On the Derandomization of Constant Depth Circuits

• Authors: Adam Klivans
• Venue: APPROX '01/RANDOM '01 Proceedings of the 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and 5th International Workshop on Randomization and Approximation Techniques in Computer Science: Approximation, Randomization and Combinatorial Optimization
• Year: 2001

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 Håstad [10] that no circuit of depth d and size 2n1/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 Håstad 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].