Lihat sumber Kalkulus diferensial Boolean

{{Short description|Bidang subjek aljabar Boolean yang membahas perubahan variabel dan fungsi Boolean}}
{{bots|deny=Citation bot}}
{{Use dmy dates|date=May 2019|cs1-dates=y}}'''Kalkulus diferensial Boole''' ({{Lang-de|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]],<ref name="Scheuring_Wehlan_1991_Petri" /> dan [[teori kontrol pengawasan]].

== Sejarah dan penerapan ==
{{Periksa terjemahan|en|Boolean differential calculus#History and applications}}
Kalkulus ini pada awalnya terinspirasi oleh sebuah desain dan uji tentang ''[[Switching circuit theorem|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]],<ref name="Reed_1954" /> [[David E. Muller]],<ref name="Muller_1954" /> [[David A. Huffman]],<ref name="Huffman_1958" /> [[Sheldon B. Akers|Sheldon B.<!-- Buckingham --> Akers Jr.]]<ref name="Akers_1959" /> dan {{lang|ru|[[A. D. Talantsev]]}},<ref name="Talantsev_1959" /> lalu dikembangkan pada tahun 1968 oleh [[Frederick F. Sellers|Frederick F. Sellers Jr.]],<ref name="Sellers_Hsiao_Bearnson_1968_1" /><ref name="Sellers_Hsiao_Bearnson_1968_2" /> [[Mu-Yue Hsiao]]<ref name="Sellers_Hsiao_Bearnson_1968_1" /><ref name="Sellers_Hsiao_Bearnson_1968_2" /> and [[Leroy W. Bearnson]].<ref name="Sellers_Hsiao_Bearnson_1968_1" /><ref name="Sellers_Hsiao_Bearnson_1968_2" />

Pada tahun 1970-an, karya-karya seperti {{lang|fr|[[André Thayse]]}},<ref name="Thayse_1970" /><ref name="Thayse_1971" /><ref name="Thayse_Davio_1973" /><ref name="Davio_Deschamps_Thayse_1978" /><ref name="Thayse_1981" /> [[Marc Davio]]<ref name="Thayse_1971" /><ref name="Thayse_Davio_1973" /><ref name="Davio_Deschamps_Thayse_1978" /> dan {{lang|fr|[[Jean-Pierre Deschamps]]}}<ref name="Davio_Deschamps_Thayse_1978" /> membentuk dasar-dasar tentang kalkulus diferensial Boole. Kalkulus tersebut dikembangkan lebih lanjut oleh {{lang|de|{{ill|Dieter Bochmann|de}}}},<ref name="Bochmann_Posthoff_1981" /> {{lang|de|[[Christian Posthoff]]}}<ref name="Bochmann_Posthoff_1981" /> dan {{lang|de|{{ill|Bernd Steinbach|de}}}}<ref name="Bochmann_Steinbach_1991" /> agar menjadi teori matematika yang mempunyai prinsip-prinsipnya tersendiri di kemudian hari.

{{anchor|Kalkulus integral Boolean}}Sebuah teori pelengkap '''kalkulus integral Boolean''' (bahasa Jerman: '''{{lang|de|Boolescher Integralkalkül}}''') telah dikembangkan juga.<ref name="Bochmann_Posthoff_1981"/><ref name="Steinbach_Posthoff_2013_2"/>

KDB juga telah menemukan kegunaan dalam [[sistem dinamis kejadian diskrit]] (SDKD)<ref name="Scheuring_Wehlan_1991_DEDS"/> dalam [[jaringan digital]] [[protokol komunikasi]].

{{anchor|Kalkulus diferensial logika}}Sementara itu, KDB telah melihat ekstensi ke [[fungsi multi-nilai|multi-nilai]] variabel dan fungsi<ref name="Bochmann_Posthoff_1981"/><ref name="Yanushkevich_1998"/><ref name="Bochmann_2006_2008"/> serta [[kekisi (modul)<!-- TBD? -->|kekisi]] dari fungsi Boolean.<ref name="Steinbach_Posthoff_2013_1"/><ref name="Steinbach_Posthoff_2017"/>

== Ikhtisar ==
[[Operator diferensial]] Boole memainkan peran penting dalam kalkulus diferensial Boole. Operator tersebut dapat memperluas kegunaan diferensial dari [[analisis (matematika)|analisis]] klasik ke fungsi logis.

Diferensial <math>dx_i</math> dari variabel Boole <math>x_i</math> menggambarkan relasi berikut:

: <math>dx_i = \begin{cases}
  0,  & \text{tidak ada perubahan } x_i\\
  1, & \text{perubahan } x_i
\end{cases}
</math>

dengan diferensial <math>dx_i</math> adalah biner, yang dapat dipakai seperti variabel biner biasa. Relasi tersebut tidak mempunyai batasan mengenai sifat, penyebab dan akibat dari suatu perubahan.

== Lihat pula ==
* [[Aljabar Boolean|Aljabar Boole]]
* [[Teorema ekspansi Boole]]
* [[Kerangka kerja Ramadge–Wonham]]

== Referensi ==
{{Reflist|refs=
<ref name="Reed_1954">{{cite journal |first=Irving Stoy |last=Reed |author-link=Irving Stoy Reed |date=1954 |title=A Class of Multiple-Error-Correcting Codes and the Decoding Scheme |work=Transactions of the [[IRE Professional Group on Information Theory]] (PGIT) |publisher=[[Institute of Radio Engineers]] (IRE) |volume=PGIT-4 |issue=4 |pages=38–49}} (12 pages)</ref>
<ref name="Muller_1954">{{cite journal |first=David Eugene |last=Muller |author-link=David Eugene Muller |title=Application of Boolean algebra to switching circuit design and to error detection |journal=Transactions of the [[IRE Professional Group on Electronic Computers]] (PGEC) |volume=PGEC-3 |date=1954 |pages=6–12}} (7 pages)</ref>
<ref name="Huffman_1958">{{cite journal |first=David Albert |last=Huffman |author-link=David Albert Huffman |title=Solvability criterion for simultaneous logical equations |publisher=MIT Research Laboratory of Electronics |location=Cambridge, MA, USA |journal=Quarterly Progress Report |number=48 |id=AD 156-161 |date=1958-01-15 |pages=87–88}} (2 pages)</ref>
<ref name="Akers_1959">{{cite journal |first=Sheldon Buckingham |last=Akers Jr. |title=On a Theory of Boolean Functions |journal=[[Journal of the Society for Industrial and Applied Mathematics]] |publisher=[[Society for Industrial and Applied Mathematics]] (SIAM) |volume=7 |number=4 |date=December 1959 |orig-year=1957-09-27 (submission), 1959-05-28 (revision) |issn=0368-4245 |doi=10.1137/0107041 |pages=487–498}} (12 pages)</ref>
<ref name="Talantsev_1959">{{cite journal |first=А. Д. [A. D.] |last=Таланцев [Talantsev<!-- Talancev -->] |script-title=ru:б анализе и синтезе некоторых электрических схем при помощи специальных логических операторов |title=Ob analize i sinteze nekotorykh električeskikh skhem pri pomośći special'nykh logičeskikh operatorov |language=ru |trans-title=Analysis and synthesis of certain electric circuits by means of special logical operators |journal=[[Автоматика и телемеханика]] ([[Avtomatika i telemekhanika]]) <nowiki>[</nowiki>[[Automation and Remote Control]]<nowiki>]</nowiki> |volume=20 |number=7 |date=1959 |orig-year=1958-11-01 (submission) |location=Moscow, Russia |pages=898–907 |id={{mathnet|at12783}} |url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=at&paperid=12783&option_lang=rus |access-date=2017-10-17 |url-status=live |archive-url=https://web.archive.org/web/20171017090358/http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=at&paperid=12783&option_lang=rus |archive-date=2017-10-17 |quote=[…] 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 {{ill|Vadim Aleksandrovich Trapeznikov{{!}}В. А. Трапезникову|ru|Трапезников, Вадим Александрович}}, [[Виктор Иванович Шестаков|В. И. Шестакову]] и [[Михаил Львович Цетлин|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 {{ill|Vadim Aleksandrovich Trapeznikov{{!}}V. A. Trapeznikov|ru|Трапезников, Вадим Александрович}}, [[Victor Ivanovich Shestakov|V. I. Shestakov]] and [[Michael Lvovitch Tsetlin|M. L. Tsetlin]] for interest in the work and valuable comments in discussing the results.[…]]}} (10 pages)</ref>
<ref name="Sellers_Hsiao_Bearnson_1968_1">{{cite journal |first1=Frederick F. |last1=Sellers Jr. |first2=Mu-Yue |last2=Hsiao |first3=Leroy W. |last3=Bearnson |title=Analyzing Errors with the Boolean Difference |journal=[[IEEE Transactions on Computers]] |volume=C-17 |issue=7 |pages=676–683 |date=July 1968 |issn=0018-9340 |doi=10.1109/TC.1968.227417}} (8 pages)</ref>
<ref name="Sellers_Hsiao_Bearnson_1968_2">{{cite book |first1=Frederick F. |last1=Sellers Jr. |first2=Mu-Yue |last2=Hsiao |first3=Leroy W. |last3=Bearnson |title=Error Detecting Logic for Digital Computers |publisher=[[McGraw-Hill Book Company]] |location=New York, USA |edition=1st |oclc=439460 |lccn=68-16491 |pages=17–37 |date=November 1968}} (21 of xviii+295 pages)</ref>
<ref name="Thayse_1970">{{cite journal |first=André |last=Thayse |title=Transient analysis of logical networks applied to hazard detection |journal=[[Philips Research Reports]] |publisher=[[Philips Research Laboratory]] |location=Brussels, Belgium |volume=25 |number=5 |date=October 1970 |orig-year=May 1970 |pages=261–336 |id=R737 |url=http://www.extra.research.philips.com/hera/people/aarts/_Philips%20Bound%20Archive/PRRep/PRRep-25-1970-261.pdf |access-date=2017-10-17 |url-status=dead |archive-url=https://web.archive.org/web/20170308203506/http://www.extra.research.philips.com/hera/people/aarts/_Philips%20Bound%20Archive/PRRep/PRRep-25-1970-261.pdf |archive-date=2017-03-08 |quote=[…] The author is indebted to Dr [[Marc Davio|M. Davio]] for his continuing interest and comments on this work. Thanks are also due to Mr [[Claude Fosséprez|C. Fosséprez]] who initially suggested the basic problem considered here. […]}} (76 pages)</ref>
<ref name="Thayse_1971">{{cite journal |first=André |last=Thayse |title=Boolean Differential Calculus |journal=[[Philips Research Reports]] |publisher=[[Philips Research Laboratory]] |location=Brussels, Belgium |volume=26 |number=2 |date=February 1971 |pages=229–246 |id=R764 |url=http://www.extra.research.philips.com/hera/people/aarts/_Philips%20Bound%20Archive/PRRep/PRRep-26-1971-229.pdf |access-date=2017-10-16 |url-status=dead |archive-url=https://web.archive.org/web/20170308203806/http://www.extra.research.philips.com/hera/people/aarts/_Philips%20Bound%20Archive/PRRep/PRRep-26-1971-229.pdf |archive-date=2017-03-08 |quote=[…] 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 [[Marc Davio|M. Davio]] for his encouragement and support and for several ideas in the presentation. […]}} (18 pages)</ref>
<ref name="Thayse_Davio_1973">{{cite journal |first1=André |last1=Thayse |first2=Marc |last2=Davio |title=Boolean Differential Calculus and its Application to Switching Theory |journal=[[IEEE Transactions on Computers]] |volume=C-22 |issue=4 |date=1973-04-01 |pages=409–420 |doi=10.1109/T-C.1973.223729}} (12 pages)</ref>
