge amount decreases when the available knowledge becomes coarser, and it increases when the available knowledge becomes finer. In other words, the more uncertain the available knowledge is, the smaller the knowledge amount becomes. Thus we can conclude that the knowledge amount proposed in Definition 5.15 can be used to evaluate the certainty degree of a given knowledge base. 5.4. Some properties Theorem 5.18. Let (U, P) be a knowledge base. Then G(P) + E(P) = 1. Proof. Denote U/P = {X1 , X2 , . . . , Xm }. Suppose Xi = {xi1 , xi2 , . . . , xisi }, |Xi | = si . Then

m

si = n.

i=1

Since {X1 , X2 , . . . , Xm } is a partition on U , for all i we have U − Xi = (

i−1

Xk )

m



( Xk ).

k=1

k=i+1

Then |U − Xi | =

i−1 

|Xk | +

k=1

m 

|Xk | = |U | − |Xi | = n − |Xi |.

k=i+1

So E(P) =

m m m m  |Xi | |U − Xi |  si (n − si )  si  si2 = = − n n n2 n n2 i=1

=1−

i=1

m  |Xi |2 i=1

n2

Thus G(P) + E(P) = 1.

= 1 − G(P). 2

i=1

i=1

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Corollary 5.19. Let (U, P) be a knowledge base. Then 1 0 ≤ E(P) ≤ 1 − . n 1 n

Proof. By Proposition 5.4,

≤ G(P) ≤ 1. By Theorem 5.18, E(P) = 1 − G(P). Thus 0 ≤ E(P) ≤ 1 − n1 .

2

Theorem 5.20. Let (U, P) be a knowledge base. Then Er (P) + H (P) = log2 n. Proof. Denote U/P = {X1 , X2 , . . . , Xm }. Suppose Xi = {xi1 , xi2 , . . . , xisi }, |Xi | = si . Then

m

si = n.

i=1

Thus Er (P) + H (P) = − =−

m  |Xi | i=1 m  i=1

=− =−

n

si 1 log2 − n si

m  si i=1 m  i=1

 |Xi | 1 |Xi | − log2 |Xi | n n m

log2

n

i=1

m  i=1

si si log2 n n

(log2 1 − log2 si + log2 si − log2 n)

 si si (log2 1 − log2 n) = − (− log2 n) n n m

i=1

= log2 n. 2 Corollary 5.21. Let (U, P) be a knowledge base. Then 0 ≤ H (P) ≤ log2 n. Proof. By Proposition 5.12, 0 ≤ Er (P) ≤ log2 n. By Theorem 5.20, H (P) = log2 n − Er (P). Thus 0 ≤ H (P) ≤ log2 n. 2 5.5. An illustrative example Example 5.22 (Continued from Example 4.6). We have (U, R) ∼ (V , fˆ(R)) and ind(R2 ) ∈ R. Note that U/R2 = {{x1 , x4 , x11 , x12 , x13 , x14 , x15 }, {x2 , x3 , x5 , x6 , x7 , x8 , x9 , x10 }}, V /fˆ(R2 ) = {{y1 , y6 }, {y2 , y3 , y4 , y5 }}. Then 2 2 1  113 1  20 2 ˆ G(R2 ) = 2 |Xi | = |Yj |2 = ; , G(f (R2 )) = 2 15 225 6 36 i=1

j =1

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Table 1 Four data sets from UCI.

