1 Nonconventional Computer Arithmetic Circuits, Systems and Applications Leonel Sousa, Senior Member, IEEE Abstract—Arithmetic plays a major role in a computer’s basic levels and leads to high power consumption. Hence, performance and efficiency. Building new computing platforms the research on unconventional number systems is of the supported by the traditional binary arithmetic and silicon-based utmost interest to explore parallelism and take advantage of technologies to meet the requirements of today’s applications is becoming increasingly more challenging, regardless whether we the characteristics of emerging technologies to improve both consider embedded devices or high-performance computers. As a the performance and the energy efficiency of computational result, a significant amount of research effort has been devoted to systems. Moreover, by avoiding the dependencies of binary the study of nonconventional number systems to investigate more systems, nonconventional number systems can also support efficient arithmetic circuits and improved computer technologies the design of reliable computing systems using the newest to facilitate the development of computational units that can meet the requirements of applications in emergent domains. available technologies, such as nanotechnologies. This paper presents an overview of the state of the art in non- conventional computer arithmetic. Several different alternative computing models and emerging technologies are analyzed, such A. Motivation as nanotechnologies, superconductor devices, and biological- and quantum-based computing, and their applications to multiple The Complementary Metal-Oxide Semiconductor (CMOS) domains are discussed. A comprehensive approach is followed transistor was invented over fifty years ago and has played in a survey of the logarithmic and residue number systems, a key role in the development of modern electronic devices the hyperdimensional and stochastic computation models, and and all that it has enabled. The CMOS transistor has evolved the arithmetic for quantum- and DNA-based computing systems into nanodevices, with characteristic dimensions less than and techniques for approximate computing. Technologies, pro- cessors and systems addressing these nonconventional computer 100 nm. The downscaling of the gate length has became arithmetic systems are also reviewed, taking into consideration one of the biggest challenges hindering progression in each some of the most prominent applications, such as deep learning new generation of CMOS transistors and integrated circuits. or postquantum cryptography. In the end, some conclusions are New device architectures and materials have been proposed to drawn, and directions for future research on nonconventional address this challenge, namely, the Fin Field-Effect transistor computer arithmetic are discussed. (FinFET) multigate devices [5]. This type of nonplanar transis- tor became the dominant gate design from the 14 nm/10 nm I. INTRODUCTION generations, used in a broad range of applications, ranging from consumer applications to embedded systems and high- The demise of Moore’s Law and the waning of Dennard performance computing [6]. scaling, which respectively stated that the number of tran- sistors on silicon chips would double every two years and that this increase in the transistor density is not achieved at constant power consumption [1], mark the end of an era in which the computational capacity growth was mainly based on the downscaling of silicon-based technology. At the same time, the demand for data processing is higher than ever before as a result of the more than 2.5 quintillion bytes that are created on a daily basis [2]. This number continues to grow exponentially, and therefore, ingenious solutions must be developed to address the limitations of traditional com- putational systems. These innovative solutions must include developments not only at the technological level but also at the arithmetic, architectural and algorithmic levels of computing. Fig. 1: Emerging technologies and CMOS speed, size, cost, Although binary arithmetic has been successfully used to and switching energy (scales are logarithmic) [7] design silicon-based computing systems, the positional nature of this representation imposes the processing of carry chains, Fig. 1 plots the cost, speed, size, and energy per operation which precludes the exploitation of parallelism at the most relationship for the CMOS and other emerging nanotechnolo- gies; all the scales are logarithmic, covering many orders of Leonel Sousa is with the the Department of Electrical and Computer magnitude. Examples of these technologies include supercon- Engineering, INESC-ID, Instituto Superior Tecnico,´ Universidade de Lis- boa; Address: Rua Alves Redol, 9, 1000-029 Lisboa, PORTUGAL; e-mail: ducting electronics, molecular electronics, resonant tunneling [email protected]. devices, quantum cellular automata, and optical switches. 2 Superconducting digital logic circuits use Single-Flux the conventional binary system has been dominant at the data Quantum (SFQ), also known as magnetic flux quanta, to representation layer. However, when considering emergent encode and process data. An Rapid Single Flux Quantum technologies or new architectures for designing these systems, (RSFQ) device is a superconducting ring with a Josephson data representation must also be reconsidered to properly adapt junction that opens and closes to admit or expel a flux each system to the features of both the bottom and top layers. quanta. The Adiabatic Quantum-Flux-Parametron (AQFP), an- Data representation and the related computer arithmetic are the other SFQ-based device not represented in the figure with motivation behind this survey, which looks beyond the CMOS characteristics that are analyzed later in this survey, drastically technology and conventional systems. reduces the energy dissipation by using Alternating Current (AC) bias/excitation currents for both the clock signal and B. Structure of the survey power supply. This is high-speed nanotechnology that offers This survey analyzes the confluence of emergent technolo- low-power dissipation and scalability. However, its main draw- gies, nonconventional computer arithmetic, innovative com- backs include the need for expensive cooling technologies and puting paradigms and new applications. Similar to a jigsaw improved techniques in manufacturing the elements. Molec- puzzle, although these pieces exist by themselves, it is only ular electronics use a single molecule as the switch. The by connecting them in the right way that the whole picture configuration of the molecule determines its state, thereby can be derived and, in this particular case, the way to design “switching” on or off the current flow through the molecule. efficient systems can be determined. Fig. 3 depicts the tight While conceptually appealing, the molecular process exhibits interconnections and dependencies of all these aspects in a poor self-assembly techniques and low reproducibility. In matrix representation. In addition, this figure shows how these Quantum Cellular Automata (QCA), cells are switched on and very important topics are addressed in this paper, highlighting off by moving the position of the electrons from one cell to the fact that nonconventional arithmetic exposes the character- another. Classical physics does not apply to these technologies, istics of the underlying technology to assist in the development which support, for example, quantum computing. While it is of efficient and reliable computing systems useful for different challenging to build general purpose computing engines based applications. on these technologies, they are much faster on certain specific tasks. A last example is the integrated optical technology, SECTION II: COMPUTER ARITHMETIC AND ARCHITECTURES [10-77] which is based on having small optical cavities that change High Performance Hyper Low Power Reliability RNS LNS Stochastic dimensional Noise Robustness their properties to either store or emit photons – a zero or a Residual Arithmetic HDC Processor one. It is a challenging approach, both from the fan-out and Nanophotonic Computer European Logarithmic StoRM cost point of view. As will be seen, computing-in-the-network Microprocessor circuits can be designed with integrated photonics switches. PIM Quantum Computing Approximate Computing DNA Computing SECTION III: NANOTECHNOLOGIES [78-119] Memristors/Hybrid Memories DNA/Molecular Biology Superconductors (AQFP) Integrated Photonics CMOS Invertible Circuits SECTION IV: APPLICATIONS [120-158] Quantum-Resistant Cryptography Machine Learning Fig. 2: Emerging research information processing devices, Fig. 3: Structure of the survey: sections and topics highlighting the non-conventional data representation (adapted from [9]) Section II introduces nonconventional number systems, the main characteristics and the associated computer arithmetic. These emergent technologies will become more and more The Logarithmic Number System (LNS) and the Residue relevant as the end of the CMOS era approaches [8]. Nev- Number System (RNS) shorten up the carry chain under- ertheless, for the time being, there is no technology better lying binary arithmetic. While LNS compresses the number than CMOS, as it provides a good balance between energy representation significantly, mitigating the problem of carry consumption, cost, speed and scalability, as shown in Fig. 1. chains, it lowers the complexity of the multiplication, division, Yet, better solutions