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JID: KNOSYS [m5G; April 21, 2016;16:5 ]

Knowledge-Based Systems 0 0 0 (2016) 1–12

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Knowle dge-Base d Systems

journal homepage: www.elsevier.com/locate/knosys

Granule description based on formal concept analysis

Huilai Zhi a , Jinhai Li b , ∗

a School of Computer Science and Technology, Henan Polytechnic University, Jiaozuo 4540 0 0, Henan, PR China b Faculty of Science, Kunming University of Science and Technology, Kunming 650500, Yunnan, PR China

a r t i c l e i n f o

Article history:

Received 31 December 2015

Revised 4 March 2016

Accepted 13 April 2016

Available online xxx

Keywords:

Granule

Granule description

Granular computing

Formal concept analysis

Stability index

a b s t r a c t

Granule description is a fundamental problem in granular computing. Although the spirit of granular

computing has been widely adopted in scientific researches, how to classify and describe granules in a

concise and apt way is still an open, interesting and important problem. The main objective of our paper

is to give a solution to this problem under the framework of granular computing. Firstly, by using stabil-

ity index, we classify the granules into three categories: atomic granules, basic granules and composite

granules. Secondly, in order to improve the conciseness and aptness of granules, we impose additional

conditions on minimal generator to define a new term which is called the most apt minimal genera-

tor. And then, based on the most apt minimal generator, we put forward methods for the description of

atomic granules and basic granules. Moreover, for composite granules, we continue to divide them into

three subcategories: ∧ -definable granules, ( ∧ , ¬)-definable granules and ( ∧ , ∨ )-definable granules, and

their respective descriptions are provided as well. Finally, some discussions are also made on indefinable

granules.

© 2016 Elsevier B.V. All rights reserved.

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. Introduction

Granular computing (GrC) is an emerging computing paradigm

f information processing, which lies in the scope of cognitive sci-

nce and cognitive informatics [2,43] . Granular computing studies

nformation and knowledge processing in an abstract way, handles

omplex information entities in different granules, and allows us to

iew a phenomenon with different levels of granularity [1,60] . The

pirit of granular computing has been adopted frequently in scien-

ific researches, such as philosophy of structured thinking, struc-

ured problem solving, and structured information processing. In

his sense, all the methods which treat information in this perspec-

ive will fall into the scope of granular computing [12,26,34,37] .

To put it simply, information granules are collections of enti-

ies which are arranged together due to their similarity, functional

r physical adjacency, coherency, and so on [29,53,69] . At present,

ranular computing is not a coherent set of methods or principles

ut rather a theoretical perspective, which encourages researchers

o deal with knowledge at different levels of abstraction or gen-

ralization [9,40,52,66,68] . It often granulates the universe of dis-

ourse into a family of disjoint or overlapping granules. Based on

his idea, different views of the universe of discourse can be linked

ogether, and a hierarchy of granulations can be established. Thus,

∗ Corresponding author. Fax: +86 871 65916703.

E-mail addresses: [email protected] (H. Zhi), [email protected] (J. Li).

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ttp://dx.doi.org/10.1016/j.knosys.2016.04.011

950-7051/© 2016 Elsevier B.V. All rights reserved.

Please cite this article as: H. Zhi, J. Li, Granule description based

http://dx.doi.org/10.1016/j.knosys.2016.04.011

ne of the main directions in the study of granular computing is

o deal with the construction, interpretation, and representation of

ranules [50] .

Rough set theory (RST), as an efficient tool of granular comput-

ng, presented by Pawlak [31] , has drawn many attentions from re-

earchers over the past thirty-four years [14,19,33,48,54,67,70,71] .

s is well known, the original idea of rough set theory is to par-

ition the universe of discourse into disjoint subsets by a given

quivalence relation, and then by using the obtained disjoint sub-

ets, target sets are characterized by means of the so-called lower

nd upper approximations.

Rough sets were used to describe a target set by the lower

nd upper approximations under one granulation, but multiple

ranulations are sometimes required to approximate a target set

hen dealing with multi-scale or multi-source data sets [35,36,51] .

nder such a circumstance, pessimistic multigranulation rough

ets and optimistic multigranulation rough sets were proposed

or applying multi-source information fusion. These information

usion strategies were soon extended to cater the cases such

s incomplete, neighborhood, covering and fuzzy environments

13,24,25,39,55,59] . Moreover, a byproduct is that “AND” and “OR”

ecision rules can be derived from decision systems with the pes-

imistic and optimistic multigranulation rough sets [35,36] , which

as further exploited by Yang et al. [58] and Li et al. [23] in terms

f local and global measurements of the “AND” and “OR” decision

ules.

on formal concept analysis, Knowledge-Based Systems (2016),


http://www.ScienceDirect.com

http://www.elsevier.com/locate/knosys

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2 H. Zhi, J. Li / Knowledge-Based Systems 0 0 0 (2016) 1–12

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Rough set theory is actually related to and complementary with

formal concept analysis (FCA) [17,27,30,49] , and more and more

attention [5,15,17,18,38,46,62] has been paid to comparing and

combining rough set theory and formal concept analysis. Object-

oriented concept lattice was introduced in [63] by incorporating

lower and upper approximation ideas into concept-forming oper-

ators, and it was further elaborated in [28,44] . In the meanwhile,

some studies were also conducted by integrating formal concept

analysis with granular computing, such as granular rule acquisition

[20,21,50,56] , concept learning [22] , fuzzy information granule de-

scription [57] , granular reduct [50] , granule transformation and ir-

reducible element judgment [47] . In addition, rough set theory has

been related to granular computing [35,36,51,61] , and vice versa.

However, as one of the most important tasks in granular com-

puting, granule description has attracted little attention. This prob-

lem deserves to be investigated since it can not only help us to

have a better understanding and comprehension of granules, but

also shed some light on the unsolved problem “why some con-

cepts are psychologically simple and easy to learn, while others

seem difficult” [3] . Motivated by this problem, the main objective

of this paper is to propose a granular description method based on

formal concept analysis. In formal concept analysis, there are two

types of granules that make sense. One is the granules formed by

the extents of formal concepts, and the other is the ones formed by

individual objects. Some studies have shown that the latter plays a

very important role and has a strong correlation with object con-

cepts [7,47] , object granules [50] and granular concepts [22] . In

fact, besides these two types of granules, there still exist other

types of granules. How to classify granules and what is the clas-

sification criteria are still open problems. We will give solutions to

these two problems in Section 3 .

Moreover, how to describe any granule in a concise and apt

way is a fundamental issue in solving granule description problem

under the framework of granular computing. As mentioned above,

there are two types of granules that make sense and have attracted

many researchers’ attentions. In fact, the descriptions of the first

type of granules can be realized by using minimal generators [8] .

However, it is worth noticing that for a given concept, there may

be more than one minimal generator, and the most apt and concise

one is the best choice. Therefore, we need additional constraints to

refine the results, and this issue will be investigated in Section 4 .

For the description of individual object, although it gives arise

to the problem of finding a reduct of the original description of the

object, it differs from the description of the first type of granules,

since at the most time it is impossible to distinguish one object

from the other by the attributes it possesses. Thus, it is necessary

to point out which attributes it does not possess. This issue will be

considered in Section 5 .

In Section 6 , we will put forward a method for the descrip-

tion of other types of granules. In Section 7 , we firstly discuss the

problem of describing indefinable granules, and then we make a

comment on the connection of RST and FCA from the viewpoint

of granule description. Some discussions and remarks are given in

Section 8 .

2. Related theoretical foundations

In this section, the involved notions are introduced briefly. In

the introduction of logic language L , we make some necessary

modifications to the definition of m ( ϕ) (i.e., the meaning of for-

mula ϕ) in order to get a finer semantic meaning.

2.1. Logic language L

Logic language L , which adopts and modifies the decision logic

language used in rough set theory, enables to formally represent



nd interpret rules in the process of knowledge discovery [32,64] .

n order to obtain stronger description ability, the logic language L

s built on a set of atomic formulas.

