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

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Logics for Rough Concept Analysis

Krishna Manoorkar1 joint work with: P . Jipsen3, G. Greco2, A. Palmigiano4,5, and

A. Tzimoulis3

1Indian Institute of Technology Kanpur, India 2Utrecht University, NL 3Chapman University, California, USA 4Delft University of Technology, NL 5Department of Pure and Applied Mathematics, University of Johannesburg, SA

ICLA , 4 March, 2019

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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)}.

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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), X := (

c∈X[[c]],( c∈X[[c]])↑)

and X := ((

c∈X([c]))↓, c∈X([c])).

◮ ⊤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]).

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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])])↑)

and Rc := ((R(0)

[[[c]]])↓,R(0) [[[c]]]).

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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)

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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 = ⊤ intermediate algebra of type 2 IA2 T5, T7: Ia ∨Ib = I(a ∨b) intermediate algebra of type 3 IA3 T5, T8: Ia ≤ Ib and Ca ≤ Cb imply a ≤ b pre-rough algebra pra T5, T6, T7, T8.

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Motivation and Aim

IA2 IA1 IA3 tqBa5 tqBa pre −rough rough

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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 (I

(0)(B),I(1)(I (0)(B))) and (I(0)(B),I(1)(I(0)(B))) respectively.

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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, sφ ⊢ φ φ ⊢ ℓφ φ ⊢ sφ ℓφ ⊢ φ, (2)

Lemma

For any amenable Rfc G = (P,E), if and R and S are defined as in (1), then R;R ⊆ R and S ⊆ S;S(3)

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Introduction

sφ ⊢ ssφ ℓℓφ ⊢ ℓφ ssφ ⊢ sφ ℓφ ⊢ ℓℓφ(4) φ ⊢ ssφ ssφ ⊢ φ φ ⊢ ℓℓφ ℓℓφ ⊢ 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, sa ≤ b iff a ≤ sb and ℓa ≤ b iff a ≤ ℓb, (6) sa ≤ a a ≤ sa a ≤ ℓa ℓa ≤ a (7) sa ≤ ssa ssa ≤ sa ℓℓa ≤ ℓa ℓa ≤ ℓℓa (8)

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Introduction

Definition

An aKa A is an aKa5’ if for any a ∈ L, ℓa ≤ sℓa sℓa ≤ ℓa sa ≤ ℓsa ℓsa ≤ sa; (9) is a K-IA3s if for any a,b ∈ L, sa ≤ sb and sa ≤ sb imply a ≤ b, (10) and is a K-IA3ℓ if for any a,b ∈ L, ℓa ≤ ℓb and ℓa ≤ ℓb imply a ≤ b. (11)

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Multi-type environment

Decompositions of unary operators s = ◦i ·i i ·◦i = idSI s = ◦c ·c c ·◦c = idSC

i : SI ֒→ L

i : L ։ SI c : L ։ SC

c : SC ֒→ L

ℓ = c ·c c

C ·c = idLC

ℓ = i ·•I

I ·i = idLI
C : L ։ LC

c : LC ֒→ L i : LI ֒→ L

I : L ։ LI

where SI := s[L], SC := s[L], LC := ℓ[L], and LI := s[L], and such that for all α ∈ SI, δ ∈ SC, a ∈ L, π ∈ LI, σ ∈ LC,

iα ≤ a iff α ≤ ia

ca ≤ δ iff a ≤ ◦cδ

C a ≤ π iff a ≤ cπ

iσ ≤ a iff σ ≤ •ia. (12)

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Multi-type environment

L SI SI LI LC I C

C
I
I
C

I C

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Multi type environment

Definition

For any aKa A, the strict interior kernel SI = (SI,∪I,∩I,tI,fI) and the strict closure kernel SC = (SC,∪C,∩C,tC,fC) are such that, for all α,β ∈ SI, and all δ,γ ∈ SC, α∪I β := i(◦iα∨◦iβ) δ∪C γ := c(◦cδ∨◦cγ) α∩I β := i(◦iα∧◦iβ) δ∩C γ := c(◦cδ∧◦cγ) ⊤i := i⊤, ⊥i := i⊥ ⊤c := c⊤, ⊥c = c⊥

