The Boolean functions may be iterated either asynchronously, when their coordinates are computed independently of each other, or synchronously, when their coordinates are computed at the same time. In Boolean Systems: Topics in Asynchronicity, a book addressed to mathematicians and computer scientists interested in Boolean systems and their use in modelling, author Serban E. Vlad presents a consistent and original mathematical theory of the discrete-time Boolean asynchronous systems. The purpose of the book is to set forth the concepts of such a theory, resulting from the synchronous Boolean system theory and mostly from the synchronous real system theory, by analogy, and to indicate the way in which known synchronous deterministic concepts generate new asynchronous nondeterministic concepts. The reader will be introduced to the dependence on the initial conditions, periodicity, path-connectedness, topological transitivity, and chaos. A property of major importance is invariance, which is present in five versions. In relation to it, the reader will study the maximal invariant subsets, the minimal invariant supersets, the minimal invariant subsets, connectedness, separation, the basins of attraction, and attractors. The stability of the systems and their time-reversal symmetry end the topics that refer to the systems without input. The rest of the book is concerned with input systems. The most consistent chapters of this part of the book refer to the fundamental operating mode and to the combinational systems (systems without feedback). The chapter Wires, Gates, and Flip-Flops presents a variety of applications. The first appendix addresses the issue of continuous time, and the second one sketches the important theory of Daizhan Cheng, which is put in relation to asynchronicity. The third appendix is a bridge between asynchronicity and the symbolic dynamics of Douglas Lind and Brian Marcus. - Presents a consistent and original theory of the discrete-time Boolean asynchronous systems, which are useful for mathematicians and computer scientists interested in Boolean Networks, dynamical systems, and modeling. - Studies the flows and equations of evolution, nullclines, dependence on initial conditions, periodicity, path-connectedness, topological transitivity, chaos, nonwandering points, invariance, connectedness, and separation, as well as the basins of attraction, attractors, stability, and time-reversal symmetry. - Explains the fundamental operating mode of the input systems and the combinational systems (systems without feedback). - Includes a chapter of applications of the Boolean systems and their modeling techniques. - Makes use of the unbounded delay model of computation of the Boolean functions.

This book provides a technical approach to a Business Resilience System with its Risk Atom and Processing Data Point based on fuzzy logic and cloud computation in real time. Its purpose and objectives define a clear set of expectations for Organizations and Enterprises so their network system and supply chain are totally resilient and protected against cyber-attacks, manmade threats, and natural disasters. These enterprises include financial, organizational, homeland security, and supply chain operations with multi-point manufacturing across the world. Market shares and marketing advantages are expected to result from the implementation of the system. The collected information and defined objectives form the basis to monitor and analyze the data through cloud computation, and will guarantee the success of their survivability's against any unexpected threats. This book will be useful for advanced undergraduate and graduate students in the field of computer engineering, engineers that work for manufacturing companies, business analysts in retail and e-Commerce, and those working in the defense industry, Information Security, and Information Technology.

Christian Herde deals with the development of decision procedures as needed, e.g., for automatic verification of hardware and software systems via bounded model checking. He provides methods for efficiently solving formulae comprising complex Boolean combinations of linear, polynomial, and transcendental arithmetic constraints, involving thousands of Boolean-, integer-, and real-valued variables.

The E M B 0 course on "Formal Analysis of Genetic Regulation" A course entitled "Formal analysis of Genetic Regulation" was held at the University of Brussels from 6 to 16 September 1977 under the auspices of EMBO (European Molecular Biology Organization). As indicated by the title of the book (but not explicitly enough by the title of the course), the main emphasis was put on a dynamic analysis of systems using logical methods, that is, methods in which functions and variables take only a limited number of values - typically two. In this respect, this course was complementary to an EMBO course using continuous methods which was held some months later in Israel by Prof. Segel. People from four very different laboratories took an active part in teaching our course in Brussels : Drs Anne LEUSSLER and Philippe VAN HAM, from the Laboratory of Prof. Jean FLORINE (Laboratoire des Systemes logiques et numeriques, Faculte des Sciences appliquees, Universite Libre de Bruxelles). Dr Stuart KAUFFMAN (Dept. of Biochemistry and Biophysics, School of Medicine, Philadelphia). Prof. Gregoire NICOLlS (Service de Biophysique Theorique, Faculte des Sciences, Universite Libre de Bruxelles) and his temporary coworker Dr David RIGNEY (presently at the Center for Statistical Mechanics and Thermodynamics of the University of Texas at Austin, Texas).

