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Formal Concept Analysis with Python

Project description

Concepts is a simple Python implementation of Formal Concept Analysis (FCA).

FCA provides a mathematical model for describing a set of objects (e.g. King Arthur, Sir Robin, and the holy grail) with a set of properties or features (e.g. human, knight, king, and mysterious) which each of the objects either has or not. A table called formal context defines which objects have a given property and vice versa which properties a given object has.

Formal contexts

With Concepts, context objects can be created from a string with an ascii-art style table. The objects and properties will simply be represented by strings. Separate the property columns with pipe symbols, create one row for each objects and indicate the presence of a property with the character X. Note that the object and property names need to be disjoint to uniquely identify them.

>>> from concepts import Context

>>> c = Context.from_string('''
               |human|knight|king |mysterious|
    King Arthur|  X  |  X   |  X  |          |
    Sir Robin  |  X  |  X   |     |          |
    holy grail |     |      |     |     X    |
    ''')

>>> c
<Context object mapping 3 objects to 4 properties at ...>

After creation, the parsed content of the table is available on the context object.

>>> c.objects  # row headings
('King Arthur', 'Sir Robin', 'holy grail')

>>> c.properties  # column headings
('human', 'knight', 'king', 'mysterious')

>>> c.bools  # data cells
[(True, True, True, False), (True, True, False, False), (False, False, False, True)]

The context object can be queried to return the common properties for a collection of objects (common intent) as well as the common objects for a collection of properties (common extent):

>>> c.intension(['King Arthur', 'Sir Robin'])  # common properties?
('human', 'knight')

>>> c.extension(['knight', 'mysterious'])  # objects with these properties?
()

In FCA these operations are called derivations and usually notated with the prime symbol(‘).

For convenience, the derivation methods automatically split string arguments on whitespace. If your names lack whitespace, you can also use them like this:

>>> c.extension('knight king')
('King Arthur',)

>>> c.extension('mysterious human')
()

Formal concepts

A pair of objects and properties such that the objects share exactly the properties and the properties apply to exactly the objects is called formal concept. Informally, they result from maximal rectangles of X-marks in the context table, when rows and columns can be reordered freely.

You can retrieve the closest matching concept corresponding to a collection of objects or properties with the __getitem__ method of the concept object:

>>> c['king']  # closest concept matching intent/extent
(('King Arthur',), ('human', 'knight', 'king'))

>>> assert c.intension(('King Arthur',)) == ('human', 'knight', 'king')
>>> assert c.extension(('human', 'knight', 'king')) == ('King Arthur',)

>>> c[('King Arthur', 'Sir Robin')]
(('King Arthur', 'Sir Robin'), ('human', 'knight'))

Within each context, there is a maximally general concept comprising all of the objects as extent and having an empty intent (supremum).

>>> c[('Sir Robin', 'holy grail')]  # maximal concept, supremum
(('King Arthur', 'Sir Robin', 'holy grail'), ())

Furthermore there is a minimally general concept comprising no object at all and having all properties as intent (infimum).

>>> c[('mysterious', 'knight')]  # minimal concept, infimum
((), ('human', 'knight', 'king', 'mysterious'))

The concepts of a context can be ordered by extent set-inclusion (or dually intent set-inclusion). With this (partial) order, they form a concept lattice having the supremum concept (i.e. the tautology) at the top, the infimum concept (i.e. the contradiction) at the bottom, and the other concepts in between.

Concept lattice

The concept lattice of a context contains all pairs of objects and properties (formal concepts) that can be retrieved from a formal context:

>>> c
<Context object mapping 3 objects to 4 properties at ...>

>>> l = c.lattice

>>> l
<Lattice object of 2 atoms 5 concepts 2 coatoms at ...>

>>> for extent, intent in l:
...     print extent, intent
() ('human', 'knight', 'king', 'mysterious')
('King Arthur',) ('human', 'knight', 'king')
('holy grail',) ('mysterious',)
('King Arthur', 'Sir Robin') ('human', 'knight')
('King Arthur', 'Sir Robin', 'holy grail') ()

Individual concepts can be retrieved by different means :

>>> l.infimum  # first concept, index 0
<Infimum {} <-> [human knight king mysterious]>

>>> l.supremum  # last concept
<Supremum {King Arthur, Sir Robin, holy grail} <-> []>

>>> l[1]
<Atom {King Arthur} <-> [human knight king] <=> King Arthur <=> king>

>>> l[('mysterious',)]
<Atom {holy grail} <-> [mysterious] <=> holy grail <=> mysterious>

The concepts form a directed acyclic graph and are linked upward (more general concepts, superconcepts) and downward (less general concepts, subconcepts):

>>> l.infimum.upper_neighbors
(<Atom {King Arthur} <-> [human knight king] <=> King Arthur <=> king>, <Atom {holy grail} <-> [mysterious] <=> holy grail <=> mysterious>)

>>> l[1].lower_neighbors
(<Infimum {} <-> [human knight king mysterious]>,)

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