Similarity relations in concept lattices 1 - Semantic Scholar
Journal of Logic and Computation Vol.10 No. 6(2000), 823–845.
Similarity relations in concept lattices
1
ˇ ´ RADIM BELOHL AVEK Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Br´ afova 7, 701 03 Ostrava, Czech Republic, [email protected] and Department of Computer Science, Technical University of Ostrava, tˇr. 17. listopadu, 708 33 Ostrava-Poruba, Czech Republic
Abstract. Studied is the issue of similarity relations in fuzzy concept lattices. Fuzzy concepts and fuzzy concept lattices represent a formal approach to the modeling of non-sharp (fuzzy) concepts and conceptual structures in the sense of traditional (PortRoyal) logic. Applications of concept lattices are in representation of conceptual knowledge and in conceptual analysis of (fuzzy) data. Similarity relations are defined and considered on three levels: similarity of objects (and similarity of attributes), similarity of concepts, and similarity of concept lattices. We show a way to factorize (simplify) concept lattices by the similarity of concepts. Also shown is how to reduce the computation of the similarity relations.
Key words: concept, similarity, fuzzy logic, residuated lattice, concept lattice, tolerance
1
Introduction
Human thinking is often identified with reasoning with concepts. The fundamental role of concepts in reasoning is reflected in their pivotal role in the early development of logic. The learning on concepts constitutes one of the three parts of traditional (or Port-Royal) logic [1]. On the intuitive level, the formation of human concepts (like MAMMAL, RED APPLE etc.) is a typical example of what is meant by information granulation [22]. Concepts are thought of as granules of information which (on a higher level of abstraction) can be taken as units for reasoning. A formal theory of concepts and conceptual structures (so called concept lattices) inspired by the traditional logic has been initiated by Wille in [18]. The theory of concept lattices is nowadays a well-elaborated one with direct applications in conceptual data analysis and conceptual knowledge representation (see e.g. [7] or the survey [20]). A generalization of this theory (Wille’s theory is, in fact, a theory of sharp 1 Supported
by grant No. 201/99/P060 of the GA CR.
1
(i.e. two-valued) concepts) from the point of view of fuzzy logic (graded truth approach [10, 11]) has been presented in [2, 3, 4]. The aim of this paper is to study the similarity phenomenon in fuzzy conceptual structures. The motivation of our study is twofold: first, the study of similarities (on various levels) is apparently natural in the context of conceptual structures and, second, it is similarity due to which a reduction of complexity (of conceptual structures, in our case) can be attained (cf. the principle of incompatibility [13, pp. 329–330]). The organization of the paper is as follows. In Section 2 we recall the fundamental notions and results. Section 3 is devoted to the study of similarity relations. Section 4 contains an illustrative example.
2
Fuzzy concepts and concept lattices
By traditional (or Port-Royal) logic [1, 16]) a concept is determined by its extent, a collection of all objects covered by the concept, and its intent, a collection of attributes (properties) covered by the concept. Thus, e.g., the extent of the concept DOG consists of all dogs (or living dogs at a fixed time at a fixed possible world to avoid the philosophical problems) while the intent of DOG consists of all attributes of all dogs (i.e. “to bark”, “to be a mammal”, “to have four extremities” etc.). The primary relation appearing on concepts is that of hierarchical ordering. For instance, the concepts DOG and CAT are not comparable w.r.t. to conceptual hierarchy, while the concept MAMMAL is more abstract (bigger in the hierarchical ordering) than both DOG and CAT. A characteristic feature of empirical concepts (e.g. BIG DOG) is that the extent and the intent are non-sharp (fuzzy) collections (some dogs are bigger than others, various dogs belong to the extent to various (not only two, i.e. fully yes or fully no) degrees). In [2, 3, 4] a theory of fuzzy concepts and hierarchical structures of fuzzy concepts (fuzzy concept lattices) in the sense of traditional logic has been initiated as a generalization of the theory developed by Wille’s group (e.g. [7, 18, 20]). The point of generalization is the non-sharpness (fuzziness) of concepts which is not taken into account in Wille’s theory. Recall that the main purpose of fuzzy logic and fuzzy set theory [13] is to model non-sharp (fuzzy) phenomena. Formally, instead of the two-element Boolean algebra, a more general structure of truth values is employed. In [8, 9], the author proposed to use a structure of a complete residuated lattice which (as turned out later on) plays an important role in fuzzy logic in narrow sense [12, 10, 14]. Definition 1 A complete residuated lattice is an algebra L = hL, ∧, ∨, ⊗, →, 0, 1i such that hL, ∧, ∨, 0, 1i is a complete lattice with the least element 0 and the greatest element 1, hL, ⊗, 1i is a commutative monoid, i.e. ⊗ is commutative, associative, and x ⊗ 1 = x holds holds for each x ∈ L, and ⊗, → form an adjoint pair, i.e. x ⊗ y ≤ z iff x ≤ y → z holds for all x, y, z ∈ L. 2
(1)
Residuated lattices have been introduced in [17]. In each residuated lattice it holds that x ≤ y implies x ⊗ z ≤ y ⊗ z, and x ≤ y implies z → x ≤ z → y (isotonicity) and x → z ≥ y → z (antitonicity in the first argument). The operation ⊗ is thus a t-norm (see e.g. [10, 13]), → is called residuum. In the following we will deal with complete residuated lattices, i.e. hL, ∧, ∨, 0, 1i is assumed to be a complete lattice. The t-norm ⊗ and the residuum → areWintended for modeling of the conjunction V and implication, respectively. Supremum ( ) and infimum ( ) are intended for modeling of the general and existential quantifier, respectively. A semantically complete first-order many-valued logic with semantics defined over complete residuated lattices can be found in [12]. Several special classes of residuated lattices serve as special structures of semantically complete logical calculi (for details see [10]). The most studied and applied set of truth values is the real interval [0, 1] with a ∧ b = min(a, b), a ∨ b = max(a, b), and with three important pairs of adjoint operations: the Lukasiewicz one (a ⊗ b = max(a + b − 1, 0), a → b = min(1 − a + b, 1)), G¨ odel one (a ⊗ b = min(a, b), a → b = 1 if a ≤ b and = b else), and product one (a⊗b = a·b, a → b = 1 if a ≤ b and = b/a else). For the role of these “building stones” in fuzzy logic see [10]. Another important set of truth values is the set (the ordering determines the complete lattice structure) {a0 = 0, a1 , . . . , an = 1} (a0 < · · · < an ). Two t-norms are often considered: ak ⊗ al = amax(k+l−n,0) and the corresponding → given by ak → al = amin(n−k+l,n) (Lukasiewicz) and ak ⊗ al = amin(k,l) and the corresponding → given by ak → al = 1 if k ≤ l and al else (G¨ odel). A special case of the latter algebras is the Boolean algebra 2 of classical logic with the support 2 = {0, 1}. It may be easily verified that the only t-norm on {0, 1} is the classical conjunction operation ∧, i.e. a ∧ b = 1 iff a = 1 and b = 1, which implies that the only residuum operation is the classical implication operation →, i.e. a → b = 0 iff a = 1 and b = 0. Note that each of the preceding residuated lattices is complete. The following identities of complete residuated lattices will be needed: a a
= ≤
a ⊗ (a → b) ≤ (a ⊗ b) → c = _ a⊗ bj = j∈J
a⊗
^
bj
≤
j∈J
_
j∈J
^
j∈J
(2) (3)
b a → (b → c) = b → (a → c) _ (a ⊗ bj )
(4) (5)
^
(a ⊗ bj )
(7)
^
(aj → b)
(8)
^
(a → bj )
(9)
(6)
j∈J
j∈J
aj → b =
a→
1→a (a → b) → b
bj
=
j∈J
j∈J
A fuzzy set (or L-set) A in a universe set X is a mapping A : X → L. The value A(x) ∈ L is interpreted as the truth value of “the element x belongs to A”. The set of all fuzzy sets in X is denoted by LX . For A1 , A2 ∈ LX we write A1 ⊆ A2 iff A1 (x) ≤ A2 (x) for all x ∈ X. Similarly, a binary fuzzy relation R between X and Y is a mapping R : X × Y → L. 3
We now recall the basic notions of fuzzy concept lattices. The primary notion is that of a fuzzy context (L-context): it is a triple hG, M, Ii, where G and M are sets interpreted as the set of objects (G) and the set of attributes (M ) to which we restrict our attention, and I ∈ LG×M is a fuzzy relation between G and M . The value I(g, m) ∈ L is interpreted as the truth value of the fact “the object g ∈ G has the attribute m ∈ M ”. In accordance to the Port-Royal definition, a (formal) fuzzy concept (L-concept) is a pair hA, Bi, A ∈ LG , B ∈ LM , A plays the role of the extent (fuzzy set of objects which determine the concept), B plays the role of the intent (fuzzy set of attributes which determine the concept), such that: (a) B is the collection of all attributes shared by all the objects of A and (b) A is the collection of all objects having all the attributes of B. There are therefore two fundamental operators: ↑ which assigns to each fuzzy set A ∈ LG of objects the fuzzy set A↑ ∈ LM of all the attributes which are common to all objects of A, and ↓ which assigns to each fuzzy set B ∈ LM of attributes the fuzzy set B ↓ ∈ LG of all the objects which which share all the attributes of B). Rewriting these linguistic descriptions on the semantical level of fuzzy logic we get formal definitions ^ A↑ (m) = (A(g) → I(g, m)) (10) g∈G
and B ↓ (g) =
^
(B(m) → I(g, m))
(11)
m∈M
which play the fundamental role. To emphasize the dependence on I, the mappings ↑ and ↓ will also be denoted by ↑I and ↓I . A (formal) fuzzy concept is therefore a pair hA, Bi ∈ LG × LM (of extent A and intent B) such that A↑ = B and B ↓ = A. Denote B (G, M, I) = {hA, Bi ∈ LG × LM | A↑ = B, B ↓ = A} the set of all fuzzy concepts given by the fuzzy context hG, M, Ii. The conceptual hierarchy discussed above is modeled by the relation ≤ defined on B (G, M, I) by hA1 , B1 i ≤ hA2 , B2 i iff A1 ⊆ A2 (iff B1 ⊇ B2 ). (12) The set B (G, M, I) equipped by ≤ is called a fuzzy concept lattice (L-concept lattice). The following theorem describes the hierarchy and characterizes fuzzy concept lattices. Theorem 2 ([4]) (1) The set B (G, M, I) is under ≤ a complete lattice where the suprema and infima are given by ^ \ [ hAj , Bj i = h Aj , ( Bj )↓↑ i , (13) j∈J
_
j∈J
j∈J
hAj , Bj i =
h(
[
j∈J
j∈J
↑↓
Aj ) ,
\
Bj i .
(14)
j∈J
(2) Moreover, an arbitrary complete lattice V = hV, ∧, ∨i is isomorphic to some B (G, M, I) iff there are mappings γ : G × L → V , µ : M × L → V such that 4
γ(G, L) is
W V -dense in V, µ(M, L) is -dense in V;
α ⊗ β ≤ I(g, m) iff γ(g, α) ≤ µ(m, β). W V Note that V ′ ⊆ V is -dense ( -dense) in V if each v ∈ V is the supremum (infimum) of some subset of V ′ . Notice that the lattice structure of concepts is very natural: to each set of concepts there is a concept which is their direct generalization (supremum) and a concept which is their direct specialization (infimum). In this perspective, the original (crisp) concepts lattices [18] are exactly L-concept lattices where L = {0, 1} is the set of truth values of classical logic. Given a context (i.e. data matrix of logical data obtained from experts), the concept lattice reveals the conceptual structure hidden in the context and provides us with formalism for representing conceptual knowledge [18, 20, 7, 4]. If the information in the context is fuzzy, the fuzzy conceptual data analysis is an appropriate tool for handling it. Fuzzy concept lattice represents a conceptual granulation both of the set of objects and attributes equipped by a hierarchical order.
3 3.1
Concepts and similarity Similarity relations
A crucial role in the way humans regard the world is played by the similarity phenomenon. In fuzzy set theory, similarity phenomenon is approached via so called similarity relations. By a ⊗-similarity relation (or fuzzy ⊗-equivalence relation, Lvalued global equality) [13, 12, 21] on a universe U it is meant a binary fuzzy relation E satisfying the following properties for all x, y, z ∈ U : E(x, x)
= 1
E(x, y) = E(y, x) E(x, y) ⊗ E(y, z) ≤ E(x, z).
(15) (16) (17)
Properties (15), (16), and (17) are called reflexivity, symmetry, and transitivity, respectively. The ⊗-similarity class of x ∈ U is the fuzzy set [x]E ∈ LU given by [x]E (y) = E(x, y) for each y ∈ U , i.e. it is a collection of elements similar to x. A fuzzy set A ∈ LG is said to be extensional w.r.t. E if for every x, y ∈ U it holds A(x) ⊗ E(x, y) ≤ A(y), i.e. if with each its element x, A contains all the elements similar to x. In this case, E is also said to be compatible with A. Nonextensional fuzzy sets are not compatible with the underlying similarity relation. It is easily seen that in the crisp case, i.e. L = {0, 1}, similarity relations are equivalence relations. For the study of similarity phenomenon, the crisp case is a degenerate one and non-interesting—two elements x and y may be “fully similar” (E(x, y) = 1) or “fully dissimilar” (E(x, y) = 0). Note that to better grasp the similarity in the crisp case, tolerance relations (i.e. reflexive and symmetric relations) are used instead [5]. Tolerance relations may be considered also from the point of view of fuzzy approach, i.e. requiring only (15) and (16). We will not follow this way explicitly. However, most of the results proved below hold also for reflexive and symmetric fuzzy relations.
