Boolean differential calculus

Boolean Differential Calculus (BDC) (German: Boolescher Differentialkalkül (BDK)) is a subject field of Boolean algebra discussing changes of Boolean variables and Boolean functions.

The development of what is now known as Boolean Differential Calculus was initiated independently by works of Sheldon B. Akers and A. D. Talantsev (A. D. Таланцев) in 1959. Since then, significant advances were accomplished in both, the theory and in the application of the BDC in switching circuit synthesis. Works of Dieter Bochmann and Christian Posthoff and of André Thayse in 1981, as well as of Dieter Bochmann and Bernd Steinbach in 1991 were of particular importance in the further development of the BDC.

Boolean differential operators play a significant role in BDC. They allow the application of differentials as known from classicial analysis to be extended to logical functions.

The differentials $dx_{i}$ of a Boolean variable $x_{i}$ models the relation:

$dx_{i}={\begin{cases}0,&{\text{no change of }}x_{i}\\1,&{\text{change of }}x_{i}\end{cases}}$

There are no constraints in regard to the nature, the causes and consequences of the change.

The differentials $dx_{i}$ are binary. They can be used just like common binary variables.

The Boolean Differential Calculus allows various aspects of discrete event system theory like

• Automata theory
• Petri nets
• Supervisory control theory

to be discussed in a united and closed form and their specific advantages to be combined.

Further reading

• Akers, Sheldon B. (1959). "On a Theory of Boolean Functions". Journal of the Society for Industrial and Applied Mathematics. 7 (4): 487–498. ISSN 0368-4245. (12 pages)
• Таланцев [Talantsev], А. Д. [A. D.] (1959) [1958-11-01]. "б анализе и синтезе некоторых электрических схем при помощи специальных логических операторов (Ob 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]. Автоматика и телемеханика (Avtomatikai telemekhanika) [Automation and Remote Control] (in Russian). 20 (7): 898–907. Mi at12783. Retrieved 2017-10-15. (10 pages)
• Thayse, André (1981). Boolean Calculus of Differences. Vol. 101. Berlin, Germany: Springer. ISBN 3-540-10286-8. {{cite book}}: |work= ignored (help) (144 pages)
• Bochmann, Dieter [in German]; Posthoff, Christian (1981). Binäre dynamische Systeme. Akademie-Verlag, Berlin / R. Oldenbourg Verlag, München. ISBN 3-486-25071-X. License number 202 100/408/81. (397 pages)
• Bochmann, Dieter [in German]; Steinbach, Bernd [in German] (1991). Logikentwurf mit XBOOLE. Algorithmen und Programme (in German). Berlin, Germany: Verlag Technik. ISBN 3-341-01006-8. (303 pages + 1 disk)
• Dresig, Frank (1992). Gruppierung – Theorie und Anwendung in der Logiksynthese. 9 (in German). Vol. 145. Düsseldorf, Germany: VDI-Verlag. ISBN 3-18-144509-6. {{cite book}}: |work= ignored (help) (NB. Also: Chemnitz, Technische Universität, Dissertation.) (147 pages)
• Posthoff, Christian; Steinbach, Bernd [in German] (2004-02-04). Logic Functions and Equations - Binary Models for Computer Science. Dordrecht, Netherlands: Springer. ISBN 1-4020-2937-3. ISBN 978-1-4020-2937-0. (392 pages)
• Steinbach, Bernd [in German]; Posthoff, Christian (2009-02-12). Logic Functions and Equations - Examples and Exercises. Dordrecht, Netherlands: Springer Science + Business Media B. V. ISBN 978-1-4020-9594-8. (232 pages)
• Steinbach, Bernd [in German]; Posthoff, Christian (2013-07-01). Boolean Differential Equations. San Rafael, USA: Morgan & Claypool Publishers. ISBN 978-1-62705-241-2. (158 pages)
• Steinbach, Bernd [in German]; Posthoff, Christian (2017-06-07). Boolean Differential Calculus. San Rafael, USA: Morgan & Claypool Publishers. doi:10.2200/S00766ED1V01Y201704DCS052. ISBN 978-1-62705-922-0. (216 pages)