Relations between Proto-fuzzy Concepts, Crisply Generated Fuzzy Concepts, and Interval Pattern Structures Vera V. Pankratieva and Sergei O. Kuznetsov State University Higher School of Economics Moscow, Russia CLA’2010, Sevilla V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 1 / 26
Motivation Recently several models for the analysis of fuzzy and numerical data within FCA were proposed: Fuzzy formal concepts [R. Bˇ elohl´ avek, 1999] Protofuzzy formal concepts [O. Kr´ ıdlo, S. Krajˇ ci, 2005] Pattern structures [B. Ganter and S.O. Kuznetsov, 2000] V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 2 / 26
Main research goal Relationships between protofuzzy concepts, fuzzy concepts, pattern structures, both from theoretical and from experimental viewpoints. V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 3 / 26
Outline Relationships between Protofuzzy [O. Kr´ ıdlo, S. Krajˇ ci] and fuzzy formal concepts [R. Bˇ elohl´ avek] Fuzzy formal concepts [R. Bˇ elohl´ avek] and pattern structures [B. Ganter and S.O. Kuznetsov] Fuzzy formal concepts [R. Bˇ elohl´ avek] and two-way pattern structures V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 4 / 26
Main definitions of FCA G is a set, called the set of objects, M is a set, called the set of attributes, I ⊆ G × M is a binary relation. The triple K := ( G , M , I ) is called a (formal) context . Galois connection between ( 2 G , ⊆ ) and ( 2 M , ⊆ ) is a pair of maps ( · ) ′ : 2 G → 2 M A ′ def = { m ∈ M | gIm for all g ∈ A } , ( · ) ′ : 2 M → 2 G B ′ def = { g ∈ G | gIm for all m ∈ B } , with the following properties: ∀ A 1 , A 2 ⊆ G , B 1 , B 2 ⊆ M 1 A 1 ⊆ A 2 ⇒ A ′ 2 ⊆ A ′ B 1 ⊆ B 2 ⇒ B ′ 2 ⊆ B ′ 1 ; 1 ; 2 A 1 ⊆ A ′′ B 1 ⊆ B ′′ 1 , 1 . V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 5 / 26
Main definitions of FCA Formal concepts are partially ordered by the relation � A 1 , B 1 � ≥ � A 2 , B 2 � ⇐ ⇒ A 1 ⊇ A 2 � B 2 ⊇ B 1 � . Implication A → B , where A , B ⊆ M , holds in the context if A ′ ⊆ B ′ . The map ( · ) ′′ is a closure operator, since it is idempotent, extensive, and monotone. A formal concept is a pair � A , B � , A ⊆ G , B ⊆ M , such that A ′ = B , B ′ = A . V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 6 / 26
Fuzzy formal concepts [R. Bˇ elohl´ avek] A fuzzy formal context is a triple � X , Y , I � , where I is a fuzzy relation between X and Y , taking a pair ( x , y ) to the truth degree I ( x , y ) ∈ L , with which object x ∈ X has attribute y ∈ Y . A fuzzy formal concept is a pair � A , B � , A ⊆ X , B ⊆ Y , such that A ↑ = B and B ↓ = A , where � � A ↑ ( y ) = B ↓ ( x ) = ( A ( x ) → I ( x , y )) and ( B ( y ) → I ( x , y )) . x ∈ X y ∈ Y Example 0.1 0.6 0.5 A = ( 0 . 3 , 0 . 9 , 1 ) is the fuzzy tuple of extent I = 1 0.2 0.4 B = ( 0 . 8 , 0 . 3 , 0 ) is the fuzzy tuple of intent 0.8 0.3 0 V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 7 / 26
Crisply generated fuzzy formal concepts [R. Bˇ elohl´ avek] c = B ↓ and B = A ↑ = A ↑↓↑ A pair � A , B � such that A = A ↑↓ is called c a crisply generated formal concept ( A c is a crisp tuple). 0.1 0.6 0.5 I = 1 0.2 0.4 0.8 0.3 0 A c = ( 0 , 1 , 0 ) is the crisp tuple of extent; A = ( 0 . 1 , 1 , 0 . 6 ) = B ↓ = A ↑↓ c ; B = ( 1 , 0 . 2 , 0 . 4 ) = A ↑ ; � A , B � is a crisply generated formal concept. A c = 1 A ↑↓ is a tuple closed by crisp components (crisply closed tuple). c V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 8 / 26
Protofuzzy formal concepts [O. Kr´ ıdlo, S. Krajˇ ci] Consider mappings ↑ l : 2 X → 2 Y and ↓ l : 2 Y → 2 X defined for a fuzzy context � X , Y , I � over lattice L , and l ∈ L . ∀ A ⊆ X let ↑ l ( A ) = { y ∈ Y : ( ∀ x ∈ A ) I ( x , y ) ≥ l } ; ∀ B ⊆ Y let ↓ l ( B ) = { x ∈ X : ( ∀ y ∈ B ) I ( x , y ) ≥ l } . Definition An l- concept is a pair � A , B � such that ↑ l ( A ) = B and ↓ l ( B ) = A, i.e., a concept in the l-cut, binary context � X , Y , I l � , where I l = { ( x , y ) ∈ X × Y : I ( x , y ) ≥ l } . By K l we denote the set of all concepts in the l-cut. Definition Triples � A , B , l � ∈ 2 X × 2 Y × L such that � A , B � ∈ � K k and k ∈ L l = sup { k ∈ L : � A , B � ∈ K k } are called protofuzzy concepts. The set of all protofuzzy concepts is denoted by K P . V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 9 / 26
