formal concept analysis: foundations and...

Formal Concept Analysis:Foundations and Applications

Philippe Balbiani

Institut de recherche en informatique de Toulouse

1

Outline

I Introduction (page 3)I Concept lattices of contexts (page 10)I Many-valued contexts (page 53)I Determination and representation (page 115)I Concept algebras (page 195)I Concepts and roles (page 231)

2

Introduction

3

IntroductionThe duality of extension and intension

A formal context

cartoon real tortoise dog cat mammalGarfield ⊗ ⊗ ⊗Snoopy ⊗ ⊗ ⊗Socks ⊗ ⊗ ⊗Bobby ⊗ ⊗ ⊗Harriet ⊗ ⊗

4


A formal context


5


A formal context


6


A formal context


The pair ({Garfield , Snoopy}, {cartoon, mammal}) is a formalconcept of the formal context

7

IntroductionFormal concept analysis in information sciences

I Formal concept analysis in information retrievalI Formal concept analysis as a tool for knowledge

representation and knowledge discoveryI Applications of formal concept analysis in logic and

artificial intelligence

8

IntroductionFormal concept analysis bibliographies and conferences

Introductions to formal concept analysisI Davey, B., Priestley, H.: Introduction to Lattices and Order.

Cambridge University Press (2002, Second Edition)I Ganter, B., Wille, R.: Formal Concept Analysis.

Mathematical Foundations. Springer-Verlag (1999)I www.fcahome.org.uk

International conferencesI International Conference on Conceptual Structures (ICCS)I International Conference on Formal Concept Analysis

(ICFCA)I Concept Lattices and their Applications (CLA)

9

Concept lattices of contexts

10

Concept lattices of contextsContext and concept

Formal context: structure of the form K = (Ob, At , I) whereI Ob is a nonempty set of formal objectsI At is a nonempty set of formal attributesI I is a binary relation between Ob and At

11


A finite context can be represented by a cross table whereI rows are headed by object namesI columns are headed by attribute names

x......

X · · · · · · ⊗ · · · · · ·......

A cross in row X and column x means thatI the object X has the attribute x

12


Example 1:

small medium large near far yes noMercury ⊗ ⊗ ⊗Venus ⊗ ⊗ ⊗Earth ⊗ ⊗ ⊗Mars ⊗ ⊗ ⊗

Jupiter ⊗ ⊗ ⊗Saturn ⊗ ⊗ ⊗Uranus ⊗ ⊗ ⊗

Neptune ⊗ ⊗ ⊗Pluto ⊗ ⊗ ⊗

13


Example 1:




14


Example 1:

small near medium large far yes noMercury ⊗ ⊗ ⊗Venus ⊗ ⊗ ⊗Earth ⊗ ⊗ ⊗Mars ⊗ ⊗ ⊗



15


Example 2:

a b c d e f g h i1 ⊗ ⊗ ⊗2 ⊗ ⊗ ⊗ ⊗3 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗4 ⊗ ⊗ ⊗ ⊗5 ⊗ ⊗ ⊗ ⊗6 ⊗ ⊗ ⊗ ⊗ ⊗7 ⊗ ⊗ ⊗ ⊗8 ⊗ ⊗ ⊗ ⊗9 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

16


Example 2:

a b c d e f g h i1 ⊗ ⊗ ⊗2 ⊗ ⊗ ⊗ ⊗3 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗4 ⊗ ⊗ ⊗ ⊗5 ⊗ ⊗ ⊗ ⊗6 ⊗ ⊗ ⊗ ⊗ ⊗7 ⊗ ⊗ ⊗ ⊗8 ⊗ ⊗ ⊗ ⊗9 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

17


Example 2:

a b c d e f g h i1 ⊗ ⊗ ⊗2 ⊗ ⊗ ⊗ ⊗3 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗4 ⊗ ⊗ ⊗ ⊗5 ⊗ ⊗ ⊗ ⊗6 ⊗ ⊗ ⊗ ⊗ ⊗7 ⊗ ⊗ ⊗ ⊗8 ⊗ ⊗ ⊗ ⊗9 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

18


Example 2:

a b c d e f g h i1 ⊗ ⊗ ⊗2 ⊗ ⊗ ⊗ ⊗3 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗4 ⊗ ⊗ ⊗ ⊗5 ⊗ ⊗ ⊗ ⊗6 ⊗ ⊗ ⊗ ⊗ ⊗7 ⊗ ⊗ ⊗ ⊗8 ⊗ ⊗ ⊗ ⊗9 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

19


Example 2:

a b c d e f g h i1 ⊗ ⊗ ⊗2 ⊗ ⊗ ⊗ ⊗3 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗4 ⊗ ⊗ ⊗ ⊗5 ⊗ ⊗ ⊗ ⊗6 ⊗ ⊗ ⊗ ⊗ ⊗7 ⊗ ⊗ ⊗ ⊗8 ⊗ ⊗ ⊗ ⊗9 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

20


For a set A ⊆ Ob of objects, we defineI A′ = {x ∈ At : X I x for every X ∈ A}

i.e. the set of attributes common to the objects in A

For a set B ⊆ At of attributes, we defineI B′ = {X ∈ Ob: X I x for every x ∈ B}

i.e. the set of objects which have all attributes in B

21


Proposition 1: If (Ob, At , I) is a context, A, A1, A2 ⊆ Ob are setsof objects and B, B1, B2 ⊆ At are sets of attributes then

I A1 ⊆ A2 ⇒ A′2 ⊆ A′1I B1 ⊆ B2 ⇒ B′

2 ⊆ B′1

I A ⊆ A′′

I B ⊆ B′′

I A′ = A′′′

I B′ = B′′′

Moreover,I A ⊆ B′ ⇔ B ⊆ A′ ⇔ A× B ⊆ I

22


A formal concept of the context (Ob, At , I) is a pair (A, B) withI A ⊆ ObI B ⊆ AtI A′ = BI B′ = A

We callI A the extent of the concept (A, B)

I B the intent of the concept (A, B)

B(Ob, At , I) denotesI the set of all concepts of the context (Ob, At , I)

23


Example 2:

a b c d e f g h i1 ⊗ ⊗ ⊗2 ⊗ ⊗ ⊗ ⊗3 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗4 ⊗ ⊗ ⊗ ⊗5 ⊗ ⊗ ⊗ ⊗6 ⊗ ⊗ ⊗ ⊗ ⊗7 ⊗ ⊗ ⊗ ⊗8 ⊗ ⊗ ⊗ ⊗9 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

24


Example 2:

a b g c d e f h i1 ⊗ ⊗ ⊗2 ⊗ ⊗ ⊗ ⊗3 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗4 ⊗ ⊗ ⊗ ⊗5 ⊗ ⊗ ⊗ ⊗6 ⊗ ⊗ ⊗ ⊗ ⊗7 ⊗ ⊗ ⊗ ⊗8 ⊗ ⊗ ⊗ ⊗9 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

25


Example 2:

a b g c d e f h i1 ⊗ ⊗ ⊗2 ⊗ ⊗ ⊗ ⊗3 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗9 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗4 ⊗ ⊗ ⊗ ⊗5 ⊗ ⊗ ⊗ ⊗6 ⊗ ⊗ ⊗ ⊗ ⊗7 ⊗ ⊗ ⊗ ⊗8 ⊗ ⊗ ⊗ ⊗

26


The extent A and the intent B of a concept (A, B) are closelyconnected by the relation I

B

⊗ · · · ⊗

A...

...⊗ · · · ⊗

27


For every set A ⊆ Ob,I A′ is an intent of some conceptI (A′′, A′) is a conceptI A′′ is the smallest extent containing AI A is an extent iff A = A′′

For every set B ⊆ At ,I B′ is an extent of some conceptI (B′, B′′) is a conceptI B′′ is the smallest intent containing BI B is an intent iff B = B′′

28


Proposition 2: If T is an index set and for every t ∈ T , At ⊆ Obis a set of objects and Bt ⊆ At is a set of attributes then

I (⋃

t∈T At)′ =

⋂t∈T A′t

I (⋃

t∈T Bt)′ =

⋂t∈T B′

t

29


If (A1, B1) and (A2, B2) are concepts of a context thenI A1 ⊆ A2 iff B2 ⊆ B1

If A1 ⊆ A2 and B2 ⊆ B1 then we say thatI (A1, B1) is a subconcept of (A2, B2)

I (A2, B2) is a superconcept of (A1, B1)

and we writeI (A1, B1) 6 (A2, B2)

The set of all concepts of (Ob, At , I) ordered in this wayI is denoted by B(Ob, At , I)I is called the concept lattice of the context (Ob, At , I)

30


Theorem 1: The concept lattice B(Ob, At , I) is a completelattice in which infimum and supremum are given by

I∧

t∈T (At , Bt) = (⋂

t∈T At , (⋃

t∈T Bt)′′)

I∨

t∈T (At , Bt) = ((⋃

t∈T At)′′,

⋂t∈T Bt)

Theorem 2: Every complete lattice (L,6) is isomorphic to theconcept lattice B(L, L,6)

31


The duality principle for concept lattices: If (Ob, At , I) is acontext then

I (At , Ob, I−1) is a contextMoreover,

I B(At , Ob, I−1) and B(Ob, At , I) are isomorphicI (B, A) 7→ (A, B) is an isomorphism

32


For an object X ∈ Ob, we writeI X ′ instead of the object intent {X}′

I γX for the object concept (X ′′, X ′)

For an attribute x ∈ At , we writeI x ′ instead of the attribute extent {x}′

I µx for the attribute concept (x ′, x ′′)

33

Concept lattices of contextsContext and concept lattice

A context can be reconstructed from its concept lattice:I Ob is the extent of the greatest concept (∅′, ∅′′)I At is the intent of the least concept (∅′′, ∅′)I I is given by

I I =⋃{A× B: (A, B) is a concept}

The contexts reconstructed from two non-isomorphic conceptlattices are non-isomorphic

