Boolean differential calculus explained

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

Boolean differential calculus concepts are analogous to those of classical differential calculus, notably studying the changes in functions and variables with respect to another/others.^[1]

The Boolean differential calculus allows various aspects of dynamical systems theory such as

• automata theory on finite automata
• Petri net theory
• supervisory control theory (SCT)

to be discussed in a united and closed form, with their individual advantages combined.

History and applications

Originally inspired by the design and testing of switching circuits and the utilization of error-correcting codes in electrical engineering, the roots for the development of what later would evolve into the Boolean differential calculus were initiated by works of Irving S. Reed, David E. Muller, David A. Huffman, Sheldon B. Akers Jr. and Russian: [[A. D. Talantsev]] (Russian: A. D. Talancev, Russian: А. Д. Таланцев) between 1954 and 1959, and of Frederick F. Sellers Jr., Mu-Yue Hsiao and Leroy W. Bearnson in 1968.

Since then, significant advances were accomplished in both, the theory and in the application of the BDC in switching circuit design and logic synthesis.

Works of French: [[André Thayse]], Marc Davio and French: [[Jean-Pierre Deschamps]] in the 1970s formed the basics of BDC on which German: {{ill|Dieter Bochmann|de, German: [[Christian Posthoff]] and German: {{ill|Bernd Steinbach|de further developed BDC into a self-contained mathematical theory later on.

A complementary theory of Boolean integral calculus (German: German: Boolescher Integralkalkül) has been developed as well.

BDC has also found uses in discrete event dynamic systems (DEDS) in digital network communication protocols.

Meanwhile, BDC has seen extensions to multi-valued variables and functions as well as to lattices of Boolean functions.

Overview

Boolean differential operators play a significant role in BDC. They allow the application of differentials as known from classical 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 , & nochangeof x _{i\\} 1 , & changeof x _{i \end{cases}}

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

The differentials

dx _{i} are binary. They can be used just like common binary variables.

See also

• Boolean Algebra
• Boole's expansion theorem
• Ramadge–Wonham framework

Further reading

• Marc . Davio . Philippe M. . Piret . Les dérivées Booléennes et leur application au diagnostic . fr . Boolean derivatives and their application and diagnosis . . Philips Research Laboratory, French: Manufacture Belge de Lampes et de Materiel Electronique (MBLE Research Laboratory) . Brussels, Belgium . 12 . 3 . 63–76 . July 1969. (14 pages)
• Book: Rudeanu, Sergiu . Boolean Functions and Equations . September 1974 . . 0-44410520-4 . . (462 pages)
• Dieter . Bochmann .

  de:Dieter Bochmann

  . Boolean differential calculus (a survey) . Engineering Cybernetics . Institute of Electrical and Electronics Engineers (IEEE) . 15 . 5 . 1977 . 0013-788X . 67–75. (9 pages) Translation of: Dieter . Bochmann .

  de:Dieter Bochmann

  . ru . [Boolean differential calculus (survey)] . Известия Академии наук СССР – Техническая кибернетика (Izvestii︠a︡ Akademii Nauk SSSR – Tekhnicheskai︠a︡ kibernetika) [Proceedings of the Academy of Sciences of the USSR – Engineering Cybernetics] . 5 . 1977 . 125–133. (9 pages)
• Martin . Kühnrich . Differentialoperatoren über Booleschen Algebren . de . Differential operators on Boolean algebras . Zeitschrift für mathematische Logik und Grundlagen der Mathematik . 1986 . Berlin, Germany (East) . 10.1002/malq.19860321703 . 32 . 17–18 .
  1. 18
  . 271–288. (18 pages)
• Book: Dresig, Frank . Gruppierung – Theorie und Anwendung in der Logiksynthese . de . Grouping – Theory and application in logic synthesis . Fortschritt-Berichte VDI, Ser. 9 . 145 . . Düsseldorf, Germany . 1992 . 3-18-144509-6 . . (NB. Also: Chemnitz, Technische Universität, Dissertation.) (147 pages)
• Book: Control of Discrete Event Systems by Means of the Boolean Differential Calculus . Rainer . Scheuring . Herbert "Hans" . Wehlan . 1993 . Silvano . Balemi . Petr . Kozák . Rein . Smedinga . Discrete Event Systems: Modeling and Control . Progress in Systems and Control Theory (PSCT) . 13 . . Basel, Switzerland . 79–93 . 10.1007/978-3-0348-9120-2_7. 978-3-0348-9916-1 . (15 pages)
• Book: Christian . Posthoff . Bernd . Steinbach .

  de:Bernd Steinbach

  . Logic Functions and Equations – Binary Models for Computer Science . . Dordrecht, Netherlands . 2004-02-04 . 1st . 1-4020-2937-3 . 254106952 . 10.1007/978-1-4020-2938-7 . . (392 pages)
• Book: Bernd . Steinbach .

  de:Bernd Steinbach

  . Christian . Posthoff . Logic Functions and Equations – Examples and Exercises . . Dordrecht, Netherlands . 2009-02-12 . 1st . 978-1-4020-9594-8 . 2008941076 . 10.1007/978-1-4020-9595-5. (xxii+232 pages) http://www.e-reading.club/bookreader.php/135805/Posthoff%2C_Steinbach_-_Logic_Functions_and_Equations_-_Examples_and_Exercises.pdf (NB. Per this hardcover edition has been rereleased as softcover edition in 2010.)
• Bernd . Steinbach .

  de:Bernd Steinbach

  . Christian . Posthoff . Boolean Differential Calculus – Theory and Applications . Journal of Computational and Theoretical Nanoscience . American Scientific Publishers . 7 . 6 . 2010-06-01 . 933–981 . 1546-1955 . 10.1166/jctn.2010.1441 . (49 pages)
• Book: Bernd . Steinbach .

  de:Bernd Steinbach

  . Christian . Posthoff . Chapter 3: Boolean Differential Calculus . Tsutomu . Sasao . Jon T. . Butler . Progress in Applications of Boolean Functions . limited . Synthesis Lectures on Digital Circuits and Systems . San Rafael, CA, USA . Morgan & Claypool Publishers . 2009 . 2010-01-15 . Lecture #26 . 1st . 978-1-60845-181-4 . 55–78, 121–126 . 10.2200/S00243ED1V01Y200912DCS026. 37053010 . (24 of 153 pages)

External links

• Encyclopedia: Boolean differential calculus . Herbert "Hans" . Wehlan . Boolean differential calculus - Encyclopedia of Mathematics . . 978-1-4020-0609-8 . Michiel . Hazewinkel . Michiel Hazewinkel. . 2010-12-06 . http://www.encyclopediaofmath.org/index.php?title=Boolean_differential_calculus&oldid=12153 . 2017-10-16 . live . https://web.archive.org/web/20171016111554/https://www.encyclopediaofmath.org/index.php/Boolean_differential_calculus . 2017-10-16.
• Web site: XBOOLE . Institut für Informatik (IfI) . . 2017 . 2017-10-31 . live . https://web.archive.org/web/20171031105416/http://www.informatik.tu-freiberg.de/xboole/index.php?language=English&aktwindow=home . 2017-10-31. with Web site: XBOOLE Monitor . 2008-07-23 . 2017-10-31 . dead . https://web.archive.org/web/20171031002108/http://www.informatik.tu-freiberg.de/xboole/XBOOLEMonitor.zip?Submit=XBOOLE+Monitor . 2017-10-31.

Notes and References

1. H. Wehlan, Boolean Algebra in Encyclopedia of Mathematics