# Switching circuit theory

**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]

From 1934 to 1936, NEC engineer Akira Nakashima published a series of papers showing that the two-valued Boolean algebra, which he discovered independently, can describe the operation of switching circuits.[2][3][4][1] His work was later cited and elaborated on in Claude Shannon's seminal 1938 paper "A Symbolic Analysis of Relay and Switching Circuits".[4] The principles of Boolean algebra are applied to switches, providing mathematical tools for analysis and synthesis of any switching system.

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

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

## Notes

- Radomir S. Stanković, Jaakko Astola (2008), Reprints from the Early Days of Information Sciences: TICSP Series On the Contributions of Akira Nakashima to Switching Theory, TICSP Series #40, Tampere International Center for Signal Processing, Tampere University of Technology
- History of Research on Switching Theory in Japan,
*IEEJ Transactions on Fundamentals and Materials*, Vol. 124 (2004) No. 8, pp. 720-726, Institute of Electrical Engineers of Japan - Switching Theory/Relay Circuit Network Theory/Theory of Logical Mathematics, IPSJ Computer Museum, Information Processing Society of Japan
- Radomir S. Stanković (University of Niš), Jaakko T. Astola (Tampere University of Technology), Mark G. Karpovsky (Boston University), Some Historical Remarks on Switching Theory, 2007, DOI 10.1.1.66.1248

## References

- Keister, William; Ritchie, Alistair E.; Washburn, Seth H. (1963) [1951].
*The Design of Switching Circuits*. The Bell Telephone Laboratories Series. Princeton, NJ: D. Van Nostrand Company. - Caldwell, Samuel H. (1965) [1958].
*Switching Circuits and Logical Design*. New York: John Wiley & Sons. - Shannon, C. E. (1938). "A Symbolic Analysis of Relay and Switching Circuits".
*Trans. AIEE*.**57**(12): 713–723. doi:10.1109/T-AIEE.1938.5057767.