H (R2 ) = −

2  |Xi |

15

i=1

H (fˆ(R2 )) = −

2  i=1

Er (fˆ(R2 )) = −

E(R2 ) =

i=1

Features

1728 12960 1389 958

6 8 10 9

log2

6

|Xi | 7 8 7 8 = − log2 − log2 , 15 15 15 15 15

log2

|Yj | 2 4 4 2 = − log2 − log2 ; 6 6 6 6 6

|Xi | 1 1 1 7 8 log2 = − log2 − log2 , 15 |Xi | 15 7 15 8 2  |Yj | j =1

2 

Objects

Car Evaluation Nursery Solar Flare Tic-Tac-Toe Endgame

2  |Yj | j =1

Er (R2 ) = −

Data sets

6

log2

1 1 4 1 2 = − log2 − log2 ; |Yj | 6 2 6 4

 |Yj | |Yj | |Xi | |Xi | 112 16 (1 − )= , E(fˆ(R2 )) = (1 − )= . 15 15 225 6 6 36 2

j =1

Thus G(R2 ) = G(fˆ(R2 )), H (R2 ) = H (fˆ(R2 )), Er (R2 ) = Er (fˆ(R2 )), E(R2 ) = E(fˆ(R2 )). Pick Q = fˆ(R2 ). By Proposition 2.18, fˆ−1 (Q) = R2 . Thus G(Q) = G(fˆ−1 (Q)), H (Q) = H (fˆ−1 (Q)), Er (Q) = Er (fˆ−1 (Q)), E(Q) = E(fˆ−1 (Q)). This example illustrates the fact that knowledge granulation, knowledge entropy, rough entropy, and the knowledge amount of knowledge bases are neither invariant nor inverse invariant under homomorphisms. 6. Numerical experiments and effectiveness analysis In this section, we conduct a numerical experiment and perform effectiveness analysis from the two aspects of dispersion and correlation in statistics. 6.1. A numerical experiment To evaluate the performance of the proposed measures for uncertainty of knowledge bases, we provide the following numerical experiment on four data sets that come from the UCI Repository of machine learning databases as shown in Table 1, and compare four tools for measuring uncertainty of knowledge bases. Car Evaluation may express an information system (U, A) with |U | = 1728, |A| = 6. Denote Pi = ind({ai }) (i = 1, . . . , 6). Pick Pi = {P1 , . . . , Pi } (i = 1, . . . , 6). Then for each i, (U, Pi ) is the knowledge base induced by Car Evaluation. Four measure sets on Car Evaluation are defined as follows:

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XG (C) = {G(P1 ), . . . , G(P6 )}, XE (C) = {E(P1 ), . . . , E(P6 )}, XEr (C) = {Er (P1 ), . . . , Er (P6 )}, XH (C) = {H (P1 ), . . . , H (P6 )}. Nursery may express an information system (V , B) with |V | = 12960, |B| = 8. Denote Qi = ind({bi }) (i = 1, . . . , 8). Pick Qi = {Q1 , . . . , Qi } (i = 1, . . . , 8). Then for each i, (V , Qi ) is the knowledge base induced by Nursery. Four measure sets on Nursery are defined as follows: XG (N ) = {G(Q1 ), . . . , G(Q8 )}, XE (N ) = {E(Q1 ), . . . , E(Q8 )}, XEr (N ) = {Er (Q1 ), . . . , Er (Q8 )}, XH (N ) = {H (Q1 ), . . . , H (Q8 )}. Solar Flare may express an information system (W, C) with |W | = 1389, |C| = 10. Denote Ri = ind({ci }) (i = 1, . . . , 10). Pick Ri = {R1 , . . . , Ri } (i = 1, . . . , 10). Then for each i, (W, Ri ) is the knowledge base induced by Solar Flare. Four measure sets on Solar Flare are defined as follows: XG (S) = {G(R1 ), . . . , G(R10 )}, XE (S) = {E(R1 ), . . . , E(R10 )}, XEr (S) = {Er (R1 ), . . . , Er (R10 )}, XH (S) = {H (R1 ), . . . , H (R10 )}. Tic-Tac-Toe Endgame may express an information system (Z, D) with |Z| = 958, |D| = 9. Denote Si = ind({di }) (i = 1, . . . , 9). Pick Si = {S1 , . . . , Si } (i = 1, . . . , 9). Then for each i, (Z, Si ) is the knowledge base induced by Tic-Tac-Toe Endgame. Four measure sets on Tic-Tac-Toe Endgame are defined as follows: XG (T ) = {G(S1 ), . . . , G(S9 )}, XE (T ) = {E(S1 ), . . . , E(S9 )}, XEr (T ) = {Er (S1 ), . . . , Er (S9 )}, XH (T ) = {H (S1 ), . . . , H (S9 )}. The experimental results are shown in Fig. 5. It is easy to see that knowledge granulation G and rough entropy Er both monotonically decrease as equivalence relations increase. Meanwhile, the knowledge amount E and knowledge entropy H both monotonically increase with the increase of equivalence relations. That means the uncertainty of a knowledge base decreases as equivalence relations increase. 6.2. Dispersion analysis In actual statistical work, we often study the dispersion degree of a data set. The amount used to measure the dispersion degree of a data set is called a difference measure. The common difference measures include range, fourpoint difference, average difference, standard deviation, and standard deviation coefficient. Here we use the standard deviation coefficient to perform effectiveness analysis of the proposed measures. Suppose we have a data set X = {x1 , · · · , xn }. Then the arithmetic average value, standard deviation, and standard deviation coefficient of X, denoted by x, σ (X), and CV (X), respectively, are defined as follows:   n n 1  1 x= xi , σ (X) =  (xi − x)2 , n n i=1

