Abstract
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 HÃ¥stad [10] that no circuit of depth d and size \( 2^{n^{1/d} } \) 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 derandomization: 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 BPAC0 can be simulated by uniform quasipolynomial size constant depth circuits, improving on results due to Nisan [18] and Nisan and Wigderson [19].
Supported in part by NSF grant CCR-97-01304.
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Klivans, A.R. (2001). On the Derandomization of Constant Depth Circuits. In: Goemans, M., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques. RANDOM APPROX 2001 2001. Lecture Notes in Computer Science, vol 2129. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44666-4_28
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