Just a Pebble Game

The two-player pebble game of Dymond-Tompa is identified as a barrier for existing techniques to save space or to speed up parallel algorithms for evaluation problems. Many combinatorial lower bounds to study I versus NI and NC versus P under different restricted settings scale in the same way as the pebbling algorithm of Dymond-Tompa. These lower bounds include, (1) the monotone separation of m-I from m-NI by studying the size of monotone switching networks in Potechin '10; (2) a new semantic separation of NC from P and of NCi from NCi+1 by studying circuit depth, based on the techniques developed for the semantic separation of NC1 from NC2 by the universal composition relation in Edmonds-Impagliazzo-Rudich-Sgall '01 and in Hastad- Wigderson '97; and (3) the monotone separation of m-NC from m-P and of m-NCi from m-NCi+1 by studying (a) the depth of monotone circuits in Raz-McKenzie '99; and (b) the size of monotone switching networks in Chan- Potechin '12. This supports the attempt to separate NC from P by focusing on depth complexity, and suggests the study of combinatorial invariants shaped by pebbling for proving lower bounds. An application to proof complexity gives tight bounds for the size and the depth of some refinements of resolution refutations.

[1]  Christoph Berkholz,et al.  On the Complexity of Finding Narrow Proofs , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

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

[3]  Emanuele Viola,et al.  One-way multiparty communication lower bound for pointer jumping with applications , 2009, Comb..

[4]  Pierre McKenzie,et al.  Oracle branching programs and Logspace versus P , 1989, Inf. Comput..

[5]  Michael Alekhnovich,et al.  Space Complexity in Propositional Calculus , 2002, SIAM J. Comput..

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

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

[8]  Avi Wigderson,et al.  Composition of the Universal Relation , 1990, Advances In Computational Complexity Theory.

[9]  Walter J. Savitch,et al.  Relationships Between Nondeterministic and Deterministic Tape Complexities , 1970, J. Comput. Syst. Sci..

[10]  Eli Ben-Sasson,et al.  Near Optimal Separation Of Tree-Like And General Resolution , 2004, Comb..

[11]  Ryan Williams,et al.  Parallelizing Time with Polynomial Circuits , 2005, SPAA '05.

[12]  Charles H. Bennett.  Time/Space Trade-Offs for Reversible Computation , 1989, SIAM J. Comput..

[13]  Noam Nisan,et al.  Constant depth circuits, Fourier transform, and learnability , 1993, JACM.

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

[15]  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.

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

[17]  Mihalis Yannakakis,et al.  On monotone formulae with restricted depth , 1984, STOC '84.

[18]  Aaron Potechin,et al.  Bounds on Monotone Switching Networks for Directed Connectivity , 2009, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[19]  Michael Alekhnovich,et al.  An Exponential Separation between Regular and General Resolution , 2007, Theory Comput..

[20]  Alasdair Urquhart.  The Depth of Resolution Proofs , 2011, Stud Logica.

[21]  Amit Chakrabarti,et al.  Lower Bounds for Multi-Player Pointer Jumping , 2007, Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).

[22]  Alasdair Urquhart,et al.  Game Characterizations and the PSPACE-Completeness of Tree Resolution Space , 2007, CSL.

[23]  Shigeki Iwata,et al.  Classes of Pebble Games and Complete Problems , 1979, SIAM J. Comput..

[24]  Johan Håstad,et al.  A Simple Lower Bound for Monotone Clique Using a Communication Game , 1992, Inf. Process. Lett..

[25]  Carl Hewitt,et al.  Comparative Schematology , 1970 .

[26]  Hilary Putnam,et al.  A Computing Procedure for Quantification Theory , 1960, JACM.

[27]  Jacobo Torán,et al.  Space Bounds for Resolution , 1999, STACS.

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

[29]  Ravi Sethi,et al.  Complete register allocation problems , 1973, SIAM J. Comput..

[30]  Ming Li,et al.  Reversible Simulation of Irreversible Computation by Pebble Games , 1997, ArXiv.

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

[32]  Harry Buhrman,et al.  Time and Space Bounds for Reversible Simulation , 2001, ICALP.

[33]  Russell Impagliazzo,et al.  Communication complexity towards lower bounds on circuit depth , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[34]  Joshua Brody,et al.  Sublinear Communication Protocols for Multi-Party Pointer Jumping and a Related Lower Bound , 2008, STACS.

[35]  Robert E. Tarjan,et al.  Space bounds for a game on graphs , 1976, STOC '76.

[36]  Toniann Pitassi,et al.  Rank Bounds and Integrality Gaps for Cutting Planes Procedures , 2006, Theory Comput..

[37]  Shigeki Iwata,et al.  Some combinatorial game problems require Ω(nk) time , 1984, JACM.

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

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

[40]  Ryan Williams,et al.  Space-Efficient Reversible Simulations , 2000 .

