Boolean Algebra Operations Research Articles

Delegation-Relegation for Boolean Matrix Factorization

The Boolean Matrix Factorization (BMF) problem aims to represent a n×m Boolean matrix as the Boolean product of two matrices of small rank k, where the product is computed using Boolean algebra operations. However, finding a BMF of minimum rank is known to be NP-hard, posing challenges for heuristic algorithms and exact approaches in terms of rank found and computation time, particularly as matrix size or the number of entries equal to 1 grows. In this paper, we present a new approach to simplifying the matrix to be factorized by reducing the number of 1-entries, which allows to directly recover a Boolean factorization of the original matrix from its simplified version. We introduce two types of simplification: one that performs numerous simplifications without preserving the original rank and another that performs fewer simplifications but guarantees that an optimal BMF on the simplified matrix yields an optimal BMF on the original matrix. Furthermore, our experiments show that our approach outperforms existing exact BMF algorithms.

Performance of 9T SRAM Cell using Boolean algebra and Arithmetic Operations

This work aims to implement the Boolean and Arithmetic functions with Static Random Access Memory (SRAM) Cell made by nine CMOS Transistor for In-Memory computation. Normally the Von-Neumann architecture is used for current computing system where data and programs are placed in a single memory by sharing and a single Bus is used for memory access, ALU and for controlling of program. This results in minimizing the efficiency of computing when there is high eminence on Data tactless applications. The Von-Neumann architecture is made more efficient by Computation in Memory (CIM) architecture which overcome the drawback of Memory barrier, the power utilization and in the resemblance of wall structure. Here is presented an inside the memory computation method and co-designs of arithmetic circuitry using 9T SRAM cell. The Boolean algebra and Arithmetic operations are illustrated with 9T SRAM cell designed by CMOS technology in 180 nm process. The Boolean algebra for AND- NAND, OR-NOR logic gates are illustrated using 9T SRAM cells with the Latch type sense amplifier to verify the capability of computation in Memory of 9T SRAM cell. To illustrate the arithmetic circuit an array of memory has been made by the 9T SRAM cell and proposed sensing amplifier which is latch type and finally is mapped with circuit of half-adder and half-subtraction.

ВПОРЯДКУВАННЯ ТА ВПОРЯДКОВУВАННЯ В ДИСКРЕТНІЙ МАТЕМАТИЦІ ТА ІНФОРМАТИЦІ

The available means of ordering and sorting in some important sections of discrete mathematics and computer science are studied, namely: in the set theory, classical mathematical logic, proof theory, graph theory, POST method, system of algorithmic algebras, algorithmic languages of object-oriented and assembly programming. The Cartesian product of sets, ordered pairs and ordered n-s, the description by means of set theory of an ordered pair, which are performed by Wiener, Hausdorff and Kuratowski, are presented. The requirements as for the relations that order sets are described. The importance of ordering in classical mathematical logic and proof theory is illustrated by the examples of calculations of the truth values of logical formulas and formal derivation of a formula on the basis of inference rules and substitution rules. Ordering in graph theory is shown by the example of a block diagram of the Euclidean algorithm, designed to find the greatest common divisor of two natural numbers. The ordering and sorting of both the instructions formed by two, three and four ordered fields and the existing ordering of instructions in the program of Post method are described. It is shown that the program is formed by the numbered instructions with unique instruction numbers and the presence of the single instruction with number 1. The means of the system of algorithmic algebras, which are used to perform the ordering and sorting in the algorithm theory, are illustrated. The operations of the system of algorithmic algebras are presented, which include Boolean algebra operations generalized to the three-digit alphabet and operator operations of operator algebra. The properties of the composition operation are described, which is intended to describe the orderings of the operators of the operator algebra in the system of algorithmic algebras. The orderings executed by means of algorithmic programming languages are demonstrated by the hypothetical application of the modern object-oriented programming language C#. The program must contain only one method Main () from which the program execution begins. The ARM microprocessor assembly program must have only one ENTRY directive from which the program execution begins.