5
To be able to model the equivalence of truth values we have at disposal the so called biresiduum (or biimplication) [12, 14] operation ↔ defined by a ↔ b = (a → b) ∧ (b → a). The following lemma will be useful in the following considerations. Lemma 3 Let E be a ⊗-similarity on U , S = {Ai ∈ LU | i ∈ I} be a family of fuzzy sets. (1) E is the largest ⊗-similarity relation compatible with all [x]E . (2) The relation ES defined by ^ ES (x, y) = (Ai (x) ↔ Ai (y)) (18) i∈I
is the largest ⊗-similarity relation compatible with all Ai ∈ S. Moreover, Ai (x) = 1 implies [x]ES ⊆ Ai . Proof (1) is an immediate consequence of (2) provided E = E{[x]E | x∈U} . Denote V E ′ = E{[x]E | x∈U} . We have E(y, z) ≤ E ′ (y, z) = x∈U (E(x, y) ↔ E(x, z)) iff for each x ∈ U it holds E(y, z) ≤ E(x, y) ↔ E(x, z) = (E(x, y) → E(x, z)) ∧ (E(x, z) → E(x, y)) iff both E(x, y) ≤ E(x, y) → E(x, z) and E(x, y) ≤ E(x, z) → E(x, y) which is evident by applying the adjointness and transitivity. On the other hand, taking x = y we get E ′ (y, z) ≤ E(y, y) ↔ E(y, z) = E(y, z), i.e. E = E ′ proving (1). Prove (2). We have to check that ES satisfies (15)–(17). (15) follows from a ↔ a = 1 and (16) follows from a ↔ b = b ↔ a. Furthermore, ES (x, y) ⊗ ES (y, z) = ^ ^ = (Ai (x) ↔ Ai (y)) ⊗ (Aj (y) ↔ Aj (z)) ≤
(19)
^ ^ Ai (x) ↔ Ai (y) ⊗ Aj (y) ↔ Aj (z) ≤
(20)
^ ^
(21)
i∈I
≤
j∈I
i∈I
≤
j∈I
i∈I j∈I
≤
Ai (x) ↔ Ai (y) ⊗ Aj (y) ↔ Aj (z) ≤
^ Ai (x) ↔ Ai (y) ⊗ Ai (y) ↔ Ai (z) =
i∈I
=
^ n
i∈I
⊗ ≤
n
(22)
o Ai (y) → Ai (z) ∧ Ai (z) → Ai (y) ≤
^ n
i∈I
o Ai (x) → Ai (y) ∧ Ai (y) → Ai (x) ⊗
o Ai (x) → Ai (y) ⊗ Ai (y) → Ai (z) ∧
n o ∧ Ai (x) → Ai (y) ⊗ Ai (z) → Ai (y) ∧ n o ∧ Ai (y) → Ai (x) ⊗ Ai (y) → Ai (z) ∧ 6
(23)
n o ∧ Ai (y) → Ai (x) ⊗ Ai (z) → Ai (y) ≤ o ^ n ≤ Ai (x) → Ai (y) ⊗ Ai (y) → Ai (z) ∧
(24)
i∈I
n o ∧ Ai (z) → Ai (y) ⊗ Ai (y) → Ai (x) ≤ ^ ≤ Ai (x) → Ai (z) ∧ Ai (z) → Ai (x) =
(25)
i∈I
=
^ Ai (x) ↔ Ai (z) = ES (x, z) ,
i∈I
V hence (17) holds. In (19)–(20), (20)–(21) and (22)–(23) we used the fact a ⊗ i∈I bi ≤ V i∈I (a ⊗ bi ). In (24)–(25) we used (a → b) ⊗ (b → c) ≤ a → c. The extensionality of each Ai ∈ S follows by Ai (x) ⊗ ES ≤ Ai (x) ⊗ (Ai (x) → Ai (y)) ≤ Ai (y). If all Ai are extensional to another ⊗-similarity E, then Ai (x) ⊗ E(x, y) ≤ Ai (y), hence E(x, y) ≤ Ai (x) → Ai (y), and, similarly, E(x, y) ≤ Ai (y) →VAi (x), i.e. E(x, y) ≤ Ai (x) ↔ Ai (y) hold for all x, y ∈ U . We conclude E(x, y) ≤ i∈I (Ai (x) ↔ Ai (y)) = ES (x, y), i.e. ES is the largest one. Finally, if Ai (x) = 1 then [x]ES (y) = ES (x, y) ≤ Ai (x) ↔ Ai (y) = 1 ↔ Ai (y) = Ai (y), i.e. [x]ES ⊆ Ai . The proof is finished. Notice that for the crisp case (i.e. L = {0, 1}), ES is a crisp equivalence relation— two elements of the universe are equivalent iff there is no set of the family which separates them.
3.2
Similarity of objects and attributes
We are going to propose a way to measure similarity of objects and similarity of attributes of a given L-context. This similarity will be induced by the structure of L-concepts determined by the L-context. We prove that the similarity may be determined directly from the L-context which is relevant from the computational point of view. We are given (in some sense relevant) objects (elements of G) and their (observed) attributes (elements of M ). The given L-context gives a rise to a complete lattice of all induced L-concepts. The idea of conceptual classification leads to the use of the induced conceptual structure B (G, M, I) to define similarity relations on G and on M. Consider the problem of similarity of objects. Informally, two objects g1 , g2 ∈ G are similar if they cannot be separated by any concept, i.e. if for each concept c it holds that g1 belongs to the extent of c iff g2 belongs to the extent of c. This leads G to the following definition of a relation EB(G,M,I) ∈ LG×G : G EB(G,M,I) (g1 , g2 ) =
^
hA,Bi∈B(G,M,I)
A(g1 ) ↔ A(g2 ) .
(26)
G The relation EB(G,M,I) will be called induced (by B (G, M, I)) similarity on G. By Lemma 3 we immediately get the following statement.
7
G Theorem 4 The relation EB(G,M,I) is the largest ⊗-similarity relation on G compatible with the extents of all concepts of B (G, M, I).
From the computational point of view (we always assume both G and M to be finite if algorithmic aspects are concerned), the foregoing definition leads to the folG lowing algorithm for computing the similarity relation EB(G,M,I) : Take an L-context, generate all the L-concepts of B (G, M, I) and determine the similarity of each pair hg1 , g2 i ∈ G × G by (26). The L-concept lattice may be, however, quite extensive. This poses the question whether the computational cost can be reduced. We give a (exact) solution which reduces the computational costs significantly. Define a relation G EhG,M,Ii ∈ LG×G by G EhG,M,Ii (g1 , g2 ) =
^
(I(g1 , m) ↔ I(g2 , m)) .
(27)
m∈M G EhG,M,Ii (g1 , g2 ) may be obtained from the L-context hG, M, Ii computing |M | times the operation ↔. Using Lemma 3 (put X = G, I = M , Ai (g) = I(g, m)) we get G Theorem 5 The relation EhG,M,Ii is the largest ⊗-similarity relation on G compatG ible with all I( , m) ∈ L , m ∈ M .
The following theorem solves the problem of finding an efficient procedure for G computing the similarity relation EB(G,M,I) . Theorem 6 Let hG, M, Ii be a L-context. Then for the similarity relations defined by (27) and (26) it holds G G EB(G,M,I) = EhG,M,Ii . (28) V V Proof We show m∈M (I(g1 , m) ↔ I(g2 , m)) = hA,Bi∈B(G,M,I) A(g1 ) ↔ A(g2 ) by checking both inequalities. Consider the concept hAm , Bm i = h{ 1 m}↓ , { 1 m}↓↑ i for m ∈ M . For each g ∈ G it holds ^ Am (g) = { { 1 m}(m′ ) → I(g, m); m′ ∈ M } = =
1 → I(g, m) = I(g, m) .
From {hAm , Bm i | m ∈ M } ⊆ B (G, M, I) and from the properties of infimum it follows that ^ ^ A(g1 ) ↔ A(g2 ) ≤ Am (g1 ) ↔ Am (g2 ) = m∈M
hA,Bi∈B(G,M,I)
=
^
(I(g1 , m) ↔ I(g2 , m))
m∈M
for all g1 , g2 ∈ G, proving the first inequality. Conversely, ^ ^ (I(g1 , m) ↔ I(g2 , m)) ≤ m∈M
hA,Bi∈B(G,M,I)
8
A(g1 ) ↔ A(g2 )
iff for each hA, Bi ∈ B (G, M, I) it holds ^ (I(g1 , m) ↔ I(g2 , m)) ≤ m∈M
≤
A(g1 ) ↔ A(g2 ) = (A(g1 ) → A(g2 )) ∧ ((A(g2 ) → A(g1 ))
which holds iff both ^
(I(g1 , m) ↔ I(g2 , m)) ≤ A(g1 ) → A(g2 )
(29)
^
(I(g1 , m) ↔ I(g2 , m)) ≤ A(g2 ) → A(g1 )
(30)
m∈M
and
m∈M
hold. (29) holds iff A(g1 ) ⊗
^
(I(g1 , m) ↔ I(g2 , m)) ≤ A(g2 ).
m∈M
Since hA, Bi ∈ B (G, M, I), i.e. A = A↑↓ , the last inequality means ^ ^ (A↑ (m) → I(g1 , m)) ⊗ (I(g1 , m) ↔ I(g2 , m)) ≤ m∈M
^
≤
m∈M
↑
(A (m) → I(g2 , m))
m∈M
which holds iff for each m′ ∈ M we have ^ ^ (A↑ (m) → I(g1 , m)) ⊗ (I(g1 , m) ↔ I(g2 , m)) ≤ m∈M ↑ ′
m∈M
′
≤ A (m ) → I(g2 , m ), i.e. A↑ (m′ ) ⊗
^
(A↑ (m) → I(g1 , m)) ⊗
m∈M
^
(I(g1 , m) ↔ I(g2 , m))
m∈M
≤ I(g2 , m).
(31)
Evidently, by (4), A↑ (m′ ) ⊗
^
(A↑ (m) → I(g1 , m)) ⊗
m∈M ↑ ′
^
(I(g1 , m) ↔ I(g2 , m)) ≤
m∈M ′
A↑ (m′ ) ⊗ (A (m ) → I(g1 , m′ )) ⊗ (I(g1 , m ) → I(g2 , m′ )) ≤ I(g2 , m′ ), thus (31) and therefore also (29) hold. (30) may be proved analogously. The second inequality and hence also the theorem is proved. From the foregoing theorem we have also the following consequence which is in accordance with our intuition: If we are given family of (elementary) properties (attributes) of objects and consider the structure of concepts which is given by these 9
properties then the similarity among objects considered w.r.t. the structure of concepts is the same as the similarity w.r.t. to only the basic properties. In a completely analogous way we may get the following results for the similarity relations on M . M M Theorem 7 For an L-context hG, M, Ii, the relations EB(G,M,I) and EhG,M,Ii defined by ^ M EB(G,M,I) (m1 , m2 ) = (B(m1 ) ↔ B(m2 )) hA,Bi∈B(G,M,I)
and M EhG,M,Ii (m1 , m2 ) =
^
(I(g, m1 ) ↔ I(g, m2 )) .
g∈G M M are ⊗-similarity relations on M and it holds EB(G,M,I) = EhG,M,Ii . They are the largest ⊗-similarity relations compatible with the intents of all concepts of B (G, M, I).
3.3
Similarity of concepts
The next level on which the similarity phenomenon will be considered is the level of concepts. Observe first the following fact. Lemma 8 For any universe U , the relation E on LU given for any A1 , A2 ∈ LU by ^ E(A1 , A2 ) = (A1 (x) ↔ A2 (x)) x∈U
is the largest ⊗-similarity relation on LU such that A1 (x) ⊗ E(A1 , A2 ) ≤ A2 (x) holds for each x ∈ U , A1 , A2 ∈ LU . Proof Putting I = U , X = LU , Ai (x) = x(i) for x ∈ LU , i ∈ U , the assertion is a direct consequence of Lemma 3. In the following, it will be clear what universe U the relation E concerns. Consider first the relations E Ext and E Int on B (G, M, I), call them induced similarity by extents and induced similarity by intents, respectively: ^ E Ext (hA1 , B1 i, hA2 , B2 i) = E(A1 , A2 ) = (A1 (g) ↔ A2 (g)), g∈G
E Int (hA1 , B1 i, hA2 , B2 i) = E(B1 , B2 ) =
^
(B1 (m) ↔ B2 (m)).
m∈M
Lemma 8 gives immediately the following statement. Theorem 9 E Ext and E Int are the largest ⊗-similarity relations on B (G, M, I) such that A1 (g)⊗E Ext (hA1 , B1 i, hA2 , B2 i) ≤ A2 (g) and B1 (m)⊗ E Int (hA1 , B1 i, hA2 , B2 i) ≤ B2 (m) hold for every g ∈ G, m ∈ M , hA1 , B1 i, hA2 , B2 i ∈ B (G, M, I). To answer the question of how the relations E Ext and E Int are related, we derive some preliminary results. The next lemma states that the operators ↑ and ↓ preserve similarity. 10
Lemma 10 Let hG, M, Ii be an L-context, A1 , A2 ∈ LG , B1 , B2 ∈ LM . Then it holds E(A1 , A2 ) ≤ E(A↑1 , A↑2 ) and E(B1 , B2 ) ≤ E(B1↓ , B2↓ ). Proof We prove only E(A1 , A2 ) ≤ E(A↑1 , A↑2 ), the second part may be obtained symmetrically. ^ (A1 (g) ↔ A2 (g)) = E(A1 , A2 ) ≤ g∈G
^
≤ E(A↑1 , A↑2 ) =
m∈M
A↑1 (m) ↔ A↑2 (m)
holds iff for each m ∈ M it holds ^ (A1 (g) ↔ A2 (g)) ≤ g∈G
≤ A↑1 (m) ↔ A↑2 (m) = A↑1 (m) → A↑2 (m) ∧ A↑2 (m) → A↑1 (m) which holds iff the left side of the inequality is less or equal than both members of the right side which are connected by ∧. We check only the first one of these inequalities, i.e. ^ (A1 (g) ↔ A2 (g)) ≤ A↑1 (m) → A↑2 (m) g∈G
By adjunction, this holds iff A↑1 (m) ⊗
^
(A1 (g) ↔ A2 (g)) ≤ A↑2 (m)
g∈G
i.e.
^
g∈G
A1 (g) → I(g, m) ⊗
^
(A1 (g) ↔ A2 (g)) ≤
g∈G
^
A2 (g) → I(g, m)
g∈G
which holds iff for each g ′ ∈ G the inequality ^ ^ (A1 (g) ↔ A2 (g)) ⊗ A1 (g) → I(g, m) ≤ A2 (g ′ ) → I(g ′ , m) g∈G
g∈G
holds. The last inequality is equivalent (by adjunction) to ^ ^ A2 (g ′ ) ⊗ (A1 (g) ↔ A2 (g)) ⊗ A1 (g) → I(g, m) ≤ I(g ′ , m) g∈G
g∈G
which holds because A2 (g ′ ) ⊗
^
g∈G
≤
′
(A1 (g) ↔ A2 (g)) ⊗
^
g∈G
′
′
A1 (g) → I(g, m) ≤
A2 (g ) ⊗ (A2 (g ) → A1 (g )) ⊗ (A1 (g ′ ) → I(g ′ , m)) ≤ I(g ′ , m)
by applying twice the rule (4). 11
The following corollary is immediate. ↑↓ Corollary 11 Under the conditions of Lemma 10, it holds E(A1 , A2 ) ≤ E(A↑↓ 1 , A2 ) and E(B1 , B2 ) ≤ E(B1↓↑ , B2↓↑ ).
Note that even without Lemma 10 we have due to the properties of E that ↑↓ ↑↓ ↑↓ ↑↓ E(A↑↓ 1 , A1 ) ⊗ E(A1 , A2 ) ⊗ E(A2 , A2 ) ≤ E(A1 , A2 ), i.e. if A1 and A1 , A1 and ↑↓ ↑↓ ↑↓ A2 , A2 and A2 are pairwise similar then also A1 and A2 are similar. Corollary 11 asserts a stronger result. Corollary 12 Under the conditions of Lemma 10, it holds E(A1 , A2 )⊗E(A1 , A↑↓ 1 )≤ ↓↑ ↓↑ E(A1 , A↑↓ ) and E(B , B ) ⊗ E(B , B ) ≤ E(B , B ). 1 2 1 1 2 1 2 ↑↓ ↑↓ ↑↓ ↑↓ Proof E(A1 , A2 ) ⊗ E(A1 , A↑↓ 1 ) ≤ E(A1 , A2 ) ⊗ E(A1 , A1 ) ≤ E(A1 , A2 ). The second part may be proved analogously.
