NDLI: Jacobian Hits Circuits: Hitting Sets, Lower Bounds for Depth-$D$ Occur-$k$ Formulas and Depth-3 Transcendence Degree-$k$ Circuits

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Agrawal, Manindra Saha, Chandan Saxena, Nitin Saptharishi, Ramprasad
Copyright Year	2016
Abstract	We present a single common tool to strictly subsume all known cases of polynomial time black box polynomial identity testing (PIT), that have been hitherto solved using diverse tools and techniques, over fields of zero or large characteristic. In particular, we show that polynomial (in the size of the circuit) time hitting-set generators for identity testing of the two seemingly different and well studied models---depth-3 circuits with bounded top fanin, and constant-depth constant-read multilinear formulas---can be constructed using one common algebraic-geometry theme: Jacobian captures algebraic independence. By exploiting the Jacobian, we design the first efficient hitting-set generators for broad generalizations of the above-mentioned models, namely, (a) depth-3 ($\Sigma \Pi \Sigma$) circuits with constant transcendence degree of the polynomials computed by the product gates (no bounded top fanin restriction), and (b) constant-depth constant-occur formulas (no multilinear restriction). Constant occur of a variable, as we define it, is a more general concept than constant read. Also, earlier work on the latter model assumed that the formula is multilinear. Thus, our work goes further beyond the related results obtained by Saxena and Seshadhri [STOC, ACM, New York, 2011, pp. 431--440], Saraf and Volkovich [STOC, ACM, New York, 2011, pp. 421--430], Anderson, van Melkebeek, and Volkovich, [IEEE Conference on Computational Complexity, IEEE, Piscataway, NJ, 2011, pp. 273--282], Beecken, Mittmann, and Saxena [ICALP, Springer, New York, 2011, pp. 134--148] and Grenet et al. [Proceedings of the 30th Foundations of Software Technology and Theoretical Computer Science (FSTTCS), Schloss Dagstuhl--Liebniz--Zentrum für Informatik, Wadern, Germany, 2011, pp. 127--139] and brings them under one unifying technique. In addition, using the same Jacobian-based approach, we prove exponential lower bounds for the immanant (which includes permanent and determinant) on the same depth-3 and depth-4 models for which we give efficient PIT algorithms. Our results reinforce the intimate connection between identity testing and lower bounds by exhibiting a concrete mathematical tool---the Jacobian. The Jacobian is equally effective in solving both the problems on certain interesting and previously well-investigated (but not well understood) models of computation.
Starting Page	1533
Ending Page	1562
Page Count	30
File Format	PDF
ISSN	00975397
DOI	10.1137/130910725
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4 (Special Section on the Forty-Fourth Annual ACM Symposium on Theory of Computing (STOC 2012))
Volume Number	45
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2016-08-30
Access Restriction	Subscribed
Subject Keyword	identity testing Analysis of algorithms and problem complexity depth Vandermonde algebraic independence Jacobian black box lower bound immanant circuits Symbolic computation and algebraic computation
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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Jacobian hits circuits: hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits

Jacobian hits circuits: Hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits

Jacobian hits circuits: Hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k

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