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Hardness-randomness tradeoffs for bounded depth arithmetic circuits

By Zeev Dvir, Amir Shpilka and Amir Yehudayoff


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 that cannot be computed by a depth d arithmetic circuit of small size then there exists an efficient deterministic black-box algorithm to test whether a given depth d − 5 circuit that computes a polynomial of relatively small individual degrees is identically zero or not. In particular, if we are guaranteed that the tested 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 Kabanets and Impagliazzo 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 (x1,..., xn, y) ≡ 0 and shows that if P has a circuit of depth d and size s and if the polynomial f(x1,..., xn) satisfies P (x1,..., xn, f) ≡ 0 then f has a circuit of depth d+3 and size poly(s, m r)

Year: 2013
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