8 - BOOLEAN ALGEBRA

CHAPTER VII - AXIOMATIC SET THEORY

In this paper a state of the art, main problems and future trends concerning evolutionary design of combinational digital circuits are presented. Several evolutionary algorithms for design and optimization of combinational digital circuits are also shortly described. In evolutionary design methods the main problem is related to "combinatorial explosion" existing in those methods. Because evolutionary design methods are based on "generate and test" model, therefore together with growing of digital circuit input number, the number of potential combinations which must be tested are growing exponentially. This problem is a "bottle-neck" of those methods, and therefore it is discussed in details in this paper. To solve this problem it becomes important to elaborate efficient decomposition techniques of the designed circuits to some less complex sub-circuits, and then to design each of them independently. Also, it is important to create novel genetic operators (as mutation and crossover) which will always lead to acceptable solutions (solutions which satisfy the truth table). Due to such kind of operators, any repair mechanisms need not necessarily be used, and therefore the "generate and test" model will be simplified only to generation of new solutions, and therefore evolutionary design methods will become more effective.

In this paper, we examine many real world examples of information granules, and construct a granular deductive reasoning system for these domains. Objects are ordered pairs; the first element is an assertion (to call logical formula) and the second is a semantic set corresponding to the assertion. So the granular language and its model involve both logic and set theory. So-called logic means that the reasoning obeys the syntax in logical language; So called set theory means that the operations corresponding to semantic sets of logical formulas obey the methods in set theory. The evaluation of truth values and the computation rule of granular formulas are established.

In this paper posibility of design and optimization of combinational digital circuits using modified evolutionary algorithm is presented. Modification of evolutionary algorithm depends on introduction of multilayer chromosomes and genetic operators operating on them. Design results for four combinational circuits obtained using this method are compared with described in literature methods: Karnaugh Maps, Quine-McCluskey and NGA and MGA genetic algorithms. Described evolutionary algorithm leads in many cases to better results.

Subject of Research.Representation features of education results for competence-based educational programs are analyzed. Solution importance of decoding and proficiency estimation for elements and components of discipline parts of competences is shown. The purpose and objectives of research are formulated. Methods. The paper deals with methods of mathematical logic, Boolean algebra, and parametrical analysis of complex diagnostic test results, that controls proficiency of some discipline competence elements. Results. The method of logical conditions analysis is created. It will give the possibility to formulate logical conditions for proficiency determination of each discipline competence element, controlled by complex diagnostic test. Normalized test result is divided into noncrossing zones; a logical condition about controlled elements proficiency is formulated for each of them. Summarized characteristics for test result zones are imposed. An example of logical conditions forming for diagnostic test with preset features is provided. Practical Relevance. The proposed method of logical conditions analysis is applied in the decoding algorithm of proficiency test diagnosis for discipline competence elements. It will give the possibility to automate the search procedure for elements with insufficient proficiency, and is also usable for estimation of education results of a discipline or a component of competence-based educational program.

Set theory is the mathematics of infinity and part of the core curriculum for mathematics majors. This book blends theory and connections with other parts of mathematics so that readers can understand the place of set theory within the wider context. Beginning with the theoretical fundamentals, the author proceeds to illustrate applications to topology, analysis and combinatorics, as well as to pure set theory. Concepts such as Boolean algebras, trees, games, dense linear orderings, ideals, filters and club and stationary sets are also developed. Pitched specifically at undergraduate students, the approach is neither esoteric nor encyclopedic. The author, an experienced instructor, includes motivating examples and over 100 exercises designed for homework assignments, reviews and exams. It is appropriate for undergraduates as a course textbook or for self-study. Graduate students and researchers will also find it useful as a refresher or to solidify their understanding of basic set theory.

Compact low-power devices with ultrafast processing speed are the fundamental building blocks for the development of the state-of-the-art logic systems and memristor prominently fulfills these demands and plays a major role in digital circuit design. In this work, design, implementation, and performance evaluation of memristor-based logic gates, such as NOT, AND, NAND, OR, NOR, XOR, and XNOR, and combinational logic circuits, such as adder, subtractor, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2\times1$ </tex-math></inline-formula> mux, are presented via SPECTRE in Cadence Virtuoso. Herein, we propose an optimized design of memristor-based logic gates and combinational logic circuits and draw a comparative analysis with the conventional 180-nm complementary metal–oxide–semiconductor (CMOS) technology. The utilized memristor model is thoroughly validated with the experimental results of a high-density <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{Y}_{{2}}\text{O}_{{3}}$ </tex-math></inline-formula> -based memristive crossbar array (MCA), which shows a significantly low values of coefficient of variabilities in device-to-device (D2D) and cycle-to-cycle (C2C) operation. The area, power, and delay calculated from these combinational circuits are found to be reduced by more than 71.4%, 40%, and 54%, respectively, as compared to the conventional 180-nm CMOS technology. The impact of multiple CMOS technology nodes (90 and 180 nm) on the power consumption at the chip-level logic circuit implementation has also been investigated. The adopted memristor-based design significantly improves the performance of various logic designs, which makes it area and power efficient and enables a major breakthrough in designing various low-power, low-cost, ultrafast, and compact circuits.

