Switching circuit theory is the mathematical study of the properties of networks of idealized switches. Such networks may be strictly combinational logic, in which their output state is only a function of the present state of their inputs; or may also contain sequential elements, where the present state depends on the present state and past states; in that sense, sequential circuits are said to include "memory" of past states. An important class of sequential circuits are state machines. Switching circuit theory is applicable to the design of telephone systems, computers, and similar systems. Switching circuit theory provided the mathematical foundations and tools for digital system design in almost all areas of modern technology.[1]

In an 1886 letter, Charles Sanders Peirce described how logical operations could be carried out by electrical switching circuits.[2] During 1880–1881 he showed that NOR gates alone (or alternatively NAND gates alone) can be used to reproduce the functions of all the other logic gates, but this work remained unpublished until 1933.[3] The first published proof was by Henry M. Sheffer in 1913, so the NAND logical operation is sometimes called Sheffer stroke; the logical NOR is sometimes called Peirce's arrow.[4] Consequently, these gates are sometimes called universal logic gates.[5]

In 1898, Martin Boda described a switching theory for signalling block systems.[6][7]

Eventually, vacuum tubes replaced relays for logic operations. Lee De Forest's modification, in 1907, of the Fleming valve can be used as a logic gate. Ludwig Wittgenstein introduced a version of the 16-row truth table as proposition 5.101 of Tractatus Logico-Philosophicus (1921). Walther Bothe, inventor of the coincidence circuit, got part of the 1954 Nobel Prize in physics, for the first modern electronic AND gate in 1924. Konrad Zuse designed and built electromechanical logic gates for his computer Z1 (from 1935 to 1938).

The theory was independently established through the works of NEC engineer Akira Nakashima in Japan,[8] Claude Shannon in the United States,[9] and Victor Shestakov in the Soviet Union.[10] The three published a series of papers showing that the two-valued Boolean algebra, can describe the operation of switching circuits.[7][11][12][13][1] However, Shannon's work has largely overshadowed the other two, and despite some scholars arguing the similarities of Nakashima's work to Shannon's, their approaches and theoretical frameworks were markedly different.[14] Also implausible is that Shestakov's influenced the other two due to the language barriers and the relative obscurity of his work abroad.[14] Furthermore, Shannon and Shestakov defended their theses the same year in 1938,[15] and Shestakov did not publish until 1941.[15]

Ideal switches are considered as having only two exclusive states, for example, open or closed. In some analysis, the state of a switch can be considered to have no influence on the output of the system and is designated as a "don't care" state. In complex networks it is necessary to also account for the finite switching time of physical switches; where two or more different paths in a network may affect the output, these delays may result in a "logic hazard" or "race condition" where the output state changes due to the different propagation times through the network.