CV (X) =

i=1

σ (X) . x

We compare CV values of four measure sets from the previous experimental The experimental results are shown in Fig. 6. “CV (XG (C)) > CV (XEr (C)) > CV (XH (C)) > CV (XE (C))” means the dispersion degree of E is minimum when Car Evaluation is selected as the test set. Similarly, the dispersion degree of E is minimum when Nursery, Solar Flare, and Tic-Tac-Toe Endgame are selected as the test sets. Thus E has much better performance for measuring uncertainty of knowledge bases.

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Fig. 5. The change of four tools for measuring uncertainty with different knowledge bases: knowledge bases induced by (a) Car Evaluation, (b) Nursery, (c) Solar Flare, and (d) Tic-Tac-Toe Endgame.

6.3. Correlation analysis In statistics, the Pearson correlation coefficient is a measure of the strength of a linear correlation between two variables or two data sets. Suppose we have two data sets X = {x1 , · · · , xn } and Y = {y1 , · · · , yn }. The Pearson correlation coefficient for X and Y , denoted by r(X, Y ) or rXY , is defined as follows: n

r(X, Y ) or rXY = 

(xi − x)(yi − y) ,  n n

(xi − x)2 (yi − y)2 i=1

i=1

where x =

1 n

n

xi , y =

i=1

1 n

n

i=1

yi .

i=1

We have to verify that n

n

xi yi −

i=1

r(X, Y ) =  n

n

i=1

xi2 − (

Obviously, −1 ≤ r(X, Y ) ≤ 1.

n

i=1

n

i=1



xi )2 n

xi

n

yi

i=1 n

i=1

yi2 − (

n

i=1

. yi )2

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Fig. 6. CV values of four measure sets: knowledge bases induced by (a) Car Evaluation, (b) Nursery, (c) Solar Flare, and (d) Tic-Tac-Toe Endgame.

Table 2 r values of 16 pairs of four measure sets on Car Evaluation. r XG (C) XE (C) XEr (C) XG (C) XE (C) XEr (C) XH (C)

