Succinct hitting sets and barriers to proving algebraic circuits lower bounds

We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike the boolean setting, there has been no concrete evidence demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little understanding whether algebraic circuits are expressive enough to support "cryptography" secure against algebraic circuits. Following a similar result of Williams in the boolean setting, we show that the existence of an algebraic natural proofs barrier is equivalent to the existence of succinct derandomization of the polynomial identity testing problem. That is, whether the coefficient vectors of polylog(N)-degree polylog(N)-size circuits is a hitting set for the class of poly(N)-degree poly(N)-size circuits. Further, we give an explicit universal construction showing that if such a succinct hitting set exists, then our universal construction suffices. Further, we assess the existing literature constructing hitting sets for restricted classes of algebraic circuits and observe that none of them are succinct as given. Yet, we show how to modify some of these constructions to obtain succinct hitting sets. This constitutes the first evidence supporting the existence of an algebraic natural proofs barrier. Our framework is similar to the Geometric Complexity Theory (GCT) program of Mulmuley and Sohoni, except that here we emphasize constructiveness of the proofs while the GCT program emphasizes symmetry. Nevertheless, our succinct hitting sets have relevance to the GCT program as they imply lower bounds for the complexity of the defining equations of polynomials computed by small circuits.

[1]  Éva Tardos,et al.  The gap between monotone and non-monotone circuit complexity is exponential , 1988, Comb..

[2]  Noam Nisan,et al.  Lower bounds for non-commutative computation , 1991, STOC '91.

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

[4]  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.

[5]  Ran Raz,et al.  Balancing Syntactically Multilinear Arithmetic Circuits , 2008, computational complexity.

[6]  Noam Nisan,et al.  Hardness vs Randomness , 1994, J. Comput. Syst. Sci..

[7]  A. Razborov.  Lower bounds on the size of bounded depth circuits over a complete basis with logical addition , 1987 .

[8]  Markus Bläser,et al.  Generalized matrix completion and algebraic natural proofs , 2018, Electron. Colloquium Comput. Complex..

[9]  Thomas Thierauf,et al.  Bipartite perfect matching is in quasi-NC , 2016, STOC.

[10]  Sébastien Tavenas,et al.  Improved bounds for reduction to depth 4 and depth 3 , 2013, Inf. Comput..

[11]  Nutan Limaye,et al.  An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[12]  Timothy Y. Chow.  Almost-natural proofs , 2011, J. Comput. Syst. Sci..

[13]  R. Solovay,et al.  Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question , 1975 .

[14]  Ryan Williams,et al.  Natural proofs versus derandomization , 2012, STOC '13.

[15]  Nitin Saxena,et al.  Identity Testing for Constant-Width, and Any-Order, Read-Once Oblivious Arithmetic Branching Programs , 2016, Theory Comput..

[16]  Nitin Saxena,et al.  Jacobian Hits Circuits: Hitting Sets, Lower Bounds for Depth-D Occur-k Formulas and Depth-3 Transcendence Degree-k Circuits , 2016, SIAM J. Comput..

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

[18]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[19]  Olaf Beyersdorff,et al.  Dependency Schemes in QBF Calculi: Semantics and Soundness , 2016, QBF@SAT.

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

[21]  Ran Raz,et al.  Separation of Multilinear Circuit and Formula Size , 2006, Theory Comput..

[22]  Neeraj Kayal.  An exponential lower bound for the sum of powers of bounded degree polynomials , 2012, Electron. Colloquium Comput. Complex..

[23]  Nitin Saxena,et al.  From Sylvester-Gallai Configurations to Rank Bounds: Improved Black-Box Identity Test for Depth-3 Circuits , 2010, FOCS.

[24]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[25]  Karl Bringmann,et al.  On Algebraic Branching Programs of Small Width , 2017, Electron. Colloquium Comput. Complex..

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

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

[28]  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.

[29]  Andrew Chi-Chih Yao,et al.  Separating the Polynomial-Time Hierarchy by Oracles (Preliminary Version) , 1985, FOCS.

[30]  Stefan Lucks,et al.  Pseudorandom functions in $ \textit{TC}^{0} $ and cryptographic limitations to proving lower bounds , 2001, computational complexity.

