A New Set of Unary and Binary Operators 
With A New Algebraic System
For Multiple-Valued Logic Systems:
The Algebra Of Priority 
(AOP)

1- INTRODUCTION

Background:   Binary logic is an area that deals with the representation of data with two values ‘0’ and ‘1’. The problem encountered with binary logic is the large number of bits that is needed to represent data. This problem is reflected at the hardware in two well-known problems: pinout problem and interconnection problem. The various efforts that were done to solve these two problems gave rise to the development of Multiple-valued Logic (MVL) field. Multiple-valued logic is an area that uses multiple values to represent data. The number of values is usually expected to be three or more. For example, in a four-valued system, MVL uses four values to represent data. If these values are to be numerical values, then 0,1,2, and 3 would be used. In this way, MVL solves (theoretically) the pinout problem and it simplifies circuit complexity of binary logic circuits. Motivation: However, MVL designs digital circuits using the traditional operators MIN, MAX, MV-NOT, and complementary operators. The problem encountered in this design, is the large number of traditional operators that is needed to build up a digital circuit. This large number increases complexity and interconnections of MVL circuits. The more operators a MVL circuit needs, the more it gets complex and its interconnections get even more complex. A solution to this problem is to increase its basic operators of design and not limit them to the traditional operators.  This approach will give rise to a new field called Multiple-Operational Logic (MOL), which uses multiple-operations from unary and binary operations to design digital circuits.  Thus, MOL is aimed on introducing, into logical systems, a variety of new operators that will make design more flexible than would be using just the MVL traditional operators.

Contributions: In just doing that, I developed a new set of unary and binary operators that will increase the number of basic operators and will make design more flexible than would be using just the traditional operators alone. The new unary operators are classified into two categories: conservative operators and orthogonal operators. The new binary operations are called prioritors. The number of these operators for a z-radix system is z²(z-1) for its orthogonal operators, z! for its conservative operators, and z! for its prioritors. The traditional operators we mentioned above are a subset of the new operators. From this point and on, we will refer to the sets of prioritors, conservative operators, and orthogonal operators by the term " AOP basic operators" or "the new operators". 

In 1938, Claude Shannon showed how the logical laws of Boolean algebra, founded by George Boole in 1849, could be used to synthesis digital circuits implemented by AND, OR, and NOT operators. Also, Post showed how the laws of his algebra, could be used to synthesis digital circuits implemented by MIN, MAX, MV-NOT, and complementary operators. Unfortunately, these algebras cannot be fully used with the new operators of AOP for all digital systems. Boolean algebra is perfect for the binary system but it does not work for other systems. Also, Post algebra is perfect for subsets of the new operators of AOP but not for all of them. Thus, it cannot be fully used to synthesis digital circuits implemented by the various combinations of prioritors, conservative operators, and orthogonal operators (see similarities and differences between AOP and Post). With no other choice is left, I developed a new algebraic system, called the Algebra of Priority (AOP), that can fully serve these new operators and be used to synthesis digital circuits implemented by the various combinations out of these new operators. Thus, AOP uses the new set of operators for circuits design and provides all the rules and procedures that lead to the design of any given digital circuit from this set of operators. The designs by the new operators, using AOP, are simpler and much more efficient than the designs obtained by the traditional operators of MVL systems. Since AOP works for large set of operators, it is a multi-operational algebra and since it works for any z-radix system, it is a multi-valued algebra. Thus, AOP is a multi-valued multi-operational algebra. 

In this paper, I solved two design problems using the traditional operators and using the new operators. In the first design, I used the traditional operators, by Post algebra, and obtained the ternary multiplication operation by sum-of-products as a composition of 9 binary operators (6 MIN, 3 MAX) and 8 unary operators and by product-of-sums as a composition of 15 binary operators (9 MIN, 6 MAX) and 14 unary operators. In the second design, I used the traditional operators, using Post algebra, and obtained the given ternary operation by sum-of-products as a composition of 16 binary operators and 12 unary operators and by product-of-sums as a composition of 20 binary operators and 16 unary operators.

For the same two problems, I obtained different designs using the new operators of AOP. For the first problem, I designed the multiplication circuit by 3 binary operators and two unary-operators. For the second example, I designed the circuit by 3 binary operators and two unary operators. 

When we compare these designs based on the traditional operators and based on the new operators of AOP we find the followings. In the first example, for the sum-of-products, the use of the new operators cut the number of binary operators by 66.66% and cut the number of unary operators by 75% and for the product-of-sums, the use of the new operators cut the number of binary operators by 80% and cut the number of unary operators by 85.7%. Thus, these cuts show that the new operators can really reduce circuit complexity of MVL circuits.

The algebra associated with these new operators was developed based on the priority concept from which its name was derived (The Algebra of Priority AOP). AOP is based on "z" logical values, z! binary operators called “prioritors”, z! unary operators called “conservative operators” and z²(z-1) unary operators called “orthogonal operators”.

It turned out that AOP is a very rich algebraic system in terms of its concepts, theorems and operators. Its results agree with the results obtained by Boolean algebra and by Post algebra and at the same time it expands the concepts and theorems of Post algebra even though it was developed totally from concepts that are completely independent of Post and Boolean concepts.  For examples:

1. AOP extends the representations of MVL functions from two representations to z! representations using its orthogonal theorems I & II. Its orthogonal theorem-II provides a much efficient representations of MVL functions for hardware implementation than Post representations because it uses a fewer number of binary operations.
   

2. AOP extends the expansion of MVL functions from two expansions to z! expansions using its expansions theorems I&II.
   

3. AOP extends DeMorgan's laws by its Image-Scaling theorem.AOP extends DeMorgan's laws by its Image-Scaling theorem. In Boolean and Post algebras we can break the image of a binary operation by DeMorgan’s laws only if we use the NOT and MV-NOT operators. However, under the Image-Scaling theorem of AOP, we can break the image of a binary operation under all conservative unary operators
   

4. AOP extends Kleene's laws by its absorption theorem III.
   

5. AOP extends the current duality theory by its degeneracy theory[1].  Instead of saying an operator has a dual we say an operator has “descendants” or “degenerate operators”. The number of descendants for an operator depends on the system radix and on the operator itself.  For prioritors, the number of descendants depends on the system radix only and it is equal to z!.  For example, it is 2 for the binary system, 6 for the ternary system and 24 for the quaternary system.
   
   The degeneracy theory of AOP agrees with the results of Boolean algebra since the number of descendants is always ‘2’.  However, it differs from Post algebra for none binary radix systems. For example, the MIN under AOP has six descendants rather than two under Post in ternary system and it has 24 descendants rather two under Post in the quaternary system.
   

6. AOP extends duality to be applied for other binary operators not just for AND (MIN) or OR (MAX). For example, AOP extends the “duality” concept in Boolean algebra to cover the XOR, and NXOR and others.
   

7. AOP extends the duality concept of equations by its degeneracy theory[2].  Instead of saying an equation has a dual we say an equation has “descendants”, “degenerate equations”, or “child-equations”.  The number of child-equations for a parent-equation depends on the system radix and on the equation itself.  For priority equations[3], the number of descendants depends on the system radix only and it is equal to z!.  For example, it is 2 for the binary system, 6 for the ternary system and 24 for the quaternary system.
   
   The degeneracy theory of AOP agrees with the results of Boolean algebra since the number of descendants is always ‘2’.  However, it differs from Post algebra for none binary radix systems.  For example, distribution equation in ternary system has six descendants in AOP rather than two in Post algebra.
   

8.