Godin, R., Missaoui, R., & Alaoui, H. (1995). Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2), 246--267.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the concept lattice is a superset lattice on attributes, if (X 0 , Y 0 ) and (X 1 , Y 1 ) are concepts with X 0 X 1 , then sim(X 0 ) sim(X 1 ) 3.1.1 Efficiency of concept analysis There are several algorithms for building concept lattices. The algorithm we use is due to Godin and others [13] (we use their Algorithm 1) Let k be an upper bound on R ( o ) where o O. That is, k is an upper bound on the number of attributes enjoyed by any object in O. Then, their algorithm runs in time O(2 2k O ) In our applications, k is typically less than ten, while O ranged up to the ....

....the number of FA transitions. The times seem to vary slightly worse than linearly, but it is hard to tell for sure, since many of the times were so short. These observations agree with the more thorough empirical evaluation that Godin and others did of their algorithm (which we use) in their paper [13]. 5.3 Traversal strategies Table 3 compares the cost of labeling by a variety of methods, where cost is defined as in Section 4.2. One of the authors (an expert user and developer of the tool) used Cable to debug each specification and create an accurate labeling. Then, we measured the cost of ....

Robert Godin, Rokia Missaoui, and Hassan Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995.

....It is convenient to model dependence and causality, and provides a vivid and concise account of the relations between the variables in the universe of discourse. Concept hierarchy is not necessarily a tree structure. Wille et al. propose to build corresponding concept lattice from binary relations [1, 2, 3]. This kind of special lattice and the corresponding Hasse diagram represent a concept hierarchy. Concept lattice reflects entity attribute relationships between objects. In some practical applications such as information management and knowledge discovery, concept lattice has gained good results ....

....This kind of special lattice and the corresponding Hasse diagram represent a concept hierarchy. Concept lattice reflects entity attribute relationships between objects. In some practical applications such as information management and knowledge discovery, concept lattice has gained good results [1, 4]. R.Missaoui et al. present algorithms for extracting rules from concept lattice [5] R.Godin et al. propose a method to build concept lattice and the corresponding Hasse diagram incrementally and compare it with other concept lattice constructing methods [1] Discovering association rules among ....

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Godin, R., Missaoui, R., Alaui, H.: Incremental Concept Formation Algorithms Based on Galois (Concept) Lattices. Computational Intelligence, 1995, 11(2): 246-267

....This algorithm is very powerful as it performs semantic classification. Topic maps are semantic structures themselves, but they may be very large and complex, so this algorithm is interesting to extract more semantics from them. The algorithm we implemented is based on the one that was proposed in [7]. Let us first introduce Galois lattices basic concepts. Let two finite sets E and E (E consists of a set of objects and E is the set of these objects properties) and a binary relation R # E x E between these two sets. Figure 2 shows an example of binary relation between two sets. According ....

Godin, R, Chau, T.-T., Incremental concept formation algorithms based on Galois Lattices, Computational intelligence, 11, n 2, p246 --267, 1998.

....[Kuz99] can help in characterizing extrinsic properties to be suggested. Then, we develop in Section 4 a more direct, local and efficient way to do it. This leads to the notion of associative concepts, which are closely related to modified and new concepts in the incremental concept formation [GMA95]. Experimental results are given in Section 5, and the limits and perspectives of the induction process are discussed in the conclusion. 2 Logical Context, Feature Context, and Sub Context This section introduces useful theoretical elements for the rest of the paper. 2.1 Logical Context We ....

....of rule conditions is helped by the justifications given for expected features. Section 5 gives more details on this interactive process. 4. 3 Connection with Incremental Concept Formation In this section, we compare our incremental building of a context to the incremental formation of concepts [GM94,GMA95,VM01]. Both methods are based on the search for specific concepts: associative concepts in our case; old, modified, generator, and new concepts in the other case. Surprisingly, we found a close relationship between associative concepts and both modified and new concepts, which can be redefined as ....

R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on Galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995.

....has entered into the mainstream of data mining. 6.2 Formal Concept Analysis Concept lattices and formal concept analysis, as developed by Ganter and Wille in [11] have gotten a lot more play in data mining than closure theory. For example, it is central in papers by Godin, Missaoui, and Zaki [13, 14, 38]. Yet, it appears that much of the formal machinery is quite unnecessary. A major reason for involving concept lattices is to reduce the huge number of rules that apriori, frequent set data mining yields. For this, we suspect the concept of closed sets and their generators is sufficient. ....

