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On heterogeneous formal contexts
Available online at www.sciencedirect.com
Fuzzy Sets and Systems 234 (2014) 22 – 33 www.elsevier.com/locate/fss
On heterogeneous formal contexts Lubomír Antoni, Stanislav Krajˇci∗ , Ondrej Krídlo, Bohuslav Macek, Lenka Pisková Institute of Computer Science, Faculty of Science, Pavol Jozef Šafárik University in Košice, Jesenná 5, 040 01 Košice, Slovakia Received 23 March 2012; received in revised form 10 April 2013; accepted 11 April 2013 Available online 23 April 2013
Abstract We propose a new type of fuzzification for formal concept analysis that works with heterogeneous values in a context and illustrate this with an example. We formulate and prove an appropriate counterpart to the so-called basic theorem of a concept lattice. We show that this is a generalization of the previous approaches: it covers the so-called generalized concept lattice and multi-adjoint t-concept lattices. © 2013 Elsevier B.V. All rights reserved. Keywords: Formal concept analysis; Galois connection; Adjointness
1. Introduction Formal concept analysis (FCA) is a data-mining method applied to a rectangular matrix of data in which each row corresponds to some object, each column corresponds to some possible attribute, and the matrix field value denotes a membership of the column attribute for row object. One of the goals of this method is to find so-called concepts, which are stable (in some sense) pairs of subsets of objects and attributes. The method can be considered a nice application of the algebraic notion of a Galois connection. It has been described in detail by Ganter and Wille [12], in particular for the so-called crisp case with binary matrix data. A natural question that arises is what happens if the matrix data are non-binary. Besides the conceptual scaling that returns concepts with crisp subsets in both coordinates [12], some other approaches return concepts with fuzzy subsets in at least one coordinate. The first was provided by Burusco and Fuentes-Gonzalez [9] and was independently improved by Bˇelohlávek [2,3] and Pollandt [25,26] using values from the same residual lattice for matrix values and for the fuzziness of subsets of the objects and attributes. Another approach independently proposed by Ben Yahia and Jaoua [8], Bˇelohlávek et al. [5] and Krajˇci [13] is not as symmetric; it considers fuzzy subsets in one coordinate and crisp (binary) subsets in another one. All these approaches are covered by a common platform, the so-called generalized concept lattice [15,16], which diversifies the fuzziness of subsets of the attributes, the fuzziness of subsets of the objects, and the fuzziness of the matrix values. Medina and Ojeda-Aciego then applied the idea of multi-adjointness used in logic-programming [20–22] to FCA [17,19]. Because of its novelty and originality, this approach is not (at least immediately) covered by generalized concept lattices. This inspired us to modify our previous approach so that it will work with different mutual relationships between the objects and attributes. We also use different lattices for the different rows and columns and the matrix data. ∗ Corresponding author. Tel.: +421 55 234 2508.
E-mail address: [email protected] (S. Krajˇci). 0165-0114/$ - see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.04.008
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The usefulness of this approach in practice is demonstrated by an example of personal accommodation requirements. In comparison with previous methods that work for attributes and objects of the same type, an important advantage of our new approach is the possibility to apply FCA to heterogeneous data. Thus, we call this new approach heterogeneous.
2. Heterogeneous formal context Consider the following situation. People who are going to stay at a cottage together have different preferences for conditions, depending on the number of days spent at the cottage. One may prefer hot water, whereas others may expect an internet connection. The following requirements are examples: • • • •
Eva will accept large discomfort regarding hot water availability and partial discomfort for an internet/TV connection. John will accept partial discomfort regarding hot water availability and large discomfort for an internet/TV connection. Eva will not accept the absence of hot water on one arbitrary day at the cottage. John will not accept the absence of hot water only on his second day.
The problem is to identify which conditions have to be fulfilled to satisfy all the people staying at the cottage for different numbers of days. We cannot cover this type of complex information using existing approaches. Therefore, we define a heterogeneous formal context to formally describe the above situation. Let A and B be non-empty sets. Let P = ((Pa,b , ≤ Pa,b ) : a ∈ A, b ∈ B) be a system of posets and let R be a function from A × B such that R(a, b) ∈ Pa,b for all a ∈ A and b ∈ B. Let C = ((Ca , ≤Ca ) : a ∈ A) and D = ((Db , ≤ Db ) : b ∈ B) be systems of complete lattices. (For simplicity, we omit the indices for all ≤? , since it is always clear which one is used.) Let = (•a,b : a ∈ A, b ∈ B) be a system of operations such that •a,b is from Ca × Db to Pa,b and is isotone and left-continuous in both arguments, that is: (1a) (1b) (2a) (2b)
c1 ≤ c2 implies c1 •a,b d ≤ c2 •a,b d for all c1 , c2 ∈ Ca and d ∈ Db , d1 ≤ d2 implies c •a,b d1 ≤ c •a,b d2 for all c ∈ Ca and d1 , d2 ∈ Db , If c •a,b d ≤ p for some d ∈ Db , p ∈ Pa,b and all c ∈ X ⊆ Ca , then sup X •a,b d ≤ p, and If c •a,b d ≤ p for some c ∈ Ca , p ∈ Pa,b and all d ∈ Y ⊆ Db , then c •a,b sup Y ≤ p.
Then we call the tuple A, B, P, R, C, D, a heterogeneous formal context. Note that if Ca = Db and •a,b is commutative, then these conditions can be reduced to the following two: (1) c1 ≤ c2 implies c1 •a,b d ≤ c2 •a,b d for all c1 , c2 , d ∈ Ca = Db , and (2) If c •a,b d ≤ p for some d ∈ C, p ∈ Pa,b and all c ∈ X ⊆ Ca = Db , then sup X •a,b d ≤ p. To illustrate such a heterogeneous formal context, let B = {Eva, John, . . .} be the set of objects consisting of people thinking about staying at a cottage. Let A = {water, services, lake} be the set of attributes in relation to hot water, services and lake availability at the cottage. For simplicity, we consider just two people and two conditions from Fig. 1, but this heterogeneous method also works for arbitrary numbers of people and conditions. Eva has three preferences for staying: not at all, one day (it does not matter which one) or both days (DEva ). John has four preferences: not at all, only Saturday, only Sunday, or both days (DJohn ). For the attributes, the water can be hot or cold (Cwater ) and there are four possibilities for services: internet and television, internet only, television only, or nothing at all (Cservices ). Each person can accept different degrees of discomfort for their preferences. Thus, the behavior of each person can be expressed as = (•water,Eva , •services,Eva , •water,John , •services,John ). Similarly P = (Pwater,Eva , Pservices,Eva , Pwater,John , Pservices,John ) expresses the degrees of discomfort for each person and each condition. For instance, •services,Eva is from Cservices × DEva to Pservices,Eva , where Pservices,Eva = {0, 1/2, 1} denotes comfort, partial discomfort and large discomfort, respectively, for Eva. The behavior of the isotone and left-continuous operations = (•water,Eva , •services,Eva , •water,John , •services,John ) with respect to the number of days and cottage conditions is shown in Fig. 2. We provide the following remarks and interpretation. • Note that c •services,Eva ∅ = 0 for all c ∈ Cservices , because no-one staying at the cottage and arbitrary conditions correspond to no discomfort.
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Fig. 1. List of possible values for objects and attributes in a heterogeneous case.
Fig. 2. Isotone and left-continuous operations.
• Note that hot •water,Eva d = 0 for all d ∈ DEva , because the availability of hot water and an arbitrary number of days correspond to no discomfort. • Note that in + tv •services,John d = 0 for all d ∈ DJohn , because the presence of all services and an arbitrary number of days correspond to no discomfort. • The following demonstrates monotonicity: staying on Saturday and cold water represent partial discomfort for John, but 2 days and cold water lead to large discomfort.
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Fig. 3. Heterogeneous formal context.
• Similarly, staying on 1 day and internet only represent partial discomfort for Eva, but 2 days and internet only or missing internet lead to large discomfort. • Missing TV and Saturday represent one-third discomfort for John; missing internet and Saturday, two-thirds discomfort; and missing internet and Sunday or 2 days, large discomfort. • The following demonstrates left-continuity: Saturday or Sunday and internet only represent one-third discomfort for John, but the supremum of these days (Saturday+Sunday) and internet only also lead to one-third discomfort. Each person defines the acceptable degree of discomfort for specific conditions: i.e., R(water, Eva) ∈ P(water, Eva), R(services, Eva) ∈ P(services, Eva), R(water, John) ∈ P(water, John) and R(services, John) ∈ P(services, John). For example, Eva can accept large discomfort for water conditions and partial discomfort for services, as shown in Fig. 3. Large discomfort for water conditions (value of 1) for Eva means that she will accept an arbitrary number of days and hot or cold water, because all values for water for Eva are less than or equal to 1 in Fig. 2. Partial discomfort for services (value of 1/2) for Eva means that she will accept neither all services nor a maximum of one arbitrary day with internet only, because the values for these cases for Eva in Fig. 2 are less than or equal to 1/2. To apply FCA and identify the cottage conditions that fulfill all personal requirements, we define the mappings
and . Let F be a set of all functions f with a domain A such that f (a) ∈ Ca for all a ∈ A (more formally, F = a∈A Ca ). Let G be a set of all functions g with a domain B such that g(b) ∈ Db for all b ∈ B (i.e., G = b∈B Db ). We define the mapping : G → F. If g ∈ G, then (g) ∈ F is defined by ( (g))(a) = sup{c ∈ Ca : (∀b ∈ B)c •a,b g(b) ≤ R(a, b)}. Symmetrically, we define the mapping : F → G. If f ∈ F, then ( f ) ∈ G is defined as ( ( f ))(b) = sup{d ∈ Db : (∀a ∈ A) f (a) •a,b d ≤ R(a, b)}. ( , ) are defined by applying the so-called (sup, •a,b )-product. The properties of such a structure are described in the literature [4,10]. Theorem 1. Let f ∈ F and g ∈ G. Then the following conditions are equivalent: (1) f ≤ (g). (2) g ≤ ( f ). (3) f (a) •a,b g(b) ≤ R(a, b) for all a ∈ A and b ∈ B. Proof. 1 → 3 Let a ∈ A and b ∈ B. Then f ≤ (g) implies that f (a) ≤ ( (g))(a). Hence, by the definition of we have f (a) ≤ sup{c ∈ Ca : (∀b ∈ B)c •a,b g(b ) ≤ R(a, b )}. Because the condition (∀b ∈ B)c •a,b g(b ) ≤ R(a, b ) implies the condition c •a,b g(b) ≤ R(a, b) for b (as a special case of b ∈ B), we have {c ∈ Ca : (∀b ∈ B)c •a,b g(b ) ≤ R(a, b )} ⊆ {c ∈ Ca : c •a,b g(b) ≤ R(a, b)}, so f (a) ≤ sup{c ∈ Ca : (∀b ∈ B)c •a,b g(b ) ≤ R(a, b )} ≤ sup{c ∈ Ca : c •a,b g(b) ≤ R(a, b)}.