1 −1 0.8178 −0.8178

−1 1 −0.8178 0.8178

0.8178 −0.8178 1 −1

XH (C) −0.8178 0.8178 −1 1

If r(X, Y ) = 0, then there is no correlation between X and Y ; if r(X, Y ) > 0, then the correlation between X and Y is positive; if r(X, Y ) < 0, then the correlation between X and Y is negative. Particularly, r(X, Y ) = 1 indicates completely positive correlation between X and Y , and r(X, Y ) = −1 means completely negative correlation between X and Y . The closer the absolute value of the Pearson correlation coefficient r is to 0, the smaller the degree of correlation between variables; conversely, the closer the absolute value of the Pearson correlation coefficient r is to 1, the greater the degree of correlation between variables. Generally speaking, the degree of correlation can be classified as follows: when |r| = 1, there is complete correlation; when 0.7 ≤ |r| < 1, there is high correlation; when 0.4 ≤ |r| < 0.7, there is moderate correlation; when 0 < |r| < 0.4, there is low correlation; when r = 0, there is no correlation. We compare r values on four measure sets from the earlier experiment. The experimental results are shown in Tables 2–5. “r(XG (C), XE (C)) = −1” means completely negative correlation between XG(C) and XG (C). There is complete negative correlation between G and E when Car Evaluation is selected as the test set. There is also complete negative correlation between G and E when Nursery, Solar Flare, and Tic-Tac-Toe Endgame are selected as the test sets. Thus

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Table 3 r values of 16 pairs of four measure sets on Nursery. r XG (N ) XE (N ) XEr (N ) XG (N ) XE (N ) XEr (N ) XH (N )

1 −1 0.7477 −0.7477

−1 1 −0.7477 0.7477

0.7477 −0.7477 1 −1

Table 4 r values of 16 pairs of four measure sets on Solar Flare. r XG (S) XE (S) XEr (S) XG (S) XE (S) XEr (S) XH (S)

1 −1 0.9461 −0.9461

−1 1 −0.9461 0.9461

0.9461 −0.9461 1 −1

XH (N ) −0.7477 0.7477 −1 1

XH (S) −0.9461 0.9461 −1 1

Table 5 r values of 16 pairs of four measure sets on Tic-Tac-Toe Endgame. r XG (T ) XE (T ) XEr (T ) XH (T ) XG (T ) XE (T ) XEr (T ) XH (T )

1 −1 0.8188 −0.8188

−1 1 −0.8188 0.8188

0.8188 −0.8188 1 −1

−0.8188 0.8188 −1 1

Table 6 Invariant and inverse invariant characteristics of knowledge bases under homomorphisms based on data compression. Characteristics Equality between knowledge bases (=) Dependency between knowledge bases () Dependency strictly between knowledge bases (≺) Partial dependency between knowledge bases () Partial dependency strictly between knowledge bases (<) Independency between knowledge bases () Knowledge distance between knowledge bases (ρ) Knowledge granulation of knowledge bases (G) Rough entropy of knowledge bases (Er ) Knowledge entropy of knowledge bases (H ) Knowledge amount of knowledge bases (E)

Invariant √ √ √ √ √ √

Inverse invariant √ √ √ √ √ √

× × × × ×

× × × × ×

there is complete negative correlation between G and E. Similarly, there is complete negative correlation between Er and H . “r(XG (C), XEr (C)) > 0.7” means there is high positive correlation between XG (C) and XEr (C). There is high positive correlation between G and Er when Car Evaluation is selected as the test set. There is also high positive correlation between G and Er when Nursery, Solar Flare, and Tic-Tac-Toe Endgame are selected as the test sets. Thus there is high positive correlation between G and Er . Similarly, there is high positive correlation between E and H . “r(XG (C), XH (C)) < −0.7” means high negative correlation between XG (C) and XH (C). There is high negative correlation between G and H when Car Evaluation is selected as the test set. There is also high negative correlation between G and H when Nursery, Solar Flare, and Tic-Tac-Toe Endgame are selected as the test sets. Thus there is high negative correlation between G and H . Similarly, there is high negative correlation between E and Er .

