A Study of a Logic for Multiple-source Approximation Systems Mohua - - PowerPoint PPT Presentation

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion

A Study of a Logic for Multiple-source Approximation Systems

Mohua Banerjee

• Md. Aquil Khan

Indian Institute of Technology Kanpur

ICLA 2009

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion

1

Basic Concepts Related to Rough Set Theory

2

Multiple-source Approximation Systems Different Notions of Lower/Upper Approximations Different Notions of Definability

3

Logic for MSAS(LMSAS) Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

4

Conclusion

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion

Out break of some disease. Totally unaffected class − → classes 1 to 8. Totally affected class − → class 9. Partially affected class − → class 10.

• Q. Is ‘A’, a class 8 student, affected?
• A. No.
• Q. Is ‘B’, a class 9 student, affected?
• A. Yes.
• Q. Is ‘C’, a class 10 student, affected?
• A. Possibly, but not certainly.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion

Pawlak Approximation space [Pawlak’82] (U, R), where R is an equivalence relation on U. U

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion

Pawlak Approximation space [Pawlak’82] (U, R), where R is an equivalence relation on U. U X

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion

Pawlak Approximation space [Pawlak’82] (U, R), where R is an equivalence relation on U. U X B(X) X R X R

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion

U X Boundary element −ve +ve

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion

U := Set of students; aRb iff a and b are in the same class; X := Set of affected students. U 1 2 3 4 5 6 7 8 9 10

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion

U := Set of students; aRb iff a and b are in the same class; X := Set of affected students. U 1 2 3 4 5 6 7 8 9 10 X

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion

U := Set of students; aRb iff a and b are in the same class; X := Set of affected students. U 1 2 3 4 5 6 7 8 9 10 X 1 2 3 4 5 6 7 8 9 10

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Different Notions of Lower/Upper Approximations Different Notions of Definability

Multiple-source Approximation Systems(MSAS) F := (U, {Ri}i∈N), where U is a non-empty set, N an initial segment of the set of positive integers, and each Ri, i ∈ N, is an equivalence relation on the domain U. |N| is referred to as the cardinality of F and is denoted by |F|.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Different Notions of Lower/Upper Approximations Different Notions of Definability

F := (U, {Ri}i∈N), X ⊆ U Definition Strong lower approximation X s :=

• i

X Ri; Weak lower approximation X w :=

• i

X Ri. Strong upper approximation X s :=

• i

X Ri; Weak upper approximation X w :=

• i

X Ri. For MSAS F := (U, {R}) X s = X w = X R and X s = X w = X R

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Different Notions of Lower/Upper Approximations Different Notions of Definability

X s ⊆ X w ⊆ X ⊆ X s ⊆ X w U

(X w)c X w \ X s X s \ X w X w \ X s X s

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Different Notions of Lower/Upper Approximations Different Notions of Definability

X s ⊆ X w ⊆ X ⊆ X s ⊆ X w U

(X w)c X w \ X s X s \ X w X w \ X s X s certain +ve possible +ve certain boundary possible -ve certain -ve

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Different Notions of Lower/Upper Approximations Different Notions of Definability

Proposition

1 X ∩ Y s = X s ∩ Y s; X ∪ Y w = X w ∪ Y w; 2 X ∩ Y s ⊆ X s ∩ Y s; X ∪ Y w ⊇ X w ∪ Y w; 3 X c

s = (X w)c; X c w = (X s)c;

4 X cs = (X w)c; X cw = (X s)c; 5 X w = (X w)w ; X s = (X s)s; 6 X w = (X w)w = (X s)w; 7 (X s)w ⊆ X w; Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Different Notions of Lower/Upper Approximations Different Notions of Definability

X ⊆ U is lower definable if X s = X w. U

(X w)c X w \ X s X s \ X w X w \ X s X s certain +ve possible +ve certain boundary possible -ve certain -ve

X is lower definable iff the sets of +ve elements in all approximations spaces are identical.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Different Notions of Lower/Upper Approximations Different Notions of Definability