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By the isotonity of •a,b in the first argument, we obtain f (a) •a,b g(b) ≤ sup{c ∈ Ca : c •a,b g(b) ≤ R(a, b)} •a,b g(b). Let X denotes the supremised set on the right-hand side of our inequality. Then c •a,b g(b) ≤ R(a, b) for every c ∈ X . Hence, by the left-continuity of •a,b in the first argument, we have sup X •a,b g(b) ≤ R(a, b). However, from the transitivity of ≤ we obtain f (a) •a,b g(b) ≤ R(a, b). 2 → 3 This can be proved analogously. 3 → 1 Let (∀b ∈ B) f (a) •a,b g(b) ≤ R(a, b) hold for all a ∈ A. This means that f (a) is an element of the set {c ∈ Ca : (∀b ∈ B)c •a,b g(b) ≤ R(a, b)} and hence f (a) ≤ sup{c ∈ Ca : (∀b ∈ B)c •a,b g(b) ≤ R(a, b)} = ( (g))(a). Because a is arbitrary, we have f ≤ (g). 3 → 2 This can be proved analogously. Corollary 1. Mappings and form a Galois connection. Proof. The proof follows from the equivalence of conditions 1 and 2 of the previous theorem. Corollary 2. (1a) g1 ≤ g2 implies (g1 ) ≥ (g2 ). (1b) f 1 ≤ f 2 implies ( f 1 ) ≥ (2 ). (2a) g ≤ ( (g)). (2b) f ≤ ( ( f )). (3a) (g) = ( ( (g))). (3b) ( f ) = ( ( ( f ))). Proof. The proof is a consequence of ( , ) being a Galois connection [12]. We use mappings ( , ) to identify the required cottage conditions as follows. Mapping ( ( f ))(b) indicates maximization of the number of days spent at the cottage for specific water and services conditions that return the greatest degree of discomfort accepted by a person. For instance, for f (water) = hot, f (services) = in we obtain ( ( f ))(Eva) = 1/2, which means that hot water and internet only correspond to a maximum stay of 1 day for Eva. Mapping ( (g))(a) indicates the worst water or services conditions at the cottage for a specific number of days that return the greatest degree of discomfort accepted. For instance, if g(Eva) = 1/2, g(John) = Sa, then we obtain ( (g))(water) = cold, which means that Eva staying for 1 day and John staying on Saturday correspond to the possibility of cold water at the cottage. In another example, if g(Eva) = Sa + Su, g(John) = Sa, then we obtain ( (g))(services) = in + tv, whereby a cottage with an internet connection and TV is the worst possible case if Eva stays on Saturday and Sunday and John stays on Saturday. 3. Heterogeneous formal concept We use the Galois connection ( , ) for concept lattice construction via the classical Ganter–Wille approach [12]. Lemma 1. (1) Let {gi : i ∈ I } ⊆ G. Then
i∈I
gi
=
i∈I
(gi ).
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Fig. 4. Heterogeneous formal concepts.
(2) Let { f i : i ∈ I } ⊆ F. Then fi = ( f i ). i∈I
i∈I
Proof. The proof is a consequence of ( , ) being a Galois connection [12]. We call a pair g, f from G × F such that (g) = f and ( f ) = g a concept. Lemma 2. If g1 , f 1 and g2 , f 2 are concepts, then g1 ≤ g2 iff f 1 ≥ f 2 . Proof. The proof is a simple consequence of Corollary 2 (parts 3a and 3b). This lemma allows us to define the following ordering of concepts: g1 , f 1 ≤ g2 , f 2 iff g1 ≤ g2 (or equivalently f 1 ≥ f 2 ). In summary, five concepts are obtained in our example, as shown in Fig. 4. Intents correspond to the worst cottage conditions that fulfill all personal requirements for a specific number of days. For example, cold water and no services at the cottage correspond to no stay for Eva and staying on Saturday for John (first concept). By contrast, hot water and full services correspond to the maximum number of days spent at the cottage for both Eva and John (last concept). The next three concepts can be interpreted similarly. For instance, for the third concept, stays of 1 day by Eva and 2 days by John correspond to hot water and an internet connection as the worst possible conditions. Note that intents do not include the possibility of hot water and no services simultaneously. In this case we obtain (hot, no) = (∅, Sa) and subsequently (∅, Sa) = (cold, no). This can be interpreted as superfluous conditions for John’s stay on Saturday and maybe a cheaper cottage can be chosen. All computations in our cottage example are for two people, but it is possible to consider more complex examples, as in Fig. 1. This case also contains the poset Pservices,Ken , where Pservices,Ken = {0, le, se, 1} denotes comfort, discomfort for length of stay, discomfort for services and large discomfort, respectively. In such cases it is possible for two people to have the same lattice structure (e.g., Eva and Lea). Nevertheless, the behavior of Eva and Lea can differ for different conditions. A stay of 1 day and cold water could correspond to discomfort for Eva but comfort for Lea or vice versa. In conclusion, this heterogeneous proposal on a formal context works with heterogeneous structures for objects, attributes and table values, so it is not possible to express this using existing approaches. 4. Heterogeneous formal concept lattice We call the poset of all concepts ordered by ≤ a heterogeneous concept lattice, denoted by HCL( A, B, P, R, C, D, , , , ≤). The following theorem shows that this is in reality a lattice. Analogous theorems have been proved for existing approaches [11,12,15]. To unify these and show that the same conditions are fulfilled for our high level of generalization, the proof is self-contained.
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Theorem 2 (Basic theorem on heterogeneous concept lattices). (1) A heterogeneous concept lattice HCL(A, B, P, R, C, D, , , , ≤) is a complete lattice in which gi , f i = gi , fi i∈I
i∈I
and
i∈I
gi , f i =
i∈I
gi
,
i∈I
fi .
i∈I
(2) For each a ∈ A and b ∈ B, let Pa,b have the least element 0 Pa,b such that 0Ca •a,b d = c •a,b 0 Db = 0 Pa,b for all c ∈ Ca , d ∈ Db . Then a complete lattice L is isomorphic to HCL(A, B, P, R, C, D, , , , ≤) if and only if there are mappings : a∈A ({a} × Ca ) → L and : b∈B ({b} × Db ) → L such that: (1a) does not increase in the second argument (for a fixed first argument); (1b) does not decrease in the second argument (for a fixed first argument); (2a) Rng() is inf-dense in L; 1 (2b) Rng() is sup-dense in L; and (3) For every a ∈ A, b ∈ B, c ∈ Ca and d ∈ Db , (a, c) ≥ (b, d) if and only if c •a,b d ≤ R(a, b). Proof. (1) We prove only the first part, since proof of the second part is analogous. Using Lemma 1 we obtain
. gi =
( f i ) = fi i∈I
i∈I
i∈I
Using Corollary 1 (part 3b) and the fact that gi , f i is a concept, we have = fi fi = ( f i ) = gi . i∈I
i∈I
i∈I
i∈I
This proves i , ( ( i∈I f i )) is a concept. that i∈I g Clearly ( j∈I g j )(b) = j∈I g j (b) ≤ gi (b), which means that j∈I g j ≤ gi for each i ∈ I . Thus, our concept is the lower bound of all gi , f i . We show that this other lower is the greatest. Let g , f be some bound. This means that g ≤ gi for all i ∈ I , and hence g ≤ i∈I gi . However, this means that g , f ≤ i∈I gi , ( ( i∈I f i )). (2) First we consider a special case, the lattice HCL( A, B, P, R, C, D, , , , ≤) or H in short, and show the implication →. We define the following singleton functions Sac ∈ F for each a ∈ A and c ∈ Ca , and Tbd ∈ G for each b ∈ B and d ∈ Db : c if x = a, d if y = b, c d Sa (x) = and Tb (y) = 0Ca elsewhere 0 Db elsewhere. Note that ( (Sac ))(b) = sup{d ∈ Db : c •a,b d ≤ R(a, b)} 1 Rng() denotes the mapping range of .
