In theoretical computer science, a circuit is a model of computation in which input values proceed through a sequence of gates, each of which computes a function. Circuits of this kind provide a generalization of Boolean circuits and a mathematical model for digital logic circuits. Circuits are defined by the gates they contain and the values the gates can produce. For example, the values in a Boolean circuit are boolean values, and the circuit includes conjunction, disjunction, and negation gates. The values in an integer circuit are sets of integers and the gates compute set union, set intersection, and set complement, as well as the arithmetic operations addition and multiplication.
A circuit is a triple , where
The vertices of the graph are called gates. For each gate of in-degree , the gate can be labeled by an element of if and only if is defined on .
The gates of in-degree 0 are called inputs or leaves. The gates of out-degree 0 are called outputs. If there is an edge from gate to gate in the graph then is called a child of . We suppose there is an order on the vertices of the graph, so we can speak of the th child of a gate when is less than the out-degree of the gate.
The size of a circuit is the number of nodes of a circuit. The depth of a gate is the length of the longest path in beginning at up to an output gate. In particular, the gates of out-degree 0 are the only gates of depth 1. The depth of a circuit is the maximum depth of any gate.
Level is the set of all gates of depth . A levelled circuit is a circuit in which the edges to gates of depth comes only from gates of depth or from the inputs. In other words, edges only exist between adjacent levels of the circuit. The width of a levelled circuit is the maximum size of any level.
The exact value of a gate with in-degree and label is defined recursively for all gates .
where each is a parent of .
The value of the circuit is the value of each of the output gates.
The labels of the leaves can also be variables which take values in . If there are leaves, then the circuit can be seen as a function from to . It is then usual to consider a family of circuits , a sequence of circuits indexed by the integers where the circuit has variables. Families of circuits can thus be seen as functions from to .
The notions of size, depth and width can be naturally extended to families of functions, becoming functions from to ; for example, is the size of the th circuit of the family.
Computing the output of a given Boolean circuit on a specific input is P-complete problem. If the input is an integer circuit, however, it is unknown whether this problem is decidable.
Circuit complexity attempts to classify Boolean functions with respect to the size or depth of circuits that can compute them.