[41]  Dustin Wehr,et al.  Lower bound for deterministic semantic-incremental branching programs solving GEN , 2011, ArXiv.

[42]  Patrick W. Dymond,et al.  Speedups of deterministic machines by synchronous parallel machines , 1983, J. Comput. Syst. Sci..

[43]  Jan Johannsen,et al.  Depth Lower Bounds for Monotone Semi-Unbounded Fan-in Circuits , 2001, RAIRO Theor. Informatics Appl..

[44]  Michael Sipser,et al.  Monotone Separation of Logarithmic Space from Logarithmic Depth , 1995, J. Comput. Syst. Sci..

[45]  Alasdair Urquhart,et al.  The Complexity of Propositional Proofs , 1995, Bulletin of Symbolic Logic.

[46]  Donald W. Loveland,et al.  A machine program for theorem-proving , 2011, CACM.

[47]  J. Kraj.  On the Weak Pigeonhole Principle , 2001 .

[48]  Walter L. Ruzzo.  On Uniform Circuit Complexity , 1981, J. Comput. Syst. Sci..

[49]  Samuel R. Buss,et al.  An Optimal Parallel Algorithm for Formula Evaluation , 1992, SIAM J. Comput..

[50]  Grant Schoenebeck,et al.  Linear Level Lasserre Lower Bounds for Certain k-CSPs , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[51]  G. S. Tseitin.  On the Complexity of Derivation in Propositional Calculus , 1983 .

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

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

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

[55]  Neil D. Jones,et al.  Complete problems for deterministic polynomial time , 1974, STOC '74.

[56]  Ming Li,et al.  Reversibility and adiabatic computation: trading time and space for energy , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[57]  Madhur Tulsiani.  CSP gaps and reductions in the lasserre hierarchy , 2009, STOC '09.

[58]  Richard J. Lipton,et al.  Non-uniform Depth of Polynomial Time and Space Simulations , 2003, FCT.

[59]  Charles H. Bennett,et al.  Logical reversibility of computation , 1973 .

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

[61]  Mark Braverman,et al.  Pebbles and Branching Programs for Tree Evaluation , 2012, TOCT.

[62]  Eli Ben-Sasson,et al.  Size space tradeoffs for resolution , 2002, STOC '02.

[63]  Russell Impagliazzo,et al.  Formula Caching in DPLL , 2010, TOCT.

[64]  Leslie G. Valiant,et al.  On Time Versus Space , 1977, JACM.

[65]  Michael Sipser,et al.  Structure in monotone complexity , 1991 .

[66]  Ravi B. Boppana,et al.  The Complexity of Finite Functions , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[67]  Russell Impagliazzo,et al.  Homogenization and the polynomial calculus , 2003, computational complexity.

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

[69]  Richard Královic.  Time and space complexity of reversible pebbling , 2004, RAIRO Theor. Informatics Appl..

[70]  Pierre McKenzie,et al.  Reversible space equals deterministic space , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.

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

[72]  Omer Reingold,et al.  Undirected connectivity in log-space , 2008, JACM.

[73]  Martin Tompa,et al.  A New Pebble Game that Characterizes Parallel Complexity Classes , 1986, FOCS.

[74]  Richard J. Lipton,et al.  Amplifying circuit lower bounds against polynomial time, with applications , 2012, computational complexity.

[75]  Aaron Potechin,et al.  Tight bounds for monotone switching networks via fourier analysis , 2012, STOC '12.

[76]  Ran Raz,et al.  Competing provers protocols for circuit evaluation , 2013, ITCS '13.

[77]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[78]  Ran Raz,et al.  Super-logarithmic depth lower bounds via the direct sum in communication complexity , 1995, computational complexity.

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

[80]  Maria Luisa Bonet,et al.  Exponential separations between restricted resolution and cutting planes proof systems , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[81]  Eli Ben-Sasson,et al.  Understanding Space in Proof Complexity: Separations and Trade-offs via Substitutions , 2011, ICS.

[82]  Stanislav Zivny,et al.  Relating Proof Complexity Measures and Practical Hardness of SAT , 2012, CP.

[83]  Jakob Nordström,et al.  Pebble Games, Proof Complexity, and Time-Space Trade-offs , 2013, Log. Methods Comput. Sci..

[84]  C. Y. Lee.  Representation of switching circuits by binary-decision programs , 1959 .

[85]  Russell Impagliazzo,et al.  A lower bound for DLL algorithms for k-SAT (preliminary version) , 2000, SODA '00.

[86]  Siu On Chan,et al.  Approximation resistance from pairwise independent subgroups , 2013, STOC '13.

[87]  Dima Grigoriev,et al.  Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity , 2001, Theor. Comput. Sci..

[88]  J. Krajícek.  On the weak pigeonhole principle , 2001 .

[89]  Jochen Messner,et al.  On Minimal Unsatisfiability and Time-Space Trade-offs for k-DNF Resolution , 2009, ICALP.