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because by the definition of Sac , for each a a, Sac (a ) •a ,b d = 0Ca •a ,b d = 0 Pa ,b , and hence the condition Sac (a ) ≤ R(a , b) is automatically fulfilled. Dually, ( (Tbd ))(a) = sup{c ∈ Ca : c •a,b d ≤ R(a, b)}. Now let H (a, c) = (Sac ), ( (Sac )) and H (b, d) = ( (Tbd )), (Tbd ). Note that both H (a, c) and H (b, d) are concepts (by Corollary 2) and hence it is enough to consider their first coordinates, for example. Now we prove that they have the desired properties. (1a) Let a ∈ A, b ∈ B, c1 , c2 ∈ Ca and c1 ≤ c2 . By the isotonity of •a,b in the first argument, we have c1 •a,b d ≤ c2 •a,b d for all d ∈ Db . This means that {d ∈ Db : c1 •a,b d ≤ R(a, b)} ⊇ {d ∈ Db : c2 •a,b d ≤ R(a, b)}, from which it follows that ( (Sac1 ))(b) = sup{d ∈ Db : c1 •a,b d ≤ R(a, b)} ≥ sup{d ∈ Db : c2 •a,b d ≤ R(a, b)} = ( (Sac2 ))(b) for all b ∈ B. This means that (Sac1 ) ≥ (Sac2 ), which implies H (a, c1 ) ≥ H (a, c2 ). (1b) This can be proved dually. (2a) We prove that if g, f is a concept, then g, f = inf{ H (a, f (a)) : a ∈ A}, which implies the required assertion. f (a) f (a) f (a) Let a ∈ A. Then Sa (a) = f (a) and Sa (a ) = 0Ca ≤ f (a ) for all a a, so Sa ≤ f . By Corollary 2, we f (a) obtain (Sa ) ≥ ( f ) = g, which means that g, f ≤ H (a, f (a)) (both are concepts, so it is enough to consider their first coordinates). This is true for all a ∈ A, and hence g, f ≤ inf{ H (a, f (a)) : a ∈ A}. Let g , f be some other lower bound of { H (a, f (a)) : a ∈ A}. Then for each a ∈ A we have g , f ≤ f (a) f (a) H (a, f (a)), so g ≤ (Sa ) (these are their first coordinates). By Theorem 1, we have Sa (a) •a,b g (b) ≤ R(a, b), that is, f (a) •a,b g (b) ≤ R(a, b) for all a ∈ A. This means that g (b) ∈ {d ∈ Db : (∀a ∈ A) f (a) •a,b d ≤ R(a, b)}, so g (b) ≤ sup{d ∈ Db : (∀a ∈ A) f (a) •a,b d ≤ R(a, b)} = ( ( f ))(b) = g(b) for all b ∈ B, that is, g ≤ g and hence g , f ≤ g, f . However, this means that g, f = inf{ H (a, f (a)) : a ∈ A}. Thus, each concept can be written as the infimum of some elements from Rng( H ), that is, the set is inf-dense. (2b) It can be proved dually that g, f = sup{(b, g(b)) : b ∈ B}, which implies the required assertion. (3) → c •a,b d ≤ R(a, b) means that c ∈ {c ∈ Ca : c •a,b d ≤ R(a, b)}, which implies that Sac (a) = c ≤ sup{c ∈ Ca : c •a,b d ≤ R(a, b)} = ( (Tbd ))(a). For a a we also have Sac (a ) = 0Ca ≤ ( (Tbd ))(a ). Hence, Sac ≤ (Tbd ). By Corollary 2, we obtain (Sac ) ≥ ( (Tbd )), which means that H (a, c) ≥ H (b, d) (both are concepts, so it is enough to consider their first coordinates).
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(3) ← H (a, c) ≥ H (b, d) means that ( (Tbd )) ≤ (Sac ) (these are their first coordinates). By Corollary 2, we have Tbd ≤ ( (Tbd )). Hence, by the transitivity of ≤ we obtain Tbd ≤ (Sac ). By Theorem 1, Sac (a ) •a ,b Tbd (b ) ≤ R(a , b ) for all a ∈ A and b ∈ B. In particular, for a and b, Sac (a) •a,b Tbd (b) ≤ R(a, b), that is, c •a,b d ≤ R(a, b). Let L be any complete lattice isomorphic to H and let be some isomorphism between them. Then we define and as follows: (a, c) = ( H (a, c)) (b, d) = ( H (b, d)). Now we show that preserves the desired properties fulfilled in H . (1a) Let a ∈ A and c1 , c2 ∈ Ca , c1 ≤ c2 . Then we know that H (a, c1 ) ≥ H (a, c2 ). Because is an isomorphism, we have ( H (a, c1 )) ≥ ( H (a, c1 )) i.e., (a, c1 ) ≥ (a, c1 ). (1b) This can be proved dually. (2a) Let ∈ L. Because is a bijection, there exist f and g such that = (g, f ). We know that g, f = inf{ H (a, f (a)) : a ∈ A} and that is an isomorphism, so we have = (g, f ) = (inf{ H (a, f (a)) : a ∈ A}) = inf{( H (a, f (a))) : a ∈ A} = inf{(a, f (a)) : a ∈ A}, which means that Rng() is inf-dense. (2b) This can be proved dually. (3) (a, c) ≥ (b, d) means that ( H (a, c)) ≥ ( H (b, d)). Because is an isomorphism, this is equivalent to H (a, c) ≥ H (b, d) and we know that this holds if and only if c •a,b d ≤ R(a, b). Thus, we have proven the implication → for all lattices. Now we prove the opposite direction, ←. Let L be an arbitrary complete lattice and let and have the assumed properties. We define a mapping from the support of H to the support of L as follows: (g, f ) = inf{(a, f (a)) : a ∈ A}. We also define a mapping from the support of L to the support of H as () = g , f , where f (a) = sup{c ∈ Ca : (a, c) ≥ } g (b) = sup{d ∈ Db : (b, d) ≤ }. First, we prove that () is a concept from the support of H . Claim 1. (a) (g ) = f ; and (b) ( f ) = g . Proof. (a) By Theorem 1 it is enough to prove f (a) •a,b g (d) ≤ R(a, b) for each a ∈ A and b ∈ B. Let c ∈ Ca be such that c ∈ {c ∈ Ca : (a, c) ≥ }, that is, (a, c) ≥ . Similarly, let d ∈ Db be such that d ∈ {d ∈ Db : (b, d) ≥ }, that is, (b, d) ≤ . By the transitivity of ≤ we obtain (a, c) ≥ (b, d), which means (by the assumptions for and ) that c •a,b d ≤ R(a, b). Because c is an arbitrary element of {c ∈ Ca : (a, c) ≥ }, by the left-continuity of •a,b in the first argument we have sup{c ∈ Ca : (a, c) ≥ } •a,b d ≤ R(a, b), that is, f (a) •a,b d ≤ R(a, b). Because d is an arbitrary element of {d ∈ Db : (b, d) ≤ }, by the left-continuity of •a,b in the second argument we have f (a) •a,b sup{d ∈ Db : (b, d) ≥ } ≤ R(a, b), that is, f (a) •a,b g (b) ≤ R(a, b), as required. (b) This can be proved analogously.
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Now we show that is the required isomorphism and is its inverse. Claim 2. preserves the ordering. Proof. Let g1 , f 1 ≤ g2 , f 2 be two concepts. Then, by definition, f 1 ≥ f 2 . Because is non-increasing in the second argument, we obtain (a, f 1 (a)) ≤ (a, f 2 (a)) for each a ∈ A. It follows that (g1 , f 1 ) = inf{(a, f 1 (a)) : a ∈ A} ≤ inf{(a, f 2 (a)) : a ∈ A} = (g2 , f 2 ).
Claim 3. preserves the ordering. Proof. Let 1 ≤ 2 . It follows from transitivity that {c ∈ Ca : (a, c) ≥ 1 } ⊇ {c ∈ Ca : (a, c) ≥ 2 } and hence f 1 (a) = sup{c ∈ Ca : (a, c) ≥ 1 } ≥ sup{c ∈ Ca : (a, c) ≥ 2 } = f 2 (a) for each a ∈ A. This means that f 1 ≥ f 2 , so (1 ) = g1 , f 1 ≤ g2 , f 2 = (2 ).
Claim 4. (()) = . Proof. Because of the sup-density of Rng(), can be written as the supremum of some set of pairs from Rng(), that is, = sup{(bi , di ) : i ∈ I }, where di ∈ Dbi for each i ∈ I . This means that (bi , di ) ≤ for each i ∈ I . If a ∈ A and c ∈ Ca are arbitrary such that (a, c) ≥ , we obtain (a, c) ≥ (bi , di ) for each i ∈ I . By property (3) for and , we obtain c •a,bi di ≤ R(a, bi ). Hence, by the left-continuity of •a,bi in the first argument, we have f (a) •a,bi di = sup{c ∈ Ca : (a, c) ≥ } •a,bi di ≤ R(a, bi ) for each i ∈ I . Again by Assumption (3) we have (a, f (a)) ≥ (bi , di ) for each i ∈ I , and hence (a, f (a)) ≥ sup{(bi , di ) : i ∈ I } = . It follows that (()) = (g , f ) = inf{(a, f (a)) : a ∈ A} ≥ . Conversely, because of the inf-density of Rng(), can be written as the infimum of some set of pairs from Rng(), that is, = inf{(ai , ci ) : i ∈ I }, where ci ∈ Cai for each i ∈ I . Obviously (ai , ci ) ≥ , so ci ∈ {c ∈ Cai : (ai , c) ≥ }, from which it follows that ci ≤ sup{c ∈ Cai : (ai , c) ≥ } = f (ai ) for each i ∈ I . Because is non-increasing in the second argument, we obtain (ai , ci ) ≥ (ai , f (ai )) ≥ inf{(a, f (a)) : a ∈ A} = (g , f ) = (()) for each i ∈ I . This means that = inf{(ai , ci ) : i ∈ I } ≥ (()), so the opposite inequality is also proven. Claim 5. ((g, f )) = g, f . Proof. We denote = (g, f ). We want to prove () = g, f , that is, g , f = g, f . Because both are concepts, it is enough to prove g = g.
