A Novel Design of Octal-Valued Logic Full Adder Using Light Color State Model

Ahmed Talal; Osama Abu-Elnasr; Samir Elmougy

doi:10.32604/cmc.2021.015759

Department of Computer Science, Faculty of Computes and Information, Mansoura University, Mansoura, 35516, Egypt
*Corresponding Author: Osama Abu-Elnasr. Email: mr_abuelnasr@mans.edu.eg
Received: 05 December 2020; Accepted: 06 January 2021

Abstract: Due to the demand of high computational speed for processing big data that requires complex data manipulations in a timely manner, the need for extending classical logic to construct new multi-valued optical models becomes a challenging and promising research area. This paper establishes a novel octal-valued logic design model with new optical gates construction based on the hypothesis of Light Color State Model to provide an efficient solution to the limitations of computational processing inherent in the electronics computing. We provide new mathematical definitions for both of the binary OR function and the PLUS operation in multi valued logic that is used as the basis of novel construction for the optical full adder model. Four case studies were used to assure the validity of the proposed adder. These cases proved that the proposed optical 8-valued logic models provide significantly more information to be packed within a single bit and therefore the abilities of data representation and processing is increased.

Keywords: Mathematical modeling; numerical simulations; optical logic; optics in computing; multi-valued logic; full adder

Arithmetic operation plays an important role in various digital systems such as computers process controller and signal processors. Nowadays, computer depends on 2-ary conventional system with 2 logical states ‘Low’ and ‘High’. Binary Logic (BL) is a classical two-valued Boolean logic that development in it be founded difficult to understand and complex.

Because the demand for high speed computational operations for solving complex problems and high bandwidth has rapidly increased to get the maximum device speed, the future of computation and communication face problems of unavoidability. This is because the conventional electronic technology will very soon reach its speed limit. So, researchers are looking at several approaches to find alternatives to address these problems. One of these approaches is the optical signal processing that refers to a wide range of optical data signals techniques [1]. It includes vastly popular systems such as optical multiplexing and demultiplexing, equalizers, optical switches, optical memories, optical amplifiers [2,3], and various areas of microwave photonics [4,5].

Another parallel alternative is Multi-Valued Logic (MVL) that can enable significantly more information to be packed within a single digit instead of BL. It is considered another way of solving many complications of transmitting, saving, and dealing with large amount of information in digital signal processing [6].

Unlike BL that contains only ‘T’ and ‘F’ states, MVL has a root greater than 2 and contains values with a finite and unlimited number of values [7]. Recently, MVL is used for high-speed arithmetic operations due to its high speed and carries less generation. A carry free arithmetic operation can be achieved using MVL. Since a huge number of gates are presented in MVL, circuit complexity will be less and require small area than binary computer [8].

Recently, Optical Multi-Valued Logic (OMVL) functions have been shown a great importance in the last years towards optical logic and information processing applications. Applying OMVL can overcome the limitations faced by traditional binary electronic technologies. This technology could be a revolution in achieving very high levels speed of data processing. Also, it provides a complexity inherent in electronics computing system [9].

Most of the existed MVL models are depending on Three-Valued Logic (3-VL) [10–18] and Four-Valued Logic (4-VL) [19,20]. They follow the mechanism of the polarization encoded optical shadow casting scheme help which vertical and horizontal strips of opaque and transparent bars represent the logical states.

We aim in this work to define a new mathematical model based on OMVL that can execute the basic computational operations for enhancing the speed of processing data and the amount of processed information. Mainly, this paper proposes new optical logic gates and reshape how to extend them to define the optical model of the PLUS operation which can derive the other operations from. It is organized as following. Section 1 presents the literature review and related works. Section 3 discusses the proposed optical binary OR gate. Section 4 presents the mathematical model of defining the PLUS operation. Section 5 provides an explanation of the construction of an optical full adder as an application of the defined mathematical model with validating it. Finally, we provide the conclusions of the work.

Most of the efforts focus on extending classic binary logic to multi-valued logic to increase their abilities in representation and data processing. These efforts concentrate on implementing MVL in the fields of electronics using Metal-Oxide Semiconductor (MOS), Carbon Nanotube Field Effect Transistors (CNFET), and Quantum-dot Cellular Automata (QCA) technologies. Also, MVL is implemented in optical field using either many technologies that will listed in this section.