The following result shows that the similarities of concepts by extents and intents are equal. Theorem 13 For any L-context hG, M, Ii it holds E Ext = E Int . Proof Let hA1 , B1 i, hA1 , B1 i ∈ B (G, M, I), i.e. A↑i = Bi and Bi↓ = Ai for i = 1, 2. By Lemma 10 we get E Ext (hA1 , B1 i, hA2 , B2 i) = E(A1 , A2 ) ≤ ≤ E(A↑1 , A↑2 ) = E(B1 , B2 ) = E Int (hA1 , B1 i, hA2 , B2 i), and analogously, E Int (hA1 , B1 i, hA2 , B2 i) ≤ E Ext (hA1 , B1 i, hA2 , B2 i). To sum up, E Ext (hA1 , B1 i, hA2 , B2 i) = E Int (hA1 , B1 i, hA2 , B2 i). We will therefore write E instead of E Ext and E Int and call it the induced similarity on concepts. Compatible similarities and factorization The primary importance of similarity relations in human reasoning is the reduction of the complexity of the outer world at a reasonable price. The complexity is reduced via considering the “collections of similar elements in concern” rather than the particular elements themselves. This is just what is in the general system theory known as the abstraction process by factorization: moving from a given level of abstraction (distinguishability) one level up where the elements are collections of elements of the lower level. Instead of the original system one therefore considers the “system modulo similarity”. The price paid is the loss of precision. Our concern in the following is the reduction of the complexity of the concept lattice by factorization modulo similarity. The concept lattice of a given context represents the overall conceptual structure which can be considerably intricate. To get an insight one has to look for methods for reducing the complexity of the structure. In the twovalued (sharp) case, a considerable attention has been paid to this problem [7, 18]. In the many-valued (fuzzy) case, one would expect methods for gradual reduction of 12
the complexity. The idea is to factorize the concept lattice by appropriate a-cut a E of the similarity E (note that a E = {hc1 , c2 i | a ≤ E(c1 , c2 )} [13]), controlling thus the complexity by a ∈ L. Clearly, the lower a ∈ L, the coarser the factorization. The process of factorization of a system consists in two steps. First, specification of the elements, and, secondly, specification of the structure of the factor system. Since both of the steps are non-standard in our case we will describe them in more detail. In general, algebraic systems can be factorized be congruences, i.e. equivalences compatible with the structure of the system. We deal with conceptual structures which are complete lattices. The a-cut a E is clearly a tolerance relation (i.e. reflexive and symmetric), not transitive in general. Compatible tolerance relations on universal algebras have been extensively studied by Chajda [5]. In general, factorization of algebras by compatible tolerances is not possible. Surprisingly, Cz´edli [6] showed a way to factorize lattices by compatible tolerance relations. The construction has been then used for the factorization of complete lattices (and hence sharp concept lattices) [19]. In the following we describe the construction of the factor lattice of an L-concept lattice by a compatible tolerance relation. Let hG, M, Ii be an L-context. A tolerance relation T on B (G, M, I) is said to be compatible if it is preserved under W W arbitrary ′ ′ suprema and infima, i.e. if hc , c i ∈ T , j ∈ J, implies both h c , j j j j∈J j∈J cj i ∈ T V V ′ ′ and h j∈J cj , j∈J cj i ∈ T for any cj , cj ∈ B (G, M, I), j ∈ J. For a compatible V W tolerance relation T on B (G, M, I) denote cT = hc,c′ i∈T c′ and cT = hc,c′ i∈T c′ . Call [c]T = [cT , (cT )T ] = {c′ ∈ B (G, M, I) | cT ≤ c′ ≤ (cT )T } a block of T and denote B (G, M, I)/T = {[c]T | c ∈ B (G, M, I)}Vthe set V of all blocks. a W Introduce W relation ≤T on B (G, M, I)/T by [c]T ≤T [c′ ]T iff [c]T ≤ [c′ ]T (iff [c]T ≤ [c′ ]T ). The justification of the construction is given by the following statement which follows immediately from [19]. Proposition 14 (1) B (G, M, I)/T is the set of all maximal tolerance blocks, i.e. B (G, M, I)/T = {B ⊆ B (G, M, I) | (B × B ⊆ T )&((∀B ′ ⊃ B)(B ′ × B ′ 6⊆ T ))}. (2) hB (G, M, I)/T, ≤T i is a complete lattice (factor lattice) where suprema and infima are described by _ _ ^ ^ [cj ]T = [ cj ]T and [cj ]T = [( cj )T ]T (32) j∈J
j∈J
j∈J
j∈J
for every cj ∈ B (G, M, I), j ∈ J. Substituing (13) and (14) into (32) we get a more concrete description of the lattice operations. Coming back to the induced similarity E on B (G, M, I), the ultimate question is that of the compatibility of the a-cuts of E. Call a ⊗-similarity relation F on B (G, M, I) compatible if a E is a compatible tolerance relation on B (G, M, I) for each a ∈ L. Notice that for the two-valued (crisp) case the situation is completely uninteresting. Namely, as one easily checks, the only cases are 0 E = B (G, M, I) × B (G, M, I) and 1 E = idB(G,M,I) = {hc, ci | c ∈ B (G, M, I)}. In the first case, B (G, M, I)/0 E = {B (G, M, I)}, i.e. the factor lattice collapsed into a one element lattice, while in the second case, B (G, M, I)/1 E = {{hA, Bi} | hA, Bi ∈ B (G, M, I)}, i.e. B (G, M, I) and B (G, M, I)/1 E are isomorphic. Note that we have in no case to confine ourselves to the induced similarity E. On the other hand, taking into account only ⊗-similarity relations F satisfying A(g) ⊗ 13
F (hA, Bi, hA′ , B ′ i) ≤ A′ (g) (which is quite natural—it reads “object belonging to the extent of some concept belongs also to the extent of any similar concept”) for each g ∈ G, Theorem 9 tells us that E provides the most extensive reduction: for any other F and each a ∈ L, a E is coarser than a F . We will make use of the following lemma.
Lemma 15 For every hAj , Bj i, A′j , Bj′ ∈ B (G, M, I), j ∈ J, it holds ^ ^
^
hAj , Bj i, A′j , Bj′ ), (33) E(hAj , Bj i, A′j , Bj′ ) ≤ E( j∈J
j∈J
^
j∈J
E(hAj , Bj i, A′j , Bj′ )
≤ E(
_
j∈J
hAj , Bj i,
j∈J
_
j∈J
A′j , Bj′ ).
(34)
Proof By the above statements, (33) holds iff ^ ^ ^ Aj , A′j ), E(Aj , A′j ) ≤ E( j∈J
j∈J
i.e.
j∈J
^ ^ ^ ^ ^ ( Aj (g) ↔ A′j (g)) ≤ ( Aj (g)) ↔ ( A′j (g))
j∈J g∈G
g∈G j∈J
j∈J
′
which holds iff for each g ∈ G we have ^ ^ ^ ^ Aj (g) ↔ A′j (g) ≤ ( Aj (g ′ )) ↔ ( A′j (g ′ )) = j∈J g∈G
=
j∈J
(
^
′
Aj (g )) → (
j∈J
^
j∈J
A′j (g ′ ))
∧(
j∈J
^
A′j (g ′ )) → (
j∈J
^
Aj (g ′ )).
j∈J
The last inequality holds iff the left side is less or equal to both of the conjuncts on the right side. As they can be handled analogously, we prove only ^ ^ ^ ^ (Aj (g) ↔ A′j (g)) ≤ ( Aj (g ′ )) ↔ ( A′j (g ′ )) j∈J g∈G
j∈J
j∈J
which is, by adjunction, equivalent to ^ ^ ^ ^ ( Aj (g ′ )) ⊗ (Aj (g) ↔ A′j (g)) ≤ ( A′j (g ′ )) j∈J
j∈J g∈G
j∈J
which holds since (
^
Aj (g ′ )) ⊗
≤
(
^
≤
^ ^ ( A′j (g ′ ) ⊗ (Aj (g ′ ) → A′j (g ′ )) ≤
j∈J
^ ^
(Aj (g) ↔ A′j (g)) ≤
j∈J g∈G
Aj (g ′ )) ⊗
j∈J
^
(Aj (g ′ ) → A′j (g ′ )) ≤
j∈J
j∈J j ′ ∈J
≤
^
(Aj (g ′ ) ⊗ (Aj (g ′ ) → A′j (g ′ )) ≤
j∈J
^
j∈J
14
A′j (g ′ ).
We have proved (33). proved D S (34) can be E symmetrically using the fact W V ↑↓ of Theorem 2 and proceeding by the simij∈J hAj , Bj i = ( j∈J Aj ) , j∈J Bj larity E on the intents. Lemma 15 has an interesting corollary stating that the similarity of any two concepts is less or equal than the similarity of any of them to their direct join or meet. Corollary 16 Let hA1 , B1 i, hA2 , B2 i ∈ B (G, M, I). The following inequalities hold for i = 1, 2: E(hA1 , B1 i, hA2 , B2 i) ≤ E(hA1 , B1 i, hA2 , B2 i) ≤
E(hAi , Bi i, hA1 , B1 i ∧ hA2 , B2 i), E(hAi , Bi i, hA1 , B1 i ∨ hA2 , B2 i).
′ ′
Proof Put J = {x, y}, hAx , Bx i = hAy , By i = hA1 , B1 i, hAx , Bx i = hA1 , B1 i, ′ ′ Ay , By = hA2 , B2 i and apply Lemma 15.
Theorem 17 The induced similarity E on B (G, M, I) is compatible. If a ∈ L is ⊗-idempotent (i.e. a ⊗ a = a) then a E is, moreover, transitive, i.e. a congruence relation on B (G, M, I). Proof The first part follows immediately from (33)
fact that if V and (34) by the a ≤ E(hAj , Bj i, A′j , Bj′ ) for j ∈ J then also a ≤ j∈J E(hAj , Bj i, A′j , Bj′ ). The second part follows from the evident fact that if a is ⊗-idempotent and a ≤ b, c then a ≤ b ⊗ c. Remark 18 Theorem 17 and the above described construction yield a method for factorizing any L-concept lattice B (G, M, I) by any a-cut a E of the induced similarity E. It is worth to notice that that the similarity E is defined “internally”, i.e. it is not supplied from the outside. Remark 19 If L the algebra of intuitionistic logic (Heyting algebra) or the algebra of G¨ odel logic [10] then each a-cut of E is indeed a congruence relation. Next we formulate a statement concerning the relation of the similarity and the hierarchy of concepts which shows that the farther are the concepts in the hierarchy, the less similar they are. Theorem 20 Let for hAi , Bi i ∈ B (G, M, I), i = 1, 2, 3, it holds hA1 , B1 i ≤ hA2 , B2 i ≤ hA3 , B3 i. Then E(hA1 , B1 i, hA3 , B3 i) ≤ E(hA1 , B1 i, hA3 , B3 i) ≤
E(hA1 , B1 i, hA2 , B2 i), E(hA2 , B2 i, hA3 , B3 i).
Proof By the assumptions, i.e.A1 (g) ≤ A2 (g) ≤ A3 (g) for all g ∈ G, V and by the antitonicity of → in the first argument we have E(hA , B i, hA , B i) = 1 1 3 3 g∈G (A3 (g) → V A1 (g)) ≤ g∈G (A2 (g) → A1 (g)) = E(hA1 , B1 i, hA2 , B2 i). The second part may be obtained symmetrically.
15
3.4
Similarity of concept lattices
Finally, we consider similarity of concept lattices. A natural way to define the similarity degree of two concept lattices over the sets G and M is based on the following intuition. Concept lattices B (G, M, I1 ) and B (G, M, I2 ) are similar iff for each concept c1 ∈ B (G, M, I1 ) there is a concept c2 ∈ B (G, M, I2 ) such that c1 and c2 are similar and, conversely, for each concept c2 ∈ B (G, M, I2 ) there is a concept c1 ∈ B (G, M, I1 ) such that c1 and c2 are similar. In the following we write B1 and B2 instead of B (G, M, I1 ) and B (G, M, I2 ), respectively. According to how the similarity of concepts is measured we distinguish two rules for the definition of the similarity degree of two concept lattices: E ∗ (B (G, M, I1 ), B (G, M, I2 )) = ^ _ = E ∗ (hA1 , B1 i, hA2 , B2 i) ∧
(35)
hA1 ,B1 i∈B1 hA2 ,B2 i∈B2
^
_
E ∗ (hA1 , B1 i, hA2 , B2 i),
hA2 ,B2 i∈B2 hA1 ,B1 i∈B1
∗ ∈ {Ext, Int}, where we put E Ext (hA1 , B1 i, hA2 , B2 i) E Int (hA1 , B1 i, hA2 , B2 i)
= E(A1 , A2 ) = E(B1 , B2 ).
Note that E Ext and E Int correspond to the cases when the similarity of concepts is measured by extents and by intents of concepts, respectively. However, Theorem 13 cannot be applied to show E Ext = E Int because concepts of different concept lattices are considered. We are going to show that all of the above relations are in fact similarity relations, that they are equal, and that they are, moreover, equal to the similarity relation E defined on the set of all contexts by ^ E(hG, M, I1 i, hG, M, I2 i) = E(I1 , I2 ) = I1 (g, m) ↔ I2 (g, m). (36) hg,mi∈G×M
The following corollary follows directly by Lemma 8. Corollary 21 The relation E defined by (36) is the largest ⊗-similarity relation on {hG, M, Ii | I ∈ LG×M } such that I1 (g, m) ⊗ E(I1 , I2 ) ≤ I2 (g, m) holds for every g ∈ G, m ∈ M . We need the following lemmata. Lemma 22 Let hG, M, I1 i, hG, M, I2 i be L-contexts, A ∈ LG , B ∈ LM . Then E(I1 , I2 ) ≤ E(A↑I1 , A↑I2 ) and E(I1 , I2 ) ≤ E(B ↓I1 , B ↓I2 ). Proof Due to symmetry we prove only the first part, i.e. E(I1 , I2 ) ≤ E(A↑I1 , A↑I2 ), 16
i.e. E(I1 , I2 ) ≤
^
(A↑I1 (m) ↔ A↑I2 (m))
m∈M
which holds iff for each m ∈ M E(I1 , I2 ) ≤ A↑I1 (m) ↔ A↑I2 (m) = A↑I1 (m) → A↑I2 (m) ∧ A↑I2 (m) → A↑I1 (m) holds. The last inequality holds iff E(I1 , I2 ) is less or equal to both A↑I1 (m) → A↑I2 (m) and A↑I2 (m) → A↑I1 (m). We check only E(I1 , I2 ) ≤ A↑I1 (m) → A↑I2 (m) which is equivalent to A↑I1 (m) ⊗ E(I1 , I2 ) ≤ A↑I2 (m) =
^
A(g) → I2 (g, m)
g∈G
iff for each g ∈ G A↑I1 (m) ⊗ E(I1 , I2 ) ≤ A(g) → I2 (g, m) iff A(g) ⊗ A↑I1 (m) ⊗ E(I1 , I2 ) ≤ I2 (g, m). We have A(g) ⊗ A↑I1 (m) ⊗ E(I1 , I2 ) = ^ = A(g) ⊗ (A(g ′ ) → I1 (g ′ , m)) ⊗ g′ ∈G
^
(I1 (g ′ , m′ ) ↔ I2 (g ′ , m′ )) ≤
hg′ ,m′ i∈G×M
≤ A(g) ⊗ (A(g) → I1 (g, m)) ⊗ (I1 (g, m) → I2 (g, m)) ≤ I2 (g, m) which had to be proved. ↑
↑
Note that by Lemma 10 and Lemma 22 we have E(A1 , A2 )⊗E(I1 , I2 ) ≤ E(A1I1 , A2I2 ) ↓ ↓ and E(B1 , B2 ) ⊗ E(I1 , I2 ) ≤ E(B1 I1 , B2 I2 ). Lemma 23 For every L-contexts hG, M, I1 i, hG, M, I2 i and ∗ ∈ {Ext, Int} it holds E(hG, M, I1 i, hG, M, I2 i) ≤ E ∗ (B (G, M, I1 ), B (G, M, I2 )). Proof We proceed only for E Ext , the second case is symmetric. We have to prove E(I1 , I2 ) ≤ E Ext (B1 , B2 ), which holds iff both ^ _ E Ext (hA1 , B1 i, hA2 , B2 i) E(I1 , I2 ) ≤ hA1 ,B1 i∈B1 hA2 ,B2 i∈B2
and E(I1 , I2 ) ≤
^
_
E Ext (hA1 , B1 i, hA2 , B2 i)
hA2 ,B2 i∈B2 hA1 ,B1 i∈B1
hold. Due to symmetry we prove only the first inequality which holds iff _ E(I1 , I2 ) ≤ E(A1 , A2 ) hA2 ,B2 i∈B2
17
↓I
↓I
1 2 holds for each hA1 , B1 i ∈ B1 . By Lemma 22 we have D E E(I1 , I2 ) ≤ E(B1 , B1 ) = ↓I ↓I ↓I ↑I E(A1 , B1 2 ), i.e. putting hA2 , B2 i = B1 2 , B1 2 2 we have
↓
_
E(I1 , I2 ) ≤ E(A1 , B1 I2 ) ≤
E(A1 , A2 )
hA2 ,B2 i∈B2
finishing the proof. Lemma 24 For every L-contexts hG, M, I1 i, hG, M, I2 i and ∗ ∈ {Ext, Int} it holds E ∗ (B (G, M, I1 ), B (G, M, I2 )) ≤ E(hG, M, I1 i, hG, M, I2 i). Proof We proceed only for E Ext , the case E Int can be handled analogously. We have to prove ^ E Ext (B1 , B2 ) ≤ E(I1 , I2 ) = I1 (g, m) ↔ I2 (g, m) hg,mi∈G×M
which holds iff E Ext (B1 , B2 ) ≤ I1 (g, m) ↔ I2 (g, m) = (I1 (g, m) → I2 (g, m)) ∧ (I2 (g, m) → I1 (g, m)) holds for every g ∈ G, m ∈ M . The inequality holds iff both E Ext (B1 , B2 ) ≤ I1 (g, m) → I2 (g, m) and E Ext (B1 , B2 ) ≤ I2 (g, m) → I1 (g, m) hold. For symmetry we show only the first inequality which is equivalent to I1 (g, m) ⊗ E Ext (B1 , B2 ) ≤ I2 (g, m).
