Journal of Advances in Mathematics and Computer Science, 2019
An -bit comparator is a celebrated combinational circuit that compares two -bit inputs and and ... more An -bit comparator is a celebrated combinational circuit that compares two -bit inputs and and produces three orthonormal outputs: G (indicating that is strictly greater than ), E (indicating that and are equal or equivalent), and L (indicating that is strictly less than ). The symbols ‘G’, ‘E’, and ‘L’ are deliberately chosen to convey the notions of ‘Greater than,’ ‘Equal to,’ and ‘Less than,’ respectively. This paper analyzes an -bit comparator in the general case of arbitrary and visualizes the analysis for on a regular and modular version of the 8-variable Karnaugh-map. The cases 3, 2, and 1 appear as special cases on 6-variable, 4-variable, and 2-variable submaps of the original map. The analysis is a tutorial exposition of many important concepts in switching theory including those of implicants, prime implicants, essential prime implicants, minimal sum, complete sum and disjoint sum of products (or probability-ready expressions).
International Journal of Mathematical, Engineering and Management Sciences
Boolean curve fitting is the process of finding a Boolean function that takes given values at cer... more Boolean curve fitting is the process of finding a Boolean function that takes given values at certain points in its Boolean domain. The problem boils down to solving a set of ‘big’ Boolean equations that may or may not be consistent. The usual formulation of the Boolean curve fitting problem is quite complicated, indeed. In this paper, we formulate the Boolean curve fitting problem using the technique of atomic decomposition of Boolean equations. This converts the problem into a set of independent switching equations. We present the solution of these switching equations and express the solution in very simple and compact forms. We also present the consistency and uniqueness conditions for this problem again in very compact forms. A few illustrative examples are given. These examples clearly pinpoint the simplicity gained by the Boolean-equation solving step within the overall Boolean-fitting procedure. The method presented here can be applied to the design of Boolean functions for c...
Any variable in Intuitionistic Fuzzy Logic (IFL) is either a Realistic Fuzzy Tautology (RFT) with... more Any variable in Intuitionistic Fuzzy Logic (IFL) is either a Realistic Fuzzy Tautology (RFT) with a Truth exceeding one half, or a non-Realistic Fuzzy Tautology (nRFT) with a Truth less than or equal to one half. This results in a dichotomy somewhat similar to that of the Excluded Middle in Ordinary Logic (OL) albeit allowing both Falsity and Hesitancy in addition to Truth in an IFL variable. Consequently, many problems (and solutions) in Boolean logic can be fuzzified without any significant change in their essence. We show herein that one such problem is that of Boolean satisfiability. We handle this problem by converting a CNF expression into a disjoint DNF one, and solving the resulting two-valued Boolean equation. This solution strategy is essentially retained in IFL, thanks to the RFT concept. All steps needed in the fuzzification process are proved, and a demonstrative example illustrates the method in both crisp and intuitionistic fuzzy cases.
This paper is a detailed tutorial exposition of the analysis of synchronous Boolean networks via ... more This paper is a detailed tutorial exposition of the analysis of synchronous Boolean networks via a particular matrix product called the Semi-Tensor Product (STP) of matrices, which multiplies two matrices and !" in which the column dimension # of the first matrix is not necessarily equal to the row dimension $ of the second matrix, but is possibly a multiple or divisor of it. The state space of a Boolean network of # nodes is denoted herein by a vector of 2 states in natural order obtained as the STP of # 2-element vectors representing the network variables. A notable contribution of the paper is that its matrix expression of logic follows the conventional truth-table order, and not the reverse unfamiliar order followed so far by the STP community. We reproduce the STP analysis of a classical example network. We include minute details that make the STP manipulations easily accessible to and more understandable by their potential users. Our analysis points to more efficient ...
Biomedical Journal of Scientific and Technical Research, 2020
The Electrocardiograph (ECG) is one of the most successful medical diagnostic tools. The ECG can ... more The Electrocardiograph (ECG) is one of the most successful medical diagnostic tools. The ECG can show, roughly speaking, all types of heart disorders that appear as ECG signal arrhythmias. In this paper, a universal ECG signal arrhythmias classification system is proposed. The proposed system is based on using the wavelet transform in two of its known forms, namely, the Discrete Wavelet Transforms (DWT) and the Wavelet Packet Transform (WPT), or a combination thereof. The purpose of the research reported herein is to find out a universal classification system; in the sense of providing a capability for simultaneous classification of all types of known heart arrhythmias.
