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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
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783:
444:
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793:
647:
615:
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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".
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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.).
986:
925:
351:
796: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
777: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
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Davio, Marc; Piret, Philippe M. (July 1969). "Les dérivées Booléennes et leur application au diagnostic" [Boolean derivatives and their application and diagnosis].
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Since then, significant advances were accomplished in both, the theory and in the application of the BDC in switching circuit design and
1222:
1927:. Synthesis Lectures on Digital Circuits and Systems (1st ed.). San Rafael, CA, USA: Morgan & Claypool Publishers. pp.
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Kühnrich, Martin (1986). "Differentialoperatoren über
Booleschen Algebren" [Differential operators on Boolean algebras].
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1626:
1614:
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1045:
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109:, the roots for the development of what later would evolve into the Boolean differential calculus were initiated by works of
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1541:. Synthesis Lectures on Digital Circuits and Systems (1st ed.). San Rafael, CA, USA: Morgan & Claypool Publishers.
1308:. Synthesis Lectures on Digital Circuits and Systems (1st ed.). San Rafael, CA, USA: Morgan & Claypool Publishers.
1919:; Posthoff, Christian (2010-01-15) . "Chapter 3: Boolean Differential Calculus". In Sasao, Tsutomu; Butler, Jon T. (eds.).
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Sellers Jr., Frederick F.; Hsiao, Mu-Yue; Bearnson, Leroy W. (July 1968). "Analyzing Errors with the
Boolean Difference".
244:
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525:
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2045:
1984:
1677:Известия Академии наук СССР – Техническая кибернетика (Izvestii︠a︡ Akademii Nauk SSSR – Tekhnicheskai︠a︡ kibernetika)
1594:
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439:{\displaystyle dx_{i}={\begin{cases}0,&{\text{no change of }}x_{i}\\1,&{\text{change of }}x_{i}\end{cases}}}
812:
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278:
82:
1043:
Thayse, André; Davio, Marc (1973-04-01). "Boolean
Differential Calculus and its Application to Switching Theory".
1883:
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61:
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At – Automatisierungstechnik – Methoden und
Anwendungen der Steuerungs-, Regelungs- und Informationstechnik
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1386:(PhD thesis) (1st ed.). Szczecin, Poland: Instytut Informatyki, Technical University of Szczecin.
737:"Ob analize i sinteze nekotorykh električeskikh skhem pri pomośći special'nykh logičeskikh operatorov"
154:
1456:] (in German) (1st ed.). Hagen, Germany: LiLoLe-Verlag GmbH (Life-Long-Learning) / BoD GmbH.
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742:[Analysis and synthesis of certain electric circuits by means of special logical operators].
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544:[The Boolean differential calculus – A method for analysis and synthesis of Petri nets].
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739:б анализе и синтезе некоторых электрических схем при помощи специальных логических операторов
645:(1954). "Application of Boolean algebra to switching circuit design and to error detection".
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542:"Der Boolesche Differentialkalkül – eine Methode zur Analyse und Synthese von Petri-Netzen"
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There are no constraints in regard to the nature, the causes and consequences of a change.
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57:, notably studying the changes in functions and variables with respect to another/others.
39:
1715:]. Fortschritt-Berichte VDI, Ser. 9 (in German). Vol. 145. Düsseldorf, Germany:
89:
to be discussed in a united and closed form, with their individual advantages combined.
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on 1958-10-02 and 1959-01-16 The author considers it his duty to express gratitude to
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16:
Subject field of
Boolean algebra discussing changes of Boolean variables and functions
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681:(48). Cambridge, MA, USA: MIT Research Laboratory of Electronics: 87–88. AD 156-161.
584:
1747:. Progress in Systems and Control Theory (PSCT). Vol. 13. Basel, Switzerland:
693:
Akers Jr., Sheldon
Buckingham (December 1959). "On a Theory of Boolean Functions".
660:
540:
Scheuring, Rainer; Wehlan, Herbert "Hans" (1991-12-01) . Bretthauer, Georg (ed.).
1227:
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for his continuing interest and comments on this work. Thanks are also due to Mr
568:
1756:
483:
1841:
1791:
1716:
888:
Sellers Jr., Frederick F.; Hsiao, Mu-Yue; Bearnson, Leroy W. (November 1968).
613:(1954). "A Class of Multiple-Error-Correcting Codes and the Decoding Scheme".
1907:
1809:
1698:
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for his encouragement and support and for several ideas in the presentation.
860:
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1140:. Lecture Notes in Computer Science. Vol. 101 (1st ed.). Berlin:
852:
53:
Boolean differential calculus concepts are analogous to those of classical
909:
819:
for interest in the work and valuable comments in discussing the results.]
677:(1958-01-15). "Solvability criterion for simultaneous logical equations".
225:
further developed BDC into a self-contained mathematical theory later on.