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7. Conclusions In this article, relationships between knowledge bases have been studied from the two aspects of dependence and separation. Measurement of uncertainty of knowledge bases was investigated, and from the viewpoint of statistics, effectiveness analysis was conducted by a numerical experiment. Invariant and inverse invariant characteristics of knowledge bases under homomorphisms based on data compression were obtained (see Table 6). These results will be important in building a framework of granular computing in a knowledge base. In the future, we will consider applications of the proposed results for dealing with knowledge discovery in a knowledge base. Acknowledgements The authors thank the editors and the anonymous reviewers for their valuable suggestions that helped immensely in improving the quality of this article. This work was supported by the National Natural Science Foundation of China (11461005), the Natural Science Foundation of Guangxi Province (2016GXNSFAA380045, 2016GXNSFAA380282, 2016GXNSFAA380286), the National Social Science Foundation of China (12BJL087), and the Philosophy and Social Sciences Planning Project of Guangxi (11BJY029). References [1] T. Beaubouef, F.E. Petry, G. Arora, Information-theoretic measures of uncertainty for rough sets and rough relational databases, Inf. Sci. 109 (1998) 185–195. [2] I. Düntsch, G. Gediga, Uncertainty measures of rough set prediction, Artif. Intell. 106 (1) (1998) 109–137. [3] M. Kryszkiewicz, Comparative study of alternative types of knowledge reduction in inconsistent systems, Int. J. Intell. Syst. 16 (2001) 105–120. [4] S. Lin, Generalized Metric Spaces and Mappings, Chinese Scientific Publishers, Beijing, 1995. [5] Z. Li, Y. Liu, Q. Li, B. Qin, Relationships between knowledge bases and related results, Knowl. Inf. Syst. 49 (2016) 171–195. [6] Z. Li, Q. Li, R. Zhang, N. Xie, Knowledge structures in a knowledge base, Expert Syst. 33 (2016) 581–591. [7] J. Li, C. Mei, Y. Lv, Knowledge reduction in decision formal contexts, Knowl.-Based Syst. 24 (2011) 709–715. [8] A.Y. Levy, M.C. Rousset, Verification of knowledge bases based on containment checking, Artif. Intell. 101 (1998) 227–250. [9] J. Liang, Y. Qian, Information granules and entropy theory in information systems, Sci. China, Ser. F 51 (2008) 1427–1444. [10] J. Liang, Z. Shi, The information entropy, rough entropy and knowledge granulation in rough set theory, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 12 (1) (2004) 37–46. [11] J. Liang, Z. Shi, D. Li, M.J. Wierman, The information entropy, rough entropy and knowledge granulation in incomplete information systems, Int. J. Gen. Syst. 35 (6) (2006) 641–654. [12] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic Publishers, Dordrecht, 1991. [13] Z. Pawlak, A. Skowron, Rudiments of rough sets, Inf. Sci. 177 (2007) 3–27. [14] Z. Pawlak, A. Skowron, Rough sets: some extensions, Inf. Sci. 177 (2007) 28–40. [15] Z. Pawlak, A. Skowron, Rough sets and Boolean reasoning, Inf. Sci. 177 (2007) 41–73. [16] B. Qin, ∗-reductions in a knowledge base, Inf. Sci. 320 (2015) 190–205. [17] Y. Qian, J. Liang, C. Dang, Knowledge structure, knowledge granulation and knowledge distance in a knowledge base, Int. J. Approx. Reason. 50 (2009) 174–188. [18] C. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 (1948) 379–423. [19] M. Wierman, Measuring uncertainty in rough set theory, Int. J. Gen. Syst. 28 (1999) 283–297. [20] C. Wang, C. Wu, D. Chen, W. Du, Some properties of relation information systems under homomorphisms, Appl. Math. Lett. 21 (2008) 940–945. [21] C. Wang, C. Wu, D. Chen, Q. Hu, C. Wu, Communicating between information systems, Inf. Sci. 178 (2008) 3228–3239. [22] Y.Y. Yao, Information granulation and rough set approximation, Int. J. Intell. Syst. 16 (1) (2001) 87–104. [23] Y.Y. Yao, Probabilistic approaches to rough sets, Expert Syst. 20 (2003) 287–297. [24] W. Zhang, G. Qiu, Uncertain Decision Making Based on Rough Set Theory, Tsinghua University Publishers, Beijing, 2005. [25] W. Zhang, W. Wu, J. Liang, D. Li, Rough Set Theory and Methods, Chinese Scientific Publishers, Beijing, 2001.