Formal Concept Analysis: Foundations and Applications

Transcription

1 Formal Concept Analysis: Foundations and Applications Philippe Balbiani Institut de recherche en informatique de Toulouse

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

3 Introduction

4 Introduction The duality of extension and intension A formal context cartoon real tortoise dog cat mammal Garfield Snoopy Socks Bobby Harriet

5 Introduction The duality of extension and intension A formal context cartoon real tortoise dog cat mammal Garfield Snoopy Socks Bobby Harriet

6 Introduction The duality of extension and intension A formal context cartoon real tortoise dog cat mammal Garfield Snoopy Socks Bobby Harriet

7 Introduction The duality of extension and intension A formal context cartoon real tortoise dog cat mammal Garfield Snoopy Socks Bobby Harriet The pair ({Garfield, Snoopy}, {cartoon, mammal}) is a formal concept of the formal context

8 Introduction Formal concept analysis in information sciences Formal concept analysis in information retrieval Formal concept analysis as a tool for knowledge representation and knowledge discovery Applications of formal concept analysis in logic and artificial intelligence

9 Introduction Formal concept analysis bibliographies and conferences Introductions to formal concept analysis Davey, B., Priestley, H.: Introduction to Lattices and Order. Cambridge University Press (2002, Second Edition) Ganter, B., Wille, R.: Formal Concept Analysis. Mathematical Foundations. Springer-Verlag (1999) International conferences International Conference on Conceptual Structures (ICCS) International Conference on Formal Concept Analysis (ICFCA) Concept Lattices and their Applications (CLA)

10 Concept lattices of contexts

11 Concept lattices of contexts Context and concept Formal context: structure of the form K = (Ob, At, I) where Ob is a nonempty set of formal objects At is a nonempty set of formal attributes I is a binary relation between Ob and At

12 Concept lattices of contexts Context and concept A finite context can be represented by a cross table where rows are headed by object names columns are headed by attribute names. X A cross in row X and column x means that the object X has the attribute x x...

13 Concept lattices of contexts Context and concept Example 1: small medium large near far yes no Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

14 Concept lattices of contexts Context and concept Example 1: small medium large near far yes no Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

15 Concept lattices of contexts Context and concept Example 1: small near medium large far yes no Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

16 Concept lattices of contexts Context and concept Example 2: a b c d e f g h i

17 Concept lattices of contexts Context and concept Example 2: a b c d e f g h i

18 Concept lattices of contexts Context and concept Example 2: a b c d e f g h i

19 Concept lattices of contexts Context and concept Example 2: a b c d e f g h i

20 Concept lattices of contexts Context and concept Example 2: a b c d e f g h i

21 Concept lattices of contexts Context and concept For a set A Ob of objects, we define 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 define B = {X Ob: X I x for every x B} i.e. the set of objects which have all attributes in B

22 Concept lattices of contexts Context and concept Proposition 1: If (Ob, At, I) is a context, A, A 1, A 2 Ob are sets of objects and B, B 1, B 2 At are sets of attributes then A 1 A 2 A 2 A 1 B 1 B 2 B 2 B 1 A A B B A = A B = B Moreover, A B B A A B I

23 Concept lattices of contexts Context and concept A formal concept of the context (Ob, At, I) is a pair (A, B) with A Ob B At A = B B = A We call A the extent of the concept (A, B) B the intent of the concept (A, B) B(Ob, At, I) denotes the set of all concepts of the context (Ob, At, I)

24 Concept lattices of contexts Context and concept Example 2: a b c d e f g h i

25 Concept lattices of contexts Context and concept Example 2: a b g c d e f h i

26 Concept lattices of contexts Context and concept Example 2: a b g c d e f h i

27 Concept lattices of contexts Context and concept The extent A and the intent B of a concept (A, B) are closely connected by the relation I B A..

28 Concept lattices of contexts Context and concept For every set A Ob, A is an intent of some concept (A, A ) is a concept A is the smallest extent containing A A is an extent iff A = A For every set B At, B is an extent of some concept (B, B ) is a concept B is the smallest intent containing B B is an intent iff B = B

29 Concept lattices of contexts Context and concept Proposition 2: If T is an index set and for every t T, A t Ob is a set of objects and B t At is a set of attributes then ( t T A t) = t T A t ( t T B t) = t T B t

30 Concept lattices of contexts Context and concept If (A 1, B 1 ) and (A 2, B 2 ) are concepts of a context then A 1 A 2 iff B 2 B 1 If A 1 A 2 and B 2 B 1 then we say that (A 1, B 1 ) is a subconcept of (A 2, B 2 ) (A 2, B 2 ) is a superconcept of (A 1, B 1 ) and we write (A 1, B 1 ) (A 2, B 2 ) The set of all concepts of (Ob, At, I) ordered in this way is denoted by B(Ob, At, I) is called the concept lattice of the context (Ob, At, I)

31 Concept lattices of contexts Context and concept Theorem 1: The concept lattice B(Ob, At, I) is a complete lattice in which infimum and supremum are given by t T (A t, B t ) = ( t T A t, ( t T B t) ) t T (A t, B t ) = (( t T A t), t T B t) Theorem 2: Every complete lattice (L, ) is isomorphic to the concept lattice B(L, L, )

32 Concept lattices of contexts Context and concept The duality principle for concept lattices: If (Ob, At, I) is a context then (At, Ob, I 1 ) is a context Moreover, B(At, Ob, I 1 ) and B(Ob, At, I) are isomorphic (B, A) (A, B) is an isomorphism

33 Concept lattices of contexts Context and concept For an object X Ob, we write X instead of the object intent {X} γx for the object concept (X, X ) For an attribute x At, we write x instead of the attribute extent {x} µx for the attribute concept (x, x )

34 Concept lattices of contexts Context and concept lattice A context can be reconstructed from its concept lattice: Ob is the extent of the greatest concept (, ) At is the intent of the least concept (, ) I is given by I = {A B: (A, B) is a concept} The contexts reconstructed from two non-isomorphic concept lattices are non-isomorphic

35 Concept lattices of contexts Context and concept lattice Example 3: Concept lattices of non-isomorphic contexts can well be isomorphic a b c d e a b c, d e 1, 3, 6,

36 Concept lattices of contexts Context and concept lattice A context (Ob, At, I) is called clarified iff for every object X, Y Ob and for every attribute x, y At, X = Y X = Y x = y x = y

