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The Space Complexity of Cutting Planes Refutations

Nicola Galesi, Pavel Pudlák, Neil Thapen, Marc Herbstritt
2015 Computational Complexity Conference  
We study the space complexity of the cutting planes proof system, in which the lines in a proof are integral linear inequalities.  ...  We show that any unsatisfiable set of inequalities has a cutting planes refutation in space five.  ...  Cutting planes has been studied from the point of view of the size complexity of proofs, usually measured as the number of lines in a refutation.  ... 
doi:10.4230/lipics.ccc.2015.433 dblp:conf/coco/GalesiPT15 fatcat:n736vrqq25d5pkys57umbmhpj4

On the virtue of succinct proofs

Trinh Huynh, Jakob Nordstrom
2012 Proceedings of the 44th symposium on Theory of Computing - STOC '12  
An active line of research in proof complexity over the last decade has been the study of proof space and trade-offs between size and space.  ...  However, for other relevant proof systems in the context of SAT solving, such as polynomial calculus (PC) and cutting planes (CP), very little has been known.  ...  during the weeks spent jointly at KTH Royal Institute of Technology.  ... 
doi:10.1145/2213977.2214000 dblp:conf/stoc/HuynhN12 fatcat:il3fpwscg5crjle2mnnbm3jihq

Stabbing Planes [article]

Paul Beame, Noah Fleming, Russell Impagliazzo, Antonina Kolokolova, Denis Pankratov, Toniann Pitassi, Robert Robere
2022 arXiv   pre-print
Finally, we prove linear lower bounds on the rank of Stabbing Planes refutations by adapting lower bounds in communication complexity and use these bounds in order to show that Stabbing Planes proofs cannot  ...  Among our results, we show that Stabbing Planes can efficiently simulate the Cutting Planes proof system, and is equivalent to a tree-like variant of the R(CP) system of Krajicek98.  ...  Recall that branch-and-cut solvers combine Stabbing Planes-style branching with additional cutting planes in order to refine the search space.  ... 
arXiv:1710.03219v2 fatcat:4dedtl7iordyvcmquixcpwmd6y

Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity [article]

Susanna F. de Rezende, Or Meir, Jakob Nordström, Toniann Pitassi, Robert Robere, Marc Vinyals
2020 arXiv   pre-print
Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting planes with unbounded coefficients, but for which there are no refutations in subexponential  ...  In particular, this implies that the standard decision tree complexity and the parity decision tree complexity of the corresponding falsified clause search problem are equal.  ...  Complexity, which is conducted with support from the National Science Foundation.  ... 
arXiv:2001.02144v1 fatcat:urgi4cd5vvczdnsqcnqloxsj6q

A Rank Lower Bound for Cutting Planes Proofs of Ramsey's Theorem

Massimo Lauria
2016 ACM Transactions on Computation Theory  
In particular we focus on the propositional proof system cutting planes; we prove that the upper bound "r(k, k) ≤ 4 k " requires cutting planes proof of high rank.  ...  We study the complexity of proving upper bounds for the number r(k, k).  ...  In this paper we prove a rank lower bound of roughly Ω( 4 √ n). The Rank of a Cutting Planes Refutation One complexity measure for cutting planes is the "rank" of an inference.  ... 
doi:10.1145/2903266 fatcat:aikdisyslzflboe65jqgi6p4l4

Some trade-off results for polynomial calculus

Chris Beck, Jakob Nordstrom, Bangsheng Tang
2013 Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13  
As such, our results cover space complexity from constant all the way up to exponential and yield mostly superpolynomial or even exponential size blow-ups.  ...  As byproducts of our analysis, we also obtain trade-offs between space and degree in PC and PCR exactly matching analogous results for space versus width in resolution, and strengthen the resolution trade-offs  ...  Acknowledgements This article is the result of a long process, and various subsets of the authors would like to acknowledge useful discussions had during the last few years with various subsets of Paul  ... 
doi:10.1145/2488608.2488711 dblp:conf/stoc/BeckNT13 fatcat:7vd2etdq7fcebbfakn5uj3gt3u

A Rank Lower Bound for Cutting Planes Proofs of Ramsey's Theorem [chapter]

Massimo Lauria
2013 Lecture Notes in Computer Science  
In particular we focus on the propositional proof system cutting planes; we prove that the upper bound "r(k, k) ≤ 4 k " requires cutting planes proof of high rank.  ...  We study the complexity of proving upper bounds for the number r(k, k).  ...  Acknowledgment Part of this work was done while the first author was at the Math Institute of the Czech Academy of Science, funded by the Eduard Čech Center.  ... 
doi:10.1007/978-3-642-39071-5_26 fatcat:tb624rsot5dtbi56vwgstndnku