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Using property (3), we obtain (∀a ∈ A) f (a) •a,b d ≤ R(a, b) if and only if (∀a ∈ A)(b, d) ≤ (a, f (a)). By the definition of the infimum, this is equivalent to (b, d) ≤ inf{(a, f (a)) : a ∈ A} = (g, f ) = . This means that {d ∈ Db : (∀a ∈ A) f (a) •a,b d ≤ R(a, b)} = {d ∈ Db : (b, d) ≤ }, from which it follows that ( ( f ))(b) = sup{d ∈ Db : (∀a ∈ A) f (a) •a,b d ≤ R(a, b)} = sup{d ∈ Db : (b, d) ≤ } = g (b), that is, g = ( f ) = g . We have proved that both and preserve the ordering and both their compositions are identities. This means that = −1 and both are isomorphisms. 5. Generalization Our previous approach called a generalized (fuzzy) concept lattice [13,14] is obviously a special case of this new approach; it is enough to take the same Ca , Db , ≤a,b and •a,b . The multi-adjointness approach of Medina and Ojeda-Aciego [17] (that generalizes the approach from [19] in which the adjoint triples are defined on (L,L,P)) has some rather strange and not very aesthetic property: it takes only •b for b ∈ B without any reference to A. We symmetrize it and take operations •a,b for each pair (a, b) ∈ A × B. This, of course, diversifies and generalizes [17]. Moreover we need not the equal lattices for all b ∈ B and/or all a ∈ A. We note one issue. The approach of Medina and Ojeda-Aciego [17] works with adjoints to • operations. Recall the notion: Let (C, ≤C ), (D, ≤ D ) and (P, ≤ P ) be posets. The triple (•, →1 , →2 ) is called an adjoint triple if • : (C × D) → P, →1 : (D × P) → C, →2 : (C × P) → D and (c • d) ≤ p iff c ≤ (d →1 p) iff d ≤ (c →2 p). (If C = D and • is commutative, both arrows are identical.) Hence, all the substantial definitions of Medina and Ojeda-Aciego [17] can be reformulated using • operations only. For each operation • that is isotone and left-continuous in both arguments, there are operations →1 and →2 such that (•, →1 , →2 ) is an adjoint triple; it is enough to define d →1 p = sup{c ∈ C : c • d ≤ P p} and symmetrically c →2 p = sup{d ∈ D : c • d ≤ P p}. Obviously, c • d ≤ P p means that c ∈ {c ∈ C : c • d ≤ P p} and hence c ≤C sup{c ∈ C : c • d ≤ P p} = d →1 p. Conversely, by the left-continuity of • in the first argument, we have sup{c ∈ C : c • d ≤ P p} • d ≤ P p, that is, (d →1 p) • d ≤ P p. Thus, if c ≤C d →1 p, then by the isotonity of • in the first argument we have c • d ≤ P p. The dual properties of →2 can be proved symmetrically. This means that working with adjoint triples is equivalent to working with isotone and left-continuous functions. 6. Conclusions and future work We introduced a new common platform for fuzzification of FCA. The main idea is diversification of all that can be diversified. The main advantage of our approach is removal of the homogeneity barrier. It intuitively follows that FCA could be used for tables with data of different types. We will consider these issues in further research. Pócs made an alternative attempt to cover all known approaches [23,24]. This also works with different lattices but it gives not just a single value but a Galois connection to fields of the matrix (i.e., certain behaviors for relationship between the object and the attribute). However, it seems rather cumbersome to work with whole Galois connections in single data fields. Our approach has no limitation regarding values in data fields. We have previously compared these approaches [1].
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Another interesting issue would be to clarify the relationship of this approach to the fuzzification of Bˇelohlávek and Vychodil for truth-stressers or so-called hedges [6,7]. Krajˇci showed that generalized concept lattices cover these in some sense [16], but our new approach seems to make this relationship more immediate. Another interesting relationship for future work is heterogeneity in multi-adjoint concept lattices that are based on heterogeneous conjunctors [18]. Acknowledgement This work was partly supported by grant VEGA 1/0832/12, by the Slovak Research and Development Agency under contract APVV-0035-10 “Algorithms, Automata, and Discrete Data Structures”, and by the Agency of the Slovak Ministry of Education for Structural Funds of the EU under project ITMS: 26220120007. References [1] L. Antoni, S. Krajˇci, O. Krídlo, B. Macek, L. Pisková, Relationship between two FCA approaches on heterogeneous formal contexts, in: L. Szathmary, U. Priss (Eds.), Proceedings of CLA, 2012, pp. 93–102. [2] R. Bˇelohlávek, Fuzzy concepts and conceptual structures: induced similarities, in: Proceedings of JCIS 98, vol. I, 1998, pp. 179–182. [3] R. Bˇelohlávek, Concept lattices and order in fuzzy logic, Ann. Pure Appl. Logic 128 (2004) 277–298. [4] R. Bˇelohlávek, Sup-t-norm and inf-residuum are one type of relational product: unifying framework and consequences, Fuzzy Sets Syst. 197 (2012) 45–58. [5] R. Bˇelohlávek, V. Sklenáˇr, J. Zacpal, Crisply generated fuzzy concepts, Lect. Notes Comput. Sci. 3403 (2005) 268–283. [6] R. Bˇelohlávek, V. Vychodil, Reducing the size of fuzzy concept lattices by hedges, in: Proceedings of FUZZ-IEEE 2005, pp. 663–668. [7] R. Bˇelohlávek, V. Vychodil, Formal concept analysis and linguistic hedges, Int. J. Gen. Syst. 41 (2012) 503–532. [8] S. Ben Yahia, A. Jaoua, Discovering knowledge from fuzzy concept lattice, in: A. Kandel, M. Last, H. Bunke (Eds.), Data Mining and Computational Intelligence, Physica-Verlag, 2001, pp. 169–190. [9] A. Burusco, R. Fuentes-Gonzalez, The study of L-fuzzy concept lattice, Mathware Soft. Comput. 3 (1994) 209–218. [10] M.E. Cornejo, J. Medina, E. Ramírez, A comparative study of adjoint triples, Fuzzy Sets Syst. 211 (2013) 1–14. [11] B.A. Davey, H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, 2002. [12] B. Ganter, R. Wille, Formal Concept Analysis Mathematical Foundation, Springer Verlag, 1999. [13] S. Krajˇci, Cluster based efficient generation of fuzzy concepts, Neural Network World 13 (2003) 521–530. [14] S. Krajˇci, A generalized concept lattice, Logic J. IGPL 13 (2005) 543–550. [15] S. Krajˇci, The basic theorem on generalized concept lattice, in: V. Snášel, R. Bˇelohlávek (Eds.), Proceedings of the 2nd International Conference on Concept Lattices and Their Applications, 2004, pp. 25–33. [16] S. Krajˇci, Every concept lattice with hedges is isomorphic to some generalized concept lattice, in: V. Snášel, R. Bˇelohlávek (Eds.), Proceedings of the 3rd International Conference on Concept Lattices and Their Applications, 2005, pp. 1–9. [17] J. Medina, M. Ojeda-Aciego, Multi-adjoint t-concept lattices, Inf. Sci. 180 (2010) 712–725. [18] J. Medina, M. Ojeda-Aciego, On multi-adjoint concept lattices based on heterogeneous conjunctors, Fuzzy Sets Syst. 208 (2012) 95–110. [19] J. Medina, M. Ojeda-Aciego, J. Ruiz-Calviño, Formal concept analysis via multi-adjoint concept lattices, Fuzzy Sets Syst. 160 (2009) 130–144. [20] J. Medina, M. Ojeda-Aciego, A. Valverde, P. Vojtáš, Towards biresiduated multi-adjoint logic programming, Lect. Notes Artif. Intell. 3040 (2004) 608–617. [21] J. Medina, M. Ojeda-Aciego, P. Vojtáš, Multi-adjoint logic programming with continuous semantics, Lect. Notes Artif. Intell. 2173 (2001) 351–364. [22] J. Medina, M. Ojeda-Aciego, P. Vojtáš, Similarity-based unification: a multi-adjoint approach, Fuzzy Set Syst. 146 (2004) 43–62. [23] J. Pócs, Note on generating fuzzy concept lattices via Galois connections, Inf. Sci. 185 (2012) 128–136. [24] J. Pócs, On possible generalization of fuzzy concept lattices, Inf. Sci. 210 (2012) 89–98. [25] S. Pollandt, Fuzzy Begriffe, Springer, 1997. [26] S. Pollandt, Datenanalyse mit Fuzzy-Begriffen, in: G. Stumme, R. Wille (Eds.), Begriffliche Wissensverarbeitung. Methoden und Anwendungen, Springer, Heidelberg, 2000, pp. 72–98.