37 Concept lattices of contexts Context and concept lattice If X Ob is an object and A Ob is a set of objects with X A but X = A then γx = Y A γy B(Ob, At, I) and B(Ob \ {X}, At, I ((Ob \ {X}) At)) are isomorphic and we say that X is a reducible object Full rows, i.e. objects X with X = At are always reducible

38 Concept lattices of contexts Context and concept lattice If x At is an attribute and B At is a set of attributes with x B but x = B then µx = y B µy B(Ob, At, I) and B(Ob, At \ {x}, I (Ob (At \ {x}))) are isomorphic and we say that x is a reducible attribute Full columns, i.e. attributes x with x = Ob are always reducible

39 Concept lattices of contexts Context and concept lattice Example 4: a b c d e f g h i

40 Concept lattices of contexts Context and concept lattice Example 4: a b c d e f g h i

41 Concept lattices of contexts Context and concept lattice The removal from context (Ob, At, I) of reducible objects and reducible attributes is called reducing the context A clarified context (Ob, At, I) is called row reduced iff every object concept is irreducible is called column reduced iff every attribute concept is irreducible A clarified context which is both row reduced and column reduced is called reduced

42 Concept lattices of contexts Context and concept lattice Every finite context can be brought into a reduced form merge objects with the same intents merge attributes with the same extents delete all reducible objects delete all reducible attributes

43 Concept lattices of contexts Context and concept lattice If (Ob, At, I) is a context, X Ob is an object and x At is an attribute then we write X x iff not X I x for every object Y Ob, if X Y then Y I x In other words, X x iff X is maximal among all object intents not containing x

44 Concept lattices of contexts Context and concept lattice If (Ob, At, I) is a context, X Ob is an object and x At is an attribute then we write X x iff not X I x for every attribute y At, if x y then X I y In other words, X x iff x is maximal among all attribute extents not containing X

45 Concept lattices of contexts Context and concept lattice Proposition 3: The following statements hold for every context: X Ob is irreducible X x for some x At x At is irreducible X x for some X Ob Proposition 4: The following statements hold for every finite context: X Ob is irreducible X x for some x At x At is irreducible X x for some X Ob

46 Concept lattices of contexts Context and concept lattice Example 5: a b c, d e 1, 3, 6,

47 Concept lattices of contexts Context and concept lattice Example 5: a b c, d e 1, 3, 6,

48 Concept lattices of contexts Context and concept lattice Example 5: a c, d e 2 4 8

49 Concept lattices of contexts Context and concept lattice A context (Ob, At, I) is called doubly founded iff for every object X Ob and for every attribute x At, if not X I x then X y and x y for some attribute y At Y x and X Y for some object Y Ob

50 Concept lattices of contexts Context and concept lattice Proposition 5: Every finite context is doubly founded Proposition 6: A context which does neither contain infinite chains X 1, X 2,... of objects with X 1 X 2... nor infinite chains x 1, x 2,... of attributes with x 1 x 2... is doubly founded Proposition 7: The following statements hold for every doubly founded context: X x X y for some y At X x Y x for some Y Ob

51 Concept lattices of contexts Context and concept lattice A complete lattice (L, ) is called doubly founded iff for every u, v L, if u < v then there exists u, v L such that u is minimal with respect to u u and u v v is maximal with respect to u v and v v

52 Concept lattices of contexts Context and concept lattice Proposition 8: If the concept lattice of the context (Ob, At, I) is doubly founded, so is (Ob, At, I) Proposition 9: If the complete lattice (L, ) is not doubly founded, neither is the context (L, L, )

53 Many-valued contexts

54 Many-valued contexts Contexts and scales Many-valued context: structure of the form (Ob, At, Va, I) where Ob is a nonempty set of formal objects At is a nonempty set of formal attributes Va is a nonempty set of formal values I is a ternary relation between Ob, At and Va A many-valued context (Ob, At, Va, I) is called a n-valued context iff Va has n elements

55 Many-valued contexts Contexts and scales Example 6: De Dl R E M Conv. poor good good good excellent Front good poor excellent excellent good Rear excellent excellent very poor poor very poor Mid excellent excellent good very poor very poor All excellent excellent good good poor De: drive efficiency empty, Dl: drive efficiency loaded, R: road handling properties, E: economy of space, M: maintainability

56 Many-valued contexts Contexts and scales The domain of an attribute x is defined to be dom(x) = {X Ob: I(X, x, v) for some v Va} The attribute x is called complete iff dom(x) = Ob A many-valued context is called complete iff all its attributes are complete

57 Many-valued contexts Contexts and scales A scale for the attribute x of a many-valued context is a (one-valued) context K x = (Ob x, At x, I x ) with {v Va: I(X, x, v) for some X Ob} Ob x If (Ob, At, Va, I) is a many-valued context and (Ob x, At x, I x ) is a scale context for every x At then the derived context with respect to plain scaling is the (one-valued) context (Ob, At, I ) with Ob = Ob At = {(x, a): x At and a At x } X I (x, a) iff I(X, x, v) and v I x a for some v Va

58 Many-valued contexts Contexts and scales Example 7: De Dl R E M Conv. poor good good good excellent Front good poor excellent excellent good Rear excellent excellent very poor poor very poor Mid excellent excellent good very poor very poor All excellent excellent good good poor ++ + excellent K De, K Dl, K R, K E, K M : good poor very poor

59 Many-valued contexts Context constructions and standard scales If K = (Ob, At, I) is a context then we define K c = (Ob, At, (Ob At) \ I) K 1 = (At, Ob, I 1 )

60 Many-valued contexts Context constructions and standard scales If K 1 = (Ob 1, At 1, I 1 ) and K 2 = (Ob 2, At 2, I 2 ) are contexts then we define for every i {1, 2}, Ob i = {i} Ob i At i = {i} At i (i, X) İi (i, x) iff X I i x

61 Many-valued contexts Context constructions and standard scales If K 1 = (Ob 1, At 1, I 1 ) and K 2 = (Ob 2, At 2, I 2 ) are contexts then we define K 1 l K 2 = ( Ob 1 Ob 2, At 1 At 2, I) with (i, X) I x iff x At i and X I i x