<ref name="Davio_Deschamps_Thayse_1978">{{cite book |first1=Marc |last1=Davio |first2=Jean-Pierre |last2=Deschamps |first3=André |last3=Thayse |title=Discrete and Switching Functions |publisher=Georgi Publishing Company / [[McGraw-Hill International Book Company]] |location=New York, USA |edition=1st |date=1978-08-01 |isbn=0-07-015509-7 |lccn=77-030718 |url-access=registration |url=https://archive.org/details/discreteswitchin0000davi}} (xx+729 pages)</ref>
<ref name="Thayse_1981">{{cite book |first=André |last=Thayse |editor-first1=Gerhard |editor-last1=Goos |editor-link1=:de:Gerhard Goos |editor-first2=Juris |editor-last2=Hartmanis |editor-link2=Juris Hartmanis |title=Boolean Calculus of Differences |series=Lecture Notes in Computer Science |volume=101 |publisher=[[Springer-Verlag]] |location=Berlin |edition=1st |date=1981 |isbn=3-540-10286-8}} (144 pages)</ref>
<ref name="Bochmann_Posthoff_1981">{{cite book |first1=Dieter |last1=Bochmann |author-link1=:de:Dieter Bochmann |first2=Christian |last2=Posthoff |title=Binäre dynamische Systeme |language=de |trans-title=Binary dynamic systems |publisher=[[Akademie-Verlag]], Berlin / {{ill|R. Oldenbourg Verlag|de}}, München |edition=1st |date=1981 |isbn=3-486-25071-X |id={{DNB-IDN|810757168|810200317|}}. {{ill|License number (East German books){{!}}License number|de|Lizenznummer}}: 202.100/408/81. Order code: 7623619&nbsp;(6391).}} (397 pages) (NB. Per {{DNB-IDN|368893146}} a Russian translation of this work was released in 1986.)</ref>
<ref name="Bochmann_Steinbach_1991">{{cite book |first1=Dieter |last1=Bochmann |author-link1=:de:Dieter Bochmann |first2=Bernd |last2=Steinbach |author-link2=:de:Bernd Steinbach |title=Logikentwurf mit XBOOLE – Algorithmen und Programme |language=de |trans-title=Logic design with XBOOLE – Algorithms and programs |publisher={{ill|Verlag Technik|de}} |location=Berlin, Germany |edition=1st |date=1991 |isbn=3-341-01006-8 |id={{DNB-IDN|911196102}}}} (303 pages + 5.25-inch floppy disk)</ref>
<ref name="Scheuring_Wehlan_1991_DEDS">{{cite journal |title=On the Design of Discrete Event Dynamic Systems by Means of the Boolean Differential Calculus |first1=Rainer |last1=Scheuring |first2=Herbert "Hans"<!-- This source identifies the author as Hans Wehlan --> |last2=Wehlan |date=1991-09-01 |work=First IFAC Symposium on Design Methods of Control Systems |publisher=[[International Federation of Automatic Control]] (IFAC) / [[Pergamon Press]] |location=Zürich, Switzerland |editor-first1=Dieter |editor-last1=Franke |editor-first2=Franta |editor-last2=Kraus |volume=2 |doi=10.1016/S1474-6670(17)54214-7 |pages=723–728}} (6 pages)</ref>
<ref name="Scheuring_Wehlan_1991_Petri">{{cite journal |title=Der Boolesche Differentialkalkül – eine Methode zur Analyse und Synthese von Petri-Netzen |language=de |trans-title=The Boolean differential calculus – A method for analysis and synthesis of Petri nets |first1=Rainer |last1=Scheuring |first2=Herbert "Hans"<!-- This source identifies the author as Herbert Wehlan --> |last2=Wehlan |editor-first=Georg |editor-last=Bretthauer |orig-year=July 1991 |date=1991-12-01 |journal=at – Automatisierungstechnik – Methoden und Anwendungen der Steuerungs-, Regelungs- und Informationstechnik |publisher={{ill|R. Oldenbourg Verlag|de}} |issn=0178-2312 |location=Stuttgart, Germany |volume=39 |number=7 |pages=226–233 |doi=10.1524/auto.1991.39.112.226 |url=https://www.degruyter.com/view/j/auto.1991.39.issue-1-12/auto.1991.39.112.226/auto.1991.39.112.226.xml |access-date=2017-10-16 |url-status=live |archive-url=https://web.archive.org/web/20171016190403/https://www.degruyter.com/view/j/auto.1991.39.issue-1-12/auto.1991.39.112.226/auto.1991.39.112.226.png |archive-date=2017-10-16}} (8 pages)</ref>