X ⊆ U is upper definable if X s = X w. U

(X w)c X w \ X s X s \ X w X w \ X s X s certain +ve possible +ve certain boundary possible -ve certain -ve

X is upper definable iff the sets of -ve elements in all approximations spaces are identical.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Different Notions of Lower/Upper Approximations Different Notions of Definability

X ⊆ U is weak definable if X s = X w. U

(X w)c X w \ X s X s \ X w X w \ X s X s certain +ve possible +ve certain boundary possible -ve certain -ve

X is weak definable iff X does not have certain boundary element.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Different Notions of Lower/Upper Approximations Different Notions of Definability

X ⊆ U is strong definable if X s = X w. U

(X w)c X w \ X s X s \ X w X w \ X s X s certain +ve possible +ve certain boundary possible -ve certain -ve

X is strong definable iff every element of U is either certain +ve or certain -ve.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Different Notions of Lower/Upper Approximations Different Notions of Definability

Proposition X is upper definable iff X c is lower definable. Arbitrary union (intersection) of upper (lower) definable sets is also upper (lower) definable. (Collection of upper (lower) definable sets is not closed under intersection (union)). Collection of all strong definable sets forms a complete field of sets.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Different Notions of Lower/Upper Approximations Different Notions of Definability

Proposition The following are equivalent: X is strong definable. X is both lower and upper definable and X is definable in some approximation space. X is definable in each approximation space. X s = X w = X = X s = X w.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Language L a non-empty countable set Var of variables, a (possibly empty) countable set Con of constants, a non-empty countable set PV of propositional variables and the propositional constants ⊤, ⊥. Terms T := Var ∪ Con. Wffs:=⊤|⊥|p|¬α|α ∧ β|tα|∀xα p ∈ PV , t ∈ T, x ∈ Var, and α, β are wffs.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Notations F− → Set of all wffs; F− → Set of all closed wffs; Con(α) − → Set of constants used in α; Var(α) − → Set of variables used in α; FV (α) − → Set of free variables used in α.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Model M := (F, V , v), where F := (U, {Ri}i∈N) is a MSAS; V : PV → 2U is a valuation function and v : Var → N is an assignment. Let α ∈ F and Γ ⊆ F. α−Model The model M := (F, V , v) is said to be an α-model if |F| ≥ k, where k is the largest integer such that ck ∈ Con(α). Γ−Model M is a Γ-model, if it is an α-model for each α ∈ Γ.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

satisfiability M, w | = p, if and only if w ∈ V (p). M, w | = ciα, if and only if there exists w′ in U such that wRiw′ and M, w′ | = α. M, w | = xα, if and only if there exists w′ in U such that wRv(x)w′ and M, w′ | = α. M, w | = ∀xα, if and only if M′, w | = α, for every model M′ := (F, V , v′) where the assignment v′ is x-equivalent to v.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Validity | = α, if and only if M, w | = α, for every α-model M and object w

• f U.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

M := (F, V , v) − → a model. V (α) := {w ∈ U : M, w | = α}, where α is a wff involving

• nly those ck which satisfy |F| ≥ k

Rough Set Interpretation V (ciα) = V (α)Ri; V ([ci]α) = V (α)Ri; Fore α which does not have free occurrence of x, V (∀x[x]α) = V (α)s; V (∃x[x]α) = V (α)w; V (∀xxα) = V (α)s; V (∃xxα) = V (α)w.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Proposition The following are valid.

1

(a) ∃x[x]α → α. (b) α → ∀xxα.

2

(a) ∀x[x](α ∧ β) ↔ ∀x[x]α ∧ ∀x[x]β. (b) ∃xx(α ∨ β) ↔ ∃xxα ∨ ∃xxβ.

3

(a) ∀xx(α ∧ β) → ∀xxα ∧ ∀xxβ. (b) ∃x[x](α ∨ β) → ∃x[x]α ∨ ∃x[x]β.

