Boolean differential calculus

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,[3] David E. Muller,[4] David A. Huffman,[5] Sheldon B. Akers Jr.[6] and A. D. Talantsev (A. D. Talancev, А. Д. Таланцев)[7] between 1954 and 1959, and of Frederick F. Sellers Jr.,[8][9] Mu-Yue Hsiao[8][9] and Leroy W. Bearnson[8][9] 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 André Thayse,[10][11][12][13][14] Marc Davio[11][12][13] and Jean-Pierre Deschamps[13] in the 1970s formed the basics of BDC on which Dieter Bochmann,[15] Christian Posthoff[15] and Bernd Steinbach[16] further developed BDC into a self-contained mathematical theory later on.

A complementary theory of Boolean integral calculus (German: Boolescher Integralkalkül) has been developed as well.[15][17]

BDC has also found uses in discrete event dynamic systems (DEDS)[18] in digital network communication protocols.

Meanwhile, BDC has seen extensions to multi-valued variables and functions[15][19][20] as well as to lattices of Boolean functions.[21][22]

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 ${\displaystyle dx_{i}}$ of a Boolean variable ${\displaystyle x_{i}}$ models the relation:

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

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

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

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

• Davio, Marc; Piret, Philippe M. (July 1969). "Les dérivées Booléennes et leur application au diagnostic" [Boolean derivatives and their application and diagnosis]. Philips Revue (in French). Brussels, Belgium: Philips Research Laboratory, Manufacture Belge de Lampes et de Materiel Electronique (MBLE Research Laboratory). 12 (3): 63–76. (14 pages)
• Rudeanu, Sergiu (September 1974). Boolean Functions and Equations. North-Holland Publishing Company/American Elsevier Publishing Company. ISBN 0-44410520-4. ISBN 0-72042082-2. (462 pages)
• Bochmann, Dieter [in German] (1977). "Boolean differential calculus (a survey)". Engineering Cybernetics. Institute of Electrical and Electronics Engineers (IEEE). 15 (5): 67–75. ISSN 0013-788X. (9 pages) Translation of: Bochmann, Dieter [in German] (1977). "[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] (in Russian) (5): 125–133. (9 pages)
• Kühnrich, Martin (1986) [1984-07-31 (submission)]. "Differentialoperatoren über Booleschen Algebren" [Differential operators on Boolean algebras]. Zeitschrift für mathematische Logik und Grundlagen der Mathematik (in German). Berlin, Germany (East). 32 (17–18): 271–288. doi:10.1002/malq.19860321703. #18. (18 pages)
• Dresig, Frank (1992). Gruppierung – Theorie und Anwendung in der Logiksynthese [Grouping – Theory and application in logic synthesis]. ISBN 3-18-144509-6. DNB-IDN 940164671. {{cite book}}: |work= ignored (help) (NB. Also: Chemnitz, Technische Universität, Dissertation.) (147 pages)
• Scheuring, Rainer; Wehlan, Herbert "Hans" (1993). "Control of Discrete Event Systems by Means of the Boolean Differential Calculus". In Balemi, Silvano; Kozák, Petr; Smedinga, Rein (eds.). Discrete Event Systems: Modeling and Control. Progress in Systems and Control Theory (PSCT). Vol. 13. Basel, Switzerland: Birkhäuser Verlag. pp. 79–93. doi:10.1007/978-3-0348-9120-2_7. (15 pages)
• Posthoff, Christian; Steinbach, Bernd [in German] (2004-02-04). Logic Functions and Equations – Binary Models for Computer Science (1st ed.). Dordrecht, Netherlands: Springer Science + Business Media B.V. doi:10.1007/978-1-4020-2938-7. ISBN 1-4020-2937-3. OCLC 254106952. ISBN 978-1-4020-2937-0. (392 pages)
• Steinbach, Bernd [in German]; Posthoff, Christian (2009-02-12). Logic Functions and Equations – Examples and Exercises (1st ed.). Dordrecht, Netherlands: Springer Science + Business Media B.V. doi:10.1007/978-1-4020-9595-5. ISBN 978-1-4020-9594-8. LCCN 2008941076. (xxii+232 pages) (NB. Per DNB-IDN 1010457748 this hardcover edition has been rereleased as softcover edition in 2010.)
• Steinbach, Bernd [in German]; Posthoff, Christian (2010-06-01). "Boolean Differential Calculus – Theory and Applications". Journal of Computational and Theoretical Nanoscience. American Scientific Publishers. 7 (6): 933–981. doi:10.1166/jctn.2010.1441. ISSN 1546-1955. (49 pages)
• Steinbach, Bernd [in German]; Posthoff, Christian (2010-01-15) [2009]. "Chapter 3: Boolean Differential Calculus". In Sasao, Tsutomu; Butler, Jon T. (eds.). Progress in Applications of Boolean Functions. Synthesis Lectures on Digital Circuits and Systems (1st ed.). San Rafael, CA, USA: Morgan & Claypool Publishers. pp. 55–78, 121–126. doi:10.2200/S00243ED1V01Y200912DCS026. ISBN 978-1-60845-181-4. Lecture #26. (24 of 153 pages)

• Wehlan, Herbert "Hans" (2010-12-06). Hazewinkel, Michiel (ed.). Boolean differential calculus. ISBN 978-1-4020-0609-8. Archived from the original on 2017-10-16. Retrieved 2017-10-16. {{cite book}}: |work= ignored (help)
• Institut für Informatik (IfI) (2017). "XBOOLE". TU Bergakademie Freiberg. Archived from the original on 2017-10-31. Retrieved 2017-10-31. with "XBOOLE Monitor". 2008-07-23. Archived from the original on 2017-10-31. Retrieved 2017-10-31.