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Boolean differential calculus

December 18, 2023

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 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, 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 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   of a Boolean variable   models the relation:

 

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

The differentials   are binary. They can be used just like common binary variables.

See also edit

References edit

  1. ^ H. Wehlan, Boolean Algebra in Encyclopedia of Mathematics
  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. S2CID 56766796. 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. IEEE. 4 (4): 38–49. doi:10.1109/TIT.1954.1057465.
  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. EC-3 (3): 6–12. doi:10.1109/IREPGELC.1954.6499441.
  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). "On a Theory of Boolean Functions". Journal of the Society for Industrial and Applied Mathematics. Society for Industrial and Applied Mathematics. 7 (4): 487–498. doi:10.1137/0107041. ISSN 0368-4245. (12 pages)
  7. ^ Таланцев [Talantsev], А. Д. [A. D.] (1959). "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] (in Russian). Moscow, Russia. 20 (7): 898–907. Mi at12783. 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. S2CID 206617023. (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]. (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). (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. S2CID 13480467. (12 pages)
  13. ^ a b c Davio, Marc; Deschamps, Jean-Pierre; Thayse, André (1978-08-01). Discrete and Switching Functions (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. S2CID 17528915. 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 (8): 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 (PhD thesis) (1st ed.). Szczecin, Poland: Instytut Informatyki, Technical University of Szczecin. ISBN 978-8-387423-16-2. ISSN 1506-3054. (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. (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. S2CID 10178376. Lecture #52. (216 pages)

Further reading edit

  • 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). "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]. 9 (in German). Vol. 145. Düsseldorf, Germany: VDI-Verlag. 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. 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. 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. S2CID 37053010. Lecture #26. (24 of 153 pages)

External links edit

December 18, 2023
boolean, differential, calculus, german, boolescher, differentialkalkül, subject, field, boolean, algebra, discussing, changes, boolean, variables, boolean, functions, concepts, analogous, those, classical, differential, calculus, notably, studying, changes, f. Boolean differential calculus BDC German Boolescher Differentialkalkul 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 Contents 1 History and applications 2 Overview 3 See also 4 References 5 Further reading 6 External linksHistory and applications editOriginally 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 A D Talancev 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 Andre 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 Integralkalkul 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 editBoolean 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 d x i displaystyle dx i nbsp of a Boolean variable x i displaystyle x i nbsp models the relation d x i 0 no change of x i 1 change of x i displaystyle dx i begin cases 0 amp text no change of x i 1 amp text change of x i end cases nbsp There are no constraints in regard to the nature the causes and consequences of a change The differentials d x i displaystyle dx i nbsp are binary They can be used just like common binary variables See also editBoolean Algebra Boole s expansion theorem Ramadge Wonham frameworkReferences edit H Wehlan Boolean Algebra in Encyclopedia of Mathematics Scheuring Rainer Wehlan Herbert Hans 1991 12 01 July 1991 Bretthauer Georg ed Der Boolesche Differentialkalkul 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 S2CID 56766796 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 IEEE 4 4 38 49 doi 10 1109 TIT 1954 1057465 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 EC 3 3 6 12 doi 10 1109 IREPGELC 1954 6499441 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 Akers Jr Sheldon Buckingham December 1959 On a Theory of Boolean Functions Journal of the Society for Industrial and Applied Mathematics Society for Industrial and Applied Mathematics 7 4 487 498 doi 10 1137 0107041 ISSN 0368 4245 12 pages Talancev Talantsev A D A D 1959 Ob analize i sinteze nekotorykh elektriceskikh skhem pri pomosci special nykh logiceskikh operatorov b analize i sinteze nekotoryh elektricheskih shem pri pomoshi specialnyh logicheskih operatorov Analysis and synthesis of certain electric circuits by means of special logical operators Avtomatika i telemehanika Avtomatika i Telemekhanika in Russian Moscow Russia 20 7 898 907 Mi at12783 Archived from the original on 2017 10 17 Retrieved 2017 10 17 Osnovnoe soderzhanie stati dolozheno na seminare po tehnicheskim prilozheniyam matematicheskoj logiki v MGU 2 H 1958 g i 16 1 1959 Avtor schitaet svoim dolgom vyrazit priznatelnost V A Trapeznikovu ru V I Shestakovu i M L Cetlinu za interes k rabote i cennye zamechaniya pri obsuzhdenii rezultatov 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 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 S2CID 206617023 8 pages 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 Thayse Andre 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 Fosseprez who initially suggested the basic problem considered here 76 pages a b Thayse Andre 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 a b Thayse Andre 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 S2CID 13480467 12 pages a b c Davio Marc Deschamps Jean Pierre Thayse Andre 1978 08 01 Discrete and Switching Functions 1st ed New York USA Georgi Publishing Company McGraw Hill International Book Company ISBN 0 07 015509 7 LCCN 77 030718 xx 729 pages Thayse Andre 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 a b c d Bochmann Dieter in German Posthoff Christian 1981 Binare dynamische Systeme Binary dynamic systems in German 1st ed Akademie Verlag Berlin R Oldenbourg Verlag de Munchen 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 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 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 amp Claypool Publishers doi 10 2200 S00511ED1V01Y201305DCS042 ISBN 978 1 62705 241 2 S2CID 17528915 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 Zurich Switzerland International Federation of Automatic Control IFAC Pergamon Press 2 8 723 728 doi 10 1016 S1474 6670 17 54214 7 6 pages Anuskevic Yanushkevich Svitlana N Svetlana N 1998 Logic Differential Calculus in Multi Valued Logic Design PhD thesis 1st ed Szczecin Poland Instytut Informatyki Technical University of Szczecin ISBN 978 8 387423 16 2 ISSN 1506 3054 326 pages 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 Binare 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 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 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 amp Claypool Publishers doi 10 2200 S00766ED1V01Y201704DCS052 ISBN 978 1 62705 922 0 S2CID 10178376 Lecture 52 216 pages Further reading editDavio Marc Piret Philippe M July 1969 Les derivees Booleennes 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 Izvestiya Akademii nauk SSSR Tehnicheskaya kibernetika Izvestii a Akademii Nauk SSSR Tekhnicheskai a kibernetika Proceedings of the Academy of Sciences of the USSR Engineering Cybernetics in Russian 5 125 133 9 pages Kuhnrich Martin 1986 Differentialoperatoren uber Booleschen Algebren Differential operators on Boolean algebras Zeitschrift fur 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 9 in German Vol 145 Dusseldorf Germany VDI Verlag ISBN 3 18 144509 6 DNB IDN 940164671 a href Template Cite book html title Template Cite book cite book a work ignored help NB Also Chemnitz Technische Universitat Dissertation 147 pages Scheuring Rainer Wehlan Herbert Hans 1993 Control of Discrete Event Systems by Means of the Boolean Differential Calculus In Balemi Silvano Kozak Petr Smedinga Rein eds Discrete Event Systems Modeling and Control Progress in Systems and Control Theory PSCT Vol 13 Basel Switzerland Birkhauser 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 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 amp 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 External links editWehlan Herbert Hans 2010 12 06 Boolean differential calculus In Hazewinkel Michiel ed Encyclopedia of Mathematics Springer Science Business Media ISBN 978 1 4020 0609 8 Archived from the original on 2017 10 16 Retrieved 2017 10 16 Institut fur 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 Retrieved from https en wikipedia org w index php title Boolean differential calculus amp oldid 1188964300, wikipedia, wiki, book, books, library,

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