4

(a) ∃x[x]α ↔ ∃x[x]∃y[y]α. (b) ∃xx∀y[y]α → ∃x[x]α.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Axioms

1 All axioms of classical propositional logic. 2 ∀xα → α[t/x], where α admits the term t for the variable x. 3 ∀x(α → β) → (α → ∀xβ), where the variable x is not free in

α.

4 ∀x[t]α → [t]∀xα, where the term t and variable x are

different.

5 [t](α → β) → ([t]α → [t]β). 6 α → tα. 7 α → [t]tα. 8 ttα → tα. Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Rules of inference ∀. α MP. α N. α ∀xα α → β [t]α β

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Soundness If ⊢ α then | = α Completeness Theorem If | = α then ⊢ α Proof Let n is the largest integer such that cn occurs in α. Ln − → obtained from the language L by restricting Con to the set {c1, c2, . . . , cn}. Ln+ − → obtained from the language Ln by adding infinitely many new variables. Var+ := {x1, x2, . . .} be an enumeration of the variables in Ln+.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Proof (Contd.) Canonical model: UC := {w : w is maximally consistent and has the ∀ property in Ln+}. V C : PV → P(UC) is such that V C(p) := {w ∈ UC : p ∈ w}, for p ∈ PV . vC : Var+ → N is such that vC(xi) := n + i. wRC

i w′, if and only if for every wff [t]α of Ln+ with

vC(t) = i, [t]α ∈ w implies α ∈ w′, where w, w′ ∈ UC. FC := (UC, {RC

i }i∈N).

MC := (FC, V C, vC).

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Proof (Contd.) Proposition Every consistent set of wffs in Ln has a maximally consistent extension in Ln+, having the ∀-property. Truth Lemma For any wff β in Ln+ and w ∈ UC, β ∈ w if and only if MC, w | = β. vc(x) = vc(y) for x = y; vc(x) ∈ {1, 2, . . . , n}.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Decidability? Some Decidable problems Proposition Given a wff α and an integer m ≥ k, where k is the largest integer such that ck ∈ Con(α), it is decidable if there exists an α-model M := (F, V , v) with |F| = m such that α is satisfiable in M. Proof Σ − → Sub-formula closed set such that r is the largest integer for which cr ∈ Con(Σ). F := (U, {Ri}1≤i≤m), m ≥ r. V : PV → 2U.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Proof(Contd.) For w, w′ ∈ U, w≡Σw′, if and only if for all β ∈ Σ and all Σ-models M := (F, V , v), M, w | = β ⇔ M, w′ | = β Filtration Model Uf := {[w] : w ∈ U}. [w]Rf

i [w′] if and only if there exist w1 ∈ [w] and w2 ∈ [w′]

such that w1Riw2; Ri f ⋆ is the transitive closure of Rf

i .

Ff := (Uf , {Ri f ⋆}1≤i≤m). V f (p) := {[w] ∈ Uf : w ∈ V (p)}.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Proof(Contd.) Filtration Theorem For all wffs β ∈ Σ, all assignment v : Var → {1, 2, . . . , m} and all

• bjects w ∈ U,

(F, V , v), w | = β iff (Ff , V f , v), [w] | = α

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Proof(Contd.) Proposition Let α be a wff such that Var(α) := {x1, x2, . . . , xn}. α is satisfiable in a model M := (F, V , v), where |F| = m. Σ is the set of all sub-formulae of α. Then α is satisfiable in a model with domain of cardinality ≤ 2|Σ|×mn. Asg := {v| v : var → {1, 2, . . . , m}}. For v1, v2 ∈ Asg, v1 ≈ v2 iff v1(x) = v2(x) x ∈ Var(α). |Asg/≈| ≤ mn.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Proof(Contd.) The mapping g : Uf → 2Σ×Asg/≈, defined as g([w]) := {(β, [v1]) ∈ Σ × Asg/ ≈: ((F, V , v1), w | = β} is injective. |Uf | ≤ 2|Σ|×mn

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Proposition Given an integer t and a wff α, it is decidable if there exists an α-model with a domain of cardinality t, in which α is satisfiable.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

R ⊆ W × W and R′ ⊆ W ′ × W ′. Z ⊆ W × W ′ Bisimulation Z : R← →R′, if the following conditions are satisfied:

