Formal $\\mathcal{F}$ -contexts and Their Induced Implication Rule Systems

$: Formal $\\mathcal{F}$ -contexts and Their Induced Implication Rule Systems$
Formal F -contexts and Their Induced

Implication Rule Systems�

Lankun Guo1,2, Qingguo Li3, Petko Valtchev1, and Robert Godin1

1 Departement d’Informatique, Universite du Quebec a MontrealCentre-Ville, Montreal, Quebec H3C 3J7, Canada

2 College of Information Science and Engineering, Hunan UniversityChangsha, Hunan 410012, P.R. China

3 College of Mathematics and Econometrics, Hunan UniversityChangsha, Hunan 410082, P.R. China

Abstract. Formal concept analysis (FCA) provides an approach to re-structuring important lattice structures such as complete lattices, dis-tributive lattices and algebraic lattices. In this paper, we focus on thetheoretical aspect of FCA and study the representation of algebraic do-mains by a special type of formal contexts. We first propose the notionof consistent F -context and investigate the detailed properties. Then westudy the induced implication rule systems of the consistent F -contextsand propose the notion of formal implication rule systems as the ax-iomatization. The results show that F -concepts inherent in consistentF -contexts can be characterized equivalently by closed subsets derivedfrom the formal implication rule systems. Furthermore, we study theorder-theoretical properties of F -concepts hierarchy (respectively, closedsubsets family) of consistent F -contexts (respectively, formal implicationrule system). It is shown that both F -contexts and formal implicationrule systems can serve as appropriate tools to concretely represent alge-braic domains.

Keywords: Formal context, Formal implication system, Algebraic do-main.

1 Introduction

Formal concept analysis (FCA) [9] has its root in applied lattice and order the-ory. This term was initiated by Rudolf Wille with the original motivation ofconcretely restructuring lattice theory and thus facilitating better communica-tion between lattice theorists and potential users of lattice theory [25]. In theclassical setting, it starts from a binary relation between a pair of sets and derivesa lattice (also called Galois lattice) which represents the hierarchy of conceptsinherent in the binary relation [9]. The well-known result in FCA is that every

� This work is partly supported by the NSERC Discovery research grants of the thirdand fourth authors, and China Postdoctoral Science Foundation (No. 2012M521515)of the first author.

P. Cellier, F. Distel, and B. Ganter (Eds.): ICFCA 2013, LNAI 7880, pp. 141–155, 2013.c© Springer-Verlag Berlin Heidelberg 2013

142 L. Guo et al.

Galois lattice is a complete lattice and, conversely, every complete lattice is orderisomorphic to the Galois lattice of an appropriate formal context. In this sense,FCA provides a concrete approach to characterize complete lattices. So far, FCAhas also been investigated from the viewpoint of other disciplines such as fuzzylogic [2, 18, 22], rough sets [17, 20, 26] and algebra [7, 8, 19, 24].

Several extensions of the classical FCA have been proposed. In [27], Zhangand Shen developed the notion of approximable concept based on the study ofcorrespondence between formal concepts of a given formal context and infor-mation states of the derived information system. Some following work furtherproved that this notion indeed provides a suitable tool to represent algebraic lat-tices [15, 21]. In [6], Ferre and Ridoux proposed a logical generalization of FCAthrough replacing the sets of attributes by expressions of an almost arbitrarylogic. In another trend, much effort has also been made to augment the classicalFCA framework by incorporating auxiliary structures. For instance, G. Hartung[12–14] proposed the notion of topological context and gave a representation ofbounded lattices. In practice, many attempts of introducing auxiliary structuresinto the FCA framework have also been made in order to improve the capabilityof FCA in dealing with real-world problems [4, 5, 11, 16]. These works suggestus to explore more potential of FCA with auxiliary structures in characterizingmore general ordered structures.

In Domain theory, a more general structure than algebraic lattices is the alge-braic domain. The only difference between them is that algebraic domains mustbe directed complete and algebraic lattices must be complete lattices. Noticingthat Zhang’s approximable concepts provide a concrete representation of alge-braic lattices, we expect to find a representation of algebraic domains by somesuitable formal contexts.

The present paper gives an answer to the above theoretical problem. We firstpropose the notion of F -context by introducing a finite-subsets-family struc-ture F into the classical FCA framework. Then we discuss the implication rulesystems induced from consistent F -contexts. The results demonstrate that theconcept hierarchy of the consistent F -contexts are just in correspondence witha special kind of subset family in the induced implication rules systems. More-over, we introduce the notion of formal implication rule system which can beviewed as the axiomatization of the induced implication rule systems of con-sistent F -contexts. A construction of consistent F -contexts from formal im-plication rule systems is given and the connection between them are discussed.Furthermore, we explore the potential of F -contexts and formal implication rulesystems in representing algebraic domain. The results show that the associatedF -concept hierarchy (respectively, closed subsets family) of every consistent F -context (respectively, formal implication rule system) is an algebraic domain andconversely, every algebraic domain is order isomorphic to the F -concept hierar-chy (respectively, closed subsets family) of an appropriate consistent F -context(respectively, formal implication rule system).

The rest of the paper is organized as follows: Section 2 briefly reviews classicalFCA and algebraic domains. In Section 3, we propose the notion of consistent

Formal F -contexts and Their Induced Implication Rule Systems 143

F -context and investigate the induced implication rule systems. In Section 4, wepropose the notion of formal implication rule system and study the capabilityof F -contexts and formal implication rule systems in characterizing algebraicdomains.

