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Short Propositional Refutations for Dense Random 3CNF Formulas [article]

Sebastian Müller, Iddo Tzameret
2011 arXiv   pre-print
Random 3CNF formulas constitute an important distribution for measuring the average-case behavior of propositional proof systems.  ...  In this paper we show that already standard propositional proof systems, within the hierarchy of Frege proofs, admit short refutations for random 3CNF formulas, for sufficiently large clause-to-variable  ...  Acknowledgments We wish to thank Jan Krajíček for very helpful discussions concerning the topic of this paper and for commenting on an earlier manuscript and Emil Jeřabek and Neil Thapen for answering  ... 
arXiv:1101.3970v2 fatcat:ljar3wx6vjbmbkqdgn57uwrkqu

Short propositional refutations for dense random 3CNF formulas

Sebastian Müller, Iddo Tzameret
2014 Annals of Pure and Applied Logic  
Random 3CNF formulas constitute an important distribution for measuring the average-case behavior of propositional proof systems.  ...  In this paper we show that already standard propositional proof systems, within the hierarchy of Frege proofs, admit short refutations for random 3CNF formulas, for sufficiently large clause-to-variable  ...  of our questions about theories of weak arithmetic and for many other insightful comments, and the anonymous reviewer who invested a lot of effort in reading the manuscript and providing us with many  ... 
doi:10.1016/j.apal.2014.08.001 fatcat:3z3cokqzqnamzphp6u4dr4pcpe

Complexity of propositional proofs under a promise

Nachum Dershowitz, Iddo Tzameret
2010 ACM Transactions on Computational Logic  
We then investigate the complexity of such systems, obtaining an exponential separation in the average-case between resolution under different size promises: (i) Resolution has polynomial-size refutations  ...  (ii) There are no sub-exponential size resolution refutations for random 3CNF formulas, when the promise is 2 δn (and the number of clauses is o(n 3/2 )), for any constant 0 < δ < 1.  ...  We also wish to thank Jan Krajíček for commenting on an earlier version of this paper and Eli Ben-Sasson and Amnon Ta-Shma for useful correspondence and conversations.  ... 
doi:10.1145/1740582.1740586 fatcat:ehy3fl6bcze3ndcpckgl535dpq

Complexity of Propositional Proofs under a Promise [article]

Nachum Dershowitz, Iddo Tzameret
2007 arXiv   pre-print
We then investigate the complexity of such systems, obtaining an exponential separation in the average-case between resolution under different size promises: 1.  ...  There are no sub-exponential size resolution refutations for random 3CNF formulas, when the promise is 2^δ n (and the number of clauses is o(n^3/2)), for any constant 0<δ<1.  ...  We wish to thank Jan Krajíček for commenting on an early version of this paper and Eli Ben-Sasson and Amnon Ta-Shma for useful correspondence and conversations.  ... 
arXiv:0707.4255v1 fatcat:5iigbvcyw5e5xp7g5z2leb23qa

Sparser Random 3SAT Refutation Algorithms and the Interpolation Problem [article]

Iddo Tzameret
2014 arXiv   pre-print
We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek (2006), as a family of unsatisfiable propositional formulas for which refutations of small size in any propositional  ...  proof system that possesses the feasible interpolation property imply an efficient deterministic refutation algorithm for random 3SAT with n variables and Ω(n^1.4) clauses.  ...  Thanks also to the anonymous reviewers of this paper who helped much in improving the exposition of this work.  ... 
arXiv:1305.0948v3 fatcat:eyybsdsry5brjgug67pkakimhi

Sparser Random 3-SAT Refutation Algorithms and the Interpolation Problem [chapter]

Iddo Tzameret
2014 Lecture Notes in Computer Science  
We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek [14], as a family of unsatisfiable propositional formulas for which refutations of small size in any propositional  ...  proof system that possesses the feasible interpolation property imply an efficient deterministic refutation algorithm for random 3SAT with n variables and Ω(n 1.4 ) clauses.  ...  Acknowledgments I wish to thank Jan Krajíček for useful comments related to this work and Albert Atserias and Neil Thapen for useful related discussions.  ... 
doi:10.1007/978-3-662-43948-7_84 fatcat:jcna7uxo3bgpplvyhbmoabzlem

Average-Case Complexity

Andrej Bogdanov
2006 Foundations and Trends® in Theoretical Computer Science  
While the relation between worst-case and average-case complexity for general NP problems remains open, there has been progress in understanding the relation between different "degrees" of average-case  ...  We survey the average-case complexity of problems in NP. We discuss various notions of good-on-average algorithms, and present completeness results due to Impagliazzo and Levin.  ...  Acknowledgments We thank Scott Aaronson, Jonathan Katz, Chris Moore and the anonymous referee for their helpful comments.  ... 
doi:10.1561/0400000004 fatcat:6emvwcgkinbgblhqzcdt7nzmou

Average-Case Complexity [article]

Andrej Bogdanov, Luca Trevisan
2021 arXiv   pre-print
While the relation between worst-case and average-case complexity for general NP problems remains open, there has been progress in understanding the relation between different "degrees" of average-case  ...  We survey the average-case complexity of problems in NP. We discuss various notions of good-on-average algorithms, and present completeness results due to Impagliazzo and Levin.  ...  Acknowledgements We thanks Scott Aaronson, Jonathan Katz, Chris Moore and the anonymous referee for their helpful comments.  ... 
arXiv:cs/0606037v3 fatcat:ishwc6ck2vhztpfb5z3bjywehi