34


Example 3:I Concept lattices of non-isomorphic contexts can well be

isomorphic

a b c d e1 ⊗ ⊗ ⊗ ⊗ ⊗2 ⊗ ⊗3 ⊗ ⊗ ⊗ ⊗ ⊗4 ⊗ ⊗ ⊗ ⊗5 ⊗6 ⊗ ⊗ ⊗ ⊗ ⊗7 ⊗ ⊗ ⊗ ⊗ ⊗8 ⊗ ⊗ ⊗

a b c, d e1, 3, 6, 7 ⊗ ⊗ ⊗ ⊗

2 ⊗ ⊗4 ⊗ ⊗ ⊗5 ⊗8 ⊗ ⊗

35


A context (Ob, At , I) is called clarified iff for every objectX , Y ∈ Ob and for every attribute x , y ∈ At ,

I X ′ = Y ′ ⇒ X = YI x ′ = y ′ ⇒ x = y

36


If X ∈ Ob is an object and A ⊆ Ob is a set of objects with X 6∈ Abut X ′ = A′ then

I γX =∨

Y∈A γYI B(Ob, At , I) and B(Ob \ {X}, At , I ∩ ((Ob \ {X})× At)) are

isomorphicand we say that

I X is a reducible object

Full rows, i.e.I objects X with X ′ = At

are always reducible

37


If x ∈ At is an attribute and B ⊆ At is a set of attributes withx 6∈ B but x ′ = B′ then

I µx =∧

y∈B µyI B(Ob, At , I) and B(Ob, At \ {x}, I ∩ (Ob × (At \ {x}))) are

isomorphicand we say that

I x is a reducible attribute

Full columns, i.e.I attributes x with x ′ = Ob

are always reducible

38


Example 4:

a b c d e f g h i1 ⊗ ⊗ ⊗2 ⊗ ⊗ ⊗ ⊗3 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗4 ⊗ ⊗ ⊗ ⊗5 ⊗ ⊗ ⊗ ⊗6 ⊗ ⊗ ⊗ ⊗ ⊗7 ⊗ ⊗ ⊗ ⊗8 ⊗ ⊗ ⊗ ⊗9 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

39


Example 4:

a b c d e f g h i1 ⊗ ⊗ ⊗2 ⊗ ⊗ ⊗ ⊗3 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗4 ⊗ ⊗ ⊗ ⊗5 ⊗ ⊗ ⊗ ⊗6 ⊗ ⊗ ⊗ ⊗ ⊗7 ⊗ ⊗ ⊗ ⊗8 ⊗ ⊗ ⊗ ⊗9 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

40


The removal from context (Ob, At , I) of reducible objects andreducible attributes is called

I reducing the context

A clarified context (Ob, At , I)I is called row reduced iff every object concept is irreducibleI is called column reduced iff every attribute concept is

irreducible

A clarified context which is both row reduced and columnreduced

I is called reduced

41


Every finite context can be brought into a reduced formI merge objects with the same intentsI merge attributes with the same extentsI delete all reducible objectsI delete all reducible attributes

42


If (Ob, At , I) is a context, X ∈ Ob is an object and x ∈ At is anattribute then we write

I X ↙ x iffI not X I xI for every object Y ∈ Ob, if X ′ ( Y ′ then Y I x

In other words,I X ↙ x iff

I X ′ is maximal among all object intents not containing x

43


If (Ob, At , I) is a context, X ∈ Ob is an object and x ∈ At is anattribute then we write

I X ↗ x iffI not X I xI for every attribute y ∈ At , if x ′ ( y ′ then X I y

In other words,I X ↗ x iff

I x ′ is maximal among all attribute extents not containing X

44


Proposition 3: The following statements hold for every context:I X ∈ Ob is irreducible ⇔ X ↙ x for some x ∈ AtI x ∈ At is irreducible ⇔ X ↗ x for some X ∈ Ob

Proposition 4: The following statements hold for every finitecontext:

I X ∈ Ob is irreducible ⇔ X ↙↗ x for some x ∈ AtI x ∈ At is irreducible ⇔ X ↙↗ x for some X ∈ Ob

45


Example 5:

a b c, d e1, 3, 6, 7 ⊗ ⊗ ⊗ ⊗

2 ⊗ ⊗4 ⊗ ⊗ ⊗5 ⊗8 ⊗ ⊗

46


Example 5:

a b c, d e1, 3, 6, 7 ⊗ ⊗ ⊗ ⊗

2 ⊗ ⊗ ↙↗ ↙4 ↙↗ ⊗ ⊗ ⊗5 ↗ ⊗ ↗8 ↗ ⊗ ⊗ ↙↗

47


Example 5:

a c, d e2 ⊗ ↙↗ ↙4 ↙↗ ⊗ ⊗8 ↗ ⊗ ↙↗

48


A context (Ob, At , I) is called doubly founded iff for every objectX ∈ Ob and for every attribute x ∈ At , if not X I x then

I X ↗ y and x ′ ⊆ y ′ for some attribute y ∈ AtI Y ↙ x and X ′ ⊆ Y ′ for some object Y ∈ Ob

49


Proposition 5: Every finite context is doubly founded

Proposition 6: A context which does neither contain infinitechains X1, X2, . . . of objects with X ′

1 ⊆ X ′2 ⊆ . . . nor infinite

chains x1, x2, . . . of attributes with x ′1 ⊆ x ′2 ⊆ . . . is doublyfounded

Proposition 7: The following statements hold for every doublyfounded context:

I X ↙ x ⇒ X ↙↗ y for some y ∈ AtI X ↗ x ⇒ Y ↙↗ x for some Y ∈ Ob

50


A complete lattice (L,6) is called doubly founded iff for everyu, v ∈ L, if u < v then there exists u′, v ′ ∈ L such that

I u′ is minimal with respect to u′ 66 u and u′ 6 vI v ′ is maximal with respect to u 6 v ′ and v 66 v ′

51


Proposition 8: If the concept lattice of the context (Ob, At , I) isdoubly founded, so is (Ob, At , I)

Proposition 9: If the complete lattice (L,6) is not doublyfounded, neither is the context (L, L,6)

52

Many-valued contexts

53

Many-valued contextsContexts and scales

Many-valued context: structure of the form (Ob, At , Va, I) where

I Ob is a nonempty set of formal objectsI At is a nonempty set of formal attributesI Va is a nonempty set of formal valuesI I is a ternary relation between Ob, At and Va

A many-valued context (Ob, At , Va, I)I is called a n-valued context iff Va has n elements

54


Example 6:

De Dl R E MConv . poor good good good excellentFront good poor excellent excellent goodRear excellent excellent very poor poor very poorMid excellent excellent good very poor very poorAll excellent excellent good good poor

De: “drive efficiency empty”, Dl : “drive efficiency loaded”, R:“road handling properties”, E : “economy of space”, M:“maintainability”

55


The domain of an attribute x is defined to beI dom(x) = {X ∈ Ob: I(X , x , v) for some v ∈ Va}

The attribute xI is called complete iff dom(x) = Ob

A many-valued contextI is called complete iff all its attributes are complete

56


A scale for the attribute x of a many-valued contextI is a (one-valued) context Kx = (Obx , Atx , Ix) with {v ∈ Va:

I(X , x , v) for some X ∈ Ob} ⊆ Obx

If (Ob, At , Va, I) is a many-valued context and (Obx , Atx , Ix) is ascale context for every x ∈ At then

I the derived context with respect to plain scaling is the(one-valued) context (Ob′, At ′, I′) with

I Ob′ = ObI At ′ = {(x , a): x ∈ At and a ∈ Atx}I X I′ (x , a) iff I(X , x , v) and v Ix a for some v ∈ Va

57


Example 7:

De Dl R E MConv . poor good good good excellentFront good poor excellent excellent goodRear excellent excellent very poor poor very poorMid excellent excellent good very poor very poorAll excellent excellent good good poor

KDe,KDl ,KR,KE ,KM :

++ + − −−excellent ⊗ ⊗

good ⊗poor ⊗

very poor ⊗ ⊗

58

Many-valued contextsContext constructions and standard scales

If K = (Ob, At , I) is a context then we defineI Kc = (Ob, At , (Ob × At) \ I)I K−1 = (At , Ob, I−1)

59


If K1 = (Ob1, At1, I1) and K2 = (Ob2, At2, I2) are contexts thenwe define for every i ∈ {1, 2},

I Obi = {i} ×Obi

I At i = {i} × AtiI (i , X ) Ii (i , x) iff X Ii x

60


If K1 = (Ob1, At1, I1) and K2 = (Ob2, At2, I2) are contexts thenwe define

I K1∪lK2 = (Ob1 ∪ Ob2, At1 ∪ At2, I)with

I (i , X ) I x iff x ∈ Ati and X Ii x

61


Example 17:

1 2a ⊗b ⊗c ⊗

∪l

1 2d ⊗ ⊗e ⊗

=

1 2a ⊗b ⊗c ⊗d ⊗ ⊗e ⊗

62



I K1∪rK2 = (Ob1 ∪Ob2, At1 ∪ At2, I)with

I X I (i , x) iff X ∈ Obi and X Ii x

63


Example 18:

1 2a ⊗b ⊗

∪r

3 4 5a ⊗ ⊗b ⊗ ⊗

=

1 2 3 4 5a ⊗ ⊗ ⊗b ⊗ ⊗ ⊗

64



I K1∪K2 = (Ob1 ∪ Ob2, At1 ∪ At2, I1 ∪ I2)

65


Example 19:

1 2a ⊗b ⊗c ⊗

∪3 4 5

d ⊗ ⊗e ⊗ ⊗

=

1 2 3 4 5a ⊗b ⊗c ⊗d ⊗ ⊗e ⊗ ⊗

66


Nominal scales: Nk = ({1, . . . , k}, {1, . . . , k},=)

Example 20:

N4:

1 2 3 41 ⊗2 ⊗3 ⊗4 ⊗

67


Ordinal scales: Ok = ({1, . . . , k}, {1, . . . , k},6)

Example 21:

O4:

1 2 3 41 ⊗ ⊗ ⊗ ⊗2 ⊗ ⊗ ⊗3 ⊗ ⊗4 ⊗

68


Interordinal scales:Ik = ({1, . . . , k}, {1, . . . , k},6)∪r ({1, . . . , k}, {1, . . . , k},>)

Example 22:

I4:

6 1 6 2 6 3 6 4 > 1 > 2 > 3 > 41 ⊗ ⊗ ⊗ ⊗ ⊗2 ⊗ ⊗ ⊗ ⊗ ⊗3 ⊗ ⊗ ⊗ ⊗ ⊗4 ⊗ ⊗ ⊗ ⊗ ⊗

69


Biordinal scales:Mk ,l = ({1, . . . , k}, {1, . . . , k},6)∪({1, . . . , l}, {1, . . . , l},>)

Example 23:

M4,2:

6 1 6 2 6 3 6 4 > 5 > 61 ⊗ ⊗ ⊗ ⊗2 ⊗ ⊗ ⊗3 ⊗ ⊗4 ⊗5 ⊗6 ⊗ ⊗

70


Dichotomic scale: D = ({0, 1}, {0, 1},=)

D:0 1

0 ⊗1 ⊗

71



I K1 +K2 = (Ob1 ∪ Ob2, At1 ∪ At2, I)with

I (i , X ) I (j , x) iff one of the following conditions holdI i = 1, j = 1 and X I1 xI i = 1 and j = 2I i = 2 and j = 1I i = 2, j = 2 and X I2 x

72


Example 24:

1 2a ⊗b ⊗c ⊗

+

3 4 5d ⊗e ⊗ ⊗

=

1 2 3 4 5a ⊗ ⊗ ⊗ ⊗b ⊗ ⊗ ⊗ ⊗c ⊗ ⊗ ⊗ ⊗d ⊗ ⊗ ⊗e ⊗ ⊗ ⊗ ⊗

73


Proposition 17: If K1 = (Ob1, At1, I1) and K2 = (Ob2, At2, I2) arecontexts then

I B(K1 +K2) and B(K1)× B(K2) are isomorphicI (A, B) 7→ ((A ∩ Ob1, B ∩ At1), (A ∩ Ob2, B ∩ At2)) is an

isomorphism

74



I K1 ./ K2 = (Ob1 ×Ob2, At1 ∪ At2, I)with

I (X1, X2) I (i , x) iff Xi Ii x

75


Example 25:

1 2a ⊗b ⊗c ⊗

./

3 4 5d ⊗e ⊗ ⊗

=

1 2 3 4 5(a, d) ⊗ ⊗(a, e) ⊗ ⊗ ⊗(b, d) ⊗ ⊗(b, e) ⊗ ⊗ ⊗(c, d) ⊗ ⊗(c, e) ⊗ ⊗ ⊗

76


Proposition 18: If K1 = (Ob1, At1, I1) and K2 = (Ob2, At2, I2) arecontexts then

I the extents of K1 ./ K2 are precisely the sets of the formA1 × A2 each set Ai being an extent of Ki

77



I K1 ×K2 = (Ob1 ×Ob2, At1 × At2, I)with

I (X1, X2) I (x1, x2) iff X1 I1 x1 or X2 I2 x2

78


Example 26:

1 2a ⊗b ⊗

×3 4

c ⊗d ⊗ ⊗

=

(1, 3) (1, 4) (2, 3) (2, 4)

(a, c) ⊗ ⊗ ⊗(a, d) ⊗ ⊗ ⊗ ⊗(b, c) ⊗ ⊗ ⊗(b, d) ⊗ ⊗ ⊗ ⊗

79


Proposition 19: If K1 = (Ob1, At1, I1), K2 = (Ob2, At2, I2) andK3 = (Ob3, At3, I3) are contexts then

I (K1 +K2)×K3 and (K1 ×K3) + (K2 ×K3) are isomorphic

80


Contranominal scales: NcS = (S, S, 6=) for every nonempty set S

Example 27:

Nc{1,2,3}:

1 2 31 ⊗ ⊗2 ⊗ ⊗3 ⊗ ⊗

Proposition 20: If S is a nonempty set thenI the concepts of Nc

S are precisely the pairs (A, S \ A) forA ⊆ S

81


General ordinal scales: OP = (P, P,6) for every ordered set(P,6)

Example 28:

O{1,2,3}:

1 2 31 ⊗ ⊗ ⊗2 ⊗ ⊗3 ⊗

Proposition 21: If (P,6) is an ordered set thenI the concepts of OP are precisely the pairs (A, B) where A

is the set of all lower bounds of B and B is the set of allupper bounds of A

82


Contraordinal scales: OcdP = (P, P, 6>) for every ordered set

(P,6)

Example 29:

Ocd{1,2,3}:

1 2 31 ⊗ ⊗2 ⊗3

Proposition 22: If (P,6) is an ordered set thenI the concepts of Ocd

P are precisely the pairs (A, P \ A) forA ⊆ P an order ideal

83


Contraordinal scales: OS = (2S, 2S, 6⊇) for every set S

Example 30:

O{a,b}:

∅ {a} {b} {a, b}∅ ⊗ ⊗ ⊗{a} ⊗ ⊗{b} ⊗ ⊗{a, b}

84


From an ordered set (P,6), we obtain the general interordinalscale

I IP = (P, P,6) ∪r (P, P,>)

and the convex-ordinal scaleI IP = (P, P, 6>) ∪r (P, P, 66)

85

Many-valued contextsIndiscernibility

If (Ob, At , Va, I) is a complete many-valued context, with everysusbset of attributes B ⊆ At , we associate a binary relationIND(B), called an indiscernibility relation and defined thus

I IND(B) = {(X , Y ) ∈ Ob ×Ob: for every x ∈ B and forevery v ∈ Va, I(X , x , v) iff I(Y , x , v)}

For an attribute x ∈ Att , we writeI IND(x) instead of IND({x})

Obviously IND(B) is an equivalence relation andI IND(B) =

⋂{IND(x): x ∈ B}

86


Example 8:

a b c d e1 1 0 2 2 02 0 1 1 1 23 2 0 0 1 14 1 1 0 2 25 1 0 2 0 16 2 2 0 1 17 2 1 1 1 28 0 1 1 0 1

Exemplary partitions generated by attributes in this contextI Ob/IND(a) = {{1, 4, 5}, {2, 8}, {3, 6, 7}}I Ob/IND(b) = {{1, 3, 5}, {2, 4, 7, 8}, {6}}

87


Approximations of sets of objects in a complete many-valuedcontext (Ob, At , Va, I): with each subset of objects A ⊆ Ob andeach subset of attributes B ⊆ At , we associate two subsets

I IND(B)(A) = {X ∈ Ob: IND(B)(X ) ⊆ A}I IND(B)(A) = {X ∈ Ob: IND(B)(X ) ∩ A 6= ∅}

called the IND(B)-lower approximation of A and theIND(B)-upper approximation of A

ObviouslyI IND(B)(A) ⊆ A ⊆ IND(B)(A)

88


Given a subset of objects A ⊆ Ob and a subset of attributesB ⊆ At , we shall say that

I A is IND(B)-definable iff IND(B)(A) = IND(B)(A)

I A is IND(B)-rough iff IND(B)(A) 6= IND(B)(A)

Given a subset of objects A ⊆ Ob and a subset of attributesB ⊆ At , let us observe that

I IND(B)(A) is the maximal IND(B)-definable set of objectscontained in A

I IND(B)(A) is the minimal IND(B)-definable set of objectscontaining A

89


Example 9:

a b c d e1 1 0 2 2 02 0 1 1 1 23 2 0 0 1 14 1 1 0 2 25 1 0 2 0 16 2 2 0 1 17 2 1 1 1 28 0 1 1 0 1

If A = {1, 2, 3, 4, 5} and B = {a, b, c} thenI IND(B)(A) = {1, 3, 4, 5}I IND(B)(A) = {1, 2, 3, 4, 5, 8}

90


Proposition 10:1. IND(B)(∅) = ∅2. IND(B)(Ob) = Ob3. IND(B)(A1 ∪ A2) ⊇ IND(B)(A1) ∪ IND(B)(A2)

4. IND(B)(A1 ∩ A2) = IND(B)(A1) ∩ IND(B)(A2)

5. IND(B)(Ob \ A) = Ob \ IND(B)(A)

6. A1 ⊆ A2 implies IND(B)(A1) ⊆ IND(B)(A2)

7. IND(B)(IND(B)(A)) = IND(B)(A)


91


Example 10:I Suppose we are given a complete many-valued context

(Ob, At , Va, I) where Ob = {1, 2, 3, 4, 5, 6, 7, 8} and letB ⊆ At be a subset of attributes defining an equivalencerelation IND(B) with the following equivalence classes:

I {1, 4, 8}I {2, 5, 7}I {3}I {6}

I If A1 = {1, 4, 7} and A2 = {2, 8} thenI IND(B)(A1) = ∅I IND(B)(A2) = ∅I IND(B)(A1 ∪ A2) = {1, 4, 8}

92


Proposition 11:1. IND(B)(∅) = ∅2. IND(B)(Ob) = Ob3. IND(B)(A1 ∪ A2) = IND(B)(A1) ∪ IND(B)(A2)

4. IND(B)(A1 ∩ A2) ⊆ IND(B)(A1) ∩ IND(B)(A2)

5. IND(B)(Ob \ A) = Ob \ IND(B)(A)

6. A1 ⊆ A2 implies IND(B)(A1) ⊆ IND(B)(A2)



93




I {1, 4, 8}I {2, 5, 7}I {3}I {6}

I If A1 = {1, 3, 5} and A2 = {2, 3, 4, 6} thenI IND(B)(A1) = {1, 2, 3, 4, 5, 7, 8}I IND(B)(A2) = {1, 2, 3, 4, 5, 6, 7, 8}I IND(B)(A1 ∩ A2) = {3}

94


Given a complete many-valued context (Ob, At , Va, I), a subsetof objects A ⊆ Ob and a subset of attributes B ⊆ At , let

I X in(B) A iff X ∈ IND(B)(A)

I X in(B) A iff X ∈ IND(B)(A)

Intuitive readingI X in(B) A: “X surely belongs to A with respect to B”

I X in(B) A: “X possibly belongs to A with respect to B”

ObviouslyI X in(B) A implies X ∈ A

I X ∈ A implies X in(B) A95


Proposition 12:1. not X in(B) ∅2. X in(B) Ob3. X in(B) (A1 ∪ A2) if X in(B) A1 or X in(B) A2