2 Preliminary

A formal context is a triplet (Po, Pa, RP ) where both Po and Pa are sets andRP ⊆ Po × Pa is a binary relation. A pair of operators (·)↑ and (·)↓ are definedas

(·)↑ : P(Po) → P(Pa) : A �→ {y ∈ Pa | (∀x ∈ A)(x, y) ∈ RP },(·)↓ : P(Pa) → P(Po) : B �→ {x ∈ Po | (∀y ∈ B)(x, y) ∈ RP }.

Usually, elements of Po and Pa are called objects and attributes of (Po, Pa, RP )respectively. Given subsets α ⊆ Po and β ⊆ Pa, the pair (α, β) is called a formalconcept of (Po, Pa, RP ) if α

↑ = β and β↓ = α. In this case, α and β are called theextent and intent of (α, β) respectively. It is easy to check that a subset A ⊆ Po

is an extent if and only if A↑↓ = A; B ⊆ Pa is an intent if and only if B↓↑ = B.The notation B(Po, Pa, RP ) is used to denote the set of all formal conceptsof (Po, Pa, RP ). Notations B(Po) and B(Pa) denote the sets of all extents andintents respectively. A partial order ≤ on B(Po, Pa, RP ) is defined by

(α1, β1) ≤ (α2, β2) ⇔ α1 ⊆ α2(⇔ β2 ⊆ β1).

One of the basic results of FCA is that (B(Po, Pa, RP ),≤) is a complete lat-tice. Moreover, it is easy to see that (B(Po, Pa, RP ),≤) is order isomorphic to(B(Po),⊆) and (B(Pa),⊇).

For more results about formal concepts, the reader can refer to [9].

Example 1. Let Table 1 be a formal context consisting of objects set O ={o1, o2, o3, o4, o5, o6} and attributes set A = {a1, a2, a3, a4, a5, a6, a7, a8}. Theincidence relation R ⊆ O × A is presented by the cross matrix in table 1 inthe sense that a cross in row oi and column aj means that the object oi has theattribute aj . Figure 1 presents the diagram of the corresponding formal conceptlattice.

Table 1. A formal context

a1 a2 a3 a4 a5 a6 a7 a8

o1 × × ×o2 × × × ×o3 × × × × ×o4 × ×o5 × ×o6 × × ×

144 L. Guo et al.

o1 · · · o6∅

��

��

��

��

��

��

��

��

o1o2o3o4a5

��

��

o1o2o3a1a5

��

��

��

��

o3o4a5a8

o3o6a2a7

��

o5o6a3a7

o1o2a1a5a6

o2a1a4a5a6

o3

a1a2a5a7a8

o6a2a3a7

��

��

∅a1 . . . a8

Fig. 1. The concept lattice of (O,A,R)

Next, we shortly review some basic notions about lattice theory and algebraicdomains. Most of them are collected from [3]. For more results about Domaintheory, we recommend the reader the reference [1, 10].

In a poset L, a ∈ L is called a lower bound (respectively, an upper bound) ofa subset X ⊆ L provided that a ≤ x (respectively, x ≤ a) for any x ∈ X . If theset of upper bounds of X has a unique smallest one, then it is called the leastupper bound of X , written as

∨X . Dually, if the set of lower bounds of X has

a unique greatest one, then it is called the greatest lower bound of X , writtenas

∧X . A poset is called a complete lattice if its every subset has a least upper

bound and a greatest lower bound.Given a subset A ⊆ L, we use the notation ↓A to denote the set {x ∈ L |

(∃a ∈ A)x ≤ a}. For any x ∈ L, ↓x represents ↓ {x} for simplicity. Dually, ↑Arepresents the subset {x ∈ L | (∃a ∈ A)a ≤ x} and ↑x denotes ↑ {x}. A subsetA ⊆ L is a down set of L if A =↓A and an upper set if A =↑A. A nonemptysubset D ⊆ L is said to be directed if for any two elements x, y ∈ D, there existsz ∈ D such that x ≤ z and y ≤ z. A directed complete poset (for short, dcpo) isa poset in which each directed subset has the least upper bound.

Let L be a poset and x, y ∈ L. We say x way below y, written as x y, iffor every directed set D ⊆ L for which

∨D exists, y ≤ ∨

D always implies theexistence of some d ∈ D such that x ≤ d. If an element x ∈ L satisfies x x,then it is said to be compact.

We use K(L) to denote the set of all compact elements of L. For any x ∈ L,the notation ⇓x is used to denote the set ↓x∩K(L), i.e., the set of all compactelements less than or equal to x. A subset B ⊆ L is called a base of L if for everyx ∈ L, the subset ↓x ∩ B is directed and x =

∨(↓x ∩ B). A dcpo is called an

algebraic domain if all its compact elements form a base.


3 F-contexts and the Induced Implication Rule Systems

In this section, we first introduce the notion of (consistent) F -context and F -concept which can be viewed as an extension of the corresponding notions in clas-sical FCA. Afterwards, we shall introduce the induced implication rule systemsof consistent F -contexts and focus on the correspondence between F -conceptsinherent in consistent F -contexts and closed subsets derived from the inducedimplication rule systems. In addition, we extract the essential characteristics ofthe induced implication rule systems and then axiomatize them to propose thenotion of formal implication rule system.

3.1 Consistent F -contexts and F -concepts

With the original motivation of exploring the contextual approach to reconstructalgebraic domains, we want to extend the classical FCA framework by incorpo-rating some additional structures as the classical one is only valid to completelattices. Noticing that every algebraic domain is completely determined by itsbase, we hope that the incorporated auxiliary structure will be competent toderive a more general concept paradigm and capture the base of the associatedconcept hierarchy. In this section, we shall propose a type of restricted formalcontexts by incorporating a finite-subset-family structure onto the attributes setof a classical formal context and investigate the basic properties in detail.

