Results of the Probability Based HeuristicMethod

symmetry detection and decision diagram variable reordering. Although some good results have been obtained thusfar, these research eeorts are ongoing. Acknowledgment The authors would like to thank Professor D. M. Miller for discussions concerning the probability calculation algorithms using DDs. been developed. The tool utilizes an OBDD representation of the logic function as input and iteratively decomposes the OBDD as the resulting logic circuit is constructed. Although a single spectral coeecient may be computed in a reasonable amount of CPU time, an entire spectral description of a circuit requires an exponential number of spectral coeecients (2 n), for this reason a subset of the Walsh coeecients, the rst order spectral coeecients referred to as the Chow parameters 4] are computed only. Depending on the Chow parameter magnitudes, a series of heuris-tic rules are invoked that govern the OBDD decompo-sition/logic realization step. Table 2 contains data regarding the synthesis of some common benchmark circuits and a comparison using the misII tool from Berkeley with the \rugged script". Table 2: Comparison of Spectral Based Heuristic Method with misII Circuit Output Number misII Spectral Name Name Inputs Size Size 5xp1 output 1 7 14 25 9sym output 1 9 262 194 alu4 output 2 14 21 13 bw output 1 5 65 14 con1 output 1 6 8 11 rd53 output 1 5 18 9 rd73 output 1 7 114 125 rd84 output 1 8 98 253 vg2 output 2 25 35 46 apex1 output 12 45 96 38 misex3c output 1 14 34 34 xor5 output 1 5 4 4 clip output 4 9 81 170 5 Boolean Function Variable Symmetry Symmetry detection in Boolean functions is important because it may lead to simpler circuits that implement the function and is also useful in nding good variable orderings for DDs 7]. General symmetries may be detected by means of a transformation approach that checks for function invariance under all possible transformations. Unfortunately the number of such tests can be exponentially large reducing the practicallity of the approach. Fortunately, the number of variables over which to search for symmetry can be reduced (sometimes substantially so) by using Theorem 1. Theorem 1 A necessary condition for symmetry under a circular permutation of k variables is: }fx i fg = }fx j fg8x i ; x j 2 fx 1 ; x 2 ; : : :x k g 6 Variable Reordering in Decision Diagrams …