Prognostics is a term that engineering borrowed from medicine to refer to the discipline concerne... more Prognostics is a term that engineering borrowed from medicine to refer to the discipline concerned with the Remaining Useful Life (RUL) of an engineering device. This paper surveys the RUL prediction techniques and classifies them into four categories of model-based techniques,knowledge-based techniques, experience-based techniques, and data-driven techniques. A comparative review is given for the main features, prominent advantages, potential shortcomings and main subcategories for each of these categories. The survey is supported by an extensive listfor up-to-date references.
Journal of Network and Computer Applications, 2020
Abstract Internet Service Providers (ISPs) are expected to continuously provide services without ... more Abstract Internet Service Providers (ISPs) are expected to continuously provide services without interruptions. However, planning for this activity requires a number of tasks such as forecasting clients' demand that needs to be considered in fulfilling their objectives particularly in face of calamity and traffic load variations. We present a Software-Defined Networking (SDN) load distribution technique for a typical ISP. We have implemented an optimization method employing linear programming to achieve this goal. A multi-objective model has been developed for this purpose. The multiple objectives are maximization of link utilization, maximization of link availability and minimization of average round-trip time for the top visited websites by users. Furthermore, an automated load distribution system has been developed for carrying out these optimization related activities. In addition, an example is presented to illustrate traffic load distribution under different operational modes (normal, critical, and catastrophic). The findings of this study should be of great help to the service provider to better plan for load distribution between international links, and help in approving down times and maintenance windows requested by cable providers.
journal of King Abdulaziz University Engineering Sciences, 2017
This paper studies Satisfiability (SAT) in finite atomic Boolean algebras larger than the two-val... more This paper studies Satisfiability (SAT) in finite atomic Boolean algebras larger than the two-valued one B2, which are named big Boolean algebras. Unlike the formula ݃(ࢄ (in the SAT problem over B2, which is either satisfiable or unsatisfiable, this formula for the SAT problem over a big Boolean algebra could be unconditionally satisfiable, conditionally satisfiable, or unsatisfiable depending on the nature of the consistency condition of the Boolean equation {݃(ࢄ = (1}, since this condition could be an identity, a genuine equation, or a contradiction. The paper handles this latter SAT problem by using a conventional method and a novel one for deriving parametric general solutions, and subsequently utilizing expansion trees for generating all particular solutions of the aforementioned Boolean equation. Each of these two methods could be cast in pure algebraic form, but becomes much easier to visualize and comprehend when presented via the natural map of a big Boolean algebra, which ...
journal of King Abdulaziz University Engineering Sciences, 2017
This paper explores the similarities and differences between two prominent problems in the mathem... more This paper explores the similarities and differences between two prominent problems in the mathematics of Boolean functions. The first of these problems is that of Boolean curve fitting (BCF), also known as Boolean interpolation, which deals with constructing a curve 𝑧 = 𝑓(𝐗) through a number of points z𝑘 = 𝑓(𝐗𝑘 ) where 𝑘 = 1,2, … , 𝑚. The second problem is the Inverse Problem of Boolean equations (IPBE), which constructs a Boolean function whose zeroes are all known. While the problem of Boolean curve fitting might require a consistency condition for its solution, the Inverse Problem of Boolean equations might use a consistency condition as an input. Without a consistency condition, the Inverse Problem of Boolean equations can be viewed as a special case of the problem of Boolean curve fitting, provided the specified points z𝑘 are the only zeros of 𝑓(𝐗). Our findings are illustrated via a detailed typical example.
journal of King Abdulaziz University Engineering Sciences, 2012
Manual methods of solving big Boolean equations include purely-algebraic methods and methods util... more Manual methods of solving big Boolean equations include purely-algebraic methods and methods utilizing the variable-entered Karnaugh map (VEKM). The VEKM techniques are demonstrated to be at least competitive with (and occasionally superior to) purelyalgebraic techniques, since the VEKM techniques (a) include purelyalgebraic methods as special cases, and (b) have a better control on the minimality of the pertinent function representations, and hence are more capable of producing more compact general solutions.