62 Many-valued contexts Context constructions and standard scales Example 17: a b c l d e = 1 2 a b c d e

63 Many-valued contexts Context constructions and standard scales If K 1 = (Ob 1, At 1, I 1 ) and K 2 = (Ob 2, At 2, I 2 ) are contexts then we define K 1 r K 2 = (Ob 1 Ob 2, At 1 At 2, I) with X I (i, x) iff X Ob i and X I i x

64 Many-valued contexts Context constructions and standard scales Example 18: a b 1 2 r a b = a b

65 Many-valued contexts Context constructions and standard scales If K 1 = (Ob 1, At 1, I 1 ) and K 2 = (Ob 2, At 2, I 2 ) are contexts then we define K 1 K 2 = ( Ob 1 Ob 2, At 1 At 2, İ1 İ2)

66 Many-valued contexts Context constructions and standard scales Example 19: a b c d e = a b c d e

67 Many-valued contexts Context constructions and standard scales Nominal scales: N k = ({1,..., k}, {1,..., k}, =) Example 20: N 4 :

68 Many-valued contexts Context constructions and standard scales Ordinal scales: O k = ({1,..., k}, {1,..., k}, ) Example 21: O 4 :

69 Many-valued contexts Context constructions and standard scales Interordinal scales: I k = ({1,..., k}, {1,..., k}, ) r ({1,..., k}, {1,..., k}, ) Example 22: I 4 :

70 Many-valued contexts Context constructions and standard scales Biordinal scales: M k,l = ({1,..., k}, {1,..., k}, ) ({1,..., l}, {1,..., l}, ) Example 23: M 4,2 :

71 Many-valued contexts Context constructions and standard scales Dichotomic scale: D = ({0, 1}, {0, 1}, =) D:

72 Many-valued contexts Context constructions and standard scales If K 1 = (Ob 1, At 1, I 1 ) and K 2 = (Ob 2, At 2, I 2 ) are contexts then we define K 1 + K 2 = ( Ob 1 Ob 2, At 1 At 2, I) with (i, X) I (j, x) iff one of the following conditions hold i = 1, j = 1 and X I1 x i = 1 and j = 2 i = 2 and j = 1 i = 2, j = 2 and X I2 x

73 Many-valued contexts Context constructions and standard scales Example 24: a b c d e = a b c d e

74 Many-valued contexts Context constructions and standard scales Proposition 17: If K 1 = (Ob 1, At 1, I 1 ) and K 2 = (Ob 2, At 2, I 2 ) are contexts then B(K 1 + K 2 ) and B(K 1 ) B(K 2 ) are isomorphic (A, B) ((A Ob 1, B At 1 ), (A Ob 2, B At 2 )) is an isomorphism

75 Many-valued contexts Context constructions and standard scales If K 1 = (Ob 1, At 1, I 1 ) and K 2 = (Ob 2, At 2, I 2 ) are contexts then we define K 1 K 2 = (Ob 1 Ob 2, At 1 At 2, I) with (X 1, X 2 ) I (i, x) iff X i I i x

76 Many-valued contexts Context constructions and standard scales Example 25: a b c d e (a, d) (a, e) (b, d) (b, e) (c, d) (c, e) =

77 Many-valued contexts Context constructions and standard scales Proposition 18: If K 1 = (Ob 1, At 1, I 1 ) and K 2 = (Ob 2, At 2, I 2 ) are contexts then the extents of K 1 K 2 are precisely the sets of the form A 1 A 2 each set A i being an extent of K i

78 Many-valued contexts Context constructions and standard scales If K 1 = (Ob 1, At 1, I 1 ) and K 2 = (Ob 2, At 2, I 2 ) are contexts then we define with K 1 K 2 = (Ob 1 Ob 2, At 1 At 2, I) (X 1, X 2 ) I (x 1, x 2 ) iff X 1 I 1 x 1 or X 2 I 2 x 2

79 Many-valued contexts Context constructions and standard scales Example 26: a c = b d (1, 3) (1, 4) (2, 3) (2, 4) (a, c) (a, d) (b, c) (b, d)

80 Many-valued contexts Context constructions and standard scales Proposition 19: If K 1 = (Ob 1, At 1, I 1 ), K 2 = (Ob 2, At 2, I 2 ) and K 3 = (Ob 3, At 3, I 3 ) are contexts then (K 1 + K 2 ) K 3 and (K 1 K 3 ) + (K 2 K 3 ) are isomorphic

81 Many-valued contexts Context constructions and standard scales Contranominal scales: N c S = (S, S, ) for every nonempty set S Example 27: N c {1,2,3} : Proposition 20: If S is a nonempty set then the concepts of N c S are precisely the pairs (A, S \ A) for A S

82 Many-valued contexts Context constructions and standard scales General ordinal scales: O P = (P, P, ) for every ordered set (P, ) Example 28: O {1,2,3} : Proposition 21: If (P, ) is an ordered set then the concepts of O P are precisely the pairs (A, B) where A is the set of all lower bounds of B and B is the set of all upper bounds of A

83 Many-valued contexts Context constructions and standard scales Contraordinal scales: O cd P (P, ) = (P, P, ) for every ordered set Example 29: O cd {1,2,3} : Proposition 22: If (P, ) is an ordered set then the concepts of O cd P are precisely the pairs (A, P \ A) for A P an order ideal

84 Many-valued contexts Context constructions and standard scales Contraordinal scales: O S = (2 S, 2 S, ) for every set S Example 30: O {a,b} : {a} {b} {a, b} {a} {b} {a, b}

85 Many-valued contexts Context constructions and standard scales From an ordered set (P, ), we obtain the general interordinal scale I P = (P, P, ) r (P, P, ) and the convex-ordinal scale I P = (P, P, ) r (P, P, )

86 Many-valued contexts Indiscernibility If (Ob, At, Va, I) is a complete many-valued context, with every susbset of attributes B At, we associate a binary relation IND(B), called an indiscernibility relation and defined thus IND(B) = {(X, Y ) Ob Ob: for every x B and for every v Va, I(X, x, v) iff I(Y, x, v)} For an attribute x Att, we write IND(x) instead of IND({x}) Obviously IND(B) is an equivalence relation and IND(B) = {IND(x): x B}

87 Many-valued contexts Indiscernibility Example 8: a b c d e Exemplary partitions generated by attributes in this context Ob/IND(a) = {{1, 4, 5}, {2, 8}, {3, 6, 7}} Ob/IND(b) = {{1, 3, 5}, {2, 4, 7, 8}, {6}}

