Kalkulus diferensial Boolean

Kalkulus diferensial Boole (bahasa Jerman: Boolescher Differentialkalkül) merupakan sebuah bidang subjek aljabar Boolean yang membahas tentang perubahan variabel Boole dan fungsi Boole. Konsep ini mirip dengan konsep kalkulus diferensial klasik, khususnya dalam mempelajari perubahan fungsi dan variabel terhadap yang lain.

Ada berbagai aspek teori sistem dinamik yang dipelajari dengan menggunakan kalkulus ini, seperti teori automata pada automata berhingga, teori jaring Petri,^[1] dan teori kontrol pengawasan.

Sejarah dan penerapan

Kalkulus ini pada awalnya terinspirasi oleh sebuah desain dan uji tentang switching circuit dan pemanfaatan sandi pengoreksian galat dalam teknik listrik. Hal tersebut merupakan akar pengembangan dari apa yang akan berkembang menjadi kalkulus diferensial Boolean. Pengembangan ini diprakarsai sekitar tahun 1954 dan 1959 oleh karya Irving S. Reed,^[2] David E. Muller,^[3] David A. Huffman,^[4] Sheldon B. Akers Jr.^[5] dan A. D. Talantsev,^[6] lalu dikembangkan pada tahun 1968 oleh Frederick F. Sellers Jr.,^[7]^[8] Mu-Yue Hsiao^[7]^[8] and Leroy W. Bearnson.^[7]^[8]

Pada tahun 1970-an, karya-karya seperti André Thayse,^[9]^[10]^[11]^[12]^[13] Marc Davio^[10]^[11]^[12] dan Jean-Pierre Deschamps^[12] membentuk dasar-dasar tentang kalkulus diferensial Boole. Kalkulus tersebut dikembangkan lebih lanjut oleh Dieter Bochmann [de],^[14] Christian Posthoff^[14] dan Bernd Steinbach [de]^[15] agar menjadi teori matematika yang mempunyai prinsip-prinsipnya tersendiri di kemudian hari.

Sebuah teori pelengkap kalkulus integral Boolean (bahasa Jerman: Boolescher Integralkalkül) telah dikembangkan juga.^[14]^[16]

KDB juga telah menemukan kegunaan dalam sistem dinamis kejadian diskrit (SDKD)^[17] dalam jaringan digital protokol komunikasi.

Sementara itu, KDB telah melihat ekstensi ke multi-nilai variabel dan fungsi^[14]^[18]^[19] serta kekisi dari fungsi Boolean.^[20]^[21]

Ikhtisar

Operator diferensial Boole memainkan peran penting dalam kalkulus diferensial Boole. Operator tersebut dapat memperluas kegunaan diferensial dari analisis klasik ke fungsi logis.

Diferensial $dx_{i}$ dari variabel Boole $x_{i}$ menggambarkan relasi berikut:

dx_{i}={\begin{cases}0,&{\text{tidak ada perubahan }}x_{i}\\1,&{\text{perubahan }}x_{i}\end{cases}}

dengan diferensial $dx_{i}$ adalah biner, yang dapat dipakai seperti variabel biner biasa. Relasi tersebut tidak mempunyai batasan mengenai sifat, penyebab dan akibat dari suatu perubahan.

