EXPANSIONS OF MVL FUNCTIONS In AOP

            AOP extends and enhances the expansions of MVL functions by two theorems called “Expansion Theorem-I” and “Expansion Theorem-II”. Both theorems extend the number of expansions of MVL functions to z!. For example, a MVL function in the quaternary system can be expanded by 24 expansions. This gives designers more alternate choices of expanding MVL functions. The enhancement in expansion theorem-II is achieved by reducing the number of prioritors by ‘z’.

Theorem 4.1 Expansion Theorem-I:- A MVL function of one variable in a z-radix digital system can be expanded by using the (a, a*) STAS systems and the orthogonal operators as

  Theorem 4.2 Expansion Theorem-II: A MVL function of one variable in a z-radix digital system can be expanded by using the (a, a*) STAS systems and the orthogonal operators as:

Due to the limited space, Example-4.2 shows only four expansions of a quaternary variable out of the 24 expansions by the orthogonal theorem-I. The first expansion is the sum-of-products by Post algebra. Also Example-4.3 shows four expansions of a quaternary variable out of the 24 expansions by the expansion theorem-II.  

Example-4.1 On Variables Expansion I & II: Using the expansion theorem for f(X)=X, any variable in AOP can be expanded by using the (a,a*) STAS systems and the orthogonal operators as:  

The variable expansion in Post algebra defined in axiom-3 by Epstein in [5] is a special case of the expansion in (I) when a=MIN and a*=MAX.  

Examples:

Example-4.2 On Variable Expansion-I: Using the expansion theorem-I, we can expand a quaternary variable by the following expansions:

Example-4.3: On Variable Expansion-II: Using the expansion theorem-II, we can expand a quaternary variable by the following expansions: