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A Sudoku-Solver for Large Puzzles using SAT

Uwe Pfeiffer, Tomas Karnagel, Guido Scheffler
<span class="release-stage">unpublished</span>
However, solving such puzzles for large n requires a significant amount of optimization.  ...  n&lt;sup&gt;2&lt;/sup&gt; x n&lt;sup&gt;2&lt;/sup&gt; Sudoku puzzles can be straightforwardly encoded as SAT problems.  ...  Acknowledgement The authors would like to thank Steffen Hlldobler for the encouragement to write this article and for providing advice.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.29007/79mc">doi:10.29007/79mc</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hqczimue7vcqziyotdifareagy">fatcat:hqczimue7vcqziyotdifareagy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180501125942/https://easychair.org/publications/open/VF3m" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a9/f8/a9f8b3add478f552153d8d6b311199c40866eb48.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.29007/79mc"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

POLYNOMIAL 3-SAT REDUCTION OF SUDOKU PUZZLE

Deepika Rai
<span title="2018-06-20">2018</span> <i title="IJARCS International Journal of Advanced Research in Computer Science"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/dbwzqwxyw5hn3aantchmoxx65q" style="color: black;">International Journal of Advanced Research in Computer Science</a> </i> &nbsp;
In this perspective, determining 3-SAT reduction from Sudoku Puzzle of size (n x n) is very helpful to obtain the solution of Sudoku Puzzle using SAT solver.  ...  Therefore, every NP-Complete problem can be reduced into 3-SAT that can be solved by a SAT solver.  ...  Sudoku Puzzle (n x n) can be reduced into 3-SAT in order to find solution using a SAT solver. There exist two SAT encodings for Sudoku Puzzle namely; extended encoding and minimal encoding.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.26483/ijarcs.v9i3.6096">doi:10.26483/ijarcs.v9i3.6096</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/p2z34m3thrglxhofkgwmrujkde">fatcat:p2z34m3thrglxhofkgwmrujkde</a> </span>
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Solving Single-digit Sudoku Subproblems [article]

David Eppstein
<span title="2012-03-04">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that single-digit "Nishio" subproblems in nxn Sudoku puzzles may be solved in time o(2^n), faster than previous solutions such as the pattern overlay method.  ...  We also show that single-digit deduction in Sudoku is NP-hard.  ...  Despite the ease of solving Sudoku using these methods, there is still a need for computer puzzle solvers that eschew both trial and error and sophisticated computer search techniques, and that instead  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1202.5074v2">arXiv:1202.5074v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jva6rwpqhfgdjlknzwmzozhonu">fatcat:jva6rwpqhfgdjlknzwmzozhonu</a> </span>
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Using Small MUSes to Explain How to Solve Pen and Paper Puzzles [article]

Joan Espasa, Ian P. Gent, Ruth Hoffmann, Christopher Jefferson, Alice M. Lynch
<span title="2021-04-30">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Pen and paper puzzles like Sudoku, Futoshiki and Skyscrapers are hugely popular. Solving such puzzles can be a trivial task for modern AI systems.  ...  We give several improvements to the existing techniques for solving puzzles with MUSes, which allow us to solve a range of significantly more complex puzzles and give higher quality explanations.  ...  For future work, we want to also explain exactly how the constraints in a MUS can be used to deduce the next step of the puzzle.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2104.15040v1">arXiv:2104.15040v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2zurl6xqpvdrtapxr3zrerzrii">fatcat:2zurl6xqpvdrtapxr3zrerzrii</a> </span>
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The Chaos Within Sudoku

Mária Ercsey-Ravasz, Zoltán Toroczkai
<span title="2012-10-11">2012</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/tnqhc2x2aneavcd3gx5h7mswhm" style="color: black;">Scientific Reports</a> </i> &nbsp;
Accordingly, η = − log 10 κ can be used to define a "Richter"-type scale for puzzle hardness, with easy puzzles falling in the range 0 < η ≤ 1, medium ones within 1 < η ≤ 2, hard in 2 < η ≤ 3 and ultra-hard  ...  Via an exact mapping of Sudoku into a deterministic, continuous-time dynamical system, here we show that the difficulty of Sudoku translates into transient chaotic behavior exhibited by the dynamical system  ...  For easy puzzles the transients are very short, p(t) decays fast resulting in large escape rates but for hard puzzles κ can be very small. Fig. 4b shows a zoom onto the p(t) of hard puzzles.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1038/srep00725">doi:10.1038/srep00725</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/23061008">pmid:23061008</a> <a target="_blank" rel="external noopener" href="https://pubmed.ncbi.nlm.nih.gov/PMC3468838/">pmcid:PMC3468838</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pqxsh7teljherj53grtvzgwtbm">fatcat:pqxsh7teljherj53grtvzgwtbm</a> </span>
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The Chaos Within Sudoku [article]

