Boolean differential calculus

The Boolean differential calculus (BDK) is a branch of Boolean algebra that deals specifically with changes to Boolean variables and Boolean functions . The development of the Boolean differential calculus was initiated by the work of Sheldon B. Akers, Jr. (1959) and AD Talancev (AD Talantsev, А. Д. Таланцев) (1959). Since that time, important results have been achieved in both theory and application in circuit design . In particular, the work of Dieter Bochmann and Christian Posthoff (1981), André Thayse (1981) and Dieter Bochmann and Bernd Steinbach (1991) were important for the development of the Boolean differential calculus.

Boolean differential operators play an important role in the BDK. They allow the application of a differential concept related to that of classical analysis to logical functions.

The differential of a Boolean variable models the relationship: ${\displaystyle dx_{i}}$ ${\displaystyle x_{i}}$

${\displaystyle dx_{i}={\begin{cases}0,&{\text{keine Ä}}\,{\text{nderung von }}x_{i}\\1,&{\text{Ä}}\,{\text{nderung von }}x_{i}\end{cases}}}$

It does not assume anything about the nature, causes or consequences of the change. The differentials are also binary quantities. You can calculate with them like with the usual binary variables. ${\displaystyle dx_{i}}$

With the help of the Boolean differential calculus, different approaches of discrete event systems theory, such as

• the automata theory ,
• the Petri net theory and
• the supervisory control theory ( s )

treated in a uniform and closed form and their specific advantages combined.

literature

• Sheldon Buckingham Akers, Jr .: On a Theory of Boolean Functions . In: Journal of the Society for Industrial and Applied Mathematics . Vol. 7, No. 4, December 1959 [1957-09-27, 1959-05-28], ISSN  0368-4245 , doi : 10.1137 / 0107041 . Pp. 487-498. (12 pages)
• А. Д. Таланцев [AD Talancev, AD Talantsev]: б анализе и синтезе некоторых электрических схем при помощи специальных логических операторов - Whether analize i sinteze nekotorykh električeskikh skhem pri pomośći special'nykh logičeskikh operatorov [On the analysis and synthesis of Certain electrical circuits by Means of special logical operators] . In: Автоматика и телемеханика - Avtomatika i telemekhanika [Automation and Remote Control] . Vol. 20, No. 7, 1959 (1958-11-01), pp. 898-907. Wed at12783. (10 pages)
• André Thayse, Marc Davio: Boolean Differential Calculus and its Application to Switching Theory . IEEE Transactions on Computers. C-22 (4). April 1973. pp. 409-420. (12 pages)
• André Thayse: Boolean Calculus of Differences (= Lecture Notes in Computer Science 101). Springer, Berlin. 1st edition. 1981. ISBN 3-540-10286-8 . (144 pages)
• Dieter Bochmann , Christian Posthoff: Binary dynamic systems . Akademie-Verlag , Berlin / R. Oldenbourg Verlag , Munich. 1st edition. 1981. ISBN 3-486-25071-X . License number : 202.100 / 408/81. Order number: 7623619 (6391) LSV 1085. (397 pages)
• Dieter Bochmann, Bernd Steinbach : logic design with XBOOLE. Algorithms and programs . VEB Verlag Technik , Berlin. 1st edition. 1991. ISBN 3-341-01006-8 . (303 pages + 5.25 ″ diskette)
• Frank Dresig: Grouping - theory and application in logic synthesis (= progress reports VDI . Series 9, Vol. 145). VDI-Verlag , Düsseldorf 1992, ISBN 3-18-144509-6 (also: Chemnitz, Techn. Univ., Diss.). (147 pages)
• Christian Posthoff, Bernd Steinbach: Logic Functions and Equations - Binary Models for Computer Science . Springer, Dordrecht, the Netherlands. 1st edition. 2004-02-04. ISBN 1-4020-2937-3 . (392 pages)
• Bernd Steinbach, Christian Posthoff: Logic Functions and Equations - Examples and Exercises . Springer Science + Business Media BV, Dordrecht, Netherlands. 1st edition. 2009-02-12. ISBN 978-1-4020-9594-8 . (232 pages)
• Bernd Steinbach, Christian Posthoff: Boolean Differential Calculus - Theory and Applications . Journal of Computational and Theoretical Nanoscience. Vol. 7 No. 6: 2010-06-01. Pp. 933-981. ISSN 1546-1955. doi : 10.1166 / jctn.2010.1441 (49 pages)
• Bernd Steinbach, Christian Posthoff: Boolean Differential Calculus . In: Tsutomu Sasao; Jon T. Butler: Progress in Applications of Boolean Functions. Synthesis Lectures on Digital Circuits and Systems. Morgan & Claypool Publishers: San Rafael, California, USA. Vol. 4, No. 1, # 26, 1st edition. (2010-01-15) [2009]. Pp. 55-78. (24 pages)
• Bernd Steinbach, Christian Posthoff: Boolean Differential Equations . Synthesis Lectures on Digital Circuits and Systems. Morgan & Claypool Publishers, San Rafael, California, USA. Vol. 8, No. 3, # 42, 1st edition. 2013-07-01, ISBN 978-1-62705-241-2 . (158 pages)
• Bernd Steinbach, Christian Posthoff: Boolean Differential Calculus . Synthesis Lectures on Digital Circuits and Systems. Morgan & Claypool Publishers, San Rafael, California, USA. Vol. 12, No. 1, # 52, 1st edition. 2017-06-07, ISBN 978-1-62705-922-0 . doi : 10.2200 / S00766ED1V01Y201704DCS052 . (216 pages)