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.

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 [

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 [

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 [

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 [

Most of the existed MVL models are depending on Three-Valued Logic (3-VL) [

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. [

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. [

Bhoi et al. [

Li et al. [

Datta et al. [

Roy et al. [

Obiniyi et al. [

Patel et al. [

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.

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

Light color | K | R | G | Y | B | M | C | W |
---|---|---|---|---|---|---|---|---|

Octal representation | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

Binary representation | 0 | 1 | 01 | 11 | 100 | 101 | 110 | 111 |

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

Where N is the number of possible functions, b is the base or radix, and ^{(2)2} = 8. The possible functions of one input variable for 8-VL is ^{(8)1} = 16,777,216, and their numbers for two variables

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

Let _{1}, _{2}), where

K | K | K | R | K | R | G | K | G | Y | K | Y | B | K | B | M | K | M | C | K | C | W | K | W | |||||||

K | R | R | R | R | R | G | R | Y | Y | R | Y | B | R | M | M | R | M | C | R | W | W | R | W | |||||||

K | G | G | R | G | Y | G | G | G | Y | G | Y | B | G | C | M | G | W | C | G | C | W | G | W | |||||||

K | Y | Y | R | Y | Y | G | Y | Y | Y | Y | Y | B | Y | W | M | Y | W | C | Y | W | W | Y | W | |||||||

K | B | B | R | B | M | G | B | W | Y | B | W | B | B | B | M | B | M | C | B | C | W | B | W | |||||||

K | M | M | R | M | M | G | M | W | Y | M | W | B | M | M | M | M | M | C | M | W | W | M | W | |||||||

K | C | C | R | C | W | G | C | C | Y | C | W | B | C | C | M | C | W | C | C | C | W | C | W | |||||||

K | W | W | R | W | W | G | W | W | Y | W | W | B | W | W | M | W | W | C | W | W | W | W | W |

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.

Let

K | R | K |

R | R | R |

G | R | K |

Y | R | R |

B | R | K |

M | R | R |

C | R | K |

W | R | R |

_{2} |
||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

K | G | K | K | Y | K | K | B | K | K | M | K | K | C | K | ||||

R | G | K | R | Y | R | R | B | K | R | M | R | R | C | K | ||||

G | G | G | G | Y | G | G | B | K | G | M | K | G | C | G | ||||

Y | G | G | Y | Y | Y | Y | B | K | Y | M | R | Y | C | G | ||||

B | G | K | B | Y | K | B | B | B | B | M | B | B | C | B | ||||

M | G | K | M | Y | R | M | B | B | M | M | M | M | C | B | ||||

C | G | G | C | Y | G | C | B | B | C | M | B | C | C | C | ||||

W | G | G | W | Y | Y | W | B | B | W | M | M | W | C | C |

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.

K | R | K |

R | R | K |

G | R | G |

Y | R | G |

B | R | B |

M | R | B |

C | R | C |

W | R | C |

Let

K | G | K | K | Y | K | K | B | K | K | M | K | K | C | K | ||||

R | G | R | R | Y | G | R | B | R | R | M | K | R | C | R | ||||

G | G | K | G | Y | R | G | B | G | G | M | G | G | C | K | ||||

Y | G | R | Y | Y | K | Y | B | Y | Y | M | G | Y | C | R | ||||

B | G | B | B | Y | B | B | B | K | B | M | K | B | C | K | ||||

M | G | M | M | Y | B | M | B | R | M | M | K | M | C | R | ||||

C | G | B | C | Y | B | C | B | G | C | M | G | C | C | K | ||||

W | G | M | W | Y | B | W | B | Y | W | M | G | W | C | R |

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

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

K | K | K | K |

R | R | K | K |

G | K | G | K |

Y | R | G | K |

B | K | K | B |

M | R | K | B |

C | K | G | B |

W | R | G | B |

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: _{n}

Where

For example: Suppose we have

The binary OR function of _{n}_{i}_{j}_{Zn} = 0 so,

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

The definition of

For example: Suppose _{4} defined as 1+1 = 2, 2+2 = 0. Thus:

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 _{n}_{n} can also depend on the sets of powers of 2 _{n}

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 [

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 _{1}, _{1}, _{2}, _{1}, _{2} and

_{1} |
_{2} |
|||
---|---|---|---|---|

K | K | K | K | K |

R | K | K | R | K |

K | R | K | R | K |

R | R | K | K | G |

K | K | R | R | K |

R | K | R | K | G |

K | R | R | K | G |

R | R | R | R | G |

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 _{1}, _{1}, _{2}, _{1}, _{2} and

_{1} |
_{2} |
|||
---|---|---|---|---|

K | K | K | K | K |

G | K | K | G | K |

K | G | K | G | K |

G | G | K | K | B |

K | K | G | G | K |

G | K | G | K | B |

K | G | G | K | B |

G | G | G | G | B |

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 _{1}, _{1}, _{2}, _{1}, _{2} and

_{1} |
_{2} |
|||
---|---|---|---|---|

K | K | K | K | K |

B | K | K | B | K |

K | B | K | B | K |

B | B | K | K | R |

K | K | B | B | K |

B | K | B | K | R |

K | B | B | K | R |

B | B | B | B | R |

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 _{O}

_{1} |
_{2} |
_{O} |
_{1} |
_{2} |
_{O} |
_{1} |
_{2} |
_{O} |
_{1} |
_{2} |
_{O} |
_{1} |
_{2} |
_{O} |
_{1} |
_{2} |
_{O} |
_{1} |
_{2} |
_{O} |
_{1} |
_{2} |
_{O} |
|||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

K | K | K | K | R | K | K | R | G | K | K | G | Y | K | K | Y | B | K | K | B | M | K | K | M | C | K | K | C | W | K | W | W | |||||||

K | R | K | R | R | R | K | G | G | R | K | Y | Y | R | K | B | B | R | K | M | M | R | K | C | C | R | K | W | W | R | R | K | |||||||

K | G | K | G | R | G | K | Y | G | G | K | B | Y | G | K | M | B | G | K | C | M | G | K | W | C | G | R | K | W | G | R | R | |||||||

K | Y | K | Y | R | Y | K | B | G | Y | K | M | Y | Y | K | C | B | Y | K | W | M | Y | R | K | C | Y | R | R | W | Y | R | G | |||||||

K | B | K | B | R | B | K | M | G | B | K | C | Y | B | K | W | B | B | R | K | M | B | R | R | C | B | R | G | W | B | R | Y | |||||||

K | M | K | M | R | M | K | C | G | M | K | W | Y | M | R | K | B | M | R | R | M | M | R | G | C | M | R | Y | W | M | R | B | |||||||

K | C | K | C | R | C | K | W | G | C | R | K | Y | C | R | R | B | C | R | G | M | C | R | Y | C | C | R | B | W | C | R | M | |||||||

K | W | K | W | R | W | R | K | G | W | R | R | Y | W | R | G | B | W | R | Y | M | W | R | B | C | W | R | M | W | W | R | C |

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 _{1} and _{2} 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 _{O}

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

C | Z | |||
---|---|---|---|---|

I | G | B | K | C |

II | Y | M | R | K |

III | W | W | R | C |

IV | C | M | R | Y |

In Case study (I), the inputs values are _{1} and _{2} 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 _{1} and _{2} inputs to (3DG1) and (3DG2) respectively the output of (3DG1) will be

_{1} |
_{2} |
||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

G | B | K | G | K | K | K | B | K | K | K | K | K | K | G | K | G | K | K | K | B | B | K | K | G | B | C | K |

Y | M | R | G | K | R | K | B | K | R | R | K | G | G | G | K | K | B | B | K | B | K | R | R | K | K | K | R |

W | W | R | G | B | R | G | B | K | R | R | K | G | G | G | G | G | B | B | B | B | B | R | R | G | B | C | R |

C | M | K | G | B | R | K | B | K | K | R | R | K | K | G | K | G | K | K | B | B | K | R | R | G | K | Y | R |

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.