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Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter

Authors:

Nitin Saxena and

C. SeshadhriAuthors Info & Claims

STOC '11: Proceedings of the forty-third annual ACM symposium on Theory of computing

June 2011

Pages 431 - 440

https://doi.org/10.1145/1993636.1993694

Published: 06 June 2011 Publication History

Abstract

Let C be a depth-3 circuit with n variables, degree d and top fanin k (called ΣΠΣ(k,d,n) circuits) over base field FF. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests if C is identically zero. Klivans & Spielman (STOC 2001) observed that the problem is open even when k is a constant. This case has been subjected to a serious study over the past few years, starting from the work of Dvir & Shpilka (STOC 2005).

We give the first polynomial time blackbox algorithm for this problem. Our algorithm runs in time poly(n)d^k, regardless of the base field. The only field for which polynomial time algorithms were previously known is FF = QQ (Kayal & Saraf, FOCS 2009, and Saxena & Seshadhri, FOCS 2010). This is the first blackbox algorithm for depth-$3$ circuits that does not use the rank based approaches of Karnin & Shpilka (CCC 2008).

We prove an important tool for the study of depth-3 identities. We design a blackbox polynomial time transformation that reduces the number of variables in a ΣΠΣ(k,d,n) circuit to k variables, but preserves the identity structure.

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References

[1]

M. Agrawal and S. Biswas. Primality and identity testing via Chinese remaindering. Journal of the ACM, 50(4):429--443, 2003. (Conference version in FOCS 1999).

Digital Library

[2]

M. Agrawal. Proving lower bounds via pseudo-random generators. In Proceedings of the 25th Annual Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pages 92--105, 2005.

Digital Library

[3]

M. Agrawal. Determinant versus permanent. In Proceedings of the 25th International Congress of Mathematicians (ICM), volume 3, pages 985--997, 2006.

[4]

L. M. Adleman and H. W. Lenstra. Finding irreducible polynomials over finite fields. In Proceedings of the 18th Annual ACM Symposium on Theory of Computing (STOC), pages 350--355, 1986.

Digital Library

[5]

V. Arvind and P. Mukhopadhyay. The monomial ideal membership problem and polynomial identity testing. In Proceedings of the 18th International Symposium on Algorithms and Computation (ISAAC), pages 800--811, 2007.

Digital Library

[6]

M. Agrawal and R. Saptharishi. Classifying polynomials and identity testing. In Current Trends in Science, Indian Academy of Sciences, pages 149--162. 2009.

[7]

M. Agrawal and V. Vinay. Arithmetic circuits: A chasm at depth four. In Proceedings of the 49th Annual Symposium on Foundations of Computer Science (FOCS), pages 67--75, 2008.

Digital Library

[8]

M. Anderson, D. van Melkebeek, and I. Volkovich. Derandomizing polynomial identity testing for multilinear constant-read formulae. In Proceedings of the 26th Conference on Computational Complexity (CCC), 2011.

Digital Library

[9]

M. Beecken, J. Mittmann, and N. Saxena. Algebraic independence and blackbox identity testing. Technical Report TR11-022, ECCC, 2011.

[10]

M. Ben-Or and P. Tiwari. A deterministic algorithm for sparse multivariate polynominal interpolation. In Proceedings of the 20th annual ACM symposium on Theory of computing (STOC), pages 301--309, 1988.

Digital Library

[11]

M. Clausen, A. W. M. Dress, J. Grabmeier, and M. Karpinski. On zero-testing and interpolation of k-sparse multivariate polynomials over finite fields. Theoretical Computer Science, 84(2):151--164, 1991.

Digital Library

[12]

Z. Chen and M. Kao. Reducing randomness via irrational numbers. In Proceedings of the 29th annual ACM symposium on Theory of computing (STOC), pages 200--209, 1997.

Digital Library

[13]

Z. Dvir and A. Shpilka. Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits. SIAM J. on Computing, 36(5):1404--1434, 2006. (Conference version in STOC 2005).

Digital Library

[14]

D. Grigoriev, M. Karpinski, and M. F. Singer. Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. SIAM J. Comput., 19(6):1059--1063, 1990.

Digital Library

[15]

A. Gabizon and R. Raz. Deterministic extractors for affine sources over large fields. Combinatorica, 28(4):415--440, 2008. (Conference version in FOCS 2005).

Digital Library

[16]

J. Heintz and C. P. Schnorr. Testing polynomials which are easy to compute (extended abstract). In Proceedings of the twelfth annual ACM Symposium on Theory of Computing, pages 262--272, 1980.

