An efficient NPN Boolean matching algorithm based on structural signature and Shannon expansion

Abstract

An efficient pairwise Boolean matching algorithm for solving the problem of matching single-output specified Boolean functions under input negation and/or input permutation and/or output negation (NPN) is proposed in this paper. We present the structural signature (SS) vector, which comprises a first-order signature value, two symmetry marks, and a group mark. As a necessary condition for NPN Boolean matching, the SS is more effective than the traditional signature. Two Boolean functions, f and g, may be equivalent when they have the same SS vector. A symmetry mark can distinguish symmetric variables and asymmetric variables and be used to search for multiple variable mappings in a single variable-mapping search operation, which reduces the search space significantly. Updating the SS vector via Shannon decomposition provides benefits in distinguishing unidentified variables, and the group mark and phase collision check can be used to discover incorrect variable mappings quickly, which also speeds up the NPN Boolean matching process. Using the algorithm proposed in this paper, we test both equivalent and non-equivalent matching speeds on the MCNC benchmark circuit sets and random circuit sets. In the experiment, our algorithm is shown to be 4.2 times faster than competitors when testing equivalent circuits and 172 times faster, on average, when testing non-equivalent circuits. The experimental results show that our approach is highly effective at solving the NPN Boolean matching problem.

A New Pairwise NPN Boolean Matching Algorithm Based on Structural Difference Signature

“…The merit of this method is that once it finds a transformation that can prove the equivalence of two Boolean functions, other transformations will not be checked. The authors of [4,5,13,14] proposed Boolean matching algorithms based on pairwise matching and used binary decision diagrams (BDDs) to represent Boolean functions. The authors of [5] proposed a structural signature vector to search the transformations between two Boolean functions and implemented NPN Boolean matching for 22 inputs.…”

“…The authors of [4,5,13,14] proposed Boolean matching algorithms based on pairwise matching and used binary decision diagrams (BDDs) to represent Boolean functions. The authors of [5] proposed a structural signature vector to search the transformations between two Boolean functions and implemented NPN Boolean matching for 22 inputs. In pairwise matching algorithms, signatures are usually used as a necessary condition for judging whether two Boolean functions are equivalent, and variable symmetry is commonly utilized to reduce the search space.…”

“…The variable symmetric attributes of Boolean function are widely used in NPN Boolean equivalence matching. In reference [5], the search space was reduced and the matching speed was improved by means of structural signatures, variable symmetry, phase collision check and variable grouping.…”

“…The new signature vector SDS is better able to distinguish variables and reduce the research space for NPN Boolean matching. Experimental results show that the search space is reduced by more than 48% compared with [5] and that the run time of our algorithm is reduced by 42% and 80% compared with [5,6], respectively.…”

“…If Boolean function f is NPN-equivalent to Boolean function g, there must be a NPN transformation that can transform f to g. On the contrary, no NPN transformation can transform f to g. The purpose of our proposed algorithm is to find the NPN transformation that can transform Boolean function f to g as early as possible. Based on the structural signature (SS) vector in our previous study [5], we proposes a new combined signature vector, i.e., SDS vector. In this paper, Boolean difference sigture is introduced into SS vector to form SDS vector.…”