4.1 Example Computation

tion 14 will generally contain more internal vertices than the AND of those same functions with product terms. Therefore, we computed the probability values of the fundamental equivalence relations for the ISCAS benchmark circuits c432 and c880 for outputs 421gat and 878gat respectively. Tables 4 and 5 contain the results of these computations. The algorithm used to compute the probabilities is a recursive, depth-rst method that actually computes path probabilities rather than node probabilities. By computing path probabilities, the temporal complexity of the algorithm is O(N), where N is the number of OBDD vertices. These computations were performed on a 100 MHz SPARCstation 20 with 160 MB of memory. All computations required 1 second or less of CPU time. The relevance of this data is that the value of the modiied Haar transform coeecients for this function may all be computed in 1 CPU second or less since the rst-ordered coeecients are algebraically related to the given probabilities. The higher ordered coee-cients are computed by ANDing a cube term with the OBDDs represented in the Table 4 and will thus generally yield smaller OBDDs allowing the corresponding probabilities to be computed with a negligible amount of increased CPU time overhead. Therefore, we have reduced the amount of time to compute a modiied Haar transform coeecient from that required to form an inner product of vectors of length 2 36 for the c432 example or 2 45 for the c880 example to a O(N) traver-sal of an OBDD. 5 Conclusion This paper has examined the relationship between the conditional output probabilities of Boolean functions and the modiied Haar transform. A methodology was implemented for the computation of the modiied Haar spectral coeecients and experimental results are given. One very interesting result is the dependence of the spectrum upon n + 1 equivalence relationships between the function to be transformed and each of its' dependent variables and the constant 0. In the future, we plan to investigate problems that can possibly be solved more eeciently by exploiting the information present in the Haar spectrum. Acknowledgment I would like to thank Roger Moore for generating the experimental results given in this paper. modiied Haar spectrum. Higher ordered coeecients are computed by partitioning the relevant equivalence relation into 2 m OBDDs covering disjoint portions of the range and nding their output probabilities. The value m ? 1 is the number of previously used equivalence …