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. One speaks of the circuit complexity of a Boolean circuit. A related notion is the circuit complexity of a recursive language that is decided by a family of circuits (see below).
A Boolean circuit with input bits is a directed acyclic graph in which every node (usually called gates in this context) is either an input node of in-degree 0 labeled by one of the input bits, an AND gate, an OR or a NOT gate. One of these gates is designated as the output gate. Such a circuit naturally computes a function of its inputs. The size of a circuit is the number of gates it contains and its depth is the maximal length of a path from an input gate to the output gate.
There are two major notions of circuit complexity (these are outlined in Sipser (1997)). The circuit-size complexity of a Boolean function is the minimal size of any circuit computing . The circuit-depth complexity of a Boolean function is the minimal depth of any circuit computing .
These notions generalize when one considers the circuit complexity of a recursive language: A formal language may contain strings with many different bit lengths. Boolean circuits, however, only allow a fixed number of input bits. Thus no single Boolean circuit is capable of deciding such a language. To account for this possibility, one considers families of circuits where each accepts inputs of size . Each circuit family will naturally generate a recursive language by outputting when a string is a member of the family, and otherwise. We say that a family of circuits is size minimal if there is no other family that decides on inputs of any size, with a circuit of smaller size than (respectively for depth minimal families).
Hence, the circuit-size complexity of a recursive language is defined as the function, that relates a bit length of an input, to the circuit-size complexity of the minimal circuit that decides whether inputs of that length are in . The circuit-depth complexity is defined similarly.
Complexity classes defined in terms of Boolean circuits include AC0, AC, TC0 and NC.
Read more about Circuit Complexity: Uniformity, History, Circuit Lower Bounds, Complexity Classes, Relation To Time Complexity
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