....algorithms is the ability to continuously update R, that is treat R as a stream of tuples. Because apriori, and similar algorithms, must make more than one sweep over R, it must be fixed. But, as Godin and Missaoui have shown, incremental creation and update of concept lattices is much faster [13, 14]. The problem is that their approach to incremental update does not update the generators as well. In [27] Theorem 6.1 is used to determine the generators of each closed set based on the faces of (sets covered by) the set. A major problem is that as more data is read, attribute combinations ....

Robert Godin, Rokia Missaoui, and Hassan Alaoui. Incremental Concept Formation Algorithms Based on Galois (Concept) Lattices. Computational Intelligence, 11(2):246--267, 1995.

....Although our measures were developed and utilized for ranking the interestingness of generalized relations using DGGs, they are more generally applicable to other problem domains. For example, alternative methods could be used to guide the generation of summaries, such as Galois lattices [6], conceptual graphs [3] or formal concept analysis [19] Also, summaries could more generally include views generated from databases or summary tables generated from data cubes. However, we do not dwell here on the methods or technical aspects of deriving summaries, views, or summary tables. ....

R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995.

....i to node D j indicates that any value in domain D i maps to a value in D j . Mitchell s version space method is an example of a learning method based on a GL. Godin, Missaoui, and Alaoui s work on incremental concept formation is based on Galois lattices, a type of GL used in discrete mathematics [5]. Bournaud and Ganascia investigated the automatic creation of a GL from a set of objects described by conceptual graphs[1] In data mining, GLs are used to conceptualize the process of generalizing data as a transformation of values from one domain to values of another, smaller domain. The ....

R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on Galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995.

....a closed itemset since it is not a maximal group of items common to some objects: all customers purchasing the items B and C also purchase the item E. The closed itemset lattice of a finite relation (the database) is dually isomorphic to the concept lattice [19, 20] also called Galois lattice [11]. Figure 4 gives the closed itemset lattice of D with frequent closed itemsets for minsupport = 2 outlined. ABCDE ABCE AC BE C ACD BCE Frequent closed itemset (minsupport=2) Infrequent closed itemset Fig. 4: Closed itemset lattice of D Using the closed itemset lattice, which is a sub order of ....

....of the closed itemset lattice is kL C k 2 K kDk. Moreover, experimental applications and theoretical results showed that the average growth factor is far less than the 2 K bound. Based on a uniform distribution hypothesis, we can observe that kL C k kDk, where is the mean value for kffogk [11]. z C1 is a sub closed itemset of C 2 and C 2 is a sup closed itemset of C 1 . Efficient Mining of Association Rules Using Closed Itemset Lattices 31 The closed itemset lattice framework leads us to the following properties (see Section 3) i) All subsets of a frequent itemset are frequent. ....

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R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on Galois (concept) lattices. Computational Intelligence, 11(2):246--267 (1995).

....with the set of attributes. This space is more general than a version space ( 15] because we allow more complex generalizations than value to ANY and our summaries contain a relation rather than a single conjunctive description. On the other hand, it is more constricted than a Galois lattice ([7]) because not every possible subset is included in the lattice. The generalization state space is obtained by determining all possible combinations of nodes from the DGGs, and then generating the summary which corresponds to each node combination. For example, the generalization state space for ....

R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995.

....Transact.Loc Cust.Loc Item 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 1 2 1 1 1 1 1 1 1 2 1 Figure 1. A data cube Of course, numerous methods could be used to guide the generation of summaries, such as concept hierarchies [5] domain generalization graphs [15] Ga lois lattices [9], conceptual graphs [4] and formal concept analysis [22] Also, summaries could more generally include many other forms of knowledge representation, such as database views, association rules, itemsets, and web search results. However, when given hundreds, or even thousands of summaries (possibly ....

R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995.

....Generalization algorithm and DGGs for mining large industrial databases is described in [15, 16] Although this example is based upon summaries generated from databases using AOG and DGGs, alternative methods could be used to guide the generation of summaries. These include Galois lattices [9], conceptual graphs [4] or formal concept analysis [24] Similarly, summaries could more generally include views generated from databases, characterized generalized association rules generated from itemsets, or summary tables (i.e. data cubes) generated from data warehouses. Regardless of the ....

R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246-- 267, 1995.

....algorithm to compute and organize formal concepts all at the same time. Each formal concept is used to compute the subsumed or subsuming ones. Then, the Galois lattice is computed when the most general or the most specific formal concept is computed. Some incremental algorithms were also proposed (Godin, 1995c; Carpineto, 1993) The lattice is then updated as soon as new objects, or features, are added in D, respectively in T. The complexity of the Galois lattice algorithms is discussed, for example, in (Carpineto, 1993) Association rules extraction The structure of a Galois lattice may be used to ....

