Karnaugh Map Research Articles

Design and Optimization of a 4-bit Absolute Value Detector

This paper presents the design and optimization of a 4-bit absolute value detector, focusing on enhancing circuit performance while minimizing delay and power consumption. A truth table was initially constructed to clearly understand the functionality of the detector. Logic expressions were simplified using Karnaugh maps, and the detector applied only NAND gates, NOR gates, and inverters. This approach not only improved the circuit's efficiency but also enabled the rapid identification of critical paths due to the design's high symmetry. The impact of input capacitance and supply voltage on delay and power consumption was analyzed. Increasing input capacitance reduced delay but increased overall power consumption, while decreasing supply voltage had the opposite effect. To further optimize performance, the internal capacitance of the last gate on the critical path was reduced, introducing a new set of constraints and power consumption equations. Minimum power consumption was calculated using MATLAB under the new constraints, resulting in a significant reduction compared to previous values. Experimental results demonstrated the effectiveness of the optimization strategy.

Genetic code, the problem of coding biological cycles, and cyclic Gray codes

This article is devoted to the problem of genetically coding of inherited cyclic structures in biological bodies, whose life activity is based on a great inherited set of mutually coordinated cyclic processes. The author puts forward and arguments the idea that the genetic coding system is capable of encoding inherited cyclic processes because it itself is a system of cyclic codes connected with Boolean algebra of logic. In other words, the physiological processes in question are cyclical because they are genetically encoded by cyclic codes. In support of this idea, the author presents a set of his results on the connection of the genetic coding system with cyclic Gray codes, which are one of many known types of cyclic codes. This opens up the possibility of using for modeling inherited cyclic biostructures those algebraic and logical theories and constructions that are associated with Gray codes and have long been used in engineering technologies: Karnaugh maps, Hilbert curve, Hadamard matrices, Walsh functions, dyadic analysis, etc. The author believes that when studying the origin, evolution and function of the genetic code, it is necessary to take into account the ability of the genetic system to encode many mutually related cyclic processes.

Задача о беспорядках

The problem of characterizing symmetric necklaces as disorders is considered. Such an idea is an alternative approach to the one using Poya's enumeration theorem, including Burnside's lemma. The relationship between partitions of numbers and permutation types is shown. Disorder types are associated with necklace types considered as permutations in which no element remains in place, as well as with disorders with fixed elements. The distributions of disorders by types for necklaces up to the seventh degree are studied. The number and enumeration of disorders of each type is carried out, and their evenness is established. Particular attention is paid to the study of the symmetric necklaces properties. Classes of symmetry and asymmetry are enumerated for each necklace type considered. At the same time, the concepts of chiral and achiral symmetry are introduced as varieties of axial symmetry. Relationships between the power of symmetry classes and the order of symmetry of necklaces are revealed. The concept of symmetry diagrams for necklaces is introduced. Diagrams could be applied for determining the properties of invariance of symmetric and asymmetric necklaces of a given length. The concept of congruence is used as an equivalence relation between necklaces. It allows implementing a geometric approach to the study of necklaces and the visualization of the results obtained. For this purpose, necklaces are associated with multiconnected non-oriented graphs, the vertices of which are the vertices of regular polygons. In this case, the number of vertices in the graph corresponds to the necklace’s length, and the number of connected components coincides with the number of bead colors. As an application of the results obtained, the possibility of studying the accent dynamics of poetic stanzas vertically using necklaces is considered. Empirical data on the use of various types of seven-line stanzas in poetic practice are presented. It is noted that approximately a quarter of the existing symmetric types of necklaces for various reasons do not find practical application.

Measuring the Impact of Augmented Reality in Flipped Learning Mode on Critical Thinking, Learning Motivation, and Knowledge of Engineering Students.

