Concept lattice

A population can be created using a collection of python objects:

[1]:

from galactic.population import Population

individuals = {48: 48, 36: 36, 64: 64, 56: 56, 84: 84}
population = Population(individuals)
population

[1]:

<galactic.population.Population at 0x7f217c1a2d90>

[2]:

list(population)

[2]:

['48', '36', '64', '56', '84']

A lattice can be created from a population using a set of strategies:

[3]:

from galactic.algebras.poset import HasseDiagram, IterativeDiagram
from galactic.concepts import (
    ConceptRenderer,
    HierarchicalRenderer,
    ConceptLatticeRenderer,
    ConceptLattice,
    ConceptClosure,
    Concept,
)

from galactic.examples.arithmetic import (
    MultipleDescription,
    DivisorDescription,
    Integer,
)

characteristic = Integer()
descriptions = [MultipleDescription(characteristic), DivisorDescription(characteristic)]
closure = ConceptClosure(population, descriptions)
concept = Concept(closure)
concept

[3]:

Predicates:

M(4)
D(4032)

Individuals:

48
36
64
56
84

[4]:

from galactic.examples.arithmetic import MultipleStrategy, DivisorStrategy

lattice = ConceptLattice.create(
    population, descriptions, [DivisorStrategy(characteristic)]
)

[5]:

lattice.global_entropy

[5]:

4.077819531114783

[6]:

lattice.global_stability

[6]:

0.28125

[7]:

lattice.global_logarithmic_stability

[7]:

0.05928469016817753

[8]:

lattice.global_information

[8]:

20.389097655573917

[9]:

list(str(concept) for concept in lattice.domain)

[9]:

['M(4) and D(4032)',
 'M(4) and D(1008)',
 'M(4) and D(1344)',
 'M(4) and D(576)',
 'M(4) and D(504)',
 'M(4) and D(336)',
 'M(12) and D(144)',
 'M(8) and D(448)',
 'M(16) and D(192)',
 'M(12) and D(252)',
 'M(28) and D(168)',
 'M(48) and D(48)',
 'M(36) and D(36)',
 'M(64) and D(64)',
 'M(56) and D(56)',
 'M(84) and D(84)',
 'M(0) and D(1)']

[10]:

for diagram in IterativeDiagram(
    lattice,
    graph_renderer=ConceptLatticeRenderer(),
    domain_renderer=ConceptRenderer(),
    edge_renderer=HierarchicalRenderer(),
):
    display(diagram)

../../_images/notebooks_concepts_notebook_12_0.png

../../_images/notebooks_concepts_notebook_12_1.png

../../_images/notebooks_concepts_notebook_12_2.png

../../_images/notebooks_concepts_notebook_12_3.png

../../_images/notebooks_concepts_notebook_12_4.png

../../_images/notebooks_concepts_notebook_12_5.png

../../_images/notebooks_concepts_notebook_12_6.png

../../_images/notebooks_concepts_notebook_12_7.png

../../_images/notebooks_concepts_notebook_12_8.png

../../_images/notebooks_concepts_notebook_12_9.png

../../_images/notebooks_concepts_notebook_12_10.png

../../_images/notebooks_concepts_notebook_12_11.png

../../_images/notebooks_concepts_notebook_12_12.png

../../_images/notebooks_concepts_notebook_12_13.png

../../_images/notebooks_concepts_notebook_12_14.png

../../_images/notebooks_concepts_notebook_12_15.png

../../_images/notebooks_concepts_notebook_12_16.png

../../_images/notebooks_concepts_notebook_12_17.png

../../_images/notebooks_concepts_notebook_12_18.png

[11]:

lattice.apply([MultipleStrategy(characteristic)])
HasseDiagram(
    lattice,
    graph_renderer=ConceptLatticeRenderer(),
    domain_renderer=ConceptRenderer(),
    edge_renderer=HierarchicalRenderer(),
)

[11]:

../../_images/notebooks_concepts_notebook_13_0.png

[12]:

from galactic.concepts import ConceptTable

ConceptTable(lattice)

[12]:

