Classification and Universality of Reversible Logic Elements with One-Bit Memory

Abstract

A reversible logic element is a model of a computing element that has an analogous property to physical reversibility. In this paper, we investigate k-symbol reversible elements with one-bit memory (i.e., two states) for k=2, 3, and 4. We classified all of them, and showed the total numbers of essentially different elements are 8 (k=2), 24 (k=3), and 82 (k=4). So far, a rotary element, a 2-state 4-symbol reversible element, has been known to be logically universal. Here, we proved that a new and interesting elements called 3- and 4-symbol left-/right-rotate elements are both universal. We also gave a new concise construction method of a Fredkin gate out of rotary elements.