2- THE UNIFORM DEGENERACY IN AOP

2.1  Definitions and Notations

The following AOP notation will be used in the following sections.  In mathematics, we usually consider variables to be the only parameters of functions.  Thus, we specify these variables in the function heading.  For example, the f(x) notation means ‘x’ is a variable parameter and the f(x,y) means ‘x’ and ‘y’ are variable parameters.  Because AOP is a multi-operational algebra, we extend the notation to specify variables, operators and constants as parameters in the function heading and at the same time use sets notations to specify such parameters.  For example, assume we have the following function in Boolean algebra G(A,B,C)= (A+B)*(A+B)+A*C+(1+C)*(B+0). In AOP, we write this as G(X,a,C) where X={A,B,C},a={+,*}, C={1,0}. Thus, in G(x,a,c) notation (1) ‘x’ is the set of all variables used in the function. (2) ‘a’ is the set of all prioritors used in the function. (3) ‘C’ is the set of all constants used in the function. (4) ‘G’ A function whose range is determined by a set of variables 'x' and a set of prioritors 'a' and a set of constants 'c'.

A function is said to be a priority function if and only if all of its binary operators are prioritors.  

Definition 2.1.2 Priority Equation: An equation is said to be a priority equation if and only if all of its binary operators are prioritors.

2.2 Uniform Degeneracy of Prioritors

            Based on the "duality" concept, in Boolean and in Post algebras, we say that the dual of MIN is MAX and the dual of AND is OR and vise versa. AOP extends the duality concept into a broader scope under the concept of "uniform degeneracy[1]". The uniform degeneracy of prioritors is defined in Definition 2.2.1

Definition 2.2.1: The uniform degeneracy of a prioritor is defined as the image of its priority-assignment under a conservative unary operator, say f, and is denoted by "a^o‎fff". Mathematically,
                                 
where a¯^f is the image of the priority-assignment of a under “f” NOT the image of the function table of a under “f” and  “o‎fff ” is called the uniform degeneracy operator.

For example, let a=Q₇=4S1023 and f=4S1023. By Definition-2.2.1, a^offf = 4S1023_¯^4S1023= 4S2301=Q_H. Thus, we say that the uniform degeneracy of a=Q₇ is Q_H.

Table-1 lists all the ^“a^offf” uniform degeneracy operations of all prioritors under all conservative unary operators in the quaternary, ternary and binary systems. To find the uniform degeneracy of a prioritor under a conservative unary operator using Table-1 , locate the row that contains the prioritor and the column that contains the conservative unary operator, which is listed in a vertical direction. The intersection of the column and row is the prioritor number that represents the uniform degeneracy. If the number is in the quaternary system, then add the “Q” prefix; in the ternary system add the “T” prefix, in the binary system add the “B” prefix. Finally, use Table-1 to determine the function table of the prioritor found.

The uniform degeneracy under f=4S3021 of a=Q_H prioritor is Q_H^offf=Q₅ (The intersection is “5” and the prefix is “Q”). The uniform degeneracy under f=3S021 of a=T₃ prioritor is T₃^offf=T₆ (The intersection is “6” and the prefix is “T”). The uniform degeneracy under f=2S01 of a=B₁ prioritor is B₁^offf=B₂. (The intersections “2” and the prefix is “B”).

The uniform degeneracy of a=Q₁ (MIN) under f=4S2301 is Q₁^offf=Q₈, of a=Q₁ (MIN) under f=4S0123 is Q₁^offf=Q_O, of a=Q_O (MAX) under f=4S0123 is Q_O^offf=Q₁ and of a=Q₁ (MIN) under f=4S1032 is Q₁^offf=Q_H.

Example-2.2.4 On Uniform Degeneracy Table: In the binary system, under f=2S01, the uniform degeneracy of the AND operator (B₁) is the OR operator (B₂) and the uniform degeneracy of the OR operator (B₂) is the AND operator (B₁). That is AND^offf=OR and OR^offf=AND.

          The duality theory is a special case of the uniform degeneracy theory of AOP. The dual operation in Boolean and Post algebras is the uniform degeneracy under the D conservative operator. For example, in the quaternary system f=D=4S0123 and MIN=Q₁=4S0123, thus the dual of MIN is MIN^offf=4S0123^¾4S0123= 4S3210=Q_O=MAX. In the binary system f=D=2S01, thus the dual of AND is AND^offf=2S01^{¾ 2S01}=2S10=OR. 

2.4 Uniform Degeneracy of Priority Functions

The duality of functions in Boolean and Post algebras is extended by AOP under the concept of "uniform degeneracy of functions" as defined by Definition 2.4.1.

Definition 2.4.1: The uniform degeneracy of a priority function, say G(x,a,c) , under a conservative unary operator, say ‘f’, is obtained by taking the uniform degeneracy of each prioritor in the function and by taking the image of each constant using the ‘f’ conservative operator where the variables of the function remain unchanged. Mathematically
                                   

The expression “G(x,a,C)^offf” is read as the uniform degeneracy of the function G. According to Definition-2.4.1, we have to take the uniform degeneracy of each prioritor and take the image of each constant and leave all variables untouched. The statement is translated symbolically as G(X,a^offf,C^─f). This means that the "^offf " uniform degeneracy is to operate on all the prioritors of the set 'a' and the '^─f' image operator is to operate on all the constants of the set 'C'. 