<ref name="Yanushkevich_1998">{{cite book |first=Svitlana N. [Svetlana N.] |last=Ânuškevič [Yanushkevich] |title=Logic Differential Calculus in Multi-Valued Logic Design |type=PhD thesis |publisher=Instytut Informatyki, Technical University of Szczecin |journal=Journal Prace Naukowe Politechniki Szczecińskiej |location=Szczecin, Poland |edition=1st |date=1998 |issue=537/1 |issn=1506-3054 |isbn=978-8-387423-16-2 |id={{ISBN|8-387423-16-5}}}} (326<!-- 322? --> pages)</ref>
<ref name="Bochmann_2006_2008">{{cite book |first=Dieter |last=Bochmann |author-link=:de:Dieter Bochmann |title=Binary Systems - A BOOLEAN Book |publisher=TUDpress Verlag der Wissenschaften |location=Dresden, Germany |edition=1st |date=2008-09-01 |isbn=978-3-940046-87-1 |id={{DNB-IDN|989771636}}}} (421 pages) Translation of: {{cite book |first=Dieter |last=Bochmann |author-link=:de:Dieter Bochmann |title=Binäre Systeme - Ein BOOLEAN Buch |language=de |trans-title=Binary systems - A Boolean book |publisher=LiLoLe-Verlag GmbH (Life-Long-Learning) / BoD GmbH |location=Hagen, Germany |edition=1st |date=February 2006 |isbn=3-934447-10-4 |id={{ISBN|978-3-934447-10-3}}. {{DNB-IDN|978899873}}}} (452 pages)</ref>
<ref name="Steinbach_Posthoff_2013_1">{{cite journal |first1=Bernd |last1=Steinbach |author-link1=:de:Bernd Steinbach |first2=Christian |last2=Posthoff |title=Derivative Operations for Lattices of Boolean Functions |journal=Proceedings Reed-Muller Workshop 2013 |location=Toyama, Japan |date=2013 |pages=110–119 |url=http://www.informatik.tu-freiberg.de/prof2/publikationen/RM_2013_dolbf.pdf |access-date=2017-10-21 |url-status=live |archive-url= https://web.archive.org/web/20171021224454/http://www.informatik.tu-freiberg.de/prof2/publikationen/RM_2013_dolbf.pdf |archive-date=2017-10-21}} (10 pages)</ref>
<ref name="Steinbach_Posthoff_2013_2">{{cite book |first1=Bernd |last1=Steinbach |author-link1=:de:Bernd Steinbach |first2=Christian |last2=Posthoff |editor-first=Mitchell A. |editor-last=Thornton |title=Boolean Differential Equations |series=Synthesis Lectures on Digital Circuits and Systems |publisher=Morgan & Claypool Publishers |location=San Rafael, CA, USA |date=2013-07-01 |id=Lecture #42 |edition=1st |isbn=978-1-62705-241-2 |doi=10.2200/S00511ED1V01Y201305DCS042}} (158 pages)</ref>
<ref name="Steinbach_Posthoff_2017">{{cite book |first1=Bernd |last1=Steinbach |author-link1=:de:Bernd Steinbach |first2=Christian |last2=Posthoff |editor-first=Mitchell A. |editor-last=Thornton |title=Boolean Differential Calculus |series=Synthesis Lectures on Digital Circuits and Systems |publisher=Morgan & Claypool Publishers |location=San Rafael, CA, USA |date=2017-06-07 |id=Lecture #52 |edition=1st |isbn=978-1-62705-922-0 |doi=10.2200/S00766ED1V01Y201704DCS052}} (216 pages)</ref>
}}

