Formal Concept Analysis: Foundations and...

Formal Concept Analysis:Foundations and Applications

Philippe Balbiani

Institut de recherche en informatique de Toulouse

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)

Introduction

IntroductionThe duality of extension and intension

A formal context

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

A formal context

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

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

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)

Concept lattices of contexts

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

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

Example 1:

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

Jupiter ⊗ ⊗ ⊗Saturn ⊗ ⊗ ⊗Uranus ⊗ ⊗ ⊗

Neptune ⊗ ⊗ ⊗Pluto ⊗ ⊗ ⊗

Example 1:

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

Example 2:

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

Example 2:

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

Example 2:

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

Example 2:

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

Example 2:

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

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

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

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)

Example 2:

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

Example 2:

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

Example 2:

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

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

⊗ · · · ⊗

...⊗ · · · ⊗

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′′

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′

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)

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

t∈T (At , Bt) = (⋂

t∈T At , (⋃

t∈T Bt)′′)

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)

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

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 ′′)

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

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 ⊗ ⊗

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

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

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

Example 4:

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

Example 4:

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

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

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

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

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

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

Example 5:

a b c, d e1, 3, 6, 7 ⊗ ⊗ ⊗ ⊗

2 ⊗ ⊗4 ⊗ ⊗ ⊗5 ⊗8 ⊗ ⊗

Example 5:

a b c, d e1, 3, 6, 7 ⊗ ⊗ ⊗ ⊗

2 ⊗ ⊗ ↙↗ ↙4 ↙↗ ⊗ ⊗ ⊗5 ↗ ⊗ ↗8 ↗ ⊗ ⊗ ↙↗

Example 5:

a c, d e2 ⊗ ↙↗ ↙4 ↙↗ ⊗ ⊗8 ↗ ⊗ ↙↗

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

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

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 ′

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)

Many-valued contexts

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

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”

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

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

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 ⊗ ⊗

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)

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

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

Example 17:

1 2a ⊗b ⊗c ⊗

1 2d ⊗ ⊗e ⊗

1 2a ⊗b ⊗c ⊗d ⊗ ⊗e ⊗

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

I X I (i , x) iff X ∈ Obi and X Ii x

Example 18:

1 2a ⊗b ⊗

3 4 5a ⊗ ⊗b ⊗ ⊗

1 2 3 4 5a ⊗ ⊗ ⊗b ⊗ ⊗ ⊗

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

Example 19:

1 2a ⊗b ⊗c ⊗

∪3 4 5

d ⊗ ⊗e ⊗ ⊗

1 2 3 4 5a ⊗b ⊗c ⊗d ⊗ ⊗e ⊗ ⊗

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

Example 20:

1 2 3 41 ⊗2 ⊗3 ⊗4 ⊗

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

Example 21:

1 2 3 41 ⊗ ⊗ ⊗ ⊗2 ⊗ ⊗ ⊗3 ⊗ ⊗4 ⊗

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

Example 22:

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

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

Example 23:

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

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

0 ⊗1 ⊗

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

Example 24:

1 2a ⊗b ⊗c ⊗

3 4 5d ⊗e ⊗ ⊗

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

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

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

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

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) ⊗ ⊗ ⊗

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

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

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

Example 26:

1 2a ⊗b ⊗

c ⊗d ⊗ ⊗

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

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

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

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

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

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

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

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

Example 30:

O{a,b}:

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

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)

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}

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}}

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)

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

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}

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)

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}

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)

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}

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)

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

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

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= ∅

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}

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

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

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

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

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

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

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)

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

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}

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) ∅

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

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′

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)

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

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

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)

Determination and representation

Determination and representationA context for the planets

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), . . .}

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

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

Determination and representationThe ordering of concepts

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)

r12 r34 r9 r56 r78

r1234 r349 r56789

r12349 r3456789

r123456789

PPPPPPPPP

��

t∈T (At , Bt) = (⋂

t∈T At , (⋃

t∈T Bt)′′)

t∈T (At , Bt) = ((⋃

t∈T At)′′,

⋂t∈T Bt)

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

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

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

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

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 ′

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

Example 32:

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

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

Example 32:

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

2. find a maximal attribute extent, say x ′

Example 32:

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

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

Example 32:

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

a STUVX

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

Example 32:

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

a STUVXb STUW

Example 32:

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

a STUVXb STUW

Example 32:

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

a STUVXbd STUW

Example 32:

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

a STUVXbd STUW

STUf SUVX

Example 32:

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

a STUVXbd STUW

STUf SUVX

Example 32:

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

a STUVXbd STUW

STUf SUVX

SUe TUV

Example 32:

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

a STUVXbd STUW

STUf SUVX

SUe TUV

Example 32:

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

a STUVXbd STUW

STUf SUVX

SUe TUV

Example 32:

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

a STUVXbd STUW

STUf SUVX

SUe TUV

c V∅

Example 32:

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

a STUVXbd STUW

STUf SUVX

SUeg TUV

c V∅

Example 32:

rSTUVX

rUV rSU

rTUV rSTU

rSTUVWX

��

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.

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.

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

Example 32:

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

Example 32:

rSTUVX

rUV rSU

rTUV rSTU

rSTUVWX

��

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

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

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}

Example 33:

a b c1 ⊗ ⊗2 ⊗ ⊗3 ⊗ ⊗ ⊗

r(3, abc)

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

r(123, c)

��

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)

��

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)

��

@I~vec(1)

6~vec(2)

��

��~vec(3)

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)

��

@I~vec(1)

6~vec(2)

��

��~vec(3)

r(3, abc)

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

r(123, c)

��

��@

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

Example 34:

a b c d1 ⊗2 ⊗ ⊗3 ⊗4 ⊗ ⊗

r∅rcd rab

��

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 ⊗ ⊗

r∅rcd rab

��

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 ⊗ ⊗

r∅rcd rab

��

rcd rd r∅

Example 34:I draw the product of these lattices

a b c d1 ⊗2 ⊗ ⊗3 ⊗4 ⊗ ⊗

r∅rcd rab

��

rcd rd r∅

r(ab, cd)

r(b, cd)

r(∅, cd)

r(ab, d)

r(b, d)

r(∅, d)

r(ab, ∅)

r(b, ∅)

r(∅, ∅)

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 ⊗ ⊗

r∅rcd rab

��

rcd rd r∅

r(ab, cd)

r(b, cd)

r(∅, cd)

r(ab, d)

r(b, d)

r(∅, d)

r(ab, ∅)

r(b, ∅)

r(∅, ∅)

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.

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, . . .”

Example 35:

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

6��

��

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

Example 36:

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

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

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

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

Example 37:

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

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

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

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

Example 38:

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

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′′

Example 39:

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

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}

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

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

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

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

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

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)

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

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

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

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.

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.

Concept algebras

Concept algebrasJoin and meet of concepts

Example 41:

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), . . .}

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

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

Example 41:

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

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)

r12 r34 r9 r56 r78

r1234 r349 r56789

r12349 r3456789

r123456789

PPPPPPPPP

��

Example 42:

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

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

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

rw re ra rf

rwe rwa ref raf

��

t∈T (At , Bt) = (⋂

t∈T At , (⋃

t∈T Bt)′′)

t∈T (At , Bt) = ((⋃

t∈T At)′′,

⋂t∈T Bt)

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” ?

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 × × ×

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

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

Jupiter × ⊗ ⊗Saturn × ⊗ ⊗Uranus × ⊗ ⊗

Neptune × ⊗ ⊗Pluto × ⊗ ⊗

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

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

Example 42:

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

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

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

rw re ra rf

rwe rwa ref raf

��

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′′

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, ∅)

��

Example 44:

a b c1 × ×2 × ×3 × × ×

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, ∅)

��

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)

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, ∅)

��

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)

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)

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)

Example 46:

a b1 ×2 × ×

r(∅, ab)

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

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

r(12, ∅)

��

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)

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)

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)

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)

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.

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.

Concepts and roles

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

Example 47: Example of a TBoxI T := {

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

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

Example 48: Example of an ABoxI A := {

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

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

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)}

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

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

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.>

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)}

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)

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}

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)

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

3 hasChild u ∃hasChild .Female))

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

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

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

Man u ∀hasChild .ManOnlyMaleDescendantsI BinaryTree ≡ Treeu 6

2 hasBranch u ∀hasBranch.BinaryTree

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 )

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

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.

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

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

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