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.

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^[2]
• supervisory control theory (SCT)

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

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 [de],^[15] Christian Posthoff^[15] and Bernd Steinbach [de]^[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

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• 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)
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• Kühnrich, Martin (1986). "Differentialoperatoren über Booleschen Algebren" [Differential operators on Boolean algebras]. Zeitschrift für mathematische Logik und Grundlagen der Mathematik (in German). 32 (17–18). Berlin, Germany (East): 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]. Fortschritt-Berichte VDI, Ser. 9 (in German). Vol. 145. Düsseldorf, Germany: VDI-Verlag. ISBN 3-18-144509-6. DNB-IDN 940164671. (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. ISBN 978-3-0348-9916-1. (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) [1] (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. 7 (6). American Scientific Publishers: 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. S2CID 37053010. Lecture #26. (24 of 153 pages)

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