Fault tree analysis of the causes of urban smog events associated with vehicle exhaust emissions: A case study in Jinan, China

The excessive consumption of fuels associated with rapid industrialization, urbanization, and modernization has caused serious smog events in many Chinese cities. Vehicle exhaust is one of the primary causes of smog events due to the rapid growth of motor vehicle ownership and increased fuel consumption. In this study, fault tree analysis (FTA) was used as a relatively simple but effective way to analyze the causes of smog associated with vehicle exhaust emissions in Jinan, China. First, after the identification of the top event, intermediate events, and basic events, a comprehensive fault tree system for urban smog associated with vehicle exhaust emissions was constructed. Then, during the qualitative analysis stage, minimal cut sets (MCSs) were grouped using Boolean algebra operations, and the original fault tree was simplified to an equivalent tree based on 6 MCSs. Finally, during the quantitative analysis stage, the effects of the 12 basic events on the top event were evaluated and ranked according to the structural importance, probabilistic importance, and critical importance of their analytical measures. Our results indicated that traffic congestion, superabundance of vehicles, poor supervision, and yellow-label vehicles with long use ages had the greatest impact on smog events, with importance degrees of 0.52930, 0.52920, 0.22719, and 0.22716, respectively. These results are consistent with common sense. Although different basic events exert different influences, all of the basic events should be comprehensively taken into consideration and corresponding precautionary measures developed. This research provides a good case study of the application of FTA in the analysis of the causes of urban smog events associated with vehicle exhaust emissions. Our study further demonstrates that FTA is a relatively simple but effective method for the causal analysis of smog, as well as an effective tool for environmental risk management.

Boolean Operations for the Simulation of Machining Processes Based on the CSG Modeling Technique

The simulation of computer numerically control (CNC) machining process is an important component of CAM, it can check errors and enhance the automation of machining process. In order to realize the material removal simulation of machining process through VC++ and OpenGL, this paper does the research on the Boolean operations based on the Constructive Solid Geometry (CSG) modeling technique. Firstly, this paper introduces the CSG technique, especially the creating complex object from simple primitives by the operations of Boolean algebra and the frame buffer. Then this paper achieves the solid modeling of cutter and stock through VC++ and OpenGL, and does the research on the theory of cutter’s swept volume generation. In terms of the theory, a cutter’s swept volume is created. Finally, the Boolean difference operation of a stock and the cutter’s swept volume is realized, and the code is also given.

Biological Sequence Matching using Boolean algebra vs. Fuzzy Logic

Biological sequence alignment is one of the crucial tasks of computational bioinformatics, and provides base for other tasks of bioinformatics. In this paper, we discuss two different approaches to sequence matching – Boolean algebra and fuzzy logic. First method is a two-valued logic whereas the second is a multi-valued logic. Both the methods perform sequence matching by direct comparison method using the operations of Boolean algebra and fuzzy logic respectively. To ensure the optimal alignment, dynamic programming is employed to align the sequences progressively. Both the methods are implemented and then tested on few sets of real biological sequences taken from NCBI bank and their performances are compared with the CLUSTALW algorithm.

Surface-Confined Assemblies and Polymers for Molecular Logic

Stimuli responsive materials are capable of mimicking the operation characteristics of logic gates such as AND, OR, NOR, and even flip-flops. Since the development of molecular sensors and the introduction of the first AND gate in solution by de Silva in 1993, Molecular (Boolean) Logic and Computing (MBLC) has become increasingly popular. In this Account, we present recent research activities that focus on MBLC with electrochromic polymers and metal polypyridyl complexes on a solid support. Metal polypyridyl complexes act as useful sensors to a variety of analytes in solution (i.e., H(2)O, Fe(2+/3+), Cr(6+), NO(+)) and in the gas phase (NO(x) in air). This information transfer, whether the analyte is present, is based on the reversible redox chemistry of the metal complexes, which are stable up to 200 °C in air. The concurrent changes in the optical properties are nondestructive and fast. In such a setup, the input is directly related to the output and, therefore, can be represented by one-input logic gates. These input-output relationships are extendable for mimicking the diverse functions of essential molecular logic gates and circuits within a set of Boolean algebraic operations. Such a molecular approach towards Boolean logic has yielded a series of proof-of-concept devices: logic gates, multiplexers, half-adders, and flip-flop logic circuits. MBLC is a versatile and, potentially, a parallel approach to silicon circuits: assemblies of these molecular gates can perform a wide variety of logic tasks through reconfiguration of their inputs. Although these developments do not require a semiconductor blueprint, similar guidelines such as signal propagation, gate-to-gate communication, propagation delay, and combinatorial and sequential logic will play a critical role in allowing this field to mature. For instance, gate-to-gate communication by chemical wiring of the gates with metal ions as electron carriers results in the integration of stand-alone systems: the output of one gate is used as the input for another gate. Using the same setup, we were able to display both combinatorial and sequential logic. We have demonstrated MBLC by coupling electrochemical inputs with optical readout, which resulted in various logic architectures built on a redox-active, functionalized surface. Electrochemically operated sequential logic systems such as flip-flops, multivalued logic, and multistate memory could enhance computational power without increasing spatial requirements. Applying multivalued digits in data storage could exponentially increase memory capacity. Furthermore, we evaluate the pros and cons of MBLC and identify targets for future research in this Account.

