Multi-valued Logic Addition and Multiplication in Galois Field

Advances in multiple-valued logic (MVL) have been inspired, in large part, by advances in integrated circuit technology. Multiple-valued logic has matured to the point where four-valued logic is now part of commercially available VLSI IC's. Besides reduction in chip area, MVL offers other benefits such as the potential for circuit test. This paper describes the historical and technical background of MVL, and areas of present and future application. It is intended, as well, to serve as a tutorial for the nonspecialist.

Many-valued logics in general and 3-valued logic in particular is an old subject which had its beginning in the work of Łukasiewicz [Łuk]. Recently there is a revived interest in this topic, both for its own sake (see, for example, [Ho]), and also because of its potential applications in several areas of computer science, such as proving correctness of programs [Jo], knowledge bases [CP] and artificial intelligence [Tu]. There are, however, a huge number of 3-valued systems which logicians have studied throughout the years. The motivation behind them and their properties are not always clear, and their proof theory is frequently not well developed. This state of affairs makes both the use of 3-valued logics and doing fruitful research on them rather difficult.Our first goal in this work is, accordingly, to identify and characterize a class of 3-valued logics which might be called natural. For this we use the general framework for characterizing and investigating logics which we have developed in [Av1]. Not many 3-valued logics appear as natural within this framework, but it turns out that those that do include some of the best known ones. These include the 3-valued logics of Łukasiewicz, Kleene and Sobociński, the logic LPF used in the VDM project, the logic RM3 from the relevance family and the paraconsistent 3-valued logic of [dCA]. Our presentation provides justifications for the introduction of certain connectives in these logics which are often regarded as ad hoc. It also shows that they are all closely related to each other. It is shown, for example, that Łukasiewicz 3-valued logic and RM3 (the strongest logic in the family of relevance logics) are in a strong sense dual to each other, and that both are derivable by the same general construction from, respectively, Kleene 3-valued logic and the 3-valued paraconsistent logic.

Traditional digital circuits use binary logic for implementation which requires more number of data lines to carry information. In contrast to binary, multiple-valued logic (MVL), digital circuits require less number of data lines to represent information. MVL uses more than two levels to represent a number; thereby reducing the number of interconnects. One of its kind is ternary logic, (known as radix-3 system) and uses three different voltage levels, whereas the other quaternary logic (radix-4 system) uses four different voltage levels to represent a number. Various digital circuits are implemented using MVL which potentially reduces the chip area, power consumption, and propagation delay. This article discusses about fundamental concepts of ternary, quaternary system and reviews some of arithmetic and logic circuits using ternary and quaternary logic. This paper also presents different methodologies used to implement arithmetic and logic operations using MVL. It is found that, due to the effective utilization of interconnects, MVL digital circuits provide better performance over binary digital circuits in terms of area, power, and delay.KeywordsNumber systemArithmetic and Logic circuitsChip areaMultiple-Valued Logic (MVL)Quaternary logicTernary logicTiming delayVLSI

Tremendous amounts of data are being generated and should be processed during the massive employment of artificial intelligence and edge/cloud computing, necessitating computers with higher capabilities. However, as Moore's Law approaches its limit, enhancing computing capability through transistor miniaturization is becoming less feasible. New strategies for “more-than-Moore” are highly preferred to drive future technological advancements. Multiple-valued logic, especially ternary logic with its superior data density compared to traditional binary logic, offers a viable route to augment computing capability while reducing chip interconnections and dynamic power consumption. Despite the growing interest in ternary logic within academia and industry, its practical application remains limited due to the increased transistor count and connections required by earlier logic gate implementations. In this thesis, we present efficient implementation methods of basic ternary logic gates based on emerging nano-electronics including two-dimensional material-based transistors. The fabricated ternary devices show high performance in DC gain and low power consumption. Additionally, compact models of the devices have been established and calibrated, facilitating the circuit design and simulation in Electronic Design Automation (EDA) tools. At the circuit level, this thesis introduces novel ternary arithmetic circuits, including adders and multipliers, that are optimized for low power consumption and minimal delay. Finally, we showcase the practical benefits of ternary logic systems in data processing through the implementation of carry-free arithmetic circuits, underscoring their potential as a transformative computing paradigm.

A lot of research has been done on multiple-valued logic (MVL) such as ternary logic in these years. MVL reduces the number of necessary operations and also decreases the chip area that would be used. Carbon nanotube field effect transistors (CNTFETs) are considered a viable alternative for silicon transistors (MOSFETs). Combining carbon nanotube transistors and MVL can produce a unique design that is faster and more flexible. In this paper, we design a new half adder and a new multiplier by nanotechnology using a ternary logic, which decreases the power consumption and chip surface and raises the speed. The presented design is simulated using CNTFET of Stanford University and HSPICE software, and the results are compared with those of other studies.