Lihat pula

Referensi

^ 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 (dalam bahasa Jerman). 39 (7). Stuttgart, Germany: R. Oldenbourg Verlag [de]: 226–233. doi:10.1524/auto.1991.39.112.226. ISSN 0178-2312. Diarsipkan dari versi aslinya tanggal 2017-10-16. Diakses tanggal 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). PGIT-4 (4). Institute of Radio Engineers (IRE): 38–49. (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: 6–12. (7 pages)
^ Huffman, David Albert (1958-01-15). "Solvability criterion for simultaneous logical equations". Quarterly Progress Report (48). Cambridge, MA, USA: MIT Research Laboratory of Electronics: 87–88. 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. 7 (4). Society for Industrial and Applied Mathematics (SIAM): 487–498. doi:10.1137/0107041. ISSN 0368-4245. (12 pages)
^ 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] (dalam bahasa Rusia). 20 (7). Moscow, Russia: 898–907. Templat:Mathnet. Diarsipkan dari versi aslinya tanggal 2017-10-17. Diakses tanggal 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)
^ ^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)
^ ^a ^b ^c Sellers Jr., Frederick F.; Hsiao, Mu-Yue; Bearnson, Leroy W. (November 1968). Error Detecting Logic for Digital Computers (Edisi 1st). New York, USA: McGraw-Hill Book Company. hlm. 17–37. LCCN 68-16491. OCLC 439460. (21 of xviii+295 pages)
^ Thayse, André (October 1970) [May 1970]. "Transient analysis of logical networks applied to hazard detection" (PDF). Philips Research Reports. 25 (5). Brussels, Belgium: Philips Research Laboratory: 261–336. R737. Diarsipkan dari asli (PDF) tanggal 2017-03-08. Diakses tanggal 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)
^ ^a ^b Thayse, André (February 1971). "Boolean Differential Calculus" (PDF). Philips Research Reports. 26 (2). Brussels, Belgium: Philips Research Laboratory: 229–246. R764. Diarsipkan dari asli (PDF) tanggal 2017-03-08. Diakses tanggal 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, 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)
^ ^a ^b ^c Davio, Marc; Deschamps, Jean-Pierre; Thayse, André (1978-08-01). Discrete and Switching Functions (Edisi 1st). New York, USA: Georgi Publishing Company / McGraw-Hill International Book Company. ISBN 0-07-015509-7. LCCN 77-030718. (xx+729 pages)
^ Thayse, André (1981). Goos, Gerhard [in Jerman]; Hartmanis, Juris (ed.). Boolean Calculus of Differences. Lecture Notes in Computer Science. Vol. 101 (Edisi 1st). Berlin: Springer-Verlag. ISBN 3-540-10286-8. (144 pages)
^ ^a ^b ^c ^d Bochmann, Dieter [in Jerman]; Posthoff, Christian (1981). Binäre dynamische Systeme [Binary dynamic systems] (dalam bahasa Jerman) (Edisi 1st). Akademie-Verlag, Berlin / R. Oldenbourg Verlag [de], München. ISBN 3-486-25071-X. Templat:DNB-IDN. License number [de]: 202.100/408/81. Order code: 7623619 (6391). (397 pages) (NB. Per Templat:DNB-IDN a Russian translation of this work was released in 1986.)
^ Bochmann, Dieter [in Jerman]; Steinbach, Bernd [in Jerman] (1991). Logikentwurf mit XBOOLE – Algorithmen und Programme [Logic design with XBOOLE – Algorithms and programs] (dalam bahasa Jerman) (Edisi 1st). Berlin, Germany: Verlag Technik [de]. ISBN 3-341-01006-8. Templat:DNB-IDN. (303 pages + 5.25-inch floppy disk)
^ Steinbach, Bernd [in Jerman]; Posthoff, Christian (2013-07-01). Thornton, Mitchell A. (ed.). Boolean Differential Equations. Synthesis Lectures on Digital Circuits and Systems (Edisi 1st). San Rafael, CA, USA: Morgan & Claypool Publishers. doi:10.2200/S00511ED1V01Y201305DCS042. ISBN 978-1-62705-241-2. Lecture #42. (158 pages)
^ Scheuring, Rainer; Wehlan, Herbert "Hans" (1991-09-01). Franke, Dieter; Kraus, Franta (ed.). "On the Design of Discrete Event Dynamic Systems by Means of the Boolean Differential Calculus". First IFAC Symposium on Design Methods of Control Systems. 2. Zürich, Switzerland: International Federation of Automatic Control (IFAC) / Pergamon Press: 723–728. doi:10.1016/S1474-6670(17)54214-7. (6 pages)
^ Ânuškevič [Yanushkevich], Svitlana N. [Svetlana N.] (1998). Logic Differential Calculus in Multi-Valued Logic Design (PhD thesis) (Edisi 1st). Szczecin, Poland: Instytut Informatyki, Technical University of Szczecin. ISBN 978-8-387423-16-2. ISSN 1506-3054. ISBN 8-387423-16-5. (326 pages)
^ Bochmann, Dieter [in Jerman] (2008-09-01). Binary Systems - A BOOLEAN Book (Edisi 1st). Dresden, Germany: TUDpress Verlag der Wissenschaften. ISBN 978-3-940046-87-1. Templat:DNB-IDN. (421 pages) Translation of: Bochmann, Dieter [in Jerman] (February 2006). Binäre Systeme - Ein BOOLEAN Buch [Binary systems - A Boolean book] (dalam bahasa Jerman) (Edisi 1st). Hagen, Germany: LiLoLe-Verlag GmbH (Life-Long-Learning) / BoD GmbH. ISBN 3-934447-10-4. ISBN 978-3-934447-10-3. Templat:DNB-IDN. (452 pages)
^ Steinbach, Bernd [in Jerman]; Posthoff, Christian (2013). "Derivative Operations for Lattices of Boolean Functions" (PDF). Proceedings Reed-Muller Workshop 2013. Toyama, Japan: 110–119. Diarsipkan (PDF) dari versi aslinya tanggal 2017-10-21. Diakses tanggal 2017-10-21. (10 pages)
^ Steinbach, Bernd [in Jerman]; Posthoff, Christian (2017-06-07). Thornton, Mitchell A. (ed.). Boolean Differential Calculus. Synthesis Lectures on Digital Circuits and Systems (Edisi 1st). San Rafael, CA, USA: Morgan & Claypool Publishers. doi:10.2200/S00766ED1V01Y201704DCS052. ISBN 978-1-62705-922-0. Lecture #52. (216 pages)

