|
|
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Table-8: Derivation of Boolean Algebra | ||
|---|---|---|
 Theorem
|
Case 1 a=·, a*=+, a#= ¾ |
Case 2 |
|
Inferiority |
1·A=A |
0+A=A |
|
Superiority |
0·A=0 |
1+A=1 |
|
Distribution Theorem |
A·(B+C)=(A·B)+(A·C) |
A+(B·C)=(A+B)·(A+C) |
|
Idempotence Theorem |
A·A=A |
A+A=A |
|
Association Theorem |
A·(B·C)=(A·B)·C |
A+(B+C)=(A+B)+C |
|
Commutation Theorem |
A·B=B·A |
A+B=B+A |
|
Absorption
Theorem-I |
A·(A+B)=A |
A+(A·B)=A |
|
Exclusion Theorems |
A¯·( A+B)=A·B |
A¯+( A ·B)=A+B |
|
Static Theorem |
A¯ ·A=0 |
A ¯+A=1 |
|
Uniform Degeneracy |
Dual ·=+ |
Dual +=· |
|
Image-Scaling |
(A·B)¯= A¯ + B¯ |
(A+B)¯ = A¯ · B¯ |
|
Expansion |
f(X)=(X¯·f(0))+(X·f(1)) |
f(X)=(X¯+f(0))·(X +f(1)) |
Derivation
of Post AlgebraPost algebra as defined by Muzio p.93 in [8] can be derived from AOP by:
Replacing the a prioritor by the MIN and the a* prioritor by the MAX operator or vise versa.
Using MIN─L=0 and MAX─ L=z-1, MIN─Ú=z-1 and MAX─Ú=0
Replacing Dm0(z-1) by Cm and Dm(z-1)0 by Jm, where Jm(X)={0 if X=m, z-1 otherwise} Cm(X)={z-1 if X=m, 0 otherwise} .
Replacing any conservative unary operation by
the MV-NOT operator.
|
|
||
|
|
Case 1 |
Case
2 |
|
Inferiority
Theorem |
(z-1)·A=A |
0+A=A |
|
Superiority
Theorem |
0·A=0 |
(z-1)+A=(z-1) |
|
Distribution
Theorem |
A· (B+C)=(A·B)+(A·C) |
A+(B·C)=(A+B)·(A+C) |
|
Idempotence
Theorem |
A·A=A |
A+A=A |
|
Association
Theorem |
A·(B·C)=(A·B)·C
|
A+(B+C)=(A+B)+C |
|
Commutation
Theorem |
A·B=B·A
|
A+B=B+A |
|
Absorption
Theorem-I |
A·(A+B)=A |
A+(A·B)=A |
|
Image-Scaling
Theorem |
(A·B)¯=
A¯+ B¯ |
(A+B)¯= A¯· B¯ |
|
Uniform
Degeneracy |
Dual
·=+ |
Dual
+=· |
|
Exclusion
Theorem |
A·(C0(A)+B)=A·B |
A+(Jz-1(A)·B)=A+B |
|
Variable
Expansion |
m=z-1 |
m=z-1 |
|
Static
Theorem |
A·C0(A)=0
|
A+Jz-1(A)=z-1 |
The table listed below shows the derivations steps for the static, quasi-static, and variable expansion theorems.
|
|
Explanation |
|
|
Letting
B=a─Ú
and using the ‘Cm(x)’ definition we replace
the orthogonal operator by Ca─L(A) |
|
AaCa─L(A)=a¾
L |
Letting a=· and using ·─L=0 |
|
A·C0(A)=0 |
Q.E.D. |
|
|
Explanation |
|
|
Letting
B=a─Ú
and using the ‘Jm(x)’ definition we replace
the orthogonal operator by Ja─L(A) |
|
AaJa─L(A)=a¾L |
Letting a=+ and using +¾ L=z-1 |
|
A+Jz-1(A)=z-1 |
Q.E.D. |
|
Explanation |
|
Aa(A─D a─L
a─L
B |
Letting
C=a─Ú
and using the ‘Cm(x)’ definition we replace
the sione orthogonal operator by Ca─L(A) |
|
Aa(
Ca─L(A)
a*
B) = AaB |
Letting a=· and using ·¾ L=0; |
|
A·(C0(A)+B)=A·B |
Q.E.D. |
|
|
Explanation |
|
Aa(A─D a─L
a─L
Ba*
B) = AaB |
Letting
C=a──Ú
and using the ‘Jm(x)’ definition we replace
the orthogonal operator by Ja─L(A) |
|
Aa(
Ja─L(A)
a*
B) = AaB |
Letting
a=+
and using +¾L=z-1
+*=· |
|
A+
(Jz-1(A) ·B)=A+B |
Q.E.D. |
|
Explanation
|
|
a*: (z-1) |
From the expansion theorem-I (theorem-16). Letting a=·
and using ·¾
L=0;
·─Ú=(z-1);
·*=+
and using the ‘Cm(x)’ definition
|
|
z-1 |
Cm(x)=
x¾
Dma¾
La
─Ú= x¾
Dm0(z-1) Q.E.D. |
|
|
Explanation
|
|
z-1 |
Letting a=+ and using +¾L=z-1;
+¾V=0;
+*=·
and using the Jm(x) definition and follow the
steps in case-1.