The essential guide showing how the unbounded delay model of computation of the Boolean functions may be used in the analysis of the Boolean networks Boolean Functions: Topics in Asynchronicity contains the most current research in several issues of asynchronous Boolean systems. In this framework, asynchronicity means that the functions which model the digital circuits from electronics iterate their coordinates independently on each other and the author—a noted expert in the field—includes a formal mathematical description of these systems. Filled with helpful definitions and illustrative examples, the book covers a range of topics such as morphisms, antimorphisms, invariant sets, path connected sets, attractors. Further, it studies race freedom, called here the technical condition of proper operation, together with some of its generalized and strengthened versions, and also time reversal, borrowed from physics and also from dynamical systems, together with the symmetry that it generates. This book: Presents up-to-date research in the field of Boolean networks, Includes the information needed to understand the construction of an asynchronous Boolean systems theory and contains proofs, Employs use of the language of algebraic topology and homological algebra. Written formathematicians and computer scientists interested in the theory and applications of Boolean functions, dynamical systems, and circuits, Boolean Functions: Topics in Asynchronicity is an authoritative guide indicating a way of using the unbounded delay model of computation of the Boolean functions in the analysis of the Boolean networks.

The Internet of Things is a great new challenge for the development of digital systems. In addition to the increasing number of classical unconnected digital systems, more people are regularly using new electronic devices and software that are controllable and usable by means of the internet. All such systems utilize the elementariness of Boolean values. A Boolean variable can carry only two different Boolean values: FALSE or TRUE (0 or 1), and has the best interference resistance in technical systems. However, a Boolean function exponentially depends on the number of its variables. This exponential complexity is the cause of major problems in the process of design and realization of circuits. According to Moore’s Law, the complexity of digital systems approximately doubles every 18 months. This requires comprehensive knowledge and techniques to solve complex Boolean problems. This book summarizes both new problems and solutions in the Boolean domain in solving such issues. Part 1 describes powerful new approaches in solving exceptionally complex Boolean problems. Efficient methods contribute to solving problems of extreme complexity. New algorithms and programs utilize the huge number of computing cores of the Graphical Processing Unit and improve the performance of calculations by several orders of magnitude. Part 2 represents several applications of digital systems. Due to the crucial role of the internet, both solutions and open problems regarding the security of these systems are discussed. The exploration of certain properties of such systems leads to a number of efficient solutions, which can be reused in a wide field of applications. Part 3 discusses the scientific basis of future circuit technologies, investigating the need for completely new design methods for the atomic level of quantum computers. This part also concerns itself with reversible circuits as the basis for quantum circuits and specifies important issues regarding future improvements.

This book is about the logic of Boolean equations. Such equations were central in the "algebra of logic" created in 1847 by Boole [12, 13] and devel oped by others, notably Schroder [178], in the remainder of the nineteenth century. Boolean equations are also the language by which digital circuits are described today. Logicians in the twentieth century have abandoned Boole's equation based logic in favor of the more powerful predicate calculus. As a result, digital engineers-and others who use Boole's language routinely-remain largely unaware of its utility as a medium for reasoning. The aim of this book, accordingly, is to is to present a systematic outline of the logic of Boolean equations, in the hope that Boole's methods may prove useful in solving present-day problems. Two Logical Languages Logic seeks to reduce reasoning to calculation. Two main languages have been developed to achieve that object: Boole's "algebra of logic" and the predicate calculus. Boole's approach was to represent classes (e. g. , happy creatures, things productive of pleasure) by symbols and to represent logical statements as equations to be solved. His formulation proved inadequate, however, to represent ordinary discourse. A number of nineteenth-century logicians, including Jevons [94], Poretsky [159], Schroder [178], Venn [210], and Whitehead [212, 213], sought an improved formulation based on ex tensions or modifications of Boole's algebra. These efforts met with only limited success.

Analysis and Control of Boolean Networks presents a systematic new approach to the investigation of Boolean control networks. The fundamental tool in this approach is a novel matrix product called the semi-tensor product (STP). Using the STP, a logical function can be expressed as a conventional discrete-time linear system. In the light of this linear expression, certain major issues concerning Boolean network topology – fixed points, cycles, transient times and basins of attractors – can be easily revealed by a set of formulae. This framework renders the state-space approach to dynamic control systems applicable to Boolean control networks. The bilinear-systemic representation of a Boolean control network makes it possible to investigate basic control problems including controllability, observability, stabilization, disturbance decoupling etc.