Fuzzy Sets and Systems 234 (2014) 22 – 33 www.elsevier.com/locate/fss
On heterogeneous formal contexts Lubomír Antoni, Stanislav Krajˇci∗ , Ondrej Krídlo, Bohuslav Macek, Lenka Pisková Institute of Computer Science, Faculty of Science, Pavol Jozef Šafárik University in Košice, Jesenná 5, 040 01 Košice, Slovakia Received 23 March 2012; received in revised form 10 April 2013; accepted 11 April 2013 Available online 23 April 2013
Abstract We propose a new type of fuzzification for formal concept analysis that works with heterogeneous values in a context and illustrate this with an example. We formulate and prove an appropriate counterpart to the so-called basic theorem of a concept lattice. We show that this is a generalization of the previous approaches: it covers the so-called generalized concept lattice and multi-adjoint t-concept lattices. © 2013 Elsevier B.V. All rights reserved. Keywords: Formal concept analysis; Galois connection; Adjointness
1. Introduction Formal concept analysis (FCA) is a data-mining method applied to a rectangular matrix of data in which each row corresponds to some object, each column corresponds to some possible attribute, and the matrix field value denotes a membership of the column attribute for row object. One of the goals of this method is to find so-called concepts, which are stable (in some sense) pairs of subsets of objects and attributes. The method can be considered a nice application of the algebraic notion of a Galois connection. It has been described in detail by Ganter and Wille [12], in particular for the so-called crisp case with binary matrix data. A natural question that arises is what happens if the matrix data are non-binary. Besides the conceptual scaling that returns concepts with crisp subsets in both coordinates [12], some other approaches return concepts with fuzzy subsets in at least one coordinate. The first was provided by Burusco and Fuentes-Gonzalez [9] and was independently improved by Bˇelohlávek [2,3] and Pollandt [25,26] using values from the same residual lattice for matrix values and for the fuzziness of subsets of the objects and attributes. Another approach independently proposed by Ben Yahia and Jaoua [8], Bˇelohlávek et al. [5] and Krajˇci [13] is not as symmetric; it considers fuzzy subsets in one coordinate and crisp (binary) subsets in another one. All these approaches are covered by a common platform, the so-called generalized concept lattice [15,16], which diversifies the fuzziness of subsets of the attributes, the fuzziness of subsets of the objects, and the fuzziness of the matrix values. Medina and Ojeda-Aciego then applied the idea of multi-adjointness used in logic-programming [20–22] to FCA [17,19]. Because of its novelty and originality, this approach is not (at least immediately) covered by generalized concept lattices. This inspired us to modify our previous approach so that it will work with different mutual relationships between the objects and attributes. We also use different lattices for the different rows and columns and the matrix data. ∗ Corresponding author. Tel.: +421 55 234 2508.
E-mail address: [email protected] (S. Krajˇci). 0165-0114/$ - see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.04.008
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The usefulness of this approach in practice is demonstrated by an example of personal accommodation requirements. In comparison with previous methods that work for attributes and objects of the same type, an important advantage of our new approach is the possibility to apply FCA to heterogeneous data. Thus, we call this new approach heterogeneous.
2. Heterogeneous formal context Consider the following situation. People who are going to stay at a cottage together have different preferences for conditions, depending on the number of days spent at the cottage. One may prefer hot water, whereas others may expect an internet connection. The following requirements are examples: • • • •
Eva will accept large discomfort regarding hot water availability and partial discomfort for an internet/TV connection. John will accept partial discomfort regarding hot water availability and large discomfort for an internet/TV connection. Eva will not accept the absence of hot water on one arbitrary day at the cottage. John will not accept the absence of hot water only on his second day.
The problem is to identify which conditions have to be fulfilled to satisfy all the people staying at the cottage for different numbers of days. We cannot cover this type of complex information using existing approaches. Therefore, we define a heterogeneous formal context to formally describe the above situation. Let A and B be non-empty sets. Let P = ((Pa,b , ≤ Pa,b ) : a ∈ A, b ∈ B) be a system of posets and let R be a function from A × B such that R(a, b) ∈ Pa,b for all a ∈ A and b ∈ B. Let C = ((Ca , ≤Ca ) : a ∈ A) and D = ((Db , ≤ Db ) : b ∈ B) be systems of complete lattices. (For simplicity, we omit the indices for all ≤? , since it is always clear which one is used.) Let = (•a,b : a ∈ A, b ∈ B) be a system of operations such that •a,b is from Ca × Db to Pa,b and is isotone and left-continuous in both arguments, that is: (1a) (1b) (2a) (2b)
c1 ≤ c2 implies c1 •a,b d ≤ c2 •a,b d for all c1 , c2 ∈ Ca and d ∈ Db , d1 ≤ d2 implies c •a,b d1 ≤ c •a,b d2 for all c ∈ Ca and d1 , d2 ∈ Db , If c •a,b d ≤ p for some d ∈ Db , p ∈ Pa,b and all c ∈ X ⊆ Ca , then sup X •a,b d ≤ p, and If c •a,b d ≤ p for some c ∈ Ca , p ∈ Pa,b and all d ∈ Y ⊆ Db , then c •a,b sup Y ≤ p.
Then we call the tuple A, B, P, R, C, D, a heterogeneous formal context. Note that if Ca = Db and •a,b is commutative, then these conditions can be reduced to the following two: (1) c1 ≤ c2 implies c1 •a,b d ≤ c2 •a,b d for all c1 , c2 , d ∈ Ca = Db , and (2) If c •a,b d ≤ p for some d ∈ C, p ∈ Pa,b and all c ∈ X ⊆ Ca = Db , then sup X •a,b d ≤ p. To illustrate such a heterogeneous formal context, let B = {Eva, John, . . .} be the set of objects consisting of people thinking about staying at a cottage. Let A = {water, services, lake} be the set of attributes in relation to hot water, services and lake availability at the cottage. For simplicity, we consider just two people and two conditions from Fig. 1, but this heterogeneous method also works for arbitrary numbers of people and conditions. Eva has three preferences for staying: not at all, one day (it does not matter which one) or both days (DEva ). John has four preferences: not at all, only Saturday, only Sunday, or both days (DJohn ). For the attributes, the water can be hot or cold (Cwater ) and there are four possibilities for services: internet and television, internet only, television only, or nothing at all (Cservices ). Each person can accept different degrees of discomfort for their preferences. Thus, the behavior of each person can be expressed as = (•water,Eva , •services,Eva , •water,John , •services,John ). Similarly P = (Pwater,Eva , Pservices,Eva , Pwater,John , Pservices,John ) expresses the degrees of discomfort for each person and each condition. For instance, •services,Eva is from Cservices × DEva to Pservices,Eva , where Pservices,Eva = {0, 1/2, 1} denotes comfort, partial discomfort and large discomfort, respectively, for Eva. The behavior of the isotone and left-continuous operations = (•water,Eva , •services,Eva , •water,John , •services,John ) with respect to the number of days and cottage conditions is shown in Fig. 2. We provide the following remarks and interpretation. • Note that c •services,Eva ∅ = 0 for all c ∈ Cservices , because no-one staying at the cottage and arbitrary conditions correspond to no discomfort.
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Fig. 1. List of possible values for objects and attributes in a heterogeneous case.
Fig. 2. Isotone and left-continuous operations.
• Note that hot •water,Eva d = 0 for all d ∈ DEva , because the availability of hot water and an arbitrary number of days correspond to no discomfort. • Note that in + tv •services,John d = 0 for all d ∈ DJohn , because the presence of all services and an arbitrary number of days correspond to no discomfort. • The following demonstrates monotonicity: staying on Saturday and cold water represent partial discomfort for John, but 2 days and cold water lead to large discomfort.
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Fig. 3. Heterogeneous formal context.
• Similarly, staying on 1 day and internet only represent partial discomfort for Eva, but 2 days and internet only or missing internet lead to large discomfort. • Missing TV and Saturday represent one-third discomfort for John; missing internet and Saturday, two-thirds discomfort; and missing internet and Sunday or 2 days, large discomfort. • The following demonstrates left-continuity: Saturday or Sunday and internet only represent one-third discomfort for John, but the supremum of these days (Saturday+Sunday) and internet only also lead to one-third discomfort. Each person defines the acceptable degree of discomfort for specific conditions: i.e., R(water, Eva) ∈ P(water, Eva), R(services, Eva) ∈ P(services, Eva), R(water, John) ∈ P(water, John) and R(services, John) ∈ P(services, John). For example, Eva can accept large discomfort for water conditions and partial discomfort for services, as shown in Fig. 3. Large discomfort for water conditions (value of 1) for Eva means that she will accept an arbitrary number of days and hot or cold water, because all values for water for Eva are less than or equal to 1 in Fig. 2. Partial discomfort for services (value of 1/2) for Eva means that she will accept neither all services nor a maximum of one arbitrary day with internet only, because the values for these cases for Eva in Fig. 2 are less than or equal to 1/2. To apply FCA and identify the cottage conditions that fulfill all personal requirements, we define the mappings
and . Let F be a set of all functions f with a domain A such that f (a) ∈ Ca for all a ∈ A (more formally, F = a∈A Ca ). Let G be a set of all functions g with a domain B such that g(b) ∈ Db for all b ∈ B (i.e., G = b∈B Db ). We define the mapping : G → F. If g ∈ G, then (g) ∈ F is defined by ( (g))(a) = sup{c ∈ Ca : (∀b ∈ B)c •a,b g(b) ≤ R(a, b)}. Symmetrically, we define the mapping : F → G. If f ∈ F, then ( f ) ∈ G is defined as ( ( f ))(b) = sup{d ∈ Db : (∀a ∈ A) f (a) •a,b d ≤ R(a, b)}. ( , ) are defined by applying the so-called (sup, •a,b )-product. The properties of such a structure are described in the literature [4,10]. Theorem 1. Let f ∈ F and g ∈ G. Then the following conditions are equivalent: (1) f ≤ (g). (2) g ≤ ( f ). (3) f (a) •a,b g(b) ≤ R(a, b) for all a ∈ A and b ∈ B. Proof. 1 → 3 Let a ∈ A and b ∈ B. Then f ≤ (g) implies that f (a) ≤ ( (g))(a). Hence, by the definition of we have f (a) ≤ sup{c ∈ Ca : (∀b ∈ B)c •a,b g(b ) ≤ R(a, b )}. Because the condition (∀b ∈ B)c •a,b g(b ) ≤ R(a, b ) implies the condition c •a,b g(b) ≤ R(a, b) for b (as a special case of b ∈ B), we have {c ∈ Ca : (∀b ∈ B)c •a,b g(b ) ≤ R(a, b )} ⊆ {c ∈ Ca : c •a,b g(b) ≤ R(a, b)}, so f (a) ≤ sup{c ∈ Ca : (∀b ∈ B)c •a,b g(b ) ≤ R(a, b )} ≤ sup{c ∈ Ca : c •a,b g(b) ≤ R(a, b)}.