MOS is one of the earlier strategies used to design ternary logic circuit. Balla et al. [10] pro-posed an MOS ternary-logic family which includes a set of inverters, NAND gates, and NOR gates that were used to design basic 3-VL arithmetic and memory circuits. The circuits thus developed are shown to have low power dissipation, and reduced propagation delay and component count compared to other known ternary circuits.

Carbon Nanotube Field Effect Transistors (CNFET) provided an alternative technique to MOS transistors in the application areas that require high performance computations and low power consumptions. Merlin et al. [14] had been utilized CNFET and introduced a strategy that enabled carry Propagate-Generate to be used in a ternary addition leading to the use of 3-VL prefix adders. Also, their results showed that the delay was reduced (up to 28%) by using binary prefix networks for calculating the carry and obtain lower power (up to 29%) by using low power encoder.

Bhoi et al. [11] proposed a redundant adder architecture using QCA technology to improve the computation speed in ternary logic-based adders. Various experimentations were conducted and proved that Ternary QCA approach assures the efficiency of the proposed computational components; adder, decoder, and binary increment and decrement.

Li et al. [13] presented the design criteria of ternary MVL optical calculators based on Decrease-Radix Design rule. The experimentations which applied on hundreds-bit reconfigurable calculator state that the formation of any logic calculator can be constructed using a few basic units and hence increase the simplicity of the structures.

Datta et al. [21] presented an optical Full Adder design using switching mechanism of Mach–Zehnder Interferometers (MZI) based on Electro-Optic effect. It accepts three digital electrical signals and produces both sum and carry bits in optical domain. The simulations were carried out to validate the ability to handle power dissipation issue by passing three input control signals to the electrodes of three distinct groups of MZIs.

Roy et al. [16] proposed 3-VL half adder circuit with Optical Nonlinear Material switch (OPNLM) schema based on manipulating polarized lights of logical states. Various ternary optical circuits were constructed on the basis of the proposed half adder including; incrementer/ decrementer (INC/DEC) component and full adder circuit. Significant advantages were provided compared to some other circuits including: reduce the time lagging in data processing, minimize the required circuit components, and increase the computational performance.

Obiniyi et al. [15] suggested a design of Ternary Arithmetic and Logic Unit based on ternary color-coded for manipulating ternary calculations. Hence, they suggested modelling a ternary half adder circuit and separate carry value.

Patel et al. [22] presented arithmetic operations using MVL in both Modulo-4 arithmetic and Galois field. Also, Quaternary-binary and binary-quaternary converters, Negation in modular arithmetic, and quaternary multiplier circuit were proposed.

This paper establishes a novel MVL model with octal optical gates based on the different eight light colors to provide an efficient solution to the limitations of computational speed and complexity inherent in the electronics computing. Section 2.1 introduces hypothesis of modelling colors as logic states. Section 2.2 presents a new construction of optical binary OR gate, Optical Binary AND, Optical NOT, and optical dispersion gate from the standpoint of physics laws.

Nowadays, most of computers and some types of printers use (Red–Green–Blue) RGB or (Cyan–Magenta–Yellow–Black) CMYK coloring systems to generate colored images and photos. In this research, we present the RGB and CMYK coloring systems not for generating images but to prove that they can be used in constructing the basis of the color numbering model that enhances the computational operations due to their abilities to generate thousands of various visible colors by changing the grey levels of basic colors processes.

Logic Color State Model (LCSM) is a numerical representation model in which a color represents an octal logic sate. Fig. 1 shows RGB light coloring model. Suppose that the red and green light colors are represented by images

and

, respectively. According to the optical observations by projecting R and G together on a surface, it gives the wave intensity of yellow light color which can be stated as images

In the same sense, if the blue light color is represented by images

, then by projecting R with B, it gives magenta light color which can be stated as images

. Also projecting G with B gives cyan light color which can be stated as images

. The white light color consists of the three basic colors Red (R), Green (G), and Blue (B), and it can be stated as images

. By considering the black color as an infrared that can be added to any light color, it is represented by images

in our color light sate model. Fig. 2 shows the color logic states by representing each color with its value. Tab. 1 illustrates the mapping between each logic color state and its corresponding basis numbers within our proposed 8-VL.

