On the relation between polynomial identity testing and finding variable disjoint factors
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Uniform derandomization from pathetic lower bounds
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Black-box identity testing of depth-4 multilinear circuits
Proceedings of the forty-third annual ACM symposium on Theory of computing
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CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
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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 $P(x_1,\dots,x_n,y)\equiv0$ and shows that if $P$ has a circuit of depth $d$ and size $s$ and if the polynomial $f(x_1,\dots,x_n)$ satisfies $P(x_1,\dots,x_n,f)\equiv0$, then $f$ has a circuit of depth $d+3$ and size $\mathrm{poly}(s,m^r)$, where $m$ is the total degree of $f$ and $r$ is the degree of $y$ in $P$. This circuit for $f$ can be found probabilistically in time $\mathrm{poly}(s,m^r)$. In the other direction we observe that the methods of Kabanets and Impagliazzo can be used to show that derandomizing identity testing for bounded depth circuits implies lower bounds for the same class of circuits. More formally, if we can derandomize polynomial identity testing for bounded depth circuits, then NEXP does not have bounded depth arithmetic circuits. That is, either $\mathrm{NEXP}\not\subseteq P/\mathrm{poly}$ or the Permanent is not computable by polynomial size bounded depth arithmetic circuits.