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By the isotonity of •a,b in the first argument, we obtain f (a) •a,b g(b) ≤ sup{c ∈ Ca : c •a,b g(b) ≤ R(a, b)} •a,b g(b). Let X denotes the supremised set on the right-hand side of our inequality. Then c •a,b g(b) ≤ R(a, b) for every c ∈ X . Hence, by the left-continuity of •a,b in the first argument, we have sup X •a,b g(b) ≤ R(a, b). However, from the transitivity of ≤ we obtain f (a) •a,b g(b) ≤ R(a, b). 2 → 3 This can be proved analogously. 3 → 1 Let (∀b ∈ B) f (a) •a,b g(b) ≤ R(a, b) hold for all a ∈ A. This means that f (a) is an element of the set {c ∈ Ca : (∀b ∈ B)c •a,b g(b) ≤ R(a, b)} and hence f (a) ≤ sup{c ∈ Ca : (∀b ∈ B)c •a,b g(b) ≤ R(a, b)} = ( (g))(a). Because a is arbitrary, we have f ≤ (g). 3 → 2 This can be proved analogously. Corollary 1. Mappings and form a Galois connection. Proof. The proof follows from the equivalence of conditions 1 and 2 of the previous theorem. Corollary 2. (1a) g1 ≤ g2 implies (g1 ) ≥ (g2 ). (1b) f 1 ≤ f 2 implies ( f 1 ) ≥ (2 ). (2a) g ≤ ( (g)). (2b) f ≤ ( ( f )). (3a) (g) = ( ( (g))). (3b) ( f ) = ( ( ( f ))). Proof. The proof is a consequence of ( , ) being a Galois connection [12]. We use mappings ( , ) to identify the required cottage conditions as follows. Mapping ( ( f ))(b) indicates maximization of the number of days spent at the cottage for specific water and services conditions that return the greatest degree of discomfort accepted by a person. For instance, for f (water) = hot, f (services) = in we obtain ( ( f ))(Eva) = 1/2, which means that hot water and internet only correspond to a maximum stay of 1 day for Eva. Mapping ( (g))(a) indicates the worst water or services conditions at the cottage for a specific number of days that return the greatest degree of discomfort accepted. For instance, if g(Eva) = 1/2, g(John) = Sa, then we obtain ( (g))(water) = cold, which means that Eva staying for 1 day and John staying on Saturday correspond to the possibility of cold water at the cottage. In another example, if g(Eva) = Sa + Su, g(John) = Sa, then we obtain ( (g))(services) = in + tv, whereby a cottage with an internet connection and TV is the worst possible case if Eva stays on Saturday and Sunday and John stays on Saturday. 3. Heterogeneous formal concept We use the Galois connection ( , ) for concept lattice construction via the classical Ganter–Wille approach [12]. Lemma 1. (1) Let {gi : i ∈ I } ⊆ G. Then
i∈I
gi
=
i∈I
(gi ).
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Fig. 4. Heterogeneous formal concepts.
(2) Let { f i : i ∈ I } ⊆ F. Then fi = ( f i ). i∈I
i∈I
Proof. The proof is a consequence of ( , ) being a Galois connection [12]. We call a pair g, f from G × F such that (g) = f and ( f ) = g a concept. Lemma 2. If g1 , f 1 and g2 , f 2 are concepts, then g1 ≤ g2 iff f 1 ≥ f 2 . Proof. The proof is a simple consequence of Corollary 2 (parts 3a and 3b). This lemma allows us to define the following ordering of concepts: g1 , f 1 ≤ g2 , f 2 iff g1 ≤ g2 (or equivalently f 1 ≥ f 2 ). In summary, five concepts are obtained in our example, as shown in Fig. 4. Intents correspond to the worst cottage conditions that fulfill all personal requirements for a specific number of days. For example, cold water and no services at the cottage correspond to no stay for Eva and staying on Saturday for John (first concept). By contrast, hot water and full services correspond to the maximum number of days spent at the cottage for both Eva and John (last concept). The next three concepts can be interpreted similarly. For instance, for the third concept, stays of 1 day by Eva and 2 days by John correspond to hot water and an internet connection as the worst possible conditions. Note that intents do not include the possibility of hot water and no services simultaneously. In this case we obtain (hot, no) = (∅, Sa) and subsequently (∅, Sa) = (cold, no). This can be interpreted as superfluous conditions for John’s stay on Saturday and maybe a cheaper cottage can be chosen. All computations in our cottage example are for two people, but it is possible to consider more complex examples, as in Fig. 1. This case also contains the poset Pservices,Ken , where Pservices,Ken = {0, le, se, 1} denotes comfort, discomfort for length of stay, discomfort for services and large discomfort, respectively. In such cases it is possible for two people to have the same lattice structure (e.g., Eva and Lea). Nevertheless, the behavior of Eva and Lea can differ for different conditions. A stay of 1 day and cold water could correspond to discomfort for Eva but comfort for Lea or vice versa. In conclusion, this heterogeneous proposal on a formal context works with heterogeneous structures for objects, attributes and table values, so it is not possible to express this using existing approaches. 4. Heterogeneous formal concept lattice We call the poset of all concepts ordered by ≤ a heterogeneous concept lattice, denoted by HCL( A, B, P, R, C, D, , , , ≤). The following theorem shows that this is in reality a lattice. Analogous theorems have been proved for existing approaches [11,12,15]. To unify these and show that the same conditions are fulfilled for our high level of generalization, the proof is self-contained.
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Theorem 2 (Basic theorem on heterogeneous concept lattices). (1) A heterogeneous concept lattice HCL(A, B, P, R, C, D, , , , ≤) is a complete lattice in which gi , f i = gi , fi i∈I
i∈I
and
i∈I
gi , f i =
i∈I
gi
,
i∈I
fi .
i∈I
(2) For each a ∈ A and b ∈ B, let Pa,b have the least element 0 Pa,b such that 0Ca •a,b d = c •a,b 0 Db = 0 Pa,b for all c ∈ Ca , d ∈ Db . Then a complete lattice L is isomorphic to HCL(A, B, P, R, C, D, , , , ≤) if and only if there are mappings : a∈A ({a} × Ca ) → L and : b∈B ({b} × Db ) → L such that: (1a) does not increase in the second argument (for a fixed first argument); (1b) does not decrease in the second argument (for a fixed first argument); (2a) Rng() is inf-dense in L; 1 (2b) Rng() is sup-dense in L; and (3) For every a ∈ A, b ∈ B, c ∈ Ca and d ∈ Db , (a, c) ≥ (b, d) if and only if c •a,b d ≤ R(a, b). Proof. (1) We prove only the first part, since proof of the second part is analogous. Using Lemma 1 we obtain
. gi =
( f i ) = fi i∈I
i∈I
i∈I
Using Corollary 1 (part 3b) and the fact that gi , f i is a concept, we have = fi fi = ( f i ) = gi . i∈I
i∈I
i∈I
i∈I
This proves i , ( ( i∈I f i )) is a concept. that i∈I g Clearly ( j∈I g j )(b) = j∈I g j (b) ≤ gi (b), which means that j∈I g j ≤ gi for each i ∈ I . Thus, our concept is the lower bound of all gi , f i . We show that this other lower is the greatest. Let g , f be some bound. This means that g ≤ gi for all i ∈ I , and hence g ≤ i∈I gi . However, this means that g , f ≤ i∈I gi , ( ( i∈I f i )). (2) First we consider a special case, the lattice HCL( A, B, P, R, C, D, , , , ≤) or H in short, and show the implication →. We define the following singleton functions Sac ∈ F for each a ∈ A and c ∈ Ca , and Tbd ∈ G for each b ∈ B and d ∈ Db : c if x = a, d if y = b, c d Sa (x) = and Tb (y) = 0Ca elsewhere 0 Db elsewhere. Note that ( (Sac ))(b) = sup{d ∈ Db : c •a,b d ≤ R(a, b)} 1 Rng() denotes the mapping range of .