Journal of Advances in Mathematics and Computer Science, 2019
An -bit comparator is a celebrated combinational circuit that compares two -bit inputs and and ... more An -bit comparator is a celebrated combinational circuit that compares two -bit inputs and and produces three orthonormal outputs: G (indicating that is strictly greater than ), E (indicating that and are equal or equivalent), and L (indicating that is strictly less than ). The symbols ‘G’, ‘E’, and ‘L’ are deliberately chosen to convey the notions of ‘Greater than,’ ‘Equal to,’ and ‘Less than,’ respectively. This paper analyzes an -bit comparator in the general case of arbitrary and visualizes the analysis for on a regular and modular version of the 8-variable Karnaugh-map. The cases 3, 2, and 1 appear as special cases on 6-variable, 4-variable, and 2-variable submaps of the original map. The analysis is a tutorial exposition of many important concepts in switching theory including those of implicants, prime implicants, essential prime implicants, minimal sum, complete sum and disjoint sum of products (or probability-ready expressions).
International Journal of Mathematical, Engineering and Management Sciences
Boolean curve fitting is the process of finding a Boolean function that takes given values at cer... more Boolean curve fitting is the process of finding a Boolean function that takes given values at certain points in its Boolean domain. The problem boils down to solving a set of ‘big’ Boolean equations that may or may not be consistent. The usual formulation of the Boolean curve fitting problem is quite complicated, indeed. In this paper, we formulate the Boolean curve fitting problem using the technique of atomic decomposition of Boolean equations. This converts the problem into a set of independent switching equations. We present the solution of these switching equations and express the solution in very simple and compact forms. We also present the consistency and uniqueness conditions for this problem again in very compact forms. A few illustrative examples are given. These examples clearly pinpoint the simplicity gained by the Boolean-equation solving step within the overall Boolean-fitting procedure. The method presented here can be applied to the design of Boolean functions for c...
Any variable in Intuitionistic Fuzzy Logic (IFL) is either a Realistic Fuzzy Tautology (RFT) with... more Any variable in Intuitionistic Fuzzy Logic (IFL) is either a Realistic Fuzzy Tautology (RFT) with a Truth exceeding one half, or a non-Realistic Fuzzy Tautology (nRFT) with a Truth less than or equal to one half. This results in a dichotomy somewhat similar to that of the Excluded Middle in Ordinary Logic (OL) albeit allowing both Falsity and Hesitancy in addition to Truth in an IFL variable. Consequently, many problems (and solutions) in Boolean logic can be fuzzified without any significant change in their essence. We show herein that one such problem is that of Boolean satisfiability. We handle this problem by converting a CNF expression into a disjoint DNF one, and solving the resulting two-valued Boolean equation. This solution strategy is essentially retained in IFL, thanks to the RFT concept. All steps needed in the fuzzification process are proved, and a demonstrative example illustrates the method in both crisp and intuitionistic fuzzy cases.
This paper is a detailed tutorial exposition of the analysis of synchronous Boolean networks via ... more This paper is a detailed tutorial exposition of the analysis of synchronous Boolean networks via a particular matrix product called the Semi-Tensor Product (STP) of matrices, which multiplies two matrices and !" in which the column dimension # of the first matrix is not necessarily equal to the row dimension $ of the second matrix, but is possibly a multiple or divisor of it. The state space of a Boolean network of # nodes is denoted herein by a vector of 2 states in natural order obtained as the STP of # 2-element vectors representing the network variables. A notable contribution of the paper is that its matrix expression of logic follows the conventional truth-table order, and not the reverse unfamiliar order followed so far by the STP community. We reproduce the STP analysis of a classical example network. We include minute details that make the STP manipulations easily accessible to and more understandable by their potential users. Our analysis points to more efficient ...
Biomedical Journal of Scientific and Technical Research, 2020
The Electrocardiograph (ECG) is one of the most successful medical diagnostic tools. The ECG can ... more The Electrocardiograph (ECG) is one of the most successful medical diagnostic tools. The ECG can show, roughly speaking, all types of heart disorders that appear as ECG signal arrhythmias. In this paper, a universal ECG signal arrhythmias classification system is proposed. The proposed system is based on using the wavelet transform in two of its known forms, namely, the Discrete Wavelet Transforms (DWT) and the Wavelet Packet Transform (WPT), or a combination thereof. The purpose of the research reported herein is to find out a universal classification system; in the sense of providing a capability for simultaneous classification of all types of known heart arrhythmias.
