This article includes a list of general references, but it remains largely unverified because it lacks sufficient corresponding inline citations. (November 2014) |

**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NAND gates alone) can be used to reproduce the functions of all the other logic gates, but this work remained unpublished until 1933.^{[3]} The first published proof was by Henry M. Sheffer in 1913, so the NAND logical operation is sometimes called Sheffer stroke; the logical NOR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]}

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).

From 1934 to 1936, NEC engineer Akira Nakashima, Claude Shannon and Viktor Shetakov published a series of papers showing that the two-valued Boolean algebra, which they discovered independently, can describe the operation of switching circuits.^{[6]}^{[7]}^{[8]}^{[1]}

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.

## See also

- Boolean circuit
- C-element
- Circuit complexity
- Circuit minimization
- Circuit switching
- Karnaugh map
- Logic design
- Logic gate
- Logic in computer science
- Nonblocking minimal spanning switch
- Programmable logic controller – computer software mimics relay circuits for industrial applications
- Quine–McCluskey algorithm
- Relay – an early kind of logic device
- Switching lemma
- Unate function

## References

- ^
^{a}^{b}Stanković, Radomir S.; Astola, Jaakko T. (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. Tampere University of Technology. #40. **^**Peirce, Charles Saunders (1993) [1886].*Letter, Peirce to A. Marquand*.*Writings of Charles S. Peirce*.**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].**^**Peirce, Charles Saunders (1933) [Winter of 1880–1881].*A Boolian Algebra with One Constant*.*Collected Papers*(manuscript).**4**. paragraphs 12–20. Reprinted in*Writings of Charles S. Peirce*.**4**(reprint ed.). 1989. pp. 218–221. ark:/13960/t11p5r61f. See also: Roberts, Don D. (2009).*The Existential Graphs of Charles S. Peirce*. p. 131.**^**Büning, Hans Kleine; Lettmann, Theodor (1999).*Propositional logic: deduction and algorithms*. Cambridge University Press. p. 2. ISBN 978-0-521-63017-7.**^**Bird, John (2007).*Engineering mathematics*. Newnes. p. 532. ISBN 978-0-7506-8555-9.**^**Yamada, Akihiko (2004). "History of Research on Switching Theory in Japan".*IEEJ Transactions on Fundamentals and Materials*. Institute of Electrical Engineers of Japan.**124**(8): 720–726. doi:10.1541/ieejfms.124.720.**^**"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.**^**Stanković, Radomir S.; Astola, Jaakko T.; Karpovsky, Mark G. (2007).*Some Historical Remarks on Switching Theory*. CiteSeerX 10.1.1.66.1248. S2CID 10029339.

## Further reading

- Keister, William; Ritchie, Alistair E.; Washburn, Seth H. (1951).
*The Design Of Switching Circuits*. The Bell Telephone Laboratories Series (1 ed.). D. Van Nostrand Company, Inc. p. 147. Archived from the original on 2020-05-09. Retrieved 2020-05-09. [1] (2+xx+556+2 pages) - Caldwell, Samuel Hawks (1958-12-01) [February 1958]. Written at Watertown, Massachusetts, USA.
*Switching Circuits and Logical Design*. 5th printing September 1963 (1st ed.). New York, USA: John Wiley & Sons Inc. ISBN 0-47112969-0. LCCN 58-7896. (xviii+686 pages) - Shannon, Claude Elwood (1938). "A Symbolic Analysis of Relay and Switching Circuits".
*Trans. AIEE*.**57**(12): 713–723. doi:10.1109/T-AIEE.1938.5057767. hdl:1721.1/11173. S2CID 51638483. - 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. CiteSeerX 10.1.1.64.1129. Archived (PDF) from the original on 2021-03-28. Retrieved 2021-03-28. (188 pages) - Stanković, Radomir S.; Sasao, Tsutomu; Astola, Jaakko T. (August 2001). "Publications in the First Twenty Years of Switching Theory and Logic Design" (PDF). Tampere International Center for Signal Processing (TICSP) Series. Tampere University of Technology / TTKK, Monistamo, Finland. ISSN 1456-2774. S2CID 62319288. #14. Archived (PDF) from the original on 2017-08-09. Retrieved 2021-03-28. (4+60 pages)