Digital Library

[17]

V. Kabanets and R. Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1):1--46, 2004. (Conference version in STOC 2003).

Digital Library

[18]

Z. Karnin, P. Mukhopadhyay, A. Shpilka, and I. Volkovich. Deterministic identity testing of depth-4 multilinear circuits with bounded top fan-in. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC), pages 649--658, 2010.

Digital Library

[19]

M. Karpinski and I. Shparlinski. On some approximation problems concerning sparse polynomials over finite fields. Theoretical Computer Science, 157(2):259--266, 1996.

Digital Library

[20]

A. Klivans and D. A. Spielman. Randomness efficient identity testing of multivariate polynomials. In Proceedings of the 33rd Symposium on Theory of Computing (STOC), pages 216--223, 2001.

Digital Library

[21]

N. Kayal and N. Saxena. Polynomial identity testing for depth 3 circuits. Computational Complexity, 16(2):115--138, 2007. (Conference version in CCC 2006).

Digital Library

[22]

Z. Karnin and A. Shpilka. Deterministic black box polynomial identity testing of depth-3 arithmetic circuits with bounded top fan-in. In Proceedings of the 23rd Annual Conference on Computational Complexity (CCC), pages 280--291, 2008.

Digital Library

[23]

Z. S. Karnin and A. Shpilka. Reconstruction of generalized depth-3 arithmetic circuits with bounded top fan-in. In Proceedings of the 24th Annual Conference on Computational Complexity (CCC), pages 274--285, 2009.

Digital Library

[24]

N. Kayal and S. Saraf. Blackbox polynomial identity testing for depth 3 circuits. In Proceedings of the 50th Annual Symposium on Foundations of Computer Science (FOCS), pages 198--207, 2009.

Digital Library

[25]

D. Lewin and S. Vadhan. Checking polynomial identities over any field: Towards a derandomization? In Proceedings of the 30th Annual Symposium on the Theory of Computing (STOC), pages 428--437, 1998.

Digital Library

[26]

R. Raz. Tensor-rank and lower bounds for arithmetic formulas. In Proceedings of the 42nd Annual ACM Symposium on Theory of Computing (STOC), 2010.

Digital Library

[27]

R. Raz and A. Shpilka. Deterministic polynomial identity testing in non-commutative models. Computational Complexity, 14(1):1--19, 2005. (Conference version in CCC 2004).

Digital Library

[28]

N. Saxena. Diagonal circuit identity testing and lower bounds. In Proceedings of the 35th Annual International Colloquium on Automata, Languages and Programming (ICALP), pages 60--71, 2008.

Digital Library

[29]

N. Saxena. Progress on polynomial identity testing. Bulletin of the European Association for Theoretical Computer Science (EATCS)- Computational Complexity Column, (99):49--79, 2009.

[30]

J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM, 27(4):701--717, 1980.

Digital Library

[31]

R. E. Schapire and L. M. Sellie. Learning sparse multivariate polynomials over a field with queries and counterexamples. Journal of Computer and System Sciences, 52(2):201--213, 1996.

Digital Library

[32]

N. Saxena and C. Seshadhri. From Sylvester-Gallai configurations to rank bounds: Improved black-box identity test for depth-3 circuits. In Proceedings of the 51st Annual Symposium on Foundations of Computer Science (FOCS), pages 21--29, 2010.

Digital Library

[33]

N. Saxena and C. Seshadhri. From Sylvester-Gallai configurations to rank bounds: Improved black-box identity test for depth-3 circuits. Technical Report TR10-013 (Revision #1), ECCC, http://eccc.hpi-web.de/report/2010/013/, 2010.

[34]

N. Saxena and C. Seshadhri. An Almost Optimal Rank Bound for Depth-$3$ Identities. SIAM J. Comp., 40(1):200--224, 2011. (Conference version in CCC 2009).

Digital Library

[35]

C. Saha, R. Saptharishi, and N. Saxena. A case of depth-3 identity testing, sparse factorization and duality. Technical Report TR11-021, ECCC, 2011.

[36]

A. Shpilka and I. Volkovich. Improved polynomial identity testing for read-once formulas. In Proceedings of the 13th International Workshop on Randomization and Computation (RANDOM), pages 700--713, 2009.

Digital Library

[37]

S. Saraf and I. Volkovich. Black-box identity testing of depth-4 multilinear circuits. In Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC), 2011.

Digital Library

[38]

A. Shpilka and A. Yehudayoff. Arithmetic Circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5(3--4):207--388, 2010.

