Arithmetic Transform of Boolean Functions

Abstract

In many applications where logic functions need to be analyzed it can be useful if we transform Boolean (or switching) functions to arithmetic functions. Such arithmetic transformations can give us new insight into solving some interesting problems. For example, the transformed functions can be easily evaluated (simulated) on integers or real numbers. Through such arithmetic simulation we can probabilistically verify a pair of functions with much more confidence than two-valued Boolean simulation. The arithmetic transform of any Boolean function can be easily computed from its BDD. To help evaluate a Boolean function on non-binary inputs, and to represent multi-variable linear polynomials with integer coefficients, a BDD like data structure snDD can be used; for many arithmetic expressions, snDDs are a very compact representation. The error in such probabilistic verification of property of a function is quantifiable and extremely low. Also, the procedures are computationally very efficient. Using a real-valued or integer-valued representation we can derive testability measures for elements of a digital circuit, or conduct the reliability analysis for various networks.