(37)
We have I1 (g, m) ⊗ E Ext (B1 , B2 ) = I1 (g, m)⊗ _ ^ E(A1 , A2 ) ∧ ⊗ hA1 ,B1 i∈B1 hA2 ,B2 i∈B2
≤ I1 (g, m) ⊗ ≤ I1 (g, m) ⊗ ≤ I1 (g, m) ⊗
hA2 ,B2 i∈B2
E({ 1 m}↓I1 , A2 ) ∧
_
E({ 1 m}↓I1 , A2 ) ≤
_
B∈LM
=
≤
hA1 ,B1 i∈B1
E(A1 , { 1 m}↓I2 ) ≤
_
I1 (g, m) ⊗ (
_
B∈LM
^ I1 (g, m) ⊗ ({ 1 m}↓I1 (g) ↔ ( B(m′ ) → I2 (g, m′ )) ≤
_
^ I1 (g, m) ⊗ ({ 1 m}↓I1 (g) → ( B(m′ ) → I2 (g, m′ )) ≤
_
I1 (g, m) ⊗
B∈LM
≤
_
E({ 1 m}↓I1 , B ↓I2 ) = ^
{ 1 m}↓I1 (g ′ ) ↔ B ↓I2 (g ′ )) ≤
g′ ∈G
B∈LM
≤
E(A1 , A2 ) ≤
hA2 ,B2 i∈B2 hA1 ,B1 i∈B1
_
hA2 ,B2 i∈B2
_
^
m′ ∈M
m′ ∈M
B∈LM
18
⊗((
^
m′ ∈M
≤
^ { 1 m}(m′ ) → I1 (g, m′ )) → ( 1 → I2 (g, m′ ))) =
I1 (g, m) ⊗ (I1 (g, m) → (
_
I1 (g, m) ⊗ (I1 (g, m) → I2 (g, m)) ≤ I2 (g, m),
B∈LM
^
m′ ∈M
B∈LM
≤
m′ ∈M
_
I2 (g, m′ )) ≤
i.e. (37) holds. The proof is complete. Theorem 25 For every L-contexts hG, M, I1 i, hG, M, I2 i it holds E(hG, M, I1 i, hG, M, I2 i) = E Ext (B1 , B2 ) = E Int (B1 , B2 ). Therefore, E Ext and E Int are similarity relations on {B (G, M, I) | I ∈ LG×M }. Proof The assertion follows immediately by Corollary 21, Lemma 23, and Lemma 24. Both E Ext and E Int may thus be denoted by E. From the computational point of view, Theorem 25 shows that the computation of the similarity of two concept lattices defined naturally by (35) may be reduced to the computation of the similarity of the corresponding contexts which is usually much more simple. Indeed, the direct computation of E(B1 , B2 ) requires |B1 | · |B2 | · (|G| + |M |) evaluation of ↔ while computing E(I1 , I2 ) requires |G|·|M | evaluation of ↔. Note that usually |G|, |M | ≪ |B (G, M, I)| holds.
4
Example
In this section we present an illustrative example. Consider L with L = {0, 12 , 1} and the Lukasiewicz structure defined on L, cf. Section 2. The context is given by the Tab. 1. The set G contains nine elements (Mercury, . . . , Pluto), the set M contains four attributes (“size small”, . . . , “near to sun”). The corresponding fuzzy concept lattice is depicted in Fig. 1. To get a deeper insight, the elements (i.e. concepts) of the lattice are identified in Tab. 2. The similarity relation E G on G (cf. Theorem 6) is figured in Tab. 3 Consider now the a-cut of the induced similarity on B (G, M, I) for a = 12 , i.e. 1 2 E. The tolerance blocks (which are, in fact, complete sublattices) are depicted in Fig. 2. Notice that each block is a maximal subset of L-concepts which are similar in 1 the degree at least 12 . The corresponding factor lattice B (G, M, I)/ 2 E is depicted in Fig. 3. A few remarks to the examples. First, note that the concepts which were found depend on the fuzzy context which is given by a subjective judgement (e.g. to what degree we consider “Mars is far from sun” to be true). Second, there are apparently natural concepts in the concept lattice (e.g. 14 (“small planet near to sun”)), as well as concepts which “were found” (e.g. 26 (“a planet far from sun which is at least partially large”)). Concept no. 1 is an example of an empirically empty concept. Concepts which do not contain any element in the degree 1 in their extents (e.g. 1, 2, 3) are partially (empirically) empty. 19
Mercury (Me) Venus (V) Earth (E) Mars (Ma) Jupiter (J) Saturn (S) Uranus (U) Neptune (N) Pluto (P)
size small large (ss) (sl) 1 0 1 0 1 0 1 0 0 1 0 1 1 2 1 2
1 2 1 2
1
0
from sun far near (df) (dn) 0 1 0 1 0 1 1 1 2 1 1 2 1 1 2 1 0 1 0 1 0
Table 1: Fuzzy context given by planets and their properties.
c38 c37
35 c
c36
29 c 30 c
c31 32 c
21 c
22 c
23 c
14 c
15 c
16 c
9 c
10 c 5 c
33 c
c24 c25 26 c 17 c
c34 c27 c28
c18 c10
c20
c12
c13
6 c
c7
c8
2 c
c3
c4
c11
c 1 Figure 1: Concept lattice of the context in Tab. 1.
20
no. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.
Me 0 0 0 0
V 0 0 0 0
E 0 0 0 0
1 2
1 2
1 2
0 0 0
0 0 0
0 0 0
1 2 1 2 1 2
1 2 1 2 1 2
1 2 1 2 1 2
0 0 1
0 0 1
0 0 1
1 2 1 2 1 2
1 2 1 2 1 2
1 2 1 2 1 2
0 0 0 1 1
0 0 0 1 1
0 0 0 1 1
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
0 0 1 1
0 0 1 1
0 0 1 1
1 2 1 2 1 2
1 2 1 2 1 2
1 2 1 2 1 2
0 1 1
0 1 1
0 1 1
1 2
1 2
1 2
1
1
1
extent Ma J 0 0 1 0 2 1 0 2 0 0 1 0 2 1 2 1 2
0 1 1 2 1 2 1 2
0 1 1 1 1 2 1 2 1 2 1 2
1 1 1 1 1 2 1 2 1 2 1 2
1 1 1 1 1 2 1 2
1 1 1 1
1 2
S 0 0 1 2
0 0 1 2
0
0
1 2
1 2
0
0
1 2
1 2
0
0
1 2
1 2
1 0
1 0
1 2
1 2
0
0
1 2
1 2
0 1
0 1
1 2 1 2
1 2 1 2
0
0
1 2
1 2
U 0 0 0
N 0 0 0
1 2
1 2
0 0
0 0
1 2 1 2
1 2 1 2
0 0
0 0
1 2 1 2 1 2
1 2 1 2 1 2
0 0
0 0
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
1 0
1 0
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
0 1
0 1
1 2 1 2
1
1 2 1 2
1
1 1 1
1 1 1
1 2
1 2
1 2 1 2 1 2
1 2 1 2 1 2
0 1
0 1
1 2
1 2
1 1 1
1 1 1
1 2
1 2
1 1
1 1
P 0 0 0 0 0 0
ss 1 1
1 2
1
0 0 0 1 2 1 2
1 2
1 2 1 2 1 2
1 2
1 0
1 2
1 2 1 2 1 2
1 1 1 2
0 1
1 2 1 2
1
1 0 0 0
1 2
1 2
1 1 2 1 2
0 1 2 1 2
1 1 2 1 2
1 1 2 1 2
1 1 2
1 1 1
1
1 2
1 2
1 2
1 1 1
1 1
0 0 0
1 1 1 1 1 1
1 2
intent sl df 1 1 1 1 2 1 1 1 1
1 2
1 1 1 1
1 2
1 0 1 2 1 2
1 1 2
1 0 1 2 1 2
0 1 2
1 0 1 2
0 0 0 1 2
0 0
0 1 2 1 2
0 0 0 0 1 2 1 2
0 1 2
0 0 0 0 1 2
0 0 0 0 0
1 2
1 1 1 1 2 1 2 1 2
1 1 0 1 2 1 2 1 2
1 1 1 0 0 1 2 1 2 1 2 1 2
1 1 0 0 1 2 1 2 1 2
1 0 0 1 2
0
Table 2: Fuzzy concepts of the context of Tab. 1.
21
dn 1 1 1 1 2
1 1 1 2 1 2
1 1 1 2 1 2 1 2
1 1 1 2 1 2
0 1 2
0 1 1 2 1 2
0 1 2
0 0 0 1 2
0 1 2
0 0 0 1 2
0 0 0
38 a a a36 a37 35 a a a 29 30 31 32 a 33 a a34 21 a 22 a 23 a a24 a25 26 a a27 a28 a a10 a20 14 a 15 a 16 a 17 q 18 a q q q12 a13 9 10 11 q q q q8 5 6 7 q3 q4 2q 1 q1
38 a a a a37 35 36 a a a a 29 30 31 32 33 a a34 21a 22 a 23 q a24 a25 26 a a27 a28 a a10 a20 14a 15 q 16 q 17 q 18 q q q q12 a13 9 10 11 q q q7 a8 5 6 a3 a4 2q 2 a1
38 a a a a37 35 36 a a a a 29 30 31 32 33 a a34 21 a 22 a 23 a a24 q25 26 a a27 a28 a q10 a20 14 a 15 a 16 a 17 q 18 a q a q12 q13 9 10 11 a q a7 q8 5 6 q3 a4 2a 3 a1
38 a a a36 a37 35 a a a 29 30 31 32 a 33 a a34 21 a 22 a 23 a a24 a25 26 q a27 a28 a a10 q20 14 a 15 a 16 a 17 q 18 a a q q12 a13 9 10 11 a a q q8 5 6 7 a3 q4 2a 4 a1
38 a a a a37 35 36 q a a a 29 30 31 32 33 a a34 21q 22 q 23 q a24 a25 26 a a27 a28 a a10 a20 14q 15 q 16 q 17 q 18 q q q a12 a13 9 10 11 q a a7 a8 5 6 a3 a4 2a 5 a1
38 a a a a37 35 36 a a q a 29 30 31 32 33 a a34 21 a 22 a 23 q a24 q25 26 a a27 a28 a q10 a20 14 a 15 q 16 a 17 q 18 a q a q12 a13 9 10 11 a q a7 a8 5 6 a3 a4 2a 6 a1
38 a a a36 a37 35 a a a 29 30 31 32 q 33 a a34 21 a 22 a 23 q q24 a25 26 q q27 a28 q a10 q20 14 a 15 a 16 q 17 q 18 a a q q12 a13 9 10 11 a a q a8 5 6 7 a3 a4 2a 7 a1
38 a a a a37 36 35 a a a a 29 30 31 32 33 q a34 21a 22 a 23 a a24 q25 26 q a27 q28 a q10 q20 14a 15 a 16 a 17 q 18 a a a q12 q13 9 10 11 a a a7 q8 5 6 a3 a4 2a 8 a1
38 a q a a37 36 35 q a q a 29 30 31 32 33 a a34 21 q 22 a 23 q a24 q25 26 a a27 a28 a a10 a20 14 a 15 q 16 a 17 q 18 a q a a12 a13 9 10 11 a a a7 a8 5 6 a3 a4 2a 9 a1
38 a a q36 a37 35 q q a 29 30 31 32 q 33 a a34 21 a 22 q 23 q q24 a25 26 q a27 a28 a a10 a20 14 a 15 a 16 q 17 q 18 a a q a12 a13 9 10 11 a a a a8 5 6 7 a3 a4 2a 10 a1
38 a a a q37 35 36 a a q q 29 30 31 32 33 q q34 21a 22 a 23 q a24 q25 26 q q27 q28 a q10 q20 14a 15 a 16 a 17 q 18 a a a q12 a13 9 10 11 a a a7 a8 5 6 a3 a4 2a 11 a1
38 q q q q37 35 36 q a q q 29 30 31 32 33 q a34 21 a 22 a 23 q a24 q25 26 q a27 a28 a a10 a20 14 a 15 a 16 a 17 q 18 a a a a12 a13 9 10 11 a a a7 a8 5 6 a3 a4 2a 12 a1
1
Figure 2: Blocks of the tolerance relation 2 E on the concept lattice of Fig. 1.
22
Me V E Ma J S U N P
Me 1
V 1 1
E 1 1 1
Ma 1 2 1 2 1 2
1
J 0 0 0 0 1
S 0 0 0 0 1 1
U 0 0 0 0
N 0 0 0 0
1 2 1 2
1 2 1 2
1
1 2 1 2
1 1
1
Table 3: Similarity relation on objects.
12 f
9
f
5
f
P 0 0 0 0 0 0
f10
f11
6 f
f7
f8
2 f
f3
f4
f1
1
Figure 3: Factor lattice B (G, M, I)/ 2 E.
23
Acknowledgement. The author is indebted to anonymous referees for valuable remarks which helped to improve the paper.
References [1] Arnauld A., Nicole P.: La logique ou l’art de penser. 1662. (Also in German: Die Logik oder die Kunst des Denkens. Darmstadt, 1972). [2] Bˇelohl´ avek R.: Lattices generated by binary fuzzy relations. Int. Conf. on Fuzzy Set Theory and Applications, Liptovsk´ y J´ an (Slovakia), 1998. To appear (1999) in Tatra Mount. Math. Publ. (Special issue on fuzzy set theory and applications). [3] Bˇelohl´ avek R.: Fuzzy Galois connections. Math. Logic Quarterly 45(1999). [4] Bˇelohl´ avek R.: Lattices of fixed points of fuzzy Galois connections. Research report no. 15, IRAFM, University of Ostrava. Submitted (1998). [5] Chajda I.: Algebraic Theory of Tolerance Relations. Palack´ y University Press, Olomouc (Czech Republic), 1995. [6] Czedli G.: Factor lattices by tolerances. Acta Sci. Math. (Szeged) 44(1982), 35– 42. [7] Ganter B., Wille R.: Formal concept analysis. Mathematical Foundations. Springer-Verlag, Berlin, 1999. [8] Goguen J. A.: L-fuzzy sets. Journal Math. Anal. Appl. 18(1967), 145–174. [9] Goguen J. A.: The logic of inexact concepts. Synthese 18(1968-9), 325–373. [10] H´ ajek P.: Metamathematics of Fuzzy Logic. Kluwer, 1998. [11] H´ ajek P.: Ten questions and one problem on fuzzy logic. Annals of Pure and Applied Logic 96(1999), 157–163. [12] H¨ ohle U.: On the fundamentals of fuzzy set theory. Journal Math. Anal. Appl. 201(1996), 786–826. [13] Klir G. J., Yuan B.: Fuzzy Sets and Fuzzy Logic. Theory and Applications. Prentice Hall, 1995. [14] Nov´ ak V., Perfilieva I., Moˇckoˇr J.: Mathematical Principles of Fuzzy Logic. Kluwer, 1999 (to appear). [15] Trillas E., Valverde L.: An inquiry on t-indistinguishability operator. In: Skala H. et al. (Ed.): Aspects of Vagueness. Reidel, Dordrecht, 1984, pp. 231–256. [16] Tugendhat E., Wolf U.: Logisch-semantische Prop¨ adautik. Philipp Reclam jun., Stutgart 1986. [17] Ward M., Dilworth R.P.: Residuated lattices. Trans. AMS 45(1939), 335–354.
