Fast Boolean matching based on NPN classification

Abstract

Abstract-This paper proposes a fast algorithm for Boolean matching of completely specified Boolean functions. The algorithm is based on the NPN classification and can be applied on-the-fly to millions of small practical functions appearing in industrial designs, leading to runtime and memory reduction in logic synthesis and technology mapping. The algorithm is conceptually simpler, faster, and more scalable than previous work. I. IntroductionAn NPN class is a set of Boolean functions derived from each other by permuting and complementing the inputs, and complementing the output. Boolean matching determines whether two Boolean functions belong to the same NPN class. In the practical applications, Boolean matching is used to match a function against a library of functions whose implementation or other properties are known or have been pre-computed.AIG This paper presents two algorithms: a fast heuristic one and a slower nearly-exact one. They are compared against [8] believed to be state-of-the-art in Boolean matching on a large set of small practical functions derived from industrial designs. The conclusion is that the proposed heuristic algorithm is often faster without degrading quality, while a more elaborate nearly-exact algorithm computes fewer classes than [8]. Moreover, [8] cannot be applied to functions of more than 9 inputs, which limits its practicality. Meanwhile, the proposed algorithm scales to 16 variables and is therefore a better fit for cut-based logic synthesis and technology mapping.The rest of the paper is organized as follows: Section II contains relevant background. Section III describes the proposed three-step algorithm. Section IV shows experimental results. Section V concludes the paper.

Fast hierarchical NPN classification

“…As the basis for our algorithm, we use the heuristic version of the state-of-the-art algorithm for NPN classification by Huang et al [12], which is used in several recent publications [2], [4]. The first two methods of the algorithm determine the polarity of the output and the polarity of each input by counting the number of 1s in the truth table and cofactors, respectively.…”

“…In this section, first, we describe the state-of-the-art heuristic algorithm [12] used to demonstrate our hierarchical approach. Then, we explain how to build a hierarchy of classes, and we propose a fast algorithm for exact NPN classification.…”

“…In this paper, we use the terms Boolean function, truth table, cofactor, and disjoint-support decomposition (DSD) as defined by Huang et al [12]. Following we give the terminology associated with NPN equivalence and canonical form.…”

“…EXPERIMENTAL RESULTS ABC [13] contains implementation of the heuristic state-ofthe-art algorithm [12] (command testnpn -A 5 <input file>) and the exhaustive exact algorithm (command testnpn -A 1 <input file>). We refer to these algorithms as OrigH and OrigE, respectively.…”

“…Due to its importance, NPN classification is a well-studied problem for which different heuristics and exact algorithms already exist [5][6][7][8][9][10][11][12]. Although some of these algorithms are composed of several methods that compute intermediate representations of the function, they do not save nor use this information.…”