== Bacaan lebih lanjut ==
* {{cite journal |first1=Marc |last1=Davio |first2=Philippe M. |last2=Piret |title=Les dérivées Booléennes et leur application au diagnostic |language=fr |trans-title=Boolean derivatives and their application and diagnosis |journal=[[Philips Revue]] |publisher=[[Philips Research Laboratory]], {{lang|fr|Manufacture Belge de Lampes et de Materiel Electronique}} (MBLE Research Laboratory) |location=Brussels, Belgium |volume=12 |issue=3  |pages=63–76 |date=July 1969}} (14 pages)
* {{cite book |first=Sergiu |last=Rudeanu |title=Boolean Functions and Equations |date=September 1974 |publisher=[[North-Holland Publishing Company]]/[[American Elsevier Publishing Company]] |isbn=0-44410520-4 |id={{isbn|0-72042082-2}}}} (462 pages)
* {{cite journal |first=Dieter |last=Bochmann |author-link=:de:Dieter Bochmann |title=Boolean differential calculus (a survey) |journal=Engineering Cybernetics |publisher=[[Institute of Electrical and Electronics Engineers]] (IEEE) |volume=15 |number=5 |date=1977 |issn=0013-788X |pages=67–75}} (9 pages) Translation of: {{cite journal |first=Dieter |last=Bochmann |author-link=:de:Dieter Bochmann  |language=ru |title=[Boolean differential calculus (survey)] |journal=Известия Академии наук СССР – Техническая кибернетика (Izvestii︠a︡ Akademii Nauk SSSR – Tekhnicheskai︠a︡ kibernetika) [Proceedings of the Academy of Sciences of the USSR – Engineering Cybernetics] |number=5 |date=1977 |pages=125–133}} (9 pages)
* {{cite journal |first=Martin |last=Kühnrich |title=Differentialoperatoren über Booleschen Algebren |language=de |trans-title=Differential operators on Boolean algebras |journal=Zeitschrift für mathematische Logik und Grundlagen der Mathematik |date=1986 |orig-year=1984-07-31 (submission) |location=Berlin, Germany (East) |doi=10.1002/malq.19860321703 |volume=32 |issue=17-18 |id=#18 |pages=271–288}} (18 pages)
* {{cite book |first=Frank |last=Dresig |title=Gruppierung – Theorie und Anwendung in der Logiksynthese |language=de |trans-title=Grouping – Theory and application in logic synthesis |work=Fortschritt-Berichte VDI |series=9 |volume=145 |publisher={{ill|VDI-Verlag|de}} |location=Düsseldorf, Germany |date=1992 |isbn=3-18-144509-6 |id={{DNB-IDN|940164671}}}} (NB. Also: Chemnitz, Technische Universität, Dissertation.) (147 pages)
* {{cite book |chapter=Control of Discrete Event Systems by Means of the Boolean Differential Calculus |first1=Rainer |last1=Scheuring |first2=Herbert "Hans"<!-- This source identifies the author variously as Hans or Herbert Wehlan --> |last2=Wehlan |date=1993 |editor-first1=Silvano |editor-last1=Balemi |editor-first2=Petr |editor-last2=Kozák |editor-first3=Rein |editor-last3=Smedinga |title=Discrete Event Systems: Modeling and Control |series=Progress in Systems and Control Theory (PSCT) |volume=13 |publisher=[[Birkhäuser Verlag]] |location=Basel, Switzerland |pages=79–93 |doi=10.1007/978-3-0348-9120-2_7}} (15 pages)
* {{cite book |first1=Christian |last1=Posthoff |first2=Bernd |last2=Steinbach |author-link2=:de:Bernd Steinbach |title=Logic Functions and Equations – Binary Models for Computer Science |publisher=[[Springer Science + Business Media B.V.]] |location=Dordrecht, Netherlands |date=2004-02-04 |edition=1st |isbn=1-4020-2937-3 |oclc=254106952 |doi=10.1007/978-1-4020-2938-7 |id={{ISBN|978-1-4020-2937-0}}}} (392 pages)
* {{cite book |first1=Bernd |last1=Steinbach |author-link1=:de:Bernd Steinbach |first2=Christian |last2=Posthoff |title=Logic Functions and Equations – Examples and Exercises |url=https://archive.org/details/logicfunctionseq0000stei |publisher=[[Springer Science + Business Media B.V.]] |location=Dordrecht, Netherlands |date=2009-02-12 |edition=1st |isbn=978-1-4020-9594-8 |lccn=2008941076 |doi=10.1007/978-1-4020-9595-5}} (xxii+232 pages) [http://www.e-reading.club/bookreader.php/135805/Posthoff%2C_Steinbach_-_Logic_Functions_and_Equations_-_Examples_and_Exercises.pdf] (NB. Per {{DNB-IDN|1010457748}} this hardcover edition has been rereleased as softcover edition in 2010.)
* {{cite journal |first1=Bernd |last1=Steinbach |author-link1=:de:Bernd Steinbach |first2=Christian |last2=Posthoff |title=Boolean Differential Calculus – Theory and Applications |journal=Journal of Computational and Theoretical Nanoscience |publisher=American Scientific Publishers |volume=7 |issue=6 |date=2010-06-01 |pages=933–981 |issn=1546-1955 |doi=10.1166/jctn.2010.1441 |url=http://www.ingentaconnect.com/content/asp/jctn/2010/00000007/00000006/art00001}} (49 pages)
* {{cite book |first1=Bernd |last1=Steinbach |author-link1=:de:Bernd Steinbach |first2=Christian |last2=Posthoff |chapter=Chapter 3: Boolean Differential Calculus |editor-first1=Tsutomu |editor-last1=Sasao |editor-first2=Jon T. |editor-last2=Butler |title=Progress in Applications of Boolean Functions |url=https://archive.org/details/progressapplicat00sasa |url-access=limited |series=Synthesis Lectures on Digital Circuits and Systems |location=San Rafael, CA, USA |publisher=Morgan & Claypool Publishers |orig-year=2009 |date=2010-01-15 |id=Lecture #26 |edition=1st |isbn=978-1-60845-181-4 |pages=[https://archive.org/details/progressapplicat00sasa/page/n69 55]–78, 121–126 |doi=10.2200/S00243ED1V01Y200912DCS026}} (24 of 153 pages)

