Randomized polynomials, threshold circuits, and the polynomial hierarchy

Abstract

A randomized polynomial over the integers is a multivariate polynomial whose coefficients are integer-valued random variables. A randomized polynomial uses m random bits if its coefficients jointly depend on m independently and uniformly distributed random bits. We show that every Boolean function family in AC ⁰ can be computed with small error by randomized polynomials over the integers that have degree (log n)^O(1) and use (log n)^O(1) random bits. Applying this result, we further show the following: (a) Every Boolean function family in AC ⁰ is computable by depth-two probabilistic threshold circuits of size $n^{(\log n)^{O(1)} }$ with one-sided error and is also computable by depth-three deterministic threshold circuits with linear number of threshold gates and $n^{(\log n)^{O(1)} }$AND gates. (b) Every language in the polynomial hierarchy is reducible to some language in the class PP (in fact, to some language in the class C=P) by a randomized polynomial-time reduction with one-sided error.

This work was supported in part by NSF grants CCR 832-0136 and CDA 882-2724