Bacaan lebih lanjut

Davio, Marc; Piret, Philippe M. (July 1969). "Les dérivées Booléennes et leur application au diagnostic" [Boolean derivatives and their application and diagnosis]. Philips Revue (dalam bahasa Prancis). 12 (3). Brussels, Belgium: Philips Research Laboratory, Manufacture Belge de Lampes et de Materiel Electronique (MBLE Research Laboratory): 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 Jerman] (1977). "Boolean differential calculus (a survey)". Engineering Cybernetics. 15 (5). Institute of Electrical and Electronics Engineers (IEEE): 67–75. ISSN 0013-788X. (9 pages) Translation of: Bochmann, Dieter [in Jerman] (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] (dalam bahasa Rusia) (5): 125–133. (9 pages)
Kühnrich, Martin (1986) [1984-07-31 (submission)]. "Differentialoperatoren über Booleschen Algebren" [Differential operators on Boolean algebras]. Zeitschrift für mathematische Logik und Grundlagen der Mathematik (dalam bahasa Jerman). 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]. 9 (dalam bahasa Jerman). Vol. 145. Düsseldorf, Germany: VDI-Verlag [de]. ISBN 3-18-144509-6. Templat:DNB-IDN. (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". Dalam Balemi, Silvano; Kozák, Petr; Smedinga, Rein (ed.). Discrete Event Systems: Modeling and Control. Progress in Systems and Control Theory (PSCT). Vol. 13. Basel, Switzerland: Birkhäuser Verlag. hlm. 79–93. doi:10.1007/978-3-0348-9120-2_7. (15 pages)
Posthoff, Christian; Steinbach, Bernd [in Jerman] (2004-02-04). Logic Functions and Equations – Binary Models for Computer Science (Edisi 1st). 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 Jerman]; Posthoff, Christian (2009-02-12). Logic Functions and Equations – Examples and Exercises (Edisi 1st). 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 Templat:DNB-IDN this hardcover edition has been rereleased as softcover edition in 2010.)
Steinbach, Bernd [in Jerman]; 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 Jerman]; Posthoff, Christian (2010-01-15) [2009]. "Chapter 3: Boolean Differential Calculus". Dalam Sasao, Tsutomu; Butler, Jon T. (ed.). Progress in Applications of Boolean Functions. Synthesis Lectures on Digital Circuits and Systems (Edisi 1st). San Rafael, CA, USA: Morgan & Claypool Publishers. hlm. 55–78, 121–126. doi:10.2200/S00243ED1V01Y200912DCS026. ISBN 978-1-60845-181-4. Lecture #26. (24 of 153 pages)