Prognostics is a term that engineering borrowed from medicine to refer to the discipline concerne... more Prognostics is a term that engineering borrowed from medicine to refer to the discipline concerned with the Remaining Useful Life (RUL) of an engineering device. This paper surveys the RUL prediction techniques and classifies them into four categories of model-based techniques,knowledge-based techniques, experience-based techniques, and data-driven techniques. A comparative review is given for the main features, prominent advantages, potential shortcomings and main subcategories for each of these categories. The survey is supported by an extensive listfor up-to-date references.
Journal of Network and Computer Applications, 2020
Abstract Internet Service Providers (ISPs) are expected to continuously provide services without ... more Abstract Internet Service Providers (ISPs) are expected to continuously provide services without interruptions. However, planning for this activity requires a number of tasks such as forecasting clients' demand that needs to be considered in fulfilling their objectives particularly in face of calamity and traffic load variations. We present a Software-Defined Networking (SDN) load distribution technique for a typical ISP. We have implemented an optimization method employing linear programming to achieve this goal. A multi-objective model has been developed for this purpose. The multiple objectives are maximization of link utilization, maximization of link availability and minimization of average round-trip time for the top visited websites by users. Furthermore, an automated load distribution system has been developed for carrying out these optimization related activities. In addition, an example is presented to illustrate traffic load distribution under different operational modes (normal, critical, and catastrophic). The findings of this study should be of great help to the service provider to better plan for load distribution between international links, and help in approving down times and maintenance windows requested by cable providers.
journal of King Abdulaziz University Engineering Sciences, 2017
This paper studies Satisfiability (SAT) in finite atomic Boolean algebras larger than the two-val... more This paper studies Satisfiability (SAT) in finite atomic Boolean algebras larger than the two-valued one B2, which are named big Boolean algebras. Unlike the formula ݃(ࢄ (in the SAT problem over B2, which is either satisfiable or unsatisfiable, this formula for the SAT problem over a big Boolean algebra could be unconditionally satisfiable, conditionally satisfiable, or unsatisfiable depending on the nature of the consistency condition of the Boolean equation {݃(ࢄ = (1}, since this condition could be an identity, a genuine equation, or a contradiction. The paper handles this latter SAT problem by using a conventional method and a novel one for deriving parametric general solutions, and subsequently utilizing expansion trees for generating all particular solutions of the aforementioned Boolean equation. Each of these two methods could be cast in pure algebraic form, but becomes much easier to visualize and comprehend when presented via the natural map of a big Boolean algebra, which ...
journal of King Abdulaziz University Engineering Sciences, 2017
This paper explores the similarities and differences between two prominent problems in the mathem... more This paper explores the similarities and differences between two prominent problems in the mathematics of Boolean functions. The first of these problems is that of Boolean curve fitting (BCF), also known as Boolean interpolation, which deals with constructing a curve 𝑧 = 𝑓(𝐗) through a number of points z𝑘 = 𝑓(𝐗𝑘 ) where 𝑘 = 1,2, … , 𝑚. The second problem is the Inverse Problem of Boolean equations (IPBE), which constructs a Boolean function whose zeroes are all known. While the problem of Boolean curve fitting might require a consistency condition for its solution, the Inverse Problem of Boolean equations might use a consistency condition as an input. Without a consistency condition, the Inverse Problem of Boolean equations can be viewed as a special case of the problem of Boolean curve fitting, provided the specified points z𝑘 are the only zeros of 𝑓(𝐗). Our findings are illustrated via a detailed typical example.
journal of King Abdulaziz University Engineering Sciences, 2012
Manual methods of solving big Boolean equations include purely-algebraic methods and methods util... more Manual methods of solving big Boolean equations include purely-algebraic methods and methods utilizing the variable-entered Karnaugh map (VEKM). The VEKM techniques are demonstrated to be at least competitive with (and occasionally superior to) purelyalgebraic techniques, since the VEKM techniques (a) include purelyalgebraic methods as special cases, and (b) have a better control on the minimality of the pertinent function representations, and hence are more capable of producing more compact general solutions.
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