Circuit complexity: Difference between revisions

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===Key results===

*The boolean function PARITY, which is 1 if and only if the sum of its input bits is odd, does not lie in <math>AC^0</math>. The result was first established independently by Ajtai (1983) and by Furst, Saxe and Sipser (1984). Later improvements by Hastad (1987) in fact establish that any family of <math>AC^0</math> computing PARITY requires exponential size. Smolensky (1986) 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 monotoneConsider the [[clique problem]] of determining whether a graph with ''n'' vertices has a clique of size ''k''. For any particular choice of ''n'' and ''k'' this can be encoded as a boolean function CLIQUEon ''n''(''n''&minus;1) inputs, one for each pair of vertices, specifying whether or not that edge is in the graph. 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 no negation). This original result of [[Alexander Razborov|Razborov]] (1985) was later improved to an exponential-size lower bound by Alon and Boppana (1987). Raz and McKenzie later showed that the monotone NC hierarchy is infinite (1999).

*Division lies in uniform [[TC0]] (Hesse 2001).