Pranala luar

Wehlan, Herbert "Hans" (2010-12-06). Hazewinkel, Michiel (ed.). Boolean differential calculus. Springer Science+Business Media. ISBN 978-1-4020-0609-8. Diarsipkan dari versi aslinya tanggal 2017-10-16. Diakses tanggal 2017-10-16.
Institut für Informatik (IfI) (2017). "XBOOLE". TU Bergakademie Freiberg. Diarsipkan dari versi aslinya tanggal 2017-10-31. Diakses tanggal 2017-10-31. with "XBOOLE Monitor". 2008-07-23. Diarsipkan dari asli tanggal 2017-10-31. Diakses tanggal 2017-10-31.

[Scheuring_Wehlan_1991_Petri-1] 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 (dalam bahasa Jerman). 39 (7). Stuttgart, Germany: R. Oldenbourg Verlag [de]: 226–233. doi:10.1524/auto.1991.39.112.226. ISSN 0178-2312. Diarsipkan dari versi aslinya tanggal 2017-10-16. Diakses tanggal 2017-10-16. (8 pages)

[Reed_1954-2] 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). PGIT-4 (4). Institute of Radio Engineers (IRE): 38–49. (12 pages)

[Muller_1954-3] 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)

[Huffman_1958-4] Huffman, David Albert (1958-01-15). "Solvability criterion for simultaneous logical equations". Quarterly Progress Report (48). Cambridge, MA, USA: MIT Research Laboratory of Electronics: 87–88. AD 156-161. (2 pages)

[Akers_1959-5] 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. 7 (4). Society for Industrial and Applied Mathematics (SIAM): 487–498. doi:10.1137/0107041. ISSN 0368-4245. (12 pages)

[Talantsev_1959-6] 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] (dalam bahasa Rusia). 20 (7). Moscow, Russia: 898–907. Templat:Mathnet. Diarsipkan dari versi aslinya tanggal 2017-10-17. Diakses tanggal 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)

[Sellers_Hsiao_Bearnson_1968_1-7] 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)

[Sellers_Hsiao_Bearnson_1968_2-8] Sellers Jr., Frederick F.; Hsiao, Mu-Yue; Bearnson, Leroy W. (November 1968). Error Detecting Logic for Digital Computers (Edisi 1st). New York, USA: McGraw-Hill Book Company. hlm. 17–37. LCCN 68-16491. OCLC 439460. (21 of xviii+295 pages)

[Thayse_1970-9] Thayse, André (October 1970) [May 1970]. "Transient analysis of logical networks applied to hazard detection" (PDF). Philips Research Reports. 25 (5). Brussels, Belgium: Philips Research Laboratory: 261–336. R737. Diarsipkan dari asli (PDF) tanggal 2017-03-08. Diakses tanggal 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)

[Thayse_1971-10] Thayse, André (February 1971). "Boolean Differential Calculus" (PDF). Philips Research Reports. 26 (2). Brussels, Belgium: Philips Research Laboratory: 229–246. R764. Diarsipkan dari asli (PDF) tanggal 2017-03-08. Diakses tanggal 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)

[Thayse_Davio_1973-11] 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)

[Davio_Deschamps_Thayse_1978-12] Davio, Marc; Deschamps, Jean-Pierre; Thayse, André (1978-08-01). Discrete and Switching Functions (Edisi 1st). New York, USA: Georgi Publishing Company / McGraw-Hill International Book Company. ISBN 0-07-015509-7. LCCN 77-030718. (xx+729 pages)

[Thayse_1981-13] Thayse, André (1981). Goos, Gerhard [in Jerman]; Hartmanis, Juris (ed.). Boolean Calculus of Differences. Lecture Notes in Computer Science. Vol. 101 (Edisi 1st). Berlin: Springer-Verlag. ISBN 3-540-10286-8. (144 pages)

[Bochmann_Posthoff_1981-14] Bochmann, Dieter [in Jerman]; Posthoff, Christian (1981). Binäre dynamische Systeme [Binary dynamic systems] (dalam bahasa Jerman) (Edisi 1st). Akademie-Verlag, Berlin / R. Oldenbourg Verlag [de], München. ISBN 3-486-25071-X. Templat:DNB-IDN. License number [de]: 202.100/408/81. Order code: 7623619 (6391). (397 pages) (NB. Per Templat:DNB-IDN a Russian translation of this work was released in 1986.)

