A lower bound on the size of resolution proofs of the Ramsey theorem

Pudlák, Pavel

doi:10.1016/j.ipl.2012.05.004

Cited by 37 publications

(5 citation statements)

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“…Put differently, a refutation of RAM n is a proof that r(k) ≤ 2 2k . This was recently shown to require exponential size (in n) resolution refutations [21]. On the other hand a refutation of Ψ G is a proof that G is c-Ramsey, and hence that G witnesses that r(k) > 2 k c .…”

Section: Definitions and Resultsmentioning

confidence: 99%

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The Complexity of Proving That a Graph Is Ramsey

Lauria

Pudlák

Rödl

et al. 2013

Automata, Languages, and Programming

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We say that a graph with n vertices is c-Ramsey if it does not contain either a clique or an independent set of size c log n. We define a CNF formula which expresses this property for a graph G. We show a superpolynomial lower bound on the length of resolution proofs that G is c-Ramsey, for every graph G. Our proof makes use of the fact that every Ramsey graph must contain a large subgraph with some of the statistical properties of the random graph.

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Section: Definitions and Resultsmentioning

confidence: 99%

“…The upper bound r(k) ≤ 4 k has short proofs in a relatively weak fragment of sequent calculus, in which every formula in a proof has small constant depth [20], [16]. Recently Pudlák [21] has shown a lower bound on proofs of r(k) ≤ 4 k in resolution. We discuss this in more detail in Section 1.…”

Section: Introductionmentioning

confidence: 99%

The Complexity of Proving That a Graph Is Ramsey

Lauria

Pudlák

Rödl

et al. 2013

Automata, Languages, and Programming

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“…The proof of our conditional lower bound in Section 4 can be seen as an extension of these ideas to the case of the Paris-Harrington principle. Recently, an unconditional lower bound of 2 n 1 4 −o(1) for the size of Resolution refutations of RAM(4 k ; k, k) has been proved by Pudlák [2013].…”

Section: Theorem 22 (Paris-harrington Theorem For Graphsmentioning

confidence: 99%

On the Proof Complexity of Paris-Harrington and Off-Diagonal Ramsey Tautologies

Carlucci¹,

Galesi²,

Lauria

2016

ACM Trans. Comput. Logic

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We study the proof complexity of Paris-Harrington's Large Ramsey Theorem for bi-colorings of graphs and of off-diagonal Ramsey's Theorem. For Paris-Harrington, we prove a non-trivial conditional lower bound in Resolution and a non-trivial upper bound in bounded-depth Frege. The lower bound is conditional on a (very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in RES(2). We show that under such an assumption, there is no refutation of the Paris-Harrington formulas of size quasipolynomial in the number of propositional variables. The proof technique for the lower bound extends the idea of using a combinatorial principle to blow up a counterexample for another combinatorial principle beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced off-diagonal Ramsey principle is established. This is obtained by adapting some constructions due to Erdős and Mills. We prove a non-trivial Resolution lower bound for a family of such off-diagonal Ramsey principles.

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“…Boolean formulas arising from this encoding have a prominent place in complexity theory because they provide hard examples for resolution. The complexity of resolution of such formulas was studied by Krishnamurthy [18], Krajíček [17], and Pudlák [22], among others.…”

Section: Introductionmentioning

confidence: 99%

Exploring the Use of Shatter for AllSAT Through Ramsey-Type Problems

Narváez¹

2017

Preprint

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In the context of SAT solvers, Shatter is a popular tool for symmetry breaking on CNF formulas. Nevertheless, little has been said about its use in the context of AllSAT problems: problems where we are interested in listing all the models of a Boolean formula. AllSAT has gained much popularity in recent years due to its many applications in domains like model checking, data mining, etc. One example of a particularly transparent application of AllSAT to other fields of computer science is computational Ramsey theory. In this paper we study the effect of incorporating Shatter to the workflow of using Boolean formulas to generate all possible edge colorings of a graph avoiding prescribed monochromatic subgraphs. Generating complete sets of colorings is an important building block in computational Ramsey theory. We identify two drawbacks in the naïve use of Shatter to break the symmetries of Boolean formulas encoding Ramsey-type problems for graphs: a "blow-up" in the number of models and the generation of incomplete sets of colorings. The issues presented in this work are not intended to discourage the use of Shatter as a preprocessing tool for AllSAT problems in combinatorial computing but to help researchers properly use this tool by avoiding these potential pitfalls. To this end, we provide strategies and additional tools to cope with the negative effects of using Shatter for AllSAT. While the specific application addressed in this paper is that of Ramsey-type problems, the analysis we carry out applies to many other areas in which highly-symmetrical Boolean formulas arise and we wish to find all of their models.

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A lower bound on the size of resolution proofs of the Ramsey theorem

Abstract: We prove an exponential lower bound on the lengths of resolution proofs of propositions expressing the finite Ramsey theorem for pairs.

Cited by 37 publications

References 5 publications

The Complexity of Proving That a Graph Is Ramsey

The Complexity of Proving That a Graph Is Ramsey

On the Proof Complexity of Paris-Harrington and Off-Diagonal Ramsey Tautologies

Exploring the Use of Shatter for AllSAT Through Ramsey-Type Problems

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