Atomic formulas, which are denoted by A = { p, q, . . . } , provide

foundation for complex knowledge representation. By using logic

onnectives such as ¬, ∧ , ∨ , → and ↔ , compound formulas can be

uilt recursively. If ϕ and ψ are formulas, then so are ¬ϕ , ϕ ∧ ψ ,

∨ ψ , ϕ → ψ and ϕ↔ ψ .

In mathematical logic, a literal is an atom or its negation. More-

ver, literals can be divided into two types: a positive literal is just

n atom; a negative literal is the negation of an atom.

For a given formula ϕ, let lit ( ϕ) denote both the positive and

egative literals contained in ϕ. Moreover, let | lit ( ϕ)| denote the

ardinality of lit ( ϕ), i.e., the number of literals contained in lit ( ϕ).

For example, let ϕ = g ∧ ¬ h . Then we have lit(ϕ) = { g, ¬ h } and

lit(ϕ) | = 2 .

The semantics of the language L is defined as a pair M = (D, K) ,

here D is a nonempty set of individuals and K is available knowl-

dge about individuals of D .

Let p be an atomic formula and x an individual. By using knowl-

dge K , if x satisfies p , then we have the denotation as x �→ p .

The meaning of the formula ϕ is the set of individuals which

atisfy this formula, and is defined by the following equation:

(ϕ) = { x | x ∈ D, x �→ ϕ} . Considering the above equation in the reverse direction, we de-

ne the description of subset A as m

−1 (A ) , where

−1 (A ) = ϕ such that m (ϕ) = A.

For a given granule A , in order to get the most concise and apt

escription, we define a new function d ( A ) instead of using the

unction m

−1 (A ) .

efinition 1. Let A be a granule. The description of A is defined by

( A ), where

(A ) = ϕ such that m (ϕ) = A and for any formula ψ, we have

(i) if m (ψ) = A, then | lit ( ψ)| ≥ | lit ( ϕ)|;

(ii) if m (ψ) = A and | lit(ψ) | = | lit(ϕ) | , then

m (∧ p) p∈ l it(ϕ) −l it(ψ) ⊆ m ( ∧ q ) q ∈ l it(ψ) −l it(ϕ) .

Condition (i) ensures the conciseness of the description. That is,

he description contains the fewest literals. Condition (ii) ensures

he aptness of the description. That is, the literals contained in the

escription have the smallest extent.

.2. Overview of granular computing and basic notions on FCA

Granular computing aims to represent and solve complicated

roblems in the procedure of granularity transformation [1] . Inter-

al structure of a granule, collective structure of a family of gran-

les and hierarchical structure of a web of granules are the three

ost important parts of a granular structure [65] .

Given a domain D , all possible granules form a power set of D ,

enoted as 2 D . Here, the part we are interested in is only subsys-

em of 2 D . For example, in FCA [7] , the granules which deserve our

ttention are the extensions of formal concepts, while in knowl-

dge spaces theory [41] , the ones turn to be the feasible knowl-

dge statements.

FCA is generally an appropriate framework for building cate-

ories which are defined as object sets sharing some attributes,

rrespectively of a particular domain of application.

Given a formal context K = (G, M, I) , where G is called a set of

bjects, M is called a set of attributes, and the binary relation I ⊆G

M specifies which objects have what attributes. Moreover, the

erivation functions f ( ·) and g ( ·) are defined for A ⊆G and B ⊆M as



H. Zhi, J. Li / Knowledge-Based Systems 0 0 0 (2016) 1–12 3

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Table 1

Formal context “Biology and water”.

a b c d e f g h i

1 ∗ ∗ ∗2 ∗ ∗ ∗ ∗3 ∗ ∗ ∗ ∗ ∗4 ∗ ∗ ∗ ∗ ∗5 ∗ ∗ ∗ ∗6 ∗ ∗ ∗ ∗ ∗7 ∗ ∗ ∗ ∗8 ∗ ∗ ∗ ∗

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ollows:

f (A ) = { b ∈ M|∀ a ∈ A, aIb} ,

(B ) = { a ∈ G |∀ b ∈ B, aIb} . Obviously, f ( A ) is the set of attributes common to all objects of

, and g ( B ) is the set of objects sharing all attributes of B .

A formal concept of the context K is a pair ( A, B ), where A ⊆G,

⊆M , f (A ) = B and g(B ) = A . The set A is called the extent and B

s called the intent of the concept ( A, B ).

A concept ( A, B ) is called a subconcept of ( C, D ) if A ⊆C , denoted

y ( A, B ) ≤ ( C, D ). In this case, ( C, D ) is called a superconcept of

A, B ) as well. We define the relations ≺ and � as usual. If ( A, B )

( C, D ) and there is no ( E, F ) such that ( A, B ) ≤ ( E, F ) ≤ ( C, D ),

hen ( A, B ) is called a lower neighbor of ( C, D ) or ( C, D ) is called an

pper neighbor of ( A, B ), which are denoted by ( A, B ) ≺( C, D ) and

C, D ) �( A, B ), respectively.

Ganter and Wille [7] have proved that the set of all formal con-

epts ordered with ≤ forms a complete lattice, called the concept

attice of K and denoted by L ( K ).

. Stability index and granule classification

In a general sense, one regards information granule as a collec-

ion of elements drawn together by their closeness, resemblance,

roximity, functionality, etc. [57] . How to measure the closeness

nd how to use the degree of closeness to classify granules are

wo important and correlated questions. In this section, we define

tability index to measure the cohesion of granules, and then we

lassify granules into different categories based on stability index.

Before proceeding, we give two symbols which will be used fre-

uently in the subsequent discussion. For a given finite set A ,

(i) | A | indicates the cardinality of A ;

(ii) 2 A indicates the power set of A .

The definition of stability index is initially proposed as a crite-

ion to measure the randomness of a data set. The following defi-

ition introduces stability index of formal concept [16] .

Let K = (G, M, I) be a formal context and ( A, B ) be a formal con-

ept of K . The stability index of ( A, B ) is defined as

(A, B ) =

|{ C ⊆ A | f (C) = B }| 2

| A | .

In [72] , concept stability is calculated by using minimal gen-

rator. Minimal generator serves as the minimal set that makes a

oncept stable when deleting some objects from the extent.

Inspired by the existing work [16,72] , we extend the notion of

tability index from formal concepts to granules.

efinition 2. Let K = (G, M, I) be a formal context and A ∈ 2 G be a

ranule. The stability index of the granule A is defined by

(A ) =

{1 , if | A | = 1 ,

|{ C ⊆ A | f (C) = f (A ) }| 2

| g( f (A )) | , otherwise.

efinition 3. Let K = (G, M, I) be a formal context and A ∈ 2 G

e a granule with | A | > 1. If there is a subset R of A such that

f (R ) =

⋂

a ∈ A f (a ) and for any R 1 ⊂ R , f (R 1 ) � =

⋂

a ∈ A f (a ) , we call R is

he minimal generator of granule A .

heorem 1. Let K = (G, M, I) be a formal context and A ∈ 2 G be a

ranule with | A | > 1 . If the granule A has a family of minimal gener-

tors { R i } i ≤n , the stability index of the granule A is

(A ) =

n ∑

i =1

2 | A −R i | − ∑

1 ≤i< j≤n

2 | A |−| R i

⋃

R j | + · · · + (−1) n 2 | A −R 1 ⋃

R 2 ⋃ ···⋃

R n |

2 | g( f (A )) | .



roof. Since { R i } i ≤n is the family of the minimal generators of A ,

e have |{ C i | R i ⊆ C i ⊆ A }| = 2 | A −R i | . According to the inclusion-exclusion principle, we can show

n ⋃

i =1

C i

∣∣∣∣∣ =

n ∑

i =1

2

| A −R i |

−∑

1 ≤i< j≤n

2

| A |−| R i ⋃

R j | + · · · + (−1) n 2

| A −R 1 ⋃

R 2 ⋃ ···⋃

R n | .

y Definition 2 , it follows

(A ) =

n ∑

i =1

2 | A −R i | − ∑

1 ≤i< j≤n

2 | A |−| R i

⋃

R j | + · · · + (−1) n 2 | A −R 1 ⋃

R 2 ⋃ ···⋃

R n |

2 | g( f (A )) | .