[Bochmann_Steinbach_1991-15] Bochmann, Dieter [in Jerman]; Steinbach, Bernd [in Jerman] (1991). Logikentwurf mit XBOOLE – Algorithmen und Programme [Logic design with XBOOLE – Algorithms and programs] (dalam bahasa Jerman) (Edisi 1st). Berlin, Germany: Verlag Technik [de]. ISBN 3-341-01006-8. Templat:DNB-IDN. (303 pages + 5.25-inch floppy disk)

[Steinbach_Posthoff_2013_2-16] Steinbach, Bernd [in Jerman]; Posthoff, Christian (2013-07-01). Thornton, Mitchell A. (ed.). Boolean Differential Equations. Synthesis Lectures on Digital Circuits and Systems (Edisi 1st). San Rafael, CA, USA: Morgan & Claypool Publishers. doi:10.2200/S00511ED1V01Y201305DCS042. ISBN 978-1-62705-241-2. Lecture #42. (158 pages)

[Scheuring_Wehlan_1991_DEDS-17] Scheuring, Rainer; Wehlan, Herbert "Hans" (1991-09-01). Franke, Dieter; Kraus, Franta (ed.). "On the Design of Discrete Event Dynamic Systems by Means of the Boolean Differential Calculus". First IFAC Symposium on Design Methods of Control Systems. 2. Zürich, Switzerland: International Federation of Automatic Control (IFAC) / Pergamon Press: 723–728. doi:10.1016/S1474-6670(17)54214-7. (6 pages)

[Yanushkevich_1998-18] Ânuškevič [Yanushkevich], Svitlana N. [Svetlana N.] (1998). Logic Differential Calculus in Multi-Valued Logic Design (PhD thesis) (Edisi 1st). Szczecin, Poland: Instytut Informatyki, Technical University of Szczecin. ISBN 978-8-387423-16-2. ISSN 1506-3054. ISBN 8-387423-16-5. (326 pages)

[Bochmann_2006_2008-19] Bochmann, Dieter [in Jerman] (2008-09-01). Binary Systems - A BOOLEAN Book (Edisi 1st). Dresden, Germany: TUDpress Verlag der Wissenschaften. ISBN 978-3-940046-87-1. Templat:DNB-IDN. (421 pages) Translation of: Bochmann, Dieter [in Jerman] (February 2006). Binäre Systeme - Ein BOOLEAN Buch [Binary systems - A Boolean book] (dalam bahasa Jerman) (Edisi 1st). Hagen, Germany: LiLoLe-Verlag GmbH (Life-Long-Learning) / BoD GmbH. ISBN 3-934447-10-4. ISBN 978-3-934447-10-3. Templat:DNB-IDN. (452 pages)

[Steinbach_Posthoff_2013_1-20] Steinbach, Bernd [in Jerman]; Posthoff, Christian (2013). "Derivative Operations for Lattices of Boolean Functions" (PDF). Proceedings Reed-Muller Workshop 2013. Toyama, Japan: 110–119. Diarsipkan (PDF) dari versi aslinya tanggal 2017-10-21. Diakses tanggal 2017-10-21. (10 pages)

[Steinbach_Posthoff_2017-21] Steinbach, Bernd [in Jerman]; Posthoff, Christian (2017-06-07). Thornton, Mitchell A. (ed.). Boolean Differential Calculus. Synthesis Lectures on Digital Circuits and Systems (Edisi 1st). San Rafael, CA, USA: Morgan & Claypool Publishers. doi:10.2200/S00766ED1V01Y201704DCS052. ISBN 978-1-62705-922-0. Lecture #52. (216 pages)

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