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Boolean Differential Equations (Synthesis Lectures on Digital Circuits and Systems), by Bernd Steinbach, Christian Posthoff
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The Boolean Differential Calculus (BDC) is a very powerful theory that extends the structure of a Boolean Algebra significantly. Based on a small number of definitions, many theorems have been proven. The available operations have been efficiently implemented in several software packages. There is a very wide field of applications. While a Boolean Algebra is focused on values of logic functions, the BDC allows the evaluation of changes of function values. Such changes can be explored for pairs of function values as well as for whole subspaces. Due to the same basic data structures, the BDC can be applied to any task described by logic functions and equations together with the Boolean Algebra. The BDC can be widely used for the analysis, synthesis, and testing of digital circuits.
Generally speaking, a Boolean differential equation (BDE) is an equation in which elements of the BDC appear. It includes variables, functions, and derivative operations of these functions. The solution of such a BDE is a set of Boolean functions. This is a significant extension of Boolean equations, which have sets of Boolean vectors as solutions. In the simplest BDE a derivative operation of the BDC on the left-hand side is equal to a logic function on the right-hand side. The solution of such a simple BDE means to execute an operation which is inverse to the given derivative. BDEs can be applied in the same fields as the BDC, however, their possibility to express sets of Boolean functions extends the application field significantly.
Table of Contents: Basics of the Binary Boolean Algebra / Summary of the Boolean Differential Calculus / Boolean Differential Equations / Solutions of the Exercises / Bibliography / Authors' Biographies / Index
- Sales Rank: #5915145 in Books
- Published on: 2013-07-01
- Original language: English
- Number of items: 1
- Dimensions: 9.25" h x .36" w x 7.50" l, .63 pounds
- Binding: Paperback
- 158 pages
Most helpful customer reviews
0 of 0 people found the following review helpful.
Review of B. Steinbach and C. Posthoff "Boolean Differential Equations"
By jbutler
The Boolean difference equation of some Boolean function expresses how changes in a function's variables cause changes in the function's value. For example, the Boolean difference of a fixed Boolean function f = 0 or f = 1 is a constant 0, since a variable change does not change the function value. Boolean difference equations are useful in specifying tests for functions, since faults, especially stuck-at faults cause the faulty function to change less than the fault-free function.
This is a new text by authors who have made seminal contributions to this field. It is a compendium of what is known, how it can be applied, and what tools are available. Chapter 1 lays the theoretical foundation of Boolean algebra on which the rest of the book is based. A welcome contribution is a description of XBOOLE, a program developed by the authors to solve Boolean equations. This application is free, and the book's description is complete, encompassing necessary details in download, installation, and use. XBOOLE allows the representation of functions, the determination of whether two expressions are equal, and if not equal, which variable assignments satisfy both expressions. One can perform various operations on functions, including differentiation.
Chapter 2 begins with basic Boolean differences and ends in Boolean calculus. Chapter 3 extends to Boolean differential equations. Each chapter has exercises, and there is a solution-to-the-exercises section at the end (Chapter 4). Throughout these chapters are insights into how XBOOLE can be used to supplement the mastering of the concepts presented.
This book is theoretical. Historical perspective is provided to help the reader to understand the context of the theory. This is not light reading; it is, however, a comprehensive explanation of an important concept in logic design by authors who have made pioneering contributions to the field.
0 of 0 people found the following review helpful.
Introduction to Boolean Differential Calculus
By rscheuring
This book is devoted to the Boolean Differential Calculus (BDC). The BDC extends the structure of a Boolean algebra. By means of the BDC properties of Boolean functions of Boolean variables can be examined.
Chapter 1 goes over basics of Boolean algebra and the notation used in this outline. Chapter 2 introduces Boolean derivative operations. Chapter 3 focuses on Boolean differential equations.
The solution of BDC equations and problems requires the usage of specific software. XBOOLE has been developed for this purpose and is able to solve problems with hundreds or even thousands of logic values. A free version of XBOOLE Monitor can be downloaded from the author's website.
In this book many examples are given. They explain how mathematical and numerical calculations can be made. However, the examples don't focus on practical applications of the BDC.
Although no specific previous knowledge is required to read this book, working through the material requires a commitment to understand more about the BDC. The BDC is not easy to understand.
0 of 0 people found the following review helpful.
Discrete and Continuous, Simultaneously
By MontereyUser
I have an interest in Boolean functions, but from a cryptographic perspective. I won't go through the summary of the book, as the reader can find that out quickly using the Table of Contents, but I want to mention that I liked the style of the book. It contains many examples, a description of a piece of software that the authors developed (I downloaded that and played with it for a bit). I was aware of many concepts from this book by Steinbach and Posthoff on Boolean Differential Calculus, as we use them in Boolean combiners, but this book contains some of these in a leisurely pace and perhaps can be used even by researchers not working in this particular area.
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