Remarks on the Number of Logic Networks with Same Complexity Derived from Spectral Transform Decision Diagrams

Decision diagrams (DDs) for representation of discrete functions permit a direct technology mapping into multi-level logic networks. In these networks, the number of logic elements is equal to the number of non-terminal nodes in DDs. Complexity of interconnections is proportional to the number of paths from the root node to the constant nodes. Therefore, the complexity of the network can be estimated through the basic characteristics of DDs. For a given function f , different DDs may result into networks of different complexity. Complexity of a network derived from a DD is expressed through the number of non-terminal nodes, the width, and the number of paths from the root node to the constant nodes. When realizations for all functions for a given number of variables are considered, the problem relates to the classification of switching functions. In this settings, DDs defined with respect to different spectral transforms may result in a different number of networks required to realize the representative functions for classes of switching functions. This paper discusses the number of different multilevel logic networks of the same complexity derived from different word-level decision diagrams for functions of three and four variables. These networks can be used as basic building blocks in realization of functions of an larger arbitrary number of variables. For uniform realizations, it might be useful to have a small number of different basic modules. 1 MULTI-LEVEL LOGIC NETWORKS Multi-level logic networks permit realizations of switching functions with fewer circuits and reduced interconnections compared to two-level logic networks. However, designs of multi-level logic networks are far more complex that those of two-level logic networks. As is noted in [9], unlike two-level networks, where there are minimization algorithms, for multilevel networks there are no established automation design algorithms and the designs are done by using combinations of ad hoc methods. The optimization of multi-level networks may be directed towards reduction of the number of elements, the number of levels, i.e., timing optimization, reduction of interconnections, and the reduction of the complexity of the design procedure. Decision diagrams (DDs) for representation of discrete functions can be directly translated into n-level networks, n the number of variables. Therefore, DDs provide for simple design procedure, at the price that the number of levels is equal to the number of Table 1: Expansion rules and their realizations.