Journal of Computer and System Sciences - 17th Annual ACM Symposium in the Theory of Computing, May 6-8, 1985
A tight bound for black and white pebbles on the pyramid
Journal of the ACM (JACM)
Journal of the ACM (JACM)
Asymptotically tight bounds on time-space trade-offs in a pebble game
Journal of the ACM (JACM)
Short proofs are narrow—resolution made simple
Journal of the ACM (JACM)
Information and Computation
Space Complexity in Propositional Calculus
SIAM Journal on Computing
On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems
SIAM Journal on Computing
Space-Time Tradeoffs for Oblivious Interger Multiplications
Proceedings of the 6th Colloquium, on Automata, Languages and Programming
Time-space tradeoffs for computing functions, using connectivity properties of their circuits
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Graph pebbling with many free pebbles can be difficult
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
Variations of a pebble game on graphs
Variations of a pebble game on graphs
Space complexity of random formulae in resolution
Random Structures & Algorithms
A combinatorial characterization of treelike resolution space
Information Processing Letters
On sufficient conditions for unsatisfiability of random formulas
Journal of the ACM (JACM)
On the complexity of resolution with bounded conjunctions
Theoretical Computer Science
A combinatorial characterization of resolution width
Journal of Computer and System Sciences
Record of the Project MAC conference on concurrent systems and parallel computation
Towards an optimal separation of space and length in resolution
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Lower bounds for natural proof systems
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
Efficient compilation of linear recursive programs
SWAT '73 Proceedings of the 14th Annual Symposium on Switching and Automata Theory (swat 1973)
Short Proofs May Be Spacious: An Optimal Separation of Space and Length in Resolution
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Size-Space Tradeoffs for Resolution
SIAM Journal on Computing
Narrow Proofs May Be Spacious:Separating Space and Width in Resolution
SIAM Journal on Computing
ACM Transactions on Computation Theory (TOCT)
Storage requirements for deterministic polynomialtime recognizable languages
Journal of Computer and System Sciences
An observation on time-storage trade off
Journal of Computer and System Sciences
SAT'05 Proceedings of the 8th international conference on Theory and Applications of Satisfiability Testing
Game characterizations and the PSPACE-completeness of tree resolution space
CSL'07/EACSL'07 Proceedings of the 21st international conference, and Proceedings of the 16th annuall conference on Computer Science Logic
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The last decade has seen a revival of interest in pebble games in the context 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. The typical approach has been to encode the pebble game played on a graph as a CNF formula and then argue that proofs of this formula must inherit (various aspects of) the pebbling properties of the underlying graph. Unfortunately, the reductions used here are not tight. To simulate resolution proofs by pebblings, the full strength of nondeterministic black-white pebbling is needed, whereas resolution is only known to be able to simulate deterministic black pebbling. To obtain strong results, one therefore needs to find specific graph families which either have essentially the same properties for black and black-white pebbling (not at all true in general) or which admit simulations of black-white pebblings in resolution. This article contributes to both these approaches. First, we design a restricted form of black-white pebbling that can be simulated in resolution and show that there are graph families for which such restricted pebblings can be asymptotically better than black pebblings. This proves that, perhaps somewhat unexpectedly, resolution can strictly beat black-only pebbling, and in particular that the space lower bounds on pebbling formulas in Ben-Sasson and Nordström [2008] are tight. Second, we present a versatile parametrized graph family with essentially the same properties for black and black-white pebbling, which gives sharp simultaneous trade-offs for black and black-white pebbling for various parameter settings. Both of our contributions have been instrumental in obtaining the time-space trade-off results for resolution-based proof systems in Ben-Sasson and Nordström [2011].