On the Hardness against Constant-Depth Linear-Size Circuits

The notion of average-case hardness is a fundamental one in complexity theory. In particular, it plays an important role in the research on derandomization, as there are general derandomization results which are based on the assumption that average-case hard functions exist. However, to achieve a complete derandomization, one usually needs a function which is extremely hard against a complexity class, in the sense that any algorithm in the class fails to compute the function on 1/2 -2-Ω(n) fraction of its n-bit inputs. Unfortunately, lower bound results are very rare and they are only known for very restricted complexity classes, and achieving such extreme hardness seems even more difficult. Motivated by this, we study the hardness against linear-size circuits of constant depth in this paper. We show that the parity function is extremely hard for them: any such circuit must fail to compute the parity function on at least 1/2 - 2-Ω(n) fraction of inputs.

[1]  J. Håstad Computational limitations of small-depth circuits , 1987 .

[2]  Avi Wigderson,et al.  P = BPP if E requires exponential circuits: derandomizing the XOR lemma , 1997, STOC '97.

[3]  Johan Håstad,et al.  Almost optimal lower bounds for small depth circuits , 1986, STOC '86.

[4]  Lance Fortnow,et al.  BPP has subexponential time simulations unlessEXPTIME has publishable proofs , 2005, computational complexity.

[5]  Christopher Umans,et al.  Simple extractors for all min-entropies and a new pseudo-random generator , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[6]  Peter Bro Miltersen,et al.  On converting CNF to DNF , 2005, Theor. Comput. Sci..

[7]  Luca Trevisan,et al.  Pseudorandom generators without the XOR lemma , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[8]  Christopher Umans Pseudo-random generators for all hardnesses , 2002, STOC '02.

[9]  Noam Nisan,et al.  Hardness vs Randomness , 1994, J. Comput. Syst. Sci..