�

Granules formed by individual objects play an important role

rom which concept learning process starts [22] . Concepts learned

rom this type of granules are deemed as granular concepts in [22] ,

nd they have a strong relationship with object concepts and at-

ribute concepts.

efinition 4 ( [7] ) . Let K = (G, M, I) be a formal context. A concept

s called an object concept if it has the form ( g ( f ( a )), f ( a )), a ∈ G ,

nd a is called its object label. Dually, a concept is called an at-

ribute concept if it has the form ( g ( b ), f ( g ( b ))), b ∈ M , and b is

alled its attribute label.

roposition 1. In a formal context with no reducible objects, gran-

les that are formed by the extents of object concepts have the stabil-

ty index 0.5.

roof. Let K = (G, M, I) be a formal context with no reducible ob-

ects and a ∈ G . Then ( g ( f ( a )), f ( a )) is an object concept with

ts minimal generator a . By Theorem 1 , we have σ (g( f (a ))) =

| g( f (a )) −{ a }| / 2 | g( f (a )) | = 0 . 5 . �

roposition 2. Let A and B be two granules with | A | > 1, | B | > 1 and

f (A ) = f (B ) . If A ⊂ B, then σ ( A ) < σ ( B ) .

roof. Since f (A ) = f (B ) , then the granules A and B have the same

inimal generators. As A ⊂ B , we have that the numerator of σ ( A )

s less than that of σ ( B ). Moreover, the equation f (A ) = f (B ) also

mplies that g( f (A )) = g( f (B )) . By Theorem 1 , we can get that

( A ) < σ ( B ). �

xample 1. Consider descriptions of several objects in Table 1

hich is the well known formal context “Biology and water” in

7] .

The corresponding concept lattice of the formal context “Biol-

gy and water” is sketched by Fig. 1 . For convenience, in the rep-

esentation of a concept, we omit the curly braces and commas.

Based on Theorem 1 , the stability indices of the granules {1, 2,

} and {1, 2} are σ ({ 1 , 2 , 3 } ) = 0 . 5 and σ ({ 1 , 2 } ) = 0 . 25 .




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Fig. 1. The concept lattice of formal context “Biology and water”.

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Hereinafter, we discuss granule classification based on stability

index. Firstly, note that the stability index of A has greatest value 1

when | A | = 1 , so we sort this type of granules into a class. Sec-

ondly, given two granules A and B with | A | > 1 and | B | > 1, if

B ⊂ A , there exists a concept C of L ( K ) with its extent equal to A ,

and there is no concept with its extent equal to B , then according

to Proposition 2 , the stability index of B is smaller than that of A .

So, we sort the granules like A into a class and others into another

class.

According to the above discussions, we formally use stability in-

dex to divide granules into three categories: atomic granules, basic

granules, and composite granules.

Definition 5. Let K = (G, M, I) be a formal context and A ∈ 2 G be

a granule. If | A | = 1 , then A is called an atomic granule; if there

exists a concept C ∈ L ( K ) with its extent equal to A , then A is called

a basic granule; if | A | > 1 and there does not exist a concept with

its extent equal to A , then A is called a composite granule.

Following Example 1 , it is obvious that {1} is an atomic granule,

{1, 2, 3} is a basic granule and {1, 2} is a composite granule. More-

over, it is necessary to point out that some granules may be atomic

and basic granules simultaneously. For example, granules {3}, {4},

{6} and {7} are atomic granules, and at the same time they are all

basic granules.

4. Descriptions of basic granules

In this section, we give a method for the descriptions of basic

granules. In order to avoid unnecessary troubles, we assume that

the formal context to be discussed is purified.

Definition 6 ( [7,11] ) . Let K = (G, M, I) be a formal context. Object

g is called a full object if f (g) = M. Dually, attribute m is called a

full attribute if g(m ) = G .

Definition 7 ( [7] ) . Let K = (G, M, I) be a formal context. Object g

is called a reducible object if there exists a set of objects { g t } t ∈ Tsuch that

⋂

t∈ T f (g t ) = f (g) , where T is an index set. Dually, attribute

m is called a reducible attribute if there exists a set of attributes

{ m t } t ∈ T such that ⋂

t∈ T g(m t ) = g(m ) .



efinition 8 ( [7] ) . Let K = (G, M, I) be a formal context. K is called

purified formal context if there does not exist full object, full at-

ribute, reducible object and reducible attribute.

heorem 2. Let K = (G, M, I) be a purified formal context. Then any

ttribute concept of K must be a meet irreducible element, and vice

ersa.

roof. Assume that an attribute concept ( A, B ) is not a meet ir-

educible element. Then, ( A, B ) has at least two upper neighbors,

nd we denote them by ( A t , B t ) t ∈ T , where T is an index set. Since

A, B ) is an attribute concept, there exists an attribute b such that

(b) = A . By the basic theorem of concept lattice, we have A =⋂

∈ T A t =

⋂

t∈ T g(B t ) . By the fact g(B ) = A, it follows g(b) =

⋂

t∈ T g(B t ) ,

hich means that b is a reducible attribute. This contradicts to the

efinition of a purified formal context. The reverse can be proved

n a similar way. Hence, this theorem holds. �

By using duality principle, the following corollary can easily be

erived.

orollary 1. Let K = (G, M, I) be a purified formal context. Then any

bject concept of K must be a join irreducible element, and vice versa.

efinition 9. Let K = (G, M, I) be a formal context and m, n ∈ M . If

( m ) ⊂ g ( n ), then we say that m is more apt than n .

For example, considering the formal context sketched by

able 1 , we have g( f ) = { 5 , 6 , 8 } and g(d) = { 5 , 6 , 7 , 8 } . Since f has

smaller extent compared with d , we say that f is more apt than

.

Let ( A, B ) be a concept with its minimal generator R = { r t } t∈ T .hen ϕ =

∧

t∈ T r t is the formula which makes m (ϕ) = A . But it is

orthwhile to note that for a given concept, it may have more

han one minimal generator. Therefore, it is necessary to add some

onditions for selecting the most apt one.

efinition 10. Let K = (G, M, I) be a formal context and ( A, B ) be

formal concept of K . If there are two minimal generators R and

satisfying g(R − S) ⊂ g(S − R ) , we say that R is more apt than S .

oreover, among all minimal generators, there must exist the most

pt one, and we call it the most apt minimal generator.




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Fig. 2. Relationships among four concepts.

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The meaning of the most apt minimal generator is that it en-

ures all of the literals contained in the generator having the small-

st extents.

roposition 3. Let K = (G, M, I) be a purified formal context and A

2 G be a granule. If there exists a concept ( A, f ( A )) ∈ L ( K ), then the

escription of A is

(A ) =

∧

l∈ M AM G

l,

here MAMG is the most apt minimal generator of ( A, f ( A )) .

roof. The theorem is immediate from Definitions 1 and 10 . �

heorem 3. Let K = (G, M, I) be a purified formal context. If a con-

ept is an attribute concept, then its most apt minimal generator con-

ists of its attribute label.

roof. The theorem is immediate from the notion of an attribute

oncept and Definition 10 . �

heorem 4. Let K = (G, M, I) be a purified formal context. If ( A, B ) is

ot an attribute concept, then its most apt minimal generator is the

nion of its any two upper neighbors’ most apt minimal generators.

roof. Since an attribute concept must be an lower bound irre-

ucible element with only one upper neighbor, then for a non at-

ribute concept ( A, B ), it must have at least two upper neighbors,

nd we denote them by ( A t , B t ) t ∈ { i, j } . Suppose the most apt min-

mal generators of ( A t , B t ) t ∈ { i, j } are R i and R j . By the definition

f most apt minimal generator, it follows R i ∪ R j ⊆B i ∪ B j . Besides,

ccording to the structure properties of concept lattice, we have

(B i ∪ B j ) = g(B ) = A . Combining R i ∪ R j ⊆B i ∪ B j and g(B i ∪ B j ) = A,

e obtain A ⊆g ( R i ∪ R j ). Finally, taking g ( R i ∪ R j ) ⊆A i and g ( R i ∪ R j ) ⊆A j

nto consideration, we derive the conclusion A = g(B i ∪ B j ) . Hence,

his theorem is proved. �

By light of Theorems 3 and 4, we get a recursive algorithm

Algorithm 1 ) to find the most apt minimal generators of all con-

epts of a given purified formal context.