4. X in(B) (A1 ∩ A2) iff X in(B) A1 and X in(B) A2

5. X in(B) (Ob \ A) iff not X in(B) A6. A1 ⊆ A2 implies X in(B) A1 only if X in(B) A2

96


Proposition 13:1. not X in(B) ∅2. X in(B) Ob3. X in(B) (A1 ∪ A2) iff X in(B) A1 or X in(B) A2

4. X in(B) (A1 ∩ A2) only if X in(B) A1 and X in(B) A2

5. X in(B) (Ob \ A) iff not X in(B) A

6. A1 ⊆ A2 implies X in(B) A1 only if X in(B) A2

97


Given a complete many-valued context (Ob, At , Va, I), a subsetof objects A ⊆ Ob and a subset of attributes B ⊆ At , let

I BN(B)(A) = IND(B)(A) \ IND(B)(A)

be the IND(B)-boundary of A

ObviouslyI A is IND(B)-definable iff BN(B)(A) = ∅I A is IND(B)-rough iff BN(B)(A) 6= ∅

98




I {1, 4, 8}I {2, 5, 7}I {3}I {6}

I If A1 = {1, 4, 7} and A2 = {2, 8} thenI BN(B)(A1) = {1, 2, 4, 5, 7, 8}I BN(B)(A2) = {1, 2, 4, 5, 7, 8}

I If A1 = {1, 3, 5} and A2 = {2, 3, 4, 6} thenI BN(B)(A1) = {1, 2, 4, 5, 7, 8}I BN(B)(A2) = {1, 2, 4, 5, 7, 8}

99


Given a complete many-valued context (Ob, At , Va, I), anonempty subset of objects A ⊆ Ob and a subset of attributesB ⊆ At , let

I α(B)(A) =Card(IND(B)(A))

Card(IND(B)(A))

be the IND(B)-accuracy measure of A

ObviouslyI 0 6 α(B)(A) 6 1

MoreoverI A is IND(B)-definable iff α(B)(A) = 1I A is IND(B)-rough iff α(B)(A) < 1

100


Given a complete many-valued context (Ob, At , Va, I), anonempty subset of objects A ⊆ Ob and a subset of attributesB ⊆ At , let

I ρ(B)(A) = Card(BN(B)(A))

Card(IND(B)(A))

be the IND(B)-roughness measure of A

ObviouslyI 0 6 ρ(B)(A) 6 1

MoreoverI A is IND(B)-definable iff ρ(B)(A) = 0I A is IND(B)-rough iff ρ(B)(A) > 0

101




I {1, 4, 8}I {2, 5, 7}I {3}I {6}

I If A = {1, 4, 5} thenI IND(B)(A) = ∅I BN(B)(A) = {1, 2, 4, 5, 7, 8}I IND(B)(A) = {1, 2, 4, 5, 7, 8}I α(B)(A) = 0

6 = 0.00I ρ(B)(A) = 6

6 = 1.00

102




I {1, 4, 8}I {2, 5, 7}I {3}I {6}

I If A = {3, 5} thenI IND(B)(A) = {3}I BN(B)(A) = {2, 5, 7}I IND(B)(A) = {2, 3, 5, 7}I α(B)(A) = 1

4 = 0.25I ρ(B)(A) = 3

4 = 0.75

103




I {1, 4, 8}I {2, 5, 7}I {3}I {6}

I If A = {3, 6, 8} thenI IND(B)(A) = {3, 6}I BN(B)(A) = {1, 4, 8}I IND(B)(A) = {1, 3, 4, 6, 8}I α(B)(A) = 2

5 = 0.40I ρ(B)(A) = 3

5 = 0.60

104


Given a complete many-valued context (Ob, At , Va, I), anonempty family of nonempty subset of objectsF = {A1, . . . , An} and a subset of attributes B ⊆ At , let

I α(B)(F ) =Σi Card(IND(B)(Ai ))

Σi Card(IND(B)(Ai ))

be the IND(B)-accuracy of approximation of F and let

I γ(B)(F ) =Σi Card(IND(B)(Ai ))

Card(Ob)

be the IND(B)-quality of approximation of F

105


Given a complete many-valued context (Ob, At , Va, I), subsetsof objects A1, A2 ⊆ Ob and a subset of attributes B ⊆ At , let

I A1 sim(B) A2 iff IND(B)(A1) = IND(B)(A2)

I A1 sim(B) A2 iff IND(B)(A1) = IND(B)(A2)

Intuitive readingI A1 sim(B) A2: “the positive examples of A1 and A2 are the

same”I A1 sim(B) A2: “the negative examples of A1 and A2 are the

same”

Obviously sim(B) and sim(B) are equivalence relations

106




I {1, 4, 5}I {2, 3}I {6}I {7, 8}

I If A1 = {1, 2, 3} and A2 = {2, 3, 7} thenI IND(B)(A1) = {2, 3}I IND(B)(A2) = {2, 3}

I If A1 = {1, 2, 7} and A2 = {2, 3, 4, 8} thenI IND(B)(A1) = {1, 2, 3, 4, 5, 7, 8}I IND(B)(A2) = {1, 2, 3, 4, 5, 7, 8}

107


Proposition 14:1. A1 sim(B) A2 iff (A1 ∩ A2) sim(B) A1 and

(A1 ∩ A2) sim(B) A2

2. if A1 sim(B) A′1 and A2 sim(B) A′2 then(A1 ∩ A2) sim(B) (A′1 ∩ A′2)

3. if A1 sim(B) A2 then (A1 ∩ (Ob \ A2)) sim(B) ∅4. if A1 ⊆ A2 then A2 sim(B) ∅ implies A1 sim(B) ∅5. if A1 ⊆ A2 then A1 sim(B) Ob implies A2 sim(B) Ob6. if A1 sim(B) ∅ or A2 sim(B) ∅ then (A1 ∩ A2) sim(B) ∅

108


Proposition 15:1. A1 sim(B) A2 iff (A1 ∪ A2) sim(B) A1 and

(A1 ∪ A2) sim(B) A2

2. if A1 sim(B) A′1 and A2 sim(B) A′2 then(A1 ∪ A2) sim(B) (A′1 ∪ A′2)

3. if A1 sim(B) A2 then (A1 ∪ (Ob \ A2)) sim(B) Ob4. if A1 ⊆ A2 then A2 sim(B) ∅ implies A1 sim(B) ∅5. if A1 ⊆ A2 then A1 sim(B) Ob implies A2 sim(B) Ob6. if A1 sim(B) Ob or A2 sim(B) Ob then (A1 ∪ A2) sim(B) Ob

109


Proposition 16:1. IND(B)(A) is the intersection of all subsets of objects

A′ ⊆ Ob such that A sim(B) A′

2. IND(B)(A) is the union of all subsets of objects A′ ⊆ Obsuch that A sim(B) A′

110

Many-valued contextsTernary contexts

Ternary context: structure of the form S = (Ob, At , Co, I) whereI Ob is a nonempty set of formal objectsI At is a nonempty set of formal attributesI Co is a nonempty set of formal conditionsI I is a ternary relation between Ob, At and Co

Ternary contexts will usually be denotedI S = (S1, S2, S3, I)

111


A ternary context can be represented by a cross cube whereI 1-rows are headed by object names (X , Y , etc)I 2-rows are headed by attribute names (x , y , etc)I 3-rows are headed by condition names (α, β, etc)

A cross in 1-row X , 2-row x and 3-row α means thatI the object X has the attribute x under the condition α

112


Given a ternary context S = (S1, S2, S3, I), i , j , k ∈ {1, 2, 3}pairwise distinct, a set Ai ⊆ Si of Si -elements and a set Aj ⊆ Sjof Sj -elements, we define

I (Ai , Aj)k = {xk ∈ Sk : I(xi , xj , xk ) for every xi ∈ Ai and for

every xj ∈ Aj}i.e. the set of Sk -elements common to the pairs (xi , xj) in Ai ×Aj

It is still a problem to generalize to ternary concepts thetechniques in formal concept analysis that are presented inthese slides

113


A ternary concept of the ternary context (S1, S2, S3, I) is a triple(A1, A2, A3) with

I A1 ⊆ S1

I A2 ⊆ S2

I A3 ⊆ S3

I (A1, A2)3 = A3

I (A1, A3)2 = A2

I (A2, A3)1 = A1

We callI A1 the extent of the concept (A1, A2, A3)

I A2 the intent of the concept (A1, A2, A3)

I A3 the mode of the concept (A1, A2, A3)

114

Determination and representation

115

Determination and representationA context for the planets




116

Determination and representationContexts and concepts

Formal context: structure of the forme K = (Ob, At , I) whereI Ob is a nonempty set of formal objectsI At is a nonempty set of formal attributesI I is a binary relation between Ob and At

Within the context of the planetsI Ob = {Mercury , Venus, . . .}I At = {small , medium, . . .}I I = {(Mercury , small), (Mercury , near), . . .}

117






Within the context of the planetsI {Earth, Mars}′ = {small , near , yes}I {small , near}′ = {Mercury , Venus, Earth, Mars}

118



Within the context of the planetsI ({Earth, Mars}, {small , near , yes})I ({Mercury , Venus, Earth, Mars}, {small , near})

119

Determination and representationThe ordering of concepts






120


Within the context of the planets (Mercury = 1, Venus = 2,Earth = 3, Mars = 4, Jupiter = 5, Saturn = 6, Uranus = 7,Neptune = 8 et Pluto = 9)

r∅

r12 r34 r9 r56 r78

r1234 r349 r56789

r12349 r3456789

r123456789

PPPPPPPPP

@@

@

��

�

��

��

�

@@

@

��

�

@@

@

��

�

@@

@�

��

@@

@

��

�

@@

@�

��

@@

@

121



I∧

t∈T (At , Bt) = (⋂

t∈T At , (⋃

t∈T Bt)′′)

I∨

t∈T (At , Bt) = ((⋃

t∈T At)′′,

⋂t∈T Bt)

122

Determination and representationThe determination problem

A simple-minded and extremely inefficient way of determiningall the concepts of a context K = (Ob, At , I)

1. choose a set A of objects2. compute the set A′ of attributes common to the objects in A3. compute the set A′′ of objects which have all attributes in A′

