[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/boolean-differential-calculus-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki40\/boolean-differential-calculus-wikipedia\/","headline":"Boolean differential calculus – Wikipedia","name":"Boolean differential calculus – Wikipedia","description":"From Wikipedia, the free encyclopedia Subject field of Boolean algebra discussing changes of Boolean variables and functions Boolean differential calculus","datePublished":"2015-01-02","dateModified":"2015-01-02","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki40\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6fcf9025bc3f34040616197a3819d50741aeb031","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6fcf9025bc3f34040616197a3819d50741aeb031","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki40\/boolean-differential-calculus-wikipedia\/","wordCount":8952,"articleBody":"From Wikipedia, the free encyclopediaSubject field of Boolean algebra discussing changes of Boolean variables and functionsBoolean differential calculus (BDC) (German: Boolescher Differentialkalk\u00fcl (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 asto be discussed in a united and closed form, with their individual advantages combined.History and applications[edit]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, \u0410. \u0414. \u0422\u0430\u043b\u0430\u043d\u0446\u0435\u0432)[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 Andr\u00e9 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\u00a0[de],[15]Christian Posthoff[15] and Bernd Steinbach\u00a0[de][16] further developed BDC into a self-contained mathematical theory later on.A complementary theory of Boolean integral calculus (German: Boolescher Integralkalk\u00fcl) 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[edit]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 dxi{displaystyle dx_{i}} of a Boolean variable xi{displaystyle x_{i}} models the relation:dxi={0,no change of\u00a0xi1,change of\u00a0xi{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 dxi{displaystyle dx_{i}} are binary. They can be used just like common binary variables.See also[edit]References[edit]^ H. Wehlan, Boolean Algebra in Encyclopedia of Mathematics^ Scheuring, Rainer; Wehlan, Herbert “Hans” (1991-12-01) [July 1991].  Bretthauer, Georg (ed.). “Der Boolesche Differentialkalk\u00fcl \u2013 eine Methode zur Analyse und Synthese von Petri-Netzen” [The Boolean differential calculus \u2013 A method for analysis and synthesis of Petri nets]. at \u2013 Automatisierungstechnik \u2013 Methoden und Anwendungen der Steuerungs-, Regelungs- und Informationstechnik (in German). Stuttgart, Germany: R. Oldenbourg Verlag\u00a0[de]. 39 (7): 226\u2013233. doi:10.1524\/auto.1991.39.112.226. ISSN\u00a00178-2312. Archived from the original on 2017-10-16. Retrieved 2017-10-16. (8 pages)^ 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\u201349. (12 pages)^ 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\u201312. (7 pages)^ 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\u201388. AD 156-161. (2 pages)^ 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\u2013498. doi:10.1137\/0107041. ISSN\u00a00368-4245. (12 pages)^ \u0422\u0430\u043b\u0430\u043d\u0446\u0435\u0432 [Talantsev], \u0410. \u0414. [A. D.] (1959) [1958-11-01 (submission)]. “Ob analize i sinteze nekotorykh elektri\u010deskikh skhem pri pomo\u015b\u0107i special’nykh logi\u010deskikh operatorov” \u0431 \u0430\u043d\u0430\u043b\u0438\u0437\u0435 \u0438 \u0441\u0438\u043d\u0442\u0435\u0437\u0435 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u044d\u043b\u0435\u043a\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0441\u0445\u0435\u043c \u043f\u0440\u0438 \u043f\u043e\u043c\u043e\u0449\u0438 \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u044c\u043d\u044b\u0445 \u043b\u043e\u0433\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u043e\u043f\u0435\u0440\u0430\u0442\u043e\u0440\u043e\u0432 [Analysis and synthesis of certain electric circuits by means of special logical operators]. \u0410\u0432\u0442\u043e\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0442\u0435\u043b\u0435\u043c\u0435\u0445\u0430\u043d\u0438\u043a\u0430 (Avtomatika i telemekhanika) [Automation and Remote Control] (in Russian). Moscow, Russia. 20 (7): 898\u2013907. Mi\u00a0at12783. Archived from the original on 2017-10-17. Retrieved 2017-10-17. [\u2026] \u041e\u0441\u043d\u043e\u0432\u043d\u043e\u0435 \u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u0441\u0442\u0430\u0442\u044c\u0438 \u0434\u043e\u043b\u043e\u0436\u0435\u043d\u043e \u043d\u0430 \u0441\u0435\u043c\u0438\u043d\u0430\u0440\u0435 \u043f\u043e \u0442\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u0438\u043c \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u044f\u043c \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u043b\u043e\u0433\u0438\u043a\u0438 \u0432 \u041c\u0413\u0423 2\/\u0425 1958 \u0433. \u0438 16\/1 1959 [\u2026] \u0410\u0432\u0442\u043e\u0440 \u0441\u0447\u0438\u0442\u0430\u0435\u0442 \u0441\u0432\u043e\u0438\u043c \u0434\u043e\u043b\u0433\u043e\u043c \u0432\u044b\u0440\u0430\u0437\u0438\u0442\u044c \u043f\u0440\u0438\u0437\u043d\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u044c \u0412. \u0410. \u0422\u0440\u0430\u043f\u0435\u0437\u043d\u0438\u043a\u043e\u0432\u0443\u00a0[ru], \u0412. \u0418. \u0428\u0435\u0441\u0442\u0430\u043a\u043e\u0432\u0443 \u0438 \u041c. \u041b. \u0426\u0435\u0442\u043b\u0438\u043d\u0443 \u0437\u0430 \u0438\u043d\u0442\u0435\u0440\u0435\u0441 \u043a \u0440\u0430\u0431\u043e\u0442\u0435 \u0438 \u0446\u0435\u043d\u043d\u044b\u0435 \u0437\u0430\u043c\u0435\u0447\u0430\u043d\u0438\u044f \u043f\u0440\u0438 \u043e\u0431\u0441\u0443\u0436\u0434\u0435\u043d\u0438\u0438 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u043e\u0432. [\u2026] [[\u2026] 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 [\u2026] The author considers it his duty to express gratitude to V. A. Trapeznikov\u00a0[ru], V. I. Shestakov and M. L. Tsetlin for interest in the work and valuable comments in discussing the results.[\u2026]] (10 pages)^ 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\u2013683. doi:10.1109\/TC.1968.227417. ISSN\u00a00018-9340. (8 pages)^ a b c Sellers Jr., Frederick F.; Hsiao, Mu-Yue; Bearnson, Leroy W. (November 1968). Error Detecting Logic for Digital Computers (1st\u00a0ed.). New York, USA: McGraw-Hill Book Company. pp.\u00a017\u201337. LCCN\u00a068-16491. OCLC\u00a0439460. (21 of xviii+295 pages)^ Thayse, Andr\u00e9 (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\u2013336. R737. Archived from the original (PDF) on 2017-03-08. Retrieved 2017-10-17. [\u2026] The author is indebted to Dr M. Davio for his continuing interest and comments on this work. Thanks are also due to Mr C. Foss\u00e9prez who initially suggested the basic problem considered here. [\u2026] (76 pages)^ a b Thayse, Andr\u00e9 (February 1971). “Boolean Differential Calculus” (PDF). Philips Research Reports. Brussels, Belgium: Philips Research Laboratory. 