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The multiple facets of the canonical direct unit implicational basis
The notion of dependencies between “attributes” arises in many areas such as relational databases, data analysis, data-mining, formal concept analysis, knowledge structures … . Formalization of dependencies leads to the notion of so-called full implicational systems (or full family of functional dep...
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Published in: | Theoretical computer science 2010-05, Vol.411 (22), p.2155-2166 |
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Main Authors: | , |
Format: | Article |
Language: | English |
Subjects: | |
Citations: | Items that this one cites Items that cite this one |
Online Access: | Get full text |
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Summary: | The notion of
dependencies between “attributes” arises in many areas such as relational databases, data analysis, data-mining, formal concept analysis, knowledge structures
…
. Formalization of dependencies leads to the notion of so-called
full implicational systems (or full family of functional dependencies) which is in one-to-one correspondence with the other significant notions of
closure operator and of
closure system. An efficient generation of a full implicational system (or a closure system) can be performed from equivalent implicational systems and in particular from the bases for such systems, for example, the so-called
canonical basis. This paper shows the equality between five other bases originating from different works and satisfying various properties (in particular they are unit implicational systems). The three main properties of this unique basis are the directness, canonical and minimal properties, whence the name
canonical direct unit implicational basis given to this unit implicational system. The paper also gives a nice characterization of this canonical basis and makes precise its link with the prime implicants of the Horn function associated to a closure operator. It concludes that it is necessary to compare more closely related works made independently, and with a different terminology, in order to take advantage of the really new results in these works. |
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ISSN: | 0304-3975 1879-2294 |
DOI: | 10.1016/j.tcs.2009.12.021 |