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Heterogeneous Algebras

Definition

A heterogeneous aKa (haKa) is a tuple H = (L,SI,SC,LI,LC,◦i,i,◦c,c,•I ,i,•C ,c) such that: H1 L,SI,SC,LI,LC are bounded lattices; H2 ◦i : SI ֒→ L, ◦c : SC ֒→ L, •I : L ։ LI, •C : L ։ LC are lattice homomorphisms; H3 ◦i ⊣ i c ⊣ ◦C

C ⊣ c

i ⊣ •i; H4 i◦i = idSI c◦C = idSC

C c = idLC
ii = idLI

1

1Condition H3 implies that i : L ։ SI and i : LI ֒→ L are ∧-hemimorphisms and

C : L ։ SC and C : LC ֒→ L are ∨-hemimorphisms; condition H4 implies that the black connectives are surjective and the white ones are injective.

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Heterogeneous Algebras

The haKas corresponding to the varieties of Definition 8 are defined as follows: Algebra Acronym Conditions heterogeneous aKa5’ haKa5’ iπ ≤ ◦iiiπ

cccσ ≤ cσ
iα ≤ i•I ◦iα

c•C ◦cδ ≤ ◦cδ heterogeneous K-IA3s hK-IA3s ia ≤ ib and ca ≤ cb imply a ≤ b heterogeneous K-IA3ℓ hK-IA3ℓ 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-IA3ℓ is converted in multitype environment to a ∧I •I b ≤ C •C a ∨b. which is also analytic inductive however corresponding inequality for hK-IA3s is not analytic inductive.

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Heterogeneous Algebras

Theorem

For every K ∈ {aKa, aKa5’, K-IA3s, K-IA3ℓ}, 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+)+

and H (H+)+.

4. The isomorphisms of the previous item restrict to perfect members
f K and HK.
5. If A ∈ K, then Aδ ((A+)δ)+ and if H ∈ HK, then Hδ ((H+)δ)+.

SLIDE 18

Multi-type calculi

Language

general lattice L A ::= p | ⊤ | ⊥ | ◦I α | ◦C δ | I π | C σ | A ∧A | A ∨A X ::= A | ˇ ⊥ | ˆ ⊤ | ˜

I Γ | ˜
C ∆ | ˆ

I Π | ˇ I Π | ˆ C Σ | ˇ C Σ | X ˆ ∧X | X ˇ ∨X strict-interior kernel SI lax-interior kernel LI α ::= I A | I A π ::= •I A Γ ::= α | ˆ I X | ˇ I X | ˇ fI | ˆ tI | Γ ˆ ∩I Γ | Γ ˇ ∪I Γ Π ::= π | ˜

I X | ˇ

0I | ˆ 1I | Π ˆ ⊓I Π | Π ˇ ⊔I Π strict-closure kernel SC lax-closure kernel LC δ ::= C A | C A σ ::= •C A ∆ ::= δ | ˆ C X | ˇ C X | ˇ fC | ˆ tC | ∆ ˆ ∩C ∆ | ∆ ˇ ∪C ∆ Σ ::= σ | ˜

C X | ˇ

0C | ˆ 1C | Σ ˆ ⊓C Σ | Σ ˇ ⊔C Σ

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Multi-type calculi

The calculus D.AKA consists of the following axiom and rules. ◮ Identity and Cut:

IdL

p ⊢ p x ⊢ a a ⊢ y

Cut

x ⊢ y

◮ Multi-type display rules:

˜

I Γ ⊢ X

adLSI

Γ ⊢ ˇ I X X ⊢ ˜

I Γ

adLSI

ˆ I X ⊢ Γ X ⊢ ˜

C ∆

adLSC

ˆ C X ⊢ ∆ ˜

C X ⊢ ∆

adLSC

X ⊢ ˇ C ∆

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Multi-type calculi

◮ Multi-type structural rules for strict-kernel operators:

˜

I ˆ

tI ⊢ X

˜

ˆ

tI

ˆ ⊤ ⊢ X X ⊢ ˜

I ˇ

fI

˜

I ˇ

fI

X ⊢ ˇ ⊥ ˜

C ˆ

tC ⊢ X

˜

C ˆ

tC

ˆ ⊤ ⊢ X X ⊢ ˜

C ˇ

fC

˜

C ˇ

fC

X ⊢ ˇ ⊥ ˆ I ˜

I Γ ⊢ Γ′

ˆ I ˜

I

Γ ⊢ Γ′ Γ′ ⊢ ˇ I ˜

I Γ

ˇ I ˜

I

Γ′ ⊢ Γ ˆ C ˜

C ∆ ⊢ ∆′

ˆ C ˜

C

∆ ⊢ ∆′ ∆′ ⊢ ˇ C ˜

C ∆

∆′ ⊢ ∆ ˜

I ˆ

I X ⊢ Y

˜

I ˆ

I

X ⊢ Y Y ⊢ ˜

I ˇ

I X

˜

I ˇ

I

Y ⊢ X ˜

C ˆ

C X ⊢ Y

˜

C ˆ

C

X ⊢ Y Y ⊢ ˜

C ˇ

C X

˜

C

Y ⊢ X

◮ Multi-type structural rules for lax-kernel operators:

˜

I ˆ

⊤ ⊢ Π

˜

I ˆ

1I

ˆ 1I ⊢ Π Π ⊢ ˜

I ˇ

⊥

˜

I ˇ

0I

Π ⊢ ˇ 0I ˜

C ˆ

⊤ ⊢ Σ

˜

ˆ

1C

ˆ 1C ⊢ Σ Σ ⊢ ˜

C ˇ

⊥

˜

C

Σ ⊢ ˇ 0C Π ⊢ Π′

ˆ I ˜

I

ˆ I ˜

I Π ⊢ Π′

Π′ ⊢ Π

ˇ I ˜

I

Π′ ⊢ ˇ I ˜

I Π

Σ ⊢ Σ′

ˆ C ˜

C

ˆ C ˜

C Σ ⊢ Σ′

Σ′ ⊢ Σ Σ′ ⊢ ˇ C ˜

C Σ

˜

I ˆ

I Π ⊢ Π′

˜

I ˆ

I

Π ⊢ Π′ Π′ ⊢ ˜

I ˇ

I Π

˜

I ˇ

I

Π′ ⊢ Π ˜

C ˆ

C Σ ⊢ Σ′

˜

C ˆ

C

Σ ⊢ Σ′ Σ′ ⊢ ˜

C ˇ

C Σ Σ′ ⊢ Σ

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Multi-type calculi

◮ Multi-type structural rules for the correspondence between kernels:

˜

I ˆ

I X ⊢ Y

˜

ˆ
˜
C ˆ

C X ⊢ Y Y ⊢ ˇ I ˜

I X

ˇ ˜

Y ⊢ ˇ

C ˜

C X

◮ Logical rules for multi-type connectives related to strict kernels:

ˆ I A ⊢ Γ

I

I A ⊢ Γ X ⊢ A

I

ˆ I X ⊢ I A A ⊢ X

C

C A ⊢ ˇ C X ∆ ⊢ ˇ C A

C

∆ ⊢ C A ˜

I α ⊢ X
I
I α ⊢ X

X ⊢ ˜

I α
I

X ⊢ ◦I α ˜

C δ ⊢ X
C
C δ ⊢ X

X ⊢ ˜

C δ
C

X ⊢ ◦C δ

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Multi-type calculi

◮ Logical rules for multi-type connectives related to lax kernels:

ˆ I π ⊢ X

I

I π ⊢ X Π ⊢ π

I

ˆ I Π ⊢ I π σ ⊢ Σ

C

C σ ⊢ ˇ C Σ X ⊢ ˇ C σ

C

X ⊢ C σ ˜

I A ⊢ Π
I
I A ⊢ Π

Π ⊢ ˜

I A
I

Π ⊢ •I A ˜

C A ⊢ Σ
C
C A ⊢ Σ

Σ ⊢ ˜

C A
C

Σ ⊢ •C A

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Multi-type calculi

The proper display calculi for the subvarieties of aKa are obtained by adding the following rules:

Logic Calculus Rules H.aKa5′ D.aKa5′ ˆ I Π ⊢ X

˜

I ˆ

I ˆ I

˜

I ˆ

I ˆ I Π ⊢ X X ⊢ ˇ C Σ

˜

C ˇ

C ˇ C

X ⊢ ˜

C ˇ

C ˇ C Σ ˆ I ˜

I ˜
I Γ ⊢ X

ˆ I ˜

I ˜
I

˜

I Γ ⊢ X

X ⊢ ˇ C ˜

C ˜
C ∆

ˇ C ˜

C ˜
C

X ⊢ ˜

C ∆

K-IA3ℓ D.K-IA3ℓ X ⊢ ˇ I ˜

I Y

ˆ C ˜

C X ⊢ Y

k-ia3ℓ

X ⊢ Y

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Properties

Theorem (Soundness)

The rules in D.K is sound w.r.t. the class of HD.

Theorem (Completeness)

D.K is complete with respect to the class of semi-De Morgan algebras.

Theorem (Conservativity)

D.K is a conservative extension of H.K.

Theorem (Cut elimination)

If X ⊢ Y is derivable in D.K, then it is derivable without (Cut).

Theorem (Subformula property)

Any cut-free proof of the sequent X ⊢ Y in D.K contains only structures

ver subformulas of formulas in X and Y.

SLIDE 25

Thank You!

SLIDE 26

References

M. Banerjee and M.K. Chakraborty.

Rough sets through algebraic logic. Fundamenta Informaticae, 28(3, 4):211–221, 1996.

W. Conradie, S. Frittella, A. Palmigiano, M. Piazzai, A. Tzimoulis, and N.M. Wijnberg.

Categories: How I Learned to Stop Worrying and Love Two Sorts. In Proc. WoLLIC 2016, volume 9803 of LNCS, pages 145–164, 2016.

W. Conradie, S. Frittella, A. Palmigiano, M. Piazzai, A. Tzimoulis, and N.M. Wijnberg.

Toward an epistemic-logical theory of categorization. In Proc. TARK 2017, volume 251 of EPTCS, pages 167–186, 2017. bibitemTrendsXIII S. Frittella, G. Greco, A. Kurz, A. Palmigiano, and

V. Sikimi´

c. Multi-type sequent calculi.

Proc. Trends in Logic XIII, A. Indrzejczak et al. eds, pages 81–93, 2014.
G. Greco, F. Liang, K. Manoorkar, and A. Palmigiano.

Proper multi-type display calculi for rough algebras. ArXiv preprint 1808.07278.

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References

R.E. Kent. Rough concept analysis: a synthesis of rough sets and formal concept analysis. Fundamenta Informaticae, 27(2, 3):169–181, 1996.

M. Ma, M.K. Chakraborty, and Z. Lin.

Sequent calculi for varieties of topological quasi-boolean algebras. In Proc. IJCRS 2018, volume 11103 of LNCS, pages 309–322, 2018.

K. Manoorkar, S. Nazari, A. Palmigiano, A. Tzimoulis, and N.M. Wijnberg.

Rough concepts.

2018. Submitted.
A. Saha, J. Sen, and M.K. Chakraborty.

Algebraic structures in the vicinity of pre-rough algebra and their logics. Information Sciences, 282:296–320, 2014.

A. Saha, J. Sen, and M.K. Chakraborty.

Algebraic structures in the vicinity of pre-rough algebra and their logics ii. Information Sciences, 333:44–60, 2016.