88 Many-valued contexts Indiscernibility Approximations of sets of objects in a complete many-valued context (Ob, At, Va, I): with each subset of objects A Ob and each subset of attributes B At, we associate two subsets IND(B)(A) = {X Ob: IND(B)(X) A} IND(B)(A) = {X Ob: IND(B)(X) A } called the IND(B)-lower approximation of A and the IND(B)-upper approximation of A Obviously IND(B)(A) A IND(B)(A)

89 Many-valued contexts Indiscernibility Given a subset of objects A Ob and a subset of attributes B At, we shall say that A is IND(B)-definable iff IND(B)(A) = IND(B)(A) A is IND(B)-rough iff IND(B)(A) IND(B)(A) Given a subset of objects A Ob and a subset of attributes B At, let us observe that IND(B)(A) is the maximal IND(B)-definable set of objects contained in A IND(B)(A) is the minimal IND(B)-definable set of objects containing A

90 Many-valued contexts Indiscernibility Example 9: a b c d e If A = {1, 2, 3, 4, 5} and B = {a, b, c} then IND(B)(A) = {1, 3, 4, 5} IND(B)(A) = {1, 2, 3, 4, 5, 8}

91 Many-valued contexts Indiscernibility Proposition 10: 1. IND(B)( ) = 2. IND(B)(Ob) = Ob 3. IND(B)(A 1 A 2 ) IND(B)(A 1 ) IND(B)(A 2 ) 4. IND(B)(A 1 A 2 ) = IND(B)(A 1 ) IND(B)(A 2 ) 5. IND(B)(Ob \ A) = Ob \ IND(B)(A) 6. A 1 A 2 implies IND(B)(A 1 ) IND(B)(A 2 ) 7. IND(B)(IND(B)(A)) = IND(B)(A) 8. IND(B)(IND(B)(A)) = IND(B)(A)

92 Many-valued contexts Indiscernibility Example 10: Suppose we are given a complete many-valued context (Ob, At, Va, I) where Ob = {1, 2, 3, 4, 5, 6, 7, 8} and let B At be a subset of attributes defining an equivalence relation IND(B) with the following equivalence classes: {1, 4, 8} {2, 5, 7} {3} {6} If A 1 = {1, 4, 7} and A 2 = {2, 8} then IND(B)(A 1 ) = IND(B)(A 2 ) = IND(B)(A1 A 2 ) = {1, 4, 8}

93 Many-valued contexts Indiscernibility Proposition 11: 1. IND(B)( ) = 2. IND(B)(Ob) = Ob 3. IND(B)(A 1 A 2 ) = IND(B)(A 1 ) IND(B)(A 2 ) 4. IND(B)(A 1 A 2 ) IND(B)(A 1 ) IND(B)(A 2 ) 5. IND(B)(Ob \ A) = Ob \ IND(B)(A) 6. A 1 A 2 implies IND(B)(A 1 ) IND(B)(A 2 ) 7. IND(B)(IND(B)(A)) = IND(B)(A) 8. IND(B)(IND(B)(A)) = IND(B)(A)

94 Many-valued contexts Indiscernibility Example 11: Suppose we are given a complete many-valued context (Ob, At, Va, I) where Ob = {1, 2, 3, 4, 5, 6, 7, 8} and let B At be a subset of attributes defining an equivalence relation IND(B) with the following equivalence classes: {1, 4, 8} {2, 5, 7} {3} {6} If A 1 = {1, 3, 5} and A 2 = {2, 3, 4, 6} then IND(B)(A1 ) = {1, 2, 3, 4, 5, 7, 8} IND(B)(A2 ) = {1, 2, 3, 4, 5, 6, 7, 8} IND(B)(A 1 A 2 ) = {3}

95 Many-valued contexts Indiscernibility Given a complete many-valued context (Ob, At, Va, I), a subset of objects A Ob and a subset of attributes B At, let X in(b) A iff X IND(B)(A) X in(b) A iff X IND(B)(A) Intuitive reading X in(b) A: X surely belongs to A with respect to B X in(b) A: X possibly belongs to A with respect to B Obviously X in(b) A implies X A X A implies X in(b) A

96 Many-valued contexts Indiscernibility Proposition 12: 1. not X in(b) 2. X in(b) Ob 3. X in(b) (A 1 A 2 ) if X in(b) A 1 or X in(b) A 2 4. X in(b) (A 1 A 2 ) iff X in(b) A 1 and X in(b) A 2 5. X in(b) (Ob \ A) iff not X in(b) A 6. A 1 A 2 implies X in(b) A 1 only if X in(b) A 2

97 Many-valued contexts Indiscernibility Proposition 13: 1. not X in(b) 2. X in(b) Ob 3. X in(b) (A 1 A 2 ) iff X in(b) A 1 or X in(b) A 2 4. X in(b) (A 1 A 2 ) only if X in(b) A 1 and X in(b) A 2 5. X in(b) (Ob \ A) iff not X in(b) A 6. A 1 A 2 implies X in(b) A 1 only if X in(b) A 2

98 Many-valued contexts Indiscernibility Given a complete many-valued context (Ob, At, Va, I), a subset of objects A Ob and a subset of attributes B At, let BN(B)(A) = IND(B)(A) \ IND(B)(A) be the IND(B)-boundary of A Obviously A is IND(B)-definable iff BN(B)(A) = A is IND(B)-rough iff BN(B)(A)

99 Many-valued contexts Indiscernibility Example 12: Suppose we are given a complete many-valued context (Ob, At, Va, I) where Ob = {1, 2, 3, 4, 5, 6, 7, 8} and let B At be a subset of attributes defining an equivalence relation IND(B) with the following equivalence classes: {1, 4, 8} {2, 5, 7} {3} {6} If A 1 = {1, 4, 7} and A 2 = {2, 8} then BN(B)(A1 ) = {1, 2, 4, 5, 7, 8} BN(B)(A2 ) = {1, 2, 4, 5, 7, 8} If A 1 = {1, 3, 5} and A 2 = {2, 3, 4, 6} then BN(B)(A 1 ) = {1, 2, 4, 5, 7, 8} BN(B)(A 2 ) = {1, 2, 4, 5, 7, 8}

