Logics for Rough Concept Analysis Krishna Manoorkar 1 . Jipsen 3 , G. - PowerPoint PPT Presentation

Logics for Rough Concept Analysis Krishna Manoorkar 1 . Jipsen 3 , G. Greco 2 , A. Palmigiano 4,5 , and joint work with: P A. Tzimoulis 3 1 Indian Institute of Technology Kanpur, India 2 Utrecht University, NL 3 Chapman University, California, USA 4 Delft University of Technology, NL 5 Department of Pure and Applied Mathematics, University of Johannesburg, SA ICLA , 4 March, 2019

Motivation and Aim ◮ Formal contexts , or polarities , are structures P = ( A , X , I ) such that A and X are sets, and I ⊆ A × X is a binary relation. Intuitively, formal contexts can be understood as abstract representations of databases. For any relation T ⊆ U × V , and any U ′ ⊆ U and V ′ ⊆ V , let T ( 0 ) [ V ′ ] := { u | ∀ v ( v ∈ V ′ ⇒ uTv ) } T ( 1 ) [ U ′ ] := { v | ∀ u ( u ∈ U ′ ⇒ uTv ) } .

Motivation and Aim Definition For every formal context P = ( A , X , I ) , a formal concept of P is a pair c = ( B , Y ) such that B ⊆ A , Y ⊆ X , and B ↑ = Y and Y ↓ = B . The set B is the extension of c , denoted by [[ c ]] , and Y is the intension of c , denoted ([ c ]) . Let L ( P ) denote the set of the formal concepts of P . Then the concept lattice of P is the complete lattice P + := ( L ( P ) , � , � ) ,where for every X ⊆ L ( P ) , c ∈X [[ c ]]) ↑ ) c ∈X ([ c ])) ↓ , � � X := ( � c ∈X [[ c ]] , ( � � X := (( � c ∈X ([ c ])) . and ◮ ⊤ P + := � ∅ = ( A , A ↑ ) and ⊥ P + := � ∅ = ( X ↓ , X ) , and the partial order underlying this lattice structure is defined as follows: for any c , d ∈ L ( P ) , c ≤ d iff [[ c ]] ⊆ [[ d ]] iff ([ d ]) ⊆ ([ c ]) .

Motivation and Aim Definition An enriched formal context is a tuple F = ( P , R � , R � ) such that P = ( A , X , I ) is a formal context, and R � ⊆ A × X and R � ⊆ X × A are I - compatible relations, that is, R ( 0 ) � [ x ] (resp. R ( 0 ) � [ a ] ) and R ( 1 ) � [ a ] (resp. R ( 1 ) � [ x ] ) are Galois-stable for all x ∈ X and a ∈ A . The complex algebra of F is F + = ( P + , [ R � ] , � R � � ) , where P + is the concept lattice of P , and [ R � ] c := ( R ( 0 ) � [([ c ])] , ( R ( 0 ) � [([ c ])]) ↑ ) � R � � c := (( R ( 0 ) � [[[ c ]]]) ↓ , R ( 0 ) � [[[ c ]]]) . and

Motivation and Aim Definition Given I -compatible relations R , T ⊆ A × X , the composition R ; T ⊆ A × X is defined as: ( R ; T ) ( 1 ) [ a ] = R ( 1 ) [ I ( 0 ) [ T ( 1 ) [ a ]]] or equivalently ( R ; T ) ( 0 ) [ x ] = R ( 0 ) [ I ( 1 ) [ T ( 0 ) [ x ]]] . Rough formal contexts are tuples G = ( P , E ) such that P = ( A , X , I ) is a polarity, and E ⊆ A × A is an equivalence relation. For every a ∈ A we let ( a ) E := { b ∈ A | aEb } . The relation E induces two relations R , S ⊆ A × I approximating I , defined as follows: aRx iff bIx for some b ∈ ( a ) E ; aSx iff bIx for all b ∈ ( a ) E . (1)