Then the pair (A′′, A′) is a concept

123


The concept ({Earth, Mars}, {small , near , yes})




124






125






126


The concept ({Mercury , Venus, Earth, Mars}, {small , near})




127






128






129

Determination and representationAn algorithm for finding all concepts of a given context

A simple-minded and extremely inefficient way of determiningall the concepts of a context K = (Ob, At , I)

1. choose a set B of attributes2. compute the set B′ of objects which have all attributes in B3. compute the set B′′ of attributes common to the objects in

B′

Then the pair (B′, B′′) is a concept

Remark that for all A ⊆ Ob and for all B ⊆ AtI A′ =

⋂X∈A X ′ and B′ =

⋂x∈B x ′

In particular, if (A, B) is a concept thenI A =

⋂x∈B x ′ and B =

⋂X∈A X ′

130


Let K = (Ob, At , I) be a given context

131


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

132


Let K = (Ob, At , I) be a given context1. draw up a table with two columns headed Attributes (A)

and Extents (E), leave the first cell of the A column emptyand write Ob in the first cell of the E column

133


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

A EOb

134




2. find a maximal attribute extent, say x ′

135


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

A EOb

136





2.1 if the set x ′ is not already in the E column, add the row[x , x ′] to the table, intersect the set x ′ with all previousextents in E , add these intersections to the E columnunless they are already in the list

2.2 if the set x ′ is already in the E column, add the label x tothe attribute cell of the rwo where x ′ previously occured

137


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

A EOb

a STUVX

138


Let K = (Ob, At , I) be a given context

1. draw up a table with two columns headed Attributes (A)and Extents (E), leave the first cell of the A column emptyand write Ob in the first cell of the E column


2.1 if the set x ′ is not already in the E column, add the row[x , x ′] to the table, intersect the set x ′ with all previousextents in E , add these intersections to the E columnunless they are already in the list

2.2 if the set x ′ is already in the E column, add the label x tothe attribute cell of the rwo where x ′ previously occured

3. delete the column below x from the context4. if the last column has been deleted, stop, otherwise return

to 2

139


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

A EOb

a STUVXb STUW

140


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

A EOb

a STUVXb STUW

STU

141


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

A EOb

a STUVXbd STUW

STU

142


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

A EOb

a STUVXbd STUW

STUf SUVX

143


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

A EOb

a STUVXbd STUW

STUf SUVX

SU

144


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

A EOb

a STUVXbd STUW

STUf SUVX

SUe TUV

145


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

A EOb

a STUVXbd STUW

STUf SUVX

SUe TUV

TUUVU

146


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

A EOb

a STUVXbd STUW

STUf SUVX

SUe TUV

TUUVU

c V

147


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

A EOb

a STUVXbd STUW

STUf SUVX

SUe TUV

TUUVU

c V∅

148


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

A EOb

a STUVXbd STUW

STUf SUVX

SUeg TUV

TUUVU

c V∅

149


Example 32:

rTU

rSUVX

rU

rSTUVX

rUV rSU

rTUV rSTU

rV

r∅

rSTUW

rSTUVWX

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

150


It is still possible to effect improvements in finding all conceptsof a given context

I Choi, V.: Faster algorithms for constructing a concept(Galois) lattice. In Butenko, S., Chaovalitwongse, W.,Pardalos, P. (Editors): Clustering Challenges in BiologicalNetworks. World Scientific (2009) 169–185.

I Kuznetsov, S., Obiedkov, S.: Comparing performance ofalgorithms for generating concept lattices. Journal ofExperimental & Theoretical Artificial Intelligence 14 (2002)189–216.

151


It is still possible to effect improvements in finding all conceptsof a given context

I Van der Merwe, D., Obiedkov, S., Kourie, D.: AddIntent: anew incremental algorithm for constructing conceptlattices. In Eklund, P. (Editor): ICFCA 2004.Springer-Verlag (2004) 372–385.

I Valtchev, P., Missaoui, R.: Building concept (Galois)lattices from parts: generalizing the incremental methods.In Delugach, H., Stumme, G. (Editors): ICCS 2001.Springer-Verlag (2001) 290–303.

152

Determination and representationDrawing the concept lattice of a given context

Given a formal context K = (Ob, At , I), the problem is toarrange the nodes and lines of the diagram of its concept latticein order to achieve

I the best visual qualityI the best visual readability

Do it fast and automatically

153


Example 32:

a b c d e f gS ⊗ ⊗ ⊗ ⊗T ⊗ ⊗ ⊗ ⊗U ⊗ ⊗ ⊗ ⊗ ⊗ ⊗V ⊗ ⊗ ⊗ ⊗W ⊗ ⊗X ⊗ ⊗

154


Example 32:

rTU

rSUVX

rU

rSTUVX

rUV rSU

rTUV rSTU

rV

r∅

rSTUW

rSTUVWX

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

155


There are several subjective human æsthetics criteriaI minimizing line crossings (planarity)I maximizing angle between incident linesI maximizing symmetriesI maximizing compactness

These criteria are often contradictory and lead tocomputationaly difficult (NP-complete) problems

How large lattices one can draw by a computer?I Up to about a hundred of nodes

156

Determination and representationA force directed approach for drawing the concept lattice of a given context

Let K = (Ob, At , I) be a given context1. within a 3-dimensional space, organize nodes of the

concept lattice in layers based on their distance from thetop node (∅′, ∅′′)

2. for each layer, randomly arrange its nodes as the verticesof a regular polygon which has a circumscribed circle ofradius 1

3. between each pair of nodes occurring in two successivelayers, calculate imaginary repulsive and attractive forcesdepending on how much this pair of nodes overlap

4. inside each layer, modify the positions of its nodesaccording to the forces calculated in step 3

5. if the resulting diagram is not ”good enough” then go tostep 3

157

Determination and representationA vectorial approach for drawing the concept lattice of a given context

Let K = (Ob, At , I) be a given context1. choose a point pos0 ∈ R× R2. associate to each object X ∈ Ob a vector

~vec(X ) ∈ R× R+?

3. for each extent A of a K-concept, computepos0 + Σ{ ~vec(X ) : X ∈ A}

158


Example 33:

a b c1 ⊗ ⊗2 ⊗ ⊗3 ⊗ ⊗ ⊗

r(3, abc)

r(13, ac) r(23, bc)

r(123, c)

@@

@

��

��

��

@@

@

159


Example 33:I choose a point pos0 ∈ R× R

a b c1 ⊗ ⊗2 ⊗ ⊗3 ⊗ ⊗ ⊗

r(3, abc)

r(13, ac) r(23, bc)

r(123, c)

@@

@

��

��

��

@@

@

rpos0

160


Example 33:I associate to each object X ∈ Ob a vector

~vec(X ) ∈ R× R+?

a b c1 ⊗ ⊗2 ⊗ ⊗3 ⊗ ⊗ ⊗

r(3, abc)

r(13, ac) r(23, bc)

r(123, c)

@@

@

��

��

��

@@

@

rpos0

@@

@I~vec(1)

6~vec(2)

��

��~vec(3)

161


Example 33:I for each extent A of a K-concept, compute

pos0 + Σ{ ~vec(X ) : X ∈ A}

a b c1 ⊗ ⊗2 ⊗ ⊗3 ⊗ ⊗ ⊗

r(3, abc)

r(13, ac) r(23, bc)

r(123, c)

@@

@

��

��

��

@@

@

rpos0

@@

@I~vec(1)

6~vec(2)

��

��~vec(3)

r(3, abc)

r(13, ac) r(23, bc)

r(123, c)

��

��@

@@I 6

6

@@

@I

162

Determination and representationA dichotomic approach for drawing the concept lattice of a given context

Let K = (Ob, At , I) be a given context1. choose At1 ⊆ At and At2 ⊆ At such that At1 ∪ At2 = At2. draw the concept lattices of the contextsK1 = (Ob, At1, I ∩ (Ob × At1)) andK2 = (Ob, At2, I ∩ (Ob × At2))

3. draw the product of these lattices4. for each K-intent B, compute the corresponding element

(B ∩ At1, B ∩ At2) in the product

163


Example 34:

a b c d1 ⊗2 ⊗ ⊗3 ⊗4 ⊗ ⊗

rabcd

rd rb

r∅rcd rab

@@

@

��

��

��

@@

@

164


Example 34:I choose At1 ⊆ At and At2 ⊆ At such that At1 ∪ At2 = At

a b c d1 ⊗2 ⊗ ⊗3 ⊗4 ⊗ ⊗

a b1 ⊗2 ⊗ ⊗34

c d123 ⊗4 ⊗ ⊗

rabcd

rd rb

r∅rcd rab

@@

@

��

��

��

@@

@

165


Example 34:I draw the concept lattices of the contextsK1 = (Ob, At1, I ∩ (Ob × At1)) andK2 = (Ob, At2, I ∩ (Ob × At2))

a b c d1 ⊗2 ⊗ ⊗3 ⊗4 ⊗ ⊗

a b1 ⊗2 ⊗ ⊗34

c d123 ⊗4 ⊗ ⊗

rabcd

rd rb

r∅rcd rab

@@

@

��

��

��

@@

@

rab

rb

r∅

rcd rd r∅

166


Example 34:I draw the product of these lattices

a b c d1 ⊗2 ⊗ ⊗3 ⊗4 ⊗ ⊗

rabcd

rd rb

r∅rcd rab

@@

@

��

��

��

@@

@

rab

rb

r∅

rcd rd r∅

r(ab, cd)

r(b, cd)

r(∅, cd)

r(ab, d)

r(b, d)

r(∅, d)

r(ab, ∅)

r(b, ∅)

r(∅, ∅)

167


Example 34:I for each K-intent B, compute the corresponding element

(B ∩ At1, B ∩ At2) in the product

a b c d1 ⊗2 ⊗ ⊗3 ⊗4 ⊗ ⊗

rabcd

rd rb

r∅rcd rab

@@

@

��

��

��

@@

@

rab

rb

r∅

rcd rd r∅

r(ab, cd)

r(b, cd)

r(∅, cd)

r(ab, d)

r(b, d)

r(∅, d)

r(ab, ∅)

r(b, ∅)

r(∅, ∅)

168


It is still possible to effect improvements in drawing the conceptlattice of a given context

I Freese, R.: Automated lattice drawing. In Eklund, P.(Editor): ICFCA 2004. Springer-Verlag (2004) 112–127.