100 Many-valued contexts Indiscernibility Given a complete many-valued context (Ob, At, Va, I), a nonempty subset of objects A Ob and a subset of attributes B At, let α(b)(a) = Card(IND(B)(A)) Card(IND(B)(A)) be the IND(B)-accuracy measure of A Obviously 0 α(b)(a) 1 Moreover A is IND(B)-definable iff α(b)(a) = 1 A is IND(B)-rough iff α(b)(a) < 1

101 Many-valued contexts Indiscernibility Given a complete many-valued context (Ob, At, Va, I), a nonempty subset of objects A Ob and a subset of attributes B At, let ρ(b)(a) = Card(BN(B)(A)) Card(IND(B)(A)) be the IND(B)-roughness measure of A Obviously 0 ρ(b)(a) 1 Moreover A is IND(B)-definable iff ρ(b)(a) = 0 A is IND(B)-rough iff ρ(b)(a) > 0

102 Many-valued contexts Indiscernibility Example 13: Suppose we are given a complete many-valued context (Ob, At, Va, I) where Ob = {1, 2, 3, 4, 5, 6, 7, 8} and let B At be a subset of attributes defining an equivalence relation IND(B) with the following equivalence classes: {1, 4, 8} {2, 5, 7} {3} {6} If A = {1, 4, 5} then IND(B)(A) = BN(B)(A) = {1, 2, 4, 5, 7, 8} IND(B)(A) = {1, 2, 4, 5, 7, 8} α(b)(a) = 0 6 = 0.00 ρ(b)(a) = 6 6 = 1.00

103 Many-valued contexts Indiscernibility Example 14: Suppose we are given a complete many-valued context (Ob, At, Va, I) where Ob = {1, 2, 3, 4, 5, 6, 7, 8} and let B At be a subset of attributes defining an equivalence relation IND(B) with the following equivalence classes: {1, 4, 8} {2, 5, 7} {3} {6} If A = {3, 5} then IND(B)(A) = {3} BN(B)(A) = {2, 5, 7} IND(B)(A) = {2, 3, 5, 7} α(b)(a) = 1 4 = 0.25 ρ(b)(a) = 3 4 = 0.75

104 Many-valued contexts Indiscernibility Example 15: Suppose we are given a complete many-valued context (Ob, At, Va, I) where Ob = {1, 2, 3, 4, 5, 6, 7, 8} and let B At be a subset of attributes defining an equivalence relation IND(B) with the following equivalence classes: {1, 4, 8} {2, 5, 7} {3} {6} If A = {3, 6, 8} then IND(B)(A) = {3, 6} BN(B)(A) = {1, 4, 8} IND(B)(A) = {1, 3, 4, 6, 8} α(b)(a) = 2 5 = 0.40 ρ(b)(a) = 3 5 = 0.60

105 Many-valued contexts Indiscernibility Given a complete many-valued context (Ob, At, Va, I), a nonempty family of nonempty subset of objects F = {A 1,..., A n } and a subset of attributes B At, let α(b)(f) = Σ i Card(IND(B)(A i )) Σ i Card(IND(B)(A i )) be the IND(B)-accuracy of approximation of F and let γ(b)(f) = Σ i Card(IND(B)(A i )) Card(Ob) be the IND(B)-quality of approximation of F

106 Many-valued contexts Indiscernibility Given a complete many-valued context (Ob, At, Va, I), subsets of objects A 1, A 2 Ob and a subset of attributes B At, let A 1 sim(b) A 2 iff IND(B)(A 1 ) = IND(B)(A 2 ) A 1 sim(b) A 2 iff IND(B)(A 1 ) = IND(B)(A 2 ) Intuitive reading A 1 sim(b) A 2 : the positive examples of A 1 and A 2 are the same A 1 sim(b) A 2 : the negative examples of A 1 and A 2 are the same Obviously sim(b) and sim(b) are equivalence relations

107 Many-valued contexts Indiscernibility Example 16: Suppose we are given a complete many-valued context (Ob, At, Va, I) where Ob = {1, 2, 3, 4, 5, 6, 7, 8} and let B At be a subset of attributes defining an equivalence relation IND(B) with the following equivalence classes: {1, 4, 5} {2, 3} {6} {7, 8} If A 1 = {1, 2, 3} and A 2 = {2, 3, 7} then IND(B)(A1 ) = {2, 3} IND(B)(A 2 ) = {2, 3} If A 1 = {1, 2, 7} and A 2 = {2, 3, 4, 8} then IND(B)(A1 ) = {1, 2, 3, 4, 5, 7, 8} IND(B)(A2 ) = {1, 2, 3, 4, 5, 7, 8}

108 Many-valued contexts Indiscernibility Proposition 14: 1. A 1 sim(b) A 2 iff (A 1 A 2 ) sim(b) A 1 and (A 1 A 2 ) sim(b) A 2 2. if A 1 sim(b) A 1 and A 2 sim(b) A 2 then (A 1 A 2 ) sim(b) (A 1 A 2 ) 3. if A 1 sim(b) A 2 then (A 1 (Ob \ A 2 )) sim(b) 4. if A 1 A 2 then A 2 sim(b) implies A 1 sim(b) 5. if A 1 A 2 then A 1 sim(b) Ob implies A 2 sim(b) Ob 6. if A 1 sim(b) or A 2 sim(b) then (A 1 A 2 ) sim(b)

109 Many-valued contexts Indiscernibility Proposition 15: 1. A 1 sim(b) A 2 iff (A 1 A 2 ) sim(b) A 1 and (A 1 A 2 ) sim(b) A 2 2. if A 1 sim(b) A 1 and A 2 sim(b) A 2 then (A 1 A 2 ) sim(b) (A 1 A 2 ) 3. if A 1 sim(b) A 2 then (A 1 (Ob \ A 2 )) sim(b) Ob 4. if A 1 A 2 then A 2 sim(b) implies A 1 sim(b) 5. if A 1 A 2 then A 1 sim(b) Ob implies A 2 sim(b) Ob 6. if A 1 sim(b) Ob or A 2 sim(b) Ob then (A 1 A 2 ) sim(b) Ob

110 Many-valued contexts Indiscernibility 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 Ob such that A sim(b) A

111 Many-valued contexts Ternary contexts Ternary context: structure of the form S = (Ob, At, Co, I) where Ob is a nonempty set of formal objects At is a nonempty set of formal attributes Co is a nonempty set of formal conditions I is a ternary relation between Ob, At and Co Ternary contexts will usually be denoted S = (S 1, S 2, S 3, I)

