Circuit Complexity, Proof Complexity, and Polynomial Identity Testing

We introduce a new and natural algebraic proof system, which has tight connections to (algebraic) circuit complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic circuits (VNP≠VP). As a corollary, super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs (as opposed to the usual measure of number of monomials) imply the Permanent versus Determinant Conjecture. Note that, prior to our work, there was no proof system for which lower bounds on an arbitrary tautology implied any computational lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems, and begins to shed light on why proof complexity lower bounds for various proof systems have been so much harder than lower bounds on the corresponding circuit classes. In doing so, we highlight the importance of polynomial identity testing (PIT) for understanding proof complexity.

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

[2]  Neeraj Kayal,et al.  Lower Bounds for Depth-Three Arithmetic Circuits with small bottom fanin , 2016, computational complexity.

[3]  Ramprasad Saptharishi,et al.  An exponential lower bound for homogeneous depth-5 circuits over finite fields , 2015, Electron. Colloquium Comput. Complex..

[4]  Jakob Nordström,et al.  A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds , 2015, CCC.

[5]  Fu Li,et al.  Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs , 2015, Computational Complexity Conference.

[6]  Russell Impagliazzo,et al.  Tighter Connections between Derandomization and Circuit Lower Bounds , 2015, APPROX-RANDOM.

[7]  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.

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

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

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

[11]  J. M. Landsberg,et al.  Geometric complexity theory: an introduction for geometers , 2013, ANNALI DELL'UNIVERSITA' DI FERRARA.

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

[13]  Ketan Mulmuley,et al.  The GCT program toward the P vs. NP problem , 2012, Commun. ACM.

[14]  Iddo Tzameret,et al.  Short proofs for the determinant identities , 2012, STOC '12.

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

[16]  Craig Huneke,et al.  Commutative Algebra I , 2012 .

[17]  Rahul Santhanam,et al.  Stronger Lower Bounds and Randomness-Hardness Tradeoffs using Associated Algebraic Complexity Classes , 2011, Electron. Colloquium Comput. Complex..

[18]  Jürgen Herzog,et al.  Grobner Bases in Commutative Algebra , 2011 .

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

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

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

[22]  K. Mulmuley,et al.  Geometric Complexity Theory VI : The flip via positivity Dedicated to Sri , 2010 .

[23]  Iddo Tzameret,et al.  The Proof Complexity of Polynomial Identities , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[24]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

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

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

[27]  Ran Raz,et al.  The Strength of Multilinear Proofs , 2008, computational complexity.

[28]  A. Wigderson,et al.  Algebrization: a new barrier in complexity theory , 2008, STOC.

[29]  Ketan Mulmuley,et al.  Geometric Complexity Theory II: Towards Explicit Obstructions for Embeddings among Class Varieties , 2006, SIAM J. Comput..

[30]  Guillaume Malod,et al.  Characterizing Valiant's algebraic complexity classes , 2006, J. Complex..

[31]  Nathan Segerlind,et al.  The Complexity of Propositional Proofs , 2007, Bull. Symb. Log..

[32]  Zbigniew Jelonek,et al.  On the effective Nullstellensatz , 2005 .

[33]  Lance Fortnow,et al.  BPP has subexponential time simulations unlessEXPTIME has publishable proofs , 2005, computational complexity.

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

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

[36]  Stephen A. Cook,et al.  The proof complexity of linear algebra , 2002, Proceedings 17th Annual IEEE Symposium on Logic in Computer Science.

[37]  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 .

[38]  Dima Grigoriev,et al.  Algebraic proof systems over formulas , 2003, Electron. Colloquium Comput. Complex..

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

[40]  Peter Bürgisser,et al.  Completeness and Reduction in Algebraic Complexity Theory , 2000, Algorithms and computation in mathematics.

[41]  Michael Alekhnovich,et al.  Space complexity in propositional calculus , 2000, STOC '00.

[42]  Ran Raz,et al.  On Interpolation and Automatization for Frege Systems , 2000, SIAM J. Comput..

[43]  Peter Bürgisser.  Cook's versus Valiant's hypothesis , 2000, Theor. Comput. Sci..

[44]  Vladimir Voevodsky,et al.  A1-homotopy theory of schemes , 1999 .

[45]  Teresa Krick,et al.  Sharp estimates for the arithmetic Nullstellensatz , 1999, math/9911094.

[46]  Toniann Pitassi,et al.  Non-Automatizability of Bounded-Depth Frege Proofs , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[47]  Ketan Mulmuley,et al.  Lower Bounds in a Parallel Model without Bit Operations , 1999, SIAM J. Comput..

[48]  R. Lazarsfeld,et al.  A geometric effective Nullstellensatz , 1998, math/9810004.

[49]  C SIAMJ..  LOWER BOUNDS IN A PARALLEL MODEL WITHOUT BIT OPERATIONS , 1999 .

[50]  W. Brownawell.  A pure power product version of the Hilbert Nullstellensatz. , 1998 .

[51]  J. Kollár.  Effective Nullstellensatz for arbitrary ideals , 1998, math/9805091.

[52]  T. Pitassi.  Unsolvable Systems of Equations and Proof Complexity , 1998 .

[53]  A. Koll.  Effective Nullstellensatz for Arbitrary Ideals , 1998 .

