Boolean Differential Calculus

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 ���� and ����ⁿ, 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.

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. Asa 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. Hehas 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.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.