Given a formal context ( G, M, I ) and let N be the number of

oncepts contained in the concept lattice under consideration. As

s well known, the best time complexity of concept lattice con-

truction is O (| G | 2 | M | N ). And in Algorithm 1 , it is easy to see

hat every concept is visited just once, so the time complexity of

lgorithm 1 is O (| G | 2 | M | N

2 ).

xample 2. Let us consider Table 1 in Example 1 . For simplicity,

e use abbreviations OC, AC and MAMG to denote object concept,

ttribute concept and most apt minimal generator, respectively.

lgorithm 1 Computing the most apt minimal generators of con-

epts of a purified formal context

equire: A purified formal context K = (G, M, I) .

nsure: The most apt minimal generators of all concepts of L (K) .

1: Construct concept lattice L (K) .

2: Deem the concept lattice L (K) as an undirected graph andtra-

verse the concept lattice upwards from the maximal concept

of L (K) by using breadth first search;

3: For eachvisited concept

4: If the current visited concept lattice node is an attribute con-

cept,

5: Then its most apt minimal generator is its attribute label;

6: Else its most apt minimal generator is theunion of its any

two upper neighbors’ most apt minimal generators.

7: End If

8: End For

9: Return the most apt minimal generators of allconcepts.

t

d

w

c

P

c

n

t

n

w

T

c

s



Moreover, when representing a set, we omit the comma among

he objects or attributes. The descriptions of basic granules are

hown in Table 2 .

At the end of this section, we explain the meaning of descrip-

ion of basic granule in decision making [20,23] .

Let K = (U, A, I, D, J) be a formal decision context, ( X, B ) ∈ L ( U,

, I ) and ( Y, C ) ∈ L ( U, D, J ), where X � = ∅ , Y � = ∅ , B � = ∅ and C � = ∅ . If ⊆ Y , we can get a decision rule, denoted by B → C . So, if we use

he description of X instead of B , it is obvious that we can get a

implified decision rule d ( X ) → C due to lit ( d ( X )) ⊆ B .

. Descriptions of atomic granules

In many cases, it is impossible to describe an object by us-

ng positive literals only. For example, considering the “Biology

nd water” formal context in Example 1 , we have f (1) = { a, b, g} ,f (2) = { a, b, g, h } . Thus, it is impossible for us to distinguish object

from object 2 by using positive literals only. Hence, the descrip-

ions of atomic granules differ from those of basic granules.

heorem 5. Let K = (G, M, I) be a formal context and { a } ∈ 2 G be

n atomic granule. If there does not exist an object b ∈ G such that

( a ) ⊂ f ( b ), then the description of { a } is

({ a } ) =

∧

t∈ T r t ,

here R = r t ( t ∈ T ) is the most apt minimal generator of the object

oncept ( g ( f ( a )), f ( a )) .

roof. This theorem is immediate from Definitions 1 and 10 . �

heorem 6. Let K = (G, M, I) be a formal context and { a } ∈ 2 G be

n atomic granule. If there exists an object b ∈ G such that f ( a ) ⊂ f ( b )

nd there does not exist an object c ∈ G such that f ( a ) ⊂ f ( c ) ⊂ f ( b ),

hen the description of granule { a } is

({ a } ) = ( ∧

t∈ T r t ) ∧ (

∧

h ∈ f (b) − f (a )

¬ h ) ,

here R = r t ( t ∈ T ) is the most apt minimal generator of the object

oncept ( g ( f ( a )), f ( a )) .

roof. In the concept lattice L ( K ), since ( g ( f ( a )), f ( a )) is an object

oncept, it is a meet irreducible element and has only one lower

eighbor. Moreover, we have made the assumption of f ( a ) ⊂ f ( b ),

hen ( g ( f ( b )), f ( b )) is the only lower neighbor of ( g ( f ( a )), f ( a )). Fi-

ally, because R is the most apt minimal generator of ( g ( f ( a ), f ( a )),

e derive the conclusion d({ a } ) = ( ∧

t∈ T r t ) ∧ (

∧

h ∈ f (b) − f (a )

¬ h ) . �

heorem 7. Let L ( K ) be a concept lattice and there exist four con-

epts ( A 1 , B 1 ), ( A 2 , B 2 ), ( A 1 f , B 1 f ) and ( A 2 f , B 2 f ) with their relationship

ketched by Fig. 2 . Then the following two equations are equivalent:

(i) A 1 f − A 1 = A 2 f − A 2 ;
(ii) A 1 f − A 2 f = A 1 − A 2 .



ARTICLE IN PRESS


Table 2

Descriptions of basic granules in “Biology and water”.

Basic granules Corresponding concepts OC AC MAMG Descriptions

{12345678} (12345678, a ) No Yes { a } a

{1234} (1234, ag ) No Yes { g } g

{34678} (34678, ac ) No Yes { c } c

{12356} (12356, ab ) No Yes { b } b

{5678} (5678, ad ) No Yes { d } d

{234} (234, agh ) No Yes { h } h

{123} (123, abg ) Yes No { b, g } b ∧ g {36} (36, abc ) No No { b, c } b ∧ c {678} (678, acd ) No No { c, d } c ∧ d {568} (568, adf ) No Yes { f } f

{5678} (5678, ad ) No Yes { d } d

{34} (34, acgh ) No No { c, h } c ∧ h {23} (23, abgh ) Yes No { b, h } b ∧ h {68} (68, acdf ) Yes No { c, f } c ∧ f {56} (56, abdf ) Yes No { b, f } b ∧ f {4} (4, acghi ) Yes Yes { i } i

{3} (3, abcgh ) Yes No { b, c, h } b ∧ c ∧ h {7} (7, acde ) Yes Yes { e } e

{6} (6, abcdf ) Yes No { b, c, f } b ∧ c ∧ f

Table 3

Descriptions of atomic granules in

the formal context “Biology and

water”.

Atomic granules Descriptions

{1} g ∧ ¬h

{2} h ∧ ¬c

{3} b ∧ c ∧ h {4} i

{5} f ∧ ¬c

{6} b ∧ c ∧ f {7} e

{8} f ∧ ¬b

t

g

s

f

Proof. Suppose A 1 f − A 1 = A 2 f − A 2 . Let A 2 f − A 2 = { x } , A 1 f − A 1 ={ x ′ } , A 1 − A 2 = { g} and A 1 f − A 2 f = { g ′ } . Then it is easy to see that

A 1 f = A 2 ∪ { x } ∪ { g ′ } = A 2 ∪ { g} ∪ { x ′ } . Since A 1 f − A 1 = A 2 f − A 2 , we

have x = x ′ , which leads to g = g ′ . Hence, (i) implies (ii). Besides,

(ii) implies (i) can be proved in a similar way. �

Theorem 7 indicates that if ( A 1 , B 1 ) is an object concept, rewrit-

ten as ( g ( f ( a )), f ( a )), and it fulfills the conditions (i) and (ii), then

we can use the minimal generators of ( A 1 f , B 1 f ) instead of ( A 1 , B 1 )

to get a simpler description.

In light of Theorems 5, 6 and 7, we derive an algorithm

( Algorithm 2 ) for atomic granule description.

Let K = (G, M, I) be a formal context and N be the num-

ber of concepts contained in the concept lattice L ( K ). Obviously,

at the worst case, the number of visited concepts in running

Algorithm 2 is less than N .

Example 3. Let us consider Table 1 , following Examples 1 and 2 .

According to Algorithm 2 , we get the descriptions of all atomic

granules, and their results are shown in Table 3 .

Algorithm 2 Computing the description of atomic granule

Require: Concept lattice L (G, M, I) and a ∈ G.