Digital electronics is a fundamental subject for engineering students, and it enables the students to learn design-based approaches and solve complex engineering problems. Students learn about minimization techniques for reducing the hardware components and size of the circuit by solving complex Boolean equations. The Karnaugh map (K-map) is one such technique utilized in digital electronics to solve complex Boolean equations and design AND-OR-INVERT (AOI) logical diagrams. The K-map technique involves several steps to solve the Boolean expression, and students often find it difficult to follow the K-map process. In this study, an AR-based learning system was developed using Unity 3D and Vuforia SDK that aimed to teach the students about the step-wise operation of the K-map technique. An experimental study was conducted with 128 undergraduate engineering students to determine the impact of the AR learning system on the critical thinking skills, learning motivation, and knowledge gain of students. The students were divided into two groups: experimental group (N = 64) and control group (N = 64). The AR learning system was implemented in flipped learning mode and utilized to provide in-class activities during the learning. The experimental group students utilized the AR learning system for in-class activities whereas control group students performed in-class activities using the traditional approach. The experimental outcomes indicate that the use of AR technology has a significant positive impact on the critical thinking skills, learning motivation, and knowledge gain of students. The study also found that critical thinking skills and learning motivation have a significant positive correlation with the knowledge gain of students in the experimental group.

A new logic circuits optimization algorithm using bipartite graph

Designing a logic circuit from the scratch requires its description in logical expression, (e.g. sum of products), and then the expression should be optimized to diminish the cost and complexity of the circuit by reducing the number of literals, the number of logical terms, and/or logical operations. Karnaugh map, K-Map, is the most popular method in the optimization process, but it suffers from many drawbacks such as its inefficiency or the inability to be used in minimizing logical expression containing more than four literals, in addition to the complexity of implementing it as a program. In this paper, we propose a new algorithm to optimize the logic circuits depending on the bipartite graph and some of the suggested mathematical operations. The proposed algorithm is simple for programming implementation, literal-unlimited number, and is easy to be visualized and understandable. Many of the logic circuits of 3, 4, 5, and 6 literals were optimized and the results were correctly matched with the results of the Karnaugh map. Also, tens of logic circuits of more than 6 literals are optimized and the results were correctly checked with their truth tables and Logic-Friday tool.

Digital Noise-Cancellation Circuit Implementation Using Proposed Algorithm and Karnaugh Map in a MASH 2-1 Delta-Sigma Modulator

This paper presents the implementation of two digital noise-cancellation circuits (DNCCs) using a proposed algorithm and a Karnaugh map for a [Formula: see text] multistage noise-shaping (MASH) delta-sigma modulator (DSM). The MASH architecture inherits a superior signal-to-noise-and-distortion ratio (SNDR) with the aid of an efficient noise-cancellation technique either in the analogue or digital domain. The key motivation of this study was to design an area-efficient DNCC. The first approach employed a proposed algorithm (Algorithm-based DNCC) to implement the DNCC and to construct a delay block with an inverter and transmission gate. The second approach involved a Karnaugh map (K-map DNCC) and a delay block with a pair of D flip-flops. A maximum simulated signal-to-noise ratio of 135[Formula: see text]dB was completed with optimal analogue scaling coefficients for the proposed [Formula: see text] MASH DSM with DNCC. The simulated SNDRs of the Algorithm-based DNCC and K-map DNCC were 91.04[Formula: see text]dB and 91.16[Formula: see text]dB, respectively. Measured results show that the SNDR of the Algorithm-based DNCC, the SNDR of the K-map DNCC, power consumption and core area are approximately 58.7[Formula: see text]dB, 62.1[Formula: see text]dB, 0.26[Formula: see text][Formula: see text]W and 2275[Formula: see text][Formula: see text]m2, respectively, for the designed DNCCs with an operating frequency of 10.24[Formula: see text]MHz and supply voltage of 1.8[Formula: see text]V. The transistor counts of the Algorithm-based DNCC are 74 transistors, while they are 106 transistors for the K-map DNCC. The proposed Algorithm-based DNCC saves 32 transistors and approximately reduces its chip area to 69.8% of the K-map DNCC.