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because by the definition of Sac , for each a a, Sac (a ) •a ,b d = 0Ca •a ,b d = 0 Pa ,b , and hence the condition Sac (a ) ≤ R(a , b) is automatically fulfilled. Dually, ( (Tbd ))(a) = sup{c ∈ Ca : c •a,b d ≤ R(a, b)}. Now let H (a, c) = (Sac ), ( (Sac )) and H (b, d) = ( (Tbd )), (Tbd ). Note that both H (a, c) and H (b, d) are concepts (by Corollary 2) and hence it is enough to consider their first coordinates, for example. Now we prove that they have the desired properties. (1a) Let a ∈ A, b ∈ B, c1 , c2 ∈ Ca and c1 ≤ c2 . By the isotonity of •a,b in the first argument, we have c1 •a,b d ≤ c2 •a,b d for all d ∈ Db . This means that {d ∈ Db : c1 •a,b d ≤ R(a, b)} ⊇ {d ∈ Db : c2 •a,b d ≤ R(a, b)}, from which it follows that ( (Sac1 ))(b) = sup{d ∈ Db : c1 •a,b d ≤ R(a, b)} ≥ sup{d ∈ Db : c2 •a,b d ≤ R(a, b)} = ( (Sac2 ))(b) for all b ∈ B. This means that (Sac1 ) ≥ (Sac2 ), which implies H (a, c1 ) ≥ H (a, c2 ). (1b) This can be proved dually. (2a) We prove that if g, f is a concept, then g, f = inf{ H (a, f (a)) : a ∈ A}, which implies the required assertion. f (a) f (a) f (a) Let a ∈ A. Then Sa (a) = f (a) and Sa (a ) = 0Ca ≤ f (a ) for all a a, so Sa ≤ f . By Corollary 2, we f (a) obtain (Sa ) ≥ ( f ) = g, which means that g, f ≤ H (a, f (a)) (both are concepts, so it is enough to consider their first coordinates). This is true for all a ∈ A, and hence g, f ≤ inf{ H (a, f (a)) : a ∈ A}. Let g , f be some other lower bound of { H (a, f (a)) : a ∈ A}. Then for each a ∈ A we have g , f ≤ f (a) f (a) H (a, f (a)), so g ≤ (Sa ) (these are their first coordinates). By Theorem 1, we have Sa (a) •a,b g (b) ≤ R(a, b), that is, f (a) •a,b g (b) ≤ R(a, b) for all a ∈ A. This means that g (b) ∈ {d ∈ Db : (∀a ∈ A) f (a) •a,b d ≤ R(a, b)}, so g (b) ≤ sup{d ∈ Db : (∀a ∈ A) f (a) •a,b d ≤ R(a, b)} = ( ( f ))(b) = g(b) for all b ∈ B, that is, g ≤ g and hence g , f ≤ g, f . However, this means that g, f = inf{ H (a, f (a)) : a ∈ A}. Thus, each concept can be written as the infimum of some elements from Rng( H ), that is, the set is inf-dense. (2b) It can be proved dually that g, f = sup{(b, g(b)) : b ∈ B}, which implies the required assertion. (3) → c •a,b d ≤ R(a, b) means that c ∈ {c ∈ Ca : c •a,b d ≤ R(a, b)}, which implies that Sac (a) = c ≤ sup{c ∈ Ca : c •a,b d ≤ R(a, b)} = ( (Tbd ))(a). For a a we also have Sac (a ) = 0Ca ≤ ( (Tbd ))(a ). Hence, Sac ≤ (Tbd ). By Corollary 2, we obtain (Sac ) ≥ ( (Tbd )), which means that H (a, c) ≥ H (b, d) (both are concepts, so it is enough to consider their first coordinates).
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(3) ← H (a, c) ≥ H (b, d) means that ( (Tbd )) ≤ (Sac ) (these are their first coordinates). By Corollary 2, we have Tbd ≤ ( (Tbd )). Hence, by the transitivity of ≤ we obtain Tbd ≤ (Sac ). By Theorem 1, Sac (a ) •a ,b Tbd (b ) ≤ R(a , b ) for all a ∈ A and b ∈ B. In particular, for a and b, Sac (a) •a,b Tbd (b) ≤ R(a, b), that is, c •a,b d ≤ R(a, b). Let L be any complete lattice isomorphic to H and let be some isomorphism between them. Then we define and as follows: (a, c) = ( H (a, c)) (b, d) = ( H (b, d)). Now we show that preserves the desired properties fulfilled in H . (1a) Let a ∈ A and c1 , c2 ∈ Ca , c1 ≤ c2 . Then we know that H (a, c1 ) ≥ H (a, c2 ). Because is an isomorphism, we have ( H (a, c1 )) ≥ ( H (a, c1 )) i.e., (a, c1 ) ≥ (a, c1 ). (1b) This can be proved dually. (2a) Let ∈ L. Because is a bijection, there exist f and g such that = (g, f ). We know that g, f = inf{ H (a, f (a)) : a ∈ A} and that is an isomorphism, so we have = (g, f ) = (inf{ H (a, f (a)) : a ∈ A}) = inf{( H (a, f (a))) : a ∈ A} = inf{(a, f (a)) : a ∈ A}, which means that Rng() is inf-dense. (2b) This can be proved dually. (3) (a, c) ≥ (b, d) means that ( H (a, c)) ≥ ( H (b, d)). Because is an isomorphism, this is equivalent to H (a, c) ≥ H (b, d) and we know that this holds if and only if c •a,b d ≤ R(a, b). Thus, we have proven the implication → for all lattices. Now we prove the opposite direction, ←. Let L be an arbitrary complete lattice and let and have the assumed properties. We define a mapping from the support of H to the support of L as follows: (g, f ) = inf{(a, f (a)) : a ∈ A}. We also define a mapping from the support of L to the support of H as () = g , f , where f (a) = sup{c ∈ Ca : (a, c) ≥ } g (b) = sup{d ∈ Db : (b, d) ≤ }. First, we prove that () is a concept from the support of H . Claim 1. (a) (g ) = f ; and (b) ( f ) = g . Proof. (a) By Theorem 1 it is enough to prove f (a) •a,b g (d) ≤ R(a, b) for each a ∈ A and b ∈ B. Let c ∈ Ca be such that c ∈ {c ∈ Ca : (a, c) ≥ }, that is, (a, c) ≥ . Similarly, let d ∈ Db be such that d ∈ {d ∈ Db : (b, d) ≥ }, that is, (b, d) ≤ . By the transitivity of ≤ we obtain (a, c) ≥ (b, d), which means (by the assumptions for and ) that c •a,b d ≤ R(a, b). Because c is an arbitrary element of {c ∈ Ca : (a, c) ≥ }, by the left-continuity of •a,b in the first argument we have sup{c ∈ Ca : (a, c) ≥ } •a,b d ≤ R(a, b), that is, f (a) •a,b d ≤ R(a, b). Because d is an arbitrary element of {d ∈ Db : (b, d) ≤ }, by the left-continuity of •a,b in the second argument we have f (a) •a,b sup{d ∈ Db : (b, d) ≥ } ≤ R(a, b), that is, f (a) •a,b g (b) ≤ R(a, b), as required. (b) This can be proved analogously.
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Now we show that is the required isomorphism and is its inverse. Claim 2. preserves the ordering. Proof. Let g1 , f 1 ≤ g2 , f 2 be two concepts. Then, by definition, f 1 ≥ f 2 . Because is non-increasing in the second argument, we obtain (a, f 1 (a)) ≤ (a, f 2 (a)) for each a ∈ A. It follows that (g1 , f 1 ) = inf{(a, f 1 (a)) : a ∈ A} ≤ inf{(a, f 2 (a)) : a ∈ A} = (g2 , f 2 ).
Claim 3. preserves the ordering. Proof. Let 1 ≤ 2 . It follows from transitivity that {c ∈ Ca : (a, c) ≥ 1 } ⊇ {c ∈ Ca : (a, c) ≥ 2 } and hence f 1 (a) = sup{c ∈ Ca : (a, c) ≥ 1 } ≥ sup{c ∈ Ca : (a, c) ≥ 2 } = f 2 (a) for each a ∈ A. This means that f 1 ≥ f 2 , so (1 ) = g1 , f 1 ≤ g2 , f 2 = (2 ).
Claim 4. (()) = . Proof. Because of the sup-density of Rng(), can be written as the supremum of some set of pairs from Rng(), that is, = sup{(bi , di ) : i ∈ I }, where di ∈ Dbi for each i ∈ I . This means that (bi , di ) ≤ for each i ∈ I . If a ∈ A and c ∈ Ca are arbitrary such that (a, c) ≥ , we obtain (a, c) ≥ (bi , di ) for each i ∈ I . By property (3) for and , we obtain c •a,bi di ≤ R(a, bi ). Hence, by the left-continuity of •a,bi in the first argument, we have f (a) •a,bi di = sup{c ∈ Ca : (a, c) ≥ } •a,bi di ≤ R(a, bi ) for each i ∈ I . Again by Assumption (3) we have (a, f (a)) ≥ (bi , di ) for each i ∈ I , and hence (a, f (a)) ≥ sup{(bi , di ) : i ∈ I } = . It follows that (()) = (g , f ) = inf{(a, f (a)) : a ∈ A} ≥ . Conversely, because of the inf-density of Rng(), can be written as the infimum of some set of pairs from Rng(), that is, = inf{(ai , ci ) : i ∈ I }, where ci ∈ Cai for each i ∈ I . Obviously (ai , ci ) ≥ , so ci ∈ {c ∈ Cai : (ai , c) ≥ }, from which it follows that ci ≤ sup{c ∈ Cai : (ai , c) ≥ } = f (ai ) for each i ∈ I . Because is non-increasing in the second argument, we obtain (ai , ci ) ≥ (ai , f (ai )) ≥ inf{(a, f (a)) : a ∈ A} = (g , f ) = (()) for each i ∈ I . This means that = inf{(ai , ci ) : i ∈ I } ≥ (()), so the opposite inequality is also proven. Claim 5. ((g, f )) = g, f . Proof. We denote = (g, f ). We want to prove () = g, f , that is, g , f = g, f . Because both are concepts, it is enough to prove g = g.