Godin R., Missaoui R. & Alaoui H. (1995). Incremental Concept Formation Algorithms Based on Galois (Concepts) Lattices. Computational Intelligence, 11(2):246--267.

....is not a closed itemset since it is not a maximal grouping of items common to some objects: all customers purchasing the items B and C also purchase the item E. The closed itemset lattice of a nite relation (the database) is isomorphic to the concept lattice [14, 15] also called Galois lattice [7]. Figure 1 gives the closed itemset lattice of D with frequent closed itemsets for minsupport = 2 outlined. ABCDE ABCE AC BE C ACD BCE Frequent closed itemset (minsupport=2) Infrequent closed itemset Figure 4: Closed itemset lattice of D 4 C 1 is a sub closed itemset of C 2 and C 2 is a ....

....closed itemset lattice is kL C k 2 K kDk. Moreover, experimental applications and theoretical results based on a uniform distribution hypothesis showed that the average growth factor is far less than the 2 K bound. Actually, we can observe that kL C k kDk, where is the mean value for kok [7]. Using the closed itemset lattice framework we can deduce the following properties (see Section 3) i) All subsets of a frequent itemset are frequent. ii) All supersets of an infrequent itemset are infrequent. iii) All sub closed itemsets 5 of a frequent closed itemset are frequent. iv) All ....

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R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on Galois (concept) lattices. In N. Cercone and G. McCalla, editors, Computational Intelligence, volume 11, pages 246267. Blackwell Publishers, May 1995.

.... Topic Maps Bndicte Desclefs Le Grand Laboratoire d Informatique de Paris 6 8, rue du Capitaine Scott 75015 Paris, France tel : 33) 1 44 27 75 12 fax : 33) 1 44 27 74 95 email : Benedicte.Le Grand lip6.fr Michel Soto Laboratoire d Informatique de Paris 6 8, rue du Capitaine Scott 75015 Paris, France tel : 33) 1 44 27 88 30 fax : 33) 1 44 27 74 95 email : Michel.Soto lip6.fr ....

.... Topic Maps Bndicte Desclefs Le Grand Laboratoire d Informatique de Paris 6 8, rue du Capitaine Scott 75015 Paris, France tel : 33) 1 44 27 75 12 fax : 33) 1 44 27 74 95 email : Benedicte.Le Grand lip6.fr Michel Soto Laboratoire d Informatique de Paris 6 8, rue du Capitaine Scott 75015 Paris, France tel : 33) 1 44 27 88 30 fax : 33) 1 44 27 74 95 email : Michel.Soto lip6.fr ....

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Godin R., Missaoui R. and Alaoui H., Incremental concept formation algorithms based on galois (concept) lattice, Computational Intelligence, 11(2), 1995.

.... subdirect decomposition of [3] di ers from the one described in this paper, as we investigate decomposition of concept lattices in terms of a set of given ones and we allow embedding of contexts (cf. section 5) Batch and incremental algorithms for building concept lattices are found in [4] and [5]. Automatic discovery of implication rules from data using concept lattices is described by Godin and Missaoui ( 6] Their approach is close to ours one, as implication rules can be used for the characterization of relations and concept lattices. The generation of sub knowledge bases, however, ....

..... The triple (G; M;R) is called a context. The lattice arises from that context by applying a Galois connection between the power sets of G and M . This Galois connection is a particular one, called the polar ( 8] Various applications of concept lattices have been reported in the literature ([9, 5, 6, 10]) Typically, such an application assumes a subset of G (resp. M) to be given as input for which a corresponding subset of M (resp. G) is computed (if such exists) by searching the concept lattice (knowledge base) for a concept having the smallest extent containing the given subset of G. So, for ....

Godin, R., Missaoui R. and Alaoui, H. (1995) Incremental concept formation algorithms based on Galois (concept) lattices. Computational Intelligence, 11(2), 246-267.

....this problem in a systematic way. This is due to, amongst others, the elegant algebraic properties, like closure and decomposition, of these lattices. Decomposition of Galois lattices is described in [Wil83] Batch and incremental algorithms for building Galois lattices are found in [CR93] and [GMA95]. A related approach using automatic discovery of implication rules from concept lattices is introduced in [GM94] 2 Galois lattices Galois (also called concept) lattices were rst de ned by R. Wille ( Wil82] Basically a concept lattice is a representation of a (e.g. binary) relation R between ....