Concept	Individuals	Predicates
0		`M(4)D(4032)`
1		`D(1008)`
2		`D(1344)`
3		`D(576)`
4		`D(504)`
5		`M(12)`
6		`D(336)`
7		`M(8)`
8		`D(144)`
9		`M(16)D(192)`
10		`D(252)`
11		`M(28)D(168)`
14		`D(448)`
15	`48`	`M(48)D(48)`
16	`36`	`M(36)D(36)`
17	`64`	`M(64)D(64)`
18	`84`	`M(84)D(84)`
19	`56`	`M(56)D(56)`
20		`M(0)D(1)`

[13]:

ConceptTable(lattice, all_individuals=True, all_predicates=True)

[13]:

Concept	Individuals	Predicates
0	`['48', '36', '64', '56', '84']`	`M(4)D(4032)`
1	`['48', '36', '56', '84']`	`M(4)D(1008)`
2	`['48', '64', '56', '84']`	`M(4)D(1344)`
3	`['48', '36', '64']`	`M(4)D(576)`
4	`['36', '56', '84']`	`M(4)D(504)`
5	`['48', '36', '84']`	`M(12)D(1008)`
6	`['48', '56', '84']`	`M(4)D(336)`
7	`['48', '64', '56']`	`M(8)D(1344)`
8	`['48', '36']`	`M(12)D(144)`
9	`['48', '64']`	`M(16)D(192)`
10	`['36', '84']`	`M(12)D(252)`
11	`['56', '84']`	`M(28)D(168)`
12	`['48', '84']`	`M(12)D(336)`
13	`['48', '56']`	`M(8)D(336)`
14	`['64', '56']`	`M(8)D(448)`
15	`48`	`M(48)D(48)`
16	`36`	`M(36)D(36)`
17	`64`	`M(64)D(64)`
18	`84`	`M(84)D(84)`
19	`56`	`M(56)D(56)`
20		`M(0)D(1)`

[14]:

display(*lattice.domain)

Predicates:

M(4)
D(4032)

Individuals:

48
36
64
56
84

Predicates:

M(4)
D(1008)

Individuals:

48
36
56
84

Predicates:

M(4)
D(1344)

Individuals:

48
64
56
84

Predicates:

M(4)
D(576)

Individuals:

48
36
64

Predicates:

M(4)
D(504)

Individuals:

36
56
84

Predicates:

M(12)
D(1008)

Individuals:

48
36
84

Predicates:

M(4)
D(336)

Individuals:

48
56
84

Predicates:

M(8)
D(1344)

Individuals:

48
64
56

Predicates:

M(12)
D(144)

Individuals:

48
36

Predicates:

M(16)
D(192)

Individuals:

48
64

Predicates:

M(12)
D(252)

Individuals:

36
84

Predicates:

M(28)
D(168)

Individuals:

56
84

Predicates:

M(12)
D(336)

Individuals:

48
84

Predicates:

M(8)
D(336)

Individuals:

48
56

Predicates:

M(8)
D(448)

Individuals:

64
56

Predicates:

M(48)
D(48)

Individuals:

48

Predicates:

M(36)
D(36)

Individuals:

36

Predicates:

M(64)
D(64)

Individuals:

64

Predicates:

M(84)
D(84)

Individuals:

84

Predicates:

M(56)
D(56)

Individuals:

56

Predicates:

M(0)
D(1)

Individuals:

Reduced context

[15]:

from galactic.algebras.lattice import ReducedContextDiagram

ReducedContextDiagram(
    lattice,
    domain_renderer=ConceptRenderer(join_irreducible=True),
    co_domain_renderer=ConceptRenderer(meet_irreducible=True),
)

[15]:

../../_images/notebooks_concepts_notebook_18_0.png

[16]:

from galactic.algebras.relational import BinaryTable

BinaryTable(
    lattice.reduced_context,
    domain_renderer=ConceptRenderer(join_irreducible=True),
    co_domain_renderer=ConceptRenderer(meet_irreducible=True),
)

[16]:

	@0	@1	@2	@3	@4	@5	@6
84		\(\checkmark\)	\(\checkmark\)			\(\checkmark\)	\(\checkmark\)
56		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
36	\(\checkmark\)	\(\checkmark\)				\(\checkmark\)	\(\checkmark\)
48	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)			\(\checkmark\)
64	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)