The ultimate usefulness of a number system depends on its implementation. Multiple-valued logic has been implemented in charge-coupled devices (CCD). In this technology, logic values are encoded as charge. For example, prototype four-valued logic devices have been implemented at the University of Twente (Enschede, Holland). Hitachi has implemented a 16-valued memory that stores the equivalent of 10/sup 6/ bits. CCD is more compact than any other VLSI technology. Although it is slower than CMOS (complementary metal oxide semiconductor), it is much faster than the disk and has the potential of replacing the disk. The use of multiple-valued logic in CCD increases its storage capacity significantly. Multiple-valued logic has also been implemented in current-mode CMOS.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The geometry dependant threshold voltage of carbon nanotube FETs (CNFETs), has been often used to design a ternary logic family. However , for the last couple of decades , multiple-valued logic (MVL) such as ternary (base=3) or quaternary (base=4) logic styles has attracted considerable attention. MVL circuits can reduce the number of operations necessary to implement a particular mathematical function and further, have an advantage in terms of reduced area. As we progress into an era of nanotechnology , molecular devices are becoming promising alternatives to the existing silicon technology. Carbon nanotube field effect transistors (CNFETs) are being extensively studied as possible successors to silicon MOSFETs. Research has started in the earnest to understand the device physics of CNFETs as well as to explore possible circuit applications. Implementable CNTFET circuits have operational characteristics to approach the advantage of using MVL in voltage mode. In this paper through using of CNTFET characteristics, we presented new CNTFET circuit design to implement optimum quaternary Galois field logic.

Background: Multiple-Valued Logic (MVL) is the non-binary-valued system, in which more than two levels of information content are available, i.e., L>2. In modern technologies, the dual level binary logic circuits have normally been used. However, these suffer from several significant issues such as the interconnection considerations including the parasitics, area and power dissipation. The MVL circuits have been proved to be consisting of reduced circuitry and increased efficiency in terms of higher utilization of the circuit resources through multiple levels of voltage. Innumerable algorithms have been developed for designing such MVL circuits. Extended form is one of the algebraic techniques used in designing these MVL circuits. Voltage mode design has also been employed for constructing various types of MVL circuits. Novelty: This paper proposes a novel MVLTRANS inverter, designed using conventional CMOS and pass transistor logic based MVLPTL inverter. Binary to MVL Converter/Encoder and MVL to binary Decoder/Converter are also presented in the paper. In addition to the proposed decoder circuit, a 4-bit novel MVL Binary decoder circuit is also proposed. Tools Used: All these circuits are designed, implemented and verified using Cadence® Virtuoso tools using 180 nm technology library.

In terms of memory, multivalued logic can be the fitting logic for the existing binary logic. Ternary logic contains three symbols in place of two symbols used in the binary logic, i.e., 0, 1, 2. More information can be stored with the help of these three symbols. SRAM cell is widely used in the digital circuit. The SRAM cell designed using the ternary logic can be used in the design of large memory arrays designed using ternary logic. The traditional ternary inverter which is used in the design of the traditional ternary SRAM cell is unable to store the proper values for the second state, there is a voltage level drop, which in turn affects the data read/write value of the SRAM cell designed using this traditional ternary inverter. Hence there is a need to design the ternary inverter cell which can give the proper output voltage level of all three states of the ternary logic. The level shift ternary inverter is designed to fulfill this disadvantage. The ternary inverter is designed in order to achieve the ideal DC characteristics, and the same level shift ternary inverter is used in the design of level shift ternary SRAM. This ternary SRAM stores the data properly at read/write signal. The traditional ternary inverter and traditional ternary SRAM, level shift ternary inverter and level shift ternary SRAM are implemented in Cadence 45 nm technology. The traditional ternary inverter consumes 2.37 μW power, and the level shift ternary SRAM consumes 2.43 μW power. The traditional ternary SRAM consumes 3.012 μW and level shift ternary SRAM consumes 3.14 μW. At the cost of a little bit increased power and the number of transistors, the traditional ternary SRAM can be replaced with level shift ternary SRAM. This level shift ternary SRAM stores all the voltage levels at all three levels. The same level shift ternary SRAM cell can be used for the design of large memory arrays.

In this paper we present a many-valued logic programming, based on reinterpreted Belnap's 4-valued bilattice: we introduce the new semantics for a 4-valued implication by relative pseudo-complement, used for intuitionistic logics. This kind of logic programming is particularly useful for data integration with possibly incomplete and inconsistent information. We define an ontological encapsulation of the epistemic many-valued logic programs with negation, based on this bilattice, into 2-valued meta logic programs. Obtained 2-valued logic semantically reflects original epistemic many-valued logic, and can be used in order to define many-valued logic entailment and inference closure for many-valued truth assignments.

