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 five-variable 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.

This paper presents a design method for three-level programmable logic arrays (PLAs), which have input decoders and two-input EXOR gates at the outputs. The PLA realizes an EXOR of two sum-of-products expressions (EX-SOP) for multiple-valued input two-valued output functions. We developed an output phase optimization method for EX-SOPs where some outputs of the function are minimized in the complemented form and presented techniques to minimize EX-SOPs for adders by using an extension of Dubrova-Miller-Muzio's AOXMIN algorithm. The proposed algorithm produces solutions with a half products of AOXMIN-like algorithm in 250 times shorter time for large adders with two-valued inputs. We also proved that an n-bit adder with two-valued inputs requires at most 32n-2+7n-5 products in an EX-SOP while it is known that a sum-of-products expression (SOP) requires 62n-4n-5 products.

This paper presents an exact minimization algorithm 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-SOP), where the two sum-of-products expressions (SOP) can share products. The objective is to minimize the total number of different products in the two SOPs. An algorithm for the exact minimization of EX-SOPs with up to five variables are shown. Up to five variables, EX-SOPs for all the representative functions of NP-equivalence classes were minimized. For five-variable functions, we confirmed that minimum EX-SOPs require up to 9 products. For n-variable functions, minimum EX-SOPs require at most 92n-5 (n6) products.