Boolean-based mathematics

After while the “Human Genome Project” proposes, the people complete measures the foreword plan after the multi-gene genome team, also starts to change to these genes and their reciprocity function understanding research. Varieties of gene regulation Boolean networks algorithms have been proposed of the gene expression profiles, however, the problem of noise could always be found in creating a Boolean network. Due to gene expression data are always noisy. In this paper, it show that after the Boolean networks logic function are learned from noisy data, some noise in the Boolean function could be restructure Karnaugh Maps. It could find logic relationships between protein and protein and restructure protein logic networks. It find logic relationship among proteins as well as COGs (clusters of orthogous groups) and build the logic network of protein.

This special issue of the VLSI DESIGN: an International Journal of Custom-Chip Design, Simulation and Testing is devoted to spectral techniques, new types of decision trees and diagrams, their applications, efficient ways of their calculation and mutual relations between spectral techniques and classical decision diagrams. Manipulations and calculations with discrete functions are fundamental tasks in many areas of Computer Science and Engineering. Many problems in digital system design, simulation and testing can be expressed as a sequence of operations on discrete functions. The performance of Computer-Aided Design systems used in solving various problems in this area strongly depends on the efficiency of representations of discrete functions. Decision Diagrams (DDs) have proved very convenient data structure for discrete function representations, permitting manipulations and calculations with large functions efficiently in terms of time and space. In many applications, as for example those involving large matrices, conventional algorithms are significantly improved by using DDs. In logic design, such applications relate to basic problems of the design, verification, simulation and testing logical networks. Besides Boolean functions, some other types of discrete representations, such as multi-valued functions, cube sets, EXOR-based spectral and probabilistic formulas have to be dealt with. Hence the numbers of different decision trees and diagrams have been introduced for these various applications and representations. The spectral or polynomial representation of Boolean function is obtained by expressing it in terms of an orthogonal basis. The most popular basis is formed by discrete Walsh functions. Note that the spectral and the polynomial representation of a Boolean function are equivalent. The discrete orthogonal functions are referred to as the basis functions when the spectral or abstract harmonic terminology is used and as monomials when the polynomial terminology is used. Such polynomial representations have extensive applications in coding theory and cryptography as well as design of nonlinear filters. Spectral techniques based on Walsh, Haar, Arithmetic and Reed–Muller transforms have been used in digital logic design for more than 30 years. These techniques have been used for Boolean function classification, disjoint decomposition, identification of symmetries, parallel and serial linear decomposition, spectral translation synthesis (extraction of linear preand post-filters), multiplexer synthesis, threshold logic synthesis, state assignment and testing, and evaluation of logic complexity. The renewed interest in applications of spectral methods in design of VLSI digital circuits is caused by their excellent design for testability properties and in their efficiency in Boolean mapping problems. Furthermore, the practical applicability of these techniques was greatly improved with the development of efficient methods that allow calculating and operating on spectra of Boolean functions directly from reduced representations of such functions in the form of arrays of cubes or decision diagrams. Following from these recent developments, techniques have been presented for efficient spectral translation to decompose a circuit into a cascade of two sub circuits: linear block composed of EXOR gates fed by the primary input of the overall circuit. Such linear decomposition drastically simplifies the synthesis task. The choice of suitable linear transformation is based on a complexity measure assigned to each Boolean function that is heuristically related to the complexity of the final circuit implementation. To avoid the experimental costs of computing the Walsh transform, Binary Decision Diagrams (BDDs) based techniques are used. Recently the idea of linear transformations is applied to decision diagrams. By combining powerful spectral linear techniques with variable reordering techniques, it is possible to synthesize large Boolean functions with standard Computer-Aided Design tools that fail otherwise. The areas addressed by this special issue span spectral transforms, their applications and efficient calculation methods through decision diagrams. New spectral decision diagrams are also emerging that support larger data structures allowing more complex discrete representations to be implemented that was previously impossible.

Boolean algebra is the primary mathematical tool used in designing the computing machines that have come into widespread use in the latter part of this century. Switching functions, described using two-valued Boolean algebra, are implemented using transistor circuits [4]. These transistors act as on-off switches, which is a natural implementation for representing the primitive operations in a Boolean algebra. Once the primitive operations of a Boolean algebra are implemented using transistor circuits, any Boolean function can be implemented using these primitive operations. Operations AND, OR and NOT form the primitive operations of Boolean algebra. Other variations of these primitive operations such and NAND and NOT can also be used as the set of primitive operations.

