An Efficient Algorithm for the Calculationof Generalized Adding and Arithmetic TransformsFrom Disjoint Cubes of Boolean Functions

Abstract

A new algorithm is given that converts a reduced representation of Boolean functions in the form of disjoint cubes to Generalized Adding and Arithmetic spectra. Since the known algorithms that generate Adding and Arithmetic spectra always start from the truth table of Boolean functions the method presented computes faster with a smaller computer memory. The method is extremely efficient for such Boolean functions that are described by only few disjoint cubes and it allows the calculation of only selected spectral coefficients, or all the coefficients can be calculated in parallel.

Cube‐Based Synthesis of ESOPs for Large Functions

“…There is an edge between a pair of vertices m a and m b if and only if the Hamming distance between their binary vectors H(m a , m b ) is equal to 1. Now we map p-degreeof -freedom cubes [8] into a lower dimensional space. As shown in Fig.1(a), a 1-degree-of-freedom cube stands for a unit interval.…”

“…Step 1: Using Eq. (3) and the rule of cube intersection [8] , an ESOP cover S can be computed directly from a SOP cover C by regarding each input and output plane of PLA as an (n + m)-cube.…”

“…However, minimization of Exclusive-or sum-of-products expansions (ESOPs) is computationally more expensive than that of Sum-of-products (SOP), which shows a double-exponential computational relationship with input variables. Thus, it is very difficult to calculate minimum ESOPs directly from non-disjoint cube set expressions of Boolean functions [7,8] , i.e., Programmable logic array (PLA) format.…”

“…Recently, many other methods have been proposed to compute canonical forms. For instance, some spectral methods are described by matrix transformation [8,16] , and some tabular techniques are extended to coefficient computation [1,15] . However, the above-mentioned algorithms have a common disadvantage that they may involve the procedure of generating the entire truth table, disjoint cubes or other canonical form coefficients, which results in an increase of procedural steps and processing time and consequently restricts their use in large Boolean functions of a SOP or ESOP.…”

Spectral Testing of Digital Circuits

“…Since the testing methods based on Walsh spectrum need the efficient way of the calculation of spectral coefficients, new methods allowing obtaining the spectral coefficients from reduced representations of Boolean functions have been introduced [13,20,23,50,68,83]. The approach based on the disjoint cube representation of Boolean function [20,23] is a general method and applies to the calculation of other transforms such as Reed -Muller [22] and Arithmetic [27]. In short, the basic advantages of methods to calculate spectral coefficients from disjoint cubes in comparison to the Fast Walsh Transform [5,6] are as follows:…”