Definition 1. Let (Po, Pa, RP ) be a formal context. If FP is a family of non-empty finite subsets of Pa, then the tuple (Po, Pa, RP ,FP ) is called a (formal)F -context. Furthermore, if for any F ∈ FP , there exists a directed family {Fi |i ∈ I} ⊆ FP such that F ↓↑ =

⋃i∈I Fi, then (Po, Pa, RP ,FP ) is called a (formal)

consistent F -context.

The following proposition provides an equivalent description to consistent F -contexts. For simplicity, we always use the notation F � X to mean that F is afinite subset of X .

Proposition 1. Let (Po, Pa, RP ) be a formal context and FP a family of non-empty finite subsets on Pa. Then (Po, Pa, RP ,FP ) is a consistent F -context ifand only if for any F ∈ FP and any subset B � F ↓↑, there exists F ′ ∈ FP suchthat B ⊆ F ′ ⊆ F ↓↑.

Proof. Straightforward. �It is worthwhile mentioning that we did not demand Pa to be finite in propos-ing the notion of F -context. But this leads to a natural question: what is thesituation when Pa is finite? In this regard, we give the following proposition.

Proposition 2. Let (Po, Pa, RP ) be a formal context and FP a family of non-empty finite subsets on Pa where Pa is finite. Then (Po, Pa, RP ,FP ) is a con-sistent F -context if and only if F ↓↑ ∈ FP for any F ∈ FP .

146 L. Guo et al.

Proof. Straightforward. �In the proposing the notion of F -context, elements of FP play the role of ap-proximating the maximal expansion of every F ∈ FP . However, only the family{F ↓↑ | F ∈ FP } is not competent to achieve our goal of reconstructing alge-braic domain. In the following, we shall propose the notion of F -concept in theframework of consistent F -context which can be regarded as the approximationof elements of {F ↓↑ | F ∈ FP }.Definition 2. Let (Po, Pa, RP ,FP ) be a consistent F -context. A pair (ρ, ω) iscalled a (formal) F -concept of P if ρ = ω↓ and for any subset B � ω, thereexists F ∈ FP such that B ⊆ F ↓↑ ⊆ ω. In this case, ρ and ω are called theextent and intent of (ρ, ω) respectively.

For convenience, we shall often use the notation P to represent a F -context(Po, Pa, RP ,FP ).

Example 2. Continue with Example 1. Let F = {A1, A2, A3, A4, A5} whereA1 = {a5}, A2 = {a2, a7}, A3 = {a1, a7, a8}, A4 = {a1, a5, a6} and A5 ={a1, a2, a5, a7, a8}. Then it is trivial to check that (O,A,R,F ) is a consistentF -context. The associated F -concepts are presented in Table 2.

Table 2. The F -concepts of (O,A,R,F )

ρ ω

{o3} {a1, a2, a5, a7, a8}{o1, o2} {a1, a5, a6}{o3, o6} {a2, a7}

{o1, o2, o3, o4} {a5}

In the case when Pa is finite, we can easily see that the intents of all F -concepts of the consistent F -context (Po, Pa, RP ,FP ) are exactly the family{F ↓↑ | F ∈ FP }. This means that our notion of F -concept is meaninglessfor consistent F -context (Po, Pa, RP ,FP ) when Pa is finite. Since the aim ofthis paper is to provide a representation of arbitrary algebraic domains, possiblereaders are recommended to keep in mind that Pa is not finite necessarily exceptin the illustrative examples.

In the following, we give an infinite example for the notions of consistentF -context and F -concept.

Example 3. Consider the real interval I = [0, 1] with the ordinary order ≤. Wecan get a formal context (I, I,≤). If we let FI be the family of all finite subsetsof I, then (I, I,≤,FI) is a consistent F -context. Consider the subset (12 , 1].We can easily check that ([0, 12 ], (

12 , 1]) is an F -concept of (I, I,≤,FI). In fact,

the reader can check that every subset (r, 1] where r ∈ [0, 1) is an an F -conceptof (I, I,≤,FI).


Proposition 3. Let P be a consistent F -context and ω ⊆ Pa. Then the follow-ing are equivalent:

(1) ω is the intent of a formal F -concept.

(2) For any B � ω, it holds that B↓↑ ⊆ ω and there exists F ∈ FP such thatB ⊆ F ⊆ ω.

(3) There exists a directed family {Fi | i ∈ I} ⊆ FP such that F ↓↑i ⊆ ω for any

i ∈ I and ω =⋃

i∈I Fi.

Proof. (1) ⇒ (2): By Definition 2, for any subset B � ω, there exists F ∈ FP

such that B ⊆ F ↓↑ ⊆ ω. This implies that B↓↑ ⊆ (F ↓↑)↓↑ = F ↓↑ ⊆ ω. Moreover,by Definition 1, there exists a directed family {Fi | i ∈ I} ⊆ FP such thatF ↓↑ =

⋃i∈I Fi. Since B is a finite subset, there exists i0 ∈ I such that B ⊆ Fi0 .

It is obvious that Fi0 ⊆ ω.The implication (2) ⇒ (1) is obvious.For (2) ⇒ (3), we only need to check that {F ∈ F | F ⊆ ω} is the desired

family.The implication (3) ⇒ (2) is obvious. �

Remark 1. (i) If we interpret every F ∈ FP as a generator of an F -concept (infact, F ↓↑ is the intent of an F -concept), then item (2) of Proposition 3 tellsthat between the intent of every F -concept and its finite segment, an elementalgenerator can be interposed.

(ii) The item (2) of Proposition 3 also shows that F -concepts are just selec-tions from approximable concepts in the sense of Zhang and herein FP plays arole of restriction during the selecting. In fact, if we replace FP in Definition 1 bythe set of all finite subsets of Pa, then the corresponding intents of F -conceptsare exactly the set of all approximable concepts.