Space Complexity in Polynomial Calculus

Yuval Filmus, Massimo Lauria, Jakob Nordstrom, Neil Thapen, Noga Ron-Zewi
2012 2012 IEEE 27th Conference on Computational Complexity  
During the last decade, an active line of research in proof complexity has been to study space complexity and timespace trade-offs for proofs.  ...  We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3-CNF in the canonical way.  ...  The work presented in this paper was initiated at the Banff International Research Station workshop on proof complexity (11w5103) in October 2011 and part of the work was also performed during the special  ... 
doi:10.1109/ccc.2012.27 dblp:conf/coco/FilmusLNTR12 fatcat:qmcbbck3fbgqhjoe5h4re3k6sm

Space Complexity in Polynomial Calculus

Yuval Filmus, Massimo Lauria, Jakob Nordström, Noga Ron-Zewi, Neil Thapen
2015 SIAM journal on computing (Print)  
During the last decade, an active line of research in proof complexity has been to study space complexity and timespace trade-offs for proofs.  ...  We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3-CNF in the canonical way.  ...  The work presented in this paper was initiated at the Banff International Research Station workshop on proof complexity (11w5103) in October 2011 and part of the work was also performed during the special  ... 
doi:10.1137/120895950 fatcat:vzsfnttcf5gnppg7vz2ij3hmvm

On the Power and Limitations of Branch and Cut [article]

Noah Fleming, Mika Göös, Russell Impagliazzo, Toniann Pitassi, Robert Robere, Li-Yang Tan, Avi Wigderson
2021 arXiv   pre-print
Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field.  ...  This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas.  ...  The resolution depth depth Res (F ) of F is the minimal depth of any resolution refutation of F . Cutting Planes and Semantic Cutting Planes.  ... 
arXiv:2102.05019v2 fatcat:2kls6apafjbzxlsw2pfpkyissm

A (Biased) Proof Complexity Survey for SAT Practitioners [chapter]

Jakob Nordström
2014 Lecture Notes in Computer Science  
For space vs. width, the answer is a strong no Theorem ([Ben09]) There are formulas for which exist refutations in width O(1) exist refutations in space O(1) optimization of one measure causes  ...  of DAG somehow carry over to resolution refutations of pebbling formulas.  ...  0 ≡ true and 1 ≡ false) Derivation rules Boolean axioms Cutting Planes Introduced in [CCT87] Clauses interpreted as linear inequalities over the reals with integer coefficients Example: x ∨ y ∨  ... 
doi:10.1007/978-3-319-09284-3_1 fatcat:a3lqtt252fgexfedgl7xbhu2by

How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity) [article]

Susanna F. de Rezende, Jakob Nordström, Marc Vinyals
2021 Electronic colloquium on computational complexity  
We obtain the first true size-space trade-offs for the cutting planes proof system, where the upper bounds hold for size and total space for derivations with constant-size coefficients, and the lower bounds  ...  These are also the first trade-offs to hold uniformly for resolution, polynomial calculus and cutting planes, thus capturing the main methods of reasoning used in current state-of-the-art SAT solvers.  ...  whom we have had many fruitful discussions on time-space trade-offs and other topics in proof complexity, and Arkadev Chattopadhyay for suggesting us to look into the monotone-AC i−1 vs monotone-NC i  ... 
dblp:journals/eccc/RezendeNV21 fatcat:5s533c7cm5aejna2p3petzrlgq

Depth lower bounds in Stabbing Planes for combinatorial principles [article]

Stefan Dantchev, Nicola Galesi, Abdul Ghani, Barnaby Martin
2021 arXiv   pre-print
The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system established via communication complexity arguments  ...  As such they work for the lifted version of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas.  ...  These methods were recently extended in [9] also to the case of Semantic Cutting Planes.  ... 
arXiv:2102.07622v2 fatcat:gjrnams6ynhf3kjj7ih67uegam

Proof Complexity (Dagstuhl Seminar 18051)

Albert Atserias, Jakob Nordström, Pavel Pudlák, Rahul Santhanam, Michael Wagner
2018 Dagstuhl Reports  
The study of proof complexity was initiated in [Cook and Reckhow 1979] as a way to attack the P vs.  ...  Proof complexity also gives a way of studying subsystems of Peano Arithmetic where the power of mathematical reasoning is restricted, and to quantify how complex different mathematical theorems are measured  ...  An exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle. Random Structures and Algorithms, 7(1):15-40, 1995.  ... 
doi:10.4230/dagrep.8.1.124 dblp:journals/dagstuhl-reports/AtseriasNPS18 fatcat:5ksfbo2ehfhspcbuw4ppcuyaqu

Page 310 of MIND Vol. 15, Issue 59 [page]

1890 MIND  
Incline the cutting plane in the slightest degree, and the circle be- comes an ellipse.  ...  Further rotation of the plane turns the parabola into an hyperbola, which changes its form with every change of position of the plane ; until, when the plane, emerging from the opposite side of the cone  ... 
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