| 1. |
- Beyersdorff, Olaf, et al.
(author)
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A Characterization of Tree-Like Resolution Size
- 2012
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In: Electronic Colloquium on Computational Complexity (ECCC). - 1433-8092. ; :161
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Journal article (peer-reviewed)abstract
- We explain an asymmetric Prover-Delayer game which precisely characterizes proof size in tree-like Resolution. This game was previously described in a parameterized complexity context to show lower bounds for parameterized formulas [BGL11] and for the classical pigeonhole principle [BGL10]. The main point of this note is to show that the asymmetric game in fact characterizes tree-like Resolution proof size, i. e. in principle our proof method allows to always achieve the optimal lower bounds. This is in contrast with previous techniques described in the literature. We also provide a very intuitive information-theoretic interpretation of the game.
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| 2. |
- Beyersdorff, Olaf, et al.
(author)
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Parameterized Bounded-Depth Frege Is not Optimal
- 2012
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In: ACM Transactions on Computation Theory. - : Association for Computing Machinery (ACM). - 1942-3454 .- 1942-3462. ; 4:3, s. 7-
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Journal article (peer-reviewed)abstract
- A general framework for parameterized proof complexity was introduced by Dantchev et al. [2007]. There, the authors show important results on tree-like Parameterized Resolution-a parameterized version of classical Resolution-and their gap complexity theorem implies lower bounds for that system. The main result of this article significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n Ω(k) in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in Dantchev et al. [2007]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNFs
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| 3. |
- Filmus, Yuval, et al.
(author)
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Space Complexity in Polynomial Calculus
- 2012
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In: Electronic Colloquium on Computational Complexity (ECCC). - 1433-8092. ; :132
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Journal article (peer-reviewed)abstract
- During the last decade, an active line of research in proof complexity has been to study space complexity and time-space trade-offs 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 k-CNF 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]:(1) We prove an Omega(n) space lower bound in PC for the canonical 3-CNF version of the pigeonhole principle formulas PHP^m_n with m pigeons and n holes, and show that this is tight.(2) For PCR, we prove an Omega(n) space lower bound for a bitwise encoding of the functional pigeonhole principle. These formulas have width O(log n), and hence this is an exponential improvement over [Alekhnovich et al. '02] measured in the width of the formulas.(3) We then present another encoding of the pigeonhole principle that has constant width, and prove an Omega(n) space lower bound in PCR for these formulas as well.(4) Finally, we prove that any k-CNF formula can be refuted in PC in simultaneous exponential size and linear space (which holds for resolution and thus for PCR, but was not obviously the case for PC). We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3-CNF in the canonical way, something that we believe can be useful when proving PCR space lower bounds for other well-studied formula families in proof complexity.
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| 4. |
- Lauria, Massimo
(author)
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A rank lower bound for cutting planes proofs of Ramsey Theorem
- 2012
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In: Electronic Colloquium on Computational Complexity (ECCC). - 1433-8092. ; :124
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Journal article (other academic/artistic)abstract
- Ramsey Theorem is a cornerstone of combinatorics and logic. In its simplest formulation it says that there is a function r such that any simple graph with r(k, s) vertices contains either a clique of size k or an independent set of size s. We study the complexity of proving upper bounds for the number r(k, k). In particular we focus on the propositional proof system cutting planes; we prove that the upper bound “r(k, k) 4k” requires cutting planes proof of high rank. In order to do that we show a protection lemma which could be of independent interest.
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