Algebraic Semantics for the Logic ofMultiple-source Approximation Systems
Mohua Banerjee Md. Aquil Khan
Indian Institute of Technology Kanpur
ICLA 2009
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Rough Set Theory
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 Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Rough Set Theory
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 Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Rough Set Theory
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 Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Rough Set Theory
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 Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Rough Set Theory
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 Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Rough Set Theory
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 Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Rough Set Theory
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 Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Rough Set TheoryPawlak Approximation Space
Approximation space [Pawlak’82]
(U,R), where R is an equivalence relation on U.
U
XX B(X )
XR
XR
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Rough Set TheoryPawlak Approximation Space
Approximation space [Pawlak’82]
(U,R), where R is an equivalence relation on U.
U
X
X B(X )
XR
XR
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Rough Set TheoryPawlak Approximation Space
Approximation space [Pawlak’82]
(U,R), where R is an equivalence relation on U.
U
XX B(X )
XR
XR
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Rough Set TheoryPawlak Approximation Space
U
X
Boundary element
−ve
+ve
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Rough Set TheoryPawlak Approximation Space
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
XX
1 2 3 4 5
6 7 8 9 10
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Rough Set TheoryPawlak Approximation Space
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 10X
X
1 2 3 4 5
6 7 8 9 10
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Rough Set TheoryPawlak Approximation Space
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 10XX
1 2 3 4 5
6 7 8 9 10
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Multiple-source Approximation Systems
Multiple-source Approximation Systems(MSAS) [Banerjee andKhan ’08]
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 Algebraic Semantics for the Logic of Multiple-source Approximation Systems
LMSAS
Language
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|〈t〉α|∀xαp ∈ PV , t ∈ T , x ∈ Var , and α, β are wffs.
F−→ Set of all wffs;
F−→ Set of all closed wffs;
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
LMSAS
Let Γ be a set of wffs of LMSAS .
Interpretation for Γ
M := (F,V , I ), where
F := (U, {Ri}i∈N) is a MSAS,
V : PV → P(U) and
I : Con(Γ) → N.
Assignment for an interpretation M is a mapv : Term(Γ) → N such that v(c) = I (c), for each c ∈ Con(Γ).
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
LMSAS
satisfiability
M, v ,w |= p, if and only if w ∈ V (p), for p ∈ PV .
M, v ,w |= 〈t〉α, if and only if there exists w ′ in U such thatwRv(t)w
′ and M, v ,w ′ |= α.
M, v ,w |= ∀xα, if and only if for every assignment v ′
x-equivalent to v , M, v ′,w |= α.
α is valid, denoted |= α, if and only if M, v ,w |= α, for everyinterpretation M := (µ,V , I ), assignment v for M and object win the domain of F.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
LMSASRough Set Interpretation
Consider F := (U, {Ri}i∈N), M := (F,V , I ), and assignmentv ;
let V (α) := {w ∈ U : M, v ,w |= α}.
V (〈t〉α) = V (α)Rv(t);
V ([t]α) = V (α)Rv(t)
V (∀x [x ]α) =⋂
i V (α)Ri
;
V (∃x [x ]α) =⋃
i V (α)Ri
;
V (∀x〈x〉α) =⋂
i V (α)Ri;
V (∃x〈x〉α) =⋃
i V (α)Ri.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
LMSAS
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 inα.
4 ∀x [t]α → [t]∀xα, where the term t and variable x aredifferent.
5 [t](α → β) → ([t]α → [t]β).
6 α → 〈t〉α.
7 α → [t]〈t〉α.
8 〈t〉〈t〉α → 〈t〉α.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
LMSAS
Rules of inference
∀. α MP. α N. α∀xα α → β [t]α
β
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
LMSAS
Soundness and Completeness Theorem
` α if and only if |= α
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Algebraic Semantics of LMSAS
Boolean Algebra with Operators(BAOs)
BAOs A := (A,∩,∼, 1, fi )i∈∆ where
(A,∩,∼, 1) is a Boolean algebra (BA) and
for each i ∈ ∆, fi is an unary operator satisfying1 f (1) = 1;2 f (a ∩ b) = fa ∩ fb.
