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Counting solutions to random CNF formulas
[article]

2021
*
arXiv
*
pre-print

We give the first efficient algorithm

arXiv:1911.07020v4
fatcat:umza5sdnqrh5bafg4vbduxswvi
*to*approximately*count*the number of*solutions*in the*random*k-SAT model when the density of the*formula*scales exponentially with k. ... Instead, our algorithm harnesses a recent technique by Moitra*to*work for*random**formulas*. ... As our goal is*to**count*satisfying assignments of sparse*random*k-*CNF**formulas*, where these degree bounds do not hold, but average degrees are small, it is natural*to*also choose Moitra's method in the ...##
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Counting Solutions to Random CNF Formulas

2020
*
International Colloquium on Automata, Languages and Programming
*

We give the first efficient algorithm

doi:10.4230/lipics.icalp.2020.53
dblp:conf/icalp/GalanisG0Y20
fatcat:qwawdewkizb2jakztnzii5bixy
*to*approximately*count*the number of*solutions*in the*random*k-SAT model when the density of the*formula*scales exponentially with k. ... Instead, our algorithm harnesses a recent technique by Moitra*to*work for*random**formulas*with much higher densities. ... Then, for a particular marked variable 53:4*Counting**Solutions**to**Random**CNF**Formulas*v, we set up an LP. ...##
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Counting Solutions to Random CNF Formulas

2021
*
SIAM journal on computing (Print)
*

We give the first efficient algorithm

doi:10.1137/20m1351527
fatcat:zzdf76xssngipefh5khhqr73kq
*to*approximately*count*the number of*solutions*in the*random*k-SAT model when the density of the*formula*scales exponentially with k. ... Instead, our algorithm harnesses a recent technique by Moitra*to*work for*random**formulas*with much higher densities. ... As our goal is*to**count*satisfying assignments of sparse*random*k-*CNF**formulas*, where these degree bounds do not hold, but average degrees are small, it is natural*to*also choose Moitra's method in the ...##
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The Hard Problems Are Almost Everywhere For Random CNF-XOR Formulas

2017
*
Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence
*

On the theoretical front, we prove that the

doi:10.24963/ijcai.2017/84
dblp:conf/ijcai/DudekMV17
fatcat:l3d4acceencftpib24rheu7n34
*solution*space of a*random**CNF*-XOR*formula*'shatters' at all nonzero XOR-clause densities into well-separated components, similar*to*the behavior seen in*random*...*CNF**formulas*known*to*be difficult for many SAT algorithms. ... We prove (in Section 4) that the*solution*space of*random*variable-width XOR*formulas*(and therefore of*random*k-*CNF*-XOR*formulas*) shatters. ...##
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On Hashing-Based Approaches to Approximate DNF-Counting
[article]

2017
*
arXiv
*
pre-print

When the constraints are expressed as DNF

arXiv:1710.05247v1
fatcat:oy5tcwuqybd7xpeia6xmfaphge
*formulas*, Monte Carlo-based techniques have been shown*to*provide a fully polynomial*randomized*approximation scheme (FPRAS). ... Given the success of hashing-based techniques for*CNF*constraints, it is natural*to*ask: Can hashing-based techniques provide an efficient FPRAS for DNF*counting*? ... Given a Boolean*formula*φ, the problem of propositional model*counting*, also referred*to*as #SAT, is*to*compute the number of*solutions*of φ [29] . ...##
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Full CNF Encoding: The Counting Constraints Case

2004
*
International Conference on Theory and Applications of Satisfiability Testing
*

Many problems are naturally expressed using

dblp:conf/sat/BailleuxB04
fatcat:i6my6motlvg25dr4cooc5j7d6q
*CNF*clauses and boolean cardinality constraints. ... We show experimentally that our encoding allows zchaff*to*be competitive with pseudo-boolean solvers on some decision and optimization problems. ... Second, we present a DLL based optimization procedure that maximizes the number of variables fixed*to*1 in a given subset of the variables belonging*to*a*CNF**formula*, while satisfying the*formula*. ...##
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Tinted, Detached, and Lazy CNF-XOR Solving and Its Applications to Counting and Sampling
[chapter]

2020
*
Lecture Notes in Computer Science
*

Given a Boolean

doi:10.1007/978-3-030-53288-8_22
fatcat:5hkp4g2yx5d2pgpzekcym44nsi
*formula*, the problem of*counting*seeks*to*estimate the number of*solutions*of F while the problem of uniform sampling seeks*to*sample*solutions*uniformly at*random*. ... The past few years have witnessed the rise of hashing-based approaches that use XORbased hashing and employ SAT solvers*to*solve the resulting*CNF**formulas*conjuncted with XOR constraints. ... The problem of propositional model*counting*is*to*compute |sol(F )| for a given*CNF**formula*F . ...##
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An Approximation Algorithm for #k-SAT

2012
*
Symposium on Theoretical Aspects of Computer Science
*

For 4-

doi:10.4230/lipics.stacs.2012.78
dblp:conf/stacs/Thurley12
fatcat:375aoaw36nclvckyjuip4bv4kq
*CNF*input the bound increases*to*O(1.6155 n ). We further show how*to*obtain upper and lower bounds on the number of*solutions**to*a*CNF**formula*in a controllable way. ... For example, even stipulating sub-exponentially small error tolerance, the number of*solutions**to*3-*CNF*input*formulas*can be approximated in time O(1.5366 n ). ... Acknowledgments I would like*to*thank Martin Grohe for bringing this problem*to*my attention and for several helpful discussions on the topic. ...##
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Combining the k-CNF and XOR Phase-Transitions
[article]