The article introduces advanced mathematical modeling methods for the automated reproduction of discrete functions of majority and Boolean logic using multiplexer systems. In this work, structural configuration is not understood as the development of specialized software implemented using microprocessors, but, on the contrary, as the transformation of technological operations for setting up and changing the configurations of large micro- and nanocircuits in such a way as to synthesize a functionally complete logical base on a structural-logical basis. The latest technology of nanoscale cellular automata (QCA or QCA-quantum-dots cellular automata) is the most promising direction of single-electron electronics, which also confirms Moore's law. Unlike traditional microtechnologies, for example, based on complementary MOS transistors, in QCA binary information is encoded by polarizations of two electrons, and not through logical voltage levels. The results of computer programming of nanodevices obtained in the article con-firmed their advantages over microelectronic equivalents in terms of increased speed and minimal energy consumption. The modern CAD QCA Designer was used for the analysis and synthesis of Boolean, majority and hybrid logic functions using micro- and nanocircuits in space applications at cryogenic temperatures to predict their trouble-free operation. The results obtained and their analysis confirm the possibility of improving the method-ology for configuring modern micro- and single-electron nanodevices.

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.

Chapter 12 - Boolean Algebra and Combinatorial Circuits

Every second year, the GEOCRYPT conference brings together researchers from arithmetic geometry and cryptography. After Guadeloupe in 2009 and Corsica in 2011, French Polynesia was the host of its 2013 edition which took place in Punaauia, on the island of Tahiti, October 7-11. The main topic of GEOCRYPT has always been the application of pure mathematical techniques to the safety and efficiency of modern communication systems, with particular interest in the fields of arithmetic and algebraic geometry over finite fields, algorithms for finite fields, error correcting codes, cryptology, boolean functions, discrete dynamical systems, and their interactions. The 2013 edition had the honour to feature six invited and eleven contributed talks from renowned international experts presenting strong results recently obtained on topics ranging from pure mathematics to cryptographic algorithms. The GEOCRYPT 2013 Program Committee carefully evaluated the submitted abstracts and selected the best contributions for presentation at the conference.

Aim/Purpose: The purpose of this study was to examine whether the authoring of computer games in a mainstream primary school setting can support the learning of game design and programming concepts. Background: Despite the benefits for students when they learn how to program and the significant body of research regarding this matter, these benefits are still under debate, and the teaching of programming has a relatively undeveloped pedagogy. With this in mind, a project was designed and implemented, having constructionism as its theoretical framework. Also, Microsoft’s Kodu Game Lab was used for the development of students’ games. Methodology: The project lasted for almost a school year (fifty two-hour sessions) and the target group was fifth-grade students (ages 10-11). A total of 138 students participated, coming from five schools in Athens, Greece. Students were divided into three groups. While all groups worked in pairs, to the first there was no teachers’ guidance, to the second there was limited teachers’ guidance, and to the third, a combination of teacher-led and pair work was used. Each group developed three games of escalating complexity and a total of 207 games were analyzed. Data were collected by analyzing students’ games and with a short questionnaire. Contribution: The study contributes to the debate surrounding the pedagogy of computer game authoring as a tool for teaching programming. That is because few studies have examined the above in mainstream settings, having as a target group primary school students. Furthermore, the present study is in contrast to most studies which involved intensive research projects, since it lasted for almost a school year. Findings: It was found that the most commonly used programming concept was conditions, followed by variables and loops, while Boolean logic and functions were the least used ones. The most problematic concepts proved to be Boolean logic and loops, closely followed by functions. The least problematic concepts were conditions and variables. Also, the number of programming concepts that were used was increasing in each game, while the errors were decreasing. All in all, students’ final games fall into the relational level according to a modified version of the SOLO taxonomy. While the findings indicated that, as well as learning some basic programming concepts, students enjoyed the activity and demonstrated positive attitudes to learning programming by developing games; it was also found that the teaching method did not have any effect on the learning outcomes nor in their views for game authoring. Recommendations for Practitioners: Extended projects can be considered for teaching programming to primary school students, using visual programming tools that allow the development of games. The classes’ teachers can undertake the task to teach programming if they are properly trained. The SOLO taxonomy can be used for assessing students’ games. Future Research: Future studies can examine a variety of game-like programming environments and the target group can be older or younger students. The assessment of students’ games is also an interesting topic. Finally, research can be conducted by using other devices and compare the results.

