NDLI: Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Shpilka, Amir Yehudayoff, Amir Dvir, Zeev
Copyright Year	2009
Abstract	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 NisanWigderson 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.
Starting Page	1279
Ending Page	1293
Page Count	15
File Format	PDF
ISSN	00975397
DOI	10.1137/080735850
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4
Volume Number	39
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2009-10-01
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems hardness polynomial identity testing Randomized algorithms derandomization randomness Computational aspects of field theory and polynomials polynomial factoring
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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