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Boolean algebra uses the AND and OR as its binary operators. Post algebra uses the MIN and MAX as its binary operators. AOP uses a more generalized set of binary operators called “prioritors” as defined in Definition 4.1. This makes AOP a multi-operational algebra. The AND, OR, MIN, MAX are a subset of the prioritors of AOP. The number of prioritors in a z-radix digital system is equal to z! as we will prove that later in this section. The number of prioritors is 2 in the binary digital system, 6 in the ternary digital system and 24 in the quaternary digital system. Table-2
Before we define prioritors in AOP, we will reanalyze mathematically the
priority-assignment represented by the priority s-code. The priority-assignment
represents a one-to-one function that maps the priority set to the logic-set. The domain of this function is the priority
set and the range of the function is the logic-set. Since xÎpriority set and f(x)Îlogic-set, then the function notation f(X) is read as the
digit that has the ‘x’ priority. For example, if f=4S3021, then f(1)
is the digit that has the "1" priority which is 2. Similarly, f(0)=1;
f(3)=3; f(2)=0. Since one-to-one functions are called unary conservative
operators by AOP, then the X─f notation is read as the digit that has the ‘x’ priority or the image of
"X" under "f". For example, if f=4S3021, then 1─f is the digit with a priority of "1" which is 2 or the image
of 1 under "f" which is 2. Similarly, 0─f =1; 3─f=3; 2─f=0.
In AOP, we are interested in the priority of a given digit. Since the
priority of a digit is the position of that digit in the priority s-code, then
we are interested in the operator which gives the priority of that digit as an
image of the digit itself. That is, if f=4S1023 and f(A)=B, then we want the
operator, say "y", that gives y(B)=A or Y(f(A))=A. Mathematically,
"y" is called the inverse function of "f". In AOP, we call
"y" the conservative inverse operator of "f"
and is denoted by "f -". For example, the
inverse of f=4S1023 is f - =4S0132. Using the inverse
operation, we can find the priority of any digit from the inverse of the
priority-assignment. For example, if y=4S1023 then y -=4S0132 and the
priority of 0 is 0─y- =2; of 1 is 1─y-=3; of 2 is 2─y-=1; of 3 is 3─y-=0. Table-1 AOP defines a prioritor as "a
processing system that defines distinct priorities for all the logical
values of its inputs (events) by its priority-assignment and
its output is equal to the input logical-value with the highest priority".
Mathematically, the prioritors of AOP are defined by Definition 4.1 as two-event
prioritors where we use the Greek alphabet "a" to refer to prioritors in general. Table-1
Definition-4.1 states that if "A"
has a priority higher than or equal to the priority of "B" then the
result is equal to "A"; if "B" has a priority higher than or
equal to the priority of "A" then the result is equal to
"B". In another words, the result of the "a" prioritor is equal to the variable
value with the highest priority. Table-1
The "a" symbol in AaB represents a binary operator, while in the
unary image operation, A─a-, it represents the prioritor
priority-assignment. For example,
in the quaternary system, let a=Q7. From Table-1 Theorem-4.1 Number of Prioritors: The number of prioritors in a z-radix digital system is equal to z!. Proof: By the priority principle all
priorities must be distinct. Thus the assignments of priorities to the
logical-values of the logic-set or vise versa is a one-to-one mapping process.
For the "0" logical-value we can assign "z" priorities, for
the "1" logical-value we can assign "z-1" priorities, for
the "2" logical-value we can assign "z-2" priorities and for
the ith logical-value we can assign
"z-i" priorities. Since each priority assignment to each logical-value
is independent from the other assignments, then using the counting principle
[14]p.3, there are z(z-1)(z-2)(z-3) ŸŸŸ 2*1=z! distinct
ways of assigning priorities to all the z-distinct logical-values. Hence, there
are z! distinct prioritors. Q.E.D. See Table-2
Since a prioritor represents a binary operation then it has two
parameters, 'A' and 'B', written in the form of AaB where ' a' is the prioritor's symbol. Using a=Q1 in Table-1
At the hardware level, prioritors are a general representation of digital
gates. They can pass and block data flow. The signal with the least
priority, which is called the prioritor infimum signal, is used to pass
data out of the prioritor and the signal with the highest priority, which is
called the prioritor supremum signal, is used to block the data flow. Definition-5.1.3
'Ñ' (Down-Del) Prioritor: The Down-Del prioirtor , denoted by "Ñ", priority assignment is given by the s-code as Ñ=zS(z-1)•••3210.
The
down-del prioritor is a prioritor with a priority assignment given by
Ñ=zS(z-1)•••3210. It corresponds to the
MAX operator in Post algebras and corresponds to the OR operator in Boolean algebra. For
example, the Down-Del prioritor is Ñ=2S10 in the binary system, Ñ=3S210 in the ternary system and Ñ=4S3210 in the quaternary system. Definition-5.1.4
'D' (Up-Del) prioritor: The
Up-Del prioritor, denoted by "D", priority assignment is given by D=zS0123•••(z-1).
The up-del prioritor is a prioritor with a priority assignment given by D=zS0123•••(z-1). It corresponds to the
MIN operator in Post algebras and to the AND operator in Boolean algebra. For
example, the Up-Del prioritor is D=2S01 in the binary system, D=3S012 in the ternary system and D=4S0123 in the quaternary system. |
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