24
[18] Wille R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival I.: Ordered Sets. Reidel, Dordrecht, Boston, 1982, 445—470. [19] Wille R.: Complete tolerance relations of concept lattices. In: Eigenthaler G. et al.: Contributions to General Algebra, vol. 3. H¨ older-Pichler-Tempsky, Wien, 1985, pp. 397–415. [20] Wille R.: Concept lattices and conceptual knowledge systems. Computers & Mathematics with Applications 23(1992), 493–515. [21] Zadeh L. A.: Similarity relations and fuzzy orderings. Information Sciences 3(1971), 159–176. [22] Zadeh L. A.: Towards a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets and Systems 90(2)(1997).
25
Similarity relations in concept lattices
1
ˇ ´ RADIM BELOHL AVEK Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Br´ afova 7, 701 03 Ostrava, Czech Republic, [email protected] and Department of Computer Science, Technical University of Ostrava, tˇr. 17. listopadu, 708 33 Ostrava-Poruba, Czech Republic
Abstract. Studied is the issue of similarity relations in fuzzy concept lattices. Fuzzy concepts and fuzzy concept lattices represent a formal approach to the modeling of non-sharp (fuzzy) concepts and conceptual structures in the sense of traditional (PortRoyal) logic. Applications of concept lattices are in representation of conceptual knowledge and in conceptual analysis of (fuzzy) data. Similarity relations are defined and considered on three levels: similarity of objects (and similarity of attributes), similarity of concepts, and similarity of concept lattices. We show a way to factorize (simplify) concept lattices by the similarity of concepts. Also shown is how to reduce the computation of the similarity relations.
Key words: concept, similarity, fuzzy logic, residuated lattice, concept lattice, tolerance
1
Introduction
Human thinking is often identified with reasoning with concepts. The fundamental role of concepts in reasoning is reflected in their pivotal role in the early development of logic. The learning on concepts constitutes one of the three parts of traditional (or Port-Royal) logic [1]. On the intuitive level, the formation of human concepts (like MAMMAL, RED APPLE etc.) is a typical example of what is meant by information granulation [22]. Concepts are thought of as granules of information which (on a higher level of abstraction) can be taken as units for reasoning. A formal theory of concepts and conceptual structures (so called concept lattices) inspired by the traditional logic has been initiated by Wille in [18]. The theory of concept lattices is nowadays a well-elaborated one with direct applications in conceptual data analysis and conceptual knowledge representation (see e.g. [7] or the survey [20]). A generalization of this theory (Wille’s theory is, in fact, a theory of sharp 1 Supported
by grant No. 201/99/P060 of the GA CR.
1
(i.e. two-valued) concepts) from the point of view of fuzzy logic (graded truth approach [10, 11]) has been presented in [2, 3, 4]. The aim of this paper is to study the similarity phenomenon in fuzzy conceptual structures. The motivation of our study is twofold: first, the study of similarities (on various levels) is apparently natural in the context of conceptual structures and, second, it is similarity due to which a reduction of complexity (of conceptual structures, in our case) can be attained (cf. the principle of incompatibility [13, pp. 329–330]). The organization of the paper is as follows. In Section 2 we recall the fundamental notions and results. Section 3 is devoted to the study of similarity relations. Section 4 contains an illustrative example.
2
Fuzzy concepts and concept lattices
By traditional (or Port-Royal) logic [1, 16]) a concept is determined by its extent, a collection of all objects covered by the concept, and its intent, a collection of attributes (properties) covered by the concept. Thus, e.g., the extent of the concept DOG consists of all dogs (or living dogs at a fixed time at a fixed possible world to avoid the philosophical problems) while the intent of DOG consists of all attributes of all dogs (i.e. “to bark”, “to be a mammal”, “to have four extremities” etc.). The primary relation appearing on concepts is that of hierarchical ordering. For instance, the concepts DOG and CAT are not comparable w.r.t. to conceptual hierarchy, while the concept MAMMAL is more abstract (bigger in the hierarchical ordering) than both DOG and CAT. A characteristic feature of empirical concepts (e.g. BIG DOG) is that the extent and the intent are non-sharp (fuzzy) collections (some dogs are bigger than others, various dogs belong to the extent to various (not only two, i.e. fully yes or fully no) degrees). In [2, 3, 4] a theory of fuzzy concepts and hierarchical structures of fuzzy concepts (fuzzy concept lattices) in the sense of traditional logic has been initiated as a generalization of the theory developed by Wille’s group (e.g. [7, 18, 20]). The point of generalization is the non-sharpness (fuzziness) of concepts which is not taken into account in Wille’s theory. Recall that the main purpose of fuzzy logic and fuzzy set theory [13] is to model non-sharp (fuzzy) phenomena. Formally, instead of the two-element Boolean algebra, a more general structure of truth values is employed. In [8, 9], the author proposed to use a structure of a complete residuated lattice which (as turned out later on) plays an important role in fuzzy logic in narrow sense [12, 10, 14]. Definition 1 A complete residuated lattice is an algebra L = hL, ∧, ∨, ⊗, →, 0, 1i such that hL, ∧, ∨, 0, 1i is a complete lattice with the least element 0 and the greatest element 1, hL, ⊗, 1i is a commutative monoid, i.e. ⊗ is commutative, associative, and x ⊗ 1 = x holds holds for each x ∈ L, and ⊗, → form an adjoint pair, i.e. x ⊗ y ≤ z iff x ≤ y → z holds for all x, y, z ∈ L. 2
(1)
Residuated lattices have been introduced in [17]. In each residuated lattice it holds that x ≤ y implies x ⊗ z ≤ y ⊗ z, and x ≤ y implies z → x ≤ z → y (isotonicity) and x → z ≥ y → z (antitonicity in the first argument). The operation ⊗ is thus a t-norm (see e.g. [10, 13]), → is called residuum. In the following we will deal with complete residuated lattices, i.e. hL, ∧, ∨, 0, 1i is assumed to be a complete lattice. The t-norm ⊗ and the residuum → areWintended for modeling of the conjunction V and implication, respectively. Supremum ( ) and infimum ( ) are intended for modeling of the general and existential quantifier, respectively. A semantically complete first-order many-valued logic with semantics defined over complete residuated lattices can be found in [12]. Several special classes of residuated lattices serve as special structures of semantically complete logical calculi (for details see [10]). The most studied and applied set of truth values is the real interval [0, 1] with a ∧ b = min(a, b), a ∨ b = max(a, b), and with three important pairs of adjoint operations: the Lukasiewicz one (a ⊗ b = max(a + b − 1, 0), a → b = min(1 − a + b, 1)), G¨ odel one (a ⊗ b = min(a, b), a → b = 1 if a ≤ b and = b else), and product one (a⊗b = a·b, a → b = 1 if a ≤ b and = b/a else). For the role of these “building stones” in fuzzy logic see [10]. Another important set of truth values is the set (the ordering determines the complete lattice structure) {a0 = 0, a1 , . . . , an = 1} (a0 < · · · < an ). Two t-norms are often considered: ak ⊗ al = amax(k+l−n,0) and the corresponding → given by ak → al = amin(n−k+l,n) (Lukasiewicz) and ak ⊗ al = amin(k,l) and the corresponding → given by ak → al = 1 if k ≤ l and al else (G¨ odel). A special case of the latter algebras is the Boolean algebra 2 of classical logic with the support 2 = {0, 1}. It may be easily verified that the only t-norm on {0, 1} is the classical conjunction operation ∧, i.e. a ∧ b = 1 iff a = 1 and b = 1, which implies that the only residuum operation is the classical implication operation →, i.e. a → b = 0 iff a = 1 and b = 0. Note that each of the preceding residuated lattices is complete. The following identities of complete residuated lattices will be needed: a a
= ≤
a ⊗ (a → b) ≤ (a ⊗ b) → c = _ a⊗ bj = j∈J
a⊗
^
bj
≤
j∈J
_
j∈J
^
j∈J
(2) (3)
b a → (b → c) = b → (a → c) _ (a ⊗ bj )
(4) (5)
^
(a ⊗ bj )
(7)
^
(aj → b)
(8)
^
(a → bj )
(9)
(6)
j∈J
j∈J
aj → b =
a→
1→a (a → b) → b
bj
=
j∈J
j∈J
A fuzzy set (or L-set) A in a universe set X is a mapping A : X → L. The value A(x) ∈ L is interpreted as the truth value of “the element x belongs to A”. The set of all fuzzy sets in X is denoted by LX . For A1 , A2 ∈ LX we write A1 ⊆ A2 iff A1 (x) ≤ A2 (x) for all x ∈ X. Similarly, a binary fuzzy relation R between X and Y is a mapping R : X × Y → L. 3
We now recall the basic notions of fuzzy concept lattices. The primary notion is that of a fuzzy context (L-context): it is a triple hG, M, Ii, where G and M are sets interpreted as the set of objects (G) and the set of attributes (M ) to which we restrict our attention, and I ∈ LG×M is a fuzzy relation between G and M . The value I(g, m) ∈ L is interpreted as the truth value of the fact “the object g ∈ G has the attribute m ∈ M ”. In accordance to the Port-Royal definition, a (formal) fuzzy concept (L-concept) is a pair hA, Bi, A ∈ LG , B ∈ LM , A plays the role of the extent (fuzzy set of objects which determine the concept), B plays the role of the intent (fuzzy set of attributes which determine the concept), such that: (a) B is the collection of all attributes shared by all the objects of A and (b) A is the collection of all objects having all the attributes of B. There are therefore two fundamental operators: ↑ which assigns to each fuzzy set A ∈ LG of objects the fuzzy set A↑ ∈ LM of all the attributes which are common to all objects of A, and ↓ which assigns to each fuzzy set B ∈ LM of attributes the fuzzy set B ↓ ∈ LG of all the objects which which share all the attributes of B). Rewriting these linguistic descriptions on the semantical level of fuzzy logic we get formal definitions ^ A↑ (m) = (A(g) → I(g, m)) (10) g∈G
and B ↓ (g) =
^
(B(m) → I(g, m))
(11)
m∈M
which play the fundamental role. To emphasize the dependence on I, the mappings ↑ and ↓ will also be denoted by ↑I and ↓I . A (formal) fuzzy concept is therefore a pair hA, Bi ∈ LG × LM (of extent A and intent B) such that A↑ = B and B ↓ = A. Denote B (G, M, I) = {hA, Bi ∈ LG × LM | A↑ = B, B ↓ = A} the set of all fuzzy concepts given by the fuzzy context hG, M, Ii. The conceptual hierarchy discussed above is modeled by the relation ≤ defined on B (G, M, I) by hA1 , B1 i ≤ hA2 , B2 i iff A1 ⊆ A2 (iff B1 ⊇ B2 ). (12) The set B (G, M, I) equipped by ≤ is called a fuzzy concept lattice (L-concept lattice). The following theorem describes the hierarchy and characterizes fuzzy concept lattices. Theorem 2 ([4]) (1) The set B (G, M, I) is under ≤ a complete lattice where the suprema and infima are given by ^ \ [ hAj , Bj i = h Aj , ( Bj )↓↑ i , (13) j∈J
_
j∈J
j∈J
hAj , Bj i =
h(
[
j∈J
j∈J
↑↓
Aj ) ,
\
Bj i .
(14)
j∈J
(2) Moreover, an arbitrary complete lattice V = hV, ∧, ∨i is isomorphic to some B (G, M, I) iff there are mappings γ : G × L → V , µ : M × L → V such that 4
γ(G, L) is
W V -dense in V, µ(M, L) is -dense in V;
α ⊗ β ≤ I(g, m) iff γ(g, α) ≤ µ(m, β). W V Note that V ′ ⊆ V is -dense ( -dense) in V if each v ∈ V is the supremum (infimum) of some subset of V ′ . Notice that the lattice structure of concepts is very natural: to each set of concepts there is a concept which is their direct generalization (supremum) and a concept which is their direct specialization (infimum). In this perspective, the original (crisp) concepts lattices [18] are exactly L-concept lattices where L = {0, 1} is the set of truth values of classical logic. Given a context (i.e. data matrix of logical data obtained from experts), the concept lattice reveals the conceptual structure hidden in the context and provides us with formalism for representing conceptual knowledge [18, 20, 7, 4]. If the information in the context is fuzzy, the fuzzy conceptual data analysis is an appropriate tool for handling it. Fuzzy concept lattice represents a conceptual granulation both of the set of objects and attributes equipped by a hierarchical order.
3 3.1
Concepts and similarity Similarity relations
A crucial role in the way humans regard the world is played by the similarity phenomenon. In fuzzy set theory, similarity phenomenon is approached via so called similarity relations. By a ⊗-similarity relation (or fuzzy ⊗-equivalence relation, Lvalued global equality) [13, 12, 21] on a universe U it is meant a binary fuzzy relation E satisfying the following properties for all x, y, z ∈ U : E(x, x)
= 1
E(x, y) = E(y, x) E(x, y) ⊗ E(y, z) ≤ E(x, z).
(15) (16) (17)
Properties (15), (16), and (17) are called reflexivity, symmetry, and transitivity, respectively. The ⊗-similarity class of x ∈ U is the fuzzy set [x]E ∈ LU given by [x]E (y) = E(x, y) for each y ∈ U , i.e. it is a collection of elements similar to x. A fuzzy set A ∈ LG is said to be extensional w.r.t. E if for every x, y ∈ U it holds A(x) ⊗ E(x, y) ≤ A(y), i.e. if with each its element x, A contains all the elements similar to x. In this case, E is also said to be compatible with A. Nonextensional fuzzy sets are not compatible with the underlying similarity relation. It is easily seen that in the crisp case, i.e. L = {0, 1}, similarity relations are equivalence relations. For the study of similarity phenomenon, the crisp case is a degenerate one and non-interesting—two elements x and y may be “fully similar” (E(x, y) = 1) or “fully dissimilar” (E(x, y) = 0). Note that to better grasp the similarity in the crisp case, tolerance relations (i.e. reflexive and symmetric relations) are used instead [5]. Tolerance relations may be considered also from the point of view of fuzzy approach, i.e. requiring only (15) and (16). We will not follow this way explicitly. However, most of the results proved below hold also for reflexive and symmetric fuzzy relations.