112 Many-valued contexts Ternary contexts A ternary context can be represented by a cross cube where 1-rows are headed by object names (X, Y, etc) 2-rows are headed by attribute names (x, y, etc) 3-rows are headed by condition names (α, β, etc) A cross in 1-row X, 2-row x and 3-row α means that the object X has the attribute x under the condition α

113 Many-valued contexts Ternary contexts Given a ternary context S = (S 1, S 2, S 3, I), i, j, k {1, 2, 3} pairwise distinct, a set A i S i of S i -elements and a set A j S j of S j -elements, we define (A i, A j ) k = {x k S k : I(x i, x j, x k ) for every x i A i and for every x j A j } i.e. the set of S k -elements common to the pairs (x i, x j ) in A i A j It is still a problem to generalize to ternary concepts the techniques in formal concept analysis that are presented in these slides

114 Many-valued contexts Ternary contexts A ternary concept of the ternary context (S 1, S 2, S 3, I) is a triple (A 1, A 2, A 3 ) with A 1 S 1 A 2 S 2 A 3 S 3 (A 1, A 2 ) 3 = A 3 (A 1, A 3 ) 2 = A 2 (A 2, A 3 ) 1 = A 1 We call A 1 the extent of the concept (A 1, A 2, A 3 ) A 2 the intent of the concept (A 1, A 2, A 3 ) A 3 the mode of the concept (A 1, A 2, A 3 )

115 Determination and representation

116 Determination and representation A context for the planets small medium large near far yes no Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

117 Determination and representation Contexts and concepts Formal context: structure of the forme K = (Ob, At, I) where Ob is a nonempty set of formal objects At is a nonempty set of formal attributes I is a binary relation between Ob and At Within the context of the planets Ob = {Mercury, Venus,...} At = {small, medium,...} I = {(Mercury, small), (Mercury, near),...}

118 Determination and representation Contexts and concepts For a set A Ob of objects, we define 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 define B = {X Ob: X I x for every x B} i.e. the set of objects which have all attributes in B Within the context of the planets {Earth, Mars} = {small, near, yes} {small, near} = {Mercury, Venus, Earth, Mars}

119 Determination and representation Contexts and concepts A formal concept of the context (Ob, At, I) is a pair (A, B) with A Ob B At A = B B = A Within the context of the planets ({Earth, Mars}, {small, near, yes}) ({Mercury, Venus, Earth, Mars}, {small, near})

120 Determination and representation The ordering of concepts If (A 1, B 1 ) and (A 2, B 2 ) are concepts of a context then A 1 A 2 iff B 2 B 1 If A 1 A 2 and B 2 B 1 then we say that (A 1, B 1 ) is a subconcept of (A 2, B 2 ) (A 2, B 2 ) is a superconcept of (A 1, B 1 ) and we write (A 1, B 1 ) (A 2, B 2 ) The set of all concepts of (Ob, At, I) ordered in this way is denoted by B(Ob, At, I) is called the concept lattice of the context (Ob, At, I)

121 Determination and representation The 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)

122 Determination and representation The ordering of concepts Theorem 3: The concept lattice B(Ob, At, I) is a complete lattice in which infimum and supremum are given by t T (A t, B t ) = ( t T A t, ( t T B t) ) t T (A t, B t ) = (( t T A t), t T B t)

123 Determination and representation The determination problem A simple-minded and extremely inefficient way of determining all the concepts of a context K = (Ob, At, I) 1. choose a set A of objects 2. compute the set A of attributes common to the objects in A 3. compute the set A of objects which have all attributes in A Then the pair (A, A ) is a concept

124 Determination and representation The determination problem The concept ({Earth, Mars}, {small, near, yes}) small medium large near far yes no Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

125 Determination and representation The determination problem The concept ({Earth, Mars}, {small, near, yes}) small medium large near far yes no Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

126 Determination and representation The determination problem The concept ({Earth, Mars}, {small, near, yes}) small medium large near far yes no Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

127 Determination and representation The determination problem The concept ({Mercury, Venus, Earth, Mars}, {small, near}) small medium large near far yes no Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

128 Determination and representation The determination problem The concept ({Mercury, Venus, Earth, Mars}, {small, near}) small medium large near far yes no Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

129 Determination and representation The determination problem The concept ({Mercury, Venus, Earth, Mars}, {small, near}) small medium large near far yes no Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

130 Determination and representation An algorithm for finding all concepts of a given context A simple-minded and extremely inefficient way of determining all the concepts of a context K = (Ob, At, I) 1. choose a set B of attributes 2. compute the set B of objects which have all attributes in B 3. 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 At A = X A X and B = x B x In particular, if (A, B) is a concept then A = x B x and B = X A X

131 Determination and representation An algorithm for finding all concepts of a given context Let K = (Ob, At, I) be a given context

132 Determination and representation An algorithm for finding all concepts of a given context Example 32: a b c d e f g S T U V W X

133 Determination and representation An algorithm for finding all concepts of a given context 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 empty and write Ob in the first cell of the E column

134 Determination and representation An algorithm for finding all concepts of a given context Example 32: a b c d e f g S T U V W X A E Ob

135 Determination and representation An algorithm for finding all concepts of a given context 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 empty and write Ob in the first cell of the E column 2. find a maximal attribute extent, say x

136 Determination and representation An algorithm for finding all concepts of a given context Example 32: a b c d e f g S T U V W X A E Ob

137 Determination and representation An algorithm for finding all concepts of a given context 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 empty and write Ob in the first cell of the E column 2. find a maximal attribute extent, say 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 previous extents in E, add these intersections to the E column unless they are already in the list 2.2 if the set x is already in the E column, add the label x to the attribute cell of the rwo where x previously occured

138 Determination and representation An algorithm for finding all concepts of a given context Example 32: a b c d e f g S T U V W X A a E Ob STUVX

139 Determination and representation An algorithm for finding all concepts of a given context 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 empty and write Ob in the first cell of the E column 2. find a maximal attribute extent, say 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 previous extents in E, add these intersections to the E column unless they are already in the list 2.2 if the set x is already in the E column, add the label x to the attribute cell of the rwo where x previously occured 3. delete the column below x from the context 4. if the last column has been deleted, stop, otherwise return to 2