Q.E.D. |
Deriving Kleenean Algebra
As Defined By [12]
|
Table-5:
Derivation of Kleenean Algebra from AOP |
|||
|
No |
Theorem
Name under AOP |
Case-1
|
Case-2 |
|
1 |
Inferiority
Theorem |
1·A=A |
0ÚA=A |
|
2 |
Superiority
Theorem |
0·A=0 |
1ÚA=1 |
|
3
|
Distribution
Theorem |
A·(BÚC)=(A·B) Ú(A·C) |
AÚ(B·C)=(AÚB)·(AÚC) |
|
4 |
Idempotence
Theorem |
A·A=A |
AÚA=A |
|
5 |
Association
Theorem |
A·(B·C)=(A·B)·C |
AÚ(BÚC)=(AÚB) ÚC |
|
6 |
Commutation
Theorem |
A·B=B·A |
AÚB=BÚA |
|
7 |
Absorption
Theorem-I |
A·(AÚB)=A |
AÚ(A·B)=A |
|
8 |
Absorption
Theorem-III |
(A¯·
A )·(B¯Ú
B)=(A¯·A)
|
(A¯·A
)Ú(B¯Ú
B) =( B¯Ú
B) |
|
9 |
Costar
Theorem |
A==A |
A==A |
|
10 |
Uniform Degeneracy |
Dual ·=Ú |
Dual Ú=· |
|
11 |
Image-Scaling Theorem |
(A·B)¯= A¯ Ú B¯ |
(AÚB)¯ = A¯ · B¯ |
Deriving
the Kleene's Laws from the Absorption Theorem III
The Kleene's laws are: (1) (A·A¯)·(BÚB¯)=
(A·A¯)
and (2) (A·A¯)Ú
(BÚB¯)=
(BÚB¯).
The '·'
operator in this law represents the MIN operator and 'Ú'
represents the MAX operator. The DnDel Ñ
prioritor in AOP corresponds to the MAX operator and the UpDel prioritor D
corresponds to the MIN operator.
Kleene's laws
are: (1) (A·A¯)·(BÚB¯)=
(A·A¯)
and (2) (A·A¯)Ú
(BÚB¯)=
(BÚB¯).
The '·'
operator in this law represents the MIN operator and 'Ú'
represents the MAX operator. The DnDel prioritor “Ñ”
in AOP corresponds to the MAX operator and the UpDel prioritor “D”
corresponds to the MIN operator.
|
|
|
|
(AaA¾ a#)a(Ba*B¾ a#)=(AaA¾ a#) |
Absorption
theorem-III |
|
(A·A¾ · #)·(B·*B¾ · #)=(A·A¾ · #) |
Letting a=· |
|
(A·A¾ · #)·(BÚB¾ · #)=(A·A¾ · #) |
Since ·=D, then ·*=D*=Ñ=Ú |
|
(A·A¾ D)·(BÚB¾ D)=(A·A¾ D) |
Since ·=D, then |
|
(A· A¯ )·(BÚ B¯)=(A·A¯) |
Since the bar '¯'
in Boolean, Post and Kleenean algebras corresponds to D
operator in AOP. Q.E.D. |
|
|
|
|
(AaA¾ a¾ )a(Ba*B¾ a#)=(AaA¾ a#) |
Absorption
theorem-III |
|
(AÚA¾Ú #)Ú (BÚ*B¾Ú #)=(AÚA¾Ú #) |
Letting a=Ú |
|
(AÚA¾Ú #)Ú (B·B¾Ú #)=(AÚA¾Ú #) |
Since Ú=Ñ, then Ú*=Ñ*=D=· |
|
(AÚA¾ D)Ú (B·B¾ D)=(AÚA¾ D) |
Since Ú=D, then Ú#=Ñ#= |
|
(AÚ A¯)Ú (B· B¯ )=(AÚ A¯ ) |
Since the bar '¯'
in Boolean, Post and Kleenean algebras corresponds to D
operator in AOP. |
|
(B· B¯ )Ú(AÚ A¯) =(AÚ A¯ ) |
Since Ú
is a commutative operator we re-order the terms on the left side |
|
(A· A¯ )Ú(BÚ B¯) =( BÚ B¯) |
By letting A=B and
B=A which does not change the equation. This is Kleene's second law. Q.E.D. |
| |||||||||