=={{anchor|XBOOLE}}Pranala luar==
* {{cite book |title=Boolean differential calculus |author-first=Herbert "Hans" |author-last=Wehlan |work=[[Encyclopedia of Mathematics]] |isbn=978-1-4020-0609-8 |editor-first=Michiel |editor-last=Hazewinkel |editor-link=Michiel Hazewinkel
|publisher=[[Springer Science+Business Media]] |date=2010-12-06 |url=http://www.encyclopediaofmath.org/index.php?title=Boolean_differential_calculus&oldid=12153 |access-date=2017-10-16 |url-status=live |archive-url=https://web.archive.org/web/20171016111554/https://www.encyclopediaofmath.org/index.php/Boolean_differential_calculus |archive-date=2017-10-16}}
* {{cite web |title=XBOOLE |author=Institut für Informatik (IfI) |publisher=[[TU Bergakademie Freiberg]] |date=2017 |url=http://www.informatik.tu-freiberg.de/xboole/index.php?language=English&aktwindow=home |access-date=2017-10-31 |url-status=live |archive-url=https://web.archive.org/web/20171031105416/http://www.informatik.tu-freiberg.de/xboole/index.php?language=English&aktwindow=home |archive-date=2017-10-31}} with {{cite web |title=XBOOLE Monitor |date=2008-07-23 |url=http://www.informatik.tu-freiberg.de/xboole/XBOOLEMonitor.zip?Submit=XBOOLE+Monitor |access-date=2017-10-31 |url-status=dead |archive-url=https://web.archive.org/web/20171031002108/http://www.informatik.tu-freiberg.de/xboole/XBOOLEMonitor.zip?Submit=XBOOLE+Monitor |archive-date=2017-10-31}}

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