Protofuzzy formal concepts [O. Kr´ ıdlo, S. Krajˇ ci] A protofuzzy formal concept is a formal concept that occurs for the “first time” in a γ -cut of a fuzzy context. Example 0.1 0.6 0.5 1 0.2 0.4 I = 0.8 0.3 0 V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 10 / 26
Protofuzzy formal concepts [O. Kr´ ıdlo, S. Krajˇ ci] A protofuzzy formal concept is a formal concept that occurs for the “first time” in a γ -cut of a fuzzy context. Example 0.1 0.6 0.5 X X 1 0.2 0.4 X I = 0.8 0.3 0 X 0.5-cut V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 11 / 26
Protofuzzy formal concepts [O. Kr´ ıdlo, S. Krajˇ ci] A protofuzzy formal concept is a formal concept that occurs for the “first time” in a γ -cut of a fuzzy context. Example 0.1 0.6 0.5 X X 1 0.2 0.4 I = 0.8 0.3 0 0.5-cut V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 12 / 26
Contractions [O. Kr´ ıdlo, S. Krajˇ ci] Let K P denote the set of all proto-fuzzy concepts. Definition Let B ⊆ Y be an arbitrary set of attributes. The contraction of the set of proto-fuzzy concepts subsistent to the set B is B = {� A , l � ∈ 2 X × L : ( ∃ B ′ ⊇ B ) � A , B ′ , l � ∈ K P } . K P V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 13 / 26
Relationship between protofuzzy formal concepts and fuzzy formal concepts Theorem Let there be a protofuzzy formal concept in the l-cut. Then there is a tuple B c that satisfies the following conditions: B ↓ c = ( a 1 , . . . , a k ) , where a i ≥ l if the ith object is contained in the given 1 protofuzzy concept, and a i < l if it is not; B c is a crisply closed tuple, i.e., 1 B ↓↑ = B c . 2 c Theorem Let A = ( a 1 , . . . , a k ) = B ↓ c , B = B ↓↑ c . Let us define a tuple Z by the formula � a j ≥ a i , 1 , z j = 0 , a j < a i . Then the context Z × 1 Y corresponds to a protofuzzy concept in the a i -cut of the initial context, which can be contracted to 1 Y . V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 14 / 26
Relationship between protofuzzy formal concepts and fuzzy concepts Results: It is shown that for every protofuzzy concept one can construct a crisply closed tuple; It is shown that for every crisply-closed tuple one can construct a contraction of a protofuzzy concept on the crisp subset of attributes of its intent (this contraction may coincide with the intent itself); There is a method for deciding whether the contraction proposed is an intent of a protofuzzy concept. V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 15 / 26
Pattern structures [B. Ganter, S. Kuznetsov] Let G be a set, called the set of objects, ( D , ⊓ ) be a lower semilattice and δ : G → D be a mapping. Definition The triple ( G , D , δ ) , where D = ( D , ⊓ ) , is called a pattern structure if the set δ ( G ) := { δ ( g ) | g ∈ G } generates a complete subsemilattice ( D δ , ⊓ ) of ( D , ⊓ ) . For a pattern structure ( G , D , δ ) we define operations A � := � g ∈ A δ ( g ) for sets A ⊆ G and d � := { g ∈ G | d ⊑ δ ( g ) } for elements of semilattice d ∈ D . A pattern concept is a pair � A , d � such that A � = d , d � = A . V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 16 / 26
Interval pattern structures object 1 0.1 0.6 0.5 object 2 1 0.2 0.4 object 3 0.8 0.3 0 { 1 , 2 } � = ([ 0 . 1 , 1 ] , [ 0 . 2 , 0 . 6 ] , [ 0 . 4 , 0 . 5 ]) ; ([ 0 . 1 , 1 ] , [ 0 . 2 , 0 . 6 ] , [ 0 . 4 , 0 . 5 ]) � = { 1 , 2 } . The pair ( { 1 , 2 } , ([ 0 . 1 , 1 ] , [ 0 . 2 , 0 . 6 ] , [ 0 . 4 , 0 . 5 ]) ) is an interval pattern concept V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 17 / 26
Relationship between fuzzy formal concepts and pattern structures Theorem If extents of interval pattern concepts coincide with crisply closed tuples, there are no nontrivial implications in the context. Theorem For any crisply-closed tuple the corresponding subset of objects is closed. Theorem If the set of crisply closed tuples coincides with the set of extents of interval pattern concepts, each interval pattern concept should satisfy the following condition: for no object from its extent the object intent majorizes the minimum of intervals. V. Pankratieva, S. Kuznetsov (SU-HSE) Relations between fuzzy concepts CLA’2010, Sevilla 18 / 26
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