1875:
this hardcover edition has been rereleased as softcover edition in 2010.)
1739:(NB. Also: Chemnitz, Technische Universität, Dissertation.) (147 pages)
1423:(1st ed.). Dresden, Germany: TUDpress Verlag der Wissenschaften.
926:"Transient analysis of logical networks applied to hazard detection"
761:
712:
1859:
1781:
Logic
Functions and Equations – Binary Models for Computer Science
648:
Transactions of the IRE Professional Group on Electronic Computers
1687:
Zeitschrift für mathematische Logik und Grundlagen der Mathematik
1537:; Posthoff, Christian (2017-06-07). Thornton, Mitchell A. (ed.).
1304:; Posthoff, Christian (2013-07-01). Thornton, Mitchell A. (ed.).
1113:
1086:
Davio, Marc; Deschamps, Jean-Pierre; Thayse, André (1978-08-01).
901:
616:
Transactions of the IRE Professional Group on Information Theory
883:
881:
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play a significant role in BDC. They allow the application of
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Journal of the Society for Industrial and Applied Mathematics
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The Boolean differential calculus allows various aspects of
1980:
Boolean differential calculus - Encyclopedia of Mathematics
1094:(1st ed.). New York, USA: Georgi Publishing Company /
969:
who initially suggested the basic problem considered here.
432:
1884:"Boolean Differential Calculus – Theory and Applications"
1499:"Derivative Operations for Lattices of Boolean Functions"
1348:
First IFAC Symposium on Design Methods of Control Systems
1242:
a Russian translation of this work was released in 1986.)
1168:
1166:
1164:
1162:
1709:
Gruppierung – Theorie und Anwendung in der Logiksynthese
1384:
Logic Differential Calculus in Multi-Valued Logic Design
1600:
Manufacture Belge de Lampes et de Materiel Electronique
1831:
Logic Functions and Equations – Examples and Exercises
458:
354:
324:
294:
1888:
Journal of Computational and Theoretical Nanoscience
1713:
Grouping – Theory and application in logic synthesis
1648:(1977). "Boolean differential calculus (a survey)".
1038:
1036:
1267:] (in German) (1st ed.). Berlin, Germany:
1261:
Logikentwurf mit XBOOLE – Algorithmen und Programme
1265:Logic design with XBOOLE – Algorithms and programs
1232:: 202.100/408/81. Order code: 7623619 (6391).
1087:
474:
438:
337:
310:
1658:Institute of Electrical and Electronics Engineers
97:Originally inspired by the design and testing of
187:in the 1970s formed the basics of BDC on which
1894:(6). American Scientific Publishers: 933–981.
980:
978:
705:Society for Industrial and Applied Mathematics
482:are binary. They can be used just like common
1923:Progress in Applications of Boolean Functions
1356:International Federation of Automatic Control
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1745:Discrete Event Systems: Modeling and Control
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890:Error Detecting Logic for Digital Computers
1693:(17–18). Berlin, Germany (East): 271–288.
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1833:(1st ed.). Dordrecht, Netherlands:
1783:(1st ed.). Dordrecht, Netherlands:
517:
2018:Institut für Informatik (IfI) (2017).
1835:Springer Science + Business Media B.V.
1785:Springer Science + Business Media B.V.
1096:McGraw-Hill International Book Company
261:variables and functions as well as to
257:Meanwhile, BDC has seen extensions to
1970:Wehlan, Herbert "Hans" (2010-12-06).
1506:Proceedings Reed-Muller Workshop 2013
285:to be extended to logical functions.
7:
1882:; Posthoff, Christian (2010-06-01).
1829:; Posthoff, Christian (2009-02-12).
1619:American Elsevier Publishing Company
1291:(303 pages + 5.25-inch floppy disk)
1609:Rudeanu, Sergiu (September 1974).
1603:(MBLE Research Laboratory): 63–76.
1184:] (in German) (1st ed.).
14:
1937:10.2200/S00243ED1V01Y200912DCS026
1547:10.2200/S00766ED1V01Y201704DCS052
1450:Binäre Systeme - Ein BOOLEAN Buch
1382:Ânuškevič , Svitlana N. (1998).
1314:10.2200/S00511ED1V01Y201305DCS042
1615:North-Holland Publishing Company
1090:Discrete and Switching Functions
145:) between 1954 and 1959, and of
2030:from the original on 2017-10-31
2006:from the original on 2017-10-16
1990:Springer Science+Business Media
1972:"Boolean differential calculus"
1611:Boolean Functions and Equations
1515:from the original on 2017-10-21
1454:Binary systems - A Boolean book
1421:Binary Systems - A BOOLEAN Book
1138:Boolean Calculus of Differences
1006:: 229–246. R764. Archived from
987:"Boolean Differential Calculus"
985:Thayse, André (February 1971).
