Reminder to all that problem set 1 is due today. And please sign up for scribing and get on the class mailing list. Note that we meet on next Tuesday instead of Monday. 2 Overview Today we show that PARITY is not in AC0. AC0 is a family of circuits with constant depth, polynomial size, and unbounded fan-in for the AND and OR gates. We establish this result through an application of the Switching Lemma. This result is the first use of randomization in its full power in complexity. Circuits were defined in previous lectures. In this lecture, we always assume that the circuits are organized into alternating levels of AND and OR gates. We can make such an assumption since we can convert circuits into this convenient form with only a constant factor of blowup. The Switching Lemma is first proved by Furst, Saxe, and Sipser in FOCS 81, and readers can find the paper in the Journal of Mathematical Systems Theory 1984. We will highlight their work by using the Lemma though the version we prove today will not be as strong as we claimed in the last lecture. Johan H˚astad, in 1986, proved a more general and powerful form of the Lemma, and interested readers can find it in his PhD thesis at MIT. There is also a surve
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