Like binary logic, many basic functions are existed in MVL. They are differing based on the base and number of used variables. Different logic functions could be constructed depending on MVL. The proposed MVL 8-VL is consists of 8 logical states K, R, Y, G, B, M, C, W. Set images

and

, a function images

, then the number of possible functions images

that can be generated is calculated according to Eq. (1), [7]:

Where N is the number of possible functions, b is the base or radix, and n is the number of variables used in the function. In binary logic of one variable (b = 2, n = 1), there are 4 possible functions. When b = 2, n = 2, then f(2, 2) = 2(2)2 = 8. The possible functions of one input variable for 8-VL is f(8, 1) = 8(8)1 = 16,777,216, and their numbers for two variables images

. These numbers are very huge. Hence, incredible numbers of functions or logical gates can possible for more variables. The next subsections discuss in detail the basic functions associated with our proposed 8-VL including: optical binary OR, Optical Binary AND, Optical NOT, and optical dispersion gates.

The behavior of the proposed optical binary OR gate simulates the behavior of the additive light property. In fact, they both obtain one optical beam in response to projection of more than one optical beam onto special types of prism or optical fibers to be one beam as shown in Fig. 3.

Let

, and X = (l1, l2), where images

, an optical binary OR function images

can be modelled as: images

. Where (

) means a binary OR operation by applying the or truth table on the binary representation of inputs, The symbolic representation of the optical binary OR gate of input X and Y is shown in Fig. 4. Tab. 2 illustrates the truth table of the possible states in the proposed 8-VL emergent from mixing two color lights A and B. It is denoted by images

and gets 64 resultant values.

The idea of the construction of light filters is to selectively transmit the desired wavelengths while restricting others. The two most common used types of filters are absorption and interference filters. Our proposed optical binary AND gate mimics the behavior of the absorption filters that absorb unwanted wavelengths, while the proposed optical binary NOT gate mimics the behavior of the interference filters in the way of removing selected wavelengths by internal destructive interference and reflection. Fig. 5 shows the mechanism of implementing a magenta filter that selectively transmits the red and blue portions of the incident white light spectrum and absorbs most of the green wavelengths.

Optical Binary AND Red Gate. Optical binary AND Red gate is built based on the work behavior of the red absorption filter. It is denoted by images

that accepts optical light X and performs binary AND operation with red light.

Let

, and X = (l) where images

, then

can be modeled as function images

,where

. Where (&) means binary AND operation by applying And truth table on the binary representation of inputs. Fig. 6 shows the symbolic representation of the optical binary AND Red gate of images

and Tab. 3 illustrates its truth table.

Another Binary AND Gates. The absorption filters can be applied to any sort of color state to design other optical AND gates. Let images

, X = (l), and l, images

then

can be modelled as function images

, where

and

are optical Binary AND gates that accept one optical light X and perform binary AND operation with the specified built-in color light. Fig. 7 shows their symbolic representations while Tab. 4 illustrates their truth tables.

Our proposed negation gate (i.e., optical binary NOT gate) is constructed based on the work behavior of the interference filters. Unlike the absorption filters, the interference filters reflect and destructively interfere with unwanted wavelengths as opposed to absorbing them. The term dichroic arises from the fact that the interference filter appears one color under illumination with transmitted light and another color with reflected light. Fig. 8 shows how the magenta dichroic filter reflects the green light and permitting the magenta light to be transmitted from the other side of the filter.

Optical NOT Red Gate. Optical binary NOT Red gate is built based on the work behavior of the cyan interference filter. The cyan dichroic filter reflects the red light and permitting the cyan light to be transmitted from the other side of the filter. The reflected light performs images

operation and is denoted by images

. Let

, and X = (l), images

, then optical binary NOT Red gate ( images

) can be modelled as function images

, where

. Fig. 9 shows the symbolic representation of the optical binary NOT Red Gate of images

and Tab. 5 illustrates its truth table.

Another Optical Binary NOT Gates. As we worked through designing the optical binary NOT Red gate by implementing the behavior of the cyan interference filter, the other colored filters can be used to design with related color the other optical NOT gates.