(iii) The item (3) of Proposition 3 means that the intent of every F -conceptcan be produced successively by merging elemental generators from FP .

Corollary 1. Let P be a consistent F -context. Then ω is the intent of an F -concept of P if and only if there exists a family {Fi | i ∈ I} ⊆ FP such that

{F ↓↑i | i ∈ I} is directed and ω =

⋃i∈I F

↓↑i .

Proof. Suppose ω is the intent of an F -concept. Then by Proposition 3(3),

there exists a directed family {Fi | i ∈ I} ⊆ FP such that F ↓↑i ⊆ ω for any i ∈ I

and ω =⋃

i∈I Fi. Thus, we have {F ↓↑i | i ∈ I} is also directed. ω =

⋃i∈I F

↓↑i is

obvious.Conversely, suppose there exists a family {Fi | i ∈ I} ⊆ FP such that {F ↓↑

i |i ∈ I} is directed and ω =

⋃i∈I F

↓↑i . For any B � ω, it is clear that there exists

i0 ∈ I such that B ⊆ F ↓↑i0

⊆ ω. Thus, by Proposition 1, ω is the intent of anF -concept. �

Corollary 2. Let P be a consistent F -context. If ρ is the extent of an F -conceptof P, then there exists a directed family {Fi | i ∈ I} ⊆ FP such that ρ =

⋂i∈I F

↓i .

148 L. Guo et al.

Proof. Suppose ρ is the extent of an F -concept. Then by Definition 2, thereexists ω ⊆ Pa such that (ω, ρ) is an F -concept of P. By Proposition 3(3), there

exists a directed family {Fi | i ∈ I} ⊆ FP such that F ↓↑i ⊆ ω for any i ∈ I and

ω =⋃

i∈I Fi. From ρ = ω↓, it follows immediately that ρ =⋂

i∈I F↓i . �

3.2 Induced Implication Rule Systems of F -contexts

FCA offers an appropriate model for the mining of a special pattern of knowl-edge which is called the implication rules (or the association rules) [23]. In thelogical form, an implication rule can be written as X → Y where X and Yare two attribute sets. They represent the dependencies among attributes (of-ten with some degree) and usually contain meaningful knowledge existing in thedatasets. Technically, in the classical FCA, an implication rule X → Y derivedfrom a formal context is said to be valid if it satisfies that Y ⊆ X↓↑. In theframework of F -context, such type of implication rules have different patterndue to the restriction of the finite-subset-family structure. Formally, we proposethe following definition.

Definition 3. Let P be a consistent F -context. Define a relation�P⊆ FP ×Pa

byX �P y ⇔ y ∈ X↓↑.

Then (FP , Pa,�P ) is called the induced implication rule system of P.

Since the operator (·)↓↑ is a closure operator on Pa, X �P y can be explained asthat y is included in the closure of X . In the sequel, given X ∈ FP and Y ⊆ Pa,we use the notation X �P Y to mean X �P y for any y ∈ Y .

Example 4. Continue with Example 2. The induced implication rules of the con-sistent F -context given in Example 2 is presented in Table 3. For simplicity, weomit the notation �P in the table.

Table 3. The implication rules induced from (O,A,R,F )

X ∈ F ai ∈ A X ∈ F ai ∈ A

{a5} a5 {a1, a5, a6} a1

{a2, a7} a2 {a1, a5, a6} a5

{a2, a7} a7 {a1, a5, a6} a6

{a1, a7, a8} a1 {a1, a2, a5, a7, a8} a1

{a1, a7, a8} a2 {a1, a2, a5, a7, a8} a2

{a1, a7, a8} a5 {a1, a2, a5, a7, a8} a5

{a1, a7, a8} a7 {a1, a2, a5, a7, a8} a7

{a1, a7, a8} a8 {a1, a2, a5, a7, a8} a8

Proposition 4. Let P be a consistent F -context and (FP , Pa,�P ) the asso-ciated induced implication rule system. Then for any z ∈ Pa, X,Y ∈ FP andB � Pa,


(1) x ∈ X ⇒ X �P x;

(2) X �P Y �P z ⇒ X �P z;

(3) X �P B ⇒ (∃F ∈ FP )(B ⊆ F & X �P F ).

Proof. (1) As X ⊆ X↓↑, it follows directly from the definition of �P thatX �P x for any x ∈ X .

(2) Suppose X �P Y �P z. By the definition of �P , we have z ∈ Y ↓↑ andY ⊆ X↓↑. Thus, we have Y ↓↑ ⊆ (X↓↑)↓↑ = X↓↑ which implies that z ∈ X↓↑, i.e.,X �P z.

(3) Suppose X �P B. By the definition of �P , it follows that B ⊆ X↓↑.Since X ∈ FP , it follows from Definition 1 that there exists a directed family{Fi | i ∈ I} ⊆ FP such that X↓↑ =

⋃i∈I Fi. Thus, we have B ⊆ ⋃

i∈I Fi. AsB is finite, there exists i0 ∈ I such that B ⊆ Fi0 . In addition, it is clear thatFi0 ⊆ X↓↑ which means X �P Fi0 . �

Theorem 1. Let P be a consistent F -context and (FP , Pa,�P ) the associatedinduced implication rule system. Then ω ⊆ Pa is the intent of an F -concept ifand only if for any x ∈ Pa, B � Pa and F ∈ FP ,

(1) (F ⊆ ω & F �P x) ⇒ x ∈ ω;

(2) B � ω ⇒ (∃F ′ ∈ FP )(F′ ⊆ ω & F ′ �P B).