[54]  Martín Sombra.  A Sparse Effective Nullstellensatz , 1997, alg-geom/9710003.

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

[56]  Mart́ın Sombra,et al.  A Sparse Effective Nullstellensatz 1 , 1997 .

[57]  Pascal Koiran.  Hilbert's Nullstellensatz Is in the Polynomial Hierarchy , 1996, J. Complex..

[58]  Russell Impagliazzo,et al.  Using the Groebner basis algorithm to find proofs of unsatisfiability , 1996, STOC '96.

[59]  Toniann Pitassi,et al.  Algebraic Propositional Proof Systems , 1996, Descriptive Complexity and Finite Models.

[60]  Toniann Pitassi,et al.  Towards lower bounds for bounded-depth Frege proofs with modular connectives , 1996, Proof Complexity and Feasible Arithmetics.

[61]  Michael Sipser,et al.  Introduction to the Theory of Computation , 1996, SIGA.

[62]  Jan Krajícek,et al.  Bounded arithmetic, propositional logic, and complexity theory , 1995, Encyclopedia of mathematics and its applications.

[63]  Miles Reid,et al.  Undergraduate Commutative Algebra , 1995 .

[64]  Ran Raz,et al.  Lower bounds for cutting planes proofs with small coefficients , 1995, STOC '95.

[65]  S. D. Chatterji.  Proceedings of the International Congress of Mathematicians , 1995 .

[66]  Jan Krajícek,et al.  An Exponenetioal Lower Bound to the Size of Bounded Depth Frege Proofs of the Pigeonhole Principle , 1995, Random Struct. Algorithms.

[67]  Jan Krajícek,et al.  Lower bounds on Hilbert's Nullstellensatz and propositional proofs , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[68]  Alexander A. Razborov,et al.  Natural Proofs , 2007 .

[69]  Jan Krajícek,et al.  Lower bounds to the size of constant-depth propositional proofs , 1994, Journal of Symbolic Logic.

[70]  Jan Kra,et al.  Lower Bounds to the Size of Constant-depth Propositional Proofs , 1994 .

[71]  Joan Feigenbaum,et al.  Random-Self-Reducibility of Complete Sets , 1993, SIAM J. Comput..

[72]  D. Mumford,et al.  What Can Be Computed in Algebraic Geometry , 1993, alg-geom/9304003.

[73]  Russell Impagliazzo,et al.  Exponential lower bounds for the pigeonhole principle , 1992, STOC '92.

[74]  Seinosuke Toda,et al.  Classes of Arithmetic Circuits Capturing the Complexity of Computing the Determinant , 1992 .

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

[76]  Noam Nisan,et al.  BPP has subexponential time simulations unless EXPTIME has publishable proofs , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

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

[78]  Volker Strassen,et al.  Algebraic Complexity Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[79]  Ernst W. Mayr,et al.  Membership in Plynomial Ideals over Q Is Exponential Space Complete , 1989, STACS.

[80]  Michael Eugene Stillman,et al.  On the Complexity of Computing Syzygies , 1988, J. Symb. Comput..

[81]  Miklós Ajtai,et al.  The complexity of the Pigeonhole Principle , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[82]  Erich Kaltofen,et al.  Improved Sparse Multivariate Polynomial Interpolation Algorithms , 1988, ISSAC.

[83]  Par Patrice Philippon,et al.  A propos du texte de W. D. Brownawell: “Bounds for the degrees in the Nullstellensatz” , 1988 .

[84]  J. Kollár.  Sharp effective Nullstellensatz , 1988 .

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

[86]  W. Brownawell.  Bounds for the degrees in the Nullstellensatz , 1987 .

[87]  Joachim von zur Gathen,et al.  Feasible Arithmetic Computations: Valiant's Hypothesis , 1987, J. Symb. Comput..

[88]  Michael Stillman,et al.  A theorem on refining division orders by the reverse lexicographic order , 1987 .

[89]  Michael Stillman,et al.  A criterion for detectingm-regularity , 1987 .

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

[91]  Leslie G. Valiant,et al.  NP is as easy as detecting unique solutions , 1985, STOC '85.

[92]  Armin Haken,et al.  The Intractability of Resolution , 1985, Theor. Comput. Sci..

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

[94]  David Masser,et al.  Fields of large transcendence degree generated by values of elliptic functions , 1983 .

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

[96]  A. Meyer,et al.  The complexity of the word problems for commutative semigroups and polynomial ideals , 1982 .

[97]  D. Mumford.  Algebraic Geometry I: Complex Projective Varieties , 1981 .

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

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

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

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

[102]  Stephen A. Cook,et al.  The Relative Efficiency of Propositional Proof Systems , 1979, Journal of Symbolic Logic.

[103]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

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

[105]  R. Y. Sharp.  ALGEBRAIC GEOMETRY I COMPLEX PROJECTIVE VARIETIES , 1978 .

[106]  D. Hilbert,et al.  Hilbert's invariant theory papers , 1978 .

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

[108]  A. Seidenberg.  Constructions in algebra , 1974 .

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

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

[111]  Michael Francis Atiyah,et al.  Introduction to commutative algebra , 1969 .

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

[113]  Grete Hermann,et al.  Die Frage der endlich vielen Schritte in der Theorie der Polynomideale , 1926 .