Simplifying Karnaugh Maps by Making Groups of Non-power-of-two Elements

When we study the Karnaugh map in the switching theory course, we learn that the ones in the map are combined in rectangles whose length and width must be a power of two. The result is the logic function described as a sum of products. This paper shows that we can also make groups where the length and width of the rectangles are equal to three. This results in a logic function that is simpler than the sum of products in terms of logic gates, leading to more hardware-efficient circuits. This idea is extended later in the paper to other groups of elements. Finally, a new perspective on the Karnaugh map that integrates the proposed approach with the conventional one is provided. This can be used in switching theory courses to improve the explanation of the Karnaugh map.

Optimized Synthesis Method for Ultra-Low Power Multi-Input Material Implication Logic With Emerging Non-Volatile Memories

In this paper, we revisit Boole’s expansion theorem to propose a new synthesis method for implication logic circuits based on memristors. By rewriting the sum-of-products form of Boole’s expansion theorem in terms that are best suited for the implication logic, we develop a generalized rule to derive the sequence of operations needed to realize any logic function written in the classical AND-OR form. The proposed method leverages on multi-input operation, minimizing both the number of steps required to compute a given Boolean function and the number of memristors involved. Moreover, it allows using well-established methods of logic circuit optimization like binary decision diagrams, Karnaugh maps, and heuristic algorithms, that are already implemented in commercial CAD software. The proposed method allows a fair comparison between the performance of CMOS and implication logic implementations of the same logic function under the same degree of optimization, and is shown to outperform existing approaches. Possible device-circuit co-design strategies to optimize circuit performance are finally discussed.

Implementation of the method of figurative transformations to minimizing symmetric Boolean functions

This paper reports a study that has established the possibility of improving the effectiveness of the method of figurative transformations in order to minimize symmetrical Boolean functions in the main and polynomial bases. Prospective reserves in the analytical method were identified, such as simplification of polynomial function conjuncterms using the created equivalent transformations based on the method of inserting the same conjuncterms followed by the operation of super-gluing the variables. The method of figurative transformations was extended to the process of minimizing the symmetrical Boolean functions with the help of algebra in terms of rules for simplifying the functions of the main and polynomial bases and developed equivalent transformations of conjuncterms. It was established that the simplification of symmetric Boolean functions by the method of figurative transformations is based on a flowchart with repetition, which is the actual truth table of the assigned function. This is a sufficient resource to minimize symmetrical Boolean functions that makes it possible to do without auxiliary objects, such as Karnaugh maps, cubes, etc. The perfect normal form of symmetrical functions can be represented by binary matrices that would represent the terms of symmetrical Boolean functions and the OR or XOR operation for them. The experimental study has confirmed that the method of figurative transformations that employs the 2-(n, b)-design, and 2-(n, x/b)-design combinatorial systems improves the efficiency of minimizing symmetrical Boolean functions. Compared to analogs, this makes it possible to enhance the productivity of minimizing symmetrical Boolean functions by 100‒200 %. There are grounds to assert the possibility of improving the effectiveness of minimizing symmetrical Boolean functions in the main and polynomial bases by the method of figurative transformations. This is ensured, in particular, by using the developed equivalent transformations of polynomial function conjuncterms based on the method of inserting similar conjuncterms followed by the operation of super-gluing the variables.

Simplifying the Boolean Equation Based on Simulation System using Karnaugh Mapping Tool in Digital Circuit Design

In computerized integrated circuits, the fundamental principle intends to avoid the multifaceted nature of the circuitry by making it as brief as attainable and minimize the expenditure. Techniques like Quine- McCluskey (QM) and Karnaugh Map (K-Map) are often used approaches of simplifying Boolean functions. This study presents a recreation framework of simplification of the Boolean capacities by the utilize of the K- Map definition for beginner-level learners. It uses the algebraic expression of the Boolean function to decrease the number of terms, generates a circuit, and does not use any redundant sets. In this way, it gets to be competent to deal with lots of parameters and minimize the computational cost. The result of the assessment is performed in this paper by contrasting it with the C- Minimizer algorithm. In computation time terms, the result appears that our comprehensive K mapping tool outflanks in current procedures, and the relative error accomplishes a lower rate of percentage (2%), which fulfills the satisfactory level.
 GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 7, Dec 2020 P 76-84

