Boolean Differential Calculus

Boolean Differential Calculus

Bernd Steinbach, Christian Posthoff
ISBN: 9781627059220 | PDF ISBN: 9781627056175
Copyright © 2017 | 215 Pages | Publication Date: June, 2017

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The Boolean Differential Calculus (BDC) is a very powerful theory that extends the basic concepts of Boolean Algebras significantly. Its applications are based on Boolean spaces B and Bn, Boolean operations, and basic structures such as Boolean Algebras and Boolean Rings, Boolean functions, Boolean equations, Boolean inequalities, incompletely specified Boolean functions, and Boolean lattices of Boolean functions. These basics, sometimes also called switching theory, are widely used in many modern information processing applications.

The BDC extends the known concepts and allows the consideration of changes of function values. Such changes can be explored for pairs of function values as well as for whole subspaces. The BDC defines a small number of derivative and differential operations. Many existing theorems are very welcome and allow new insights due to possible transformations of problems. The available operations of the BDC have been efficiently implemented in several software packages.

The common use of the basic concepts and the BDC opens a very wide field of applications. The roots of the BDC go back to the practical problem of testing digital circuits. The BDC deals with changes of signals which are very important in applications of the analysis and the synthesis of digital circuits. The comprehensive evaluation and utilization of properties of Boolean functions allow, for instance, to decompose Boolean functions very efficiently; this can be applied not only in circuit design, but also in data mining. Other examples for the use of the BDC are the detection of hazards or cryptography. The knowledge of the BDC gives the scientists and engineers an extended insight into Boolean problems leading to new applications, e.g., the use of Boolean lattices of Boolean functions.

Table of Contents

Basics of Boolean Structures
Derivative Operations of Boolean Functions
Derivative Operations of Lattices of Boolean Functions
Differentials and Differential Operations
Solutions of the Exercises
Authors' Biographies

About the Author(s)

Bernd Steinbach, Freiberg University of Mining and Technology
From 1973–1977, Bernd Steinbach studied Information Technology at the University of Technology in Chemnitz (Germany) and graduated with an M.Sc. in 1977. He graduated with a Ph.D. and with a Dr. sc. techn. (Doctor scientiae technicarum) for his second doctoral thesis from the Faculty of Electrical Engineering of the Chemnitz University of Technology in 1981 and 1984, respectively. In 1991, Steinbach obtained the habilitation (Dr.-Ing. habil.) from the same faculty. Topics of his theses involved Boolean equations, Boolean differential equations, and their application in the field of circuit design using efficient algorithms and data structures on computers.

Steinbach worked in industry as an electrician, where he had tested professional controlling systems at the Niles Company. After his studies he taught as Assistant Lecturer at the Department of Information Technology of the Chemnitz University of Technology. As a research engineer he developed programs for test pattern generation for computer circuits at the company Robotron. He later returned to the Department of Information Technology of the Chemnitz University of Technology as Associate Professor for design automation in logic design. Since 1992 he has worked as a Full Professor of Computer Science/Software Engineering and Programming at the Freiberg University of Mining and Technology, Department of Computer Science. He has served as Head of the Department of Computer Science and Vice-Dean of the Faculty of Mathematics and Computer Science. His research areas include logic functions and equations and their application in many fields, such as artificial intelligence, UML-based testing of software, and UML-based hardware/software co-design. He is the head of a group that developed the XBOOLE software system. He published three books about logic synthesis. The first one (together with D. Bochmann) covers Logic Design using XBOOLE (in German), Technik 1991. The following two, co-authored by Christian Posthoff, are Logic Functions and Equations-Binary Models for Computer Science and Logic Functions and Equations - Examples and Exercises, Springer 2004, and 2009, respectively. As one application of the Boolean Differential Calculus, he co-authored another book with Christian Posthoff, Boolean Differential Equations, Morgan & Claypool Publishers 2013. He is the editor and co-author of several sections of the books Recent Problems in the Boolean Domain and Problems and New Solutions in the Boolean Domain, both of which were published by Cambridge Scholars Publishing in 2014 and 2016, respectively. Again co-authored by Christian Posthoff, he published three textbooks in German: Logic Functions-Boolean Models, Efficient Calculations Using XBOOLE, and Java Programming for Beginners EAGLE 2014, 2015, and 2016. He published more than 250 chapters in books, complete issues of journals, and papers in journals and proceedings.

He has served as Program Chairman for the IEEE International Symposium on Multiple-Valued Logic (ISMVL), and as guest editor of the Journal of Multiple-Valued Logic and Soft Computing. He is the initiator and general chair of a biennial series of International Workshops on Boolean Problems (IWSBP) which started in 1994, now with 12 workshops. He received the Barkhausen Award from the University of Technology Dresden in 1983.

Christian Posthoff, The University of West Indies
From 1963-1968, Christian Posthoff studied Mathematics at the University of Leipzig. His thesis was titled: "Axiomatic Description of a Finite Class Calculus" (Prof. Dr. Klaua). From 1968-1972, he worked as a programmer and in the field of Operations Research; simultaneously, he did his Ph.D. in 1975 with the thesis "Application of Mathematical Methods in Communicative Psychotherapy." In 1972, he joined the Department of Information Technology at the Chemnitz University of Technology; until 1983, his research activities concentrated on logic design, particularly on numerical methods for Boolean problems. His cooperation with B. Steinbach goes back to these days. Important results have been algorithms and programs for solving Boolean equations with a high number of variables and the Boolean Differential Calculus for the analytical treatment of different problems in the field of logic design.These results have been collected in a monograph Binary Dynamic Systems published simultaneously in Akademie-Verlag Berlin, Oldenbourg-Verlag Munich-Vienna and in the Soviet Union, and allowed the habilitation (Dr.-Ing. habil.) at the Faculty of Electrical Engineering in 1979 and the promotion to Associate Professor. He wrote two textbooks at this time, aimed at a higher level in the theoretical and mathematical training of graduate engineers of information technology. About 1976, he started research activities in Artificial Intelligence.

In 1983, Posthoff started as Full Professor of Computer Science in the Department of Computer Science at the same university, with the aim of starting the program in Computer Science in 1984. In 1984, he became Head of the Institute of Theoretical Computer Science and Artificial Intelligence and Research Director of the Department of Computer Science. An independent direction of research activities within AI, investigations of computer chess and other strategic games, arose from his love for chess. His research activities concentrated on the application of fuzzy logic for the modeling of human-like "thinking" methods, the learning from examples, the construction of intelligent tutoring systems, the application of inference mechanisms, the construction of systems for diagnosis, and configuration. In cooperation with colleagues from different areas of mechanical engineering and medicine, he has been supervising the construction of several expert systems. He received the Scientific Award of the Chemnitz University of Technology four times.

In 1994, he moved to the Chair of Computer Science at the University of The West Indies, St. Augustine, Trinidad & Tobago. From 1996-2002 he was Head of the Department of Mathematics & Computer Science. His main focus was the development of Computer Science education at the undergraduate and graduate level to attain international standard. In 2001, he received the Vice-Chancellor's Award of Excellence. He is the author or co-author of 19 books and many publications in journals and conference proceedings.

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