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NAND gates alone) can be used to reproduce the functions of all the other logic gates, but this work remained unpublished until 1933.[3] The first published proof was by Henry M. Sheffer in 1913, so the NAND logical operation is sometimes called Sheffer stroke; the logical NOR is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]
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] Shannon and Shestakov defended their theses the same year in 1938.[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.
^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. ISBN9780253372017. ark:/13960/t11p5r61f. See also: Roberts, Don D. (2009). The Existential Graphs of Charles S. Peirce. p. 131.
^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].)
^ abKlir, 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. ISBN0-442-24463-0. LCCN72-181095. 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)
^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.
Perkowski, Marek A.; Grygiel, Stanislaw (1995-11-20). "6. Historical Overview of the Research on Decomposition". A Survey of Literature on Function Decomposition(PDF). Version IV. Functional Decomposition Group, Department of Electrical Engineering, Portland University, Portland, Oregon, USA. CiteSeerX10.1.1.64.1129. Archived(PDF) from the original on 2021-03-28. Retrieved 2021-03-28. (188 pages)
