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 DSOPs 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 NP-hard. 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 cube c as a functions of the number of fragments induced on other cubes by the selection of c, and show how cube 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.
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Notes
The minimal exact cover problem is as follows: given a family of subsets S of a set U and a positive integer k, is there a subset family T⊆S such that the subsets in T are k in number, are disjoint, and their union is the entire set U? The minimal exact cover is NP-hard since it can be easily reduced by the “exact cover problem” introduced by Karp in 1972 [11] setting k to the cardinality of U.
An alternative reduction from Circuit SAT has been proposed in [20].
The characteristic vector of a set A⊆{1,2,…,n} is the n-bit vector whose ith bit is 1 if and only if i∈A.
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Bernasconi, A., Ciriani, V., Luccio, F. et al. Compact DSOP and Partial DSOP Forms. Theory Comput Syst 53, 583–608 (2013). https://doi.org/10.1007/s00224-013-9447-2
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DOI: https://doi.org/10.1007/s00224-013-9447-2