Pebble Games, Proof Complexity, and Time-Space Trade-offs

Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen a revival of interest in pebble games coming from the field of proof complexity. Pebbling has proven to be a useful tool for studying resolution-based proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing size-space trade-offs. This is a survey of research in proof complexity drawing on results and tools from pebbling, with a focus on proof space lower bounds and trade-offs between proof size and proof space.

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

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

[3]  Toniann Pitassi,et al.  Simplified and improved resolution lower bounds , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[4]  Archie Blake.  Canonical expressions in Boolean algebra , 1938 .

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

[6]  Adnan Darwiche,et al.  On the power of clause-learning SAT solvers as resolution engines , 2011, Artif. Intell..

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

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

[9]  Eli Ben-Sasson,et al.  Random Cnf’s are Hard for the Polynomial Calculus , 2010, computational complexity.

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

[11]  Roberto J. Bayardo,et al.  Using CSP Look-Back Techniques to Solve Real-World SAT Instances , 1997, AAAI/IAAI.

[12]  Yuval Filmus,et al.  Space Complexity in Polynomial Calculus , 2015, SIAM J. Comput..

[13]  Maria Luisa Bonet,et al.  On the automatizability of resolution and related propositional proof systems , 2002, Inf. Comput..

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

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

[18]  Samuel R. Buss,et al.  Resolution Trees with Lemmas: Resolution Refinements that Characterize DLL Algorithms with Clause Learning , 2008, Log. Methods Comput. Sci..

[19]  Michael Alekhnovich,et al.  Minimum propositional proof length is NP-hard to linearly approximate , 1998, Journal of Symbolic Logic.

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

[21]  Maria M. Klawe.  A Tight Bound for Black and White Pebbles on the Pyramid , 1983, FOCS.

[22]  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..

[23]  Stephen A. Cook,et al.  Storage Requirements for Deterministic Polynomial Time Recognizable Languages , 1976, J. Comput. Syst. Sci..

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

[25]  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).

[26]  Michael Alekhnovich,et al.  Lower bounds for polynomial calculus: non-binomial case , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[27]  J. P. Marques,et al.  GRASP : A Search Algorithm for Propositional Satisfiability , 1999 .

[28]  Maria Luisa Bonet,et al.  Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence (2008) , 2022 .

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

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

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

[32]  Toniann Pitassi,et al.  The complexity of resolution refinements , 2007, Journal of Symbolic Logic.

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

[34]  Michael Alekhnovich.  Lower bounds for k-DNF resolution on random 3-CNFs , 2005, STOC.

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

[36]  Michael Alekhnovich,et al.  Pseudorandom generators in propositional proof complexity , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

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

[38]  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.

[39]  Jakob Nordstr,et al.  New Wine into Old Wineskins: A Survey of Some Pebbling Classics with Supplemental Results , 2015 .

[40]  Samuel R. Buss,et al.  Linear gaps between degrees for the polynomial calculus modulo distinct primes , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

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

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

[43]  Joao Marques-Silva.  Practical applications of Boolean Satisfiability , 2008, 2008 9th International Workshop on Discrete Event Systems.

[44]  Toby Walsh,et al.  Handbook of Satisfiability: Volume 185 Frontiers in Artificial Intelligence and Applications , 2009 .

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

[46]  Maria Luisa Bonet,et al.  On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems , 2000, SIAM J. Comput..

[47]  Eli Ben-Sasson,et al.  Lower Bounds for Width-Restricted Clause Learning on Small Width Formulas , 2010, SAT.

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

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

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

[51]  Oliver Kullmann,et al.  Upper and Lower Bounds on the Complexity of Generalised Resolution and Generalised Constraint Satisfaction Problems , 2004, Annals of Mathematics and Artificial Intelligence.

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

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

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

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

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

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

[58]  Phokion G. Kolaitis,et al.  Constraint Propagation as a Proof System , 2004, CP.

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

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

[61]  Maria Luisa Bonet,et al.  Lower Bounds for the Weak Pigeonhole Principle and Random Formulas beyond Resolution , 2002, Inf. Comput..

[62]  John E. Savage,et al.  Extreme Time-Space Tradeoffs for Graphs with Small Space Requirements , 1982, Inf. Process. Lett..

[63]  Michael E. Saks,et al.  The Efficiency of Resolution and Davis--Putnam Procedures , 2002, SIAM J. Comput..

[64]  Eli Ben-Sasson,et al.  Space complexity of random formulae in resolution , 2003, Random Struct. Algorithms.

[65]  Ran Raz,et al.  Lower bounds for cutting planes proofs with small coefficients , 1995, Symposium on the Theory of Computing.

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

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

[68]  Endre Szemerédi,et al.  Many hard examples for resolution , 1988, JACM.

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

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

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

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

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

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

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

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

[77]  Albert Atserias,et al.  On sufficient conditions for unsatisfiability of random formulas , 2004, JACM.

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

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

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

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

[82]  Chris Beck,et al.  Some trade-off results for polynomial calculus: extended abstract , 2013, STOC '13.

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

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

[85]  Andreas Goerdt.  Regular Resolution Versus Unrestricted Resolution , 1993, SIAM J. Comput..

[86]  Stephen A. Cook,et al.  An observation on time-storage trade off , 1973, J. Comput. Syst. Sci..

[87]  Ashok K. Chandra.  Efficient Compilation of Linear Recursive Programs , 1973, SWAT.

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

[89]  Johannes Klaus Fichte,et al.  Clause-Learning Algorithms with Many Restarts and Bounded-Width Resolution , 2011, J. Artif. Intell. Res..

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

[91]  Maria Luisa Bonet,et al.  Optimality of size-width tradeoffs for resolution , 2001, computational complexity.

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

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

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

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

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

[97]  John E. Savage,et al.  Graph pebbling with many free pebbles can be difficult , 1980, STOC '80.

[98]  Eli Ben-Sasson,et al.  Short Proofs May Be Spacious: An Optimal Separation of Space and Length in Resolution , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

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

[100]  Henry A. Kautz,et al.  Towards Understanding and Harnessing the Potential of Clause Learning , 2004, J. Artif. Intell. Res..

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

[102]  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.

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

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

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

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

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

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

[109]  William J. Cook.  Cutting-plane proofs in polynomial space , 1990, Math. Program..

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

[111]  Russell Impagliazzo,et al.  Time-space tradeoffs in resolution: superpolynomial lower bounds for superlinear space , 2012, STOC '12.

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

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

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

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

[116]  Jakob Nordström.  On the Relative Strength of Pebbling and Resolution , 2010, 2010 IEEE 25th Annual Conference on Computational Complexity.