|
AOP
Unary Operators and Operations
AOP unary operators are operations that operate on variables. In a z-radix digital system, there are zz unary operations. The set of unary operators is partitioned into "m" partitions, where "m" is the number of integer partitions of system radix
'z'. AOP uses two out of these
"m" partitions in a z-radix digital system. The first partition is called the conservative partition and its operators are called "conservative operators", the second partition is called the orthogonal partition and its operators are called "orthogonal operators".
In this section, we will describe only these two types of operators.
Unary operators are one-variable functions that map the logic-set into itself. Because we will use unary functions as unary operators, we modified the "f(x)" functional notation to the
"x─f
" operational notation as defined by Definition 3.1.
Definition 3.1 Unary Image Operator: The unary image operator, denoted by ─f, is defined as x─f =f(x)
where x is a parameter and f is a unary operator.
In AOP, we identify unary operators by the unary s-code. The s-code lists the function table of a unary operator in a string of digits starting from right to left prefixed with 'zS' where 'z' is the system
radix and 'S' is a character stands for labeling the code as a system-code. Figure-1 shows the format of the unary s-code using the 4S1023-unary
operator
According to Definition 3.1, the unary operator must be shown to the right of the image ─
operator. Using f=2S01 (NOT) operator in the binary system, the followings are unary image operations 0─f
=1, 1─f=0. Using f=4S1023 0─f=3, 1─f=2, 2─f=0, 3─f=1; f=4S1122 0─f=2, 1─f=2, 2─f=1, 3─f=1, f=4S3023 0─f=3, 1─f=2, 2─f=0, 3─f=3.
At some point in time, in MVL digital systems we have to take the image of a data set and then retrieve this data at another point in time. Such cases are seen in data encryption at the hardware and software levels and in data storage devices like MVL
flip-flops. Boolean algebra uses the NOT operator to convert and reconvert data. Post algebra uses the MV-NOT operator. AOP uses a more generalized set of operators to convert and reconvert data, which are called conservative operators as defined by Definition 3.2. AOP uses this name, because these operators preserve data and the converted data can be
retrieved without any loss.
Definition 3.2 Unary Conservative Operators: A conservative unary operator is an operator that is represented by a one-to-one function that maps the logic-set into itself.
The number of conservative operators in a z-radix digital system is equal to z! [14]p.172.
Table-1 shows a list of all conservative operators in the binary, ternary and quaternary systems. The table lists
conservative unary operators using the unary s-code under the "a" column. The MSD is called the most significant digit in the
s-code and is defined as MSD=(z-1)─f. The LSD is called the least significant digit in the s-code and is
defined as MSD=0─f.
Definition-3.3 'Ñ'
(Down-Del) Operator: The Down-Del operator, denoted by "Ñ",
is given by the s-code as Ñ=zS(z-1)3210.
The Down-Del operator is a unary operator where the image of a variable "A" under it is always equal to the variable itself. That is, A─Ñ =A (this property is called the identity property of the Down-Del operator). For example, the Down-Del operator is Ñ=2S10 in the binary system, Ñ=3S210
in the ternary system and Ñ=4S3210 in the quaternary system.
Definition-3.4 'D'
(Up-Del) Operator: The Up-Del operator, denoted by "D", is given
by D=zS0123(z-1).
The Up-Del operator is a unary operator where the image of a variable "A" under it is always equal to Z-A-1. That is, A─D=Z-A-1.
This operator corresponds to the MV-NOT or complement operator in Post algebra and for NOT operator in Boolean algebra. For example, the Up-Del operator is D=2S01
in the binary system, D=3S012 in the ternary system and D=4S0123
in the quaternary system.
In AOP, data retrieval is done by the use of unary operators called the inverse unary operators as defined by Definition-3.5.
Definition-3.5 Inverse Operator: If "y" is a conservative unary operator, then y- is the inverse of "y" if and only if (A─y
)─ = (A─y-)─y
=Ñ.
Where Ñ is called the Down-Del unary operator and is defined by the unary s-code as Ñ=zS(z-1)...3210 (see
Definition 3.3) and - is the inverse operation operator. If y=y- then 'y' is called a self-inverse operator. At the hardware level in AOP, conservative operators are called converters and self-inverse
operators are called inverters. Table-1 a"
and "a-" columns.
Post algebra uses the generalized complementation operators C0, ... Cn-1 where "n" is the order of Post algebra [4]. Boolean algebra uses the NOT complement operator. AOP
uses a more generalized set of unary operators called the orthogonal operators as defined in Definition 3.6. The generalized complementation operators in Post algebra and the NOT operator in Boolean algebra are a subset of the orthogonal operators of AOP.
Definition 3.6 Unary Orthogonal Operator: The unary orthogonal operator in AOP is defined as x─Dabc={
c if x=a and b otherwise}.
The c is called the active-state digit, b is called the inactive-state digit, a is called the activating-digit, x is a parameter and ─D
is the orthogonal operator symbol. The image of x under a unary orthogonal operator is equal to the active-state digit when x=a and is equal to the inactive state digit when x is not equal to a. For example, 0─D301=0,
1─D301=0, 2─D301=0,
3─D301=1.
The number of unary orthogonal
operators in a z-radix digital system is given by j(z)=zē(z-1).
Table-1 shows a list of all orthogonal operators of AOP in
the quaternary, ternary and binary systems using the W-code whose format is shown in Figure-2.
There are two major unary operations in AOP: the star operation and the costar operation.
Table-1 lists the star under the a* column.
Definition-4.3 Star Operation: The star operation of a unary operator is obtained by flipping the function table in the unary s-code so that the digit at the ith
position becomes at the "(z-i-1)th" position and it is denoted by 'f*' and is read as the star of a. The image under f* is given by xū
f*= (z-1-x) ū f.
Example-4.1 On The Star Operation: In the binary system, for f=2S01 f*=2S10; f=2S10 f*=2S01. In the ternary system, for f=3S012 f*=3S210; f=3S021 f*=3S120; In the quaternary system, for f=4S0123, f*=4S3210; f=4S3210, f*=4S0123.
Another operation which is related to the star operation is the costar operation. The "costar" operation generates a unary operator, say 'y', from a unary operator, say 'u', such that u─y=u*.
Table-1 lists the costar under the a# column of each conservative operator.
Definition-4.4 The Costar Operation: The costar operation, denoted by #, is defined as f#=f-─
f* where
f is a conservative unary operator.
Example-4.2 On The Costar Operation: Let f=4S3021=QK.
The f# is obtained by taking the image of its inverse by its star. Using
Table-1, f-=4S3102 and f*=4S1203=Q9 . Thus, the costar of a
is f#=f- ─f*=4S3102─4S1203=
4S1032=Q8.
| 