945:: 261–336. R737. Archived from
924:Thayse, André (October 1970) .
892:(1st ed.). New York, USA:
768:from the original on 2017-10-17
591:from the original on 2017-10-16
1497:; Posthoff, Christian (2013).
1306:Boolean Differential Equations
1176:; Posthoff, Christian (1981).
1046:IEEE Transactions on Computers
840:IEEE Transactions on Computers
756:(7). Moscow, Russia: 898–907.
245:discrete event dynamic systems
240:) has been developed as well.
1:
1539:Boolean Differential Calculus
1368:10.1016/S1474-6670(17)54214-7
961:The author is indebted to Dr
661:10.1109/IREPGELC.1954.6499441
30:Boolescher Differentialkalkül
20:Boolean differential calculus
2044:. 2008-07-23. Archived from
1443:(421 pages) Translation of:
569:10.1524/auto.1991.39.112.226
1985:Encyclopedia of Mathematics
1757:10.1007/978-3-0348-9120-2_7
1595:Philips Research Laboratory
1004:Philips Research Laboratory
943:Philips Research Laboratory
528:Encyclopedia of Mathematics
243:BDC has also found uses in
2106:
1679:(in Russian) (5): 125–133.
1670:(9 pages) Translation of:
1508:. Toyama, Japan: 110–119.
1354:(8). Zürich, Switzerland:
1023:especially grateful to Dr
749:Avtomatika i Telemekhanika
735:Таланцев , А. Д. (1959).
228:A complementary theory of
83:supervisory control theory
1842:10.1007/978-1-4020-9595-5
1792:10.1007/978-1-4020-2938-7
1178:Binäre dynamische Systeme
744:Автоматика и телемеханика
679:Quarterly Progress Report
552:(7). Stuttgart, Germany:
501:Boole's expansion theorem
236:Boolescher Integralkalkül
230:Boolean integral calculus
141:
38:)) is a subject field of
2024:TU Bergakademie Freiberg
1699:10.1002/malq.19860321703
1593:(3). Brussels, Belgium:
1002:(2). Brussels, Belgium:
995:Philips Research Reports
941:(5). Brussels, Belgium:
934:Philips Research Reports
894:McGraw-Hill Book Company
738:
629:10.1109/TIT.1954.1057465
506:Ramadge–Wonham framework
281:as known from classical
147:Frederick F. Sellers Jr.
93:History and applications
62:dynamical systems theory
1650:Engineering Cybernetics
1059:10.1109/T-C.1973.223729
914:(21 of xviii+295 pages)
798:Moscow State University
252:communication protocols
101:and the utilization of
2075:Automata (computation)
1900:10.1166/jctn.2010.1441
1707:Dresig, Frank (1992).
1599:
1182:Binary dynamic systems
1128:Thayse, André (1981).
853:10.1109/TC.1968.227417
476:
475:{\displaystyle dx_{i}}
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318:of a Boolean variable
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265:of Boolean functions.
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103:error-correcting codes
42:discussing changes of
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1234:(397 pages) (NB. Per
675:Huffman, David Albert
477:
441:
345:models the relation:
340:
338:{\displaystyle x_{i}}
313:
183:Jean-Pierre Deschamps
55:differential calculus
1190:R. Oldenbourg Verlag
752:] (in Russian).
643:Muller, David Eugene
554:R. Oldenbourg Verlag
456:
352:
322:
292:
123:Sheldon B. Akers Jr.
1976:Hazewinkel, Michiel
526:Boolean Algebra in
2080:Mathematical logic
1751:. pp. 79–93.
896:. pp. 17–37.
779:В. А. Трапезникову
623:(4). IEEE: 38–49.
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452:The differentials
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389:no change of
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288:The differentials
206:Christian Posthoff
99:switching circuits
1999:978-1-4020-0609-8
1959:(24 of 153 pages)
1946:978-1-60845-181-4
1864:(xxii+232 pages)
1851:978-1-4020-9594-8
1818:978-1-4020-2937-0
1766:978-3-0348-9916-1
1749:Birkhäuser Verlag
1556:978-1-62705-922-0
1472:978-3-934447-10-3
1448:(February 2006).
1430:978-3-940046-87-1
1393:978-8-387423-16-2
1323:978-1-62705-241-2
802:V. A. Trapeznikov
611:Reed, Irving Stoy
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44:Boolean variables
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73:finite automata
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40:Boolean algebra
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1957:. Lecture #26.
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1779:(2004-02-04).
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1567:. Lecture #52.
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1419:(2008-09-01).
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1360:Pergamon Press
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1334:. Lecture #42.
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2052:2017-10-31
2034:2017-10-31
2010:2017-10-16
1912:(49 pages)
1873:1010457748
1860:2008941076
1771:(15 pages)
1717:VDI-Verlag
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1519:2017-10-21
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512:References
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177:Marc Davio
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263:lattices
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