On the derandomization of constant depth circuits

Nisan [18] and Nisan and Wigderson [19] have constructed a pseudo-random generator which fools any family of polynomial-size constant depth circuits. At the core of their construction is the result due to Hastad [10] that no circuit of depth d and size 2(n1/d) can even weakly approximate (to within an inverse exponential factor) the parity function. We give a simpler proof of the inapproximability of parity by constant depth circuits which does not use the Hastad Switching Lemma. Our proof uses a well-known hardness amplification technique from de-randomization: the XOR lemma. This appears to be the first use of the XOR lemma to prove an unconditional inapproximability result for an explicit function (in this case parity). In addition, we prove that BPAC(0) can be simulated by uniform quasipolynomial size constant depth circuits, improving on results due to Nisan [18] and Nisan and Wigderson [19].