140 Determination and representation An algorithm for finding all concepts of a given context Example 32: a b c d e f g S T U V W X A a b E Ob STUVX STUW

141 Determination and representation An algorithm for finding all concepts of a given context Example 32: a b c d e f g S T U V W X A a b E Ob STUVX STUW STU

142 Determination and representation An algorithm for finding all concepts of a given context Example 32: a b c d e f g S T U V W X A a bd E Ob STUVX STUW STU

143 Determination and representation An algorithm for finding all concepts of a given context Example 32: a b c d e f g S T U V W X A a bd f E Ob STUVX STUW STU SUVX

144 Determination and representation An algorithm for finding all concepts of a given context Example 32: a b c d e f g S T U V W X A a bd f E Ob STUVX STUW STU SUVX SU

145 Determination and representation An algorithm for finding all concepts of a given context Example 32: a b c d e f g S T U V W X A a bd f e E Ob STUVX STUW STU SUVX SU TUV

146 Determination and representation An algorithm for finding all concepts of a given context Example 32: a b c d e f g S T U V W X A a bd f e E Ob STUVX STUW STU SUVX SU TUV TU UV U

147 Determination and representation An algorithm for finding all concepts of a given context Example 32: a b c d e f g S T U V W X A a bd f e c E Ob STUVX STUW STU SUVX SU TUV TU UV U V

148 Determination and representation An algorithm for finding all concepts of a given context Example 32: a b c d e f g S T U V W X A a bd f e c E Ob STUVX STUW STU SUVX SU TUV TU UV U V

149 Determination and representation An algorithm for finding all concepts of a given context Example 32: a b c d e f g S T U V W X A a bd f eg c E Ob STUVX STUW STU SUVX SU TUV TU UV U V

150 Determination and representation An algorithm for finding all concepts of a given context Example 32: V TUV SUVX STU UV TU SU U STUVWX STUVX STUW

151 Determination and representation An algorithm for finding all concepts of a given context It is still possible to effect improvements in finding all concepts of a given context Choi, V.: Faster algorithms for constructing a concept (Galois) lattice. In Butenko, S., Chaovalitwongse, W., Pardalos, P. (Editors): Clustering Challenges in Biological Networks. World Scientific (2009) Kuznetsov, S., Obiedkov, S.: Comparing performance of algorithms for generating concept lattices. Journal of Experimental & Theoretical Artificial Intelligence 14 (2002)

152 Determination and representation An algorithm for finding all concepts of a given context It is still possible to effect improvements in finding all concepts of a given context Van der Merwe, D., Obiedkov, S., Kourie, D.: AddIntent: a new incremental algorithm for constructing concept lattices. In Eklund, P. (Editor): ICFCA Springer-Verlag (2004) Valtchev, P., Missaoui, R.: Building concept (Galois) lattices from parts: generalizing the incremental methods. In Delugach, H., Stumme, G. (Editors): ICCS Springer-Verlag (2001)

153 Determination and representation Drawing the concept lattice of a given context Given a formal context K = (Ob, At, I), the problem is to arrange the nodes and lines of the diagram of its concept lattice in order to achieve the best visual quality the best visual readability Do it fast and automatically

154 Determination and representation Drawing the concept lattice of a given context Example 32: a b c d e f g S T U V W X

155 Determination and representation Drawing the concept lattice of a given context Example 32: V TUV SUVX STU UV TU SU U STUVWX STUVX STUW

156 Determination and representation Drawing the concept lattice of a given context There are several subjective human æsthetics criteria minimizing line crossings (planarity) maximizing angle between incident lines maximizing symmetries maximizing compactness These criteria are often contradictory and lead to computationaly difficult (NP-complete) problems How large lattices one can draw by a computer? Up to about a hundred of nodes

157 Determination and representation A force directed approach for drawing the concept lattice of a given context Let K = (Ob, At, I) be a given context 1. within a 3-dimensional space, organize nodes of the concept lattice in layers based on their distance from the top node (, ) 2. for each layer, randomly arrange its nodes as the vertices of a regular polygon which has a circumscribed circle of radius 1 3. between each pair of nodes occurring in two successive layers, calculate imaginary repulsive and attractive forces depending on how much this pair of nodes overlap 4. inside each layer, modify the positions of its nodes according to the forces calculated in step 3 5. if the resulting diagram is not good enough then go to step 3

158 Determination and representation A vectorial approach for drawing the concept lattice of a given context Let K = (Ob, At, I) be a given context 1. choose a point pos 0 R R 2. associate to each object X Ob a vector vec(x) R R + 3. for each extent A of a K-concept, compute pos 0 + Σ{ vec(x) : X A}

159 Determination and representation A vectorial approach for drawing the concept lattice of a given context Example 33: a b c (123, c) (13, ac) (23, bc) (3, abc)

160 Determination and representation A vectorial approach for drawing the concept lattice of a given context Example 33: choose a point pos 0 R R a b c (123, c) (13, ac) (23, bc) (3, abc) pos 0

161 Determination and representation A vectorial approach for drawing the concept lattice of a given context Example 33: associate to each object X Ob a vector vec(x) R R + a b c vec(1) vec(2) vec(3) (123, c) (13, ac) (23, bc) (3, abc) pos 0

162 Determination and representation A vectorial approach for drawing the concept lattice of a given context Example 33: for each extent A of a K-concept, compute pos 0 + Σ{ vec(x) : X A} a b c (123, c) (13, ac) (23, bc) (3, abc) vec(1) vec(2) vec(3) (123, c) (13, ac) (23, bc) (3, abc) pos 0

163 Determination and representation A dichotomic approach for drawing the concept lattice of a given context Let K = (Ob, At, I) be a given context 1. choose At 1 At and At 2 At such that At 1 At 2 = At 2. draw the concept lattices of the contexts K 1 = (Ob, At 1, I (Ob At 1 )) and K 2 = (Ob, At 2, I (Ob At 2 )) 3. draw the product of these lattices 4. for each K-intent B, compute the corresponding element (B At 1, B At 2 ) in the product

164 Determination and representation A dichotomic approach for drawing the concept lattice of a given context Example 34: a b c d d b cd ab abcd

165 Determination and representation A dichotomic approach for drawing the concept lattice of a given context Example 34: choose At 1 At and At 2 At such that At 1 At 2 = At a b c d a b c d d b cd ab abcd

