Space Bounds for Resolution

Esteban, Juan Luis; Torán, Jacobo

doi:10.1006/inco.2001.2921

Cited by 84 publications

(49 citation statements)

References 6 publications

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“…The only relation that we can still show to hold is the classical size-space relation (Theorem 3.4), which we transfer from Res T to Q-Res T . In classical resolution, this relationship was obtained using pebbling games [Esteban and Torán 2001]. We observe that the same approach works for Q-Res T as well, giving the analogous relationship.…”

Section: The Size-space Relation In Tree-like Q-resolutionmentioning

confidence: 55%

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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Section: The Size-space Relation In Tree-like Q-resolutionmentioning

confidence: 55%

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

“…3) The equivalent approach via tree-resolution space was introduced in [55,26,27,56]. These approaches concern only unsatisfiable clause-sets; the extension to satisfiable clause-sets considered in [44,48] generalises the reduction-rules-based approach, and is essentially different from the general extension process as discussed above; the extension as in this paper was first considered in [4].…”

Section: Tree-hardnessmentioning

confidence: 99%

“…An equivalent bound is shown in [53] (see Subsection 4.1). In [25,26] the lower bound 2 crs(F )−1 is shown. 5) In [33] the notation "whd" was used, to emphasise that we have an extension of "hardness"; but now we consider the relation to "width" as more important.…”

Section: Asymmetric Widthmentioning

confidence: 99%

“…By definition we have css(ϕ * F ) ≤ css(F ) for F ∈ USAT and ϕ ∈ PASS, and thus css(F ) is naturally defined for all F ∈ CLS. We come to the notion of resolution space originating in [41,42] and [55,26]. This measure was intensively studied during the last decade (cf.…”

Section: Space Complexitymentioning

confidence: 99%

“…Indeed, the discovery of this relation between width and size of resolution proofs was a milestone in our understanding of resolution. Around the same time (tree) resolution space was investigated, and first lower bounds were obtained [55,25,26,27,56]. 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].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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The impact of heterogeneity and geometry on the proof complexity of random satisfiability

Bläsius

Friedrich

Göbel

et al. 2023

Random Struct Algorithms

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Satisfiability is considered the canonical NP‐complete problem and is used as a starting point for hardness reductions in theory, while in practice heuristic SAT solving algorithms can solve large‐scale industrial SAT instances very efficiently. This disparity between theory and practice is believed to be a result of inherent properties of industrial SAT instances that make them tractable. Two characteristic properties seem to be prevalent in the majority of real‐world SAT instances, heterogeneous degree distribution and locality. To understand the impact of these two properties on SAT, we study the proof complexity of random ‐SAT models that allow to control heterogeneity and locality. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random ‐SAT instances have superpolynomial resolution size. This implies intractability of these instances for modern SAT‐solvers. In contrast, modeling locality with underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time.

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Space Bounds for Resolution

Cited by 84 publications

References 6 publications

Are Short Proofs Narrow? QBF Resolution Is Not So Simple

Are Short Proofs Narrow? QBF Resolution Is Not So Simple

Unified Characterisations of Resolution Hardness Measures

The impact of heterogeneity and geometry on the proof complexity of random satisfiability

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