R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on Galois (concept) lattices. Computational Intelligence, 11(2):246-267, 1995.

....rarely occurs and a non exponential behavior is usual. Godin et al. 1993) and Lindig fischer.tex; 9 08 1999; 17:16; p.15 16 Bernd Fischer (1995a) give more experimental evidence for this. Moreover, algorithms to construct the concept lattice incrementally are known (Godin and Missaoui, 1994; Godin et al. 1995). For our example library, the concept lattices derived from the full (i.e. manually computed) and the approximated (i.e. automatically computed using SPASS) contexts contained 153 and 180 concepts, respectively. Their computation took approximately a second and is thus negligible compared to ....

Godin, R., R. Missaoui, and H. Alaoui: 1995, `Incremental Concept Formation Algorithms based on Galois (Concept) Lattices'. Computational Intelligence 11(2), 246--267.

....concepts in the diagram can then be determined by examining and comparing the results of queries that are joins of the SQL statements in the individual scales. For a large database calculating a large number of SQL joins can be compuationally expensive. 3. 2 Large Lattice Approaches Godin et al. [11] and Carpineto et al. 3] both advocate the use of the concept lattice generated from a single valued context containing a large number of attributes and objects. They both concentrated in the speci c domain of analyzing text collections. Text collections as a source of data are much more speci c ....

....1 1 v p 1 m 1 1 v m(p 1) Experimental results with several data sets (Voting Records, Breast Cancer, and INSPECT AI) showed that in general the number of concepts varies as the square of the number of objects. These results supported the previous similar nding of Godin et al. [11]. Both Godin and Carpineto experimented with lattice determination algorithms, in each case advocating the use of incremental algorithms that compute the set of intents of the lattice. The algorithm presented by Carpineto is at least O(jLj 2 ) in its complexity, where jLj is the number of ....

R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on Galois (Concept) lattices. Computational Inteligence, 11(2):246-267, 1995.

....itemsets. Finally some duplicate closures are removed (e.g. both AT and TW produce the same closure) We will show that while AClose is much better than Apriori, it is uncompetitive with CHARM. A number of previous algorithms have been proposed for generating the Galois lattice of concepts [5, 6]. These algorithms will have to be adapted to enumerate only the frequent concepts. Further, they have only been studied on very small datasets. Finally the problem of generating a basis (a minimal non redundant rule set) for association rules was discussed in [18] but no algorithms were given) ....

R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1991.

....is covered by every test in T and there is no entity outside of E covered by every test in T . Stated yet another way, concepts determine maximal sets of tests covering identical entities (and maximal sets of entities covered by identical tests) Concepts can be computed by a variety of algorithms [12, 18]. In the worst case, for a table of size n rows by n columns, there may be 2 n concepts, so the worst case running time of any batch algorithm that computes all concepts is exponential in n. In practice, concept lattices have O(n 2 ) concepts and sometimes even O(n) concepts [18] The table ....

R. Godin and R. Missaoui H. Alaoui. Incremental concept formation algorithms based on Galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995.

....to find all the objects that have this attribute. Here, for an attribute A and a qualified value A , we select all the entities of which the qualified value associated with the attribute A is less general than A . The algorithm we present 1 is inspired by the incremental algorithm of Gaudin [3]. Our aim is to integrate conceptual clustering methods in data base systems, and because this algorithm is incremental it is interesting to update the concept lattice when new entities are available. 5.2.1 General principle of the algorithm Let G be the Galois lattice of the connexion (int; ....

....of the sets F and f(feg) the set of the attributes present in e) The nodes are sorted in ascending cardinality of their intension sets. Thus, 1 This algorithm computes the concept lattice contrary to the one proposed in [5] that only finds the nodes of the lattice. 2 In the classical case [3], X; Y ) 2 G is the generator of (X 0 ; Y 0 ) 2 G e iff (X; Y ) inff(Z; Z 0 ) 2 GjY 0 = Z 0 f(feg)g. It is the smallest pair that produces the intersection Y 0 . Procedure 1 Add(e; P e ) fThe Lev[i] contain the nodesg fwith a generalization level ig Classify Level(Lev) fM ....

