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

Nordström, Jakob

doi:10.2168/lmcs-9(3:15)2013

Cited by 82 publications

(47 citation statements)

References 125 publications

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“…also the survey by Nordström [2013]), and we here use the pebbling number as the formal definition of the space used by the proof. We first define the pebbling game on graphs.…”

Section: Defining Size Width and Space For Resolutionmentioning

confidence: 99%

Are Short Proofs Narrow? QBF Resolution Is Not So Simple

Beyersdorff

Chew

Mahajan

et al. 2017

ACM Trans. Comput. Logic

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The groundbreaking paper 'Short proofs are narrow -resolution made simple' by Ben-Sasson and Wigderson (J. ACM 2001) introduces what is today arguably the main technique to obtain resolution lower bounds: to show a lower bound for the width of proofs. Another important measure for resolution is space, and in their fundamental work, Atserias and Dalmau (J. Comput. Syst. Sci. 2008) show that lower bounds for space again can be obtained via lower bounds for width.In this paper we assess whether similar techniques are effective for resolution calculi for quantified Boolean formulas (QBF). There are a number of different QBF resolution calculi like Q-resolution (the classical extension of propositional resolution to QBF) and the more recent calculi A Exp+Res and IR-calc. For these systems a mixed picture emerges. Our main results show that both the relations between size and width as well as between space and width drastically fail in Q-resolution, even in its weaker tree-like version. On the other hand, we obtain positive results for the expansion-based resolution systems A Exp+Res and IR-calc, however only in the weak tree-like models.Technically, our negative results rely on showing width lower bounds together with simultaneous upper bounds for size and space. For our positive results we exhibit space and width-preserving simulations between QBF resolution calculi.

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“…also the survey by Nordström [2013]), and we here use the pebbling number as the formal definition of the space used by the proof. We first define the pebbling game on graphs.…”

Section: Defining Size Width and Space For Resolutionmentioning

confidence: 99%

Are Short Proofs Narrow? QBF Resolution Is Not So Simple

Beyersdorff

Chew

Mahajan

et al. 2017

ACM Trans. Comput. Logic

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show abstract

“…[18]. We assume familiarity with CNF formulas, which are conjunctions of clauses, where a clause is a disjunction of literals (unnegated or negated variables, with negation denoted by overbar).…”

Section: Proof Complexity Preliminariesmentioning

confidence: 99%

Relating Proof Complexity Measures and Practical Hardness of SAT

Järvisalo

Matsliah

Nordström

et al. 2012

Lecture Notes in Computer Science

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Abstract. Boolean satisfiability (SAT) solvers have improved enormously in performance over the last 10-15 years and are today an indispensable tool for solving a wide range of computational problems. However, our understanding of what makes SAT instances hard or easy in practice is still quite limited. A recent line of research in proof complexity has studied theoretical complexity measures such as length, width, and space in resolution, which is a proof system closely related to state-of-the-art conflict-driven clause learning (CDCL) SAT solvers. Although it seems like a natural question whether these complexity measures could be relevant for understanding the practical hardness of SAT instances, to date there has been very limited research on such possible connections. This paper sets out on a systematic study of the interconnections between theoretical complexity and practical SAT solver performance. Our main focus is on space complexity in resolution, and we report results from extensive experiments aimed at understanding to what extent this measure is correlated with hardness in practice. Our conclusion from the empirical data is that the resolution space complexity of a formula would seem to be a more fine-grained indicator of whether the formula is hard or easy than the length or width needed in a resolution proof. On the theory side, we prove a separation of general and tree-like resolution space, where the latter has been proposed before as a measure of practical hardness, and also show connections between resolution space and backdoor sets.

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“…These unified characterisations allow elegant proofs of basic relations between the different hardness measures. Unlike in the works [8,12,50], our emphasis here is not on trade-off results, but on exact relations between the different measures. For a clause-set F we will consider the following measures: (i) size measures: the depth dep(F ) and the hardness hd(F ) (of best resolution refutations of F ); (ii) width measures: the symmetric and asymmetric width wid(F ) and awid(F ); (iii) clause-space measures: semantic space css(F ), resolution space crs(F ) and tree-resolution space cts(F ).…”

Section: Introductionmentioning

confidence: 99%

“…The primary method to obtain lower bounds on resolution space is based on width, and the general bound was shown in the fundamental paper by Atserias and Dalmau [5]. Since then the relations between size, width and space have been intensely investigated, resulting in particular in sharp trade-off results [11,8,12,50,51,9]. Independently, in [44,47,48] the concept of "hardness" has been introduced, with an algorithmic focus (as shown in [44], equivalent to tree resolution space; one can also say "tree-hardness"), together with a generalised form of width, which we call "asymmetric width" in this paper.…”

Section: Introductionmentioning

confidence: 99%

Unified Characterisations of Resolution Hardness Measures

Beyersdorff

Kullmann

2014

Lecture Notes in Computer Science

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Abstract. Various "hardness" measures have been studied for resolution, providing theoretical insight into the proof complexity of resolution and its fragments, as well as explanations for the hardness of instances in SAT solving. In this paper we aim at a unified view of a number of hardness measures, including different measures of width, space and size of resolution proofs. Our main contribution is a unified game-theoretic characterisation of these measures. As consequences we obtain new relations between the different hardness measures. In particular, we prove a generalised version of Atserias and Dalmau's result on the relation between resolution width and space from [5].

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

Cited by 82 publications

References 125 publications

Are Short Proofs Narrow? QBF Resolution Is Not So Simple

Are Short Proofs Narrow? QBF Resolution Is Not So Simple

Relating Proof Complexity Measures and Practical Hardness of SAT

Unified Characterisations of Resolution Hardness Measures

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