Hardness-randomness tradeoffs for bounded depth arithmetic circuits

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(x1,...,xm) that cannot be computed by a depth d arithmetic circuit of small size then there exists an efficient deterministic algorithm to test whether a given depth d-8 circuit is identically zero or not (assuming the individual degrees of the tested circuit are not too high). In particular, if we are guaranteed that the 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 Nisan-Wigderson generator of Impagliazzo and Kabanets 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(x1,...,xn,y) ≡ 0 and shows that if P has a circuit of depth d and size s and if the polynomial f(x1,...,xn) satisfies P(x1,...,xn,f(x1,...,xn))≡ 0 then f has a circuit of depth d+3 and size O(s • r + mr), where m is the degree of f and r is the highest degree of the variable y appearing in P. In the other direction we observe that the methods of Impagliazzo and Kabanets imply that if we can derandomize polynomial identity testing for bounded depth circuits then NEXP does not have bounded depth arithmetic circuits. That is, either NEXP ⊄ P/poly or the Permanent is not computable by polynomial size bounded depth arithmetic circuits.

[1]  B. M. Fulk MATH , 1992 .

[2]  Avi Wigderson,et al.  In search of an easy witness: exponential time vs. probabilistic polynomial time , 2001, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[3]  Luca Trevisan,et al.  Pseudorandom generators without the XOR lemma , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[4]  Ran Raz Elusive functions and lower bounds for arithmetic circuits , 2008, STOC '08.

[5]  László Lovász,et al.  On determinants, matchings, and random algorithms , 1979, FCT.

[6]  Vikraman Arvind,et al.  The ideal membership problem and polynomial identity testing , 2010, Inf. Comput..

[7]  M. Ben-Or,et al.  A Deterministic Algorithm for Sparse Multivariate Polynominal Interpolation (Extended Abstract) , 1988, Symposium on the Theory of Computing.

[8]  K. Ramachandra,et al.  Vermeidung von Divisionen. , 1973 .

[9]  Aravind Srinivasan,et al.  Randomness-Optimal Unique Element Isolation with Applications to Perfect Matching and Related Problems , 1995, SIAM J. Comput..

[10]  Vikraman Arvind,et al.  The Monomial Ideal Membership Problem and Polynomial Identity Testing , 2007, ISAAC.

[11]  Amir Shpilka,et al.  Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[12]  Christopher Umans,et al.  Simple extractors for all min-entropies and a new pseudorandom generator , 2005, JACM.

[13]  Zeev Dvir,et al.  Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits , 2005, STOC '05.

[14]  Vijay V. Vazirani,et al.  Matching is as easy as matrix inversion , 1987, STOC.

[15]  Daniel A. Spielman,et al.  Randomness efficient identity testing of multivariate polynomials , 2001, STOC '01.

[16]  Manindra Agrawal,et al.  PRIMES is in P , 2004 .

[17]  Avi Wigderson,et al.  Extractors and pseudo-random generators with optimal seed length , 2000, STOC '00.

[18]  V. Vinay,et al.  Arithmetic Circuits: A Chasm at Depth Four , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[19]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[20]  Avi Wigderson,et al.  P = BPP if E requires exponential circuits: derandomizing the XOR lemma , 1997, STOC '97.

[21]  Nitin Saxena,et al.  An Almost Optimal Rank Bound for Depth-3 Identities , 2008, 2009 24th Annual IEEE Conference on Computational Complexity.

[22]  Victor Shoup,et al.  New algorithms for finding irreducible polynomials over finite fields , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[23]  Noam Nisan,et al.  Hardness vs. randomness , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[24]  Marek Karpinski,et al.  An exponential lower bound for depth 3 arithmetic circuits , 1998, STOC '98.

[25]  Marek Karpinski,et al.  Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields , 1988, SIAM J. Comput..

[26]  Salil P. Vadhan,et al.  Checking polynomial identities over any field: towards a derandomization? , 1998, STOC '98.

[27]  Amir Shpilka Affine projections of symmetric polynomials , 2001, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[28]  Seinosuke Toda,et al.  PP is as Hard as the Polynomial-Time Hierarchy , 1991, SIAM J. Comput..

[29]  Amir Shpilka,et al.  Black Box Polynomial Identity Testing of Depth-3 Arithmetic Circuits with Bounded Top Fan-in , 2007, Electron. Colloquium Comput. Complex..

[30]  Amir Yehudayoff,et al.  Arithmetic Circuits: A survey of recent results and open questions , 2010, Found. Trends Theor. Comput. Sci..

[31]  Ran Raz,et al.  Deterministic polynomial identity testing in non-commutative models , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[32]  Zeev Dvir,et al.  Locally Decodable Codes with Two Queries and Polynomial Identity Testing for Depth 3 Circuits , 2007, SIAM J. Comput..

[33]  Erich Kaltofen,et al.  Factorization of Polynomials Given by Straight-Line Programs , 1989, Adv. Comput. Res..

[34]  Ran Raz,et al.  Lower bounds for matrix product, in bounded depth circuits with arbitrary gates , 2001, STOC '01.

[35]  Manindra Agrawal,et al.  Proving Lower Bounds Via Pseudo-random Generators , 2005, FSTTCS.

[36]  Neeraj Kayal,et al.  Polynomial Identity Testing for Depth 3 Circuits , 2006, 21st Annual IEEE Conference on Computational Complexity (CCC'06).

[37]  Shubhangi Saraf,et al.  Blackbox Polynomial Identity Testing for Depth 3 Circuits , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[38]  Walter Baur,et al.  The Complexity of Partial Derivatives , 1983, Theor. Comput. Sci..

[39]  Alexander A. Razborov,et al.  Exponential complexity lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[40]  Alexander A. Razborov,et al.  Exponential Lower Bounds for Depth 3 Arithmetic Circuits in Algebras of Functions over Finite Fields , 2000, Applicable Algebra in Engineering, Communication and Computing.

[41]  V. Strassen Die Berechnungskomplexität von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten , 1973 .

[42]  Avi Wigderson,et al.  Depth-3 arithmetic circuits over fields of characteristic zero , 2002, computational complexity.

[43]  R. Impagliazzo,et al.  Subexponential Circuits : Derandomizing the XOR Lemma , 2003 .

[44]  Aravind Srinivasan,et al.  Randomness-optimal unique element isolation, with applications to perfect matching and related problems , 1993, SIAM J. Comput..

[45]  Russell Impagliazzo,et al.  Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds , 2003, STOC '03.

[46]  Manindra Agrawal,et al.  Primality and identity testing via Chinese remaindering , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[47]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[48]  Pavel Pudlák,et al.  Communication in bounded depth circuits , 1994, Comb..