Ensure: Description of the granule { a } . 1: Locate the object concept (g( f (a )) , f (a )) in L (G, M, I) andde-

note it as C = (A, B ) ;

2: If there does not exist an object b ∈ G such that f (a ) ⊂ f (b) ,

3: Then compute the most apt minimal generator R of C =

(A, B ) ;

4: d({ a } ) =

∧

t∈ T

r t ,where R = r t (t ∈ T ) ;

5: Else denote the lower neighbor of C = (A, B ) as S = (A s , B s ) ;

6: While there exists (A f , B f ) � Cand (A f s , B f s ) � S and A f −A f s = A − A s

7: Perform C ← (A f , B f ) , S ← (A f s , B f s )

8: End While

9: Compute the most apt minimal generator R of Cand denote

it as { r t } t∈ T ;

10: d({ a } ) = ( ∧

t∈ T r t ) ∧ (

∧

s ∈ S ¬ h s ) , where { h s } s ∈ S is the intent dif-

ference between C and S.

11: End If

12: Return d({ a } ) .

g

a

6

u

i

u

u

t

6

d

o

(

D

{

s

o

K



At the end of this section, we explain the meaning of descrip-

ion of atomic granule in decision making [20,23,50] .

Let K = (U, A, I, D, J) be a formal decision context and x ∈ U . If

I ( f I ( x )) ⊆g J ( f J ( x )), we can get a granule rule f I ( x ) → f J ( x ), where the

ubscripts I and J represent the operations f and g carried on the

ormal contexts ( U, A, I ) and ( U, D, J ), respectively. Apparently, the

ranular rule becomes much more concise if we use d ( x ) as the

ntecedent of f I ( x ) → f J ( x ).

. Descriptions of composite granules

In this section, we discuss the descriptions of composite gran-

les. As the structures of composite granules vary dramatically, we

nitially investigate some important properties of composite gran-

les and then classify them into three categories: ∧ -definable gran-

le, ( ∧ , ¬)-definable granule and ( ∧ , ∨ )-definable granule. After

hat, we propose the descriptions of them, respectively.

.1. ∧ -definable granule, ( ∧ , ¬)-definable granule and ( ∧ , ∨ )-

efinable granule

In this subsection, we initially study some important features

f composite granules for dividing them into ∧ -definable granule,

∧ , ¬)-definable granule and ( ∧ , ∨ )-definable granule.

efinition 11 ( [47] ) . Let K = (G, M, I) be a formal context, H a = (a, f (a )) | a ∈ G } . For any a i , a j ∈ G , if f ( a i ) ⊆f ( a j ) or f ( a i ) ⊇f ( a j ), we

ay that a i and a j are comparable, denoted by ( a i , f ( a i )) ≤ ( a j , f ( a j ))

r ( a i , f ( a i )) ≥ ( a j , f ( a j )). ( H a , ≤) is called the object diagram of

= (G, M, I) , and ( a, f ( a )) is called a labeled object class on G .




ARTICLE IN PRESS


Fig. 3. ( H a , ≤) of formal context “Biology and water”.

i

↓ .

D

↑ .

D

a

D

a

a

o

t

c

A

t

c

p

o

i

t

D

b

p

A

t

b

c

o

D

b

o

A

A

A

E

D

i

{

c

a

t

6

c

a

c

h

u

u

D

a

t

l

A

t

t

t

i

s

m

t

t

g

g

d

t

P

g

d

≥P

i

t

A

Furthermore, for each a ∈ G , the down-set of a , denoted by ↓ a ,

s defined as

a = { a j ∈ G | (a j , f (a j )) ≤ (a, f (a )) , (a j , f (a j )) ∈ H a , (a, f (a )) ∈ H a }ually, the up-set of a , denoted by ↑ a , is defined as

a = { a j ∈ G | (a j , f (a j )) ≥ (a, f (a )) , (a j , f (a j )) ∈ H a , (a, f (a )) ∈ H a }efinition 12. Let K = (G, M, I) be a formal context and A ∈ 2 G . If

ny two elements of A are comparable, we say that A is a list.

efinition 13. Let K = (G, M, I) be a formal context and A ∈ 2 G be

list. If A =

⋃

a ∈ A ↓ a , we say that A is a down-set complete list. Du-

lly, if A =

⋃

a ∈ A ↑ a , we say that A is an up-set complete list. More-

ver, if A is both down-set complete and up-set complete, we say

hat A is a complete list.

Furthermore, if A =

⋃

t∈ T B t , where each B t ( t ∈ T ) is a down-set

omplete list, we say that A is a down-set complete set. Dually, if

=

⋃

t∈ T B t , where each B t ( t ∈ T ) is an up-set complete list, we say

hat A is an up-set complete set. Moreover, if A is both down-set

omplete and up-set complete, we say that A is a complete set.

Apparently, a down-set complete list is a special down-set com-

lete set, and a down-set complete set can be treated as the union

f several down-set complete lists. Dually, an up-set complete list

s a special up-set complete set, and an up-set complete set can be

reated as the union of several up-set complete lists.


e not a down-set complete set. We call A

d+ the down-set com-

lement of A , if A

d+ ∪ A is a down-set complete set but for any

d ⊂ A

d+ , A

d ∪ A is not a down-set complete set.

Dually, let A ∈ 2 G be not an up-set complete set. We call A

u +

he up-set complement of A , if A

u + ∪ A is an up-set complete set

ut for any A

u ⊂ A

u + , A

u ∪ A is not a down-set complete set.

In order to make the expression of this paper clear, for a given

oncept C , we use ext ( C ) and int ( C ) to denote the extent and intent

f C , respectively.


e a granule. C ∈ L ( K ) is called the upper approximate concept

f A , if A ⊆ext ( C ) and there does not exist D ∈ L ( K ) such that

⊂ ext ( D ) ⊂ ext ( C ).

Dually, C ∈ L ( K ) is called the lower approximate concept of

, if A ⊇ext ( C ) and there does not exist D ∈ L ( K ) such that

⊃ext ( D ) ⊃ext ( C ).

xample 4. Consider the formal context in Table 1 . According to

efinition 11 , we have the object pictorial diagram ( H a , ≤) which

s shown in Fig. 3 .

In Fig. 3 , there are 9 down-set complete lists, namely, {1}, {1, 2},

1, 2, 3}, {5}, {8}, {5, 6}, {6, 8}, {4} and {7}. And there are 8 up-set

omplete lists, namely, {1, 2, 3}, {2, 3}, {3}, {4}, {7}, {6, 8}, {5, 6}

nd {6}.



Moreover, {1, 2, 3, 5, 6} is a down-set complete set since it is

he union of two down-set complete lists {1, 2, 3} and {5, 6}; {3, 4,

, 8} is an up-set complete set since it is the union of three up-set

omplete lists {3}, {4} and {6, 8}.

Let A = { 1 , 2 } be a granule. Then ({1, 2, 3}, { a, b, g }) is the upper

pproximate concept of A , but there does not exist a concept that

an serve as the lower approximate concept of A . Furthermore, we

ave A

u + = { 3 } and { 3 } d+ = A .

Based on the logic connectives used in the descriptions of gran-

les, we define three types of granules which are ∧ -definable gran-

les, ( ∧ , ¬)-definable granules and ( ∧ , ∨ )-definable granules.

efinition 16. Let K = (G, M, I) be a formal context and A ∈ 2 G be

granule. If only the logic connective ∧ is enough in the descrip-

ion of granule A , we say that it is a ∧ -definable granule; if the

ogic connectives ∧ and ¬ are enough in the description of granule

, we say that it is a ( ∧ , ¬)-definable granule; if the logic connec-

ives ∧ and ∨ are enough in the description of granule A , we say

hat it is a ( ∧ , ∨ )-definable granule.

Actually, ( ∧ , ∨ )-definable granule refers to the extent of mono-

one concept, which was initially proposed in [4] and further stud-

ed in [42] . A monotone concept is the one whose intent is de-

cribed by positive literals connected by ∧ and ∨ . The description

ethods proposed in [4] and [42] have an obvious defect. That is,

he intent of a monotone concept may contain redundant informa-

ion. Therefore, we give a new method to describe ( ∧ , ∨ )-definable

ranule in the forthcoming subsection.

Although we can define ∨ -definable granule, ( ∨ , ¬)-definable

ranule and ( ∧ , ∨ , ¬)-definable granule likewise, we do not do this

ue to the following two obvious reasons:

(i) ∨ -definable granules and ( ∨ , ¬)-definable granules may have

no common attributes. Thus, it has no theoretical and prac-

tical value in researches and applications.