Motivation and Aim Definition T = ( L , I ) is a topological quasi-Boolean algebra (tqBa) if L = ( L , ∨ , ∧ , ¬ , ⊤ , ⊥ ) is a De Morgan algebra and for all a , b ∈ L , T1. I ( a ∧ b ) = Ia ∧ Ib , T2. IIa = Ia , T3. Ia ≤ a , T4. I ⊤ = ⊤ . Algebras Acronyms Axioms topological quasi Boolean algebra 5 tqBa5 T5: CIa = Ia intermediate algebra of type 1 IA1 T5, T6: Ia ∨¬ Ia = ⊤ T5, T7: Ia ∨ Ib = I ( a ∨ b ) intermediate algebra of type 2 IA2 intermediate algebra of type 3 IA3 T5, T8: Ia ≤ Ib and Ca ≤ Cb imply a ≤ b pre-rough algebra pra T5, T6, T7, T8.

Motivation and Aim tqBa tqBa 5 IA 1 IA 2 IA 3 pre − rough rough

Motivation and Aim Kent’s rough formal concepts Consider a formal context F = ( G , M , I ) with an approximation space ( G , E ) on objects. We define upper and lower approximations for F as ( G , M , I ) and ( G , M , I ) respectively defined as 1. gIm iff there exists m ′ ∈ M such that, mEm ′ and gIm 2. gIm iff for all m ′ ∈ M , mEm ′ implies gIm Upper and lower approximations for concept ( A , B ) are defined as ( 0 ) ( B ) , I ( 1 ) ( I ( 0 ) ( B ))) and ( I ( 0 ) ( B ) , I ( 1 ) ( I ( 0 ) ( B ))) respectively. ( I

Introduction Definition An Rfc G = ( P , E ) is amenable if E , R and S are I -compatible. ◮ R - lax operators - � ℓ , � ℓ ◮ S - strict operators - � s , � s ◮ For any amenable Rfc, � s φ ⊢ φ φ ⊢ � ℓ φ φ ⊢ � s φ � ℓ φ ⊢ φ, (2) Lemma For any amenable Rfc G = ( P , E ) , if and R and S are defined as in (1) , then R ; R ⊆ R S ⊆ S ; S (3) and

Introduction � s φ ⊢ � s � s φ � ℓ � ℓ φ ⊢ � ℓ φ � s � s φ ⊢ � s φ � ℓ φ ⊢ � ℓ � ℓ φ (4) φ ⊢ � s � s φ � s � s φ ⊢ φ φ ⊢ � ℓ � ℓ φ � ℓ � ℓ φ ⊢ φ (5) We define Kent algebras as motivated from above: Definition A basic Kent algebra is a structure A = ( L , � s , � s , � ℓ , � ℓ ) such that L is a complete lattice, and � s , � s , � ℓ , � ℓ are unary operations on L such that for all a , b ∈ L , � s a ≤ b iff a ≤ � s b and � ℓ a ≤ b iff a ≤ � ℓ b , (6) � s a ≤ a a ≤ � s a a ≤ � ℓ a � ℓ a ≤ a (7) � s a ≤ � s � s a � s � s a ≤ � s a � ℓ � ℓ a ≤ � ℓ a � ℓ a ≤ � ℓ � ℓ a (8)

Introduction Definition An aKa A is an aKa5’ if for any a ∈ L , � ℓ a ≤ � s � ℓ a � s � ℓ a ≤ � ℓ a � s a ≤ � ℓ � s a � ℓ � s a ≤ � s a ; (9) is a K-IA3 s if for any a , b ∈ L , � s a ≤ � s b and � s a ≤ � s b imply a ≤ b , (10) and is a K-IA3 ℓ if for any a , b ∈ L , � ℓ a ≤ � ℓ b and � ℓ a ≤ � ℓ b imply a ≤ b . (11)