5
To be able to model the equivalence of truth values we have at disposal the so called biresiduum (or biimplication) [12, 14] operation ↔ defined by a ↔ b = (a → b) ∧ (b → a). The following lemma will be useful in the following considerations. Lemma 3 Let E be a ⊗-similarity on U , S = {Ai ∈ LU | i ∈ I} be a family of fuzzy sets. (1) E is the largest ⊗-similarity relation compatible with all [x]E . (2) The relation ES defined by ^ ES (x, y) = (Ai (x) ↔ Ai (y)) (18) i∈I
is the largest ⊗-similarity relation compatible with all Ai ∈ S. Moreover, Ai (x) = 1 implies [x]ES ⊆ Ai . Proof (1) is an immediate consequence of (2) provided E = E{[x]E | x∈U} . Denote V E ′ = E{[x]E | x∈U} . We have E(y, z) ≤ E ′ (y, z) = x∈U (E(x, y) ↔ E(x, z)) iff for each x ∈ U it holds E(y, z) ≤ E(x, y) ↔ E(x, z) = (E(x, y) → E(x, z)) ∧ (E(x, z) → E(x, y)) iff both E(x, y) ≤ E(x, y) → E(x, z) and E(x, y) ≤ E(x, z) → E(x, y) which is evident by applying the adjointness and transitivity. On the other hand, taking x = y we get E ′ (y, z) ≤ E(y, y) ↔ E(y, z) = E(y, z), i.e. E = E ′ proving (1). Prove (2). We have to check that ES satisfies (15)–(17). (15) follows from a ↔ a = 1 and (16) follows from a ↔ b = b ↔ a. Furthermore, ES (x, y) ⊗ ES (y, z) = ^ ^ = (Ai (x) ↔ Ai (y)) ⊗ (Aj (y) ↔ Aj (z)) ≤
(19)
^ ^ Ai (x) ↔ Ai (y) ⊗ Aj (y) ↔ Aj (z) ≤
(20)
^ ^
(21)
i∈I
≤
j∈I
i∈I
≤
j∈I
i∈I j∈I
≤
Ai (x) ↔ Ai (y) ⊗ Aj (y) ↔ Aj (z) ≤
^ Ai (x) ↔ Ai (y) ⊗ Ai (y) ↔ Ai (z) =
i∈I
=
^ n
i∈I
⊗ ≤
n
(22)
o Ai (y) → Ai (z) ∧ Ai (z) → Ai (y) ≤
^ n
i∈I
o Ai (x) → Ai (y) ∧ Ai (y) → Ai (x) ⊗
o Ai (x) → Ai (y) ⊗ Ai (y) → Ai (z) ∧
n o ∧ Ai (x) → Ai (y) ⊗ Ai (z) → Ai (y) ∧ n o ∧ Ai (y) → Ai (x) ⊗ Ai (y) → Ai (z) ∧ 6
(23)
n o ∧ Ai (y) → Ai (x) ⊗ Ai (z) → Ai (y) ≤ o ^ n ≤ Ai (x) → Ai (y) ⊗ Ai (y) → Ai (z) ∧
(24)
i∈I
n o ∧ Ai (z) → Ai (y) ⊗ Ai (y) → Ai (x) ≤ ^ ≤ Ai (x) → Ai (z) ∧ Ai (z) → Ai (x) =
(25)
i∈I
=
^ Ai (x) ↔ Ai (z) = ES (x, z) ,
i∈I
V hence (17) holds. In (19)–(20), (20)–(21) and (22)–(23) we used the fact a ⊗ i∈I bi ≤ V i∈I (a ⊗ bi ). In (24)–(25) we used (a → b) ⊗ (b → c) ≤ a → c. The extensionality of each Ai ∈ S follows by Ai (x) ⊗ ES ≤ Ai (x) ⊗ (Ai (x) → Ai (y)) ≤ Ai (y). If all Ai are extensional to another ⊗-similarity E, then Ai (x) ⊗ E(x, y) ≤ Ai (y), hence E(x, y) ≤ Ai (x) → Ai (y), and, similarly, E(x, y) ≤ Ai (y) →VAi (x), i.e. E(x, y) ≤ Ai (x) ↔ Ai (y) hold for all x, y ∈ U . We conclude E(x, y) ≤ i∈I (Ai (x) ↔ Ai (y)) = ES (x, y), i.e. ES is the largest one. Finally, if Ai (x) = 1 then [x]ES (y) = ES (x, y) ≤ Ai (x) ↔ Ai (y) = 1 ↔ Ai (y) = Ai (y), i.e. [x]ES ⊆ Ai . The proof is finished. Notice that for the crisp case (i.e. L = {0, 1}), ES is a crisp equivalence relation— two elements of the universe are equivalent iff there is no set of the family which separates them.
3.2
Similarity of objects and attributes
We are going to propose a way to measure similarity of objects and similarity of attributes of a given L-context. This similarity will be induced by the structure of L-concepts determined by the L-context. We prove that the similarity may be determined directly from the L-context which is relevant from the computational point of view. We are given (in some sense relevant) objects (elements of G) and their (observed) attributes (elements of M ). The given L-context gives a rise to a complete lattice of all induced L-concepts. The idea of conceptual classification leads to the use of the induced conceptual structure B (G, M, I) to define similarity relations on G and on M. Consider the problem of similarity of objects. Informally, two objects g1 , g2 ∈ G are similar if they cannot be separated by any concept, i.e. if for each concept c it holds that g1 belongs to the extent of c iff g2 belongs to the extent of c. This leads G to the following definition of a relation EB(G,M,I) ∈ LG×G : G EB(G,M,I) (g1 , g2 ) =
^
hA,Bi∈B(G,M,I)
A(g1 ) ↔ A(g2 ) .
(26)
G The relation EB(G,M,I) will be called induced (by B (G, M, I)) similarity on G. By Lemma 3 we immediately get the following statement.
7
G Theorem 4 The relation EB(G,M,I) is the largest ⊗-similarity relation on G compatible with the extents of all concepts of B (G, M, I).
From the computational point of view (we always assume both G and M to be finite if algorithmic aspects are concerned), the foregoing definition leads to the folG lowing algorithm for computing the similarity relation EB(G,M,I) : Take an L-context, generate all the L-concepts of B (G, M, I) and determine the similarity of each pair hg1 , g2 i ∈ G × G by (26). The L-concept lattice may be, however, quite extensive. This poses the question whether the computational cost can be reduced. We give a (exact) solution which reduces the computational costs significantly. Define a relation G EhG,M,Ii ∈ LG×G by G EhG,M,Ii (g1 , g2 ) =
^
(I(g1 , m) ↔ I(g2 , m)) .
(27)
m∈M G EhG,M,Ii (g1 , g2 ) may be obtained from the L-context hG, M, Ii computing |M | times the operation ↔. Using Lemma 3 (put X = G, I = M , Ai (g) = I(g, m)) we get G Theorem 5 The relation EhG,M,Ii is the largest ⊗-similarity relation on G compatG ible with all I( , m) ∈ L , m ∈ M .
The following theorem solves the problem of finding an efficient procedure for G computing the similarity relation EB(G,M,I) . Theorem 6 Let hG, M, Ii be a L-context. Then for the similarity relations defined by (27) and (26) it holds G G EB(G,M,I) = EhG,M,Ii . (28) V V Proof We show m∈M (I(g1 , m) ↔ I(g2 , m)) = hA,Bi∈B(G,M,I) A(g1 ) ↔ A(g2 ) by checking both inequalities. Consider the concept hAm , Bm i = h{ 1 m}↓ , { 1 m}↓↑ i for m ∈ M . For each g ∈ G it holds ^ Am (g) = { { 1 m}(m′ ) → I(g, m); m′ ∈ M } = =
1 → I(g, m) = I(g, m) .
From {hAm , Bm i | m ∈ M } ⊆ B (G, M, I) and from the properties of infimum it follows that ^ ^ A(g1 ) ↔ A(g2 ) ≤ Am (g1 ) ↔ Am (g2 ) = m∈M
hA,Bi∈B(G,M,I)
=
^
(I(g1 , m) ↔ I(g2 , m))
m∈M
for all g1 , g2 ∈ G, proving the first inequality. Conversely, ^ ^ (I(g1 , m) ↔ I(g2 , m)) ≤ m∈M
hA,Bi∈B(G,M,I)
8
A(g1 ) ↔ A(g2 )
iff for each hA, Bi ∈ B (G, M, I) it holds ^ (I(g1 , m) ↔ I(g2 , m)) ≤ m∈M
≤
A(g1 ) ↔ A(g2 ) = (A(g1 ) → A(g2 )) ∧ ((A(g2 ) → A(g1 ))
which holds iff both ^
(I(g1 , m) ↔ I(g2 , m)) ≤ A(g1 ) → A(g2 )
(29)
^
(I(g1 , m) ↔ I(g2 , m)) ≤ A(g2 ) → A(g1 )
(30)
m∈M
and
m∈M
hold. (29) holds iff A(g1 ) ⊗
^
(I(g1 , m) ↔ I(g2 , m)) ≤ A(g2 ).
m∈M
Since hA, Bi ∈ B (G, M, I), i.e. A = A↑↓ , the last inequality means ^ ^ (A↑ (m) → I(g1 , m)) ⊗ (I(g1 , m) ↔ I(g2 , m)) ≤ m∈M
^
≤
m∈M
↑
(A (m) → I(g2 , m))
m∈M
which holds iff for each m′ ∈ M we have ^ ^ (A↑ (m) → I(g1 , m)) ⊗ (I(g1 , m) ↔ I(g2 , m)) ≤ m∈M ↑ ′
m∈M
′
≤ A (m ) → I(g2 , m ), i.e. A↑ (m′ ) ⊗
^
(A↑ (m) → I(g1 , m)) ⊗
m∈M
^
(I(g1 , m) ↔ I(g2 , m))
m∈M
≤ I(g2 , m).
(31)
Evidently, by (4), A↑ (m′ ) ⊗
^
(A↑ (m) → I(g1 , m)) ⊗
m∈M ↑ ′
^
(I(g1 , m) ↔ I(g2 , m)) ≤
m∈M ′
A↑ (m′ ) ⊗ (A (m ) → I(g1 , m′ )) ⊗ (I(g1 , m ) → I(g2 , m′ )) ≤ I(g2 , m′ ), thus (31) and therefore also (29) hold. (30) may be proved analogously. The second inequality and hence also the theorem is proved. From the foregoing theorem we have also the following consequence which is in accordance with our intuition: If we are given family of (elementary) properties (attributes) of objects and consider the structure of concepts which is given by these 9
properties then the similarity among objects considered w.r.t. the structure of concepts is the same as the similarity w.r.t. to only the basic properties. In a completely analogous way we may get the following results for the similarity relations on M . M M Theorem 7 For an L-context hG, M, Ii, the relations EB(G,M,I) and EhG,M,Ii defined by ^ M EB(G,M,I) (m1 , m2 ) = (B(m1 ) ↔ B(m2 )) hA,Bi∈B(G,M,I)
and M EhG,M,Ii (m1 , m2 ) =
^
(I(g, m1 ) ↔ I(g, m2 )) .
g∈G M M are ⊗-similarity relations on M and it holds EB(G,M,I) = EhG,M,Ii . They are the largest ⊗-similarity relations compatible with the intents of all concepts of B (G, M, I).
3.3
Similarity of concepts
The next level on which the similarity phenomenon will be considered is the level of concepts. Observe first the following fact. Lemma 8 For any universe U , the relation E on LU given for any A1 , A2 ∈ LU by ^ E(A1 , A2 ) = (A1 (x) ↔ A2 (x)) x∈U
is the largest ⊗-similarity relation on LU such that A1 (x) ⊗ E(A1 , A2 ) ≤ A2 (x) holds for each x ∈ U , A1 , A2 ∈ LU . Proof Putting I = U , X = LU , Ai (x) = x(i) for x ∈ LU , i ∈ U , the assertion is a direct consequence of Lemma 3. In the following, it will be clear what universe U the relation E concerns. Consider first the relations E Ext and E Int on B (G, M, I), call them induced similarity by extents and induced similarity by intents, respectively: ^ E Ext (hA1 , B1 i, hA2 , B2 i) = E(A1 , A2 ) = (A1 (g) ↔ A2 (g)), g∈G
E Int (hA1 , B1 i, hA2 , B2 i) = E(B1 , B2 ) =
^
(B1 (m) ↔ B2 (m)).
m∈M
Lemma 8 gives immediately the following statement. Theorem 9 E Ext and E Int are the largest ⊗-similarity relations on B (G, M, I) such that A1 (g)⊗E Ext (hA1 , B1 i, hA2 , B2 i) ≤ A2 (g) and B1 (m)⊗ E Int (hA1 , B1 i, hA2 , B2 i) ≤ B2 (m) hold for every g ∈ G, m ∈ M , hA1 , B1 i, hA2 , B2 i ∈ B (G, M, I). To answer the question of how the relations E Ext and E Int are related, we derive some preliminary results. The next lemma states that the operators ↑ and ↓ preserve similarity. 10
Lemma 10 Let hG, M, Ii be an L-context, A1 , A2 ∈ LG , B1 , B2 ∈ LM . Then it holds E(A1 , A2 ) ≤ E(A↑1 , A↑2 ) and E(B1 , B2 ) ≤ E(B1↓ , B2↓ ). Proof We prove only E(A1 , A2 ) ≤ E(A↑1 , A↑2 ), the second part may be obtained symmetrically. ^ (A1 (g) ↔ A2 (g)) = E(A1 , A2 ) ≤ g∈G
^
≤ E(A↑1 , A↑2 ) =
m∈M
A↑1 (m) ↔ A↑2 (m)
holds iff for each m ∈ M it holds ^ (A1 (g) ↔ A2 (g)) ≤ g∈G
≤ A↑1 (m) ↔ A↑2 (m) = A↑1 (m) → A↑2 (m) ∧ A↑2 (m) → A↑1 (m) which holds iff the left side of the inequality is less or equal than both members of the right side which are connected by ∧. We check only the first one of these inequalities, i.e. ^ (A1 (g) ↔ A2 (g)) ≤ A↑1 (m) → A↑2 (m) g∈G
By adjunction, this holds iff A↑1 (m) ⊗
^
(A1 (g) ↔ A2 (g)) ≤ A↑2 (m)
g∈G
i.e.
^
g∈G
A1 (g) → I(g, m) ⊗
^
(A1 (g) ↔ A2 (g)) ≤
g∈G
^
A2 (g) → I(g, m)
g∈G
which holds iff for each g ′ ∈ G the inequality ^ ^ (A1 (g) ↔ A2 (g)) ⊗ A1 (g) → I(g, m) ≤ A2 (g ′ ) → I(g ′ , m) g∈G
g∈G
holds. The last inequality is equivalent (by adjunction) to ^ ^ A2 (g ′ ) ⊗ (A1 (g) ↔ A2 (g)) ⊗ A1 (g) → I(g, m) ≤ I(g ′ , m) g∈G
g∈G
which holds because A2 (g ′ ) ⊗
^
g∈G
≤
′
(A1 (g) ↔ A2 (g)) ⊗
^
g∈G
′
′
A1 (g) → I(g, m) ≤
A2 (g ) ⊗ (A2 (g ) → A1 (g )) ⊗ (A1 (g ′ ) → I(g ′ , m)) ≤ I(g ′ , m)
by applying twice the rule (4). 11
The following corollary is immediate. ↑↓ Corollary 11 Under the conditions of Lemma 10, it holds E(A1 , A2 ) ≤ E(A↑↓ 1 , A2 ) and E(B1 , B2 ) ≤ E(B1↓↑ , B2↓↑ ).
Note that even without Lemma 10 we have due to the properties of E that ↑↓ ↑↓ ↑↓ ↑↓ E(A↑↓ 1 , A1 ) ⊗ E(A1 , A2 ) ⊗ E(A2 , A2 ) ≤ E(A1 , A2 ), i.e. if A1 and A1 , A1 and ↑↓ ↑↓ ↑↓ A2 , A2 and A2 are pairwise similar then also A1 and A2 are similar. Corollary 11 asserts a stronger result. Corollary 12 Under the conditions of Lemma 10, it holds E(A1 , A2 )⊗E(A1 , A↑↓ 1 )≤ ↓↑ ↓↑ E(A1 , A↑↓ ) and E(B , B ) ⊗ E(B , B ) ≤ E(B , B ). 1 2 1 1 2 1 2 ↑↓ ↑↓ ↑↓ ↑↓ Proof E(A1 , A2 ) ⊗ E(A1 , A↑↓ 1 ) ≤ E(A1 , A2 ) ⊗ E(A1 , A1 ) ≤ E(A1 , A2 ). The second part may be proved analogously.
