Lower bounds for the polynomial calculus and the Gröbner basis algorithm

Impagliazzo, Russell; Pudlák, Pavel; Sgall, Jiřį́

doi:10.1007/s000370050024

Cited by 123 publications

(106 citation statements)

References 14 publications

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“…The paper [IPS97] proves that linear lower bound on the proof degree in Polynomial Calculus implies the exponential lower bound on the proof size in Polynomial Calculus under fields. The paper [BGIP99] presents a linear lower bound on the degree of proofs of Tseitin formulas in Polynomial Calculus under fields and rings.…”

Section: Introductionmentioning

confidence: 89%

Graph Expansion, Tseitin Formulas and Resolution Proofs for CSP

Itsykson

Oparin

2013

Computer Science – Theory and Applications

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Abstract. We study the resolution complexity of Tseitin formulas over arbitrary rings in terms of combinatorial properties of graphs. We give some evidence that an expansion of a graph is a good characterization of the resolution complexity of Tseitin formulas. We extend the method of Ben-Sasson and Wigderson of proving lower bound for the size of resolution proofs to constraint satisfaction problems under an arbitrary finite alphabet. For Tseitin formulas under the alphabet of cardinality d we provide a lower bound d e(G)−k for the tree-like resolution complexity that is stronger than the one that can be obtained by the Ben-Sasson and Wigderson method. Here k is an upper bound on the degree of the graph and e(G) is the graph expansion that is equal to the minimal cut such that both its parts are no more than twice bigger than each other. We give a formal argument why a large graph expansion is necessary for lower bounds. Let G = V, E be the dependency graph of the CSP, vertices of G correspond to constraints; two constraints are connected by an edge for every common variable. We prove that the tree-like resolution complexity of the CSP is at most d e(H)·log 3 2 |V | for some subgraph H of G.

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Section: Introductionmentioning

confidence: 89%

Graph Expansion, Tseitin Formulas and Resolution Proofs for CSP

Itsykson

Oparin

2013

Computer Science – Theory and Applications

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“…This is the main hurdle in performing our computations. However, for certain problems, lower bounds on the algorithm can be shown to be polynomial [40]. We hope that the specific nature of our problem may induce a non-prohibitive Gröbner reduction.…”

Section: A3 a Brief Glossarymentioning

confidence: 99%

Exploring the vacuum geometry of gauge theories

Gray

Jejjala

et al. 2006

Nuclear Physics B

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Using techniques of algorithmic algebraic geometry, we present a new and efficient method for explicitly computing the vacuum space of N = 1 gauge theories. We emphasize the importance of finding special geometric properties of these spaces in connecting phenomenology to guiding principles descending from high-energy physics. We exemplify the method by addressing various subsectors of the MSSM. In particular the geometry of the vacuum space of electroweak theory is described in detail, with and without right-handed neutrinos. We discuss the impact of our method on the search for evidence of underlying physics at a higher energy. Finally we describe how our results can be used to rule out certain top-down constructions of electroweak physics.

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“…This is a parallel to the size-width relation for resolution in [11] discussed above, and in fact [11] can be seen as a translation of the bound in [26] from PC to resolution. This size-degree relation has been used to prove exponential lower bounds on size in a number of papers, with [2] perhaps providing the most general setting.…”

Section: Polynomial Calculusmentioning

confidence: 63%

“…Impagliazzo et al [26] showed that strong degree lower bounds imply strong size lower bounds. This is a parallel to the size-width relation for resolution in [11] discussed above, and in fact [11] can be seen as a translation of the bound in [26] from PC to resolution.…”

Section: Polynomial Calculusmentioning

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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Lower bounds for the polynomial calculus and the Gröbner basis algorithm

Cited by 123 publications

References 14 publications

Graph Expansion, Tseitin Formulas and Resolution Proofs for CSP

Graph Expansion, Tseitin Formulas and Resolution Proofs for CSP

Exploring the vacuum geometry of gauge theories

A (Biased) Proof Complexity Survey for SAT Practitioners

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