Concepts formal analysis — Wikipedia
L’ formal analysis of concepts (in English Formal Concept Analysis , FCA ) endeavors to study the concepts when they are formally described, that is to say that the context and the concepts are completely and precisely defined.
It was introduced by Rudolf Wille in 1982 [ first ] As an application of trellis theory (see Trellis of Galois).
It is based on the previous work of Mr. Barbut and B. Monjardet [ 2 ] , all theory of trellis [ 3 ] and also has a solid philosophical base [ 4 ] .
A concept can be defined by its intension and its extension: the extension is the set of objects that belong to the concept while the intension is the set of attributes shared by these objects.
And context is a triplet
Or
And
are sets and
.
The elements of
are called objects and those of
attributes.
The couple set
is considered a relationship and is therefore noted
instead of
What is said: “The object
has the attribute
». Letters
And
Proviennent de l’Allemand objects et characteristics.
We define the operators of derivation For
And
about
And
. All
is the set of attributes shared by all objects of
and the whole
is the set of objects that have all the attributes of
.
And concept context
is a couple
Or
And
checking
And
. For a concept
, where he said that
is his extension And
son intension .
We define a order (partial) on the concepts by
.
You can use derivation operators to build a concept from a set of objects
or attributes
Considering concepts
And
respectively. In particular for an object
we call
the Object concept
and for an attribute
we call
the Concept attribute
.
Example [ modifier | Modifier and code ]
Let us consider as set of objects whole numbers from 1 to 10:
and as a set of mathematical properties attributes:
.
The impact relationship
Can be represented in the form of a table where the lines correspond to the objects and the columns correspond to the attributes.
name | compound | pair | impair | premier | edge |
---|---|---|---|---|---|
first | x | x | |||
2 | x | x | |||
3 | x | x | |||
4 | x | x | x | ||
5 | x | x | |||
6 | x | x | |||
7 | x | x | |||
8 | x | x | |||
9 | x | x | x | ||
ten | x | x |
On a
And
. SO
is a formal concept.
Each pair of concepts has a unique lower terminal and upper terminal. Given the concepts
And
, their lower terminal is
and their upper terminal is
.
Due to the partial order between concepts and terminals, the conditions are respected to build a trellis of concepts.
- Wille, R. (1982) Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered Sets.445-470. Dordrecht-Boston, Reidel.
- M. Barbut & B. Monjardet, Order and classification (2 volumes) , Paris, Hachette Universite, , 176 p.
- (in) G. Birkhoff, Lattice theory , American Mathematical Soc., , 418 p. (ISBN 978-0-8218-1025-5 , read online )
- A. Arnauld & P. Nicole, Logic or art of thinking , Gallimard, , 406 p. (ISBN 978-2-07-072726-1 )
(in) Bernhard Ganter et Rudolf Wille (in) , Formal Concept Analysis : Mathematical Foundations , Berlin, Springer Verlag, , 284 p. (ISBN 978-3-540-62771-5 )
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