A Novel Bottom-Up/Top-Down Hybrid Strategy-Based Fast Sequential Fault Diagnosis Method

Sequential fault diagnosis is a kind of important fault diagnosis method for large scale complex systems, and generating an excellent fault diagnosis strategy is critical to ensuring the performance of sequential diagnosis. However, with the system complexity increasing, the complexity of fault diagnosis tree increases sharply, which makes it extremely difficult to generate an optimal diagnosis strategy. Especially, because the existing methods need massive redundancy iteration and repeated calculation for the state parameters of nodes, the resulting diagnosis strategy is often inefficient. To address this issue, a novel fast sequential fault diagnosis method is proposed. In this method, we present a new bottom-up search idea based on Karnaugh map, SVM and simulated annealing algorithm. It combines failure sources to generate states and a Karnaugh map is used to judge the logic of every state. Eigenvalues of SVM are obtained quickly through the simulated annealing algorithm, then SVM is used to eliminate the less useful state. At the same time, the bottom-up method and cost heuristic algorithms are combined to generate the optimal decision tree. The experiments show that the calculation time of the method is shorter than the time of previous algorithms, and a smaller test cost can be obtained when the number of samples is sufficient.

Utilization of the Karnaugh Map in Exploring Cause-effect Relations Modeled by Partially-defined Boolean Functions

This paper utilizes a modern regular and modular eight-variable Karnaugh map in a systematic investigation of cause-effect relationships modeled by partially-defined Boolean functions (PDBF) (known also as incompletely specified switching functions). First, we present a Karnaugh-map test that can decide whether a certain variable must be included in a set of supporting variables of the function, and, otherwise, might enforce the exclusion of this variable from such a set. This exclusion is attained via certain don’t-care assignments that ensure the equivalence of the Boolean quotient w.r.t. the variable, and that w.r.t. its complement, i.e., the exact matching of the half map representing the internal region of the variable, and the remaining half map representing the external region of the variable, in which case any of these two half maps replaces the original full map as a representation of the function. Such a variable exclusion might be continued w.r.t. other variables till a minimal set of supporting variables is reached. The paper addresses a dominantly-unspecified PDBF to obtain all its minimal sets of supporting variables without resort to integer programming techniques. For each of the minimal sets obtained, standard map methods for extracting prime implicants allow the construction of all irredundant disjunctive forms (IDFs). According to this scheme of first identifying a minimal set of supporting variables, we avoid the task of drawing prime-implicant loops on the initial eight-variable map, and postpone this task till the map is dramatically reduced in size. The procedure outlined herein has important ramifications for the newly-established discipline of Qualitative Comparative Analysis (QCA). These ramifications are not expected to be welcomed by the QCA community, since they clearly indicate that the too-often strong results claimed by QCA adherents need to be checked and scrutinized.

Using Eight-Variable Karnaugh Maps to Unravel Hidden Technicalities in Qualitative Comparative Analysis

We use a regular and modular eight-variable Karnaugh map to reveal some technical details of Boolean minimization usually employed in solving problems of Qualitative Comparative Analysis (QCA). We utilize as a large running example a prominent eight-variable political-science problem of sparse diversity (involving a partially-defined Boolean function (PDBF), that is dominantly unspecified). We recover the published solution of this problem, showing that it is merely one candidate solution among a set of many equally-likely competitive solutions. We immediately obtain one of these rival solutions, that looks better than the published solution in two aspects, namely: (a) it is based on a smaller minimal set of supporting variables, and (b) it provides a more compact Boolean formula. However, we refrain from labelling our solution as a better one, but instead we stress that it is simply un-comparable with the published solution. The comparison between any two rival solutions should be context-specific and not tool-specific. In fact, the Boolean minimization technique, borrowed from the area of digital design, cannot be used as is in QCA context. A more suitable paradigm for QCA problems is to identify all minimal sets of supporting variables (possibly via integer programming), and then obtain all irredundant disjunctive forms (IDFs) for each of these sets. Such a paradigm stresses inherent ambiguity, and does not seem appealing as the QCA one, which usually provides a decisive answer (irrespective of whether it is justified or not).The problem studied herein is shown to have at least four distinct minimal sets of supporting variables with various cardinalities. Each of the corresponding functions does not have any non-essential prime implicants, and hence each enjoys the desirable feature of having a single IDF that is both a unique minimal sum and the complete sum. Moreover, each of them is unate as it is expressible in terms of un-complemented literals only. Political scientists are invited to investigate the meanings of the (so far) abstract formulas we obtained, and to devise some context-specific tool to assess and compare them.