[31]  Noam Nisan,et al.  Pseudorandom bits for constant depth circuits , 1991, Comb..

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

[33]  Thomas Thierauf,et al.  Linear Matroid Intersection is in Quasi-NC , 2017, computational complexity.

[34]  Ketan Mulmuley,et al.  Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems , 2002, SIAM J. Comput..

[35]  Moni Naor,et al.  Number-theoretic constructions of efficient pseudo-random functions , 2004, JACM.

[36]  Zeev Dvir,et al.  Hardness-randomness tradeoffs for bounded depth arithmetic circuits , 2008, SIAM J. Comput..

[37]  Neeraj Kayal,et al.  Approaching the Chasm at Depth Four , 2013, 2013 IEEE Conference on Computational Complexity.

[38]  Russell Impagliazzo,et al.  Learning Algorithms from Natural Proofs , 2016, CCC.

[39]  Noam Nisan,et al.  Lower bounds on arithmetic circuits via partial derivatives , 2005, computational complexity.

[40]  Venkatesan Guruswami,et al.  Dimension Expanders via Rank Condensers , 2014, Electron. Colloquium Comput. Complex..

[41]  Ryan Williams.  Nonuniform ACC Circuit Lower Bounds , 2014, JACM.

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

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

[44]  Mark Braverman.  Poly-logarithmic Independence Fools AC0 Circuits , 2009, Computational Complexity Conference.

[45]  Ramprasad Saptharishi,et al.  Hitting sets for multilinear read-once algebraic branching programs, in any order , 2014, STOC.

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

[47]  Joshua A. Grochow.  Unifying Known Lower Bounds via Geometric Complexity Theory , 2013, 2014 IEEE 29th Conference on Computational Complexity (CCC).

[48]  Mark Braverman,et al.  Poly-logarithmic Independence Fools AC^0 Circuits , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[49]  Adam R. Klivans,et al.  Learning Arithmetic Circuits via Partial Derivatives , 2003, COLT.

[50]  John Gill,et al.  Relativizations of the P =? NP Question , 1975, SIAM J. Comput..

[51]  Amir Shpilka,et al.  Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in , 2011, Comb..

[52]  Joshua A. Grochow,et al.  Boundaries of VP and VNP , 2016, ICALP.

[53]  Ryan Williams,et al.  The Circuit-Input Game, Natural Proofs, and Testing Circuits With Data , 2015, ITCS.

[54]  Nitin Saxena,et al.  Progress on Polynomial Identity Testing , 2009, Bull. EATCS.

[55]  Stuart J. Berkowitz,et al.  On Computing the Determinant in Small Parallel Time Using a Small Number of Processors , 1984, Inf. Process. Lett..

[56]  Amir Shpilka,et al.  Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas , 2016, computational complexity.

[57]  Shubhangi Saraf,et al.  On the Power of Homogeneous Depth 4 Arithmetic Circuits , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[58]  NaorMoni,et al.  Number-theoretic constructions of efficient pseudo-random functions , 2004 .

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

[60]  Neeraj Kayal,et al.  An exponential lower bound for homogeneous depth four arithmetic circuits with bounded bottom fanin , 2012, Electron. Colloquium Comput. Complex..

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

[62]  Vikraman Arvind,et al.  Perspectives in Computational Complexity: The Somenath Biswas Anniversary Volume , 2014 .

[63]  Noga Alon,et al.  Simple construction of almost k-wise independent random variables , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

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

[65]  Silvio Micali,et al.  How to construct random functions , 1986, JACM.

[66]  Nutan Limaye,et al.  Lower bounds for depth 4 formulas computing iterated matrix multiplication , 2014, STOC.

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

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

[69]  LundCarsten,et al.  Algebraic methods for interactive proof systems , 1992 .

[70]  Michael A. Forbes.  Deterministic Divisibility Testing via Shifted Partial Derivatives , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[71]  V. Vinay,et al.  The Chasm at Depth Four, and Tensor Rank : Old results, new insights , 2016, Electron. Colloquium Comput. Complex..

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

[73]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[74]  Johan Håstad,et al.  Almost optimal lower bounds for small depth circuits , 1986, STOC '86.

