Boolean circuit

In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for digital logic circuits. A formal language can be decided by a family of Boolean circuits, one circuit for each possible input length. Boolean circuits are also used as a formal model for combinational logic in digital electronics.

Boolean circuits are defined in terms of the logic gates they contain. For example, a circuit might contain binary AND and OR gates and unary NOT gates, or be entirely described by binary NAND gates. Each gate corresponds to some Boolean function that takes a fixed number of bits as input and outputs a single bit.

Boolean circuits provide a model for many digital components used in computer engineering, including multiplexers, adders, and arithmetic logic units.

In giving a formal definition of Boolean circuits, Vollmer starts by defining a basis set B of Boolean functions, corresponding to the gates allowable in the circuit model. A Boolean circuit over a basis B, with n inputs and m outputs, is then defined as a finite directed acyclic graph. Each vertex corresponds to either a basis function or one of the inputs, and there is a set of exactly m nodes which are labeled as the outputs. The edges must also have some ordering, to distinguish between different arguments to the same Boolean function.

As a special case, a propositional formula or Boolean expression is a Boolean circuit with a single output node in which every other node has fan-out of 1. Thus, a Boolean circuit can be regarded as a generalization that allows shared subformulas and multiple outputs.

...
Wikipedia