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

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, [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]

## 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 ${\displaystyle dx_{i}}$ 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.

## References

1. ^
2. ^ Scheuring, Rainer; Wehlan, Herbert "Hans" (1991-12-01) [July 1991]. Bretthauer, Georg (ed.). "Der Boolesche Differentialkalkül – eine Methode zur Analyse und Synthese von Petri-Netzen" [The Boolean differential calculus – A method for analysis and synthesis of Petri nets]. at – Automatisierungstechnik – Methoden und Anwendungen der Steuerungs-, Regelungs- und Informationstechnik (in German). Stuttgart, Germany: R. Oldenbourg Verlag [ de]. 39 (7): 226–233. doi: 10.1524/auto.1991.39.112.226. ISSN  0178-2312. Archived from the original on 2017-10-16. Retrieved 2017-10-16. (8 pages)
3. ^ Reed, Irving Stoy (1954). "A Class of Multiple-Error-Correcting Codes and the Decoding Scheme". Transactions of the IRE Professional Group on Information Theory (PGIT). Institute of Radio Engineers (IRE). PGIT-4 (4): 38–49. (12 pages)
4. ^ Muller, David Eugene (1954). "Application of Boolean algebra to switching circuit design and to error detection". Transactions of the IRE Professional Group on Electronic Computers (PGEC). PGEC-3: 6–12. (7 pages)
5. ^ Huffman, David Albert (1958-01-15). "Solvability criterion for simultaneous logical equations". Quarterly Progress Report. Cambridge, MA, USA: MIT Research Laboratory of Electronics (48): 87–88. AD 156-161. (2 pages)
6. ^ Akers Jr., Sheldon Buckingham (December 1959) [1957-09-27 (submission), 1959-05-28 (revision)]. "On a Theory of Boolean Functions". Journal of the Society for Industrial and Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM). 7 (4): 487–498. doi: 10.1137/0107041. ISSN  0368-4245. (12 pages)
7. ^ Таланцев [Talantsev], А. Д. [A. D.] (1959) [1958-11-01 (submission)]. "Ob analize i sinteze nekotorykh električeskikh skhem pri pomośći special'nykh logičeskikh operatorov" б анализе и синтезе некоторых электрических схем при помощи специальных логических операторов [Analysis and synthesis of certain electric circuits by means of special logical operators]. Автоматика и телемеханика ( Avtomatika i telemekhanika) [ Automation and Remote Control (in Russian). Moscow, Russia. 20 (7): 898–907. Mi  at12783. Archived from the original on 2017-10-17. Retrieved 2017-10-17. […] Основное содержание статьи доложено на семинаре по техническим приложениям математической логики в МГУ 2/Х 1958 г. и 16/1 1959 […] Автор считает своим долгом выразить признательность В. А. Трапезникову [ ru], В. И. Шестакову и М. Л. Цетлину за интерес к работе и ценные замечания при обсуждении результатов. […] [[…] The main content of the article was presented at the technical application workshop on mathematical logic at the Moscow State University on 1958-10-02 and 1959-01-16 […] The author considers it his duty to express gratitude to V. A. Trapeznikov [ ru], V. I. Shestakov and M. L. Tsetlin for interest in the work and valuable comments in discussing the results.[…]] (10 pages)
8. ^ a b c Sellers Jr., Frederick F.; Hsiao, Mu-Yue; Bearnson, Leroy W. (July 1968). "Analyzing Errors with the Boolean Difference". IEEE Transactions on Computers. C-17 (7): 676–683. doi: 10.1109/TC.1968.227417. ISSN  0018-9340. (8 pages)
9. ^ a b c Sellers Jr., Frederick F.; Hsiao, Mu-Yue; Bearnson, Leroy W. (November 1968). Error Detecting Logic for Digital Computers (1st ed.). New York, USA: McGraw-Hill Book Company. pp. 17–37. LCCN  68-16491. OCLC  439460. (21 of xviii+295 pages)
10. ^ Thayse, André (October 1970) [May 1970]. "Transient analysis of logical networks applied to hazard detection" (PDF). Philips Research Reports. Brussels, Belgium: Philips Research Laboratory. 25 (5): 261–336. R737. Archived from the original (PDF) on 2017-03-08. Retrieved 2017-10-17. […] The author is indebted to Dr M. Davio for his continuing interest and comments on this work. Thanks are also due to Mr C. Fosséprez who initially suggested the basic problem considered here. […] (76 pages)