Chapter 12 - Boolean Algebra and Combinatorial Circuits

Boolean algebra as algebra is value indifferent. Although it is very important, the classical two-valued realization of the Boolean algebra is only a special case. The real valued realization of the Boolean algebra is a frame for the Boolean consistent fuzzy logic in wider sense, such as the two valued realization is a frame for the classical case. This paper presents the brief review of the real-valued realization of the finite (atomic) Boolean algebra and its application. The real-valued implication is a Boolean consistent generalization of the classical binary implication. The real-valued implication plays important roles in the real-valued set theory as a generalization of the classical set theory, as well as, in many applications such as morphology in image processing, association rules in data mining and decision making generally.

With the affordable advanced computerized equipments and tools, systems engineering and software systems engi- neering become closely related. Software engineering to the low level computer software and systems is not suffi- cient enough for systems engineers' needs to build com- plex computerized systems. Every engineering system has its own systems engineering's theory and technology. Are there general principles, theories, and technology for two or a collection of systems? In this article a new inte- grated field called Systems Software Engineering (SSE) is proposed. SSE research is to study the common paradigms, theories, technology of the systems among all or simply a collection of engineering fields, applied science fields, com- puter science, software engineering, and mathematical sci- ences. According to the author's opinion, none except few like Lofti Zadeh's soft computing and granular computing are towards this trend that they have integrated fuzzy set the- ory and granular computing as an example. An integrated "Systems Software Engineering" theory based on Boolean logic, Hilbert logic and set theory is introduced. Based on Boolean logic and Hilbert logic as the foundation for the de- cision systems and control systems for information systems, we can develop integrated foundations, theories, principles, and tools for information science. We adopt bioinformat- ics, bioengineering and medical information systems as a special case to describe our integrated "Software Systems Engineering". As a matter of fact, we can choose any other integrated modern related fields as our special case to study in this paper.

In this paper we show that the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras. In every combinatory algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings), whose operations are defined by suitable combinators. Central elements are used to represent any combinatory algebra as a Boolean product of directly indecomposable combinatory algebras (i.e., algebras which cannot be decomposed as the Cartesian product of two other nontrivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e., the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible lambda theories. In one of the main results of the paper we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible: for every recursively enumerable lambda theory Tscr, there is a continuum of lambda theories including Tscr which are omitted by the indecomposable semantics

This paper presents a fully digital implementation of a memristor hardware (HW) simulator, as the core of an emulator, based on a behavioral model of voltage-controlled threshold-type bipolar memristors. Compared to other analog solutions, the proposed digital design is compact, easily reconfigurable, demonstrates very good matching with the mathematical model on which it is based, and complies with all the required features for memristor emulators. We validated its functionality using Altera Quartus II and ModelSim tools targeting low-cost yet powerful field-programmable gate array families. We tested its suitability for complex memristive circuits as well as its synapse functioning in artificial neural networks, implementing examples of associative memory and unsupervised learning of spatiotemporal correlations in parallel input streams using a simplified spike-timing-dependent plasticity. We provide the full circuit schematics of all our digital circuit designs and comment on the required HW resources and their scaling trends, thus presenting a design framework for applications based on our HW simulator.

This chapter addresses Boolean algebra, which is based on Boolean logic. In the social sciences, Boolean algebra comes under different labels. It is often used in set-theoretic and qualitative comparative analysis to assess complex causation that leads to particular outcomes involving different combinations of conditions. The basic features of Boolean algebra are the use of binary data, combinatorial logic, and Boolean minimization to reduce the expressions of causal complexity. By calculating the intersection between the final Boolean equation and the hypotheses formulated in Boolean terms, three subsets of causal combinations emerge: hypothesized and empirically confirmed; hypothesized, but not detected within the empirical evidence; and causal configurations found empirically, but not hypothesized. This approach is both holistic and analytic because it examines cases as a whole and in parts.

As has been shown by the investigations of some mathematicians, there are numerous analogies between forcing and omitting types. Evidence of that can also be found in the application of forcing to model theory. Both methods use the Rasiowa-Sikorski lemma on the existence in a Boolean algebra of an ultrafilter intersecting every dense set from a given denumerable family.In this paper we will not use the terms Cohen forcing and omitting types in their original sense. We shall deal mainly with the Scott-Boolean kind of forcing as a transformation of Cohen's idea and with methods used in logic that consist in finding a suitable ultrafilter in the Lindenbaum-Tarski algebra for a given theory and defining canonical models—under the name of omitting types.The two methods will be confronted to show that Cohen's forcing stems from logical methods. Attention will also be drawn to some differences between forcing in set theory and the general methods of logic.In logic we construct a model by defining relations in a set of constants. In particular when defining a model for set theory we define a certain relation E in a set of constants. It is usually immaterial what E is, in particular whether it is the true relation of membership up to isomorphism.

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.