The problem of finding “highly probable” approximations of Boolean functions consists in generating a list of all linear Boolean functions that agree with a given Boolean function in at least a specified number of binary sets. If a Boolean function is given by its truth table, the most well-known algorithm for solving this problem is the Fast Hadamard Transform. However, this method is not optimal in terms of complexity, even among deterministic algorithms. If the Boolean function is given by an oracle and depends on hundreds or thousands of variables, the application of deterministic algorithms for finding its linear approximations becomes practically infeasible. In this case, polynomial probabilistic algorithms are used, such as the Goldreich–Levin algorithm and its modifications. One of the fastest among them currently is the improved Levin’s algorithm. In the language of coding theory, this algorithm performs list decoding of the Hadamard code, which involves the value vectors of all n-variable linear Boolean functions. This paper presents a generalization of the improved Levin’s algorithm to the case of constrained probabilistic pseudo-Boolean functions. The main result is a theorem establishing a lower bound on the probability that each sought approximation appears in a random list generated by the proposed algorithm. The consideration of such functions is necessary to extend the applicability of the improved Levin’s algorithm (in place of the original Goldreich–Levin algorithm) within the well-known framework for proving the security of stream ciphers. In particular, the results of this paper enable a more efficient reduction of problems in proofs of pseudo-randomness for certain well-known keystream generators, assuming high computational complexity of decoding random linear block codes or solving random systems of nonlinear Boolean equations. The obtained results can also be used for finding linear approximations of encryption transformations of block ciphers, which is important for constructing linear attacks against them.

Boolean algebra operating on the binary numbers 0 and 1 was first developed by George Boolean in 19th century [1]. A Boolean function is an expression using binary variables. We review basic Boolean algebra laws, duals of these laws, functions and a visual method called Karnaugh-maps to simplify a Boolean expression in the first part of this chapter. We then review combinational circuits in the second part and describe simple structures called logic gates to build a logic circuit to represent a Booelan function. We conclude this chapter by arithmetic circuits to add binary numbers.

The random logic portion of a chip Implementing a set of Boolean functions and sequential circuits usually represents a major contribution to chip area. Obviously, there are many circuits which realize the same Boolean function. Unfortunately, at present there is no general theory that provides designers (and design automation programs) with lower bounds for total area, for gate oxide area, and for delay time of logic Implementations in Integrated systems. Therefore, the main task for the computer optimization program appears In choice of the circuit with the most convenient layout.DIADES is a design automation system with register-transfer level description on its Input and CIF file on output [5]. The digital circuit can be described in both behavioral and structural mode. A set of successive compilations and hardware implementing and optimizing transformations create the description of the network on the level of logic gates and pass transistors. As the output of hardware compilation from the higher level, this description is usually nonoptimal and thus is next optimized by recursive technology independent transformations based on Boolean algebra (like A*O = O, A*A = A, etc.). Inverters are also inserted into long chains of AND or OR gates, being the results of iterative circuits' compilation [4].The next stages are: technology dependant optimization of the logic network, and network's layout. It is assumed that the resultant network is multilevel, consists of complex negative gates, and is realized in semiregular Weinberger-style gate matrix layout. The logic minimization method intended for layout minimization is described in this paper.It is assumed in our silicon compiler, that logic is constructed of n-channel, polysilicon gate MOSFET ratioless complex gates, performing any negative function. By negative function we understand negation of positive function, while positive function is any combination of AND and OR functors [1]. Functions for evaluation of circuit's performance parameters such as total area, area of gate oxide, and gate delay time are used. These functions are defined in terms of basic technology and selected topology parameters. The method can be adapted to other technologies.