(ii) it is also obvious that by combination of logic connectives

∧ , ∨ and ¬, we can describe any given granules in theory.

In other words, all of the granules are ( ∧ , ∨ , ¬)-definable.

So, it is unnecessary to deliberately study ( ∧ , ∨ , ¬)-definable

granules.

At end of this subsection, we discuss the stability index of these

hree kinds of composite granules.

roposition 4. Let A be a ∧ -definable granule, B be a ( ∧ , ¬) -definable

ranule and C be a ( ∧ , ∨ ) -definable granule with their respective

escriptions being d(A ) =

∧

r∈ R r, d(B ) = (

∧

r∈ R r) ∧ (

∧

s ∈ S ¬ s ) and d(C) =

( ∧

r∈ R r) ∨ (

∨

t∈ T t) , where R, S and T are index sets. Then, σ ( A ) ≥ σ ( B )

σ ( C ) .

roof. For simplicity, we denote #(X ) = { Y ⊆ X| f (Y ) = f (X ) } . That

s, | #(X ) | is the numerator of σ ( X ). In order to prove this proposi-

ion, we divide it into two steps:

On one hand, since the description of B is stricter than that of

, it follows that for any u ∈ #(B ) , we have u ∈ #(A ) . However, the




ARTICLE IN PRESS


T

g

P

m

c

R

o

E

d

d

∧

R

(

E

u

¬

∧

T

g

∅

P

a

f

H

C

g

∅

P

C

u

g

P

(

n

i

A

reverse may not be true. So, we obtain | #(A ) | ≥ | #(B ) | . Besides,

| #(B ) | ≥ | #(C) | can be proved in a similar way.

On the other hand, from the descriptions of A and B , we can get

f (A ) = f (B ) = { r} r∈ R , which leads to | g( f (A )) | = | g( f (B )) | . Besides,

from the descriptions of B and C , we have f (C) ⊂ f (B ) = { r} r∈ R ,which yields | g ( f ( B ))| ≤ | g ( f ( C ))|.

To sum up, σ ( A ) ≥ σ ( B ) ≥ σ ( C ) is at hand. �

In conclusion, ∧ -definable granules have the greatest stability

indices, while ( ∧ , ∨ )-definable granules have the smallest stability

indices, if they share a common part ∧

r∈ R r in their descriptions.

6.2. Properties and descriptions of ( ∧ , ¬)-definable granules

In this subsection, we put forward sufficient conditions to de-

termine ( ∧ , ¬)-definable granules, and then we give a method for

the descriptions of ( ∧ , ¬)-definable granules.

Theorem 8. Let K = (G, M, I) be a formal context and A ∈ 2 G be a

granule. If A is a down-set complete list, then A is ( ∧ , ¬) -definable.

Proof. It is easy to show that ( g ( f ( A )), f ( A )) is the upper approxi-

mate concept of A . Since ( g ( f ( A )), f ( A )) is a concept, then g ( f ( A )) is

∧ -definable.

(i) If g( f (A )) = A, then A is also ∧ -definable;

(ii) If g ( f ( A )) ⊃A , we suppose C = g( f (A )) − A . Then, A possesses

all attributes in f ( A ) but it does not possess the attributes in⋃

c∈ C f (c) − f (A ) . Therefore, A can be described by the follow-

ing formula

( ∧

b∈ f (A )

b) ∧ ( ∧

d∈ ⋃

c∈ C f (c) − f (A )

¬ d ) .

Hence, A is ( ∧ , ¬)-definable.

Combining (i) with (ii), this theorem is proved. �

Example 5. Let us consider Table 1 , following Example 4 . In order

to discuss the description of granule {1, 2}, there are two steps.

Step 1. According to Definition 13 , we know that {1, 2} is a

down-set complete list.

Step 2. Explore the concept lattice, and we get that the up-

per approximate concept of {1, 2} is ({1, 2, 3}, { a, b, g }). More-

over, since {1, 2, 3} is a basic granule with d({ 1 , 2 , 3 } ) = b ∧ g, and

f (3) − f (1) ∪ f (2) = { c} , we have d({ 1 , 2 } ) = b ∧ g ∧ ¬ c.

Theorem 9. Let K = (G, M, I) be a formal context and A ∈ 2 G be a

granule. If A is an up-set complete list, then A is a basic granule and

∧ -definable.

Proof. Suppose A = { a t | t ∈ T } and f (a i ) ⊆ f (a i +1 ) . Then, it is trivial

to show f (A ) = f (a 0 ) and A = g( f (A )) , which means that ( A, f ( A ))

is a concept. Hence, A is a basic granule and ∧ -definable. �

Corollary 2. Let K = (G, M, I) be a formal context and A ∈ 2 G be

a granule. If A is a complete list, then A is a basic granule and ∧ -

definable.

Proof. By Definition 13 , a complete list must be an up-set com-

plete list, and hence this corollary is proved. �

Lemma 1. Let K = (G, M, I) be a formal context and A ∈ 2 G be a

granule. If A is a complete set, then G − A is also a complete set.

Proof. Assume that G − A is not a complete set yet. Then there

must exist two objects a ∈ G − A and b / ∈ G − A such that a and b

are comparable. Note that b / ∈ G − A implies b ∈ A . And there exists

an object a ∈ G − A such that b and a are comparable, which means

that A is not a complete set. This is contradict to the premise.

Hence, this corollary is proved. �




ranule. If A is a complete set, then A is ( ∧ , ¬) -definable.

roof. It is easy to show that ( g ( f ( A )), f ( A )) is the upper approxi-

ate concept of A .

(i) If g( f (A )) = A, then A is obviously ∧ -definable;

(ii) If g ( f ( A )) ⊃A , let C = g( f (A )) − A . By Lemma 1 , C is also a

complete set. Suppose C =

⋃

t∈ T B t , where each B t ( t ∈ T ) is

a complete list. According to Corollary 2 , each B t ( t ∈ T )

is ∧ -definable. We suppose the description of each B t is

d ( B t ). Then, A can be described by the formula ( ∧

b∈ f (A )

b) ∧

( ∧

l∈ d(B t ) ,t∈ T ¬ l ) . Hence, A is ( ∧ , ¬)-definable.

Combining (i) with (ii), the proof of Theorem 10 is

ompleted. �

emark 1. Down-set complete sets can either be ( ∧ , ¬)-definable

r ( ∧ , ¬)-indefinable.

xample 6. Discuss the formal context in Table 1 . Consider two

own-set complete sets {4, 7, 8} and {1, 2, 4}.

Since d({ 4 , 7 , 8 } ) = c ∧ ¬ b, we conclude that {4, 7, 8} is ( ∧ , ¬)-

efinable.

Note that we cannot find a formula containing the connectives

and ¬ only to describe {1, 2, 4}. So, {1, 2, 4} is ( ∧ , ¬)-indefinable.

emark 2. Up-set complete sets can either be ( ∧ , ¬)-definable or

∧ , ¬)-indefinable.

xample 7. Discuss the formal context in Table 1 . Consider two

p-set complete sets {3, 4, 6, 8} and {2, 3, 6}.

Since d({ 3 , 4 , 6 , 8 } ) = c ∧ ¬ e, we conclude that {3, 4, 6, 8} is ( ∧ ,

)-definable.


and ¬ only to describe {2, 3, 6}. So, {2, 3, 6} is ( ∧ , ¬)-indefinable.


ranule. If for any object b ∈ g( f (A )) − A, we have f (b) − ⋃

a ∈ A f (a ) � =

, then A is ( ∧ , ¬) -definable.

roof. According to the assumption, it follows that A has common

ttributes f ( A ) and it can be differentiate from g( f (A )) − A . There-

ore, it is trivial to show that A can be described by the formula

( ∧

i ∈ I r i ) ∧ (

∧

j∈ J ¬ s j ) , where r i ∈ f ( A ), s j ∈

⋂

b∈ g( f (A )) −A

f (b) − ⋃

a ∈ A f (a ) .

ence, this theorem is proved. �

orollary 3. Let K = (G, M, I) be a formal context and A ∈ 2 G be a

ranule. If for any object b ∈ g( f (A )) − A, we have f (b) − ⋃

a ∈ A f (a ) =

, then A is ( ∧ , ¬) -indefinable.

roof. This corollary follows immediately from Theorem 11 . �

orollary 4. Let K = (G, M, I) be a formal context and A ∈ 2 G be an

p-set complete set. If there exists an object e ∈ A

d+ such that e ∈ ( f ( A )), then A is ( ∧ , ¬) -indefinable.

roof. This corollary follows immediately from Corollary 3 . �

At the end of this subsection, we give an algorithm

Algorithm 3 ) which has two types of possible outputs: if A is

ot ( ∧ , ¬)-definable, then this algorithm outputs “A is ( ∧ , ¬)-

ndefinable”; otherwise, this algorithm outputs the description of

.




