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Circuit complexity goes back to [[Claude Shannon|Shannon]] (1949), who proved that almost all Boolean functions on ''n'' variables require circuits of size Θ(2<sup>''n''</sup>/''n''). Despite this fact, complexity theorists were unable to prove circuit lower bounds for specific Boolean functions.
The first function for which superpolynomial circuit lower bounds could be shown was the [[parity function]], which computes the sum of its input bits modulo 2. The fact that parity is not contained in [[AC0|AC<sup>0</sup>]] was first established independently by Ajtai (1983) and by Furst, Saxe and Sipser (1984). Later improvements by [[Johan Håstad|Håstad]] (1987) in fact establish that any family of constant-depth circuits computing the parity function requires exponential size. Smolensky (1987) proved that this is true even if the circuit is augmented with gates computing the sum of its input bits modulo some odd prime p.
The [[clique problem|''k''-clique problem]] is to decide whether a given graph on ''n'' vertices has a clique of size ''k''. For any particular choice of the constants ''n'' and ''k'', the graph can be encoded in binary using <math>{n \choose 2}</math> bits which indicate for each possible edge whether it is present. Then the ''k''-clique problem is formalized as a function <math>f_k:\{0,1\}^{{n \choose 2}}\to\{0,1\}</math> such that <math>f_k</math> outputs ''1'' if and only if the graph encoded by the string contains a clique of size ''k''. This family of functions is monotone and can be computed by a family of circuits, but it has been shown that it cannot be computed by a polynomial-size family of monotone circuits (that is, circuits with AND and OR gates but without negation). The original result of [[Alexander Razborov|Razborov]] (1985) was later improved to an exponential-size lower bound by Alon and Boppana (1987). Rossman (2008) shows that constant-depth circuits with AND, OR, and NOT gates require size <math>\Omega(n^{k/4})</math> to solve the ''k''-clique problem even in the [[average-case complexity|average case]]. Moreover, there is a circuit of size <math>n^{k/4+O(1)}</math> which computes <math>f_k</math>.
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