Babbitt's Canonical Form Revisited:Codes and Metaphors for Epistemology Joshua Banks Mailman (bio) Even among Milton Babbitt's composition titles (such as It takes Twelve to Tango, Whirled Series, and the Joy of More Sextets) his title Canonical Form is unusually mysterious, not for being abstruse or impenetrable, but rather for being so multiply penetrable; that is, for being so easily unwrapped to reveal several possible meanings. Until recently, the titles of Babbitt's compositions have not been subject to much scrutiny, usually being treated merely as casual diversions, prompting mere casual speculation. Alison Maggart's (2017; 2021) recent work, however, delves into aspects of Babbitt's biography, identity, and personality to develop an unprecedented hermeneutical account of his music, foregrounding Babbitt's predilection for wordplay in the form of puns, considering the possible power dynamics of humor, Babbitt's Jewish identity, and his nostalgia for baseball. The present essay pursues such a hermeneutical account, focusing on Babbitt's solo piano work Canonical Form (1983). The third of Joseph Dubiel's "Three Essays on Milton Babbitt" (1990; 1991; 1992) addresses the overall grandeur of Babbitt's Canonical Form, suggesting some possible meanings of the title, which [End Page 215] provide part of the foundation that my remarks build on. Yet since the time of Dubiel's writing, not only have Babbitt's titles begun to be scrutinized more closely (Maggart 2017; 2021), but also a number of analytical technologies and attitudes have arisen which enable, or underwrite, an even broader spectrum of interpretations than was perhaps possible, or tenable. For instance, Gilles Fauconnier and Mark Turner's (1998; 2002) conceptual blending theory, including the analytical technology of conceptual integration networks (CINs), is enlisted to model the multifaceted, multivalent nature of Babbitt's music and titles.1 It provides a framework for interpreting various analytical results, as I've demonstrated before. For instance, as I showed previously (Mailman 2020) and show below, Schenkerian analysis can be deployed to reveal tonality in Babbitt's music, such that it can be read as a conceptual blend of serial and tonal syntaxes. Later in the analyses below I also apply, to pitch registers, both oscillator modeling and David Lewin's (1995) technique of binary state operators, which he had applied specifically to rhythms and instrumentation in two other Babbitt works. Finally I present a technique of cumulative pattern matching (across domains of pitch class and pitch register) based on Boolean functions (Boolean algebra) which reveals a hidden meaning in Babbitt's title. Yet another arises from an important matrix pattern in linear algebra. Ultimately, I suggest, as one aspect of Babbitt's compositional poetics, a certain pragmatism regarding ontologies in music theoretical discourse. That is, this piano work and its title, taken together, endorse a certain pragmatism as instrumental to the progress of musical inquiry. Canonicity One of the possible meanings of the title "canonical form" that is at least worth considering is the work's engagement with canonical repertoire, not in terms of anything as overt as quoting, but rather through subtler means. In this work, and others from the 1980s, Babbitt's pitch configurations dance with the tonal languages of canonic repertoire, by eluding to brief triadic chord progressions and even longer voice-leading motions. Canonical Form is based on the same array as Whirled Series, Quartet No. 5, and Around the Horn (Mead 1994; Dubiel 1997), an array whose row features three consonant triads as consecutive segments. Furthermore these works are based on superarrays, which enable a surprisingly vast flexibility in aligning pitches to create various configurations (Mailman 2019). This is because these superarrays feature three or more [End Page 216] simultaneous arrays whose ordering relative to each other is undetermined. Thus Babbitt can choose to align pitches from different array lynes, not only to forge consonant triads but also to suggest cadential tonal gestures and, in some cases, longer chromatic tonal progressions. A passage near the beginning of Canonical Form exemplifies this strikingly. Shown in Example 1, a perfect authentic (I-V-I) cadence in F# minor launches m. 21, prepared by a melodic lead-in (A and G# from m. 20). In m. 22, six pitches, drawn...

The orthogonalization of Boolean functions in disjunctive form, that means a Boolean function formed by sum of products, is a classical problem in the Boolean algebra. In this work, the novel methodologyORTH[ⴱ] of orthogonalization which is an universally valid formula based on the combination technique »orthogonalizing difference-building ⴱ« is presented. Therefore, the technique ⴱ is used to transform Sum of Products into disjoint Sum of Products. The scope of orthogonalization will be solved by a novel formula in a mathematically easier way. By a further procedure step of sorting product terms, a minimized disjoint Sum of Products can be reached. Compared to other methods or heuristicsORTH[ⴱ] provides a faster computation time.

Lattices can be defined either as special partially ordered sets or as algebras. In this chapter, we present both definitions and show their equivalence. We study several special classes of lattices: modular and distributive lattices and complete lattices. The last part of the chapter is dedicated to Boolean algebras and Boolean functions.

We present a new technique for defining, analysing, and simplifying digital functions, through hand-calculations, easily demonstrable therefore in the classrooms. It can be extended to represent discrete systems beyond the Boolean logic. The method is graphical in nature and provides complete ‘‘implementation-free” description of the logical functions, similar to binary decision diagrams (BDDs) and Karnaugh-maps (K-maps). Transforming a function into the proposed representations (also the inverse) is a very intuitive process, easy enough that a person can hand-calculate these transformations. The algorithmic nature allows for its computing-based implementations. Because the proposed technique effectively transforms a function into a scatter plot, it is possible to represent multiple functions simultaneously. Usability of the method, therefore, is constrained neither by the number of inputs of the function nor by its outputs in theory. This, being a new paradigm, offers a lot of scope for further research. Here, we put forward a few of the strategies invented so far for using the proposed representation for simplifying the logic functions. Finally, we present extensions of the method: one that extends its applicability to multivalued discrete systems beyond Boolean functions and the other that represents the variants in terms of the coordinate system in use.