Utilization of Eight-Variable Karnaugh Maps in the Exploration of Problems of Qualitative Comparative Analysis

Qualitative Comparative Analysis (QCA) is an emergent methodology of diverse applications in many disciplines. However, its premises and techniques are continuously subject to discussion, debate, and (even) dispute. We use a regular and modular Karnaugh map to explore a prominent recently-posed eight-variable QCA problem. This problem involves a partially-defined Boolean function (PDBF), that is dominantly unspecified. Without using the algorithmic integer-programming approach, we devise a simple heuristic map procedure to discover minimal sets of supporting variables. The eight-variable problem studied herein is shown to have at least two distinct such sets, with cardinalities of 4 and 3, respectively. For these two sets, the pertinent function is still a partially-defined Boolean function (PDBF), equivalent to 210 = 1024 completely-specified Boolean functions (CSBFs) in the first case, and to four CSBFs only in the second case. We obtained formulas for the four functions of the second case, and a formula for a sample fifth function in the first case. Although only this fifth function is unate, each of the five functions studied does not have any non-essential prime implicant, and hence each of them enjoys the desirable feature of having a single IDF that is both a unique minimal sum and the complete sum. According to our scheme of first identifying a minimal set of supporting variables, we avoided the task of drawing prime-implicant loops on the initial eight-variable map, and postponed this task till the map became dramatically reduced in size. Our map techniques and results are hopefully of significant utility in future QCA applications.

Research on a design method of pneumatic logic control system

Aiming at the complex characteristics of the sequential logic control system design, such as X-D diagram and karnaugh map method, a design method based on stepper module was proposed, the procedure of which does not need to be checked and corrected. The design of different pneumatic circuits was carried out with the automatic drilling machine as the application object. First, a stepper module composed of a two-position three-way valve and a dual-pressure valve was constructed, and its structure principle was analyzed in detail. Next, three kinds of stroke programs were listed by analyzing the working process of automatic drilling machine, which were multi-cylinder single reciprocating, multi-cylinder multi reciprocating and multi-cylinder multi-section reciprocating stroke program, and the pneumatic circuit control system was designed by the stepper module method. Finally, the pneumatic circuit was simulated and analyzed using the software Fluid-SIM. The simulation results showed that each actuator can complete the corresponding actions according to the design requirements, which verified the correctness and reliability of the stepper module design method. This work would provide a fast and effective method for the design of the sequential logic control system.

Mathematical Knot Sematic Logic:Theory and Example in Surah Al-Kahf

Start your abstract here Knot semantic logic theory is a new idea to explain the symmetry structure of a literature. The methodology used in this research is the combination of Theme-Rheme framework, Paltridge’s classification of thematic progression and symmetry diagram mapping. The results of this research are the establishment of knot semantic logic theory and the application to analyze the symmetrical structure in Surah Al-Kahf. After applying this theory to Surah Al-Kahf resulting that the chapter contains 9 groups form symmetrical structure. The first theme on this Surah is about Believers, non-believers and consequences connected to the last group with the similar theme. The theme of the second group is about self-quarantine mirrors to the second to the last group about lockdown. The third theme is about Patience and its reward related to the seventh group. The test of wealthyis the theme of the fourth group which mirrors to the sixth group talking about provision as a sign. Finally, the fifth group as the central of this chapter states that Quran has provided many lessons and guidance for human beings. Surah Al-Kahf is knot chapter with a mathematical symbol :