A novel Boolean algebraic framework for association and pattern mining

Data mining has been defined as the non- trivial extraction of implicit, previously unknown and potentially useful information from data. Association mining and sequential mining analysis are considered as crucial components of strategic control over a broad variety of disciplines in business, science and engineering. Association mining is one of the important sub-fields in data mining, where rules that imply certain association relationships among a set of items in a transaction database are discovered. In Sequence mining, data are represented as sequences of events, where order of those events is important. Finding patterns in sequences is valuable for predicting future events. In many applications such as the WEB applications, stock market, and genetic analysis, finding patterns in a sequence of elements or events, helps in predicting what could be the next event or element. At the conceptual level, association mining and sequence mining are two similar processes but using different representations of data. In association mining, items are distinct and the order of items in a transaction is not important. While in sequential pattern mining, the order of elements (events) in transactions (sequences) is important, and the same event may occur more than once. In this paper, we propose a new mapping function that maps event sequences into itemsets. Based on the unified representation of the association mining and the sequential pattern, a new approach that uses the Boolean representation of input database D to build a Boolean matrix M. Boolean algebra operations are applied on M to generate all frequent itemsets. Finally, frequent items or frequent sequential patterns are represented by logical expressions that could be minimized by using a suitable logical function minimization technique.

Localization of light energy of a femtosecond laser pulse upon generation of the second harmonic in a one-dimensional nonlinear photonic crystal with alternating nonlinear response

A possibility of localization of the light energy upon generation of the second harmonic of a femtosecond laser pulse in a one-dimensional photonic crystal, some layers of which are either quadratic or cubic nonlinear, is demonstrated in computer simulation. It is shown that there exist light energy localization modes at one of the interacting frequencies and at both interacting frequencies simultaneously. The formation of solitons in separate layers of the photonic crystal is the basis of light energy localization. Logical operations of Boolean algebra can be implemented based on the discovered effect.

A BOOLEAN ALGEBRA BASED RULE EXTRACTION ALGORITHM FOR NEURAL NETWORKS WITH BINARY OR BIPOLAR INPUTS

ABSTRACT Neural networks have been applied in various domain including science, commerce, medicine, and industry. However, The knowledge learned by a trained neural network is difficult to understand. This paper proposes a Boolean algebra based algorithm to extract comprehensible Boolean rules from supervised feed-forward neural networks to uncover the black-boxed knowledge. This algorithm is called the BAB-BB rule extraction algorithm, which stands for a Boolean algebra based rule extraction algorithm for neural networks with binary and bipolar inputs. Decomposition techniques and interval arithmetic are used in the algorithm. First, each neuron associated with its inputs is analyzed and a Boolean function, describing the activation rule from its inputs to the neuron, is derived. These Boolean functions are merged into an aggregated Boolean rule according to the network topology. The Boolean rule is then further simplified by Boolean algebra operations. During the rule extraction procedure, redundant hidden neurons can be detected and removed without affecting the original function of the neural network. Examples of unipolar and bipolar inputs are presented to demonstrate the use of our algorithm. Finally, the Exclusive OR problem is presented and solved by our algorithm. Results show that our BAB-BB algorithm is practicable and of high efficiency.

Boolean algebra operations performed on optical bits by the generation of holographic fields through second-order nonlinear interactions

Boolean algebra operations such as AND or XOR are performed on optical bits encoded as amplitude modulations in two wave fronts that are made interact in a crystal (β-BaB2O4) endowed with second-order nonlinearity. If the corresponding wave fields are at the same frequency ω and the crystal is tuned for the phase-matched generation of 2ω, we show that the generated wave front reconstructs a holographic image containing the outputs of the desired operations. Since a nearly diffraction-limited optical resolution can be easily achieved in the holographic image at 2ω, a correspondingly high density of data is encodeable in the wave front at ω playing the role of object wave front. The experiments demonstrate the feasibility of the operation of a parallel half-adder performing the sum of multiple data with a one-digit binary number, which is encoded in the reference wave front.

