JACOBIAN HITS CIRCUITS: HITTING SETS, LOWER BOUNDS FOR DEPTH-D OCCUR-k FORMULAS AND DEPTH-3 TRANSCENDENCE DEGREE-k CIRCUITS

Agrawal, Manindra and Saha, Chandan and Saptharishi, Ramprasad and Saxena, Nitin (2016) JACOBIAN HITS CIRCUITS: HITTING SETS, LOWER BOUNDS FOR DEPTH-D OCCUR-k FORMULAS AND DEPTH-3 TRANSCENDENCE DEGREE-k CIRCUITS. In: SIAM JOURNAL ON COMPUTING, 45 (4, SI). pp. 1533-1562.

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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 constantread 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 fur 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.

Item Type:	Journal Article
Publication:	SIAM JOURNAL ON COMPUTING
Additional Information:	Copy right for this article belongs to the SIAM PUBLICATIONS, 3600 UNIV CITY SCIENCE CENTER, PHILADELPHIA, PA 19104-2688 USA
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	03 Dec 2016 06:31
Last Modified:	03 Dec 2016 06:31
URI:	http://eprints.iisc.ac.in/id/eprint/55293

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