New Wine into Old Wineskins: A Survey of Some Pebbling Classics with Supplemental Results

Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen renewed interest in pebbling in the field of proof complexity. This is a survey of some classical theorems in pebbling, as well as a couple of new ones, with a focus on results that have proven relevant in proof complexity applications. THIS IS A MANUSCRIPT IN PREPARATION. See Section 1.4 on page 6 for a report on the status of the different sections of the paper. Questions, corrections, clarifications or any other comments are most welcome!

[1]  Robert E. Tarjan,et al.  Variations of a pebble game on graphs , 1978 .

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

[3]  Henry A. Kautz,et al.  Using Problem Structure for Efficient Clause Learning , 2003, SAT.

[4]  J. A. Robinson,et al.  A Machine-Oriented Logic Based on the Resolution Principle , 1965, JACM.

[5]  Bart Selman,et al.  The state of SAT , 2007, Discret. Appl. Math..

[6]  Andrzej Lingas.  A PSPACE Complete Problem Related to a Pebble Game , 1978, ICALP.

[7]  Uwe Schöning.  Smaller superconcentrators of density 28 , 2006, Inf. Process. Lett..

[8]  David S. Johnson,et al.  The NP-Completeness Column: An Ongoing Guide , 1982, J. Algorithms.

[9]  Oliver Kullmann,et al.  Investigating a general hierarchy of polynomially decidable classes of CNF's based on short tree-like resolution proofs , 1999, Electron. Colloquium Comput. Complex..

[10]  N. S. Narayanaswamy,et al.  An Optimal Lower Bound for Resolution with 2-Conjunctions , 2002, MFCS.

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

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

[13]  John E. Savage,et al.  Space-time trade-offs on the FFT algorithm , 1978, IEEE Trans. Inf. Theory.

[14]  Michael C. Loui.  A Note on the Pebble Game , 1980, Inf. Process. Lett..

[15]  Zvi Galil.  On Resolution with Clauses of Bounded Size , 1977, SIAM J. Comput..

[16]  Neil Thapen,et al.  Resolution and Pebbling Games , 2005, SAT.

[17]  Friedhelm Meyer auf der Heide,et al.  A Comparison of two Variations of a Pebble Game on Graphs , 1981, Theor. Comput. Sci..

[18]  Toniann Pitassi,et al.  The PSPACE-Completeness of Black-White Pebbling , 2010, SIAM J. Comput..

[19]  Bala Kalyanasundaram,et al.  On the power of white pebbles , 1991, Comb..

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

[21]  Jacobo Torán,et al.  A combinatorial characterization of treelike resolution space , 2003, Inf. Process. Lett..

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

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

[24]  Jakob Nordström.  A simplified way of proving trade-off results for resolution , 2009, Inf. Process. Lett..

[25]  Robert E. Tarjan,et al.  The Space Complexity of Pebble Games on Trees , 1980, Inf. Process. Lett..

[26]  Zvi Galil,et al.  Explicit Constructions of Linear-Sized Superconcentrators , 1981, J. Comput. Syst. Sci..

[27]  Alasdair Urquhart,et al.  The Resolution Width Problem is EXPTIME-Complete , 2006, Electron. Colloquium Comput. Complex..

[28]  Hans K. Buning,et al.  Propositional Logic: Deduction and Algorithms , 1999 .

[29]  Martin Tompa.  Time-Space Tradeoffs for Computing Functions, Using Connectivity Properties of Their Circuits , 1980, J. Comput. Syst. Sci..

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

[31]  Robert E. Wilber.  White Pebbles Help , 1988, J. Comput. Syst. Sci..

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

[33]  Alexander A. Razborov,et al.  Lower bounds for the polynomial calculus , 1998, computational complexity.

[34]  Johan Håstad,et al.  Towards an optimal separation of space and length in resolution , 2008, Theory Comput..

[35]  Robert E. Tarjan,et al.  The Pebbling Problem is Complete in Polynomial Space , 1980, SIAM J. Comput..

[36]  Nicholas Pippenger.  A Time-Space Trade-Off , 1978, JACM.