R. Gaudin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246-- 267, 1995.

....functions that define a Galois connexion [8] int : P(E) A] fg E 7 int(E) e2E ffi(e) ext : A] fg P(E) I 7 ext(I) fe 2 E j I 6 ffi(e)g 5. Construction of the Galois lattice To compute the concept lattice, we propose an extension of the incremental algorithm of Gaudin [5] to take into account fuzzy structured data. Our aim is to integrate our conceptual clustering method in Data Base Management Systems, and it is important to have an incremental algorithm to make easy the update of the concept lattice when a new data is available. 5.1. General principle of the ....

....; V D 4 ) c.f. figure 4) When the observation e 3 occurs, both the two nodes 2 and 3 produce a new intension by generalization with e 3 : a, b, c, but the generator of the new concept is the node 3. It is the more general node that produces the new intension. 1 In the classical case [5], X; Y ) 2 G is the generator of (X 0 ; Y 0 ) 2 G e iff (X; Y ) inff(Z; Z 0 ) 2 GjY 0 = Z 0 f(feg)g Gamma Gamma (a, b, c, 0 Gamma Gamma fe1 , e2 g (a, b, c, 3 Gamma Gamma (a, b, c, 4 Gamma Gamma fe1 g (a, b, c, 1 Gamma Gamma fe2 ....

R. Gaudin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995.

.... lattice framework can not only aid in the development of efficient algorithms, but can also help in the visualization of discovered associations, and can provide a unifying framework for reasoning about associations, and supervised (classification) and unsupervised (clustering) concept learning [5, 6, 10]. The rest of the paper is organized as follows. We present the association rule problem statement in Section 2. A graph theoretic view of the problem is given in Section 3. Section 4 casts association mining as a search for frequent concepts, and Section 5 looks at the problem of generating rule ....

....13] the connection between associations and hypergraph transversals was made. They also presented a model of association mining as the discovery of maximal elements of theories, and gave some complexity bounds. A lot of algorithms have been proposed for generating the Galois lattice of concepts [5, 9, 10, 11, 18]. An incremental approach for building the concepts was studied in [6, 10] These algorithms will have to be adapted to enumerate only the frequent concepts. Further, they have only been studied on small datasets. It remains to be seen how scalable these approaches are compared to the association ....

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R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1991.

....marineg) fintelligentg) fintelligentg) fchimpanzees; humans; dolphins; whalesg; fintelligentg) This computation corresponds to the fact that c 1 t c 2 = c 5 in the lattice shown in Figure 2. There are several algorithms for computing the concept lattice for a given context [6, 18]. We describe a simple bottom up algorithm here. An important fact about concepts and contexts used in the algorithm is that, given a set of objects X, the smallest concept with extent containing X is ( oe(X) oe(X) Thus, the bottom element of the concept lattice is ( oe( oe( the ....

R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995.

....Sub structures of L of lesser size may be used by the algorithm as reported in [7] but their detection within L x requires extra computations. Finally, for terminal x, algorithms computing DM(L x) as the concept lattice of the context K = L [ fxg; L [ fxg; L x ) 5] may be found in [8]. In our completion strategy the lattice L is still isomorphic to DM(L x) However, we design a new procedure for building DM(L x) from L and x which takes the greatest advantage of the existing lattice structure. Thus, subsets of X which have no GLB (LUB) in L x are detected together ....

....in Sup(x) called upper odds, whereas ODDL (x) fajakx; 9b; c 2 Inf(x) with a = b L cg will denote the set of LUB of elements in Inf(x) called lower odds. The duality principle, allows us to consider only ODDU (x) Odds give rise to auxiliary elements so they may be compared to generators in [8] and to canonical representatives in [1] The existing strategies for lattice insertion differ as to the way odds are detected and the number of auxiliaries per odd. Thus, the algorithm in [2] checks the GLB of all couples of elements in Sup(x) and inserts an auxiliary each time this GLB is ....

[Article contains additional citation context not shown here]

R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995.

....of a set of concepts can be computed by intersecting their intents, and by finding the common objects of the resulting intersection. An example of the application of the fundamental theorem is shown in Figure 4. There are several algorithms for computing the concept lattice for a given context [9, 21]. We describe a simple bottom up algorithm here. An important fact about concepts and contexts used in the algorithm is that, given a set of objects X , the smallest concept with extent containing X is ( oe(X) oe(X) Thus, the bottom element of the concept lattice is ( oe( oe( the ....

....the for loop in line [6] c refers to c 0 . The covering set of c 0 is the singleton set consisting of c 4 , so c 0 is assigned c 4 in line [7] In line [8] p 0 is assigned the value of p minus the subordinate concepts of c 4 (i.e. c 1 , c 0 , and bottom) so p 0 is fc 2 ; c 6 g. In line [9], the union of the extents of c 2 and c 6 is disjoint with the extent of c 4 ; thus, in line [10] the partition p 00 = fc 2 ; c 6 g [ fc 4 g is formed. p 00 is added to the set of partitions and to the worklist in line [12] and line [13] In the worst case, the number of partitions can be ....