I Tilley, T.: Tool support for FCA. In Eklund, P. (Editor):ICFCA 2004. Springer-Verlag (2004) 104–111.

169

Determination and representationImplications between attributes

It isI often necessary to classify a large number of objects with

respect to a relatively small number of attributesI frequently useless or impracticable to write down the whole

context

In such casesI the concept lattice can be inferred from the implication

between the attributesI the concept lattice can be inferred from statements of the

kind “every object with the attributes x1, y1, . . . also has theattributes x2, y2, . . .”

170


Example 35:

concave square rectangle equilateral parallelogram1 ×23 × ×4 × × × ×5 × ×6 ×

1

2

3 4 5

JJJ

JJJ

6��

��

AA

A

BB

BBB

171


Implication between attributes in a given context K = (Ob, At , I)

I implication B1 −→ B2 where B1 and B2 are sets ofK-attributes

Let B be a set of K-attributes, B1 −→ B2 a K-implication and La set of K-implications

I B respects B1 −→ B2 iff B1 6⊆ B or B2 ⊆ BI B respects L iff B respects every K-implication

B1 −→ B2 ∈ L

172


Example 36:


{concave, parallelogram} −→ {square, rectangle, equilateral}{square} −→ {rectangle, equilateral , parallelogram}{rectangle} −→ {parallelogram}{rectangle, equilateral , parallelogram} −→ {square}{equilateral} −→ {parallelogram}

173


Implication between attributes in a given context K = (Ob, At , I)

I implication B1 −→ B2 where B1 and B2 are sets ofK-attributes

Let B1 −→ B2 a K-implication and L a set of K-implicationsI K respects B1 −→ B2 iff B respects B1 −→ B2 for eachK-concept (A, B)

I K respects L iff K respect every K-implicationB1 −→ B2 ∈ L

Implicational theory of KI Set Imp(K) of all K-implications that K respects

174


Suppose thatI K = (Ob, At , I) is a contextI B1 −→ B2 is a K-implication

Then the following conditions are equivalentI K respects B1 −→ B2

I B′1 ⊆ B′

2I B′′

1 ⊇ B2

175


Implicational closure of a set L of K-implications : mappingClL(·) : 2At −→ 2At such that for all B ⊆ At , ClL(B)is the smallest set of K-attributes containing B andrespecting L

176


Example 37:


If L contains the implications {rectangle} −→ {parallelogram}and {rectangle, equilateral , parallelogram} −→ {square} then

I ClL({rectangle, equilateral}) ={square, rectangle, equilateral , parallelogram}

177


Let B1 −→ B2 be a K-implication and L be a set ofK-implications

I B1 −→ B2 is a consequence of L iff ClL(B1) ⊇ B2

Let L and M be sets of K-implicationsI L is sound for M iff every implication that follows from L is

in MI L is complete for M iff every implication in M follows fromL

178


Let L be a set of K-implicationsI L is a base for K iff L is sound and complete for the set of

all K-implications that K respectsI L is a Duquenne-Guigues base for K iff L is a base for K

that is of minimum cardinality

179


Example 38:


{concave, parallelogram} −→ {square, rectangle, equilateral}{square} −→ {rectangle, equilateral , parallelogram}{rectangle} −→ {parallelogram}{rectangle, equilateral , parallelogram} −→ {square}{equilateral} −→ {parallelogram}

180


Suppose thatI K = (Ob, At , I) is a contextI B ⊆ At

Then B is a good attribute subset of K iffI B′′ ) BI for all C ( B, if C′′ ) C then B ) C′′

181


Example 39:


{concave, parallelogram}{square}{rectangle}{rectangle, equilateral , parallelogram}{equilateral}

182


Suppose thatI K = (Ob, At , I) is a context

ThenI {B −→ B′′ : B ⊆ At is a good attribute subset of K} is a

Duquenne-Guigues base for K

Example 40: Within the context of the quadrilateralsI {concave, parallelogram} −→{square, rectangle, equilateral}

I {square} −→ {rectangle, equilateral , parallelogram}I {rectangle} −→ {parallelogram}I {rectangle, equilateral , parallelogram} −→ {square}I {equilateral} −→ {parallelogram}

183


We consider the following problemI Deciding whether a set of attributes is a good attribute

subset of a contextInput A context K = (Ob, At , I) and a set of

attributes B ⊆ AtOutput Decide whether B is a good attribute subset

of K

184


Suppose thatI K = (Ob, At , I) is a contextI B ⊆ At is a set of K-attributes

We shall say thatI B is closed iff B′′ = BI B is quasi-closed iff for all sets C ( B of K-attributes,

C′′ ⊆ B or C′′ = B′′

I B is pseudo-closed iff B is not closed, B is quasi-closedand for all quasi-closed sets C ( B of K-attributes, C′′ ( B

Note thatI if B is closed then B is quasi-closed

185


Proposition 23: If K = (Ob, At , I) is a context and B ⊆ At is aset of K-attributes then

1. B is quasi-closed iff B ∩ C is closed for every closed set Cwith B 6⊆ C

2. B is quasi-closed iff B ∩ X ′ is closed or B ∩ X ′ = B for anyobject X ∈ Ob

3. B is pseudo-closed iff B is a good attribute subset of K

Proposition 24: If K = (Ob, At , I) is a context and B1, B2 ⊆ Atare sets of K-attributes then

I if B1, B2 are quasi-closed then B1 ∩ B2 is quasi-closed

186


Proposition 25: Testing whether B ⊆ At is quasi-closed in thecontext K = (Ob, At , I) may be performed inO(Card(Ob)× Card(At)) time

Proposition 26: The following problem is in coNP:Input A context K = (Ob, At , I) and a set of attributes

B ⊆ AtOutput Decide whether B is a good attribute subset of K

187


A hypergraph H = (V , E) is a pair consisting ofI a finite nonempty set V (vertices)I a set E of subsets of V (edges)

A hypergraph H = (V , E) is called simple iffI none of H’s edges contains another edge of H

A hypergraph H = (V , E) is called saturated iffI every subset of V is contained in at least one edge of H or

it contains at least one edge of H

188


Proposition 27: The following problem is coNP-complete:Input A hypergraph H = (V , E)

Output Decide whether H is saturated

Proposition 28: The following problem is in coNP:Input A simple hypergraph H = (V , E)


189


A set of vertices W ⊆ V is called a transversal of a hypergraphH = (V , E) iff

I W intersects every edge of H

A set of vertices W ⊆ V is called a minimal transversal of ahypergraph H = (V , E) iff

I W is a transversal of HI no proper subset of W is a transversal of H

The set of all minimal transversals of a hypergraph H = (V , E)constitutes another hypergraph on V called the transversal of H

190


Proposition 28: The following problem is in coNP:Input A simple hypergraph H = (V , E)


Proposition 29: The following problem is under polynomialtransformations computationally equivalent to the problemconsidered in Proposition 28:

Input Two hypergraphs G = (V , EG) and H = (V , EH)

Output Decide whether G is the transversal of H

191


Proposition 30: The following problem is under polynomialtransformations at least as hard as the problems considered inProposition 28 and Proposition 29:

Input A context K = (Ob, At , I) and a set of attributesB ⊆ At

Output Decide whether B is a good attribute subset of K

192


It is still possible to effect improvements in deciding if a givenset of attributes is a good attribute subset of a given context

I Distel, F., Sertkaya, B.: On the complexity of enumeratingpseudo-intents. Discrete Applied Mathematics 159 (2011)450–466.

I Kuznetsov, S., Obiedkov, S.: Counting pseudo-intents and]P-completeness. In Missaoui, R., Schmid, J. (Editors):ICFCA 2006. Springer-Verlag (2006) 306–308.

I Sertkaya, B.: Some computational problems related topseudo-intents. In Ferre, S., Rudolph, S. (Editors): ICFCA2009. Springer-Verlag (2009) 130–145.

193


It is still possible to effect improvements in enumerating the setof all good attribute subsets of a given context

I Distel, F., Sertkaya, B.: On the complexity of enumeratingpseudo-intents. Discrete Applied Mathematics 159 (2011)450–466.

I Kuznetsov, S., Obiedkov, S.: Counting pseudo-intents and]P-completeness. In Missaoui, R., Schmid, J. (Editors):ICFCA 2006. Springer-Verlag (2006) 306–308.

I Sertkaya, B.: Some computational problems related topseudo-intents. In Ferre, S., Rudolph, S. (Editors): ICFCA2009. Springer-Verlag (2009) 130–145.

194

Concept algebras

195

Concept algebrasJoin and meet of concepts

Example 41:




196


Formal context: structure of the forme K = (Ob, At , I) whereI Ob is a nonempty set of formal objectsI At is a nonempty set of formal attributesI I is a binary relation between Ob and At

Example 41: Within the context of the planetsI Ob = {Mercury , Venus, . . .}I At = {small , medium, . . .}I I = {(Mercury , small), (Mercury , near), . . .}

197






Example 41: Within the context of the planetsI {Earth, Mars}′ = {small , near , yes}I {small , near}′ = {Mercury , Venus, Earth, Mars}

198



Example 41: Within the context of the planetsI ({Earth, Mars}, {small , near , yes})I ({Mercury , Venus, Earth, Mars}, {small , near})

199







200


Example 41:




Mercury = 1, Venus = 2, Earth = 3, Mars = 4, Jupiter = 5,Saturn = 6, Uranus = 7, Neptune = 8 et Pluto = 9

201


Example 41: Within the context of the planets (Mercury = 1,Venus = 2, Earth = 3, Mars = 4, Jupiter = 5, Saturn = 6,Uranus = 7, Neptune = 8 et Pluto = 9)

r∅

r12 r34 r9 r56 r78

r1234 r349 r56789

r12349 r3456789

r123456789

PPPPPPPPP

@@

@

��

�

��

��

�

@@

@

��

�

@@

@

��

�

@@

@�

��

@@

@

��

�

@@

@�

��

@@

@

202


Example 42:

cold moist dry warmwater × ×earth × ×air × ×fire × ×

water = w , earth = e, air = a et fire = f

203


Example 42: water = w , earth = e, air = a et fire = f

r∅

rw re ra rf

rwe rwa ref raf

rweaf

QQ

QQ

QQ

QQQ

AA

AA

AA

��

��

��

��

��

��

��

��

@@

@@

@@

��

��

��

@@

@@

@@

��

��

��

@@

@@

@@

��

��

��

��

��

AA

AA

AA

QQ

QQ

QQ

QQQ

204



I∧

t∈T (At , Bt) = (⋂

t∈T At , (⋃

t∈T Bt)′′)

I∨

t∈T (At , Bt) = ((⋃

t∈T At)′′,

⋂t∈T Bt)

205

Concept algebrasJoin, meet and complement of concepts

Example: the concept “piano”extent : the piano of Ray Charles, the piano of Diana Krall,

etcintent : to have a keyboard, to have pedals, etc

What is the negation of the concept “piano” ?extent : the objects that do not possess one of the

attributes of the concept “piano” ?intent : the attributes that are not possessed by one of the

objects of the concept “piano” ?