Proof. Suppose ω is the intent of an F -concept. For any F ∈ FP and x ∈ Pa

satisfying F ⊆ ω and F �P x, it immediately follows from the definition of �P

that x ∈ F ↓↑. As F � ω, it follows from Proposition 3(2) that F ↓↑ ⊆ ω. Thisimplies x ∈ ω. Moreover, for any B � ω, by Definition 2 there exists F ′ ∈ FP

such that B ⊆ F ′↓↑ ⊆ ω. It is clear that F ′ ⊆ ω. In addition, by the definitionof �P , we have F ′ �P B.

Conversely, suppose ω satisfies conditions (1) and (2) above. For any B � ω,by condition (2) there exists F ′ ∈ FP such that F ′ ⊆ ω and F ′ �P B. Forthis F ′, we have B ⊆ F ′↓↑ by the definition of �P . Moreover, it is obvious thatF ′ �P x for any x ∈ F ′↓↑. Thus, it follows from condition (1) that F ′↓↑ ⊆ ω.As a conclusion, we obtain that there exists F ′ ∈ FP such that B ⊆ F ′↓↑ ⊆ ω,which means that ω is the intent of an F -concept. �

3.3 Formal Implication Rule Systems

From Theorem 1, we can see that the F -concepts of a consistent F -contextcan be equivalently represented by the closed subsets in some appropriate im-plication rule system. In the following, we extract the essential characteristics ofthe induced implication rule systems of consistent F -contexts and employ theaxiomatic approach to develop the notion of formal implication rule system.

Definition 4. A (formal) implication rule system is a tuple (U , U,�) where Uis a non-empty set, U is a family of non-empty finite subsets of U , and � is abinary relation from U to U , i.e., �⊆ U ×U , which satisfies: for any X,Y ∈ U ,u ∈ U and B � U ,

150 L. Guo et al.

(1) x ∈ X ⇒ X � x;

(2) X � Y � u⇒ X � u;

(3) X � B ⇒ (∃Z ∈ U )(B ⊆ Z & X � Z).

Definition 5. Let (U , U,�) be an implication rule system. A subset D ⊆ U issaid to be closed in (U , U,�) if for any u ∈ U , B � U and F ∈ U ,

(1) (F ⊆ D & F � u) ⇒ u ∈ D;

(2) B � D ⇒ (∃F ′ ∈ U )(F ′ ⊆ D & F ′ � B).

In the following, we always use C(U , U,�) to denote the set of all closed subsetsof an implication rule system (U , U,�).

Proposition 5. Let (U , U,�) be an implication rule system and D ⊆ U . If Dis a closed subset, then

D =⋃

{F | F ∈ U & F ⊆ D} (1)

where F = {u ∈ U | F � u}.Proof. Suppose D is a closed subset of (U , U,�). Given any F ∈ U satisfyingF ⊆ D, we have u ∈ D for any u ∈ F by Definition 5(1). This implies that⋃{F | F ∈ U & F ⊆ D} ⊆ D. On the other hand, for any u ∈ D, by Definition5(2) there exists F ′ ∈ U such that F ′ ⊆ D and F ′ � u, i.e., u ∈ F ′. It followsthat D ⊆ ⋃{F | F ∈ U & F ⊆ D}. �

Theorem 2. Let (U , U,�) be an implication rule system and D ⊆ U . Thenthe following are equivalent:

(1) D is a closed subset of (U , U,�).

(2) D =⋃{F | F ∈ U & F ⊆ D} where {F | F ∈ U & F ⊆ D} is directed.

(3) There exists a family {Fi | i ∈ I} ⊆ U such that {Fi | i ∈ I} is directed andD =

⋃i∈I Fi.

Proof. (1) ⇒ (2): From Proposition 5, we have D =⋃{F | F ∈ U & F ⊆ D}.

It is sufficient to prove that the family {F | F ∈ U & F ⊆ D} is directed:Indeed, suppose F1, F2 ∈ U and they satisfy that Fi ⊆ D(i = 1, 2). Thenwe have F1 ∪ F2 ⊆ D. Since F1 ∪ F2 is finite, by Definition 5(2) there existsF ′ ∈ U such that F ′ ⊆ D and F ′ � F1 ∪ F2. Then by Definition 4(2), we haveFi ⊆ F ′(i = 1, 2).

(2) ⇒ (3): Straightforward.(3) ⇒ (1): Suppose F ∈ U and u ∈ U which satisfy that F ⊆ D and F � u.

Because F is finite and {Fi | i ∈ I} is directed, there exists i0 ∈ I such thatF ⊆ Fi0 , i.e., Fi0 � F . By Definition 4(2), we have Fi0 � u, i.e., u ∈ Fi0 andthus u ∈ D. Moreover, for any B � D, because {Fi | i ∈ I} is directed, thereexists i1 ∈ I such that B ⊆ Fi1 , i.e., Fi1 � B. In addition, by Definition 4(1),we have Fi1 ⊆ Fi1 and thus Fi1 ⊆ D. Therefore, D is a closed subset. �


Proposition 6. Given an implication rule system (U , U,�), construct a for-mal context (C(U , U,�), U,RU ) where the binary relation RU ⊆ C(U , U,�)×Uis defined by

(c, u) ∈ RU ⇔ u ∈ c.