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R. Godin and R. Missaoui H. Alaoui. Incremental concept formation algorithms based on Galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995.

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R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995.

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R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995.

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R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995. 23

....from o = 0, A) 0) and gradually incorporating a new object oi into the lattice i which corresponds to a table ]i = o, oi ) A, I) Each incorporation involves a set of structural updates [17] 3.2. 1 Principles of the incremental approach The basic approach initially described in [10] and then improved in [18] follows a fundamental property of the Galois connection established by f and g on (P(O) P(A) both families of closed subsets are themselves closed under set intersection [3] Thus, the integration of a new object transaction is mainly aimed at the insertion into i ....

....12 then 10: c NEw CoNCEPT(Extent( U o , Int) 5) is a generator 11: UPDATE OEDEE(c,5) ADD(12, c) Algorithm 1. Update of a Galois (concept) lattice upon an insertion of a new object. 3.2. 2 Description of the algorithm In the sequel, we consider the subset of the algorithm described in [10] which deals with the recog nition of the above three concept sets and the creation of new concepts only. Details about the lattice order updates (primitive UPDAT, OaD,a) are skipped since they are irrelevant to our purposes. Thus, the concepts are first sorted in increasing order with respect to ....

R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246-267, 1995.

.... Missaoui Mili, 1995c) One interesting subset is the pruned concept (Galois) hierarchy (also called Galois subhierarchy in (Dicky et al. 1994) which has been introduced in the more general context of conceptual clustering of conceptual graphs, under the name knowledge space (Mineau, Gecsei Godin, 1990). We describe the pruned concept hierarchy for the case of binary relations below. The pruned concept hierarchy (PCH) can be generated from the concept lattice by eliminating the pairs which have empty N(A) and N(B) sets. In the example of Figure 2.2.1, two pairs would be eliminated, and the ....

Godin, R., Missaoui, R. & Alaoui, H. (1995d). Incremental Concept Formation Algorithms Based on Galois (Concept) Lattices. Computational Intelligence, 11(2), 246-267.

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Godin, R., Missaoui, R. & Alaoui, H. (1995b). Incremental Concept Formation Algorithms Based on Galois (Concept) Lattices. Computational Intelligence, 11(2), 246-267.

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Godin, R., Missaoui, R., & Alaoui, H. (1995). Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2), 246--267.

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GODIN R., MISSAOUI R., ALAOUI H., "Incremental Concept Formation Algorithms Based on Galois (Concept) Lattices", Computational Intelligence, vol. 11, num. 2, 1995, p. 246-267.

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R. Godin, R. Missaoui, H. Alaoui, Incremental Concept Formation Algorithms Based on Galois (Concept) Lattices. Computational Intelligence,11(2), 246-267, 1995.

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R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1991.

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R. Godin, R. Missaoui, and H. Alaoui. Incremental concept formation algorithms based on galois (concept) lattices. Computational Intelligence, 11(2):246--267, 1995.

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Godin, R., Missaoui, R., and Alaoui, H., Incremental Concept Formation Algorithms Based on Galois Lattices, Computation Intelligence, 1995.

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R. Godin, R. Missaoui and H. Alaoui, Incremental Concept Formation Algorithms Based on Galois (Concept) Lattices, Computational Intelligence, 11(2): 246-267, 1995.

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R. Godin, G. Mineau, and R. Missaoui, "Incremental concept formation algorithms based on galois (concept) lattices," Computational Intelligence, vol 11, 1995.

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Godin, R. & Missaoui, R. & Alaoui, H. Incremental Concept Formation Algorithms based on Galois (concept) Lattices. Computational Intelligence, 11(2):246-267, 1995

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Godin, R., Missaoui, R. & Alaoui, H. (1995). Incremental concept formation algorithms based on Galois (concept) lattices. Computational Intelligence, 11, 246--267.

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Incremental concept formation algorithms based on galois (concept) lattice, Godin R., Missaoui R. and Alaoui H., Computational Intelligence, 11(2), 1995.

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R. Godin, and Missaoui, R. and Alaoui, H. Incremental Concept Formation Algorithms based on Galois (Concept) Lattices, Computational Intelligence, Vol. 11, number 2, pp. 246-267, 1995.

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