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Space Bounds for Resolution [chapter]

Juan Luis Esteban, Jacobo Torán
1999 Lecture Notes in Computer Science  
Measured on the number of clauses, this result is the best possible.  ...  We also give upper and lower bounds on the space needed for the resolution of unsatis able formulas.  ...  Acknowledgment: The authors would like to thank Jochen Messner for helpful discussions on earlier versions of the paper.  ... 
doi:10.1007/3-540-49116-3_52 fatcat:iu5a7m5svfaxhefvhobweftzcy

Lower Bounds for Width-Restricted Clause Learning on Small Width Formulas [chapter]

Eli Ben-Sasson, Jan Johannsen
2010 Lecture Notes in Computer Science  
This experience is supported by lower bound theorems. It is shown that lower bounds on the runtime of width-restricted clause learning follow from resolution width lower bounds.  ...  This yields the first lower bounds on width-restricted clause learning for formulas in 3-CNF.  ...  In this work we prove lower bounds on the size of RTL(k)-refutations, which thus yield lower bounds on the runtime of DLL algorithms with width-restricted clause-learning.  ... 
doi:10.1007/978-3-642-14186-7_4 fatcat:c4k72ngmqze3pa4hohv3lxitie

Resolution and the weak pigeonhole principle [chapter]

Sam Buss, Toniann Pitassi
1998 Lecture Notes in Computer Science  
We give new upper bounds for resolution proofs of the weak pigeonhole principle. We also give lower bounds for tree-like resolution proofs.  ...  We present a normal form for resolution proofs of pigeonhole principles based on a new monotone resolution rule. n ; in other words,  ...  In section 5, we prove an exponential lower bound on the size of tree-like resolution proofs of PHP m n . Definitions This paper deals exclusively with propositional logic.  ... 
doi:10.1007/bfb0028012 fatcat:yg6rzz6ekrdnlicasswsggy4de

Number of Variables for Graph Identification and the Resolution of GI Formulas [article]

Jacobo Torán, Florian Wörz
2021 Electronic colloquium on computational complexity  
We also show a resolution size lower bound of exp Ω(k 2 /n) for the case of colored graphs with constant color class size.  ...  isomorphism formula, as well as lower bounds of 2 k−1 and k for the tree-like resolution size and resolution clause space for this formula.  ...  Acknowledgments We thank Pascal Schweitzer for helpful discussion regarding the topics of this paper.  ... 
dblp:journals/eccc/ToranW21 fatcat:572wgmfwnngojjrmpypomagari

Width versus size in resolution proofs

Alasdair Urquhart
2007 Theoretical Computer Science  
A general theme in the proof of lower bounds on the size of resolution refutations in propositional logic has been that of basing size lower bounds on lower bounds on the width of refutations.  ...  Ben-Sasson and Wigderson have proved a general width-size tradeoff result that can be used to prove many of the lower bounds on resolution complexity in a uniform manner.  ...  Acknowledgement The author gratefully acknowledges the support of the Natural Sciences and Engineering Research Council of Canada.  ... 
doi:10.1016/j.tcs.2007.05.022 fatcat:oypxlxhvtrgndfxskf4b73f6vi

The Efficiency of Resolution and Davis--Putnam Procedures

Paul Beame, Richard Karp, Toniann Pitassi, Michael Saks
2002 SIAM journal on computing (Print)  
ACM, 35 (1988), pp. 759-768] had shown that random 3-CNF formulas with a linear number of clauses require exponential size resolution proofs, and Fu [On the Complexity of Proof Systems, Ph.D. thesis, University  ...  Finally, we give new lower bounds on res(F ), the size of the smallest resolution refutation of F , for a class of formulas representing the pigeonhole principle and for randomly generated formulas.  ...  ., by listing the b-bounded clauses of F and, for each clause on the list, resolve it with each clause preceding it (if possible) and add the resolvent to the end of the list, if it is of size at most  ... 
doi:10.1137/s0097539700369156 fatcat:ezkcs6xthncspcc3dgkd525bfy

On the Resolution Complexity of Graph Non-isomorphism [chapter]

Jacobo Torán
2013 Lecture Notes in Computer Science  
These graphs pairs are colored with color multiplicity bounded by 4.  ...  Using this fact, we provide an explicit family of non-isomorphic graph pairs for which any resolution refutation requires an exponential number of clauses in the size of the initial formula.  ...  On the other hand we prove an exponential lower bound for the resolution refutation of certain pairs of non-isomorphic graphs with color classes of size 4 or larger.  ... 
doi:10.1007/978-3-642-39071-5_6 fatcat:aafq24tq2vgbhij4qafzb5bnea