For example, let G(X,a,C)= (A+B)*(A+B)+A*C+(1+C)*(B+0) where x={A,B,C},a={+,*}, and C={1,0}.  Using the uniform degeneracy definition, G(A,B,C)^offf=G(X,a^offf,C^─f)= G({A,B,C},{+,*}^offf,{1,0}^─f)= G({A,B,C},{+^offf,*^offf},{1^─f,0^─f})= (A+^offf B)* ^offf (A+^offf B)+ ^offf A* ^offf C+^offf (1^─f +^offf C)* ^offf (B+^offf 0^─f

Example-1: Let a=Q₁ (MIN) in the quaternary system and let g(X,a,C)=Aa(2a*B). The set of all variables in the function is X={A, B}, the set of all constants in the function is C={2}, and the set of all prioritors is a={a, a*}. The “offf” uniform degeneracy of g(X,a,C) is given by g(X,a,C)^offf=A a^offf(2^─f a*^offf B). In this example, we took the uniform degeneracy of each prioritor and took the image of the constant '2'. If we let f=4S0123, we get g(X,a,C)^offf=A MAX (1 MIN B) or g(X,a,C)^offf=A+(1·B)=AÚ (1ÙB) using Boolean and Post notations.

Let g(X,a,C)=Aa(Ba*C). The “offf” uniform degeneracy of g(X,a,C) is g(X,a,C)^offf=A a^offf(Ba*^offf C).  This example dose not have constants.

Example-3: Let a=Q₁=MIN in the quaternary system and let g(X,a,C)=(Aa(2a*B)a(Ca*3))a*(0aC). The “offf” uniform degeneracy of g(X,a,C) is given by  g(X,a,C)^offf=(Aa^offf(2^─fofff(Ca*^offf 3^─fofffC). 
If f=4S1302, we obtain g(X,a,C)^offf=(AQ_E(3Q_BB)Q_E(CQ_B1))Q_B(2Q_EC).  Note that a^offf= Q₁^offf=4S0123^─4S1302= 4S2031=Q_E and a*^offf=Q_O^offf=MAX^offf=4S3210^─4S1302=4S1302=Q_B. 

Theorem-2.4.1 shows another important theorem in AOP.  This theorem gives us another way of obtaining the uniform degeneracy without using Definition-2.4.1. 

Example-4: Let g(X,a,C)=Aa(Ba*C).
Let g(X,a,C)=Aa(2a*C). 

2.5 Uniform Degeneracy of Priority Equations

The “duality” theory in Boolean and Post algebras shows us how to obtain the dual of equations. In Boolean algebra, the dual of an equation is obtained by substituting an AND for OR, an OR for AND, 0 for 1 and 1 for 0. In Post algebra, the dual of an equation is obtained by substituting a MIN for MAX, a MAX for MIN, 0 for ‘u’ and ‘u’ for 0 [5]. According to this theory, there is a dual for every equation in Boolean and Post algebras. AOP extends the duality theory into a new theory (Theorem-2.5.1) called “Equations Uniform Degeneracy Theorem”. This new theory tells us that each MVL equation has z! degenerate forms. It simply states, "give me a MVL equation and I will generate for you z! equations". For example, we can generate 24 equations from a MVL equation in the quaternary system by AOP (see Example-2.5.3) but on the other hand we can generate only two equations by Post algebra. AOP in this example extends the number of equations to 24 equations [1].

Example-2.5.1 In Boolean algebra the dual of A+(B+1)=1 is A·(B·0)=0. In AOP, the uniform degeneracy of A+(B+1)=1 under the unary f=2S01 (NOT) is “A+^offf(B+^offf 1^─f)=1^─f ”. Using Table-1, +^offf=B₂^offf=B₁=AND=· and 1^─f =0, we get A·(B·0)=0, which is the same result obtained by the dual operation.

In the quaternary system, let a=Q₁ =MIN. The uniform degeneracy of the 2a (Ba*C)=(2aB) a*(2aC) MVL equation is given by 2^─f a^offf(Ba*^offfC)=( 2^─f a^offf B) a*^offf(2^─f a^offfC). If f=4S2301, then we obtain 3Q₈(BQ_HC)=( 3Q₈B)Q_H(3Q₈C). Note that the Q₈ and Q_H form a STAS system that satisfies the distribution theorem. Also, note that a^offf=Q₁^offf=4S0123^─4S2301=4S1032=Q₈ and a*^offf=Q_O^offf= MAX^offf=4S3210^─4S2301=4S2301=Q_H.

Given this equation "AÙ(BÚC)=(AÙB)Ú(AÙC)" in Post notation, give all the 24 degenerate equations in the quaternary system where Ù=MIN and Ú=MAX. Using AOP notation, we can express AÙ(BÚC)=(AÙB)Ú(AÙC) as Aa1 (B a_OC)=(A a₁B) a_O(A a₁C) where Ù=MIN=Q₁ , Ú=MAX= Q_O. The 24 degenerate equations are listed in Table-5. The index of each a represents the prioritor number as listed in Table-1.  For example, a₇=Q₇, a_M=Q_M …etc. Note that the 24 uniform degeneracy operators are obtained directly from Table-3 by picking all the prioritors in a row. For example, for the z! degeneracy forms of Q₁ we pick its row in Table-3 which reads "OIMCGA-NHK6E4-LBJ582-F9D371" and the same for Q_O which reads "123456-789ABC-DEFGHI-JKLMNO".  Note that the first equation is the dual of the given equation in Post algebra.  This shows that the dual operation in Boolean and Post algebras is equivalent to the uniform degeneracy under the “D” operator.  Also, note that equation No 24 is equal to the given equation, because the uniform degeneracy of any equation under the “Ñ” operator is equal to the equation itself.