∆ = Neach fk satisfies the following additional conditions:
(B1) fka ≤ fk fka;(B2) fka ≤ a;(B3) a ≤ fkgka, where gk :=∼ fk ∼.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Algebraic Semantics of LMSAS
Boolean Algebra with Operators(BAOs)
BAOs A := (A,∩,∼, 1, fi )i∈∆ where
(A,∩,∼, 1) is a Boolean algebra (BA) and
for each i ∈ ∆, fi is an unary operator satisfying1 f (1) = 1;2 f (a ∩ b) = fa ∩ fb.
∆ = Neach fk satisfies the following additional conditions:
(B1) fka ≤ fk fka;(B2) fka ≤ a;(B3) a ≤ fkgka, where gk :=∼ fk ∼.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Algebraic Semantics of LMSAS
Definition
Let A := (A,∩,∼, 1, fi )i∈N −→ be a BAOs.
Assignment θ : PV → A.
θ : F → A where
θ([ci ]α) = fi (θ(α)), i ∈ N,
θ(∀xα) =⋂{θ(α(cj/x)) : j ∈ N}, provided the g.l.b. exists.
For α ∈ F , θ(α) := θ(cl(α)).
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Algebraic Semantics of LMSAS
A := (A,∩,∼, 1, fi )i∈N −→ be a BAOs satisfying (B1)-(B3).
Assignment θ : PV → A.
θ(α) exists for all α ∈ F if and only if⋂{θ(α(cj/x)) : j ∈ N}
exists for all α with only free variable x .
fi (a ∩ b) = fia ∩ fib;fi (
⋂X ) 6=
⋂fiX ;
fk⋂j
θ(α(cj/x)) 6=⋂j
fk θ(α(cj/x));
(∀x [t]α → [t]∀xα, t and x are different)
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Algebraic Semantics of LMSAS
A := (A,∩,∼, 1, fi )i∈N −→ be a BAOs satisfying (B1)-(B3).
Assignment θ : PV → A.
θ(α) exists for all α ∈ F if and only if⋂{θ(α(cj/x)) : j ∈ N}
exists for all α with only free variable x .
fi (a ∩ b) = fia ∩ fib;
fi (⋂
X ) 6=⋂
fiX ;
fk⋂j
θ(α(cj/x)) 6=⋂j
fk θ(α(cj/x));
(∀x [t]α → [t]∀xα, t and x are different)
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Algebraic Semantics of LMSAS
A := (A,∩,∼, 1, fi )i∈N −→ be a BAOs satisfying (B1)-(B3).
Assignment θ : PV → A.
θ(α) exists for all α ∈ F if and only if⋂{θ(α(cj/x)) : j ∈ N}
exists for all α with only free variable x .
fi (a ∩ b) = fia ∩ fib;fi (
⋂X ) 6=
⋂fiX ;
fk⋂j
θ(α(cj/x)) 6=⋂j
fk θ(α(cj/x));
(∀x [t]α → [t]∀xα, t and x are different)
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Algebraic Semantics of LMSAS
A := (A,∩,∼, 1, fi )i∈N −→ be a BAOs satisfying (B1)-(B3).
Assignment θ : PV → A.
θ(α) exists for all α ∈ F if and only if⋂{θ(α(cj/x)) : j ∈ N}
exists for all α with only free variable x .
fi (a ∩ b) = fia ∩ fib;fi (
⋂X ) 6=
⋂fiX ;
fk⋂j
θ(α(cj/x)) 6=⋂j
fk θ(α(cj/x));
(∀x [t]α → [t]∀xα, t and x are different)
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Algebraic Semantics of LMSAS
Realization
A BAOs A := (A,∩,∼, 1, fi )i∈N satisfying (B1)-(B3) such that forevery assignment θ : PV → A the following is satisfied:
(R1) θ(α) exists for all α ∈ F ,
(R2) fk⋂j
θ(α(cj/x)) =⋂j
fk θ(α(cj/x)), where α has
only one free variable x .
