Optimality of size-width tradeoffs for resolution

Bonet, María Luisa; Galesi, Nicola

doi:10.1007/s000370100000

Cited by 56 publications

(48 citation statements)

References 16 publications

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“…If a formula has a narrow proof then this proof must also be short (simply by counting the total number of distinct clauses). The opposite does not necessarily hold as proven in [11] (although very strong lower bounds on width do imply strong lower bounds on length by [21]). …”

Section: Proof Complexity Preliminariesmentioning

confidence: 95%

See 1 more Smart Citation

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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Section: Proof Complexity Preliminariesmentioning

confidence: 95%

“…However, the fact that a resolution proof needs to be wide, i.e., has to contain some large clause, does not necessarily imply that the minimum length is large [11]. Width is thus a stricter hardness measure than length.…”

Section: Introductionmentioning

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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“…More interestingly, [11] established the converse that strong enough lower bounds on width imply strong lower bounds on size, and used this to rederive essentially all known size lower bounds in terms of width. The relation between size and width was elucidated further in [4,15].…”

Section: Resolutionmentioning

confidence: 99%

A (Biased) Proof Complexity Survey for SAT Practitioners

Nordström

2014

Lecture Notes in Computer Science

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Abstract. This talk is intended as a selective survey of proof complexity, focusing on some comparatively weak proof systems that are of particular interest in connection with SAT solving. We will review resolution, polynomial calculus, and cutting planes (related to conflict-driven clause learning, Gröbner basis computations, and pseudo-Boolean solvers, respectively) and some proof complexity measures that have been studied for these proof systems. We will also briefly discuss if and how these proof complexity measures could provide insights into SAT solver performance.Proof complexity studies how hard it is to find succinct certificates for the unsatisfiability of formulas in conjunctive normal form (CNF), i.e., proofs that formulas always evaluate to false under any truth value assignment, where these proofs should be efficiently verifiable. It is generally believed that there cannot exist a proof system where such proofs can always be chosen of size at most polynomial in the formula size. If this belief could be proven correct, it would follow that NP = coNP, and hence P = NP, and this was the original reason research in proof complexity was initiated by Cook and Reckhow [18]. However, the goal of separating P and NP in this way remains very distant.Another, perhaps more recent, motivation for proof complexity is the connection to applied SAT solving. Any algorithm for deciding SAT defines a proof system in the sense that the execution trace on an unsatisfiable instance is itself a polynomial-time verifiable witness (often referred to as a refutation rather than a proof ). In the other direction, most SAT solvers in effect search for proofs in systems studied in proof complexity, and upper and lower bounds for these proof systems hence give information about the potential and limitations of such SAT solvers.In addition to running time, an important concern in SAT solving is memory consumption. In proof complexity, time and memory are modelled by proof size and proof space. It therefore seems interesting to understand these two complexity measures and how they are related to each other, and such a study reveals intriguing connections that are also of intrinsic interest to proof complexity. In this context, it is natural to concentrate on comparatively weak proof systems that are, or could plausibly be, used as a basis for SAT solvers. This talk will focus on such proof systems, and the purpose of these notes is to summarize the main points. Readers interested in more details can refer to, e.g, the survey [31].

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“…One of the fundamental facts about resolution is that a partial converse is also true: building on the work of Clegg, Edmonds, and Impagliazzo [19] and Beame and Pitassi [8], Ben-Sasson and Wigderson [12] proved if a contradictory set of clauses has a resolution refutation of size s, then it also has a resolution refutation of width O( √ n log s + w), where n is again the number of variables, and w is the width of the widest clause in the given set of clauses. To appreciate the depth of this result let us look at the case of polynomial s and constant w. In that case the width becomes O( √ n log n) which is very significantly smaller than the maximum possible width n. It is also known that this trade-off is worstcase optimal (up to logarithmic factors): there exist n-variable sets of 3-clauses that have polynomial-size resolution refutations but that do not have resolution refutations of width o( √ n) (see [16]). Among other applications, the fundamental size-width tradeoff result for resolution can be used to argue that, for contradictory sets of w-clauses with constant w, resolution is automatizable in non-trivial time.…”

Section: Width-related Algorithmsmentioning

confidence: 99%

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs

Atserias

2013

Theory and Applications of Satisfiability Testing – SAT 2013

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Abstract. In recent years there has been some progress in our understanding of the proof-search problem for very low-depth proof systems, e.g. proof systems that manipulate formulas of very low complexity such as clauses (i.e. resolution), DNF-formulas (i.e. R(k) systems), or polynomial inequalities (i.e. semi-algebraic proof systems). In this talk I will overview this progress. I will start with bounded-width resolution, whose specialized proof-search algorithm is as easy as uninteresting, but whose proof-search problem is unintentionally solved by certain versions of conflict-driven clause-learning algorithms with restarts. I will continue with R(k) systems, whose proof-search problem turned out to hide the complexity of certain two-player games of interest in the area of systems synthesis and verification. And I will close with bounded-degree semialgebraic proof systems, whose proof-search problem turned out to hide the complexity of systems of linear equations over finite fields, among other problems.

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Optimality of size-width tradeoffs for resolution

Cited by 56 publications

References 16 publications

Relating Proof Complexity Measures and Practical Hardness of SAT

Relating Proof Complexity Measures and Practical Hardness of SAT

A (Biased) Proof Complexity Survey for SAT Practitioners

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs

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