26 (2): 229\u2013246. R764. Archived from the original (PDF) on 2017-03-08. Retrieved 2017-10-16. [\u2026] 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. [\u2026] Acknowledgement: The author is especially grateful to Dr M. Davio for his encouragement and support and for several ideas in the presentation. [\u2026] (18 pages)^ a b Thayse, Andr\u00e9; Davio, Marc (1973-04-01). “Boolean Differential Calculus and its Application to Switching Theory”. IEEE Transactions on Computers. C-22 (4): 409\u2013420. doi:10.1109\/T-C.1973.223729. (12 pages)^ a b c Davio, Marc; Deschamps, Jean-Pierre; Thayse, Andr\u00e9 (1978-08-01). Discrete and Switching Functions (1st\u00a0ed.). New York, USA: Georgi Publishing Company \/ McGraw-Hill International Book Company. ISBN\u00a00-07-015509-7. LCCN\u00a077-030718. (xx+729 pages)^ Thayse, Andr\u00e9 (1981).  Goos, Gerhard [in German]; Hartmanis, Juris (eds.). Boolean Calculus of Differences. Lecture Notes in Computer Science. Vol.\u00a0101 (1st\u00a0ed.). Berlin: Springer-Verlag. ISBN\u00a03-540-10286-8. (144 pages)^ a b c d Bochmann, Dieter [in German]; Posthoff, Christian (1981). Bin\u00e4re dynamische Systeme [Binary dynamic systems] (in German) (1st\u00a0ed.). Akademie-Verlag, Berlin \/ R. Oldenbourg Verlag\u00a0[de], M\u00fcnchen. ISBN\u00a03-486-25071-X. DNB-IDN\u00a0810757168, 810200317. License number\u00a0[de]: 202.100\/408\/81. Order code: 7623619\u00a0(6391). (397 pages) (NB. Per DNB-IDN\u00a0368893146 a Russian translation of this work was released in 1986.)^ Bochmann, Dieter [in German]; Steinbach, Bernd [in German] (1991). Logikentwurf mit XBOOLE \u2013 Algorithmen und Programme [Logic design with XBOOLE \u2013 Algorithms and programs] (in German) (1st\u00a0ed.). Berlin, Germany: Verlag Technik. ISBN\u00a03-341-01006-8. DNB-IDN\u00a0911196102. (303 pages + 5.25-inch floppy disk)^ Steinbach, Bernd [in German]; Posthoff, Christian (2013-07-01).  Thornton, Mitchell A. (ed.). Boolean Differential Equations. Synthesis Lectures on Digital Circuits and Systems (1st\u00a0ed.). San Rafael, CA, USA: Morgan & Claypool Publishers. doi:10.2200\/S00511ED1V01Y201305DCS042. ISBN\u00a0978-1-62705-241-2. Lecture #42. (158 pages)^ 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. Z\u00fcrich, Switzerland: International Federation of Automatic Control (IFAC) \/ Pergamon Press. 2: 723\u2013728. doi:10.1016\/S1474-6670(17)54214-7. (6 pages)^ \u00c2nu\u0161kevi\u010d [Yanushkevich], Svitlana N. [Svetlana N.] (1998). Logic Differential Calculus in Multi-Valued Logic Design. Journal Prace Naukowe Politechniki Szczeci\u0144skiej (PhD thesis) (1st\u00a0ed.). Szczecin, Poland: Instytut Informatyki, Technical University of Szczecin. ISBN\u00a0978-8-387423-16-2. ISSN\u00a01506-3054. ISBN\u00a08-387423-16-5. (326 pages)^ Bochmann, Dieter [in German] (2008-09-01). Binary Systems – A BOOLEAN Book (1st\u00a0ed.). Dresden, Germany: TUDpress Verlag der Wissenschaften. ISBN\u00a0978-3-940046-87-1. DNB-IDN\u00a0989771636. (421 pages) Translation of: Bochmann, Dieter [in German] (February 2006). Bin\u00e4re Systeme – Ein BOOLEAN Buch [Binary systems – A Boolean book] (in German) (1st\u00a0ed.). Hagen, Germany: LiLoLe-Verlag GmbH (Life-Long-Learning) \/ BoD GmbH. ISBN\u00a03-934447-10-4. ISBN\u00a0978-3-934447-10-3. DNB-IDN\u00a0978899873. (452 pages)^ Steinbach, Bernd [in German]; Posthoff, Christian (2013). “Derivative Operations for Lattices of Boolean Functions” (PDF). Proceedings Reed-Muller Workshop 2013. Toyama, Japan: 110\u2013119. Archived (PDF) from the original on 2017-10-21. Retrieved 2017-10-21. (10 pages)^ Steinbach, Bernd [in German]; Posthoff, Christian (2017-06-07).  