The article proposes a method for synthesizing ternary binary logic functions using a multi-threshold element, which significantly extends the capabilities of ternary logic compared to traditional binary logic. The aim of the work is to develop an algorithm for synthesizing ternary logic functions, based on the use of a multi-threshold element, which simplifies the design of logical circuits and reduces energy consumption while maintaining high data processing speed. The study includes a comparative analysis of ternary and binary logic functions, allowing an assessment of the advantages of ternary logic, particularly its ability to process more values than binary logic, which opens up new possibilities for creating more efficient computing devices. Specifically, ternary logic can reduce the complexity of circuits and hardware resource usage, while also enhancing performance in tasks where energy efficiency is critical. An essential part of the research is the development of an algorithm for synthesizing ternary binary logic functions based on a four-threshold multi-valued element. Furthermore, a software module was implemented to automate the synthesis of ternary functions. The program not only generates a graphical representation of the logical circuit but also formulates the analytical expression corresponding to the given logical function. This significantly simplifies the design and analysis of logical systems and enhances the efficiency of computations. Considering the high potential of ternary logic, the results of this work are significant not only for theoretical research but also for practical applications in the development of new computing devices and technologies. The further development of these ideas could contribute to the progress in computational systems working on new principles and open up new possibilities for optimizing technological processes in various fields of science and engineering.

Multi-valued logic (MVL) is that logic which has two or more logic values. In complex digital circuits, MVL (mainly Ternary logic) offers several advantages over binary logic. Carbon Nanotube Field Effect Transistor (CNTFET) technology is ideal to implement ternary logic circuits because of the threshold voltage of CNTFETs depends on the physical dimensions (chirality) of their channel. This work presents the implementation of a three-bit Ternary Prefix Adder using CNTFET technology. In this paper, a carry propagate-generate concept is used in order to implement the ternary prefix adder. A Kogge-Stone based prefix network is preferred for carry computation due to its high performance. A low power enoder is used for reducing the overall power of the adder. HSpice tool is chosen for designing this system. Simulation results show that there is a significant reduction in Power Delay Product (PDP) by 28 % compared to all other previous works.

The quest for efficient technologies beyond the traditional CMOS (Complementary-Metal-Oxide-Semiconductor) technology has led researchers to explore newer technologies like the CNFET(Carbon-Nanotube-Field-Effect-Transistor). CNFETs are being used to design and implement various ternary logic circuits. Multivalued logic (MVL) requires multiple threshold voltages that are possible to obtain using CNFETs by modifying the physical dimensions of their CNTs (Carbon Nano Tubes). The ternary sequential logic circuits like ternary D-latch, D-flipflop, and counters existing in literature have been implemented either using STIs (Standard Ternary Inverters) or successor-predecessor circuits. This paper proposes a new design methodology for a ternary D-flipflop that uses a ternary buffer and two STIs designed using two power supplies. Two hybrid architectures for ternary D-flipflop designs have also been proposed that are implemented using a combination of ternary buffer, STIs and successor-predecessor circuits. Using these ternary D-flipflop designs, ternary 3- trit synchronous and asynchronous counters are designed in this paper. The proposed designs are simulated using HSPICE and a standard Stanford CNFET model. The proposed ternary D-flipflops show an improvement of up to 93.5&#x0025; in power, up to 44.7&#x0025; in delay and up to 94.3&#x0025; in PDP compared to existing designs.

Reversible quantum computer system is one of the best choices for future computer systems. Multiple-valued logic especially ternary logic is a good candidate for the realization of reversible quantum computer. An efficient logic synthesis mechanism is essential for the low-cost realization. Toffoli gate is an important gate for quantum logic synthesis. It is the basic element for the Galois Field Sum of Product (GFSOP) expression based logic synthesis mechanism. So, for low-cost realization of any ternary logic function, a low-cost implementation of Toffoli gate is very necessary. This paper shows a low-cost, practically realizable, and efficient realization of 3-qutrit ternary Toffoli gate by using ion-trap realizable Muthukrishnan-Stroud gate. This realization is more efficient and less costly than other realizations.

In this scientific paper, a software library for Ternary logic gates will be built based on VHDL language to be used in the implementation of combinational Ternary logic circuits. The VHDL language had been designed to be used to automate and design Binary electronic logic systems. Therefore, several problems have been arised in adapting the VHDL language to be used with Ternary electronic logic systems. However, these problems have been solved and the result were encouragement to continue the study with the other Ternary components, so that a complete software library based on VHDL language for different Ternary components will be used as a tool in the design and simulation of Ternary electronic logic circuits and systems. &#x0D; Keywords: Ternary logic, Ternary logic gates, VHDL language.