Average-case Complexity

Luca Trevisan
2008 2008 49th Annual IEEE Symposium on Foundations of Computer Science  
We survey the theory of average-case complexity, with a focus on problems in NP.  ...  For the case of 3CNF, there is evidence suggesting that refuting random formulas is hard for m 3 (n) < n 3/2−ε for every ε > 0.  ...  This short discussion anticipates the formal definition of one-way function and the difficulties in defining a robust notion of "average-case tractability" in Levin's theory of average-case complexity.  ... 
doi:10.1109/focs.2008.90 dblp:conf/focs/Trevisan08 fatcat:6rp35qbwpvhvjpzcxw4uufxg3m

Semialgebraic Proofs and Efficient Algorithm Design

Noah Fleming, Pravesh Kothari, Toniann Pitassi
2019 Foundations and Trends® in Theoretical Computer Science  
The authors would like to thank Ian Mertz for his suggestions on the presentation of the 3XOR lower bound in Section 5.1.  ...  Finally, we are grateful to Paul Beame for correcting several historical remarks made in an earlier draft of this manuscript.  ...  Recently there has been a surge of works for showing SoS lower bounds for average-case settings.  ... 
doi:10.1561/0400000086 fatcat:sotovlogzjfy7kxxloglpu5tym

Complexity of Propositional Proofs Under a Promise [chapter]

Nachum Dershowitz, Iddo Tzameret
Lecture Notes in Computer Science  
We then investigate the complexity of such systems, obtaining an exponential separation in the average-case between resolution under different size promises: (i) Resolution has polynomial-size refutations  ...  (ii) There are no sub-exponential size resolution refutations for random 3CNF formulas, when the promise is 2 δn (and the number of clauses is o(n 3/2 )), for any constant 0 < δ < 1.  ...  We also wish to thank Jan Krajíček for commenting on an earlier version of this paper and Eli Ben-Sasson and Amnon Ta-Shma for useful correspondence and conversations.  ... 
doi:10.1007/978-3-540-73420-8_27 fatcat:3w77gvjfg5aernsvrfc5uorxpq

A Resolution-Based Theorem Prover for $${\textsf {K}}_{n}^{}$$Kn: Architecture, Refinements, Strategies and Experiments

Cláudia Nalon, Ullrich Hustadt, Clare Dixon
2018 Journal of automated reasoning  
In this paper we describe the implementation of K S P, a resolution-based prover for the basic multimodal logic K n .  ...  We describe these options in detail and discuss their implications. We provide experiments comparing some of these options and comparing the prover with other provers for this logic.  ...  In this case, a refutation for a set of clauses Φ is either a global refutation or a derivation Φ 0 , . . . , Φ k , k ∈ N, where * : start ⇒ false ∈ Φ k .  ... 
doi:10.1007/s10817-018-09503-x fatcat:rhvht7pvnnfp3ggvvaq6d6whdm

Polylogarithmic Cuts in Models of V0

Sebastian Müller, S. Barry Cooper
2013 Logical Methods in Computer Science  
As an interesting observation we also obtain an average case separation of Resolution from AC0-Frege by applying a recent result with Tzameret.  ...  We can then exploit our result in Proof Complexity to observe that Frege proof systems can be sub exponentially simulated by bounded depth Frege proof systems.  ...  Thus, this yields an average case separation between Resolution and TC 0 -Frege. We can now extend this result to an average case separation between Resolution and AC 0 -Frege as follows. Proof.  ... 
doi:10.2168/lmcs-9(1:16)2013 fatcat:3jfrecll2vaejmnhew5iymzwx4

Small Clique Detection and Approximate Nash Equilibria [chapter]

Lorenz Minder, Dan Vilenchik
2009 Lecture Notes in Computer Science  
(2 + δ) log n in G n,1/2 in polynomial time for arbitrarily small δ > 0.  ...  Recently, Hazan and Krauthgamer showed [12] that if, for a fixed small ε, an ε-best ε-approximate Nash equilibrium can be found in polynomial time in two-player games, then it is also possible to find  ...  -The planted k-clique problem is (for certain values of k) related to the assumption that refuting low-density 3CNF formulas is hard on the average.  ... 
doi:10.1007/978-3-642-03685-9_50 fatcat:dv4xm3ewpvcebghe6f3p2zhkdi

Proof Complexity Lower Bounds from Algebraic Circuit Complexity [article]

Michael A. Forbes, Amir Shpilka, Iddo Tzameret, Avi Wigderson
2016 pre-print
We complement our lower bounds by giving short refutations of the previously-studied subset-sum axiom using IPS subsystems, allowing us to conclude strict separations between some of these subsystems.  ...  This mimics an established research direction in the boolean setting for subsystems of Extended Frege proofs, where proof-lines are circuits from restricted boolean circuit classes.  ...  in this work, as well as Avishay Tal for pointing out how Proposition B.1 implies an optimal functional lower bound for sparsity (Corollary B.3).  ... 
doi:10.4230/lipics.ccc.2016.32 arXiv:1606.05050v1 fatcat:t3i35lsfrnhcjmqcvcsuudgmwa
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