[75]  Moni Naor,et al.  Small-bias probability spaces: efficient constructions and applications , 1990, STOC '90.

[76]  Noga Alon,et al.  The monotone circuit complexity of boolean functions , 1987, Comb..

[77]  Markus Bläser,et al.  Explicit tensors , 2015 .

[78]  Miklós Ajtai,et al.  ∑11-Formulae on finite structures , 1983, Ann. Pure Appl. Log..

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

[80]  J. Hartmanis,et al.  On the Computational Complexity of Algorithms , 1965 .

[81]  Roman Smolensky,et al.  Algebraic methods in the theory of lower bounds for Boolean circuit complexity , 1987, STOC.

[82]  CRAIG ALAN FEINSTEIN,et al.  P = NP , 2003 .

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

[84]  Avi Wigderson,et al.  Algebrization: A New Barrier in Complexity Theory , 2009, TOCT.

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

[86]  Amir Shpilka,et al.  On identity testing of tensors, low-rank recovery and compressed sensing , 2011, STOC '12.

[87]  Meena Mahajan,et al.  Non-Commutative Arithmetic Circuits: Depth Reduction and Size Lower Bounds , 1998, Theor. Comput. Sci..

[88]  Nitin Saxena,et al.  Progress on Polynomial Identity Testing - II , 2014, Electron. Colloquium Comput. Complex..

[89]  Alexander A. Razborov,et al.  Natural Proofs , 1997, J. Comput. Syst. Sci..

[90]  Neeraj Kayal,et al.  A super-polynomial lower bound for regular arithmetic formulas , 2014, STOC.

[91]  Ran Raz,et al.  Lower Bounds and Separations for Constant Depth Multilinear Circuits , 2008, Computational Complexity Conference.

[92]  Michael E. Saks,et al.  Towards an algebraic natural proofs barrier via polynomial identity testing , 2017, Electron. Colloquium Comput. Complex..

[93]  Avi Wigderson,et al.  Proof Complexity Lower Bounds from Algebraic Circuit Complexity , 2016, Electron. Colloquium Comput. Complex..

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

[95]  Avi Wigderson,et al.  Barriers for Rank Methods in Arithmetic Complexity , 2017, Electron. Colloquium Comput. Complex..

[96]  Dominic Welsh,et al.  COMPLETENESS AND REDUCTION IN ALGEBRAIC COMPLEXITY THEORY (Algorithms and Computation in Mathematics 7) By PETER BÜRGISSER: 168 pp., $44.50, ISBN 3-540-66752-0 (Springer, Berlin, 2000). , 2002 .

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

[98]  Noam Nisan,et al.  Pseudorandom generators for space-bounded computation , 1992, Comb..

[99]  Neeraj Kayal,et al.  Arithmetic Circuits: A Chasm at Depth 3 , 2016, SIAM J. Comput..

[100]  Adi Shamir,et al.  IP = PSPACE , 1992, JACM.

[101]  Leonid A. Levin,et al.  A Pseudorandom Generator from any One-way Function , 1999, SIAM J. Comput..

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

[103]  Nitin Saxena,et al.  Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs , 2014, computational complexity.

[104]  Joshua A. Grochow.  Unifying and generalizing known lower bounds via geometric complexity theory , 2013, ArXiv.

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

[106]  Ran Raz,et al.  Lower Bounds and Separations for Constant Depth Multilinear Circuits , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[107]  Ramprasad Saptharishi,et al.  Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity , 2016, Computational Complexity Conference.

[108]  Meena Mahajan,et al.  A combinatorial algorithm for the determinant , 1997, SODA '97.

[109]  Ran Raz,et al.  Multi-linear formulas for permanent and determinant are of super-polynomial size , 2004, STOC '04.

[110]  Nitin Saxena,et al.  Hitting-Sets for ROABP and Sum of Set-Multilinear Circuits , 2014, SIAM J. Comput..

[111]  Nitin Saxena,et al.  Quasi-polynomial Hitting-set for Set-depth-Delta Formulas , 2012, Electron. Colloquium Comput. Complex..

[112]  Carsten Lund,et al.  Algebraic methods for interactive proof systems , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.