Thornton, Mitchell A. (ed.). Boolean Differential Calculus. Synthesis Lectures on Digital Circuits and Systems (1st\u00a0ed.). San Rafael, CA, USA: Morgan & Claypool Publishers. doi:10.2200\/S00766ED1V01Y201704DCS052. ISBN\u00a0978-1-62705-922-0. Lecture #52. (216 pages)Further reading[edit]Davio, Marc; Piret, Philippe M. (July 1969). “Les d\u00e9riv\u00e9es Bool\u00e9ennes 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\u201376. (14 pages)Rudeanu, Sergiu (September 1974). Boolean Functions and Equations. North-Holland Publishing Company\/American Elsevier Publishing Company. ISBN\u00a00-44410520-4. ISBN\u00a00-72042082-2. (462 pages)Bochmann, Dieter [in German] (1977). “Boolean differential calculus (a survey)”. Engineering Cybernetics. Institute of Electrical and Electronics Engineers (IEEE). 15 (5): 67\u201375. ISSN\u00a00013-788X. (9 pages) Translation of: Bochmann, Dieter [in German] (1977). “[Boolean differential calculus (survey)]”. \u0418\u0437\u0432\u0435\u0441\u0442\u0438\u044f \u0410\u043a\u0430\u0434\u0435\u043c\u0438\u0438 \u043d\u0430\u0443\u043a \u0421\u0421\u0421\u0420 \u2013 \u0422\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u043a\u0438\u0431\u0435\u0440\u043d\u0435\u0442\u0438\u043a\u0430 (Izvestii\ufe20a\ufe21 Akademii Nauk SSSR \u2013 Tekhnicheskai\ufe20a\ufe21 kibernetika) [Proceedings of the Academy of Sciences of the USSR \u2013 Engineering Cybernetics] (in Russian) (5): 125\u2013133. (9 pages)K\u00fchnrich, Martin (1986) [1984-07-31 (submission)]. “Differentialoperatoren \u00fcber Booleschen Algebren” [Differential operators on Boolean algebras]. Zeitschrift f\u00fcr mathematische Logik und Grundlagen der Mathematik (in German). Berlin, Germany (East). 32 (17\u201318): 271\u2013288. doi:10.1002\/malq.19860321703. #18. (18 pages)Dresig, Frank (1992). Gruppierung \u2013 Theorie und Anwendung in der Logiksynthese [Grouping \u2013 Theory and application in logic synthesis]. Fortschritt-Berichte VDI. 9 (in German). Vol.\u00a0145. D\u00fcsseldorf, Germany: VDI-Verlag. ISBN\u00a03-18-144509-6. DNB-IDN\u00a0940164671. (NB. Also: Chemnitz, Technische Universit\u00e4t, Dissertation.) (147 pages)Scheuring, Rainer; Wehlan, Herbert “Hans” (1993). “Control of Discrete Event Systems by Means of the Boolean Differential Calculus”.  In Balemi, Silvano; Koz\u00e1k, Petr; Smedinga, Rein (eds.). Discrete Event Systems: Modeling and Control. Progress in Systems and Control Theory (PSCT). Vol.\u00a013. Basel, Switzerland: Birkh\u00e4user Verlag. pp.\u00a079\u201393. doi:10.1007\/978-3-0348-9120-2_7. (15 pages)Posthoff, Christian; Steinbach, Bernd [in German] (2004-02-04). Logic Functions and Equations \u2013 Binary Models for Computer Science (1st\u00a0ed.). Dordrecht, Netherlands: Springer Science + Business Media B.V. doi:10.1007\/978-1-4020-2938-7. ISBN\u00a01-4020-2937-3. OCLC\u00a0254106952. ISBN\u00a0978-1-4020-2937-0. (392 pages)Steinbach, Bernd [in German]; Posthoff, Christian (2009-02-12). Logic Functions and Equations \u2013 Examples and Exercises (1st\u00a0ed.). Dordrecht, Netherlands: Springer Science + Business Media B.V. doi:10.1007\/978-1-4020-9595-5. ISBN\u00a0978-1-4020-9594-8. LCCN\u00a02008941076. (xxii+232 pages) [1] (NB. Per DNB-IDN\u00a01010457748 this hardcover edition has been rereleased as softcover edition in 2010.)Steinbach, Bernd [in German]; Posthoff, Christian (2010-06-01). “Boolean Differential Calculus \u2013 Theory and Applications”. Journal of Computational and Theoretical Nanoscience. American Scientific Publishers. 7 (6): 933\u2013981. doi:10.1166\/jctn.2010.1441. ISSN\u00a01546-1955. (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\u00a0ed.). San Rafael, CA, USA: Morgan & Claypool Publishers. pp.\u00a055\u201378, 121\u2013126. doi:10.2200\/S00243ED1V01Y200912DCS026. ISBN\u00a0978-1-60845-181-4. Lecture #26. (24 of 153 pages)External links[edit]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/boolean-differential-calculus-wikipedia\/#breadcrumbitem","name":"Boolean differential calculus – Wikipedia"}}]}]