Detecting symmetries have many applications in logic synthesis, which include, among other things, technology mapping, deciding equivalence of Boolean functions when the input correspondence is unknown, and finding support-reducing bound sets. Mishchenko showed how to efficiently detect symmetries in reduced ordered binary decision diagrams (ROBDDs) without the need for checking equivalence of all cofactor pairs. This work resulted in practical algorithms for detecting classical and generalized symmetries. Both the classical and generalized symmetry-detection algorithms are monolithic in the sense that they only return a meaningful answer when they are left to run to completion. In this paper, we present anytime algorithms for detecting both classical and generalized symmetries, which output pairs of symmetric variables until a prescribed time bound is exceeded. These anytime algorithms are complete in that, given sufficient time, they are guaranteed to find all symmetric pairs. Anytime generality is not gained at the expense of efficiency because this approach requires only very modest data-structure support and offers unique opportunities for optimization, so that the resulting algorithms are competitive with their monolithic counterparts.

Detecting symmetries is crucial to logic synthesis, technology mapping, detecting function equivalence under unknown input correspondence, and ROBDD minimization. State-of-the-art is represented by Mishchenko's algorithm. In this paper, we present an efficient anytime algorithm for detecting symmetries in Boolean functions represented as ROBDDs, that output pairs of symmetric variables until a prescribed time bound is exceeded. The algorithm is complete in that given sufficient time it is guaranteed to find all symmetric pairs. The complexity of this algorithm is in O(n4+ n|G| + |G|3) where n is the number of variables and |G| the number of nodes in the ROBDD, and it is thus competitive with Mishchenko's O(|G|3) algorithm in the worst-case since n Lt |G|. However, our algorithm performs significantly better because the anytime approach only requires lightweight data structure support and it offers unique opportunities for optimization