2017
*
arXiv
*
pre-print

Recent universal hashing-based approaches

arXiv:1702.08392v1
fatcat:vyap6gbgnjedtm6t2vsbebljqi
*to*sampling and*counting*crucially depend on the runtime performance of SAT solvers on*formulas*expressed as the conjunction of both k-*CNF*and XOR constraints ... (known as k-*CNF*-XOR*formulas*), but the behavior of*random*k-*CNF*-XOR*formulas*is unexplored in prior work. ... Acknowledgments The authors would like*to*thank Dimitris Achlioptas for helpful discussions in the early stages of this project. ...##
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An Approximation Algorithm for #k-SAT
[article]

2011
*
arXiv
*
pre-print

For 4-

arXiv:1107.2001v1
fatcat:4ti42kporvef7kgmm2qhlqotli
*CNF*input the bound increases*to*O(1.6155^n). We further show how*to*obtain upper and lower bounds on the number of*solutions**to*a*CNF**formula*in a controllable way. ... For example, even stipulating sub-exponentially small error tolerance, the number of*solutions**to*3-*CNF*input*formulas*can be approximated in time O(1.5366^n). ... Preliminaries For a*CNF**formula*F , let sat(F ) be the set of its*solutions*and #F = |sat(F )|. We shall always use n*to*denote the number of variables of a*CNF**formula*under consideration. ...##
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Computing the Density of States of Boolean Formulas
[chapter]

2010
*
Lecture Notes in Computer Science
*

In this paper we consider the problem of computing the density of states of a Boolean

doi:10.1007/978-3-642-15396-9_6
fatcat:ba4b72rtbzayhg2bc3bfugse5i
*formula*in*CNF*, a generalization of both MAX-SAT and model*counting*. ... We propose a novel Markov Chain Monte Carlo algorithm based on flat histogram methods that, despite the hardness of the problem, converges quickly*to*a very accurate*solution*. ...*Random**Formulas*In this section we present a detailed study of the behavior of the DOS for*random*3-SAT*formulas*as a function of the ratio clauses*to*variables α. ...##
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Solving and Sampling with Many Solutions: Satisfiability and Other Hard Problems
[article]

2017
*
arXiv
*
pre-print

We investigate parameterizing hard combinatorial problems by the size of the

arXiv:1708.01122v1
fatcat:dk4o2sovnvhb7anxj7iagu2era
*solution*set compared*to*all*solution*candidates. ... Our main result is a uniform sampling algorithm for satisfying assignments of 2-*CNF**formulas*that runs in expected time O^*(ε^-0.617) where ε is the fraction of assignments that are satisfying. ... Acknowledgments We would like*to*thank Noga Alon and József Solymosi for discussions on the problem. We also thank the reviewers of IPEC 2017 for valuable remarks that improved the exposition. ...##
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Distribution-Aware Sampling and Weighted Model Counting for SAT

2014
*
PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE
*

Given a

doi:10.1609/aaai.v28i1.8990
fatcat:474lujaf4zccrdy56rlclkskqi
*CNF**formula*and a weight for each assignment of values tovariables, two natural problems are weighted model*counting*anddistribution-aware sampling of satisfying assignments. ... Due*to*the inherentcomplexity of the exact versions of the problems, interest has focusedon solving them approximately. ... Acknowledgments: The authors would like*to*thank Armando Solar-Lezama for generously providing a large set of benchmarks. ...##
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Solving Stochastic Boolean Satisfiability under Random-Exist Quantification

2017
*
Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence
*

Experimental results show that our method outperforms the state-of-the-art algorithm on

doi:10.24963/ijcai.2017/96
dblp:conf/ijcai/LeeWJ17
fatcat:lcgtk2zz75gqvcz2jwzzikvq3q
*random*k-*CNF**formulas*and has effective application*to*approximate SSAT on circuit benchmarks. ... This paper focuses on*random*-exist quantified SSAT*formulas*, and proposes an algorithm combining binary decision diagram (BDD), logic synthesis, and modern SAT techniques*to*improve computational efficiency ...*Random*k-*CNF**Formulas*The*random*k-*CNF**formulas*are generated using the cnfgen command [Lauria, 2012] . ...##
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BIRD: Engineering an Efficient CNF-XOR SAT Solver and Its Applications to Approximate Model Counting

2019
*
PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE
*

Given a Boolean

doi:10.1609/aaai.v33i01.33011592
fatcat:jfq2fifo6vemxmggx5sagyukp4
*formula*φ, the problem of model*counting*, also referred*to*as #SAT is*to*compute the number of*solutions*of φ. ... The primary contribution of this paper is an affirmative answer*to*the above question. We present a novel architecture, called BIRD,*to*handle*CNF*-XOR*formulas*arising from hashingbased techniques. ... Acknowledgements We are grateful*to*the anonymous Reviewer #3 for the excellent suggestions*to*rewrite the Introduction. ...
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