symmetry SS Article Synthesis Strategy of Reversible Circuits on DNA Computers Mirna Rofail 1, * and Ahmed Younes 1,2   Citation: Rofail, M.; Younes, A. Synthesis Strategy of Reversible Circuits on DNA Computers. Symmetry 2021, 13, 1242. https:// doi.org/10.3390/sym13071242 Academic Editors: Enrique Maciá Barber and Simone Fiori Received: 3 June 2021 Accepted: 8 July 2021 Published: 10 July 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria 21568, Egypt; ayounes@alexu.edu.eg 2 School of Computer Science, University of Birmingham, Birmingham B15 2TT, UK * Correspondence: mirna.hosny.comp@alexu.edu.eg Abstract: DNA computers and quantum computers are gaining attention as alternatives to classical digital computers. DNA is a biological material that can be reprogrammed to perform computing functions. Quantum computing performs reversible computations by nature based on the laws of quantum mechanics. In this paper, DNA computing and reversible computing are combined to propose novel theoretical methods to implement reversible gates and circuits in DNA computers based on strand displacement reactions, since the advantages of reversible logic gates can be exploited to improve the capabilities and functionalities of DNA computers. This paper also proposes a novel universal reversible gate library (URGL) for synthesizing n-bit reversible circuits using DNA to reduce the average length and cost of the constructed circuits when compared with previous methods. Each n-bit URGL contains building blocks to generate all possible permutations of a symmetric group of degree n. Our proposed group (URGL) in the paper is a permutation group. The proposed implementation methods will improve the efficiency of DNA computer computations as the results of DNA implementations are better in terms of quantum cost, DNA cost, and circuit length. Keywords: DNA computing; reversible logic; quantum circuits; logic gates; nanotechnology Highlights This paper presents three main points: First, Improving the functionalities of DNA computers by constructing reversible gates and reversible circuits using strand displace- ment reactions (SDR) and toehold exchange principle. Second, Proposing two theoretical methods for implementing reversible circuits on DNA computers based on dual-rail logic (DLG) and switching circuits (DCS) which are experimented successfully in vitro in Ref. [1]. Third, Proposing a novel universal reversible gate library (URGL) for synthesizing n-bit reversible circuits using DNA with better average length and cost than relevant reversible gate libraries. 1. Introduction Deoxyribonucleic acid (DNA) in living cells is one of three essential cores for life (DNA, RNA, and Proteins) [2]. Based on experimental results, the efficiency of using DNA as a computer for storing and manipulating data at the molecular level was proven [3] when Leonard M. Adleman [4] and Richard J Lipton [5] solved the directed Hamiltonian path problem and the Satisfaction problem, respectively, in polynomial time using DNA computers. DNA computing has many applications in bioinformatics, biomedical, bio- electronics, bioengineering, and biocomputers areas because of its parallelism in data processing, density, and cheap cost properties [3,6]. In biocomputers, DNA is used to synthesize logic circuits instead of electronic chips [7–9], and is used as a storage medium instead of hard drives [10–12]. To implement logic gates and circuits on a DNA computer, DNA strand displacement reaction system (SDR) [1,6,9,13–15] is used because DNA sequences using SDR can be resynthesized to perform specific functions such as logic gate or logic circuit function. Symmetry 2021, 13, 1242. https://doi.org/10.3390/sym13071242 https://www.mdpi.com/journal/symmetry