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

R ⊆ W × W and R′ ⊆ W ′ × W ′. Z ⊆ W × W ′ Bisimulation Z : R← →R′, if the following conditions are satisfied: A w u w′ Z R

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

R ⊆ W × W and R′ ⊆ W ′ × W ′. Z ⊆ W × W ′ Bisimulation Z : R← →R′, if the following conditions are satisfied: A w u w′ Z R u′

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

R ⊆ W × W and R′ ⊆ W ′ × W ′. Z ⊆ W × W ′ Bisimulation Z : R← →R′, if the following conditions are satisfied: A w u w′ Z R u′ Z R′

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

R ⊆ W × W and R′ ⊆ W ′ × W ′. Z ⊆ W × W ′ Bisimulation Z : R← →R′, if the following conditions are satisfied: A w u w′ Z R u′ Z R′ B w w′ u′ Z R′

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

R ⊆ W × W and R′ ⊆ W ′ × W ′. Z ⊆ W × W ′ Bisimulation Z : R← →R′, if the following conditions are satisfied: A w u w′ Z R u′ Z R′ B w w′ u′ Z R′ u R Z

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C ⊆ Con, V 1, V 2 ⊆ Var. Γ := {α : Con(α) ⊆ C, FV (α) ⊆ V 1 and Var(α) ⊆ V 2}. Theorem M := (F, V , v) M′ := (F′, V ′, v′) Z ⊆ W × W ′

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C ⊆ Con, V 1, V 2 ⊆ Var. Γ := {α : Con(α) ⊆ C, FV (α) ⊆ V 1 and Var(α) ⊆ V 2}. Theorem M := (F, V , v) M′ := (F′, V ′, v′) Z ⊆ W × W ′ C A ci

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C ⊆ Con, V 1, V 2 ⊆ Var. Γ := {α : Con(α) ⊆ C, FV (α) ⊆ V 1 and Var(α) ⊆ V 2}. Theorem M := (F, V , v) M′ := (F′, V ′, v′) Z ⊆ W × W ′ C A ci Ri R′i

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C ⊆ Con, V 1, V 2 ⊆ Var. Γ := {α : Con(α) ⊆ C, FV (α) ⊆ V 1 and Var(α) ⊆ V 2}. Theorem M := (F, V , v) M′ := (F′, V ′, v′) Z ⊆ W × W ′ C A ci Ri R′i Z :

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C ⊆ Con, V 1, V 2 ⊆ Var. Γ := {α : Con(α) ⊆ C, FV (α) ⊆ V 1 and Var(α) ⊆ V 2}. Theorem M := (F, V , v) M′ := (F′, V ′, v′) Z ⊆ W × W ′ C A ci Ri R′i Z : B V 1 x

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C ⊆ Con, V 1, V 2 ⊆ Var. Γ := {α : Con(α) ⊆ C, FV (α) ⊆ V 1 and Var(α) ⊆ V 2}. Theorem M := (F, V , v) M′ := (F′, V ′, v′) Z ⊆ W × W ′ C A ci Ri R′i Z : B V 1 x Rv(x) R′v′(x)

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C ⊆ Con, V 1, V 2 ⊆ Var. Γ := {α : Con(α) ⊆ C, FV (α) ⊆ V 1 and Var(α) ⊆ V 2}. Theorem M := (F, V , v) M′ := (F′, V ′, v′) Z ⊆ W × W ′ C A ci Ri R′i Z : B V 1 x Rv(x) R′v′(x) Z :

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C V 2 x Rv(x)

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C V 2 x Rv(x) R′j

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C V 2 x Rv(x) R′j Z :

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C V 2 x Rv(x) R′j Z : R′v′(x)

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C V 2 x Rv(x) R′j Z : R′v′(x) Rj

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C V 2 x Rv(x) R′j Z : R′v′(x) Rj Z :

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C V 2 x Rv(x) R′j Z : R′v′(x) Rj Z : D u Z u′