Then for any F ∈ U , we have

F ↓↑ = F =⋃

{F ′ ∈ U | F � F ′}. (2)

Proof.We first prove F =⋃{F ′ ∈ U | F � F ′} for any F ∈ U in the following:

On the one hand, for any u ∈ F , it follows from Definition 4(3) that there existsF ′ ∈ U such that u ∈ F ′ and F � F ′. This means F ⊆ ⋃{F ′ ∈ U | F � F ′}.On the other hand, the inclusion

⋃{F ′ ∈ U | F � F ′} ⊆ F is obvious.Next, we prove F ↓↑ = F : By the definition of RU we haveF ↓↑ = {c ∈ C(U , U,�) | (∀u ∈ F )u ∈ c}↑

= {c ∈ C(U , U,�) | F ⊆ c}↑= {u ∈ U | (∀c ∈ C(U , U,�))(F ⊆ c⇒ u ∈ c)}=

⋂{c ∈ C(U , U,�) | F ⊆ c}.We show that F ∈ C(U , U,�) in the following: (i) Suppose F ′ ∈ U and u′ ∈ Uwhich satisfy that F ′ ⊆ F and F ′ � u. It immediately follows that F � F ′ � u.By Definition 4(2), F � u′ and thus u′ ∈ F . (ii) Suppose B � F . Then we haveF � B. By Definition 4(3), there exists Z ∈ U such that B ⊆ Z and F � Z.Since Z ∈ U , it follows from Definition 4(1) that Z � B. In addition, it isobviously follows from F � Z that Z ⊆ F .

To finish the proof, we only need to check that F is the least element of{c ∈ C(U , U,�) | F ⊆ c}. Indeed, as F ∈ U , by Definition 4(1) we haveF ⊆ F . This means that F ∈ {c ∈ C(U , U,�) | F ⊆ c}. Moreover, supposec ∈ C(U , U,�) satisfying F ⊆ c. For any u ∈ F , by Definition 5(1) we haveu ∈ c. This implies that F ⊆ c. �

Proposition 7. For any implication rule system (U , U,�), the induced tuple(C(U , U,�), U,RU ,U ) is a consistent F -context.

Proof. Based on Proposition 6, we only need to prove that {F ′ ∈ U | F � F ′}is directed. Indeed, suppose F1, F2 ∈ U satisfying F � F1 and F � F2. Then wehave F � F1 ∪F2. Since F1 ∪F2 is finite, by Definition 4(3) there exists Z ∈ Usuch that F1 ∪ F2 ⊆ Z and F � Z which means Z ∈ {F ′ ∈ U | F � F ′}. �

Theorem 3. Let (U , U,�) be an implication rule system and D ⊆ U . Then Dis a closed subset of (U , U,�) if and only if it is the intent of an F -concept of(C(U , U,�), U,RU ,U ).

Proof. Suppose D is a closed subset of (U , U,�). For any B � D, by Definition5(2) there exists F ∈ U such that F ⊆ D and F � B, i.e., B ⊆ F . Since B isfinite, by Proposition 6 there exists F ′ ∈ U such that F � F ′ and B ⊆ F ′. AsF ′ ∈ U , by Definition 4(1) we have F ′ � F ′. Thus, by Definition 4(2), F ′ � B,i.e., B ⊆ F ′ = F ′↓↑. Moreover, for any u ∈ F ′, i.e., F ′ � u, as F � F ′ we have

152 L. Guo et al.

F � u by Definition 4(2). Thus, by Definition 5(1) we have u ∈ D. This meansthat F ′ ⊆ D, i.e., F ′↓↑ ⊆ D. Therefore, we obtain that D is the intent of anF -concept.

Conversely, suppose D is the intent of an F -concept. For any F ∈ U andu ∈ U which satisfy that F ⊆ D and F � u, i.e., u ∈ F , by Proposition 3(2) wehave F ↓↑ ⊆ D. By Proposition 6 we have F ⊆ D and thus u ∈ D. Furthermore,for any B � D. By Proposition 3(2) there exists F ′ ∈ U such that B ⊆ F ′ ⊆ D.As F ′ ∈ U , by Definition 4(1), F ′ � F ′, which implies that F ′ � B. Thus, weobtain that D is a closed subset of (U , U,�). �

4 Representation of Algebraic Domains

In this section, we shall focus on the order-theoretical properties of the associ-ated F -concepts hierarchy (respectively, closed subsets family) of F -contexts(respectively, formal implication rule systems). Throughout this section, we de-note the set of all F -concepts of the consistent F -context P by the notationA(P). Moreover, we use Ao(P) and Aa(P) to denote the sets of extents and in-tents of F -concepts of P respectively.

Given a consistent F -context P, define a relation � on A(P) by

(ρ1, ω1) � (ρ2, ω2) ⇔ ω1 ⊇ ω2.

Then it is trivial to check that � is a partial order and (A(P),�) is order iso-morphic to (Aa(P),⊇).

Since the extent of every F -concept is uniquely determined by its intent andthere exists bijective correspondence between F -concepts and closed subsets,we shall concentrate on the order-theoretical properties of Aa(P) under the set-inclusion order. The following proposition shows that (Aa(P),⊆) is a dcpo forany F -context P.

Proposition 8. Let P be a consistent F -context. If {ωj}j∈J is a directed familyof intents of F -concepts of P, then

⋃j∈J ωj is also the intent of an F -concept.

Proof. Suppose B � ⋃j∈J ωj . Then there exists j0 ∈ J such that B ⊆ ωj0 since

B is a finite subset and {ωj}j∈J is directed. As ωj0 is the intent of an F -concept,by Definition 2 there exists F ∈ FP such that B ⊆ F ↓↑ ⊆ ωj0 ⊆ ⋃

j∈J ωj . Thisimplies that

⋃j∈J ωj is the intent of an F -concept. �

The following proposition characterizes the way-below relation in (Aa(P),⊆).

Proposition 9. Let P be a consistent F -context. Then ω ω′ in (Aa(P),⊆)if and only if there exists F ∈ FP such that ω ⊆ F ↓↑ ⊆ ω′.