Digital Library

[39]

K. Werther. The complexity of sparse polynomial interpolation over finite fields. Applicable Algebra in Engineering, Communication and Computing, 5:91--103, 1994.

[40]

R. Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposium on Symbolic and Algebraic Manipulation (EUROSAM), pages 216--226, 1979.

Digital Library

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Oliveira RShpilka AVolk B(2016)Subexponential Size Hitting Sets for Bounded Depth Multilinear FormulasComputational Complexity10.1007/s00037-016-0131-125:2(455-505)Online publication date: 1-Jun-2016
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Oliveira RShpilka AVolk BZuckerman D(2015)Subexponential size hitting sets for bounded depth multilinear formulasProceedings of the 30th Conference on Computational Complexity10.5555/2833227.2833242(304-322)Online publication date: 17-Jun-2015
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Index Terms

Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter
1. Theory of computation
  1. Randomness, geometry and discrete structures

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cover image ACM Conferences

STOC '11: Proceedings of the forty-third annual ACM symposium on Theory of computing

June 2011

840 pages

ISBN:9781450306911

DOI:10.1145/1993636

General Chair:
Lance Fortnow
Northwestern University
,
Program Chair:
Salil Vadhan
Harvard University

Copyright © 2011 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 06 June 2011

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STOC'11: Symposium on Theory of Computing

June 6 - 8, 2011

California, San Jose, USA

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STOC '11 Paper Acceptance Rate 84 of 304 submissions, 28%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Cited By

Anderson MForbes MSaptharishi RShpilka AVolk B(2016)Identity testing and lower bounds for read-k oblivious algebraic branching programsProceedings of the 31st Conference on Computational Complexity10.5555/2982445.2982475(1-25)Online publication date: 29-May-2016
https://dl.acm.org/doi/10.5555/2982445.2982475
Oliveira RShpilka AVolk B(2016)Subexponential Size Hitting Sets for Bounded Depth Multilinear FormulasComputational Complexity10.1007/s00037-016-0131-125:2(455-505)Online publication date: 1-Jun-2016
https://dl.acm.org/doi/10.1007/s00037-016-0131-1
Oliveira RShpilka AVolk BZuckerman D(2015)Subexponential size hitting sets for bounded depth multilinear formulasProceedings of the 30th Conference on Computational Complexity10.5555/2833227.2833242(304-322)Online publication date: 17-Jun-2015
https://dl.acm.org/doi/10.5555/2833227.2833242
Forbes MSaptharishi RShpilka AShmoys D(2014)Hitting sets for multilinear read-once algebraic branching programs, in any orderProceedings of the forty-sixth annual ACM symposium on Theory of computing10.1145/2591796.2591816(867-875)Online publication date: 31-May-2014
https://dl.acm.org/doi/10.1145/2591796.2591816
Saxena NSeshadhri C(2013)From sylvester-gallai configurations to rank boundsJournal of the ACM10.1145/252840360:5(1-33)Online publication date: 28-Oct-2013
https://dl.acm.org/doi/10.1145/2528403
Agrawal MSaha CSaxena NBoneh DRoughgarden TFeigenbaum J(2013)Quasi-polynomial hitting-set for set-depth-Δ formulasProceedings of the forty-fifth annual ACM symposium on Theory of Computing10.1145/2488608.2488649(321-330)Online publication date: 1-Jun-2013
https://dl.acm.org/doi/10.1145/2488608.2488649
Beecken MMittmann JSaxena N(2013)Algebraic independence and blackbox identity testingInformation and Computation10.1016/j.ic.2012.10.004222(2-19)Online publication date: 1-Jan-2013
https://dl.acm.org/doi/10.1016/j.ic.2012.10.004
Bshouty N(2013)Exact Learning from Membership Queries: Some Techniques, Results and New DirectionsAlgorithmic Learning Theory10.1007/978-3-642-40935-6_4(33-52)Online publication date: 2013
https://doi.org/10.1007/978-3-642-40935-6_4
Agrawal MSaha CSaptharishi RSaxena NKarloff HPitassi T(2012)Jacobian hits circuitsProceedings of the forty-fourth annual ACM symposium on Theory of computing10.1145/2213977.2214033(599-614)Online publication date: 19-May-2012
https://dl.acm.org/doi/10.1145/2213977.2214033
Forbes MShpilka AKarloff HPitassi T(2012)On identity testing of tensors, low-rank recovery and compressed sensingProceedings of the forty-fourth annual ACM symposium on Theory of computing10.1145/2213977.2213995(163-172)Online publication date: 19-May-2012
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