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C V 2 x Rv(x) R′j Z : R′v′(x) Rj Z : D u Z u′ ⇓ u ∈ V (p) iff u′ ∈ V ′(p) for all p ∈ V (p)

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C V 2 x Rv(x) R′j Z : R′v′(x) Rj Z : D u Z u′ ⇓ u ∈ V (p) iff u′ ∈ V ′(p) for all p ∈ V (p) E w Z w′

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Then for all α ∈ Γ, M, w α ⇔ M′, w′ α.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C ⊆ Con, V 2 ⊆ Var. Γ := {α : Con(α) ⊆ C, Var(α) ⊆ V 2}. Corollary(*) M := (F, V , v) and M′ := (F′, V ′, v′), where F := (W , {Ri}i∈N), F′ := (W ′, {R′

i }i∈N′) be two Γ-models.

Z ⊆ W × W ′ satisfying the following:

• a. Z : Ri←

→R′

i for all i such that ci ∈ C;

• b. Z : Rv(x)←

→R′

v′(x) for all x ∈ V 2;

• c. If uZu′, then u ∈ V (p) if and only if u′ ∈ V ′(p)

for all p ∈ PV ;

• d. wZw ′.

Then for all α ∈ Γ, M, w α ⇔ M′, w′ α.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Converse of Corollary(*)? C ⊆ Con, V 1 ⊆ Var. Let Γ := {α : Con(α) ⊆ C, Var(α) ⊆ V 1} Theorem

1 M := (F, V , v) and M′ := (F′, V ′, v′), where

F := (W , {Ri}i∈N), F′ := (W ′, {R′

i }i∈N′), be two Γ-models.

2 Equivalence classes of the relations Ri, Ri ′ for ci ∈ C and

Rv(x), R′

v′(x), x ∈ V 1 are all finite.

3 w ∈ W and w′ ∈ W ′ such that M, w α ⇔ M′, w′ α for

all α ∈ Γ.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Then there exists a relation Z ⊆ W × W ′ satisfying the following:

• a. Z : Ri←

→R′

i for all i such that ci ∈ C;

• b. Z : Rv(x)←

→R′

v′(x) for all x ∈ V 1;

• c. If uZu′, then u ∈ V (p) if and only if u′ ∈ V ′(p) for

all p ∈ PV ;

• d. wZw ′.

Result does not hold if condition (2) is removed.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

P ⊆f PV and α be a wff which involves only the propositional variables from the set P. Proposition M := (F, V , v) M′ := (F′, V ′, v′) f : W → W ′ (Surjective)

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

P ⊆f PV and α be a wff which involves only the propositional variables from the set P. Proposition M := (F, V , v) M′ := (F′, V ′, v′) f : W → W ′ (Surjective) Con(α) A ci

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

P ⊆f PV and α be a wff which involves only the propositional variables from the set P. Proposition M := (F, V , v) M′ := (F′, V ′, v′) f : W → W ′ (Surjective) Con(α) A ci s Ri r ⇔ f (s) R′ f (r)

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

P ⊆f PV and α be a wff which involves only the propositional variables from the set P. Proposition M := (F, V , v) M′ := (F′, V ′, v′) f : W → W ′ (Surjective) Con(α) A ci s Ri r ⇔ f (s) R′ f (r) B FV (α) x

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

P ⊆f PV and α be a wff which involves only the propositional variables from the set P. Proposition M := (F, V , v) M′ := (F′, V ′, v′) f : W → W ′ (Surjective) Con(α) A ci s Ri r ⇔ f (s) R′ f (r) B FV (α) x s Rv(x) r ⇔ f (s) R′v′(x) f (r)

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C Var(α) x Rv(x) Then we have, M, w α ⇔ M′, w′ α.

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C Var(α) x Rv(x) R′j Then we have, M, w α ⇔ M′, w′ α.

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C Var(α) x Rv(x) R′j s r ⇔ f (s) f (r) Then we have, M, w α ⇔ M′, w′ α.

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C Var(α) x Rv(x) R′j s r ⇔ f (s) f (r) R′v′(x) Then we have, M, w α ⇔ M′, w′ α.

