Monotone circuit lower bounds from resolution

For any unsatisfiable CNF formula F that is hard to refute in the Resolution proof system, we show that a gadget-composed version of F is hard to refute in any proof system whose lines are computed by efficient communication protocols—or, equivalently, that a monotone function associated with F has large monotone circuit complexity. Our result extends to monotone real circuits, which yields new lower bounds for the Cutting Planes proof system.

[1]  Toniann Pitassi,et al.  Communication lower bounds via critical block sensitivity , 2013, STOC.

[2]  Pavel Pudlák.  On extracting computations from propositional proofs (a survey) , 2010, FSTTCS.

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

[4]  Avi Wigderson,et al.  Monotone circuits for connectivity require super-logarithmic depth , 1990, STOC '88.

[5]  Ran Raz,et al.  Resolution over linear equations and multilinear proofs , 2007, Ann. Pure Appl. Log..

[6]  Pritish Kamath,et al.  Adventures in Monotone Complexity and TFNP , 2018, Electron. Colloquium Comput. Complex..

[7]  Akira Maruoka,et al.  The Potential of the Approximation Method , 2004, SIAM J. Comput..

[8]  Toniann Pitassi,et al.  Lower Bounds for Lovász-Schrijver Systems and Beyond Follow from Multiparty Communication Complexity , 2005, ICALP.

[9]  Christer Berg,et al.  Symmetric approximation arguments for monotone lower bounds without sunflowers , 1999, computational complexity.

[10]  Jakob Nordström,et al.  On the virtue of succinct proofs: amplifying communication complexity hardness to time-space trade-offs in proof complexity , 2012, STOC '12.

[11]  Alexander A. Razborov,et al.  Proof Complexity and Beyond , 2018 .

[12]  Armin Haken,et al.  Counting bottlenecks to show monotone P/spl ne/NP , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[13]  Albert Atserias,et al.  A combinatorial characterization of resolution width , 2008, J. Comput. Syst. Sci..

[14]  Ran Raz,et al.  Separation of the Monotone NC Hierarchy , 1999, Comb..

[15]  Dmitry Itsykson,et al.  Lower Bounds for Splittings by Linear Combinations , 2014, MFCS.

[16]  Alexander A. Razborov.  On Small Size Approximation Models , 2013, The Mathematics of Paul Erdős I.

[17]  Dmitry Sokolov.  Dag-Like Communication and Its Applications , 2016, CSR.

[18]  Anna Gál.  A characterization of span program size and improved lower bounds for monotone span programs , 1998, STOC '98.

[19]  Moni Naor,et al.  Search problems in the decision tree model , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[20]  A. Razborov.  Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic , 1995 .

[21]  Toniann Pitassi,et al.  Random CNFs are Hard for Cutting Planes , 2017, Electron. Colloquium Comput. Complex..

[22]  Xiaodi Wu,et al.  Raz-McKenzie simulation with the inner product gadget , 2017, Electron. Colloquium Comput. Complex..

[23]  Pavel Pudlák,et al.  Random Formulas, Monotone Circuits, and Interpolation , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[24]  A. Razborov.  Communication Complexity , 2011 .

[25]  Toniann Pitassi,et al.  Hardness amplification in proof complexity , 2009, STOC '10.

[26]  Ketan Mulmuley,et al.  A fast parallel algorithm to compute the rank of a matrix over an arbitrary field , 1986, STOC '86.

[27]  Shi-Chun Tsai,et al.  A note on the bottleneck counting argument , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.

[28]  Alexander A. Razborov,et al.  On the method of approximations , 1989, STOC '89.

[29]  Toniann Pitassi,et al.  Propositional Proof Complexity: Past, Present and Future , 2001, Bull. EATCS.

[30]  Ran Raz,et al.  Higher lower bounds on monotone size , 2000, STOC '00.

[31]  A. Wigderson.  The Fusion Method for Lower Bounds in Circuit Complexity , 2003 .

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

[33]  Benjamin Rossman.  The Monotone Complexity of k-clique on Random Graphs , 2010, FOCS.

[34]  Eli Ben-Sasson,et al.  Short proofs are narrow—resolution made simple , 2001, JACM.

[35]  Stasys Jukna.  Finite limits and monotone computations: the lower bounds criterion , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.

[36]  William J. Cook,et al.  On the complexity of cutting-plane proofs , 1987, Discret. Appl. Math..

[37]  Alasdair Urquhart,et al.  Formal Languages]: Mathematical Logic--mechanical theorem proving , 2022 .

[38]  Pavel Pudlák.  Proofs as Games , 2000, Am. Math. Mon..

[39]  Shi-Chun Tsai,et al.  On the bottleneck counting argument , 2000, Theor. Comput. Sci..

[40]  Marc Vinyals,et al.  How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity) , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[41]  Arkadev Chattopadhyay,et al.  Simulation Theorems via Pseudo-random Properties , 2017, computational complexity.

[42]  Toniann Pitassi,et al.  Random Θ(log n)-CNFs Are Hard for Cutting Planes , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[43]  Stephen A. Cook,et al.  An Exponential Lower Bound for the Size of Monotone Real Circuits , 1999, J. Comput. Syst. Sci..

[44]  Alexander A. Razborov.  Guest Column: Proof Complexity and Beyond , 2016, SIGA.

[45]  Czech Republickrajicek.  Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic , 2007 .

[46]  Toniann Pitassi,et al.  Query-to-Communication Lifting for PNP , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[47]  Pavel Pudlák,et al.  Lower bounds for resolution and cutting plane proofs and monotone computations , 1997, Journal of Symbolic Logic.

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

[49]  Shachar Lovett,et al.  Rectangles Are Nonnegative Juntas , 2015, SIAM J. Comput..

[50]  Discretely Ordered Modules as a First-Order Extension of The Cutting Planes Proof System , 1998, J. Symb. Log..

[51]  Stasys Jukna,et al.  Boolean Function Complexity Advances and Frontiers , 2012, Bull. EATCS.

[52]  Alexander A. Razborov,et al.  A New Kind of Tradeoffs in Propositional Proof Complexity , 2016, J. ACM.

[53]  Pavel Pudlák,et al.  A note on monotone real circuits , 2018, Inf. Process. Lett..

[54]  HierarchyRan Raz,et al.  Separation of the Monotone NC , 1999 .

[55]  Benjamin Rossman,et al.  The Monotone Complexity of k-clique on Random Graphs , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[56]  Toniann Pitassi,et al.  Query-to-Communication Lifting for BPP , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[57]  Toniann Pitassi,et al.  Deterministic Communication vs. Partition Number , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[58]  Toniann Pitassi,et al.  Query-to-Communication Lifting for P^NP , 2017, Electron. Colloquium Comput. Complex..