The following result shows that the similarities of concepts by extents and intents are equal. Theorem 13 For any L-context hG, M, Ii it holds E Ext = E Int . Proof Let hA1 , B1 i, hA1 , B1 i ∈ B (G, M, I), i.e. A↑i = Bi and Bi↓ = Ai for i = 1, 2. By Lemma 10 we get E Ext (hA1 , B1 i, hA2 , B2 i) = E(A1 , A2 ) ≤ ≤ E(A↑1 , A↑2 ) = E(B1 , B2 ) = E Int (hA1 , B1 i, hA2 , B2 i), and analogously, E Int (hA1 , B1 i, hA2 , B2 i) ≤ E Ext (hA1 , B1 i, hA2 , B2 i). To sum up, E Ext (hA1 , B1 i, hA2 , B2 i) = E Int (hA1 , B1 i, hA2 , B2 i). We will therefore write E instead of E Ext and E Int and call it the induced similarity on concepts. Compatible similarities and factorization The primary importance of similarity relations in human reasoning is the reduction of the complexity of the outer world at a reasonable price. The complexity is reduced via considering the “collections of similar elements in concern” rather than the particular elements themselves. This is just what is in the general system theory known as the abstraction process by factorization: moving from a given level of abstraction (distinguishability) one level up where the elements are collections of elements of the lower level. Instead of the original system one therefore considers the “system modulo similarity”. The price paid is the loss of precision. Our concern in the following is the reduction of the complexity of the concept lattice by factorization modulo similarity. The concept lattice of a given context represents the overall conceptual structure which can be considerably intricate. To get an insight one has to look for methods for reducing the complexity of the structure. In the twovalued (sharp) case, a considerable attention has been paid to this problem [7, 18]. In the many-valued (fuzzy) case, one would expect methods for gradual reduction of 12
the complexity. The idea is to factorize the concept lattice by appropriate a-cut a E of the similarity E (note that a E = {hc1 , c2 i | a ≤ E(c1 , c2 )} [13]), controlling thus the complexity by a ∈ L. Clearly, the lower a ∈ L, the coarser the factorization. The process of factorization of a system consists in two steps. First, specification of the elements, and, secondly, specification of the structure of the factor system. Since both of the steps are non-standard in our case we will describe them in more detail. In general, algebraic systems can be factorized be congruences, i.e. equivalences compatible with the structure of the system. We deal with conceptual structures which are complete lattices. The a-cut a E is clearly a tolerance relation (i.e. reflexive and symmetric), not transitive in general. Compatible tolerance relations on universal algebras have been extensively studied by Chajda [5]. In general, factorization of algebras by compatible tolerances is not possible. Surprisingly, Cz´edli [6] showed a way to factorize lattices by compatible tolerance relations. The construction has been then used for the factorization of complete lattices (and hence sharp concept lattices) [19]. In the following we describe the construction of the factor lattice of an L-concept lattice by a compatible tolerance relation. Let hG, M, Ii be an L-context. A tolerance relation T on B (G, M, I) is said to be compatible if it is preserved under W W arbitrary ′ ′ suprema and infima, i.e. if hc , c i ∈ T , j ∈ J, implies both h c , j j j j∈J j∈J cj i ∈ T V V ′ ′ and h j∈J cj , j∈J cj i ∈ T for any cj , cj ∈ B (G, M, I), j ∈ J. For a compatible V W tolerance relation T on B (G, M, I) denote cT = hc,c′ i∈T c′ and cT = hc,c′ i∈T c′ . Call [c]T = [cT , (cT )T ] = {c′ ∈ B (G, M, I) | cT ≤ c′ ≤ (cT )T } a block of T and denote B (G, M, I)/T = {[c]T | c ∈ B (G, M, I)}Vthe set V of all blocks. a W Introduce W relation ≤T on B (G, M, I)/T by [c]T ≤T [c′ ]T iff [c]T ≤ [c′ ]T (iff [c]T ≤ [c′ ]T ). The justification of the construction is given by the following statement which follows immediately from [19]. Proposition 14 (1) B (G, M, I)/T is the set of all maximal tolerance blocks, i.e. B (G, M, I)/T = {B ⊆ B (G, M, I) | (B × B ⊆ T )&((∀B ′ ⊃ B)(B ′ × B ′ 6⊆ T ))}. (2) hB (G, M, I)/T, ≤T i is a complete lattice (factor lattice) where suprema and infima are described by _ _ ^ ^ [cj ]T = [ cj ]T and [cj ]T = [( cj )T ]T (32) j∈J
j∈J
j∈J
j∈J
for every cj ∈ B (G, M, I), j ∈ J. Substituing (13) and (14) into (32) we get a more concrete description of the lattice operations. Coming back to the induced similarity E on B (G, M, I), the ultimate question is that of the compatibility of the a-cuts of E. Call a ⊗-similarity relation F on B (G, M, I) compatible if a E is a compatible tolerance relation on B (G, M, I) for each a ∈ L. Notice that for the two-valued (crisp) case the situation is completely uninteresting. Namely, as one easily checks, the only cases are 0 E = B (G, M, I) × B (G, M, I) and 1 E = idB(G,M,I) = {hc, ci | c ∈ B (G, M, I)}. In the first case, B (G, M, I)/0 E = {B (G, M, I)}, i.e. the factor lattice collapsed into a one element lattice, while in the second case, B (G, M, I)/1 E = {{hA, Bi} | hA, Bi ∈ B (G, M, I)}, i.e. B (G, M, I) and B (G, M, I)/1 E are isomorphic. Note that we have in no case to confine ourselves to the induced similarity E. On the other hand, taking into account only ⊗-similarity relations F satisfying A(g) ⊗ 13
F (hA, Bi, hA′ , B ′ i) ≤ A′ (g) (which is quite natural—it reads “object belonging to the extent of some concept belongs also to the extent of any similar concept”) for each g ∈ G, Theorem 9 tells us that E provides the most extensive reduction: for any other F and each a ∈ L, a E is coarser than a F . We will make use of the following lemma.
Lemma 15 For every hAj , Bj i, A′j , Bj′ ∈ B (G, M, I), j ∈ J, it holds ^ ^
^
hAj , Bj i, A′j , Bj′ ), (33) E(hAj , Bj i, A′j , Bj′ ) ≤ E( j∈J
j∈J
^
j∈J
E(hAj , Bj i, A′j , Bj′ )
≤ E(
_
j∈J
hAj , Bj i,
j∈J
_
j∈J
A′j , Bj′ ).
(34)
Proof By the above statements, (33) holds iff ^ ^ ^ Aj , A′j ), E(Aj , A′j ) ≤ E( j∈J
j∈J
i.e.
j∈J
^ ^ ^ ^ ^ ( Aj (g) ↔ A′j (g)) ≤ ( Aj (g)) ↔ ( A′j (g))
j∈J g∈G
g∈G j∈J
j∈J
′
which holds iff for each g ∈ G we have ^ ^ ^ ^ Aj (g) ↔ A′j (g) ≤ ( Aj (g ′ )) ↔ ( A′j (g ′ )) = j∈J g∈G
=
j∈J
(
^
′
Aj (g )) → (
j∈J
^
j∈J
A′j (g ′ ))
∧(
j∈J
^
A′j (g ′ )) → (
j∈J
^
Aj (g ′ )).
j∈J
The last inequality holds iff the left side is less or equal to both of the conjuncts on the right side. As they can be handled analogously, we prove only ^ ^ ^ ^ (Aj (g) ↔ A′j (g)) ≤ ( Aj (g ′ )) ↔ ( A′j (g ′ )) j∈J g∈G
j∈J
j∈J
which is, by adjunction, equivalent to ^ ^ ^ ^ ( Aj (g ′ )) ⊗ (Aj (g) ↔ A′j (g)) ≤ ( A′j (g ′ )) j∈J
j∈J g∈G
j∈J
which holds since (
^
Aj (g ′ )) ⊗
≤
(
^
≤
^ ^ ( A′j (g ′ ) ⊗ (Aj (g ′ ) → A′j (g ′ )) ≤
j∈J
^ ^
(Aj (g) ↔ A′j (g)) ≤
j∈J g∈G
Aj (g ′ )) ⊗
j∈J
^
(Aj (g ′ ) → A′j (g ′ )) ≤
j∈J
j∈J j ′ ∈J
≤
^
(Aj (g ′ ) ⊗ (Aj (g ′ ) → A′j (g ′ )) ≤
j∈J
^
j∈J
14
A′j (g ′ ).
We have proved (33). proved D S (34) can be E symmetrically using the fact W V ↑↓ of Theorem 2 and proceeding by the simij∈J hAj , Bj i = ( j∈J Aj ) , j∈J Bj larity E on the intents. Lemma 15 has an interesting corollary stating that the similarity of any two concepts is less or equal than the similarity of any of them to their direct join or meet. Corollary 16 Let hA1 , B1 i, hA2 , B2 i ∈ B (G, M, I). The following inequalities hold for i = 1, 2: E(hA1 , B1 i, hA2 , B2 i) ≤ E(hA1 , B1 i, hA2 , B2 i) ≤
E(hAi , Bi i, hA1 , B1 i ∧ hA2 , B2 i), E(hAi , Bi i, hA1 , B1 i ∨ hA2 , B2 i).
′ ′
Proof Put J = {x, y}, hAx , Bx i = hAy , By i = hA1 , B1 i, hAx , Bx i = hA1 , B1 i, ′ ′ Ay , By = hA2 , B2 i and apply Lemma 15.
Theorem 17 The induced similarity E on B (G, M, I) is compatible. If a ∈ L is ⊗-idempotent (i.e. a ⊗ a = a) then a E is, moreover, transitive, i.e. a congruence relation on B (G, M, I). Proof The first part follows immediately from (33)
fact that if V and (34) by the a ≤ E(hAj , Bj i, A′j , Bj′ ) for j ∈ J then also a ≤ j∈J E(hAj , Bj i, A′j , Bj′ ). The second part follows from the evident fact that if a is ⊗-idempotent and a ≤ b, c then a ≤ b ⊗ c. Remark 18 Theorem 17 and the above described construction yield a method for factorizing any L-concept lattice B (G, M, I) by any a-cut a E of the induced similarity E. It is worth to notice that that the similarity E is defined “internally”, i.e. it is not supplied from the outside. Remark 19 If L the algebra of intuitionistic logic (Heyting algebra) or the algebra of G¨ odel logic [10] then each a-cut of E is indeed a congruence relation. Next we formulate a statement concerning the relation of the similarity and the hierarchy of concepts which shows that the farther are the concepts in the hierarchy, the less similar they are. Theorem 20 Let for hAi , Bi i ∈ B (G, M, I), i = 1, 2, 3, it holds hA1 , B1 i ≤ hA2 , B2 i ≤ hA3 , B3 i. Then E(hA1 , B1 i, hA3 , B3 i) ≤ E(hA1 , B1 i, hA3 , B3 i) ≤
E(hA1 , B1 i, hA2 , B2 i), E(hA2 , B2 i, hA3 , B3 i).
Proof By the assumptions, i.e.A1 (g) ≤ A2 (g) ≤ A3 (g) for all g ∈ G, V and by the antitonicity of → in the first argument we have E(hA , B i, hA , B i) = 1 1 3 3 g∈G (A3 (g) → V A1 (g)) ≤ g∈G (A2 (g) → A1 (g)) = E(hA1 , B1 i, hA2 , B2 i). The second part may be obtained symmetrically.
15
3.4
Similarity of concept lattices
Finally, we consider similarity of concept lattices. A natural way to define the similarity degree of two concept lattices over the sets G and M is based on the following intuition. Concept lattices B (G, M, I1 ) and B (G, M, I2 ) are similar iff for each concept c1 ∈ B (G, M, I1 ) there is a concept c2 ∈ B (G, M, I2 ) such that c1 and c2 are similar and, conversely, for each concept c2 ∈ B (G, M, I2 ) there is a concept c1 ∈ B (G, M, I1 ) such that c1 and c2 are similar. In the following we write B1 and B2 instead of B (G, M, I1 ) and B (G, M, I2 ), respectively. According to how the similarity of concepts is measured we distinguish two rules for the definition of the similarity degree of two concept lattices: E ∗ (B (G, M, I1 ), B (G, M, I2 )) = ^ _ = E ∗ (hA1 , B1 i, hA2 , B2 i) ∧
(35)
hA1 ,B1 i∈B1 hA2 ,B2 i∈B2
^
_
E ∗ (hA1 , B1 i, hA2 , B2 i),
hA2 ,B2 i∈B2 hA1 ,B1 i∈B1
∗ ∈ {Ext, Int}, where we put E Ext (hA1 , B1 i, hA2 , B2 i) E Int (hA1 , B1 i, hA2 , B2 i)
= E(A1 , A2 ) = E(B1 , B2 ).