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Algebraic Semantics of LMSAS
Definition
A BAOs A := (A,∩,∼, 1, fi )i∈∆ satisfying (B1)-(B3), is said to bea complete BAOs (CBAOs) if it satisfies the following propertiesfor all X ⊆ A:
(A1)⋂
X and⋃
X exists,
(A2) fk⋂
X =⋂
fkX , k ∈ ∆,
Every CBAOs is a realization but not conversely.
Consider the BAOs which is not complete but which has only afinite number of distinct operators fi .
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Algebraic Semantics of LMSAS
Definition
A BAOs A := (A,∩,∼, 1, fi )i∈∆ satisfying (B1)-(B3), is said to bea complete BAOs (CBAOs) if it satisfies the following propertiesfor all X ⊆ A:
(A1)⋂
X and⋃
X exists,
(A2) fk⋂
X =⋂
fkX , k ∈ ∆,
Every CBAOs is a realization but not conversely.
Consider the BAOs which is not complete but which has only afinite number of distinct operators fi .
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Algebraic Semantics of LMSAS
Definition
A BAOs A := (A,∩,∼, 1, fi )i∈∆ satisfying (B1)-(B3), is said to bea complete BAOs (CBAOs) if it satisfies the following propertiesfor all X ⊆ A:
(A1)⋂
X and⋃
X exists,
(A2) fk⋂
X =⋂
fkX , k ∈ ∆,
Every CBAOs is a realization but not conversely.
Consider the BAOs which is not complete but which has only afinite number of distinct operators fi .
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Algebraic Semantics of LMSAS
Definition
Let A := (A,∩,∼, 1, fi )i∈N be a realization.
A α ≈ β if and only if for every assignmentθ : PV → A, θ(α) = θ(β).
C α if A α ≈ > for all A ∈ C (Class of all CBAOs).
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Algebraic Semantics of LMSAS
Soundness
If ` α then C α
Completeness
If C α then ` α
If 6` α, then there exists a CBAOs A := (A,∩,∼, 1, fi )i∈N and anassignment θ : PV → A such that θ(α) 6= 1.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Algebraic Semantics of LMSAS
Soundness
If ` α then C α
Completeness
If C α then ` α
If 6` α, then there exists a CBAOs A := (A,∩,∼, 1, fi )i∈N and anassignment θ : PV → A such that θ(α) 6= 1.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Lindenbaum Algebra
Equivalence relation ≡ on F :
α ≡ β if and only if ` α ↔ β.
F|≡ := {[α] : α ∈ F}.
Operations on F|≡:
[α] ∩ [β] := [α ∧ β];∼ [α] := [¬α];fi [α] := [[ci ]α].
L := (F|≡,∩,∼, 1, fi )i∈N, where 1 = [>] is a BAOs.
Question : Is L a realization?Yes
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Lindenbaum Algebra
Equivalence relation ≡ on F :
α ≡ β if and only if ` α ↔ β.
F|≡ := {[α] : α ∈ F}.
Operations on F|≡:
[α] ∩ [β] := [α ∧ β];∼ [α] := [¬α];fi [α] := [[ci ]α].
L := (F|≡,∩,∼, 1, fi )i∈N, where 1 = [>] is a BAOs.
Question : Is L a realization?
Yes
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Lindenbaum Algebra
Equivalence relation ≡ on F :
α ≡ β if and only if ` α ↔ β.
F|≡ := {[α] : α ∈ F}.
Operations on F|≡:
[α] ∩ [β] := [α ∧ β];∼ [α] := [¬α];fi [α] := [[ci ]α].
L := (F|≡,∩,∼, 1, fi )i∈N, where 1 = [>] is a BAOs.
Question : Is L a realization?Yes
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
(R1) θ(α) exists for all α ∈ F .