A feasibly constructive lower bound for resolution proofs

Stephen Cook, Toniann Pitassi
1990 Information Processing Letters  
Other applications Urquhart [6] obtained an exponential lower bound on the size of resolution proofs for a family of graph-based clauses.  ...  Later, several authors [6,1,2] extended Haken's techniques to obtain exponential lower bounds on the size of resolution proofs.  ... 
doi:10.1016/0020-0190(90)90141-j fatcat:yrscyxntibg67kvh5p72vvrdum

Narrow Proofs May Be Maximally Long

Albert Atserias, Massimo Lauria, Jakob Nordström
2016 ACM Transactions on Computational Logic  
Proof size and space in PCR are defined in analogy with resolution, but counting monomials rather than clauses, and the measure corresponding to width of clauses is (total) degree of polynomials.  ...  relation has been employed to prove exponential lower bounds on size in a number of papers, with the most general setting perhaps provided in [AR03, MN15].  ...  in the overview of previous work in the introduction.  ... 
doi:10.1145/2898435 fatcat:fyixt7jbinastnhcaltahbrjmi

Space in weak propositional proof systems

Ilario Bonacina
2016 Bulletin of the European Association for Theoretical Computer Science  
Ilario's thesis contributes to a classic and deep topic in theoretical computer science, and settles natural questions on the space complexity of proofs using Resolution and the Polynomial Calculus that  ...  The award is presented at the annual Italian Conference on Theoretical Computer Science (ICTCS), where the award recipient delivers a presentation on her/his work.  ...  Size and width in Resolution Together with the main results about space this thesis contains also a detour on size, cf. Section 5 of this abstract.  ... 
dblp:journals/eatcs/Bonacina16 fatcat:7oxgzcxtdnf4bfpncyd27bhshe

Reflections on Proof Complexity and Counting Principles [article]

Noah Fleming, Toniann Pitassi
2021 Electronic colloquium on computational complexity  
We focus on the central role of counting principles, and in particular Tseitin's graph tautologies, to most of the key advances in lower bounds in proof complexity.  ...  We reflect on a couple of key ideas that Urquhart pioneered: (i) graph expansion as a tool for distinguishing between easy and hard principles, and (ii) "reductive" lower bound arguments, proving via a  ...  That is, ( ) is the size of the minimal set of vertices ⊆ such that the clauses associated with imply . Since Resolution is a sound procedure, is subadditive.  ... 
dblp:journals/eccc/FlemingP21 fatcat:3kswt2vybrgd5k6xfkqbeiv24u

Trade-offs Between Time and Memory in a Tighter Model of CDCL SAT Solvers [chapter]

Jan Elffers, Jan Johannsen, Massimo Lauria, Thomas Magnard, Jakob Nordström, Marc Vinyals
2016 Lecture Notes in Computer Science  
With a view towards obtaining results with tighter relations between CDCL and resolution, we introduce a more fine-grained model of CDCL that captures not only time but also memory usage and number of  ...  We show how previously established strong size-space trade-offs for resolution can be transformed into equally strong trade-offs between time and memory usage for CDCL, where the upper bounds hold for  ...  The third author performed this work while at KTH Royal Institute of Technology, and most of the work of the second and fourth author was done while visiting KTH.  ... 
doi:10.1007/978-3-319-40970-2_11 fatcat:4jiqks3o7rgsrim2njfx36rt54

The Resolution Complexity of Independent Sets and Vertex Covers in Random Graphs

Paul Beame, Russell Impagliazzo, Ashish Sabharwal
2007 Computational Complexity  
We consider the problem of providing a resolution proof of the statement that a given graph with n vertices and average degree roughly ∆ does not contain an independent set of size k.  ...  Further, it provides an unconditional exponential lower bound on the running time of resolution-based search algorithms for finding a maximum independent set or approximating it within a factor of ∆/(6  ...  A preliminary version of this paper appeared in the Proceedings of the 16 th Annual IEEE Conference on Computational Complexity, Chicago, June 2001.  ... 
doi:10.1007/s00037-007-0230-0 fatcat:aolzyvo4v5g3xd4wxq2usgz4oi

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  
Since the upper bounds in our trade-offs hold for resolution, our work shows that there are formulas for which adding algebraic reasoning on top of resolution does not improve the trade-off properties  ...  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

Narrow Proofs May Be Maximally Long

Albert Atserias, Massimo Lauria, Jakob Nordstrom
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
Proof size and space in PCR is defined in analogy with resolution, and the measure corresponding to width of clauses is (total) degree of polynomials.  ...  been employed to prove exponential lower bounds on size in a number of papers, with [AR03] perhaps providing the most general setting.  ...  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
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