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C Var(α) x Rv(x) R′j s r ⇔ f (s) f (r) R′v′(x) Rj Then we have, M, w α ⇔ M′, w′ α.

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C Var(α) x Rv(x) R′j s r ⇔ f (s) f (r) R′v′(x) Rj s r ⇔ f (s) f (r) Then we have, M, w α ⇔ M′, w′ α.

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C Var(α) x Rv(x) R′j s r ⇔ f (s) f (r) R′v′(x) Rj s r ⇔ f (s) f (r) D u ∈ V (p) ⇔ f (u) ∈ V ′(p) fora all p ∈ P Then we have, M, w α ⇔ M′, w′ α.

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

C Var(α) x Rv(x) R′j s r ⇔ f (s) f (r) R′v′(x) Rj s r ⇔ f (s) f (r) D u ∈ V (p) ⇔ f (u) ∈ V ′(p) fora all p ∈ P E f (w) = w′ Then we have, M, w α ⇔ M′, w′ α.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Example M M′ W := {wi : i ∈ N} ∪ {w, d} U := {ui : i ∈ N} ∪ {u} Ri = W × W R′i = U × U V (pj) := W \ {w, wj} V (pj) := U \ {u, uj} v v Obs1: d ∈ V (p) for all p ∈ PV . Obs2: For each P ⊆f PV , there exists a t ∈ U such that t ∈ V ′(p) for all p ∈ P. Claim 1: M, w | = α ⇔ M′, u | = α for all α. P = {pj1, pj1, . . . , pjn} consists of propositional variables which occurs in α. f (d) = ujn+1, f (w) = u, f (wi) = ui. Use previous result.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Obs3: There is no t ∈ U such that t ∈ V ′(p) for all p ∈ PV . Claim 2: There is Z satisfying (a)-(d). If possible, let there exists such a Z wRu and wZu. There exists t ∈ U such that dZt. t ∈ V ′(p) for all p ∈ PV .

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Bisimulation Invariance Result and the Hennessy-Milner Theorem for S5 Γ := {α : Con(α) ⊆ {c1}, Var(α) = ∅} corresponds to the set

• f all wffs of a normal modal logic, where [c1] and c1 are

considered as and ♦ respectively. Φ : (W , {Ri}i∈N) → (W , R1) (Surjective). (F, V , v), w α if and only if (Φ(F), V ), w S5 α, for all α ∈ Γ. From this observation and choice of Γ, we obtain the bisimulation invariance result and the Hennessy-Milner theorem for S5.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Proposition S5 ⇀ LMSAS. Kn ⇀ LMSAS. LMSAS ⇀ SOL. B ⇀ LMSAS. Proof

• L be the set of basic modal logic wffs with modal operator L.

Choose a variable x and fix it. TB : L → L be the translation such that:

TB(Lα) = ∀x[x]TB(α).

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion Axiomatization Some Decidable Problems Bisimulation Relationship with Other Logics

Proof(Contd.) Proposition α ∈ L is satisfiable in a reflexive, symmetric model if and only if TB(α) is a satisfiable LMSAS wff. F := (W , {Ri}i∈N), F∗ := (W , R :=

i Ri), where R is

reflexive and symmetric. (F, V , v), w | = TB(α) iff (F∗, V ), w | = α Given reflexive and symmetric frame (W , R) with finite domain, there exists MSAS F such that F∗ = (W , R).

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion

Decidability?

1

There exists a function f such that if α is satisfiable, then it is satisfiable in a model based on a MSAS F with |F| ≤ f (|α|).

2

There exists a function f such that if α is satisfiable, then it is satisfiable in a model with finite domain W such that |W | ≤ f (|α|).

LMSAS corresponds to which fragment of SOL? Extension of LMSAS to capture the group knowledge of the sources.

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems

Outline Basic Concepts Related to Rough Set Theory Multiple-source Approximation Systems Logic for MSAS(LMSAS) Conclusion

Thank you

Mohua Banerjee, Md. Aquil Khan A Study of a Logic for Multiple-source Approximation Systems