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20
Towards an Optimal Separation of Space and Length in Resolution
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY
, 2008
"... Most stateoftheart satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory corre ..."
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Cited by 13 (10 self)
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Most stateoftheart satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory correspond to the length and space of resolution proofs. There has been a long line of research trying to understand these proof complexity measures, as well as relating them to the width of proofs, i.e., the size of the largest clause in the proof, which has been shown to be intimately connected with both length and space. While strong results have been proven for length and width, our understanding of space is still quite poor. For instance, it has remained open whether the fact that a formula is provable in short length implies that it is also provable in small space (which is the case for length versus width), or whether on the contrary these measures are completely unrelated in the sense that short proofs can be arbitrarily complex with respect to space. In this paper, we present some evidence that the true answer should be that the latter case holds and provide a possible roadmap for how such an optimal separation result could be obtained. We do this by proving a tight bound of Θ ( √ n) on the space needed for socalled pebbling contradictions over pyramid graphs of size n. This yields the first polynomial lower bound on space that is not a consequence of a corresponding lower bound on width, as well as an improvement of the weak separation of space and width in (Nordström 2006) from logarithmic to polynomial. Also, continuing the line of research initiated by (BenSasson 2002) into tradeoffs between different proof complexity measures, we present a simplified proof of the recent lengthspace tradeoff result in (Hertel and Pitassi 2007), and show how our ideas can be used to prove a couple of other exponential tradeoffs in resolution.
Short proofs may be spacious: An optimal separation of space and length in resolution
, 2008
"... A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering it negat ..."
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Cited by 11 (8 self)
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A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering it negatively in the strongest possible way. We show that there are families of 6CNF formulas of size n, for arbitrarily large n, that have resolution proofs of length O(n) but for which any proof requires space Ω(n / log n). This is the strongest asymptotic separation possible since any proof of length O(n) can always be transformed into a proof in space O(n / log n). Our result follows by reducing the space complexity of so called pebbling formulas over a directed acyclic graph to the blackwhite pebbling price of the graph. The proof is somewhat simpler than previous results (in particular, those reported in [Nordström 2006, Nordström and H˚astad 2008]) as it uses a slightly different flavor of pebbling formulas which allows for a rather straightforward reduction of proof space to standard blackwhite pebbling price.
PEBBLE GAMES, PROOF COMPLEXITY AND TIMESPACE TRADEOFFS
, 2010
"... 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 resolutionbased proof systems when compari ..."
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Cited by 7 (4 self)
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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 resolutionbased proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing sizespace tradeoffs. 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 tradeoffs between proof size and proof space.
A Lower Bound for the Pigeonhole Principle in Treelike Resolution by Asymmetric ProverDelayer Games
"... In this note we show that the asymmetric ProverDelayer game developed in [3] for Parameterized Resolution is also applicable to other treelike proof systems. In particular, Ω( ..."
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Cited by 6 (1 self)
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In this note we show that the asymmetric ProverDelayer game developed in [3] for Parameterized Resolution is also applicable to other treelike proof systems. In particular, Ω(
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a pla ..."
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Cited by 6 (0 self)
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
On the Virtue of Succinct Proofs: Amplifying Communication Complexity Hardness to TimeSpace Tradeoffs in Proof Complexity [Extended Abstract]
, 2012
"... An active line of research in proof complexity over the last decade has been the study of proof space and tradeoffs between size and space. Such questions were originally motivated by practical SAT solving, but have also led to the development of new theoretical concepts in proof complexity of intr ..."
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Cited by 4 (2 self)
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An active line of research in proof complexity over the last decade has been the study of proof space and tradeoffs between size and space. Such questions were originally motivated by practical SAT solving, but have also led to the development of new theoretical concepts in proof complexity of intrinsic interest and to results establishing nontrivial relations between space and other proof complexity measures. By now, the resolution proof system is fairly well understood in this regard, as witnessed by a sequence of papers leading up to [BenSasson and Nordström 2008, 2011] and [Beame, Beck, and Impagliazzo 2012]. However, for other relevant proof systems in the context of SAT solving, such as polynomial calculus (PC) and cutting planes (CP), very little has been known. Inspired by [BN08, BN11], we consider CNF encodings of socalled pebble games played on graphs and the approach of making such pebbling formulas harder by simple syntactic modifications. We use this paradigm of hardness amplification to make progress on the relatively longstanding open question of proving timespace tradeoffs for PC and CP. Namely, we exhibit a family of modified pebbling formulas {Fn} ∞ n=1 such that: • The formulas Fn have size Θ(n) and width O(1). • They have proofs in length O(n) in resolution, which generalize to both PC and CP. • Any refutation in CP or PCR (a generalization of PC) in length L and space s must satisfy s log L � 4 √ n. A crucial technical ingredient in these results is a new twoplayer communication complexity lower bound for composed search problems in terms of block sensitivity, a contribution that we believe to be of independent interest.