[37]  Jakob Nordström.  Short Proofs May Be Spacious : Understanding Space in Resolution , 2008 .

[38]  Robert E. Tarjan,et al.  Asymptotically tight bounds on time-space trade-offs in a pebble game , 1982, JACM.

[39]  Jacobo Torán.  Lower Bounds for Space in Resolution , 1999, CSL.

[40]  Jan Johannsen.  Exponential Incomparability of Tree-like and Ordered Resolution , 2013 .

[41]  Ralph E. Gomory,et al.  Outline of an Algorithm for Integer Solutions to Linear Programs and An Algorithm for the Mixed Integer Problem , 2010, 50 Years of Integer Programming.

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

[43]  Allen Van Gelder,et al.  Pool Resolution and Its Relation to Regular Resolution and DPLL with Clause Learning , 2005, LPAR.

[44]  Jacobo Torán,et al.  Space and Width in Propositional Resolution (Column: Computational Complexity) , 2004, Bull. EATCS.

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

[46]  Alexander A. Razborov,et al.  Pseudorandom generators hard for $k$-DNF resolution and polynomial calculus resolution , 2015 .

[47]  Kazuo Iwama.  Complexity of Finding Short Resolution Proofs , 1997, MFCS.

[48]  Russell Impagliazzo,et al.  Lower bounds for the polynomial calculus and the Gröbner basis algorithm , 1999, computational complexity.

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

[50]  Friedhelm Meyer auf der Heide.  A Comparison Between Two Variations of a Pebble Game on Graphs , 1979, ICALP.

[51]  Jakob Nordström.  Narrow Proofs May Be Spacious: Separating Space and Width in Resolution , 2009, SIAM J. Comput..

[52]  Alasdair Urquhart,et al.  Comments on ECCC Report TR06-133: The Resolution Width Problem is EXPTIME-Complete , 2009, Electron. Colloquium Comput. Complex..

[53]  Massimo Lauria,et al.  Optimality of size-degree tradeoffs for polynomial calculus , 2010, TOCL.

[54]  Samuel R. Buss,et al.  A Switching Lemma for Small Restrictions and Lower Bounds for k-DNF Resolution , 2004, SIAM J. Comput..

[55]  Jakob Nordstr.  PEBBLE GAMES, PROOF COMPLEXITY, AND TIME-SPACE TRADE-OFFS ∗ , 2013 .

[56]  Thomas Lengauer.  Black-white pebbles and graph separation , 2004, Acta Informatica.

[57]  John E. Savage,et al.  Space-time tradeoffs for linear recursion , 2005, Mathematical systems theory.

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

[59]  Toniann Pitassi,et al.  Clause Learning Can Effectively P-Simulate General Propositional Resolution , 2008, AAAI.

[60]  Leslie G. Valiant,et al.  Graph-Theoretic Properties in computational Complexity , 1976, J. Comput. Syst. Sci..

[61]  Alexander Hertel,et al.  Applications of Games to Propositional Proof Complexity , 2008 .

[62]  Leslie G. Valiant,et al.  Size Bounds for Superconcentrators , 1983, Theor. Comput. Sci..

[63]  Rüdiger Reischuk.  Improved bounds on the problem of time-space trade-off in the pebble game , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[64]  John E. Savage,et al.  Space-Time Tradeoffs for Oblivious Interger Multiplications , 1979, ICALP.

[65]  Maria M. Klawe,et al.  A tight bound for black and white pebbles on the pyramid , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[66]  Robert E. Tarjan,et al.  Time-Space Trade-Offs in a Pebble Game , 1977, ICALP.

[67]  Joao Marques-Silva,et al.  GRASP-A new search algorithm for satisfiability , 1996, Proceedings of International Conference on Computer Aided Design.

[68]  Dexter Kozen,et al.  Lower bounds for natural proof systems , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[69]  Toniann Pitassi,et al.  Exponential Time/Space Speedups for Resolution and the PSPACE-completeness of Black-White Pebbling , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

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

[71]  John E. Savage,et al.  Models of computation - exploring the power of computing , 1998 .