Let

, X = (l), and images

then

can be modelled as function images

, where

and

are optical Binary NOT gates that accept optical light X and perform negation operation of the specified built-in color light. Fig. 10 shows their symbolic representations and Tab. 6 illustrates their truth tables.

The behavior of our proposed optical dispersion gate simulates the behavior of the dispersion light property. In fact, they both accept one optical beam and disperse the projected light to its basic constituent colors as shown below in Fig. 11.

So far, there are two types of the optical dispersion gate based on their ability to disperse lights. The first one can disperse the projected light color to its all constituent logical states as shown in Fig. 11. The other type is one that disperses the input projected light color to its three basic constituent logical states as called shown in Fig. 12. Our proposed optical 3-Dispersion Gate (3DG) follows the characteristics of the second type. Fig. 13 shows the symbolic representation of the optical dispersion gate and Tab. 7 illustrates its truth table.

The proposed model provides a new definition for the PLUS operation based on the definition of binary OR gate in MVL. Section 4.1 defines the binary OR function. Section 4.2 defines the PLUS operation for any MVL. Section 4.3 highlights the definition of the plus-carry function.

Binary OR function defined in binary logic as: images

is True when A OR B, or both, are True but in MVL has another definition. Let images

is the set of powers of 2 where images

, if we have Zn is MVL with n states where images

, and

then

are the set of powers of 2 that found in a, b respectively, these sets A, B calculated according to Eq. (2).

Where i is the position of the index of element of 2’s power in set A. The Binary OR function in MVL images

is calculated by modulo n of summation of not repeated powers of 2 that found in inputs a and b in other words images

calculated by Eq. (3).

For example: Suppose we have images

and let a = 3, b = 1 and the sets of powers of 2 for a, b images

then

and

The binary OR function of images

can work as PLUS function on Zn of a, b if there is no repeated powers of 2 appear in input sets A, B and that means images

. So, binary OR function will not equal PLUS function in cases that powers of 2 is repeated in the set A, B; in other words: images

, Ai = Bj. for example: Suppose we have images

as MVL and let a = 3, b = 1 and images

then

where

and (a + b)Zn = 0 so, images

Form binary OR operation definition, we can add a new definition of PLUS operation for any MVL as binary OR operation with replacing the cases that has repeated Powers of 2 of inputs with another value images

that matched with PLUS operation, by define images

where n is the number of states in MVL. This images

value will be a new input to the recursive PLUS function with images

, until there is no any repeated item then applying binary or operation, the final images

definition of can defined by Eq. (4).

The definition of images

Value is changed from MVL to another. In our case of 8-VL, images

have the values: images

For example: Suppose images

as MVL , let a = 3, b = 1 and images

, and

value in Z4 defined as 1+1 = 2, 2+2 = 0. Thus: images

and

. Here,

by setting images

and applying the Eq. (4) then images

by applying the recursive Eq. (4) again on the result by replacing images

then

and

then

will be 0.

The carry value C of PLUS operation is the next state of the result if the summation greater than or equals the number of states in MVL. For any MVL Zn for a, b images

Zn can also depend on the sets of powers of 2 images

that found in a, b. The C value will be the first state of Zn if there exist a summation of two or more powers of 2 is greater than n so C can be calculated according to Eq. (5)

For example: In binary logic, the only state that gives carry value is: (1+1), in 3-VL (1+2, 2+2), in 4-VL there is one case (2+2) and so on. In our case study 8-VL, there is only one case can give carry value if exist (4+4) in the set of powers of 2 in PLUS operation.

The full adder in the binary logic circuits represents the main core of any micro-controller or processor. The subsequent sub-sections implement the defined mathematical model of PLUS operation by representing an optical full adder construction based on proposed optical 8-VL gates. Section 5.1 uncovers the principle and the implementation of constructing the optical tiny adders. Section 5.2 discusses in detail the design of the proposed optical full adder. Section 5.3 introduces set of case studies to validate the functionality of the optical full adder.

The architecture of the proposed optical adder is consisting of three tiny adders: Optical Red Tiny Adder (ORTA), Optical Green Tiny Adder (OGTA), and Optical Blue Tiny Adder (OBTA). The idea behind constructing the tiny adders is built on extending the abilities of manufactured nonlinear materials that can double the frequency of the light passes through them and generates new light [23]. Each tiny adder may be constructed using one optical component or sort of simple optical gate.

