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

${displaystyle (G,M,I)}$

Or

${displaystyle G}$

And

${displaystyle M}$

are sets and

${displaystyle Isubseteq Gtimes M}$

.
The elements of

${displaystyle G}$

are called objects and those of

${displaystyle M}$

attributes.
The couple set

${displaystyle I}$

is considered a relationship and is therefore noted

${displaystyle gim}$

instead of

${displaystyle (g,m)in I}$

What is said: “The object

${displaystyle g}$

has the attribute

${displaystyle m}$

». Letters

${displaystyle G}$

And

${displaystyle M}$

Proviennent de l’Allemand objects et characteristics.

We define the operators of derivation For

${displaystyle Asubseteq G}$

And

${displaystyle Bsubseteq M}$

about

${displaystyle A’={min Mmid forall gin Acdot gIm}}$

And

${displaystyle B’={gin Gmid forall min Bcdot gIm}}$

. All

${displaystyle A’}$

is the set of attributes shared by all objects of

${displaystyle A}$

and the whole

${displaystyle B’}$

is the set of objects that have all the attributes of

${displaystyle B}$

.

And concept context

${displaystyle (G,M,I)}$

is a couple

${displaystyle (A,B)}$

Or

${displaystyle Asubseteq G}$

And

${displaystyle Bsubseteq M}$

checking

${displaystyle A’=B}$

And

${displaystyle B’=A}$

. For a concept

${displaystyle (A,B)}$

, where he said that

${displaystyle A}$

is his extension And

${displaystyle B}$

son intension .

We define a order (partial) on the concepts by

${displaystyle (A_{1},B_{1})leq (A_{2},B_{2})Leftrightarrow A_{1}subseteq A_{2}(Leftrightarrow B_{2}subseteq B_{1})}$

.

You can use derivation operators to build a concept from a set of objects

${displaystyle X}$

or attributes

${displaystyle Y}$

Considering concepts

${displaystyle (X”,X’)}$

And

${Displaystyle (and ‘, and’ ‘)}$

respectively. In particular for an object

${displaystyle g}$

we call

${displaystyle gamma g}$

the Object concept

${displaystyle ({g}”,{g}’)}$

and for an attribute

${displaystyle m}$

we call

${displaystyle mu m}$

the Concept attribute

${displaystyle ({m}’,{m}”)}$

.

Example [ modifier | Modifier and code ]

Let us consider as set of objects whole numbers from 1 to 10:

${displaystyle G={1,2,3,4,5,6,7,8,9,10}}$

and as a set of mathematical properties attributes:

${displaystyle M={pair,impair,compos{acute {e}},premier,carr{acute {e}}}}$

.

The impact relationship

${displaystyle Isubseteq Gtimes M}$

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

${displaystyle {3}’={impair,premier}}$

And

${displaystyle {impair,premier}’={3,5,7}}$

. SO

${displaystyle ({3,5,7},{impair,premier})}$

is a formal concept.

Each pair of concepts has a unique lower terminal and upper terminal. Given the concepts

${displaystyle (G_{i},M_{i})}$

And

${displaystyle (G_{j},M_{j})}$

, their lower terminal is

${displaystyle (G_{i}cap G_{j},(M_{i}cup M_{j})”)}$

and their upper terminal is

${displaystyle ((G_{i}cup G_{j})”,M_{i}cap M_{j})}$

.

Due to the partial order between concepts and terminals, the conditions are respected to build a trellis of concepts.

(in) Bernhard Ganter et Rudolf Wille  (in) , Formal Concept Analysis : Mathematical Foundations , Berlin, Springer Verlag, 1999 , 284 p. (ISBN  978-3-540-62771-5 )