166 Determination and representation A dichotomic approach for drawing the concept lattice of a given context Example 34: draw the concept lattices of the contexts K 1 = (Ob, At 1, I (Ob At 1 )) and K 2 = (Ob, At 2, I (Ob At 2 )) a b c d d b cd ab abcd a b b ab c d cd d

167 Determination and representation A dichotomic approach for drawing the concept lattice of a given context Example 34: draw the product of these lattices a b c d 1 b 2 3 ab 4 cd d d b cd ab abcd (, cd) (b, cd) (ab, cd) (, d) (b, d) (ab, d) (, ) (b, ) (ab, )

168 Determination and representation A dichotomic approach for drawing the concept lattice of a given context Example 34: for each K-intent B, compute the corresponding element (B At 1, B At 2 ) in the product a b c d 1 b cd d 2 3 ab 4 d b cd ab abcd (, cd) (b, cd) (ab, cd) (, d) (b, d) (ab, d) (, ) (b, ) (ab, )

169 Determination and representation A dichotomic approach for drawing the concept lattice of a given context It is still possible to effect improvements in drawing the concept lattice of a given context Freese, R.: Automated lattice drawing. In Eklund, P. (Editor): ICFCA Springer-Verlag (2004) Tilley, T.: Tool support for FCA. In Eklund, P. (Editor): ICFCA Springer-Verlag (2004)

170 Determination and representation Implications between attributes It is often necessary to classify a large number of objects with respect to a relatively small number of attributes frequently useless or impracticable to write down the whole context In such cases the concept lattice can be inferred from the implication between the attributes the concept lattice can be inferred from statements of the kind every object with the attributes x 1, y 1,... also has the attributes x 2, y 2,...

171 Determination and representation Implications between attributes Example 35: concave square rectangle equilateral parallelogram

172 Determination and representation Implications between attributes Implication between attributes in a given context K = (Ob, At, I) implication B 1 B 2 where B 1 and B 2 are sets of K-attributes Let B be a set of K-attributes, B 1 B 2 a K-implication and L a set of K-implications B respects B 1 B 2 iff B 1 B or B 2 B B respects L iff B respects every K-implication B 1 B 2 L

173 Determination and representation Implications between attributes Example 36: concave square rectangle equilateral parallelogram {concave, parallelogram} {square, rectangle, equilateral} {square} {rectangle, equilateral, parallelogram} {rectangle} {parallelogram} {rectangle, equilateral, parallelogram} {square} {equilateral} {parallelogram}

174 Determination and representation Implications between attributes Implication between attributes in a given context K = (Ob, At, I) implication B 1 B 2 where B 1 and B 2 are sets of K-attributes Let B 1 B 2 a K-implication and L a set of K-implications K respects B 1 B 2 iff B respects B 1 B 2 for each K-concept (A, B) K respects L iff K respect every K-implication B 1 B 2 L Implicational theory of K Set Imp(K) of all K-implications that K respects

175 Determination and representation Implications between attributes Suppose that K = (Ob, At, I) is a context B 1 B 2 is a K-implication Then the following conditions are equivalent K respects B 1 B 2 B 1 B 2 B 1 B 2

176 Determination and representation Implications between attributes Implicational closure of a set L of K-implications : mapping Cl L ( ) : 2 At 2 At such that for all B At, Cl L (B) is the smallest set of K-attributes containing B and respecting L

177 Determination and representation Implications between attributes Example 37: concave square rectangle equilateral parallelogram If L contains the implications {rectangle} {parallelogram} and {rectangle, equilateral, parallelogram} {square} then Cl L ({rectangle, equilateral}) = {square, rectangle, equilateral, parallelogram}

178 Determination and representation Implications between attributes Let B 1 B 2 be a K-implication and L be a set of K-implications B 1 B 2 is a consequence of L iff Cl L (B 1 ) B 2 Let L and M be sets of K-implications L is sound for M iff every implication that follows from L is in M L is complete for M iff every implication in M follows from L

179 Determination and representation Implications between attributes Let L be a set of K-implications L is a base for K iff L is sound and complete for the set of all K-implications that K respects L is a Duquenne-Guigues base for K iff L is a base for K that is of minimum cardinality

180 Determination and representation Implications between attributes Example 38: concave square rectangle equilateral parallelogram {concave, parallelogram} {square, rectangle, equilateral} {square} {rectangle, equilateral, parallelogram} {rectangle} {parallelogram} {rectangle, equilateral, parallelogram} {square} {equilateral} {parallelogram}

181 Determination and representation Implications between attributes Suppose that K = (Ob, At, I) is a context B At Then B is a good attribute subset of K iff B B for all C B, if C C then B C

182 Determination and representation Implications between attributes Example 39: concave square rectangle equilateral parallelogram {concave, parallelogram} {square} {rectangle} {rectangle, equilateral, parallelogram} {equilateral}

183 Determination and representation Implications between attributes Suppose that K = (Ob, At, I) is a context Then {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 quadrilaterals {concave, parallelogram} {square, rectangle, equilateral} {square} {rectangle, equilateral, parallelogram} {rectangle} {parallelogram} {rectangle, equilateral, parallelogram} {square} {equilateral} {parallelogram}

184 Determination and representation Implications between attributes We consider the following problem Deciding whether a set of attributes is a good attribute subset of a context Input A context K = (Ob, At, I) and a set of attributes B At Output Decide whether B is a good attribute subset of K

185 Determination and representation Implications between attributes Suppose that K = (Ob, At, I) is a context B At is a set of K-attributes We shall say that B is closed iff B = B B is quasi-closed iff for all sets C B of K-attributes, C B or C = B B is pseudo-closed iff B is not closed, B is quasi-closed and for all quasi-closed sets C B of K-attributes, C B Note that if B is closed then B is quasi-closed

186 Determination and representation Implications between attributes Proposition 23: If K = (Ob, At, I) is a context and B At is a set of K-attributes then 1. B is quasi-closed iff B C is closed for every closed set C with B C 2. B is quasi-closed iff B X is closed or B X = B for any object 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 B 1, B 2 At are sets of K-attributes then if B 1, B 2 are quasi-closed then B 1 B 2 is quasi-closed

187 Determination and representation Implications between attributes Proposition 25: Testing whether B At is quasi-closed in the context K = (Ob, At, I) may be performed in O(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 At Output Decide whether B is a good attribute subset of K