ORTA is one type of OTA that can only makes PLUS operation on 3 Red beams as input values and the output may be red or green or both. Let images

, and L1,

where

, and

and l1, l2, images

, Then, ORTA can be modelled as the function images

where f defined as in Eq. (6). Fig. 14 shows the simple circuit design of ORTA given inputs; L1, L2 and images

and Tab. 8 illustrate its truth table.

OGTA is another type of OTA that can only make PLUS operation on 3 green beams as input values and the output may be either green or blue or both. Let images

, and L1,

where

, and

and l1, l2, images

, Then OGTA can be modelled as the function images

where f defined as in Eq. (7). Fig. 15 shows the simple circuit design of OGTA given inputs; L1, L2 and images

where Tab. 9 illustrate its truth table.

OBTA is another type of OTA that can only make PLUS operation on 3 blue beams as input values and the output may be blue or red or both. Let images

, and L1,

where

, and

, and l1, l2, images

, Then, OBTA can be modelled as the function images

where f is defined as given in Eq. (8). Fig. 16 shows the simple circuit design of OBTA given inputs: L1, L2 and images

where Tab. 10 illustrates its truth table.

Optical Full Adder (OFA) is an optical device that implements the PLUS operation of two octal values. It consists of the three types of the tiny adders; ORTA, OGTA, and OBTA. OFA has three input values: L1, L2, and empty value of Cin and generate Z as a summation result of the PLUS operation and CO as the resultant carry. Tab. 11 illustrates the truth table of OFA.

The design of OFA is consisting of two optical 3-dispersion gates followed by three optical tiny adders; ORTA, OGTA, OBTA and then an optical binary OR gate as shown in Fig. 13. The projected light colors L1 and L2 are dispersed to their basic constituent colors using the first and the second optical 3-dispersion gates respectively. The dispersed lights are propagated to their related optical tiny adders. The resultant values of the three tiny adders are passed through the optical binary OR gate to produce Z as the result of the PLUS operations. The carry CO is emitted at the end from OBTA as a red value as illustrated above in Tab. 11. The resultant carry value is connected as a carry input to another OFA in the chain of computation as shown in Fig. 17.

In order to assure the validity of the defined mathematical model of the PLUS operation and hence the proposed optical full adder model, four case studies are used according to the description illustrated in Tab. 12.

In Case study (I), the inputs values are G and B as L1 and L2 inputs to optical 3-dispersion gates (3DG1) and (3DG2) respectively, the output of (3DG1) will be G, that will connected as first input of OGTA, and the output of (3DG2) will be B connected as a second input of OBTA, the output of OGTA is G, and the output of OBTA is B, the output values G and B will be inputs to binary OR Gate and the output will be C. In Case study IV, the inputs values are C and M as L1 and L2 inputs to (3DG1) and (3DG2) respectively the output of (3DG1) will be G and B that will connected as first inputs of OGTA and OBTA respectively, the output of (3DG2) will be R and B connected as a second inputs of ORTA and OBTA respectively, the output of ORTA is R and output of OGTA is G, the output of OBTA is R, the output values R and G will be inputs to the binary OR Gate and the output will be Y and the carry value is R. Tab. 13 display the inputs and outputs of each case for all components in OFA.

In this paper, a new optical 8-VL model was proposed to provide an efficient approach for handling the computational speed and complexity issues in the classical logic. CLSM was defined as a numerical color representation model for describing the optical logical states. The proposed optical gates in 8-VL were constructed and their truth table were defined. The optical binary OR gate mimics the behavior of the additive light property. The optical binary AND gate mimics the behavior of the of the absorption filter. The optical binary NOT gate mimics the behavior of the of the interference filter. The optical dispersion gate mimics the behavior of the of the dispersion light property filter. We introduced a new definition for the binary OR function that led us to establish a new mathematical representation for the PLUS operation in MVL. Also, a novel construction for the full adder that sum octal numbers in optical MVL is proposed. Four case studies were used and proved the validity of the proposed models.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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Computers, Materials & Continua DOI:10.32604/cmc.2021.015759
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