A New Equivalence Relation of Logic Functions and Its Application in the Design of AND-OR-EXOR Networks

This paper presents a design method for AND-OR-EXOR three-level networks, where a single two-input exclusive-OR (EXOR) gate is used. The network realizes an EXOR of two sum-of-products expressions (EX-SOPs). The problem is to minimize the total number of products in the two sum-of-products expressions (SOPs). We introduce the notion of μ-equivalence of logic functions to develop exact minimization algorithms for EX-SOPs with up to five variables. We minimized all the NP-representative functions for up to five variables and showed that fivevariable functions require 9 or fewer products in minimum EX-SOPs. For n-variable functions, minimum EX-SOPs require at most 9 · 2n−5 (n ≥ 6) products. This upper bound is smaller than 2n−1, which is the upper bound for SOPs. We also found that, for five-variable functions, on the average, minimum EX-SOPs require about 40% fewer literals than minimum SOPs. key words: three-level networks, AND-EXOR, NP-equivalence, coordinate representation, μ-equivalence, spectral method, logic minimization