An Improved Separation of Regular Resolution from Pool Resolution and Clause Learning

We prove that the graph tautology principles of Alekhnovich, Johannsen, Pitassi and Urquhart have polynomial size pool resolution refutations that use only input lemmas as learned clauses and without degenerate resolution inferences. We also prove that these graph tautology principles can be refuted by polynomial size DPLL proofs with clause learning, even when restricted to greedy, unit-propagating DPLL search

On the Relative Strength of Pebbling and Resolution

The last decade has seen a revival of interest in pebble games in the context of proof complexity. Pebbling has proven 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 paper 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 Nordstrom 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 Nordstrom 2009].Comment: Full-length version of paper to appear in Proceedings of the 25th Annual IEEE Conference on Computational Complexity (CCC '10), June 201

Understanding Space in Proof Complexity: Separations and Trade-offs via Substitutions

For current state-of-the-art DPLL SAT-solvers the two main bottlenecks are the amounts of time and memory used. In proof complexity, these resources correspond to the length and space of resolution proofs. There has been a long line of research investigating these proof complexity measures, but while strong results have been established for length, our understanding of space and how it relates to length has remained quite poor. In particular, the question whether resolution proofs can be optimized for length and space simultaneously, or whether there are trade-offs between these two measures, has remained essentially open. In this paper, we remedy this situation by proving a host of length-space trade-off results for resolution. Our collection of trade-offs cover almost the whole range of values for the space complexity of formulas, and most of the trade-offs are superpolynomial or even exponential and essentially tight. Using similar techniques, we show that these trade-offs in fact extend to the exponentially stronger k-DNF resolution proof systems, which operate with formulas in disjunctive normal form with terms of bounded arity k. We also answer the open question whether the k-DNF resolution systems form a strict hierarchy with respect to space in the affirmative. Our key technical contribution is the following, somewhat surprising, theorem: Any CNF formula F can be transformed by simple variable substitution into a new formula F' such that if F has the right properties, F' can be proven in essentially the same length as F, whereas on the other hand the minimal number of lines one needs to keep in memory simultaneously in any proof of F' is lower-bounded by the minimal number of variables needed simultaneously in any proof of F. Applying this theorem to so-called pebbling formulas defined in terms of pebble games on directed acyclic graphs, we obtain our results.Comment: This paper is a merged and updated version of the two ECCC technical reports TR09-034 and TR09-047, and it hence subsumes these two report

Towards a Theoretical Understanding of the Power of Restart in SAT solvers

Restart policy is a widely used class of techniques integral to the efficiency of conflict-driven clause-learning (CDCL) SAT solvers. While the utility of such policies has been well-established, to-date we still lack a deep theoretical understanding of why restart policies are crucial to the power of CDCL SAT solvers. In this paper, we provide a series of results that theoretically establish the power of restarts for various models of Boolean SAT solvers. More precisely, we make the following contributions. First, we show that certain model of CDCL solvers with restarts are no more powerful from a proof-complexity theoretic point of view than the same configurations without restarts. Second, we define \textit{decision depth} for DPLL proofs of an unsatisfiable formula $\varphi$, and then we relate decision depth of $\varphi$ and size of DPLL proofs (or the running time of a DPLL based solver) for $\varphi$. Third, we introduce a new class of satisfiable instances called $Ladder_n$, then we use decision depth as a tool to proved that a drunk style DPLL solver with restarts can solve $Ladder_n$ in polynomial time with high probability while the solvers with the same configuration without restarts have exponential run time with high probability. Finally, the crucial insight that drives this line of research is the fact that restarts add proof-theoretic or algorithmic power to solver configurations by compensating for the weaknesses of some other important heuristics like branching or value selection or clause learning

A Theoretical Comparison of Resolution Proof Systems for CSP Algorithms

Many problems from a variety of applications such as graph coloring and circuit design can be modelled as constraint satisfaction problems (CSPs). This provides strong motivation to develop effective algorithms for CSPs. In this thesis, we study two resolution-based proof systems, NG-RES and C-RES, for finite-domain CSPs which have a close connection to common CSP algorithms. We give an almost complete characterization of the relative power among the systems and their restricted tree-like variants. We demonstrate an exponential separation between NG-RES and C-RES, improving on the previous super-polynomial separation, and present other new separations and simulations. We also show that most of the separations are nearly optimal. One immediate consequence of our results is that simple backtracking with 2-way branching is exponentially more powerful than simple backtracking with d-way branching

An Exponential Separation between Regular and General Resolution

This paper gives two distinct proofs of an exponential separation between regular resolution and unrestricted resolution. The previous best known separation between these systems was quasi-polynomial