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

We present a single common tool to strictly subsume all known cases of polynomial time blackbox 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 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: - depth-3 (Ω Π Ω) circuits with constant transcendence degree of the polynomials computed by the product gates (no bounded top fanin restriction), and - constant-depth constant-occur formulas (no multilinear restriction). Constant-occur of a variable, as we define it, is a much 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 & Seshadhri (STOC 2011), Saraf & Volkovich (STOC 2011), Anderson et al. (CCC 2011), Beecken et al. (ICALP 2011) and Grenet et al. (FSTTCS 2011), 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 - that is equally effective in solving both the problems on certain interesting and previously well-investigated (but not well understood) models of computation.

[1]  Marek Karpinski,et al.  Deterministic Polynomial Time Algorithms for Matrix Completion Problems , 2010, SIAM J. Comput..

[2]  Ilya Volkovich,et al.  Derandomizing Polynomial Identity Testing for Multilinear Constant-Read Formulae , 2011, 2011 IEEE 26th Annual Conference on Computational Complexity.

[3]  Neeraj Kayal,et al.  Polynomial Identity Testing for Depth 3 Circuits , 2006, Computational Complexity Conference.

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

[5]  Nitin Saxena,et al.  Algebraic Independence in Positive Characteristic -- A p-Adic Calculus , 2012, Electron. Colloquium Comput. Complex..

[6]  Adam R. Klivans,et al.  Learning Restricted Models of Arithmetic Circuits , 2006, Theory Comput..

[7]  Leslie G. Valiant,et al.  A complexity theory based on Boolean algebra , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

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

[9]  A. Wigderson,et al.  Partial Derivatives in Arithmetic Complexity and Beyond (Foundations and Trends(R) in Theoretical Computer Science) , 2011 .

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

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

[12]  D. E. Littlewood,et al.  Group Characters and Algebra , 1934 .

[13]  Leslie G. Valiant,et al.  Fast Parallel Computation of Polynomials Using Few Processors , 1983, SIAM J. Comput..

[14]  Avi Wigderson,et al.  Partial Derivatives in Arithmetic Complexity and Beyond , 2011, Found. Trends Theor. Comput. Sci..

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

[16]  Joos Heintz,et al.  Testing polynomials which are easy to compute (Extended Abstract) , 1980, STOC '80.

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

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

[19]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[20]  Margo McCall,et al.  IEEE Computer Society , 2019, Encyclopedia of Software Engineering.

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

[22]  Ketan Mulmuley,et al.  On P vs. NP and geometric complexity theory: Dedicated to Sri Ramakrishna , 2011, JACM.

[23]  Neeraj Kayal,et al.  Arithmetic Circuits: A Chasm at Depth Three , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

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

[25]  Marek Karpinski,et al.  On Zero-Testing and Interpolation of k-Sparse Multivariate Polynomials Over Finite Fields , 1991, Theor. Comput. Sci..

[26]  Ilya Volkovich,et al.  On the Relation between Polynomial Identity Testing and Finding Variable Disjoint Factors , 2010, ICALP.

[27]  Ryan Williams,et al.  Non-uniform ACC Circuit Lower Bounds , 2011, 2011 IEEE 26th Annual Conference on Computational Complexity.

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

[29]  Bruno Grenet,et al.  The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent , 2011, FSTTCS.

[30]  James G. Oxley,et al.  Matroid theory , 1992 .

[31]  Michael Ben-Or,et al.  A deterministic algorithm for sparse multivariate polynomial interpolation , 1988, STOC '88.

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

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

[34]  Ilya Volkovich,et al.  Black-Box Identity Testing of Depth-4 Multilinear Circuits , 2011, Combinatorica.

[35]  Prasad Raghavendra,et al.  Optimal algorithms and inapproximability results for every CSP? , 2008, STOC.

[36]  Nitin Saxena,et al.  Blackbox Identity Testing for Bounded Top-Fanin Depth-3 Circuits: The Field Doesn't Matter , 2012, SIAM J. Comput..

[37]  Sergey Yekhanin,et al.  Detecting Rational Points on Hypersurfaces over Finite Fields , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[38]  Pascal Koiran,et al.  Arithmetic circuits: The chasm at depth four gets wider , 2010, Theor. Comput. Sci..

[39]  Richard J. Lipton,et al.  A Probabilistic Remark on Algebraic Program Testing , 1978, Inf. Process. Lett..

[40]  Nitin Saxena,et al.  Algebraic independence and blackbox identity testing , 2011, Inf. Comput..

[41]  Leslie G. Valiant,et al.  Completeness classes in algebra , 1979, STOC.

[42]  Amir Shpilka,et al.  Reconstruction of Generalized Depth-3 Arithmetic Circuits with Bounded Top Fan-in , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[43]  Nitin Saxena,et al.  From Sylvester-Gallai Configurations to Rank Bounds: Improved Black-Box Identity Test for Depth-3 Circuits , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[44]  Ketan Mulmuley,et al.  Geometric Complexity Theory V: Equivalence between Blackbox Derandomization of Polynomial Identity Testing and Derandomization of Noether's Normalization Lemma , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

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

[46]  K. Kalorkoti,et al.  A Lower Bound for the Formula Size of Rational Functions , 1982, SIAM J. Comput..

[47]  Nitin Saxena,et al.  An Almost Optimal Rank Bound for Depth-3 Identities , 2011, SIAM J. Comput..

[48]  Carsten Lund,et al.  Algebraic methods for interactive proof systems , 1992, JACM.

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

[50]  Christopher Umans,et al.  Pseudo-random generators for all hardnesses , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[51]  Nitin Saxena,et al.  Quasi-polynomial hitting-set for set-depth-Δ formulas , 2012, STOC '13.

[52]  Ilya Volkovich,et al.  Deterministic Identity Testing of Depth-4 Multilinear Circuits with Bounded Top Fan-in , 2013, SIAM J. Comput..

[53]  Dieter van Melkebeek,et al.  A note on circuit lower bounds from derandomization , 2010, Electron. Colloquium Comput. Complex..

[54]  Michael E. Saks,et al.  Exponential lower bounds for depth three Boolean circuits , 2000, computational complexity.

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

[56]  Ran Raz,et al.  Deterministic extractors for affine sources over large fields , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[57]  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.

[58]  Amir Shpilka,et al.  Quasipolynomial-Time Identity Testing of Non-commutative and Read-Once Oblivious Algebraic Branching Programs , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[59]  Ilya Volkovich,et al.  Improved Polynomial Identity Testing for Read-Once Formulas , 2009, APPROX-RANDOM.

[60]  Nitin Saxena,et al.  Algebraic independence and blackbox identity testing , 2013, Inf. Comput..