Information Data Structures
Data structures, orderings and applied classifications are generally defined on finite sets or sets of relations, which supposes that we know what are such entities (Sect. 2.3 ). But a part of classification research deals with data mining and the constit
- PDF / 433,478 Bytes
- 34 Pages / 477 x 684 pts Page_size
- 49 Downloads / 184 Views
Information Data Structures
2.1 Overview Data structures, orderings and applied classifications are generally defined on finite sets or sets of relations, which supposes that we know what are such entities (Sect. 2.3). But a part of classification research deals with data mining and the constitution of structured domains of concepts or objects. The fact that a mathematical structure, the Galois connection, contains a quasi-exhaustive information about the correspondence of two sets (Sect. 2.4) has suggested to use this structure in association with an order relation (Sect. 2.5) to initiate formal conceptual analysis (Sect. 2.6). But formal concepts are not real concepts. The exploration of concrete structures of objects has then led to the construction of formal (Sect. 2.7) and regional (Sect. 2.8) ontologies, using sometimes, as Barry Smith does, non-classical logics (the mereology of Lesniewski). In all this chapter, we study these models in relation to the main problems of classification and, finally, discuss (Sect. 2.9) the theories and results that have been introduced.
2.2 Historical Notes Finite sets were first studied by Alfred Tarski (see [479]), a polish logician and one of the father founders of modern mathematical logic. From a set theoretic viewpoint, they have been investigated by the French mathematician Claude Frasnay (see [180, 181]) and it is now, obviously, the true domain of combinatorics (see [4]). Galois connection, the basic model for FCA (Formal Concept Analysis), has been rediscovered many times. Studied by Barbut and Monjardet (see [24]), and already used, with the Galois lattice, to structure philosophical data information in the 1980s by Parrochia (see [375]), it has already been anticipated in Salton’s work (see [441]), where document/term lattices are essentially Galois lattices in the sense of FCA. Similarly, during the 1980s, Yuli Schreider (see [444]), from the Russian school of taxonomy (see [220]), seems to have developed a categorical formalization of the D. Parrochia, P. Neuville, Towards a General Theory of Classifications, Studies in Universal Logic, DOI 10.1007/978-3-0348-0609-1_2, © Springer Basel 2013
23
24
2
Information Data Structures
Galois lattice. We can also mention that feature structure lattices, as used in linguistic Componential Analysis at the end of the 1990s, were very similar to concept lattices (see [146]). But as far as we know, the use of Galois lattices (now more often called “conceptual lattices”) in Data Analysis goes back to Barbut (see [22]), which proved also, in the same paper, that every lattice is Galois. “Formal concepts” have been popularized by Rudolf Wille (see [502]), but they were already known, under the name of “maximal rectangles”, by Kaufmann and Pichat (see [271]), and also by Parrochia (see [378]). Since that time, lattice theory and conceptual graphs (see [471]) have been used in many applied fields, from information science (see [403]) to technology design (see [384]), going through most of social sciences, medicine, biology,
Data Loading...