θ(pi ) = [αi ], where p1, p2, . . . is an enumerationof the set PV.β∗ is obtained from β by uniform replacement ofpi ’s by αi ’s.(α(cj/x))∗ = α∗(cj/x), j ∈ N.⋂j
[α(cj/x)] exists and is given by [∀xα], where
α has only x as free variable.θ(α) := [α∗].
(R2) fk⋂j
θ(α(cj/x)) =⋂j
fk θ(α(cj/x)).
∀x [t]α → [t]∀xα, t and x are different.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
(R1) θ(α) exists for all α ∈ F .
θ(pi ) = [αi ], where p1, p2, . . . is an enumerationof the set PV.
β∗ is obtained from β by uniform replacement ofpi ’s by αi ’s.(α(cj/x))∗ = α∗(cj/x), j ∈ N.⋂j
[α(cj/x)] exists and is given by [∀xα], where
α has only x as free variable.θ(α) := [α∗].
(R2) fk⋂j
θ(α(cj/x)) =⋂j
fk θ(α(cj/x)).
∀x [t]α → [t]∀xα, t and x are different.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
(R1) θ(α) exists for all α ∈ F .
θ(pi ) = [αi ], where p1, p2, . . . is an enumerationof the set PV.β∗ is obtained from β by uniform replacement ofpi ’s by αi ’s.
(α(cj/x))∗ = α∗(cj/x), j ∈ N.⋂j
[α(cj/x)] exists and is given by [∀xα], where
α has only x as free variable.θ(α) := [α∗].
(R2) fk⋂j
θ(α(cj/x)) =⋂j
fk θ(α(cj/x)).
∀x [t]α → [t]∀xα, t and x are different.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
(R1) θ(α) exists for all α ∈ F .
θ(pi ) = [αi ], where p1, p2, . . . is an enumerationof the set PV.β∗ is obtained from β by uniform replacement ofpi ’s by αi ’s.(α(cj/x))∗ = α∗(cj/x), j ∈ N.⋂j
[α(cj/x)] exists and is given by [∀xα], where
α has only x as free variable.
θ(α) := [α∗].
(R2) fk⋂j
θ(α(cj/x)) =⋂j
fk θ(α(cj/x)).
∀x [t]α → [t]∀xα, t and x are different.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
(R1) θ(α) exists for all α ∈ F .
θ(pi ) = [αi ], where p1, p2, . . . is an enumerationof the set PV.β∗ is obtained from β by uniform replacement ofpi ’s by αi ’s.(α(cj/x))∗ = α∗(cj/x), j ∈ N.⋂j
[α(cj/x)] exists and is given by [∀xα], where
α has only x as free variable.θ(α) := [α∗].
(R2) fk⋂j
θ(α(cj/x)) =⋂j
fk θ(α(cj/x)).
∀x [t]α → [t]∀xα, t and x are different.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
(R1) θ(α) exists for all α ∈ F .
θ(pi ) = [αi ], where p1, p2, . . . is an enumerationof the set PV.β∗ is obtained from β by uniform replacement ofpi ’s by αi ’s.(α(cj/x))∗ = α∗(cj/x), j ∈ N.⋂j
[α(cj/x)] exists and is given by [∀xα], where
α has only x as free variable.θ(α) := [α∗].
(R2) fk⋂j
θ(α(cj/x)) =⋂j
fk θ(α(cj/x)).
∀x [t]α → [t]∀xα, t and x are different.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
(R1) θ(α) exists for all α ∈ F .
θ(pi ) = [αi ], where p1, p2, . . . is an enumerationof the set PV.β∗ is obtained from β by uniform replacement ofpi ’s by αi ’s.(α(cj/x))∗ = α∗(cj/x), j ∈ N.⋂j
[α(cj/x)] exists and is given by [∀xα], where
α has only x as free variable.θ(α) := [α∗].
(R2) fk⋂j
θ(α(cj/x)) =⋂j
fk θ(α(cj/x)).