## References

1. ^ a b Stanković, Radomir S. [in German]; Astola, Jaakko Tapio [in Finnish], eds. (2008). Reprints from the Early Days of Information Sciences: TICSP Series on the Contributions of Akira Nakashima to Switching Theory (PDF). Tampere International Center for Signal Processing (TICSP) Series. Vol. 40. Tampere University of Technology, Tampere, Finland. ISBN 978-952-15-1980-2. ISSN 1456-2774. Archived from the original (PDF) on 2021-03-08.`((cite book))`: CS1 maint: location missing publisher (link) (3+207+1 pages) 10:00 min
2. ^ Peirce, Charles Sanders (1993) [1886]. "Letter, Peirce to A. Marquand". Writings of Charles S. Peirce. Vol. 5. pp. 421–423. See also: Burks, Arthur Walter (1978). "Review: Charles S. Peirce, The new elements of mathematics". Bulletin of the American Mathematical Society (review). 84 (5): 913–918 [917]. doi:10.1090/S0002-9904-1978-14533-9.
3. ^ Peirce, Charles Sanders (1933) [Winter of 1880–1881]. "A Boolian Algebra with One Constant". Collected Papers (manuscript). Vol. 4. paragraphs 12–20. Reprinted in Writings of Charles S. Peirce. Vol. 4 (reprint ed.). 1989. pp. 218–221. ISBN 9780253372017. ark:/13960/t11p5r61f. See also: Roberts, Don D. (2009). The Existential Graphs of Charles S. Peirce. p. 131.
4. ^ Kleine Büning, Hans; Lettmann, Theodor (1999). Propositional logic: deduction and algorithms. Cambridge University Press. p. 2. ISBN 978-0-521-63017-7.
5. ^ Bird, John (2007). Engineering mathematics. Newnes. p. 532. ISBN 978-0-7506-8555-9.
6. ^ Boda, Martin (1898). "Die Schaltungstheorie der Blockwerke" [The switching theory of block systems]. Organ für die Fortschritte des Eisenbahnwesens in technischer Beziehung – Fachblatt des Vereins deutscher Eisenbahn-Verwaltungen (in German). Neue Folge XXXV (1–7). Wiesbaden, Germany: C. W. Kreidel's Verlag: 1–7, 29–34, 49–53, 71–75, 91–95, 111–115, 133–138. [1][2][3][4][5][6][7] (NB. This series of seven articles was republished in a 91-pages book in 1899 with a foreword by Georg Barkhausen [de].)
7. ^ a b Klir, George Jiří (May 1972). "Reference Notations to Chapter 1". Introduction to the Methodology of Switching Circuits (1 ed.). Binghamton, New York, USA: Litton Educational Publishing, Inc. / D. van Nostrand Company. p. 19. ISBN 0-442-24463-0. LCCN 72-181095. C4463-000-3. p. 19: Although the possibility of establishing a switching theory was recognized by M. Boda[A] as early as in the 19th century, the first important works on this subject were published by A. Nakashima[B] and C. E. Shannon[C] shortly before World War II. (xvi+573+1 pages)
8. ^ Nakashima, Akira (May 1936). "Theory of Relay Circuit Composition". Nippon Electrical Communication Engineering (3): 197–226. (NB. Translation of an article which originally appeared in Japanese in the Journal of the Institute of Telegraph and Telephone Engineers of Japan (JITTEJ) September 1935, 150 731–752.)
9. ^ Shannon, Claude Elwood (1938). "A Symbolic Analysis of Relay and Switching Circuits". Transactions of the American Institute of Electrical Engineers. 57 (12). American Institute of Electrical Engineers (AIEE): 713–723. doi:10.1109/T-AIEE.1938.5057767. hdl:1721.1/11173. S2CID 51638483. (NB. Based on Shannon's master thesis of the same title at Massachusetts Institute of Technology in 1937.)
10. ^ Shestakov [Шестаков], Victor Ivanovich [Виктор Иванович] (1938). Некоторые математические методы кон-струирования и упрощения двухполюсных электрических схем класса А [Some mathematical methods for the construction and simplification of two-terminal electrical networks of class A] (PhD thesis) (in Russian). Lomonosov State University.
11. ^ Yamada [山田], Akihiko [彰彦] (2004). "History of Research on Switching Theory in Japan". IEEJ Transactions on Fundamentals and Materials. 124 (8). Institute of Electrical Engineers of Japan: 720–726. Bibcode:2004IJTFM.124..720Y. doi:10.1541/ieejfms.124.720. Archived from the original on 2022-07-10. Retrieved 2022-10-26.
12. ^ "Switching Theory/Relay Circuit Network Theory/Theory of Logical Mathematics". IPSJ Computer Museum. Information Processing Society of Japan. 2012. Archived from the original on 2021-03-22. Retrieved 2021-03-28.
13. ^ Stanković, Radomir S. [in German]; Astola, Jaakko Tapio [in Finnish]; Karpovsky, Mark G. (2007). Some Historical Remarks on Switching Theory (PDF). Niš, Serbia; Tampere, Finland; Boston, Massachusetts, USA. CiteSeerX 10.1.1.66.1248. S2CID 10029339. Archived (PDF) from the original on 2022-10-25. Retrieved 2022-10-25.`((cite book))`: CS1 maint: location missing publisher (link) (8 pages)
14. ^ a b Kawanishi, Toma (2019). "Prehistory of Switching Theory in Japan: Akira Nakashima and His Relay-circuit Theory". Historia Scientiarum. Second Series: International Journal of the History of Science Society of Japan. 29 (1): 136–162. doi:10.34336/historiascientiarum.29.1_136.
15. ^ a b Moisil, GR. C. (1969). The Algebraic Theory of Switching Circuits. Pergamon Press. pp. 12, 17. ISBN 9781483160764.