Note that E Ext and E Int correspond to the cases when the similarity of concepts is measured by extents and by intents of concepts, respectively. However, Theorem 13 cannot be applied to show E Ext = E Int because concepts of different concept lattices are considered. We are going to show that all of the above relations are in fact similarity relations, that they are equal, and that they are, moreover, equal to the similarity relation E defined on the set of all contexts by ^ E(hG, M, I1 i, hG, M, I2 i) = E(I1 , I2 ) = I1 (g, m) ↔ I2 (g, m). (36) hg,mi∈G×M
The following corollary follows directly by Lemma 8. Corollary 21 The relation E defined by (36) is the largest ⊗-similarity relation on {hG, M, Ii | I ∈ LG×M } such that I1 (g, m) ⊗ E(I1 , I2 ) ≤ I2 (g, m) holds for every g ∈ G, m ∈ M . We need the following lemmata. Lemma 22 Let hG, M, I1 i, hG, M, I2 i be L-contexts, A ∈ LG , B ∈ LM . Then E(I1 , I2 ) ≤ E(A↑I1 , A↑I2 ) and E(I1 , I2 ) ≤ E(B ↓I1 , B ↓I2 ). Proof Due to symmetry we prove only the first part, i.e. E(I1 , I2 ) ≤ E(A↑I1 , A↑I2 ), 16
i.e. E(I1 , I2 ) ≤
^
(A↑I1 (m) ↔ A↑I2 (m))
m∈M
which holds iff for each m ∈ M E(I1 , I2 ) ≤ A↑I1 (m) ↔ A↑I2 (m) = A↑I1 (m) → A↑I2 (m) ∧ A↑I2 (m) → A↑I1 (m) holds. The last inequality holds iff E(I1 , I2 ) is less or equal to both A↑I1 (m) → A↑I2 (m) and A↑I2 (m) → A↑I1 (m). We check only E(I1 , I2 ) ≤ A↑I1 (m) → A↑I2 (m) which is equivalent to A↑I1 (m) ⊗ E(I1 , I2 ) ≤ A↑I2 (m) =
^
A(g) → I2 (g, m)
g∈G
iff for each g ∈ G A↑I1 (m) ⊗ E(I1 , I2 ) ≤ A(g) → I2 (g, m) iff A(g) ⊗ A↑I1 (m) ⊗ E(I1 , I2 ) ≤ I2 (g, m). We have A(g) ⊗ A↑I1 (m) ⊗ E(I1 , I2 ) = ^ = A(g) ⊗ (A(g ′ ) → I1 (g ′ , m)) ⊗ g′ ∈G
^
(I1 (g ′ , m′ ) ↔ I2 (g ′ , m′ )) ≤
hg′ ,m′ i∈G×M
≤ A(g) ⊗ (A(g) → I1 (g, m)) ⊗ (I1 (g, m) → I2 (g, m)) ≤ I2 (g, m) which had to be proved. ↑
↑
Note that by Lemma 10 and Lemma 22 we have E(A1 , A2 )⊗E(I1 , I2 ) ≤ E(A1I1 , A2I2 ) ↓ ↓ and E(B1 , B2 ) ⊗ E(I1 , I2 ) ≤ E(B1 I1 , B2 I2 ). Lemma 23 For every L-contexts hG, M, I1 i, hG, M, I2 i and ∗ ∈ {Ext, Int} it holds E(hG, M, I1 i, hG, M, I2 i) ≤ E ∗ (B (G, M, I1 ), B (G, M, I2 )). Proof We proceed only for E Ext , the second case is symmetric. We have to prove E(I1 , I2 ) ≤ E Ext (B1 , B2 ), which holds iff both ^ _ E Ext (hA1 , B1 i, hA2 , B2 i) E(I1 , I2 ) ≤ hA1 ,B1 i∈B1 hA2 ,B2 i∈B2
and E(I1 , I2 ) ≤
^
_
E Ext (hA1 , B1 i, hA2 , B2 i)
hA2 ,B2 i∈B2 hA1 ,B1 i∈B1
hold. Due to symmetry we prove only the first inequality which holds iff _ E(I1 , I2 ) ≤ E(A1 , A2 ) hA2 ,B2 i∈B2
17
↓I
↓I
1 2 holds for each hA1 , B1 i ∈ B1 . By Lemma 22 we have D E E(I1 , I2 ) ≤ E(B1 , B1 ) = ↓I ↓I ↓I ↑I E(A1 , B1 2 ), i.e. putting hA2 , B2 i = B1 2 , B1 2 2 we have
↓
_
E(I1 , I2 ) ≤ E(A1 , B1 I2 ) ≤
E(A1 , A2 )
hA2 ,B2 i∈B2
finishing the proof. Lemma 24 For every L-contexts hG, M, I1 i, hG, M, I2 i and ∗ ∈ {Ext, Int} it holds E ∗ (B (G, M, I1 ), B (G, M, I2 )) ≤ E(hG, M, I1 i, hG, M, I2 i). Proof We proceed only for E Ext , the case E Int can be handled analogously. We have to prove ^ E Ext (B1 , B2 ) ≤ E(I1 , I2 ) = I1 (g, m) ↔ I2 (g, m) hg,mi∈G×M
which holds iff E Ext (B1 , B2 ) ≤ I1 (g, m) ↔ I2 (g, m) = (I1 (g, m) → I2 (g, m)) ∧ (I2 (g, m) → I1 (g, m)) holds for every g ∈ G, m ∈ M . The inequality holds iff both E Ext (B1 , B2 ) ≤ I1 (g, m) → I2 (g, m) and E Ext (B1 , B2 ) ≤ I2 (g, m) → I1 (g, m) hold. For symmetry we show only the first inequality which is equivalent to I1 (g, m) ⊗ E Ext (B1 , B2 ) ≤ I2 (g, m).
(37)
We have I1 (g, m) ⊗ E Ext (B1 , B2 ) = I1 (g, m)⊗ _ ^ E(A1 , A2 ) ∧ ⊗ hA1 ,B1 i∈B1 hA2 ,B2 i∈B2
≤ I1 (g, m) ⊗ ≤ I1 (g, m) ⊗ ≤ I1 (g, m) ⊗
hA2 ,B2 i∈B2
E({ 1 m}↓I1 , A2 ) ∧
_
E({ 1 m}↓I1 , A2 ) ≤
_
B∈LM
=
≤
hA1 ,B1 i∈B1
E(A1 , { 1 m}↓I2 ) ≤
_
I1 (g, m) ⊗ (
_
B∈LM
^ I1 (g, m) ⊗ ({ 1 m}↓I1 (g) ↔ ( B(m′ ) → I2 (g, m′ )) ≤
_
^ I1 (g, m) ⊗ ({ 1 m}↓I1 (g) → ( B(m′ ) → I2 (g, m′ )) ≤
_
I1 (g, m) ⊗
B∈LM
≤
_
E({ 1 m}↓I1 , B ↓I2 ) = ^
{ 1 m}↓I1 (g ′ ) ↔ B ↓I2 (g ′ )) ≤
g′ ∈G
B∈LM
≤
E(A1 , A2 ) ≤
hA2 ,B2 i∈B2 hA1 ,B1 i∈B1
_
hA2 ,B2 i∈B2
_
^
m′ ∈M
m′ ∈M
B∈LM
18
⊗((
^
m′ ∈M
≤
^ { 1 m}(m′ ) → I1 (g, m′ )) → ( 1 → I2 (g, m′ ))) =
I1 (g, m) ⊗ (I1 (g, m) → (
_
I1 (g, m) ⊗ (I1 (g, m) → I2 (g, m)) ≤ I2 (g, m),
B∈LM
^
m′ ∈M
B∈LM
≤
m′ ∈M
_
I2 (g, m′ )) ≤
i.e. (37) holds. The proof is complete. Theorem 25 For every L-contexts hG, M, I1 i, hG, M, I2 i it holds E(hG, M, I1 i, hG, M, I2 i) = E Ext (B1 , B2 ) = E Int (B1 , B2 ). Therefore, E Ext and E Int are similarity relations on {B (G, M, I) | I ∈ LG×M }. Proof The assertion follows immediately by Corollary 21, Lemma 23, and Lemma 24. Both E Ext and E Int may thus be denoted by E. From the computational point of view, Theorem 25 shows that the computation of the similarity of two concept lattices defined naturally by (35) may be reduced to the computation of the similarity of the corresponding contexts which is usually much more simple. Indeed, the direct computation of E(B1 , B2 ) requires |B1 | · |B2 | · (|G| + |M |) evaluation of ↔ while computing E(I1 , I2 ) requires |G|·|M | evaluation of ↔. Note that usually |G|, |M | ≪ |B (G, M, I)| holds.
4
Example
In this section we present an illustrative example. Consider L with L = {0, 12 , 1} and the Lukasiewicz structure defined on L, cf. Section 2. The context is given by the Tab. 1. The set G contains nine elements (Mercury, . . . , Pluto), the set M contains four attributes (“size small”, . . . , “near to sun”). The corresponding fuzzy concept lattice is depicted in Fig. 1. To get a deeper insight, the elements (i.e. concepts) of the lattice are identified in Tab. 2. The similarity relation E G on G (cf. Theorem 6) is figured in Tab. 3 Consider now the a-cut of the induced similarity on B (G, M, I) for a = 12 , i.e. 1 2 E. The tolerance blocks (which are, in fact, complete sublattices) are depicted in Fig. 2. Notice that each block is a maximal subset of L-concepts which are similar in 1 the degree at least 12 . The corresponding factor lattice B (G, M, I)/ 2 E is depicted in Fig. 3. A few remarks to the examples. First, note that the concepts which were found depend on the fuzzy context which is given by a subjective judgement (e.g. to what degree we consider “Mars is far from sun” to be true). Second, there are apparently natural concepts in the concept lattice (e.g. 14 (“small planet near to sun”)), as well as concepts which “were found” (e.g. 26 (“a planet far from sun which is at least partially large”)). Concept no. 1 is an example of an empirically empty concept. Concepts which do not contain any element in the degree 1 in their extents (e.g. 1, 2, 3) are partially (empirically) empty. 19
Mercury (Me) Venus (V) Earth (E) Mars (Ma) Jupiter (J) Saturn (S) Uranus (U) Neptune (N) Pluto (P)
size small large (ss) (sl) 1 0 1 0 1 0 1 0 0 1 0 1 1 2 1 2
1 2 1 2
1
0
from sun far near (df) (dn) 0 1 0 1 0 1 1 1 2 1 1 2 1 1 2 1 0 1 0 1 0
Table 1: Fuzzy context given by planets and their properties.
c38 c37
35 c
c36
29 c 30 c
c31 32 c
21 c
22 c
23 c
14 c
15 c
16 c
9 c
10 c 5 c
33 c
c24 c25 26 c 17 c
c34 c27 c28
c18 c10
c20
c12
c13
6 c
c7
c8
2 c
c3
c4
c11
c 1 Figure 1: Concept lattice of the context in Tab. 1.
20
no. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.
Me 0 0 0 0
V 0 0 0 0
E 0 0 0 0
1 2
1 2
1 2
0 0 0
0 0 0
0 0 0
1 2 1 2 1 2
1 2 1 2 1 2
1 2 1 2 1 2
0 0 1
0 0 1
0 0 1
1 2 1 2 1 2
1 2 1 2 1 2
1 2 1 2 1 2
0 0 0 1 1
0 0 0 1 1
0 0 0 1 1
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
0 0 1 1
0 0 1 1
0 0 1 1
1 2 1 2 1 2
1 2 1 2 1 2
1 2 1 2 1 2
0 1 1
0 1 1
0 1 1
1 2
1 2
1 2
1
1
1
extent Ma J 0 0 1 0 2 1 0 2 0 0 1 0 2 1 2 1 2
0 1 1 2 1 2 1 2
0 1 1 1 1 2 1 2 1 2 1 2
1 1 1 1 1 2 1 2 1 2 1 2
1 1 1 1 1 2 1 2
1 1 1 1
1 2
S 0 0 1 2
0 0 1 2
0
0
1 2
1 2
0
0
1 2
1 2
0
0
1 2
1 2
1 0
1 0
1 2
1 2
0
0
1 2
1 2
0 1
0 1
1 2 1 2
1 2 1 2
0
0
1 2
1 2
U 0 0 0
N 0 0 0
1 2
1 2
0 0
0 0
1 2 1 2
1 2 1 2
0 0
0 0
1 2 1 2 1 2
1 2 1 2 1 2
0 0
0 0
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
1 0
1 0
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
0 1
0 1
1 2 1 2
1
1 2 1 2
1
1 1 1
1 1 1
1 2
1 2
1 2 1 2 1 2
1 2 1 2 1 2
0 1
0 1
1 2
1 2
1 1 1
1 1 1
1 2
1 2
1 1
1 1
P 0 0 0 0 0 0
ss 1 1
1 2
1
0 0 0 1 2 1 2
1 2
1 2 1 2 1 2
1 2
1 0
1 2
1 2 1 2 1 2
1 1 1 2
0 1
1 2 1 2
1
1 0 0 0
1 2
1 2
1 1 2 1 2
0 1 2 1 2
1 1 2 1 2
1 1 2 1 2
1 1 2
1 1 1
1
1 2
1 2
1 2
1 1 1
1 1
0 0 0
1 1 1 1 1 1
1 2
intent sl df 1 1 1 1 2 1 1 1 1
1 2
1 1 1 1
1 2
1 0 1 2 1 2
1 1 2
1 0 1 2 1 2
0 1 2
1 0 1 2
0 0 0 1 2
0 0
0 1 2 1 2
0 0 0 0 1 2 1 2
0 1 2
0 0 0 0 1 2
0 0 0 0 0
1 2
1 1 1 1 2 1 2 1 2
1 1 0 1 2 1 2 1 2
1 1 1 0 0 1 2 1 2 1 2 1 2
1 1 0 0 1 2 1 2 1 2
1 0 0 1 2
0
Table 2: Fuzzy concepts of the context of Tab. 1.
21
dn 1 1 1 1 2
1 1 1 2 1 2
1 1 1 2 1 2 1 2
1 1 1 2 1 2
0 1 2
0 1 1 2 1 2
0 1 2
0 0 0 1 2
0 1 2
0 0 0 1 2
0 0 0
38 a a a36 a37 35 a a a 29 30 31 32 a 33 a a34 21 a 22 a 23 a a24 a25 26 a a27 a28 a a10 a20 14 a 15 a 16 a 17 q 18 a q q q12 a13 9 10 11 q q q q8 5 6 7 q3 q4 2q 1 q1
38 a a a a37 35 36 a a a a 29 30 31 32 33 a a34 21a 22 a 23 q a24 a25 26 a a27 a28 a a10 a20 14a 15 q 16 q 17 q 18 q q q q12 a13 9 10 11 q q q7 a8 5 6 a3 a4 2q 2 a1
38 a a a a37 35 36 a a a a 29 30 31 32 33 a a34 21 a 22 a 23 a a24 q25 26 a a27 a28 a q10 a20 14 a 15 a 16 a 17 q 18 a q a q12 q13 9 10 11 a q a7 q8 5 6 q3 a4 2a 3 a1
38 a a a36 a37 35 a a a 29 30 31 32 a 33 a a34 21 a 22 a 23 a a24 a25 26 q a27 a28 a a10 q20 14 a 15 a 16 a 17 q 18 a a q q12 a13 9 10 11 a a q q8 5 6 7 a3 q4 2a 4 a1
38 a a a a37 35 36 q a a a 29 30 31 32 33 a a34 21q 22 q 23 q a24 a25 26 a a27 a28 a a10 a20 14q 15 q 16 q 17 q 18 q q q a12 a13 9 10 11 q a a7 a8 5 6 a3 a4 2a 5 a1
38 a a a a37 35 36 a a q a 29 30 31 32 33 a a34 21 a 22 a 23 q a24 q25 26 a a27 a28 a q10 a20 14 a 15 q 16 a 17 q 18 a q a q12 a13 9 10 11 a q a7 a8 5 6 a3 a4 2a 6 a1
38 a a a36 a37 35 a a a 29 30 31 32 q 33 a a34 21 a 22 a 23 q q24 a25 26 q q27 a28 q a10 q20 14 a 15 a 16 q 17 q 18 a a q q12 a13 9 10 11 a a q a8 5 6 7 a3 a4 2a 7 a1
38 a a a a37 36 35 a a a a 29 30 31 32 33 q a34 21a 22 a 23 a a24 q25 26 q a27 q28 a q10 q20 14a 15 a 16 a 17 q 18 a a a q12 q13 9 10 11 a a a7 q8 5 6 a3 a4 2a 8 a1
38 a q a a37 36 35 q a q a 29 30 31 32 33 a a34 21 q 22 a 23 q a24 q25 26 a a27 a28 a a10 a20 14 a 15 q 16 a 17 q 18 a q a a12 a13 9 10 11 a a a7 a8 5 6 a3 a4 2a 9 a1
38 a a q36 a37 35 q q a 29 30 31 32 q 33 a a34 21 a 22 q 23 q q24 a25 26 q a27 a28 a a10 a20 14 a 15 a 16 q 17 q 18 a a q a12 a13 9 10 11 a a a a8 5 6 7 a3 a4 2a 10 a1
38 a a a q37 35 36 a a q q 29 30 31 32 33 q q34 21a 22 a 23 q a24 q25 26 q q27 q28 a q10 q20 14a 15 a 16 a 17 q 18 a a a q12 a13 9 10 11 a a a7 a8 5 6 a3 a4 2a 11 a1
38 q q q q37 35 36 q a q q 29 30 31 32 33 q a34 21 a 22 a 23 q a24 q25 26 q a27 a28 a a10 a20 14 a 15 a 16 a 17 q 18 a a a a12 a13 9 10 11 a a a7 a8 5 6 a3 a4 2a 12 a1
1
Figure 2: Blocks of the tolerance relation 2 E on the concept lattice of Fig. 1.
22
Me V E Ma J S U N P
Me 1
V 1 1
E 1 1 1
Ma 1 2 1 2 1 2
1
J 0 0 0 0 1
S 0 0 0 0 1 1
U 0 0 0 0
N 0 0 0 0
1 2 1 2
1 2 1 2
1
1 2 1 2
1 1
1
Table 3: Similarity relation on objects.
12 f
9
f
5
f
P 0 0 0 0 0 0
f10
f11
6 f
f7
f8
2 f
f3
f4
f1
1
Figure 3: Factor lattice B (G, M, I)/ 2 E.
23
Acknowledgement. The author is indebted to anonymous referees for valuable remarks which helped to improve the paper.
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