11. ^ a b Thayse, André (February 1971). "Boolean Differential Calculus" (PDF). Philips Research Reports. Brussels, Belgium: Philips Research Laboratory. 26 (2): 229–246. R764. Archived from the original (PDF) on 2017-03-08. Retrieved 2017-10-16. […] Abstract: After a brief outline of classical concepts relative to Boolean differential calculus, a theoretical study of various differential operators is undertaken. Application of these concepts to several important problems arising in switching practice is mentioned. […] Acknowledgement: The author is especially grateful to Dr M. Davio for his encouragement and support and for several ideas in the presentation. […] (18 pages)
12. ^ a b Thayse, André; Davio, Marc (1973-04-01). "Boolean Differential Calculus and its Application to Switching Theory". IEEE Transactions on Computers. C-22 (4): 409–420. doi: 10.1109/T-C.1973.223729. (12 pages)
13. ^ a b c Davio, Marc; Deschamps, Jean-Pierre; Thayse, André (1978-08-01). (1st ed.). New York, USA: Georgi Publishing Company / McGraw-Hill International Book Company. ISBN  0-07-015509-7. LCCN  77-030718. (xx+729 pages)
14. ^ Thayse, André (1981). Goos, Gerhard [in German]; Hartmanis, Juris (eds.). Boolean Calculus of Differences. Lecture Notes in Computer Science. Vol. 101 (1st ed.). Berlin: Springer-Verlag. ISBN  3-540-10286-8. (144 pages)
15. ^ a b c d Bochmann, Dieter [in German]; Posthoff, Christian (1981). Binäre dynamische Systeme [Binary dynamic systems] (in German) (1st ed.). Akademie-Verlag, Berlin / R. Oldenbourg Verlag [ de], München. ISBN  3-486-25071-X. DNB-IDN  810757168, 810200317. License number [ de]: 202.100/408/81. Order code: 7623619 (6391). (397 pages) (NB. Per DNB-IDN  368893146 a Russian translation of this work was released in 1986.)
16. ^ Bochmann, Dieter [in German]; Steinbach, Bernd [in German] (1991). Logikentwurf mit XBOOLE – Algorithmen und Programme [Logic design with XBOOLE – Algorithms and programs] (in German) (1st ed.). Berlin, Germany: Verlag Technik. ISBN  3-341-01006-8. DNB-IDN  911196102. (303 pages + 5.25-inch floppy disk)
17. ^ Steinbach, Bernd [in German]; Posthoff, Christian (2013-07-01). Thornton, Mitchell A. (ed.). Boolean Differential Equations. Synthesis Lectures on Digital Circuits and Systems (1st ed.). San Rafael, CA, USA: Morgan & Claypool Publishers. doi: 10.2200/S00511ED1V01Y201305DCS042. ISBN  978-1-62705-241-2. Lecture #42. (158 pages)
18. ^ Scheuring, Rainer; Wehlan, Herbert "Hans" (1991-09-01). Franke, Dieter; Kraus, Franta (eds.). "On the Design of Discrete Event Dynamic Systems by Means of the Boolean Differential Calculus". First IFAC Symposium on Design Methods of Control Systems. Zürich, Switzerland: International Federation of Automatic Control (IFAC) / Pergamon Press. 2: 723–728. doi: 10.1016/S1474-6670(17)54214-7. (6 pages)
19. ^ Ânuškevič [Yanushkevich], Svitlana N. [Svetlana N.] (1998). Logic Differential Calculus in Multi-Valued Logic Design. Journal Prace Naukowe Politechniki Szczecińskiej (PhD thesis) (1st ed.). Szczecin, Poland: Instytut Informatyki, Technical University of Szczecin. ISBN  978-8-387423-16-2. ISSN  1506-3054. ISBN  8-387423-16-5. (326 pages)
20. ^ Bochmann, Dieter [in German] (2008-09-01). Binary Systems - A BOOLEAN Book (1st ed.). Dresden, Germany: TUDpress Verlag der Wissenschaften. ISBN  978-3-940046-87-1. DNB-IDN  989771636. (421 pages) Translation of: Bochmann, Dieter [in German] (February 2006). Binäre Systeme - Ein BOOLEAN Buch [Binary systems - A Boolean book] (in German) (1st ed.). Hagen, Germany: LiLoLe-Verlag GmbH (Life-Long-Learning) / BoD GmbH. ISBN  3-934447-10-4. ISBN  978-3-934447-10-3. DNB-IDN  978899873. (452 pages)
21. ^ Steinbach, Bernd [in German]; Posthoff, Christian (2013). "Derivative Operations for Lattices of Boolean Functions" (PDF). Proceedings Reed-Muller Workshop 2013. Toyama, Japan: 110–119. Archived (PDF) from the original on 2017-10-21. Retrieved 2017-10-21. (10 pages)
22. ^ Steinbach, Bernd [in German]; Posthoff, Christian (2017-06-07). Thornton, Mitchell A. (ed.). Boolean Differential Calculus. Synthesis Lectures on Digital Circuits and Systems (1st ed.). San Rafael, CA, USA: Morgan & Claypool Publishers. doi: 10.2200/S00766ED1V01Y201704DCS052. ISBN  978-1-62705-922-0. Lecture #52. (216 pages)