Space Complexity in Polynomial Calculus
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 132
, 2012
"... During the last decade, an active line of research in proof complexity has been to study space complexity and timespace tradeoffs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving, and so research has mostly focused on wea ..."
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Cited by 4 (2 self)
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During the last decade, an active line of research in proof complexity has been to study space complexity and timespace tradeoffs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving, and so research has mostly focused on weak systems that are used by SAT solvers. There has been a relatively long sequence of papers on space in resolution, which is now reasonably well understood from this point of view. For other natural candidates to study, however, such as polynomial calculus or cutting planes, very little has been known. We are not aware of any nontrivial space lower bounds for cutting planes, and for polynomial calculus the only lower bound has been for CNF formulas of unbounded width in [Alekhnovich et al. ’02], where the space lower bound is smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent with current knowledge that polynomial calculus could be able to refute any kCNF formula in constant space. In this paper, we prove several new results on space in polynomial calculus (PC), and in the extended proof system polynomial calculus resolution (PCR) studied in [Alekhnovich et al. ’02]:
Hardness of Parameterized Resolution
"... Parameterized Resolution and, moreover, a general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [16] (FOCS’07). In that paper, Dantchev et al. show a complexity gap in treelike Parameterized Resolution for propositional formulas arising from translatio ..."
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Cited by 2 (2 self)
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Parameterized Resolution and, moreover, a general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [16] (FOCS’07). In that paper, Dantchev et al. show a complexity gap in treelike Parameterized Resolution for propositional formulas arising from translations of firstorder principles. We broadly investigate Parameterized Resolution obtaining the following main results: ∙ We introduce a purely combinatorial approach to obtain lower bounds to the proof size in treelike Parameterized Resolution. For this we devise a new asymmetric ProverDelayer game which characterizes proofs in (parameterized) treelike Resolution. By exhibiting good Delayer strategies we then show lower bounds for the pigeonhole principle as well as the order principle. ∙ Interpreting a wellknown FPT algorithm for vertex cover as a DPLL procedure for Parameterized Resolution, we devise a proof search algorithm for Parameterized Resolution and show that treelike Parameterized Resolution allows short refutations of all parameterized contradictions given as boundedwidth CNF’s. ∙ We answer a question posed by Dantchev, Martin, and Szeider in [16] showing that daglike Parameterized Resolution is not fptbounded. We obtain this result by proving that the pigeonhole principle requires proofs of size
Optimality of sizedegree tradeoffs for Polynomial Calculus
"... We answer this question by showing a polynomial encoding of Graph Ordering Principle on m variables which requires PC and PCR refutations of degree Ω ( √ m). Tradeoffs optimality follows from our result and from the short refutations of Graph Ordering Principle in [Bonet and Galesi 1999; 2001]. We ..."
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Cited by 2 (0 self)
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We answer this question by showing a polynomial encoding of Graph Ordering Principle on m variables which requires PC and PCR refutations of degree Ω ( √ m). Tradeoffs optimality follows from our result and from the short refutations of Graph Ordering Principle in [Bonet and Galesi 1999; 2001]. We then introduce the algebraic proof system PCRk which combines together Polynomial Calculus (PC) and kDNF Resolution (RESk). We show a size hierarchy theorem for PCRk: PCRk is exponentially separated from PCRk+1. This follows from the previous degree lower bound and from techniques developed for RESk. Finally we show that random formulas in conjunctive normal form (3CNF) are hard to refute in PCRk.
Parameterized complexity of DPLL search procedures
 In Proc. 14th International Conference on Theory and Applications of Satisfiability Testing. Lecture Notes in Computer Science Series
, 2011
"... We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a ProverDelayer game which models the running time of DPL ..."
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Cited by 2 (2 self)
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We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a ProverDelayer game which models the running time of DPLL procedures and we establish an informationtheoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a kclique requires n Ω(k) steps for a nontrivial distribution of graphs close to the critical threshold. For the restricted case of treelike Parameterized Resolution, this result answers a question asked by Beyersdorff et al. [2012] of understanding the Resolution complexity of this family of formulas.