Switching circuit theory is the mathematical study of the properties of networks of idealized switches.

Such networks may be strictly combinational logic

Combinational logic

In digital circuit theory, combinational logic is a type of digital logic which is implemented by boolean circuits, where the output is a pure function of the present input only. This is in contrast to sequential logic, in which the output depends not only on the present input but also on the...

, in which their output state is only a function of the present state of their inputs; or may also contain sequential elements

Sequential logic

In digital circuit theory, sequential logic is a type of logic circuit whose output depends not only on the present input but also on the history of the input. This is in contrast to combinational logic, whose output is a function of, and only of, the present input...

, 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.

In the paper A Symbolic Analysis of Relay and Switching Circuits

A Symbolic Analysis of Relay and Switching Circuits

A Symbolic Analysis of Relay and Switching Circuits is the title of a master's thesis written by computer science pioneer Claude E. Shannon while attending the Massachusetts Institute of Technology in 1937...

 of 1938, Claude Shannon showed that the two-valued Boolean algebra can describe the operation of switching circuits. The principles of Boolean algebra

Boolean algebra

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets...

 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"

Hazard (logic)

In digital logic, a hazard in a system is an undesirable effect caused by either a deficiency in the system or external influences. Logic hazards are manifestations of a problem in which changes in the input variables do not change the output correctly due to some form of delay caused by logic...

  or "race condition

Race condition

A race condition or race hazard is a flaw in an electronic system or process whereby the output or result of the process is unexpectedly and critically dependent on the sequence or timing of other events...

" where the output state changes due to the different propagation times through the network.

See also

• Karnaugh map
  Karnaugh map
  The Karnaugh map , Maurice Karnaugh's 1953 refinement of Edward Veitch's 1952 Veitch diagram, is a method to simplify Boolean algebra expressions...
  
  
• Boolean circuit
  Boolean circuit
  A Boolean circuit is a mathematical model of computation used in studying computational complexity theory. Boolean circuits are the main object of study in circuit complexity and are a special kind of circuits; a formal language can be decided by a family of Boolean circuits, one circuit for each...
  
  
• C-element
  C-element
  The Muller C-element, or Muller C-gate, is a commonly used asynchronous logic component originally designed by David E. Muller. It applies logical operations on the inputs and has hysteresis. The output of the C-element reflects the inputs when the states of all inputs match. The output then...
  
  
• Circuit minimization
  Circuit minimization
  In Boolean algebra, circuit minimization is the problem of obtaining the smallest logic circuit that represents a given Boolean function or truth table...
  
  
• Circuit complexity
  Circuit complexity
  In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according to the size or depth of Boolean circuits that compute them....
  
  
• Circuit switching
  Circuit switching
  Circuit switching is a methodology of implementing a telecommunications network in which two network nodes establish a dedicated communications channel through the network before the nodes may communicate. The circuit guarantees the full bandwidth of the channel and remains connected for the...
  
  
• Logic design
• Logic in computer science
• Logic gate
  Logic gate
  A logic gate is an idealized or physical device implementing a Boolean function, that is, it performs a logical operation on one or more logic inputs and produces a single logic output. Depending on the context, the term may refer to an ideal logic gate, one that has for instance zero rise time and...
  
  
• Nonblocking minimal spanning switch
  Nonblocking minimal spanning switch
  A nonblocking minimal spanning switch is a device that can connect N inputs to N outputs in any combination. The most familiar use of switches of this type is in a telephone exchange. The term "non-blocking" means that if it is not defective, it can always make the connection...
  
  
• Relay
  Relay
  A relay is an electrically operated switch. Many relays use an electromagnet to operate a switching mechanism mechanically, but other operating principles are also used. Relays are used where it is necessary to control a circuit by a low-power signal , or where several circuits must be controlled...
  
   - the kind of logic device Shannon was concerned with in 1938
• Switching lemma
  Switching lemma
  In computational complexity theory, Håstad's switching lemma is a key tool for proving lower bounds on the size of constant-depth Boolean circuits....
  
  
• Unate function
  Unate function
  A unate function is a type of boolean function which has monotonic properties.They have been studied extensively in switching theory.A function f is said to be positive unate in x_i...