∀x [t]α → [t]∀xα, t and x are different.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Question
Given a realization A := (A,∩,∼, 1, fi )i∈N, does there exist aCBAOs B := (B,∩,∼, gi )i∈N and a BAOs monomorphism
r : A → B which also preserves all of⋂j
θ(α(cj/x)) and⋃j
θ(α(cj/x))?
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Complex Algebra
F := (U, {Ri}i∈∆) −→ frame.The complex algebra of F (notation F+) is the expansion of thepower set algebra P(U) with operators mRi
: 2U → 2U , i ∈ ∆,where
mRi(X ) := {x ∈ U : For all y such that xRiy , y ∈ X}.
If F is MSAS, then F+ is a CBAOs satisfying (B1)-(B3)
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Complex Algebra
F := (U, {Ri}i∈∆) −→ frame.The complex algebra of F (notation F+) is the expansion of thepower set algebra P(U) with operators mRi
: 2U → 2U , i ∈ ∆,where
mRi(X ) := {x ∈ U : For all y such that xRiy , y ∈ X}.
If F is MSAS, then F+ is a CBAOs satisfying (B1)-(B3)
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Definition
A := (A,∩,∼, 1, fi )i∈∆ −→ BAOs.A+ := (FP(A),Qfi )i∈∆, where
FP(A) is the set of all prime filters of the Boolean algebra(A,∩,∼, 1);
F Qfi G if and only if fia ∈ F implies a ∈ G .
If ∆ = N and each fi satisfies (B1)-(B3), then A+ is a MSAS.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Definition
A := (A,∩,∼, 1, fi )i∈∆ −→ BAOs.A+ := (FP(A),Qfi )i∈∆, where
FP(A) is the set of all prime filters of the Boolean algebra(A,∩,∼, 1);
F Qfi G if and only if fia ∈ F implies a ∈ G .
If ∆ = N and each fi satisfies (B1)-(B3), then A+ is a MSAS.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Stone Representation Theorem
A := (A,∩,∼, 1) −→ Boolean algebra.r : A → 2FP(A) defined by
r(x) := {F ∈ FP(A) : x ∈ F}
is a monomorphism.
Jonsson-Tarski Theorem
A := (A,∩,∼, 1, fi )i∈∆ −→ BAOs.r : A → 2FP(A) defined by
r(x) := {F ∈ FP(A) : x ∈ F}
is a BAOs embedding of A into (A+)+
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Stone Representation Theorem
A := (A,∩,∼, 1) −→ Boolean algebra.r : A → 2FP(A) defined by
r(x) := {F ∈ FP(A) : x ∈ F}
is a monomorphism.
Jonsson-Tarski Theorem
A := (A,∩,∼, 1, fi )i∈∆ −→ BAOs.r : A → 2FP(A) defined by
r(x) := {F ∈ FP(A) : x ∈ F}
is a BAOs embedding of A into (A+)+
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Is Jonsson-Tarski Theorem enough?
Monomorphism r given in Stone representation theorem andJonsson-Tarski theorem does not always preserve infinite joinsand meets in A.
We want r to preserve⋂j
θ(α(cj/x)) and⋃j
θ(α(cj/x)).
It is impossible to extend Stone Representation Theorem insuch a way that the mapping r preserves all existing infinitejoins and meets in A (Tanaka and Ono).
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
A := (A,∩,∼, 1) −→ Boolean algebra.Q := {Qn ⊆ A : n ∈ N}, where each Qn is non-empty.
Q-Filter [Tanaka and Ono’01]
A prime filter F of A is called Q − filter , if it satisfies the followingfor each n ∈ N.
1 If Qn ⊆ F and⋂
Qn exists then⋂
Qn ∈ F .
2 If⋃
Qn exists and belongs to F then Qn ∩ F 6= ∅.
FQ(A) −→ Set of all Q − filters.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Definition
A := (A,∩,∼, 1, fi )i∈∆ −→ BAOs.AQ+ := (FQ(A),Qfi )i∈∆, where
F Qfi G if and only if fia ∈ F implies a ∈ G .
