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Revisiting Graph Width Measures for CNF-Encodings

Romain Wallon, Stefan Mengel
2020 The Journal of Artificial Intelligence Research  
We consider bounded width CNF-formulas where the width is measured by popular graph width measures on graphs associated to CNF-formulas.  ...  Such restricted graph classes, in particular those of bounded treewidth, have been extensively studied for their uses in the design of algorithms for various computational problems on CNF-formulas.  ...  Acknowledgements The authors are grateful to the anonymous reviewers for their comments, which greatly helped to improve the presentation of the paper.  ... 
doi:10.1613/jair.1.11750 fatcat:6jhrmdr6ojfhpdp22ib6uy5thm

Graph Width Measures for CNF-Encodings with Auxiliary Variables [article]

Stefan Mengel, Romain Wallon
2020 arXiv   pre-print
We consider bounded width CNF-formulas where the width is measured by popular graph width measures on graphs associated to CNF-formulas.  ...  Such restricted graph classes, in particular those of bounded treewidth, have been extensively studied for their uses in the design of algorithms for various computational problems on CNF-formulas.  ...  The authors are grateful to the anonymous reviewers for their comments, which greatly helped to improve the presentation of the paper.  ... 
arXiv:1905.05290v2 fatcat:c5c34hybozf2bphhernvd65vaq

On sufficient conditions for unsatisfiability of random formulas

Albert Atserias
2004 Journal of the ACM  
As a consequence to our result and the known size-width relationship in Resolution, we obtain new proofs of the exponential lower bounds for Resolution refutations of random 3-CNF formulas and the Pigeonhole  ...  Our result is a consequence of designing a winning strategy for one of the players in the existential pebble game.  ...  Acknowledgments I am grateful to Phokion Kolaitis for his constructive criticism and suggestions.  ... 
doi:10.1145/972639.972645 fatcat:mt4ad4lbivfijafeusxqq73tla

A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds [article]

Mladen Mikša, Jakob Nordström
2015 arXiv   pre-print
This answers an open question in [Razborov '02], and also implies that the standard CNF encoding of the FPHP formulas require exponential proof size in polynomial calculus resolution.  ...  [Alekhnovich and Razborov '03] established that if the clause-variable incidence graph of a CNF formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree.  ...  Acknowledgements We are grateful to Ilario Bonacina, Yuval Filmus, Nicola Galesi, Massimo Lauria, Alexander Razborov, and Marc Vinyals for numerous discussions on proof complexity in general and polynomial  ... 
arXiv:1505.01358v1 fatcat:u5f3ihuflfcspmqgza7e3rl6ai

SAT and Interactions (Dagstuhl Seminar 20061)

Olaf Beyersdorff, Uwe Egly, Meena Mahajan, Cláudia Nalon
2020 Dagstuhl Reports  
lambda-calculus, a typed lambda-calculus with explicit sharing and atomic duplication that achieves spinal full laziness: duplicating only the direct paths between a binder and bound variables is enough for  ...  bounded width CNF-formulas where the width is measured by popular graph width measures on graphs associated to CNF-formulas.  ...  Practical MaxSAT Solving: A Survey Revisiting Graph Width Measures for CNF-Encodings Stefan Mengel (CNRS, CRIL -Lens FR) License Creative Commons BY 3.0 Unported license © Stefan Mengel We consider  ... 
doi:10.4230/dagrep.10.2.1 dblp:journals/dagstuhl-reports/BeyersdorffEMN20 fatcat:q6vjxmce4bd2nkm3vym2qkvqam

Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling

Susanna F. De Rezende, Jakob Nordström, Or Meir, Robert Robere, Michael Wagner
2019 Computational Complexity Conference  
We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t and space  ...  We use this correspondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to the best of our knowledge are the first such results for this proof system.  ...  Acknowledgements We are grateful for many interesting discussions about matters pebbling-related (and not-so-pebbling-related) with Arkadev Chattopadhyay, Toniann Pitassi, and Marc Vinyals.  ... 
doi:10.4230/lipics.ccc.2019.18 dblp:conf/coco/RezendeNMR19 fatcat:wlibxrycy5hirmyvxb3yvgjal4

Proof Complexity Meets Algebra

Albert Atserias, Joanna Ochremiak
2018 ACM Transactions on Computational Logic  
SOS unsatisfiability proofs exist only for problems of bounded width [11, 25] .  ...  The former holds exactly for the widely studied class of constraint languages of bounded width.  ...  In [7] , such a reduction was obtained for the so-called Tseitin formulas, that encode a certain system of linear equations over Z 2 that is derived from an expander graph.  ... 
doi:10.1145/3265985 fatcat:hhamzk75vfhrbj4vuf7eo5lzmm

Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling [article]

Susanna F. de Rezende, Or Meir, Jakob Nordström, Robert Robere
2020 arXiv   pre-print
We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t and space  ...  We use this correspondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to the best of our knowledge are the first such results for this proof system.  ...  Acknowledgements We are grateful for many interesting discussions about matters pebbling-related (and not-so-pebblingrelated) with Arkadev Chattopadhyay, Toniann Pitassi, and Marc Vinyals.  ... 
arXiv:2001.02481v1 fatcat:65zt7y2dlnag3kjwir4s62afl4

Learning to Reason: Leveraging Neural Networks for Approximate DNF Counting

Ralph Abboud, Ismail Ceylan, Thomas Lukasiewicz
2020 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
In this paper, we propose a neural model counting approach for weighted #DNF that combines approximate model counting with deep learning, and accurately approximates model counts in linear time when width  ...  Weighted model counting (WMC) has emerged as a prevalent approach for probabilistic inference. In its most general form, WMC is #P-hard.  ...  Experiments for this work were conducted on servers provided by the Advanced Research Computing (ARC) cluster administered by the University of Oxford.  ... 
doi:10.1609/aaai.v34i04.5705 fatcat:xqx4sazhybasfgy3cxti5yaoce

Narrow Proofs May Be Maximally Long

Albert Atserias, Massimo Lauria, Jakob Nordström
2016 ACM Transactions on Computational Logic  
The first lower bounds on space in PCR were obtained in [ABRW02], but only worked for CNF formulas of unbounded width.  ...  We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n Ω(w) .  ...  Acknowledgments The authors would like to thank Mladen Mikša and Marc Vinyals for interesting discussions related to the topics of this work.  ... 
doi:10.1145/2898435 fatcat:fyixt7jbinastnhcaltahbrjmi

Narrow Proofs May Be Maximally Long [article]

Albert Atserias, Massimo Lauria, Jakob Nordström
2014 arXiv   pre-print
We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n^Omega(w).  ...  This shows that the simple counting argument that any formula refutable in width w must have a proof in size n^O(w) is essentially tight.  ...  Acknowledgments The authors would like to thank Mladen Mikša and Marc Vinyals for interesting discussions related to the topics of this work. Part  ... 
arXiv:1409.2731v1 fatcat:luvihb4my5dnnmegzob5lomsne

Narrow Proofs May Be Maximally Long

Albert Atserias, Massimo Lauria, Jakob Nordstrom
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
For all of these results, the concept of width, measured as the size of a largest clause in a resolution proof, has turned out to play a key role.  ...  We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n Ω(w) .  ...  Acknowledgments The authors would like to thank Mladen Mikša and Marc Vinyals for interesting discussions related to the topics of this work. Part  ... 
doi:10.1109/ccc.2014.36 dblp:conf/coco/AtseriasLN14 fatcat:uek3kmbdobaehdcplxysvsmoaa

Are Short Proofs Narrow? QBF Resolution Is Not So Simple

Olaf Beyersdorff, Leroy Chew, Meena Mahajan, Anil Shukla
2017 ACM Transactions on Computational Logic  
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 tree-like Q-Res (Theorem 4.2), thus refuting the size-width relation for tree-like Q-Res as well as the space-width relation for general dag-like Q-Res.  ... 
doi:10.1145/3157053 fatcat:yr3iizyeofdunebajxyi45ei6q

The complexity of first-order and monadic second-order logic revisited

Markus Frick, Martin Grohe
2004 Annals of Pure and Applied Logic  
We establish a number of similar lower bounds for the model-checking problem for first-order logic, for example, on the class of all trees.  ...  In this paper, we give super-exponential lower bounds for fixed-parameter tractable model-checking problems for first-order and monadic second-order logic.  ...  is also fixed-parameter tractable on trees and graphs of bounded tree-width [3] .  ... 
doi:10.1016/j.apal.2004.01.007 fatcat:ujya375lw5a7vdx7njjjgur5qy

Learning to Reason: Leveraging Neural Networks for Approximate DNF Counting [article]

Ralph Abboud, Ismail Ilkan Ceylan, Thomas Lukasiewicz
2020 arXiv   pre-print
In this paper, we propose a neural model counting approach for weighted #DNF that combines approximate model counting with deep learning, and accurately approximates model counts in linear time when width  ...  Weighted model counting (WMC) has emerged as a prevalent approach for probabilistic inference. In its most general form, WMC is #P-hard.  ...  Model Setup We encode a DNF formula as a graph with 3 layers as shown in Figure 2 : a literal layer, a conjunction layer, and a disjunction layer.  ... 
arXiv:1904.02688v5 fatcat:sakdlw6jyfbdtcqzwrc7j3mwua
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