206


The negation of the concept({Earth, Mars}, {small , near , yes}) :({Mercury , Venus, Jupiter , Saturn, Uranus, Neptune, Pluto}, ?)

small medium large near far yes noMercury × × ×Venus × × ×Earth × × ×Mars × × ×

Jupiter × × ×Saturn × × ×Uranus × × ×

Neptune × × ×Pluto × × ×

207


The negation of the concept({Earth, Mars}, {small , near , yes}) :(?, {medium, large, far , no})




208


The negation of the concept({Mercury , Venus, Earth, Mars}, {small , near}) :({Jupiter , Saturn, Uranus, Neptune, Pluto}, ?)


Jupiter × ⊗ ⊗Saturn × ⊗ ⊗Uranus × ⊗ ⊗

Neptune × ⊗ ⊗Pluto × ⊗ ⊗

209


The negation of the concept({Mercury , Venus, Earth, Mars}, {small , near}) :(?, {medium, large, far , yes, no})




210


Join of concepts (A1, B1) and (A2, B2)

I ((A1 ∪ A2)′′, B1 ∩ B2)

Meet of concepts (A1, B1) and (A2, B2)

I (A1 ∩ A2, (B1 ∪ B2)′′)

Complement of concept (A, B)

I (Obj \A,−) ? No since • is not always an extentI (−,Att \ B) ? No since • is not always an intentI ((Obj \ A)′′, (Obj \ A)′) ? No since • may intersect AI ((Att \ B)′,(Att \ B)′′) ? No since • may intersect B

211


Example 42:

cold moist dry warmwater × ×earth × ×air × ×fire × ×

water = w , earth = e, air = a et fire = f

212


Example 42: water = w , earth = e, air = a et fire = f

r∅

rw re ra rf

rwe rwa ref raf

rweaf

QQ

QQ

QQ

QQQ

AA

AA

AA

��

��

��

��

��

��

��

��

@@

@@

@@

��

��

��

@@

@@

@@

��

��

��

@@

@@

@@

��

��

��

��

��

AA

AA

AA

QQ

QQ

QQ

QQQ

213

Concept algebrasSemiconcepts and protoconcepts

ContextsI K = (Ob, At , I) be a contextI A ⊆ Ob be a set of objectsI B ⊆ At be a set of attributes

Concepts

I (A, B) is a H-concept iff B′ = A and A′ = B

Semiconcepts

I (A, B) is a H-semiconcept iff B′ = A or A′ = B

Protoconcepts

I (A, B) is a H-protoconcept iff B′ = A′′ or A′ = B′′

214


Example 43:

a b1 ×2 × ×

r(∅, ab)

r(∅, a) r(1, b) r(2, ab)

r(1, ∅) r(2, a) r(12, b)

r(12, ∅)

@@

@@

@@

��

��

��

��

��

��

@@

@@

@@

��

��

��

@@

@@

@@

��

��

��

@@

@@

@@

215


Example 44:

a b c1 × ×2 × ×3 × × ×

216


Example 44:

r(∅, abc)

r(1, ac) r(2, bc)

r(12, c)

@@

@

��

��

��

@@

@

r(∅, ab)

r(1, a) r(2, b)

r(12, ∅)

@@

@

��

��

��

@@

@

r(∅, ab)

r(13, ac) r(23, bc)

r(123, c)

@@

@

��

��

��

@@

@

r(3, ab)

r(13, a) r(23, b)

r(123, ∅)

@@

@

��

��

��

@@

@

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

QQ

217

Concept algebrasProtoconcept algebras

StructureA(H) = (AH,⊥Hl ,>Hr ,>Hl ,⊥Hr ,¬Hl ,¬Hr ,∨Hl ,∧Hr ,∧Hl ,∨Hr ) whereAH is the set of all H’s protoconcepts and

I ⊥Hl = (∅, At)I >Hr = (Ob, ∅)I >Hl = (Ob, Ob′)I ⊥Hr = (At ′, At)I ¬Hl (A, B) = (Ob \ A, (Ob \ A)′)

I ¬Hr (A, B) = ((At \ B)′, At \ B)

I (A1, B1)∨Hl (A2, B2) = (A1 ∪ A2, (A1 ∪ A2)′)

I (A1, B1)∧Hr (A2, B2) = ((B1 ∪ B2)′, B1 ∪ B2)

I (A1, B1)∧Hl (A2, B2) = (A1 ∩ A2, (A1 ∩ A2)′)

I (A1, B1)∨Hr (A2, B2) = ((B1 ∩ B2)′, B1 ∩ B2)

218


Example 45:

a b1 ×2 × ×

r(∅, ab)

r(∅, a) r(1, b) r(2, ab)

r(1, ∅) r(2, a) r(12, b)

r(12, ∅)

@@

@@

@@

��

��

��

��

��

��

@@

@@

@@

��

��

��

@@

@@

@@

��

��

��

@@

@@

@@

219


I ∧l is AC ∨r is ACI ∧l distributes over ∨l ∨r distributes over ∧r

I ¬l(x ∧l x) = ¬lx ¬r (x ∨r x) = ¬r xI x ∧l (y ∧l y) = x ∧l y x ∨r (y ∨r y) = x ∨r yI x ∧l (x ∨l y) = x ∧l x x ∨r (x ∧r y) = x ∨r xI x ∧l (x ∨r y) = x ∧l x x ∨r (x ∧l y) = x ∨r xI ¬l(¬lx ∧l ¬ly) = x ∨l y ¬r (¬r x ∨r ¬r y) = x ∧r yI ¬l⊥l = >l ¬r>r = ⊥r

I ¬l>r = ⊥l ¬r⊥l = >r

I >r ∧l >r = >l ⊥l ∨r ⊥l = ⊥r

I x ∧l ¬lx = ⊥l x ∨r ¬r x = >r

I ¬l¬l(x ∧l y) = x ∧l y ¬r¬r (x ∨r y) = x ∨r yI (x ∨r x) ∧l (x ∨r x) = (x ∧l x) ∨r (x ∧l x)

220


Let H = (Ob, At , I) be a contextI If AH is the set of all H’s protoconcepts then the structureA(H) = (AH,⊥Hl ,>Hr ,¬Hl ,¬Hr ,∨Hl ,∧Hr ) is a protoconceptalgebra

Let A = (A,⊥l ,>r ,¬l ,¬r ,∨l ,∧r ) be a protoconcept algebraI There exists a context H(A) = (ObA, AtA, IA) such that A

is embeddable into the structure A(H(A)) =

(AH(A),⊥H(A)l ,>H(A)

r ,¬H(A)l ,¬H(A)

r ,∨H(A)l ,∧H(A)

r )

221

Concept algebrasSemiconcept algebras

StructureA(H) = (AH,⊥Hl ,>Hr ,>Hl ,⊥Hr ,¬Hl ,¬Hr ,∨Hl ,∧Hr ,∧Hl ,∨Hr ) whereAH is the set of all H’s semiconcepts and

I ⊥Hl = (∅, At)I >Hr = (Ob, ∅)I >Hl = (Ob, Ob′)I ⊥Hr = (At ′, At)I ¬Hl (A, B) = (Ob \ A, (Ob \ A)′)

I ¬Hr (A, B) = ((At \ B)′, At \ B)

I (A1, B1)∨Hl (A2, B2) = (A1 ∪ A2, (A1 ∪ A2)′)

I (A1, B1)∧Hr (A2, B2) = ((B1 ∪ B2)′, B1 ∪ B2)

I (A1, B1)∧Hl (A2, B2) = (A1 ∩ A2, (A1 ∩ A2)′)

I (A1, B1)∨Hr (A2, B2) = ((B1 ∩ B2)′, B1 ∩ B2)

222


Example 46:

a b1 ×2 × ×

r(∅, ab)

r(1, b) r(2, ab)

r(2, a) r(12, b)

r(12, ∅)

��

��

��

��

��

��

@@

@@

@@

@@

@@

@@

223


I ∧l is AC ∨r is ACI ∧l distributes over ∨l ∨r distributes over ∧r

I ¬l(x ∧l x) = ¬lx ¬r (x ∨r x) = ¬r xI x ∧l (y ∧l y) = x ∧l y x ∨r (y ∨r y) = x ∨r yI x ∧l (x ∨l y) = x ∧l x x ∨r (x ∧r y) = x ∨r xI x ∧l (x ∨r y) = x ∧l x x ∨r (x ∧l y) = x ∨r xI ¬l(¬lx ∧l ¬ly) = x ∨l y ¬r (¬r x ∨r ¬r y) = x ∧r yI ¬l⊥l = >l ¬r>r = ⊥r

I ¬l>r = ⊥l ¬r⊥l = >r

I >r ∧l >r = >l ⊥l ∨r ⊥l = ⊥r

I x ∧l ¬lx = ⊥l x ∨r ¬r x = >r

I ¬l¬l(x ∧l y) = x ∧l y ¬r¬r (x ∨r y) = x ∨r yI (x ∨r x) ∧l (x ∨r x) = (x ∧l x) ∨r (x ∧l x)

