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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 the Boolean circuits that compute them. A related notion is the circuit complexity of a recursive language that is decided by a uniform family of circuits (see below). Proving lower bounds on size of Boolean circuits computing explicit Boolean functions is a popular approach to separating complexity classes. For example, a prominent circuit class P/poly consists of Boolean functions computable by circuits of polynomial size. Proving that would separate P and NP (see below). Complexity classes defined in terms of Boolean circuits include AC0, AC, TC0, NC1, NC, and P/poly. 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 labelled by one of the input bits, an AND gate, an OR gate, 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 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 any language that contains strings with different bit lengths, especially infinite formal languages. 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 the language by circuit outputting when a length string is a member of the family, and otherwise.

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