Proof. Suppose ω ω′. By Corollary 1, ω′ =⋃

i∈I F↓↑i for some directed family

{F ↓↑i | Fi ∈ FP , i ∈ I}. By the definition of way below, there exists i0 ∈ I such

that Fi0 ∈ FP and ω ⊆ F ↓↑i0

. In addition, it is easy to see that F ↓↑i0

⊆ ω′.Conversely, suppose that there exists a directed family of intents of F -concepts

{ωj | j ∈ J} such that ω′ ⊆ ∨j∈J ωj . By Proposition 8, we have

∨j∈J ωj =


⋃j∈J ωj and thus ω′ ⊆ ⋃

j∈J ωj. Moreover, there exists F ∈ FP such that

ω ⊆ F ↓↑ ⊆ ω′ by hypothesis. Since {ωj | j ∈ J} is directed and F ⊆ F ↓↑ ⊆ ω′ ⊆⋃j∈J ωj, it holds that there exists j0 ∈ J such that F ⊆ ωj0 . By Proposition

3(2), we have F ↓↑ ⊆ ωj0 and thus ω ⊆ ωj0 . This means that ω ω′. �As an immediate consequence, we show that FP just determines all compact

elements of (Aa(P),⊆).

Corollary 3. Let P be consistent F -context. Then ω ⊆ Pa is a compact elementof (Aa(P),⊆) if and only if there exists F ∈ FP such that ω = F ↓↑.

Now we can obtain the following core properties of F -context which shows thatthis notion provides a concrete approach to reconstruct algebraic domains.

Theorem 4. For any consistent F -context P, (Aa(P),⊆) is an algebraic do-main. Conversely, for any algebraic domain (L,≤), there exists a consistentF -context Q such that (L,≤) is order isomorphic to (Aa(Q),⊆).

Proof. Suppose ω is the intent of an F -concept of P. Consider the set {F ↓↑ |F ∈ FP & F ↓↑ ⊆ ω}. Given any F1, F2 ∈ FP with F ↓↑

i ⊆ ω (i = 1, 2), we

have F1 ∪ F2 ⊆ ω since Fi ⊆ F ↓↑i (i = 1, 2). As ω is the intent of an F -concept,

by Definition 2, there exists F3 ∈ FP such that F1 ∪ F2 ⊆ F ↓↑3 ⊆ ω. Thus, we

have F ↓↑1 ⊆ F ↓↑

3 and F ↓↑2 ⊆ F ↓↑

3 . It implies that {F ↓↑ | F ∈ FP & F ↓↑ ⊆ ω} isdirected. Moreover, it is trivial to check that ω =

⋃{F ↓↑ | F ∈ FP & F ↓↑ ⊆ ω}.Therefore, {F ↓↑ | F ∈ FP } is a base of (Aa(P),⊆). Since every such F ↓↑ iscompact by Corollary 3, we obtain that (Aa(P),⊆) is an algebraic domain.

Conversely, given an algebraic domain (L,≤), construct a tupleQL = (L,K(L),≥,FL) where FL is the family of non-empty finite subsetsof compact elements which has greatest elements. For convenience, we denoteby cF the greatest element of such F . Then for any F ∈ FL, we have F ↓↑ =↓cF∩K(L) = c↓↑

F. Given any B � F ↓↑, because

∨B ∈ K(L) and

∨B ≤ c

F, we

have B ∪ {∨B} ∈ FL and B ⊆ B ∪ {∨B} ⊆↓cF∩K(L). Thus, by Proposition

1, QL is indeed a consistent F -context.Suppose ω ∈ Aa(QL). Then by Corollary 1 there exists a directed family {c↓↑

F|

F ∈ FL} such that ω =⋃

F∈FLc↓↑F. For any x, y ∈ ω, there exist F1, F2 ∈ FL

such that x ∈ c↓↑F1and y ∈ c↓↑F2

. Since {c↓↑F

| F ∈ FL} is directed, there exists

F3 ∈ FL such that x, y ∈ c↓↑F3. Thus we have x, y ≤ cF3 . Because cF3 ∈ ω, it

follows that ω is a directed subset of L and thus∨ω exists in L.

Now we prove ω = (↓ ∨ω) ∩ K(L). On the one hand, it is obvious that

ω ⊆ (↓∨ω)∩K(L). On the other hand, for any k ∈ K(L) with k ≤ ∨ω, because

ω is directed and k is a compact element, there exists d ∈ ω such that k ≤ d. Asω =

⋃F∈FL

c↓↑F

for some directed family {c↓↑F

| F ∈ FL}, there exists c ∈ K(L)

such that k ≤ d ∈ c↓↑ ⊆ ω. As c↓↑ is a down set of K(L), we have k ∈ c↓↑ andthus k ∈ ω, which means (↓∨ω) ∩K(L) ⊆ ω.

Finally, we show the order isomorphism between (L,≤) and (Aa(QL),⊆).Define a function ϕ : L→ Aa(QL) by ϕ(x) =↓x ∩K(L). Because ↓x ∩K(L) =⋃{↓ y ∩ K(L) | y ∈ x ∩ K(L)} and {↓ y ∩ K(L) | y ∈ x ∩ K(L)} is directed,

154 L. Guo et al.

we have ↓x ∩K(L) is the intent of an F -concept, which means that ϕ is well-defined. Define a function ψ : Aa(QL) → L by ψ(ω) =

∨ω. It is trivial to check

that they are inverse with each other. �Combing Theorem 3 and Theorem 4, we obtain the following result.

Corollary 4. For any implication rule system (U , U,�), (C(U , U,�),⊆) isan algebraic domain. Conversely, for any algebraic domain (L,≤), there existsan implication rule system (U , U,�) such that (L,≤) is order isomorphic to(C(U , U,�),⊆).