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Using property (3), we obtain (∀a ∈ A) f (a) •a,b d ≤ R(a, b) if and only if (∀a ∈ A)(b, d) ≤ (a, f (a)). By the definition of the infimum, this is equivalent to (b, d) ≤ inf{(a, f (a)) : a ∈ A} = (g, f ) = . This means that {d ∈ Db : (∀a ∈ A) f (a) •a,b d ≤ R(a, b)} = {d ∈ Db : (b, d) ≤ }, from which it follows that ( ( f ))(b) = sup{d ∈ Db : (∀a ∈ A) f (a) •a,b d ≤ R(a, b)} = sup{d ∈ Db : (b, d) ≤ } = g (b), that is, g = ( f ) = g . We have proved that both and preserve the ordering and both their compositions are identities. This means that = −1 and both are isomorphisms. 5. Generalization Our previous approach called a generalized (fuzzy) concept lattice [13,14] is obviously a special case of this new approach; it is enough to take the same Ca , Db , ≤a,b and •a,b . The multi-adjointness approach of Medina and Ojeda-Aciego [17] (that generalizes the approach from [19] in which the adjoint triples are defined on (L,L,P)) has some rather strange and not very aesthetic property: it takes only •b for b ∈ B without any reference to A. We symmetrize it and take operations •a,b for each pair (a, b) ∈ A × B. This, of course, diversifies and generalizes [17]. Moreover we need not the equal lattices for all b ∈ B and/or all a ∈ A. We note one issue. The approach of Medina and Ojeda-Aciego [17] works with adjoints to • operations. Recall the notion: Let (C, ≤C ), (D, ≤ D ) and (P, ≤ P ) be posets. The triple (•, →1 , →2 ) is called an adjoint triple if • : (C × D) → P, →1 : (D × P) → C, →2 : (C × P) → D and (c • d) ≤ p iff c ≤ (d →1 p) iff d ≤ (c →2 p). (If C = D and • is commutative, both arrows are identical.) Hence, all the substantial definitions of Medina and Ojeda-Aciego [17] can be reformulated using • operations only. For each operation • that is isotone and left-continuous in both arguments, there are operations →1 and →2 such that (•, →1 , →2 ) is an adjoint triple; it is enough to define d →1 p = sup{c ∈ C : c • d ≤ P p} and symmetrically c →2 p = sup{d ∈ D : c • d ≤ P p}. Obviously, c • d ≤ P p means that c ∈ {c ∈ C : c • d ≤ P p} and hence c ≤C sup{c ∈ C : c • d ≤ P p} = d →1 p. Conversely, by the left-continuity of • in the first argument, we have sup{c ∈ C : c • d ≤ P p} • d ≤ P p, that is, (d →1 p) • d ≤ P p. Thus, if c ≤C d →1 p, then by the isotonity of • in the first argument we have c • d ≤ P p. The dual properties of →2 can be proved symmetrically. This means that working with adjoint triples is equivalent to working with isotone and left-continuous functions. 6. Conclusions and future work We introduced a new common platform for fuzzification of FCA. The main idea is diversification of all that can be diversified. The main advantage of our approach is removal of the homogeneity barrier. It intuitively follows that FCA could be used for tables with data of different types. We will consider these issues in further research. Pócs made an alternative attempt to cover all known approaches [23,24]. This also works with different lattices but it gives not just a single value but a Galois connection to fields of the matrix (i.e., certain behaviors for relationship between the object and the attribute). However, it seems rather cumbersome to work with whole Galois connections in single data fields. Our approach has no limitation regarding values in data fields. We have previously compared these approaches [1].
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Another interesting issue would be to clarify the relationship of this approach to the fuzzification of Bˇelohlávek and Vychodil for truth-stressers or so-called hedges [6,7]. Krajˇci showed that generalized concept lattices cover these in some sense [16], but our new approach seems to make this relationship more immediate. Another interesting relationship for future work is heterogeneity in multi-adjoint concept lattices that are based on heterogeneous conjunctors [18]. Acknowledgement This work was partly supported by grant VEGA 1/0832/12, by the Slovak Research and Development Agency under contract APVV-0035-10 “Algorithms, Automata, and Discrete Data Structures”, and by the Agency of the Slovak Ministry of Education for Structural Funds of the EU under project ITMS: 26220120007. References [1] L. Antoni, S. Krajˇci, O. Krídlo, B. Macek, L. Pisková, Relationship between two FCA approaches on heterogeneous formal contexts, in: L. Szathmary, U. Priss (Eds.), Proceedings of CLA, 2012, pp. 93–102. [2] R. Bˇelohlávek, Fuzzy concepts and conceptual structures: induced similarities, in: Proceedings of JCIS 98, vol. I, 1998, pp. 179–182. [3] R. Bˇelohlávek, Concept lattices and order in fuzzy logic, Ann. Pure Appl. Logic 128 (2004) 277–298. [4] R. Bˇelohlávek, Sup-t-norm and inf-residuum are one type of relational product: unifying framework and consequences, Fuzzy Sets Syst. 197 (2012) 45–58. [5] R. Bˇelohlávek, V. Sklenáˇr, J. Zacpal, Crisply generated fuzzy concepts, Lect. Notes Comput. Sci. 3403 (2005) 268–283. [6] R. Bˇelohlávek, V. Vychodil, Reducing the size of fuzzy concept lattices by hedges, in: Proceedings of FUZZ-IEEE 2005, pp. 663–668. [7] R. Bˇelohlávek, V. Vychodil, Formal concept analysis and linguistic hedges, Int. J. Gen. Syst. 41 (2012) 503–532. [8] S. Ben Yahia, A. Jaoua, Discovering knowledge from fuzzy concept lattice, in: A. Kandel, M. Last, H. Bunke (Eds.), Data Mining and Computational Intelligence, Physica-Verlag, 2001, pp. 169–190. [9] A. Burusco, R. Fuentes-Gonzalez, The study of L-fuzzy concept lattice, Mathware Soft. Comput. 3 (1994) 209–218. [10] M.E. Cornejo, J. Medina, E. Ramírez, A comparative study of adjoint triples, Fuzzy Sets Syst. 211 (2013) 1–14. [11] B.A. Davey, H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, 2002. [12] B. Ganter, R. Wille, Formal Concept Analysis Mathematical Foundation, Springer Verlag, 1999. [13] S. Krajˇci, Cluster based efficient generation of fuzzy concepts, Neural Network World 13 (2003) 521–530. [14] S. Krajˇci, A generalized concept lattice, Logic J. IGPL 13 (2005) 543–550. [15] S. Krajˇci, The basic theorem on generalized concept lattice, in: V. Snášel, R. Bˇelohlávek (Eds.), Proceedings of the 2nd International Conference on Concept Lattices and Their Applications, 2004, pp. 25–33. [16] S. Krajˇci, Every concept lattice with hedges is isomorphic to some generalized concept lattice, in: V. Snášel, R. Bˇelohlávek (Eds.), Proceedings of the 3rd International Conference on Concept Lattices and Their Applications, 2005, pp. 1–9. [17] J. Medina, M. Ojeda-Aciego, Multi-adjoint t-concept lattices, Inf. Sci. 180 (2010) 712–725. [18] J. Medina, M. Ojeda-Aciego, On multi-adjoint concept lattices based on heterogeneous conjunctors, Fuzzy Sets Syst. 208 (2012) 95–110. [19] J. Medina, M. Ojeda-Aciego, J. Ruiz-Calviño, Formal concept analysis via multi-adjoint concept lattices, Fuzzy Sets Syst. 160 (2009) 130–144. [20] J. Medina, M. Ojeda-Aciego, A. Valverde, P. Vojtáš, Towards biresiduated multi-adjoint logic programming, Lect. Notes Artif. Intell. 3040 (2004) 608–617. [21] J. Medina, M. Ojeda-Aciego, P. Vojtáš, Multi-adjoint logic programming with continuous semantics, Lect. Notes Artif. Intell. 2173 (2001) 351–364. [22] J. Medina, M. Ojeda-Aciego, P. Vojtáš, Similarity-based unification: a multi-adjoint approach, Fuzzy Set Syst. 146 (2004) 43–62. [23] J. Pócs, Note on generating fuzzy concept lattices via Galois connections, Inf. Sci. 185 (2012) 128–136. [24] J. Pócs, On possible generalization of fuzzy concept lattices, Inf. Sci. 210 (2012) 89–98. [25] S. Pollandt, Fuzzy Begriffe, Springer, 1997. [26] S. Pollandt, Datenanalyse mit Fuzzy-Begriffen, in: G. Stumme, R. Wille (Eds.), Begriffliche Wissensverarbeitung. Methoden und Anwendungen, Springer, Heidelberg, 2000, pp. 72–98.