An Anytime Algorithm for Generalized Symmetry Detection in ROBDDs

Abstract

Abstract-Detecting symmetries has many applications in logic synthesis that include, amongst 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 ROBDDs without the need for checking equivalence of all co-factor 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, that 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 since this approach requires only very modest data structure support and offers unique opportunities for optimization so the resulting algorithms are competitive with their monolithic counterparts.

Extending symmetric variable-pair transitivities using state-space transformations

“…(The term anti-symmetry was introduced in [16]. Anti-symmetry is also known as skew symmetry [17] and negative symmetry [18].) These six new relations are given in Figure 2 along with their associated symmetries.…”

“…However, symmetry detection can be terminated at any point using either time or space constraints. This enables the detection process to be run in "anytime" fashion as in [18]. The process is adaptable to many existing algorithms, particularly those that already use cofactor relations for symmetry detection.…”

A universal symmetry detection algorithm

“…The importance such functions was first recognized by Shannon in (Shannon 1949 ), who characterized function symmetries using permutations of the input variables. Since that time, the detection and exploitation of symmetric Boolean functions has been of recurring interest in the field of design automation (Abdollahi 2006 ; Biswas 1970 ; Born and Scidmore 1968 ; Butler et al 2000 ; Chrzanowska-Jeske 2001 ; Chung and Liu 1998 ; Darga et al 2008 ; Drechsler and Becker 1995 ; Hu and Marek-Sadowska 2001 ; Hu et al 2008 ; Ke and Menon 1995 ; Kettle and King 2008 ; Kravets and Sakallah 2002 ; Maurer 2011 ; Mohnke et al 2002 ; Moller et al 1993 ; Muzio et al 2008 ; Rice and Muzio 2002 ; Scholl et al 1997 ; Tsai and Marek-Sadowska 1996 ; Wang and Chen 2004 ; Zhang et al 2004 ). Virtually all of these algorithms are based on Shannon’s Theorem (Shannon 1949 ) which detects symmetry by comparison of two-variable cofactors.…”

Fault Detection of Bridging Faults in Digital Circuits by Shared Binary Decision Diagram

“…Every nonterminal node is labeled with a variable and has edges directed toward children nodes: the 0-branch (also called '0' edge) corresponds to the case where the variable is assigned 0, and the 1-branch (also called '1' edge) corresponds to the case where the variable is assigned 1. Each leaf node is labeled 0 or 1 to correspond to the value of the function [10][11][12].…”