Multilevel logic synthesis with extended arrays

The authors offer extended algebraic operations and strategies for se in algebraic approaches. A novel cube/array representation that directly shows some legal expressional variations from a minimal expression written in the conventional three symbol (0, 1, x) representation is defined, along with certain operations on those cube/arrays. Conversions between the new and conventional representations are explained. Division methods are presented that are based on expressions like those used to describe algebraic division but use more operations of Boolean algebra. Factoring on the new representations is explained. A multilevel logic synthesis algorithm with a control strategy is presented and compared with published systems. >

Image analysis of electron microscope images using Boolean algebra operations

For real time television image processing of electron microscope images it is inefficient to have image arithmetic performed on a host computer which requires considerable CPU and transfer time overhead. It is usually more efficient to use an analog preprocessor before image display such as the shading correctors used for particle analysis. These techniques can also be applied to digital TV frame stores which have simple arithmetic units as preprocessors to a digital frame store. In this case sampled images are either added, subtracted or divided much as is done in many STEM type instruments at slower scan rates. Alternatively, the digital frame store can have an arithmetic and Boolean algebra processor capability which operates on digitized images in the frame store in real time. Such processors are available commercially and one has been adapted for use on various electron microscopes at the IBM Research Center.

Modularizing, Minimizing, and Interpreting the K&H Fault-Tree

The verification by Worrell & Stack (W&S) of results previously obtained by Kumamoto & Henley (K&H) for a fault tree of an s-noncoherent system and its inverse and their correction of the three errors in the tree makes it possible to simplify the analysis by forming modules; this facilitates Boolean algebraic operations, so that both sets are described economically in their minimal forms, a subset of the prime implicants (p.i.'s). Quine's consensus operation is used to minimize and to find the p.i.'s. Corresponding to the MOCUS output for the inverse reported by K&H, which is neither minimal nor the set of p.i.'s, instead of 32 terms, there are 15 in the modularized set. Instead of the 42 p.i.'s obtained by both K&H and W&S, we have 17; 13 of these are a unique minimal form. Instead of 352 p.i.'s for the tree per both K&H and W&S, we have a 15-term minimal form, identical to the list of p.i.'s. The results are further analyzed as a contribution to the continuing discussion of the utility of minimal forms vis-a-vis the p.i.'s and of the consensus method.

Development Of An Optical Parallel Logic Device And A Half-Adder Circuit For Digital Optical Processing

An Optical Parallel Logic (OPAL) device which performs Boolean algebraic operations on two binary images has been developed. This device consists of a photoconductor and an electro-optic light modulating material appropriately arranged to bring about an in-teraction between the input signals. Two such OPAL devices have been interconnected to form a half-adder circuit (one of the essential components in the CPU of a futuristic digital optical processor). Operation of an 8 x 8 OPAL device containing CdS (photoconductor) and twisted nematic liquid crystal (electro-optic light modulating material) is reported. A contrast ratio of 20:1 was obtained. A circuit composed of two such devices is shown to perform addition of two 8 x 8 binary images generating SUM and CARRY images.

Probabilistic Treatment of General Combinational Networks

In this correspondence two methods are given for calculating the probability that the output of a general combinational network is 1 given the probabilities for each input being 1. We define the notions of the probability of a signal and signal independence. Then several proofs are given to show the relationship between Boolean operations and algebraic operations upon probabilities. As a result of these, two simple algorithms are presented for calculating output probabilities. An example of the usefulness of these results is given with respect to the generation of tests for the purpose of fault detection.

Possibility of construction of an arithmetic unit based on controlled optical transparencies

An analysis is made of the possibility of construction of arithmetic units based on the passage of light signals through electric-field-controlled optical transparencies. Such arithmetic units can perform any Boolean algebraic operations. Methods for performing the basic arithmetic operations (addition, subtraction, multiplication, and division) are described. A brief discussion is given of the block diagram of an optoelectronic arithmetic unit with a speed of several tens of millions of algorithmic operations per second.