Developing the minimization of a polynomial normal form of boolean functions by the method of figurative transformations

This paper reports a study that has established the possibility of improving the effectiveness of the method of figurative transformations in order to minimize Boolean functions on the Reed-Muller basis. Such potential prospects in the analytical method have been identified as a sequence in the procedure of inserting the same conjuncterms of polynomial functions followed by the operation of super-gluing the variables. The extension of the method of figurative transformations to the process of simplifying the functions of the polynomial basis involved the developed algebra in terms of the rules for simplifying functions in the Reed-Muller basis. It was established that the simplification of Boolean functions of the polynomial basis by a figurative transformation method is based on a flowchart with repetition, which is actually the truth table of the predefined function. This is a sufficient resource to minimize functions that makes it possible not to refer to such auxiliary objects as Karnaugh maps, Weich charts, cubes, etc. A perfect normal form of the polynomial basis functions can be represented by binary sets or a matrix that would represent the terms of the functions and the addition operation by module two for them. The experimental study has confirmed that the method of figurative transformations that employs the systems of 2-(n, b)-design, and 2-(n, x/b)-design in the first matrix improves the efficiency of minimizing Boolean functions. That also simplifies the procedure for finding a minimum function on the Reed-Muller basis. Compared to analogs, this makes it possible to enhance the performance of minimizing Boolean functions by 100‒200 %. There is reason to assert the possibility of improving the efficiency of minimizing Boolean functions in the Reed-Muller basis by a method of figurative transformations. This is ensured by using more complex algorithms to simplify logical expressions involving a procedure of inserting the same function terms in the Reed-Muller basis, followed by the operation of super-gluing the variables.

Construction of three associate PBIB designs using K - maps

A new series of s (s ≥ 3) associate class Partially Balanced Incomplete Block (PBIB) designs have been constructed by applying hamming distance on Karnaugh maps. In this series, every character is expressed in binary set and calculate the weight, then hamming distance for each literal of k – map. We considersꞌ different factors (s≥3) simultaneously and each factor has two levels, it means each factor is introduced in primal as well as dual form.

Reliability Evaluation of Rooftop Solar Photovoltaics Using Coherent Threshold Systems

The trend for use of rooftop solar photovoltaics (PV) is rising due to their promising economic potential as a source of clean renewable energy. In general, a source of renewable solar energy consists of solar PV, an automatic charge controller, a battery pack, and an inverter. The reliability of a rooftop solar PV system is evaluated herein as that of a coherent threshold system (CTS). First, we utilize the unit-gap method and the fair-power method to verify that a given Boolean function is a threshold one and to identify its threshold and component weights. Both methods utilize specific features of the Karnaugh map (K-map). The unit-gap method uses the map to list all necessary inequalities by inspection, and then reduce them significantly by omitting dominated ones. The fair-power method uses the Karnaugh map to compute Banzhaf indices by appropriate map folding followed by XORing of true cells and false cells. We evaluate the CTS reliability via a recursive algorithm based on the Boole-Shannon’s expansion in the switching domain, which is transformed via the real transform to the total probability law in the probability domain.

Design of Compact S Box for Resource Constrained Applications

The era of IoT has brought huge impact in the lives of people. It changed the way the people interact with the outside world and led to the development of many smart devices. The important factor to be considered in smart devices is security as the sensor nodes are prone to vulnerable attacks. To incorporate security in IoT lightweight block ciphers are preferred. Lightweight block ciphers can be tuned according to applications. This can be motivated by optimizing the S-box since S-box has significant impact on area. Reducing the area produces an impact on power, delay and memory. Here area optimized S-box of lightweight ciphers that serves for IoT devices is proposed. The S-box of PRESENT, GIFT and LED are optimized using karnaugh mapping. Further factorization is carried out to utilize the redundant terms which in turn reduces the gate count. The proposed method is found to be optimum when compared with previous works and it also consumes less area and power.