5 Conclusions

In this paper, we first proposed the notion of F -context which can be seen asan extension of classical FCA by introducing a finite-subsets-family structure Fonto the attributes set. In the setting of consistent F -context, we proposed thenotion of F -concept which can be seen as an extension of the classical formalconcept. Moreover, we discussed the implication rule systems which can be in-duced from consistent F -contexts in a natural way. The results demonstrate thatthe F -concepts of the original consistent F -context are just in correspondencewith a special kind of closed subsets in the induced implication rules systems.Based on this result, we introduced the notion of formal implication rule sys-tem which can be viewed as the axiomatization of the induced implication rulesystems of consistent F -contexts. We also presented an approach to constructconsistent F -contexts from formal implication rule systems and obtained theone-to-one correspondence between the closed subsets of the former structuresand the F -concepts of the latter ones.

Finally, we discussed the detailed order-theoretical properties of the F -concepthierarchy of a consistent F -context. We showed that every such F -concept hi-erarchy is an algebraic domain and conversely, every algebraic domain is orderisomorphic to the F -concept hierarchy of an appropriate consistent F -context.The results demonstrate that our notions of consistent F -context and formal im-plication rule system can serve as alternative approaches to reconstruct algebraicdomains.

References

1. Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., et al. (eds.) Handbookof Logic in Computer Science, vol. 3, pp. 1–168. Clarendon Press (1995)

2. Belohlavek, R.: Concept lattices and order in fuzzy logic. Annals of Pure andApplied Logic 128, 277–298 (2004)

3. Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cam-bridge University Press (2002)

4. Dolques, X., Huchard, M., Nebut, C., Saada, H.: Formal and relational conceptanalysis approaches in software engineering: An overview and an application tolearn model transformation patterns in examples. In: The First ICESE VirtualWorkshop, Search-based Model-Driven Engineering, Qatar (2011)


5. Faid, M., Missaoui, R., Godin, R.: Knowledge discovery in complex objects. Com-putational Intelligence 15(1), 28–49 (1999)

6. Ferre, S., Ridoux, O.: A logical generalization of formal concept analysis. In: Gan-ter, B., Mineau, G.W. (eds.) ICCS 2000. LNCS, vol. 1867, pp. 371–385. Springer,Heidelberg (2000)

7. Ganter, B., Kwuida, L.: Dicomplemented lattices: towards a representation theo-rem. Technical Report, MATH-AL-20-2002 (2002)

8. Ganter, B., Kwuida, L.: Representable weak dicomplementations on finite lattices.Contributions to General Algebra 14, J. Heyn, Klagenfurt, 63–72 (2004)

9. Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations.Springer, Berlin (1999)

10. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.:Continuous Lattices and Domains. Cambridge University Press (2003)

11. Guo, L.K., Huang, F.P., Li, Q.G., Zhang, G.-Q.: Power contexts and their conceptlattices. Discrete Mathematics 311, 2049–2063 (2011)

12. Hartung, G.: A topological representation of lattices. Algebra Universalis 29(2),273–299 (1992)

13. Hartung, G.: Sublattices of topologically represented lattices. Acta Math. Univ.Comenianae LXIV(1), 25–41 (1995)

14. Hartung, G., Kamara, M., Sacarea, C.: Topological representation of polarity lat-tices. Acta Math. Univ. Comenianae LXVIII(1), 49–70 (1999)

15. Hitzler, P., Krotzsch, M., Zhang, G.-Q.: A categorical view on algebraic lattices informal concept analysis. Fundamenta Informaticae 74, 1–29 (2006)

16. Huchard, M., Rouane Hacene, M., Roume, C., Valtchev, P.: Relational conceptdiscovery in structured datasets. Ann. Math. Artif. Intell. 49, 39–76 (2007)

17. Kent, R.E.: Rough concept analysis: A synthesis of rough sets and formal conceptanalysis. Fundamenta Informaticae 27(2), 169–181 (1996)

18. Krajci, S.: A generalized concept lattice. Logic Journal of the IGPL 13(5), 543–550(2005)

19. Kwuida, L.: Dicomplemented Lattices: A Contextual Generalization of BooleanAlgebras. Shaker (2004)

20. Lai, H., Zhang, D.: Concept lattices of fuzzy contexts: Formal concept analysis vs.rough set theory. International Journal of Approximate Reasoning 50(5), 695–707(2009)

21. Lei, Y., Luo, M.: Rough concept lattices and domains. Annals of Pure and AppliedLogic 159, 333–340 (2009)

22. Popescu, A.: A general approach to fuzzy concepts. Mathematical Logic Quar-terly 50, 265–280 (2004)

23. Valtchev, P., Missaoui, R., Godin, R., Meridji, M.: Generating Frequent ItemsetsIncrementally: Two Novel Approaches Based on Galois Lattice Theory. Journal ofExperimental & Theoretical Artificial Intelligence 14, 115–142 (2002)

24. Wang, L.D., Liu, X.D.: Concept analysis via rough set and AFS algebra. Informa-tion Sciences 178(21), 4125–4137 (2008)

25. Wille, R.: Restructuring lattice theory: An approach based on hierarchies of concepts.In: Rival, I. (ed.) Ordered Sets, Reidel, Dordrecht, Boston, pp. 445–470 (1982)

26. Yao, Y.: A comparative study of formal concept analysis and rough set theory indata analysis. In: Tsumoto, S., S�lowinski, R., Komorowski, J., Grzyma�la-Busse, J.W.(eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 59–68. Springer, Heidelberg (2004)

27. Zhang, G.-Q., Shen, G.Q.: Approximable concepts, Chu spaces, and informationsystems. Theory and Applications of Categories 17(5), 80–102 (2006)

Formal $\\mathcal{F}$ -contexts and Their Induced Implication Rule Systems

Documents

Transcript of Formal $\\mathcal{F}$ -contexts and Their Induced Implication Rule Systems