If ∆ = N and each fi satisfies (B1)-(B3), then AQ+ is aMSAS.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Definition
A := (A,∩,∼, 1, fi )i∈∆ −→ BAOs.AQ+ := (FQ(A),Qfi )i∈∆, where
F Qfi G if and only if fia ∈ F implies a ∈ G .
If ∆ = N and each fi satisfies (B1)-(B3), then AQ+ is aMSAS.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Theorem
A := (A,∩,∼, 1) −→ Boolean algebra.Q := {Qn ⊆ A : n ∈ N}, where each Qn is non-empty.r : A → 2FQ(A) defined by
r(x) := {F ∈ FQ(A) : x ∈ F}
is a monomorphism which preserves all⋂
Qn and⋃
Qn if theyexist.
Extension of the above theorem to BAOs ?
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Theorem
A := (A,∩,∼, 1) −→ Boolean algebra.Q := {Qn ⊆ A : n ∈ N}, where each Qn is non-empty.r : A → 2FQ(A) defined by
r(x) := {F ∈ FQ(A) : x ∈ F}
is a monomorphism which preserves all⋂
Qn and⋃
Qn if theyexist.
Extension of the above theorem to BAOs ?
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
{Xn}n∈N and {Yn}n∈N be an enumeration of the setsQ∗ := {Qm ∈ Q :
⋂Qm ∈ A} and
Q∗ := {Qm ∈ Q :⋃
Qm ∈ A}.Following conditions are satisfied for each i ∈ N:
(C1) for any n,⋂
fiXn exists and satisfies that⋂fiXn = fi
⋂Xn,
(C2) for any z ∈ A and n, there exists m such that{fi (z → x) : x ∈ Xn} = Xm, wherez → x :=∼ z ∪ x ,
(C3) for any z ∈ A and n, there exists m such that{fi (y → z) : y ∈ Yn} = Ym.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Theorem
Suppose (C1)-(C3) are satisfied. Then r : A → 2FQ(A) defined by
r(x) := {F ∈ FQ(A) : x ∈ F}
is a BAOs embedding of A into the complex algebra (AQ+)+
which also preserves all of⋂
Xn and⋃
Yn.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Can we use the above result to obtain the following ?
“Given a realization A := (A,∩,∼, 1, fi )i∈N, there exist aCBAOs B := (B,∩,∼, gi )i∈N and a BAOs monomorphism
r : A → B which also preserves all of⋂j
θ(α(cj/x)) and⋃j
θ(α(cj/x))”
No, as the set Q := {Qθα}, where Qθ
α := θ(α(cj/x)) is notcountable.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Can we use the above result to obtain the following ?
“Given a realization A := (A,∩,∼, 1, fi )i∈N, there exist aCBAOs B := (B,∩,∼, gi )i∈N and a BAOs monomorphism
r : A → B which also preserves all of⋂j
θ(α(cj/x)) and⋃j
θ(α(cj/x))”
No, as the set Q := {Qθα}, where Qθ
α := θ(α(cj/x)) is notcountable.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Can we use the above result to obtain the following ?
“If 6` α, then there exists a CBAOs A := (A,∩,∼, 1, fi )i∈N andan assignment θ : PV → A such that θ(α) 6= 1”
Yes.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Can we use the above result to obtain the following ?
“If 6` α, then there exists a CBAOs A := (A,∩,∼, 1, fi )i∈N andan assignment θ : PV → A such that θ(α) 6= 1”
Yes.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Proposition
Consider the Lindenbaum algebra L and canonical assignmentθc : PV → F|≡ which maps p to [p]. Then there exists a CBAOsB := (B,∩,∼, 1, gi )i∈N and a BAOs monomorphism r : F|≡ → Bsuch that
r(⋂
Qθc
α ) =⋂
r(Qθc
α )
Proof.
Qθc:= {Qθc
α } is countable.
Take any enumeration of Q.