Multi-type environment Decompositions of unary operators � s = ◦ i · � i � i ·◦ i = id S I � s = ◦ c · � c � c ·◦ c = id S C ◦ i : S I ֒ → L � i : L ։ S I � c : L ։ S C ◦ c : S C ֒ → L � ℓ = � c · c c • C · � c = id L C � ℓ = � i ·• I • I · � i = id L I • C : L ։ L C � c : L C ֒ → L � i : L I ֒ → L • I : L ։ L I where S I := � s [ L ] , S C := � s [ L ] , L C := � ℓ [ L ] , and L I := � s [ L ] , and such that for all α ∈ S I , δ ∈ S C , a ∈ L , π ∈ L I , σ ∈ L C , ◦ i α ≤ a iff α ≤ � i a � c a ≤ δ iff a ≤ ◦ c δ • C a ≤ π iff a ≤ � c π � i σ ≤ a iff σ ≤ • i a . (12)

Multi-type environment S I ◦ I � I • I • C L C L I L � C � I ◦ C � C S I

Multi type environment Definition For any aKa A , the strict interior kernel S I = ( S I , ∪ I , ∩ I , t I , f I ) and the strict closure kernel S C = ( S C , ∪ C , ∩ C , t C , f C ) are such that, for all α,β ∈ S I , and all δ,γ ∈ S C , α ∪ I β := � i ( ◦ i α ∨◦ i β ) δ ∪ C γ := � c ( ◦ c δ ∨◦ c γ ) α ∩ I β := � i ( ◦ i α ∧◦ i β ) δ ∩ C γ := � c ( ◦ c δ ∧◦ c γ ) ⊤ i := � i ⊤ , ⊥ i := � i ⊥ ⊤ c := � c ⊤ , ⊥ c = � c ⊥

Heterogeneous Algebras Definition A heterogeneous aKa (haKa) is a tuple H = ( L , S I , S C , L I , L C , ◦ i , � i , ◦ c , � c , • I , � i , • C , � c ) such that: H1 L , S I , S C , L I , L C are bounded lattices; H2 ◦ i : S I ֒ → L , ◦ c : S C ֒ → L , • I : L ։ L I , • C : L ։ L C are lattice homomorphisms; H3 ◦ i ⊣ � i � c ⊣ ◦ C • C ⊣ � c � i ⊣ • i ; 1 H4 � i ◦ i = id S I � c ◦ C = id S C • C � c = id L C • i � i = id L I 1 Condition H3 implies that � i : L ։ S I and � i : L I ֒ → L are ∧ -hemimorphisms and � C : L ։ S C and � C : L C ֒ → L are ∨ -hemimorphisms; condition H4 implies that the black connectives are surjective and the white ones are injective.

Heterogeneous Algebras The haKas corresponding to the varieties of Definition 8 are defined as follows: Algebra Acronym Conditions heterogeneous aKa5’ haKa5’ � i π ≤ ◦ i � i � i π ◦ c � c � c σ ≤ � c σ ◦ i α ≤ � i • I ◦ i α � c • C ◦ c δ ≤ ◦ c δ heterogeneous K-IA3 s hK-IA3 s � i a ≤ � i b and � c a ≤ � c b imply a ≤ b heterogeneous K-IA3 ℓ hK-IA3 ℓ � C • a ≤ � c • b and � I • a ≤ � I • b imply a ≤ b Notice that the inequalities defining haKa5’ are all analytic inductive. Defining rule for hK-IA3 ℓ is converted in multitype environment to a ∧ � I • I b ≤ � C • C a ∨ b . which is also analytic inductive however corresponding inequality for hK-IA3 s is not analytic inductive.

Heterogeneous Algebras Theorem For every K ∈ { aKa, aKa5’, K-IA3 s , K-IA3 ℓ } , letting HK denote its corresponding class of heterogeneous algebras, the following holds: 1. If A ∈ K , then A + ∈ HK ; 2. If H ∈ HK , then H + ∈ K ; 3. A � ( A + ) + H � ( H + ) + . and 4. The isomorphisms of the previous item restrict to perfect members of K and HK . 5. If A ∈ K , then A δ � (( A + ) δ ) + and if H ∈ HK , then H δ � (( H + ) δ ) + .