I x ∧l x = x or x ∨r x = x224


Let H = (Ob, At , I) be a contextI If AH is the set of all H’s semiconcepts then the structureA(H) = (AH,⊥Hl ,>Hr ,¬Hl ,¬Hr ,∨Hl ,∧Hr ) is a semiconceptalgebra

Let A = (A,⊥l ,>r ,¬l ,¬r ,∨l ,∧r ) be a semiconcept algebraI There exists a context H(A) = (ObA, AtA, IA) such that A

is embeddable into the structure A(H(A)) =

(AH(A),⊥H(A)l ,>H(A)

r ,¬H(A)l ,¬H(A)

r ,∨H(A)l ,∧H(A)

r )

225

Concept algebrasThe word problem

We define terms as followsI s ::= x | 0l | 1r | −ls | −r s | (s tl t) | (s ur t)

We define the following abbreviationsI 1l ::= −l0l

I 0r ::= −r 1r

I (s ul t) ::= −l(−ls tl −l t)I (s tr t) ::= −r (−r s ur −r t)

226


A valuation based on a protoconcept algebra / semiconceptalgebra A = (A,⊥l ,>r ,¬l ,¬r ,∨l ,∧r ) is a function

I θ: x 7→ θ(x) ∈ A

θ induces a function θ: s 7→ θ(s) ∈ A as follows:I θ(x) = θ(x)

I θ(0l) = ⊥l

I θ(1r ) = >r

I θ(−ls) = ¬l θ(s)

I θ(−r s) = ¬r θ(s)

I θ(s tl t) = θ(s) ∨l θ(t)I θ(s ur t) = θ(s) ∧r θ(t)

227


We consider the following problem: Deciding whether twoterms are equivalent in every protoconcept algebras /semiconcept algebras

Input Terms s, tOutput Decide whether s 6' t , i.e. whether there exists a

valuation θ based on a protoconcept algebra /semiconcept algebra A = (A,⊥l ,>r ,¬l ,¬r ,∨l ,∧r )such that θ(s) 6= θ(t)

228


The exact computational complexity of the above problem isunknown

I Herrmann, C., Luksch, P., Skorsky, M., Wille, R.: Algebrasof semiconcepts and double Boolean algebras. TechnischeUniversitat Darmstadt (2000).

I Vormbrock, B.: A solution of the word problem for freedouble Boolean algebras. In Kuznetsov, S., Schmidt, S.(Editors): ICFCA 2007. Springer-Verlag (2007) 240–270.

229


The exact computational complexity of the above problem isunknown

I Vormbrock, B., Wille, R.: Semiconcept and protoconceptalgebras: the basic theorems. In Ganter, B., Stumme, G.,Wille, R. (Editors): Formal Concept Analysis.Springer-Verlag (2005) 34–48.

I Wille, R.: Boolean concept logic. In Eklund, P. (Editor):ICFCA 2004. Springer-Verlag (2004) 1–13.

230

Concepts and roles

231

Concepts and rolesDescription logics

Description logics: syntaxKnowledge base terminological box (TBox) + assertional box

(ABox)TBox

I terminology of an application domainI set of concept definitions of the form A ≡ C

and general concept inclusion of the formC v D

I A ≡ C assigns the concept name A to theconcept description C

I C v D states a subconcept/superconceptrelationship between C and D

232


Example 47: Example of a TBoxI T := {

LandlockedCountry ≡ Country u ∀hasBorderTo.Land ,OceanCountry ≡ Country u ∃hasBorderTo.Ocean}

233


Description logics: syntaxKnowledge base terminological box (TBox) + assertional box

(ABox)ABox

I facts about a specific worldI set of concept assertions of the form C(a)

and role assertions of the form R(a, b)I C(a) means that the individual a is an

instance of the concept CI R(a, b) means that the individual a is in

R-relation with individual b

234


Example 48: Example of an ABoxI A := {

LandlockedCountry(Austria),Country(Portugal),Ocean(Atlantic),hasBorderTo(Portugal , Atlantic)}

235


Description logics: semanticsInterpretation I = (∆I , ·I)

Domain ∆I is a nonempty setInterpretation function ·I maps

I every concept occurring in the TBox to asubset of the domain

I every individual name occurring in the ABoxto an element of the domain

I every role to a binary relation on the domain

236


Example 49:I T := {

LandlockedCountry ≡ Country u ∀hasBorderTo.Land ,OceanCountry ≡ Country u ∃hasBorderTo.Ocean}

I A := {LandlockedCountry(Austria),Country(Portugal),Ocean(Atlantic),hasBorderTo(Portugal , Atlantic)}

237


Description logics: inferencesI Given an explicit TBox and an explicit ABox, deduce

implicit consequences such asSubsumption problem C v DInstance checking C(a)

238

Concepts and rolesThe basic description language AL

Let A1, A2, . . . be atomic concepts and A1, A2, . . . atomic roles

Concepts are formed by means of the ruleI C ::= A | > | ⊥ | ¬A | (C u D) | ∀R.C | ∃R.>

239


InterpretationInterpretation I = (∆I , ·I)

Domain ∆I is a nonempty setInterpretation function ·I maps

I every atomic concept A to a subset AI of ∆I

I every atomic role R to a binary relation RI on∆I

I > to ∆I and ⊥ to ∅I ¬A to ∆I \ AI and C u D to CI ∩ DI

I ∀R.C to {a ∈ ∆I : for all b ∈ ∆I , if RI(a, b)then b ∈ CI} and ∃R.> to {a ∈ ∆I : thereexists b ∈ ∆I such that RI(a, b)}

240


We say that two concepts C and D are equivalent, in symbolsC ≡ D, iff

I CI = DI for all interpretations I

Example 50:I ∀hasChild .Female u ∀hasChild .Student ≡∀hasChild .(Female u Student)

241

Concepts and rolesThe family of AL-languages

We obtain more expressive languages if we add furtherconstructors

negation ¬C interpreted in I by ∆I \ CI

union (C t D) interpreted in I by CI ∪ DI

full existential quantification ∃R.C interpreted in I by {a ∈ ∆I :there exists b ∈ ∆I such that RI(a, b) andb ∈ CI}

at-least restriction (> n R) interpreted in I by {a ∈ ∆I : thereexists at least n b ∈ ∆I such that RI(a, b) andb ∈ CI}

at-most restriction (6 n R) interpreted in I by {a ∈ ∆I : thereexists at most n b ∈ ∆I such that RI(a, b) andb ∈ CI}

242


We obtain more expressive languages if we add furtherconstructorsrole intersection R u S interpreted in I by RI ∩ SI

role composition R; S interpreted in I by RI ◦ SI

transitive closure of a role R+ interpreted in I by thetransitiveclosure of RI

role inverse R−1 interpreted in I by the inverse of RI

Example:I ∃(hasSon t hasDaughter)+−1

.(Woman uMathematician)

243


Example 51:I Person u (6 1 hasChild t (>

3 hasChild u ∃hasChild .Female))

244

Concepts and rolesTerminologies

Terminological axioms have the formI C v D, C ≡ D, R v S, R ≡ S

Concept definitions have the formI A ≡ C

Example 52:I Mother ≡ Woman u ∃hasChild .PersonI Parent ≡ Mother t Father

245


Example 53: A terminology (TBox) with concepts about familyrelationships

I Woman ≡ Person u FemaleI Man ≡ Person u ¬FemaleI Mother ≡ Woman u ∃hasChild .PersonI Father ≡ Man u ∃hasChild .PersonI Parent ≡ Mother t FatherI Grandmother ≡ Mother u ∃hasChild .ParentI MotherWithManyChildren ≡ Motheru > 3hasChildI MotherWithoutDaughter ≡ Mother u ∀hasChild .¬WomanI Wife ≡ Woman u ∃hasHusband .Man

246


Cyclic definitions in TBoxI Human ≡ Animal u ∀hasParent .HumanI ManOnlyMaleDescendants ≡

Man u ∀hasChild .ManOnlyMaleDescendantsI BinaryTree ≡ Treeu 6

2 hasBranch u ∀hasBranch.BinaryTree

247

Concepts and rolesWorld descriptions

The second component of a knowledge base, in addition to theTBox, is the world description or ABox

I C(a), R(a, b)

Example 54: A world descrption (ABox)I MotherWithoutDaughter(MARY )

I hasChild(MARY , PAUL)

I hasChild(MARY , PETER)

I Father(PETER)

I hasChild(PETER, HARRY )

248

Concepts and rolesInferences

The different kinds of reasoning performed by a DL systemsare

I checking satisfiability of concepts: a concept C issatisfiable with respect to T if there exists a modelI = (∆I , ·I) of T such that CI is nonempty

I checking subsumption of concepts: a concept C issubsumed by a concept D with respect to T if for everymodel I = (∆I , ·I) of T , CI ⊆ DI

I checking equivalence of concepts: a concept C isequivalent to a concept D with respect to T if for everymodel I = (∆I , ·I) of T , CI = DI

249

Concepts and rolesReferences

I Baader, F., Calvanese, D., McGuinness, D., Nardi, D.,Patel-Schneider, P. (Editors): The Description LogicHandbook. Cambridge University Press (2003).

I Sertkaya, B.: A survey on how description logic ontologiesbenefit from FCA. In Kryszkiewicz, M., Obiedkov, S.(Editors): Proceedings of the 7th International Conferenceon Concept Lattices and Their Applications (CLA 2010).Volume 672 of CEUR Workshop Proceedings (2010) 2–21.

250

Research problems

Generalize to ternary contexts the techniques in formal conceptanalysis that are presented in these slides

Generalize to fuzzy/probabilistic/possibilistic contexts thetechniques in formal concept analysis that are presented inthese slides

Effect improvements in finding all concepts of a given context

Effect improvements in drawing the concept lattice of a givencontext

251

Research problems

Effect improvements in deciding if a given set of attributes is agood attribute subset of a given context

Effect improvements in enumerating the set of all good attributesubsets of a given context

Exact computational complexity of deciding whether two termsare equivalent in every protoconcept algebras / semiconceptalgebras

Enrich formal concept analysis with description logicconstructors and apply formal concept analysis methods indescription logics

252

formal concept analysis: foundations and...

Documents