Majority Logic Formulations for Parallel Adder Designs at Reduced Delay and Circuit Complexity

Abstract

The design of high-performance adders has experienced a renewed interest in the last few years; among high performance schemes, parallel prefix adders constitute an important class. They require a logarithmic number of stages and are typically realized using AND-OR logic; moreover with the emergence of new device technologies based on majority logic, new and improved adder designs are possible. However, the best existing majority gate-based prefix adder incurs a delay of 2log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (n) - 1 (due to the nth carry); this is only marginally better than a design using only AND-OR gates (the latter design has a 2log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (n) + 1 gate delay). This paper initially shows that this delay is caused by the output carry equation in majority gate-based adders that is still largely defined in terms of AND-OR gates. In this paper, two new majority gate-based recursive techniques are proposed. The first technique relies on a novel formulation of the majority gate-based equations in the used group generate and group propagate hardware; this results in a new definition for the output carry, thus reducing the delay. The second contribution of this manuscript utilizes recursive properties of majority gates (through a novel operator) to reduce the circuit complexity of prefix adder designs. Overall, the proposed techniques result in the calculation of the output carry of an n-bit adder with only a majority gate delay of log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (n) + 1. This leads to a reduction of 40percent in delay and 30percent in circuit complexity (in terms of the number of majority gates) for multi-bit addition in comparison to the best existing designs found in the technical literature.