ARTICLE IN PRESS


Algorithm 3 Computing the description of ( ∧ , ¬)-definable granule

Require: Concept lattice L (G, M, I) , and A ⊆ G .

Ensure: If A is not (∧ , ¬ ) -definable, then thisalgorithm outputs “A

is (∧ , ¬ ) -indefinable”; otherwise,this algorithm outputs the de-

scription of A .

1: Compute the most apt minimal generator of g( f (A )) anddenote

it by { r i } i ∈ I . 2: If g( f (A )) − A = ∅ 3: Then A is ∧ -definable;

4: Return d(A ) =

∧

i ∈ I r i ;

5: Else If there exists an object b ∈ g( f (A )) − A with f (b) −⋃

a ∈ A f (a ) = ∅

6: Then Return “A is not (∧ , ¬ ) -definable”.

7: End If

8: End If

9: If A is a down-set complete list

10: Then denote (g( f (A )) , f (A )) by C = (A, B ) ;

11: Let D = g( f (A )) − A , and denote (g( f (D )) , f (D )) by S =

(A s , B s ) ;

12: While there exists (A f , B f ) � Cand (A f s , B f s ) � S and A f −A f s = A − A s

13: Perform C ← (A f , B f ) and S ← (A f s , B f s ) ;

14: End While

15: Compute the most apt minimal generator R of Cand denote

it by { r t } t∈ T ; 16: Return d({ A } ) = (

∧

t∈ T r t ) ∧ (

∧

s ∈ S ¬ h s ) , where { h s } s ∈ S is the in-

tent difference between C and S.

17: End If

18: If A does not fall into the categories describedin the former

steps

19: Then Return d(A ) = ( ∧

i ∈ I r i ) ∧ (

∧

j∈ J ¬ s j ) , where s j ∈

⋂

b∈ g( f (A )) −A

f (b) − ⋃

a ∈ A f (a ) .

20: End If

6

t

t

T

g

t

P

b

p

∨

T

g

s

P

b

e

B

s

E

g

Algorithm 4 Computing the description of ( ∧ , ∨ )-definable gran-

ule

Require: Concept lattice L (G, M, I) , and A ⊆ G .

Ensure: If A is not (∧ , ∨ ) -definable, then thisalgorithm outputs “A

is not (∧ , ∨ ) -definable”;otherwise, this algorithm outputs the

description of A .

1: Establish the object pictorial diagram (H a , ≤) .

2: If there exist two objects a ∈ A and b ∈ G − A with a < b

3: Then Return “A is not (∧ , ∨ ) -definable”;

4: Else find up-set complete lists B t (t ∈ T ) in the diagram

(H a , ≤) such that A =

⋃

t∈ T B t ;

5: Repeatedly explore B t (t ∈ T ) , and use Cinstead of B i and

B j if there is a basic granule C = B i ∪ B j ;

6: Suppose the results we get from the former step are B s (s ∈

S) with their respective descriptions being d(B s )(s ∈ S) , then

d(A ) =

∨

s ∈ S d(B s ) ;

7: End If

8: Return d(A ) .

∨

∧

(

u

6

e

e

c

t

p

fi

p

s

t

d

c

fi

b

f

c

t

e

[

g

l

c

l

v

t

k

l

i

i

s

i

e

.3. Descriptions of ( ∧ , ∨ )-definable granules

In this subsection, we put forward sufficient conditions to de-

ermine ( ∧ , ∨ )-definable granules, and then we give a method for

he descriptions of ( ∧ , ∨ )-definable granules.


ranule. If there exist two objects a ∈ A and b ∈ G − A with a < b,

hen granule A is ( ∧ , ∨ ) -indefinable.

roof. According to the premise there exist two objects a ∈ A and

∈ G − A with a < b , so a cannot be distinguished from b by using

ositive literals and negative literals are needed. Hence, A is ( ∧ ,

)-indefinable. �


ranule. If for any object a ∈ A there does not exist an object b ∈ G − A

uch that a < b, then A is ( ∧ , ∨ ) -definable.

roof. Since there does not exist an object b ∈ G − A such that a <

, A is an up-set complete set. So, we can suppose A =

⋃

t∈ T B t , where

ach B t ( t ∈ T ) is an up-set complete list. According to Theorem 9 ,

t ( t ∈ T ) is a basic granule and ∧ -definable. Then it is trivial to

how m ( ∨

t∈ T d(B t ) ) = A . As a result, A is ( ∧ , ∨ )-definable. �

xample 8. For the formal context in Table 1 , we consider the

ranules {2, 3, 4} and {1, 2, 5}.



Since d({ 2 , 3 , 4 } ) = (b ∧ h ) ∨ i, we conclude that {2, 3, 4} is ( ∧ ,

)-definable.


and ∨ only to describe {1, 2, 5}. So, {1, 2, 5} is ( ∧ , ∨ )-indefinable.

At the end of this subsection, we give an algorithm

Algorithm 4 ) to explore the descriptions of ( ∧ , ∨ )-definable gran-

les.

.4. Some further discussions

The problem why some concepts are psychologically simple and

asy to learn, while others seem difficult, complex and incoher-

nt, is one of the unsolved problems in the field of human-level

oncept. Many effort s have been made to find the factors that de-

ermine the subjective difficulty of a concept since 1960, but this

roblem was never settled [3] .

A series of experiments have revealed that the subjective dif-

culty of a concept is directly proportional to its Boolean com-

lexity, i.e., the length of the shortest logically equivalent propo-

itional formula. Among all the empirical researches, majorities of

hem just considered two types of logical rules: conjunction and

isjunction, and an empirical result is that conjunctive (‘and’) con-

epts are learned more easily than disjunctive (‘or’) ones [6] . This

nding may be a satisfying explanation for the phenomenon that

oth basic granules and atomic granules attracted much attention

rom researchers, while the granules contained in monotone con-

epts [4,42] were rarely mentioned in the existing studies.

Compared with the granules contained in monotone concepts,

he importance of atomic granules was highly pointed out in sev-

ral recent researches. For example, in concept learning system

22] , information granules were built from a finite set of atomic

ranules by using logic connective ‘ ∩ ’. In the process of granu-

ar transformation [47] , atomic granules were employed to define

oncepts in object-oriented, attribute-oriented and formal concept

attices by using partial orders ≤ between objects, where ≤ can be

iewed as a ranking relation according to the number of attributes

hat the objects possess. In setting of formal concept analysis, the

nowledge reduction gained a granular perspective and a granu-

ar consistent attribute reduction was proposed and well-studied

n [50] . Moreover, in fuzzy data environment, a granular comput-

ng approach was presented by using a pair of object and attribute

ets to define different types of granules, namely, necessary fuzzy

nformation granule, sufficient fuzzy information granule, and nec-

ssary and sufficient fuzzy information granule [57] .




ARTICLE IN PRESS


fi

n

f

w

s

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I

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o

D

I

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F

In the aforementioned researches, composite granules were

built by using a set of atomic granules. However, how to delineate

these obtained composite granules by using the simplest form, is

not considered yet, and this is exactly what we have done in this

paper.

Another reason accounting for the importance of our study is

that the Boolean complexity of a propositional concept lies in the

length of the shortest Boolean formula logically equivalent to the

concept [10,45] . Obviously, short descriptions are always better

than longer ones. And all our effort s spent in this paper is just

to find the shortest descriptions of all kinds of granules.

