📘 Formal convex concepts#
Formal convex concepts are the building blocks of formal concept analysis (FCA),
representing the relationships between a set of items and their patterns.
In the GALACTIC framework, formal convex concepts are implemented using the
Concept class of the
galactic.algebras.convex.descriptions.core module, which allows for the creation,
manipulation, and analysis of these concepts within a given antitone Galois connection.
Creating convex concepts#
The constructor of the Concept
class takes an instance of the
GaloisConnection class as input,
which defines the context in which the concepts are formed and optionally an iterable
of items or attributes to initialise the concept. When neither items nor attributes
are provided, the concept is initialised as the top concept (with full extent and
smallest intent).
from galactic.algebras.concept.core import ItemUniverse
from galactic.algebras.convex.characteristics.core import Component, Integer
from galactic.algebras.convex.descriptions.core import (
Concept,
Context,
GaloisConnection,
PredicateUniverse,
)
from galactic.algebras.convex.descriptions.examples.arithmetic.core import (
MultipleDescription,
)
characteristic = Integer(components=(Component(),))
description = MultipleDescription(space=(characteristic,))
dataset = [36, 48, 16, 32]
domain = ItemUniverse(dataset)
co_domain = PredicateUniverse(descriptions=(description,))
context = Context(domain, co_domain)
connection = GaloisConnection(context)
top = Concept(connection=connection)
top
bottom = Concept(connection=connection, items=[])
bottom
<galactic.algebras.convex.descriptions.core.Concept object at 0x718cf3258ff0>
The extent and intent of the concept can be accessed using the
extent and
intent attributes,
respectively. They are closed sets of items and attributes within the context of the
antitone Galois connection.
<galactic.algebras.convex.descriptions.core.Extent object at 0x718cf3248980>
[Item(key=0, value=36),
Item(key=1, value=48),
Item(key=2, value=16),
Item(key=3, value=32)]
<galactic.algebras.convex.descriptions.core.Intent object at 0x718cf3248840>
['M(int(@),4)']
<galactic.algebras.convex.descriptions.core.Extent object at 0x718cf324b040>
[]
<galactic.algebras.convex.descriptions.core.Intent object at 0x718cf3248fc0>
['⊥']
Operating on concepts#
The Concept class implements the
Element protocol of the
galactic.algebras.lattice.core module, which
also provides methods to create concepts using the join (\(\vee\)) or
meet (\(\wedge\)) operators.
concept_1 = Concept(
connection=connection,
items=[context.domain[1], context.domain[2]],
)
concept_2 = Concept(
connection=connection,
items=[context.domain[0]],
)
join = concept_1 | concept_2
meet = concept_1 & concept_2
display(concept_1.extent, list(concept_1.extent))
display(concept_1.intent, [str(attr) for attr in concept_1.intent])
display(concept_2.extent, list(concept_2.extent))
display(concept_2.intent, [str(attr) for attr in concept_2.intent])
display(concept_2.intent, [str(attr) for attr in concept_2.intent.convexes[0]])
display(meet.extent, list(meet.extent))
display(meet.intent, [str(attr) for attr in meet.intent])
display(join.extent, list(join.extent))
display(join.intent, [str(attr) for attr in join.intent])
display(meet <= join, meet.extent <= join.extent, meet.intent >= join.intent)
<galactic.algebras.convex.descriptions.core.Extent object at 0x718cf3279300>
[Item(key=1, value=48), Item(key=2, value=16), Item(key=3, value=32)]
<galactic.algebras.convex.descriptions.core.Intent object at 0x718cf3270880>
['M(int(@),16)']
<galactic.algebras.convex.descriptions.core.Extent object at 0x718cf32788c0>
[Item(key=0, value=36)]
<galactic.algebras.convex.descriptions.core.Intent object at 0x718cf324ac00>
['M(int(@),4)', 'M(int(@),9)']
<galactic.algebras.convex.descriptions.core.Intent object at 0x718cf324ac00>
['M(int(@),4)', 'M(int(@),9)']
<galactic.algebras.convex.descriptions.core.Extent object at 0x718cf327ab40>
[]
<galactic.algebras.convex.descriptions.core.Intent object at 0x718cf327ac00>
['⊥']
<galactic.algebras.convex.descriptions.core.Extent object at 0x718cf32791c0>
[Item(key=0, value=36),
Item(key=1, value=48),
Item(key=2, value=16),
Item(key=3, value=32)]
<galactic.algebras.convex.descriptions.core.Intent object at 0x718cf32783c0>
['M(int(@),4)']
True
True
True