Hardness-randomness tradeoffs for bounded depth arithmetic circuits

In this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits. More formally, if there exists an explicit polynomial f(x₁,...., x_m) that cannot be computed by a depth d arithmetic circuit of small size then there exists an efficient deterministic algorithm to test whether a given depth d - 8 circuit is identically zero or not (assuming the individual degrees of the tested circuit are not too high). In particular, if we are guaranteed that the circuit computes a multilinear polynomial then we can perform the identity test efficiently. To the best of our knowledge this is the first hardness-randomness tradeoff for bounded depth arithmetic circuits. The above results are obtained using the arithmetic Nisan-Wigderson generator of Impagliazzo and Kabanets together with a new theorem on bounded depth circuits, which is the main technical contribution of our work. This theorem deals with polynomial equations of the form P(x ₁,..., x_n, y) =≡ 0 and shows that if P has a circuit of depth d and size s and if the polynomial f(x₁,..., x_n) satisfies P(x₁,..., x_n, f(x₁,..., x _n)) = 0≡ then f has a circuit of depth d + 3 and size O(s r + m^r), where m is the degree of f and r is the highest degree of the variable y appearing in P. In the other direction we observe that the methods of Impagliazzo and Kabanets imply that if we can derandomize polynomial identity testing for bounded depth circuits then NEXP does not have bounded depth arithmetic circuits. That is, either NEXP ⊈ P/poly or the Permanent is not computable by polynomial size bounded depth arithmetic circuits.