In addition, the study of finding the most concise and apt de-

scriptions of granules makes sense not only in theoretical research

of granular computing, but also in the field of human-level con-

cept learning. In fact, it will give some hints on many real-life ap-

plications. For example, in dictionary compilation, one of the most

important problems is to give words the most concise and easily

understood explanations, which shares a lot of common features

with the one we discussed in this paper. Another highly related

example is ontology engineering. In the establishment of ontology,

one issue that costs our tremendous effort s is to delineate each

concept within the given domain, which is similar to the problem

of granule description. Undoubtedly, we can list many more such

application fields, but we omit them here.

7. Some necessary discussions on indefinable granules

In this section, we first give a superfluous study on the descrip-

tions of indefinable granules and then give our comments on the

connection of RST and FCA from the viewpoint of granule descrip-

tion.

7.1. Descriptions of indefinable granules

As was shown in the previous sections, there often exist inde-

finable granules. So, how to approximately describe these granules

may be a more interesting and appealing issue. In what follows,

we firstly give an index to measure the description quality of a

formula for a given granule. And then we make some necessary

discussions on the optimal approximate description of indefinable

granules.

Definition 17. Let A be a granule and ϕ be a formula. The recall

and precision indices of ϕ with respect to A are respectively de-

fined by

α(ϕ, A ) =

| m (ϕ) ∩ A | | A | and β( ϕ, A ) =

| m ( ϕ) ∩ A | | m ( ϕ) | .

In most occasions, if it makes no confusion, we will use α( ϕ)

and β( ϕ) instead of α( ϕ, A ) and β( ϕ, A ) for the sake of conve-

nience.

By Definition 17 , we can obtain the following proposition im-

mediately.

Proposition 5. If A is a definable granule and d(A ) = ϕ, then

α(ϕ) = 1 and β(ϕ) = 1 .

By using the recall and precision of a formula, we can define the

notion of an optimal approximate description of an indefinable gran-

ule.

Definition 18. Let A be a granule and � be the set of all formulas.

If both α( ϕ) and β( ϕ) have the greatest values among all formulas

in �, then we call ϕ the optimal approximate description of A .

Obviously, for a definable granule A, d ( A ) must be the optimal

approximate description. For an indefinable granule, it may cannot



nd a formula that achieves the greatest α and β values simulta-

eously. So, we must resort to the optimal approximate description

or an indefinable granule. In our opinion, there are two simple

ays. For instance, we can take α into consideration first and βecond, or β first and α second.

efinition 19. Let A be a granule and � be the set of all formulas.

f α(ϕ) = 1 and β( ϕ) has the greatest value among all formulas in

, then we call ϕ the α−priored optimal approximate description

f A .

efinition 20. Let A be a granule and � be the set of all formulas.

f β(ϕ) = 1 and α( ϕ) has the greatest value among all formulas in

, then we call ϕ the β−priored optimal approximate description

f A .

Note that the aim of this paper is to give a description method

or definable granules. Although we have slightly discussed the ap-

roximate description of indefinable granules, it is a challenging

ssue and cannot be settled in this paper. We will investigate this

ssue in depth in our forthcoming work.

.2. Connection of RST and FCA from the viewpoint of granule

escription

As is well-known, rough set theory (RST) also established the

otion of definable set, and it has frequently been related to formal

oncept analysis (FCA) in recent years [15,17,18,22,23,44,46,62,63] .

n this subsection, we discuss the relations between RST and FCA

rom the perspective of granule description.

The discussion of RST starts with the notion of an information

ystem IS = (U, AT ) [31] , where U is the universe of discourse, AT is

set of attributes, and the value of an object x under an attribute

is often denoted by a ( x ). Note that a formal context is a special

nformation system and in reverse an information system can be

ransformed into a formal context by the scaling method [7] .

In order to define rough set, we need to introduce an equiva-

ence relation IND ( A ) ( A ⊆AT ) in advance

ND (A ) = { (x, y ) ∈ U × U|∀ a ∈ A, a (x ) = a (y ) } . hen, [ x ] A = { y ∈ U| (x, y ) ∈ IND (A ) } is called equivalence class.

oreover, a collection of them {[ x ] A | x ∈ U } forms a partition of U

nd it is denoted by U / IND ( A ).

Pawlak’s rough set of X ⊆U with respect to A ⊆AT is a pair

A (X ) , A (X )] , where

(X ) = { x ∈ U| [ x ] A ⊆ X } and A (X ) = { x ∈ U| [ x ] A ∩ X � = ∅} . urthermore, if A (X ) = A (X ) , then X is said to be a crisp set (or

efinable set); otherwise, it is a rough set.

If for any x ∈ U, [ x ] AT = { x } , then this information system is

quivalent to a formal context with no reducible object. Under

uch circumstance, for any granule X ∈ 2 U , we have A (X ) = A (X ) =. Aided by RST, we know that nothing about the common at-

ributes shared by the elements of X . But when we resort to FCA,

e can get such information. From the discussion in the previous

ections, it follows the following remark.

emark 3. A crisp set of RST can either be a definable granule or

n indefinable granule under the framework of granule description

f FCA.

The following proposition points out that a rough set of RST

s also an indefinable granule under the framework of granule de-

cription of FCA.

roposition 6. Let X ∈ 2 U be a granule. If there exists an object x ∈ such that 0 <

| X∩ [ x ] AT | | [ x ] AT | < 1 , then X is a rough set of RST and it is

n indefinable granule under the framework of granule description of

CA.




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[

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[

[[

roof. Since there exists an object x ∈ U such that 0 <

| X∩ [ x ] AT | | [ x ] AT | <

, there must be two objects u, v ∈ [ x ] AT such that u ∈ X and v �∈ X .

ccording to the definition of rough set approximation, it follows

∈ A (X ) but v �∈ A (X ) . Hence, A (X ) � = A (X ) and X is a rough set

ith respect to AT .

On the other hand, note that u and v belong to the same equiv-

lence class [ x ] AT . So, it is impossible to distinguish u from v by the

ttributes contained in AT . Moreover, considering that u is an ob-

ect contained in X while v is outside of X , we draw the conclusion

hat X is indefinable. Therefore, the proof is completed. �

Intuitively, granule description with RST is realized by using a

et of resemblance objects, while it is achieved by using a set of

iterals connected by logical connectives in FCA. In our opinion,

ST-based granule description may be more effective in the re-

rieval of similar objects, while FCA-based granule description may

e more effective in describing one’s intrinsic features. So, as men-

ioned in the introduction, RST is actually related to and comple-

entary with FCA. In other words, both of them have advantages

nd disadvantages within their own tracks.

. Conclusions

The theory of GrC concerns information and knowledge pro-

essing in an abstract way, and one of the main directions in the

tudy of granular computing is to deal with construction, interpre-

ation, and representation of granules. In this study, the descrip-

ion of granules has been investigated from the perspective of FCA.

ore specifically, stability index has been introduced and used to

lassify granules into atomic granules, basic granules and compos-

te granules. And then based on the most apt minimal generator,

e have put forward a method for the description of atomic gran-

les and basic granules. Thereafter, composite granules have been

urther divided into ∧ -definable granules, ( ∧ , ¬)-definable gran-

les and ( ∧ , ∨ )-definable granules, and their respective description

ethods have also been explored. Finally, some discussions have

een made on indefinable granules, including approximate descrip-

ion of indefinable granules and the relationship between the inde-

nable granules in RST and FCA.

Some interesting and unsolved problems in granule description

till deserve to be studied. For example, how to establish some nice

pproximate description methods for indefinable granules, what

re the differences and similarities between RST and FCA in terms

f granule description, and whether it is possible to extend the ob-

ained results to the generalized formal contexts.

cknowledgments

The authors would like to thank anonymous reviewers for their

aluable comments and helpful suggestions which lead to a signifi-

ant improvement on the manuscript. This work was supported by

he National Natural Science Foundation of China (Nos. 61502150 ,

1305057 and 61562050 ) and the Doctorial Foundation of Henan

olytechnic University (No. B2011-102 ).

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