Hardness-randomness tradeoffs for bounded depth arithmetic circuits

SIAM Journal on Computing

Volumn 39, Issue 4, 2009, Pages 1279-1293

  Dvir, Zeev ^a   Shpilka, Amir ^b   Yehudayoff, Amir ^a  

^a WEIZMANN INSTITUTE OF SCIENCE   (Israel)

^b TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Derandomization; Hardness; Polynomial factoring; Polynomial identity testing; Randomness

Indexed keywords

ARITHMETIC CIRCUIT; BLACK BOXES; BOUNDED DEPTH CIRCUITS; DERANDOMIZATION; DERANDOMIZING; HARDNESS-RANDOMNESS TRADEOFFS; IDENTITY TESTING; LOWER BOUNDS; MULTILINEAR POLYNOMIALS; POLYNOMIAL EQUATION; POLYNOMIAL FACTORING; POLYNOMIAL IDENTITY TESTING; POLYNOMIAL SIZE; RANDOMNESS; SMALL SIZE; TECHNICAL CONTRIBUTION;

CONTROL THEORY; HARDNESS; HARDNESS TESTING; INTEGRATING CIRCUITS; LOGIC CIRCUITS;

POLYNOMIALS;

EID: 70350173835     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/080735850     Document Type: Article

References (35)

1. M. Agrawal and S. Biswas, Primality and identity testing via Chinese remaindering, J. ACM, 50 (2003), pp. 429-443.
2. M. Agrawal, N. Kayal, and N. Saxena, Primes is in P, Ann. Math., 160 (2004), pp. 781-793.
3. V. Arvind and P. Mukhopadhyay, The monomial ideal membership problem and polynomial identity testing, in Proceedings of the 18th ISAAC, 2007, ACM, pp. 800-811.
4. M. Agrawal and V. Vinay, Arithmetic circuits: A chasm at depth four, in Proceedings of the 49th Annual FOCS, 2008, IEEE, pp. 67-75.
5. M. Ben-Or and P. Tiwari, Adeterministic algorithmfor sparse multivariate polynominal interpolation, in Proceedings of the 20th Annual STOC, 1988, ACM, pp. 301-309.
6. W. Baur and V. Strassen, The complexity of partial derivatives, Theoret. Comput. Sci., 22 (1983), pp. 317-330.
7. S. Chari, P. Rohatgi, and A. Srinivasan, Randomness-optimal unique element isolation with applications to perfect matching and related problems, SIAM J. Comput., 24 (1995), pp. 1036-1050.
8. Z. Dvir and A. Shpilka, Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits, SIAM J. Comput., 36 (2007), pp. 1404-1434. (Pubitemid 47632453)
9. D. Grigoriev and M. Karpinski, An exponential lower bound for depth 3 arithmetic circuits, in Proceedings of the 30th Annual STOC, 1988, ACM, pp. 577-582.
10. D. Y. Grigoriev, M. Karpinski, and M. F. Singer, Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields, SIAM J. Comput., 19 (1990), pp. 1059-1063.
11. D. Grigoriev and A. A. Razborov, Exponential complexity lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields, Appl. Algebra Engrg., Comm. Comput., 10 (2000), pp. 465-487.
12. R. Impagliazzo, V. Kabanets, and A. Wigderson, In search of an easy witness: Exponential time vs. probabilistic polynomial time, J. Comput. System Sci., 65 (2002), pp. 672-694.
13. R. Impagliazzo, R. Shaltiel, and A. Wigderson, Extractors and pseudo-random generators with optimal seed length, in Proceedings of the 32nd STOC Conference, 2001, ACM, pp. 1-10.
14. R. Impagliazzo and A. Wigderson, P=BPP unless E has subexponential circuits: Derandomizing the xor lemma, in Proceedings of the 29th STOC, 1997, ACM, pp. 220-229.
15. E. Kaltofen, Factorization of polynomials given by straight-line programs, in Randomness in Computation, Adv. Comput. Res. 5, S. Micali, ed., JAI Press, Greenwich, CT, 1989, pp. 375-412.
16. V. Kabanets and R. Impagliazzo, Derandomizing polynomial identity tests means proving circuit lower bounds, Comput. Complexity, 13 (2004) pp. 1-46.
17. A. Klivans and D. Spielman, Randomness effcient identity testing of multivariate polynomials, in Proceedings of the 33rd Annual STOC, 2001, ACM, pp. 216-223.
18. 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 CCC, 2008, IEEE, pp. 280-291.
19. N. Kayal and S. Saraf, Blackbox Polynomial Identity Testing for Depth 3 Circuits, manusript, 2009.
20. N. Kayal and N. Saxena, Polynomial identity testing for depth 3 circuits, Comput. Complexity, 16 (2007), pp. 115-138.
21. L. Lovasz, On determinants, matchings, and random algorithms, in Fundamentals of Computing Theory, L. Budach, ed., Akademia-Verlag, Budapest, 1979.
22. D. Lewin and S. Vadhan, Checking polynomial identities over any field: Towards a derandomization?, in Proceedings of the 30th Annual STOC, 1998, ACM, pp. 428-437.
23. K. Mulmuley, U. Vazirani, and V. Vazirani, Matching is as easy as matrix inversion, Combinatorica, 7 (1987), pp. 105-113.
24. N. Nisan and A. Wigderson, Hardness vs. randomness, J. Comput. System Sci., 49 (1994), pp. 149-167.
25. P. Pudlak, Communication in bounded depth circuits, Combinatorica, 14 (1994), pp. 203-216.
26. R. Raz, Elusive functions and lower bounds for arithmetic circuits, in Proceedings of the 40th Annual STOC, 2008, ACM, pp. 711-720.
27. R. Raz and A. Shpilka, Deterministic polynomial identity testing in non commutative models, Comput. Complexity, 14 (2005), pp. 1-19.
28. N. Saxena and C. Seshadri, An almost optimal rank bound for depth-3 identities, in Proceedings of CCC 2009, IEEE, to appear.
29. J. T. Schwartz, Fast probabilistic algorithms for veri.cation of polynomial identities, J. ACM, 27 (1980), pp. 701-717.
30. V. Strassen, Vermeidung von divisionen, J. Reine Angew. Math., 264 (1973), pp. 182-202.
31. M. Sudan, L. Trevisan, and S. Vadhan, Pseudorandom generators without the XOR lemma, J. Comput. System Sci., 62 (2001), pp. 236-266.
32. R. Shaltiel and C. Umans, Simple extractors for all min-entropies and a new pseudorandom generator, J. ACM, 52 (2005), pp. 172-216.
33. A. Shpilka and A. Wigderson, Depth-3 arithmetic circuits over fields of characteristic zero, Comput. Complexity, 10 (2001), pp. 1-27.
34. S. Toda, PP is as hard as the polynomial-time hierarchy, SIAM J. Comput., 20 (1991), pp. 865-877.
35. R. Zippel, Probabilistic algorithms for sparse polynomials, in Symbolic and Algebraic Computation, Springer-Verlag, Berlin, 1979, pp. 216-226.