Compact DSOP and Partial DSOP Forms

Abstract

Given a Boolean function f on n variables, a Disjoint Sum-of-Products (DSOP) of f is a set of products (ANDs) of subsets of literals whose sum (OR) equals f , such that no two products cover the same minterm of f . DSOP forms are a special instance of partial DSOPs, i.e. the general case where a subset of minterms must be covered exactly once and the other minterms (typically corresponding to don't care conditions of f ) can be covered any number of times. We discuss finding DSOPs and partial DSOP with a minimal number of products, a problem theoretically connected with various properties of Boolean functions and practically relevant in the synthesis of digital circuits. Finding an absolute minimum is hard, in fact we prove that the problem of absolute minimization of partial DSOPs is NPhard. Therefore it is crucial to devise a polynomial time heuristic that compares favorably with the known minimization tools. To this end we develop a further piece of theory starting from the definition of the weight of a product p as a functions of the number of fragments induced on other cubes by the selection of p, and show how product weights can be exploited for building a class of minimization heuristics for DSOP and partial DSOP synthesis. A set of experiments conducted on major benchmark functions show that our method, with a family of variants, always generates better results than the ones of previous heuristics, including the method based on a BDD representation of f .

Approximate Logic Synthesis by Symmetrization

“…Let f : {0, 1} n → {0, 1} be a completely specified Boolean function depending on n binary variables. Our algorithm for computing the closest symmetric approximation of f starts from a minimal disjoint sum of products form (DSOP) representing f [4]. Recall that in a DSOP representation, each minterm in the on-set of f is covered by exactly one product.…”

“…Recall that in a DSOP representation, each minterm in the on-set of f is covered by exactly one product. We refer the reader to [4] for more details and references on DSOP forms and their applications.…”

“…For instance, considering all the products in the previous example, we have that T (0) = 1, T (1) = 3, T (2) = 2, T (3) = 1, and T (4) = 0. We can note that f is not symmetric since some of the non-zero values of T are not equal to the corresponding binomial: i.e., T (1) = 4 1 = 4, T (2) = 4 2 = 6, T (3) = 4 3 = 4. The second phase of the algorithm consists in deriving the totally symmetric function f closest to f .…”

Asynchronous sum‐of‐products logic minimization and orthogonalization

“…Any minimization tool can be used. However, we propose to use ESPRESSO or BOOM , where the most effective minimization heuristics are implemented. To satisfy conditions 3, 4, partial DSOP is produced. This is the step where the reachable and unreachable DCs are distinguished.…”

“…In the latter case, all the SOP terms are required to be orthogonal. For this purpose, a tool DSOP was used instead of the partial DSOP . The complexities are represented in columns ‘ Full orth .’ and ‘ Part.…”