(C1)-(C3) are satisfied.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Proposition
Consider the Lindenbaum algebra L and canonical assignmentθc : PV → F|≡ which maps p to [p]. Then there exists a CBAOsB := (B,∩,∼, 1, gi )i∈N and a BAOs monomorphism r : F|≡ → Bsuch that
r(⋂
Qθc
α ) =⋂
r(Qθc
α )
Proof.
Qθc:= {Qθc
α } is countable.
Take any enumeration of Q.
(C1)-(C3) are satisfied.
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Completeness Theorem
If C α then ` α
Proof
Consider Lindenbaum algebra L.
Canonical assignment θc : PV → F|≡ where θc(p) = [p].
There exists a CBAOs B := (B,∩,∼, 1, gi )i∈N and a BAOsmonomorphism r : A → B as above.
Consider the assignment γ : PV → B such that γ(p) = r([p]).
γ(α) = r([α]).
If 6` α, then [α] 6= 1 and hence γ(α) = r([α]) 6= 1
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Completeness Theorem
If C α then ` α
Proof
Consider Lindenbaum algebra L.
Canonical assignment θc : PV → F|≡ where θc(p) = [p].
There exists a CBAOs B := (B,∩,∼, 1, gi )i∈N and a BAOsmonomorphism r : A → B as above.
Consider the assignment γ : PV → B such that γ(p) = r([p]).
γ(α) = r([α]).
If 6` α, then [α] 6= 1 and hence γ(α) = r([α]) 6= 1
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Completeness Theorem
If C α then ` α
Proof
Consider Lindenbaum algebra L.
Canonical assignment θc : PV → F|≡ where θc(p) = [p].
There exists a CBAOs B := (B,∩,∼, 1, gi )i∈N and a BAOsmonomorphism r : A → B as above.
Consider the assignment γ : PV → B such that γ(p) = r([p]).
γ(α) = r([α]).
If 6` α, then [α] 6= 1 and hence γ(α) = r([α]) 6= 1
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Completeness Theorem
If C α then ` α
Proof
Consider Lindenbaum algebra L.
Canonical assignment θc : PV → F|≡ where θc(p) = [p].
There exists a CBAOs B := (B,∩,∼, 1, gi )i∈N and a BAOsmonomorphism r : A → B as above.
Consider the assignment γ : PV → B such that γ(p) = r([p]).
γ(α) = r([α]).
If 6` α, then [α] 6= 1 and hence γ(α) = r([α]) 6= 1
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Completeness Theorem
If C α then ` α
Proof
Consider Lindenbaum algebra L.
Canonical assignment θc : PV → F|≡ where θc(p) = [p].
There exists a CBAOs B := (B,∩,∼, 1, gi )i∈N and a BAOsmonomorphism r : A → B as above.
Consider the assignment γ : PV → B such that γ(p) = r([p]).
γ(α) = r([α]).
If 6` α, then [α] 6= 1 and hence γ(α) = r([α]) 6= 1
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Completeness Theorem
If C α then ` α
Proof
Consider Lindenbaum algebra L.
Canonical assignment θc : PV → F|≡ where θc(p) = [p].
There exists a CBAOs B := (B,∩,∼, 1, gi )i∈N and a BAOsmonomorphism r : A → B as above.
Consider the assignment γ : PV → B such that γ(p) = r([p]).
γ(α) = r([α]).
If 6` α, then [α] 6= 1 and hence γ(α) = r([α]) 6= 1
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems
Completeness Theorem
Completeness Theorem
If C α then ` α
Proof
Consider Lindenbaum algebra L.
Canonical assignment θc : PV → F|≡ where θc(p) = [p].
There exists a CBAOs B := (B,∩,∼, 1, gi )i∈N and a BAOsmonomorphism r : A → B as above.
Consider the assignment γ : PV → B such that γ(p) = r([p]).
γ(α) = r([α]).
If 6` α, then [α] 6= 1 and hence γ(α) = r([α]) 6= 1
Mohua Banerjee, Md. Aquil Khan Algebraic Semantics for the Logic of Multiple-source Approximation Systems