Maria Ercsey-Ravasz, Zoltan Toroczkai
<span title="2012-08-01">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Accordingly, η = -_10κ can be used to define a "Richter"-type scale for puzzle hardness, with easy puzzles falling in the range 0 < η≤ 1, medium ones within 1 < η≤ 2, hard in 2 < η≤ 3 and ultra-hard with  ...  Via an exact mapping of Sudoku into a deterministic, continuous-time dynamical system, here we show that the difficulty of Sudoku translates into transient chaotic behavior exhibited by the dynamical system  ...  For easy puzzles the transients are very short, p(t) decays fast resulting in large escape rates but for hard puzzles κ can be very small. Fig. 4b shows a zoom onto the p(t) of hard puzzles.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1208.0370v1">arXiv:1208.0370v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5ixlij5nhfh6hpuzfkzaajyffe">fatcat:5ixlij5nhfh6hpuzfkzaajyffe</a> </span>
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Multiple-Valued Problem Solvers -- Comparison of Several Approaches

Bernd Steinbach, Christian Posthoff
<span title="">2014</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/je7g4kpl6jbpxdooxmljnhyubm" style="color: black;">2014 IEEE 44th International Symposium on Multiple-Valued Logic</a> </i> &nbsp;
Utilizing XBOOLE on a GPU, all 44,664 solutions of a 25-clue Sudoku were found 20 times faster than by a SAT-solver.  ...  These approaches are applied to the well-known Sudoku problem of 81 9-valued variables. Hence, the search space for this finite problem has a size of 9 81 (approximately 1.9 * 10 77 ).  ...  Table III RUNTIME III FOR THE 25-CLUE SUDOKU OF FIGURE 7 USING A SAT-SOLVER OR XBOOLE EITHER ON A CPU OR ON A GPU method hardware 32 bits 64 bits SAT-solver CPU 14.29 seconds XBOOLE CPU 4.76  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/ismvl.2014.13">doi:10.1109/ismvl.2014.13</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/ismvl/SteinbachP14.html">dblp:conf/ismvl/SteinbachP14</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ocvlgdo2qfayldpikjy5hayxti">fatcat:ocvlgdo2qfayldpikjy5hayxti</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170705121438/http://www.informatik.tu-freiberg.de/prof2/publikationen/ISMVL_2014_%20mvps_csa.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/50/d9/50d9b2e183a2ded803c1cb73dbff1eba25dca3fa.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/ismvl.2014.13"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Problem Solving at the Edge of Chaos: Entropy, Puzzles and the Sudoku Freezing Transition [article]

Marcelo Prates, Luis Lamb
<span title="2018-10-08">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Sudoku is a widely popular NP-Complete combinatorial puzzle whose prospects for studying human computation have recently received attention, but the algorithmic hardness of Sudoku solving is yet largely  ...  In this paper, we study the statistical mechanical properties of random Sudoku grids, showing that puzzles of varying sizes attain a hardness peak associated with a critical behavior in the constrainedness  ...  For a large number of N P-Hard problems, it has been shown that this boundary is sharp, meaning that statistical variables such as the satisfiability probability of a random k-SAT formula undergo abrupt  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1810.03742v1">arXiv:1810.03742v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ajs6jk5evjefphxju3opefttoe">fatcat:ajs6jk5evjefphxju3opefttoe</a> </span>
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Effective problem solving using SAT solvers [article]

Curtis Bright, Jürgen Gerhard, Ilias Kotsireas, Vijay Ganesh
<span title="2019-09-16">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this article we demonstrate how to solve a variety of problems and puzzles using the built-in SAT solver of the computer algebra system Maple.  ...  In particular, we describe how to solve the n-queens problem, how to generate and solve Sudoku puzzles, how to solve logic puzzles like the Einstein riddle, how to solve the 15-puzzle, how to solve the  ...  Sudoku puzzles Sudoku is a popular puzzle that appears in many puzzle books and newspapers.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1906.06251v2">arXiv:1906.06251v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/onppeogpsfhe3hpa6ssgoytm54">fatcat:onppeogpsfhe3hpa6ssgoytm54</a> </span>
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Difficulty Rating of Sudoku Puzzles: An Overview and Evaluation [article]

Radek Pelánek
<span title="2014-03-28">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
How can we predict the difficulty of a Sudoku puzzle?  ...  We give an overview of difficulty rating metrics and evaluate them on extensive dataset on human problem solving (more then 1700 Sudoku puzzles, hundreds of solvers).  ...  Acknowledgement The author thanks to webmasters of fed-sudoku.eu and czech-sudoku.com portals for providing the data, JiříŠimša and Petr Jarušek for inspiring discussions, and Martin Křivánek for carrying  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1403.7373v1">arXiv:1403.7373v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rsirx5bfmfegfdtikkglhwbheq">fatcat:rsirx5bfmfegfdtikkglhwbheq</a> </span>
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Functional SMT solving with Z3 and racket

Siddharth Agarwal, Amey Karkare
<span title="">2013</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wvv27s77dvd5flktsj246kcxwu" style="color: black;">2013 1st FME Workshop on Formal Methods in Software Engineering (FormaliSE)</a> </i> &nbsp;
We present an attempt to make using SMT solvers simpler by integrating the Z3 solver into a host language, Racket.  ...  This system is expected to be useful for a wide variety of applications, from simple constraint solving to writing tools for debugging, verification, and automatic test generation for functional programs  ...  ACKNOWLEDGEMENTS We thank Leonardo de Moura at Microsoft Research for his help in understanding the Z3 C API.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/formalise.2013.6612272">doi:10.1109/formalise.2013.6612272</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/icse/AgarwalK13.html">dblp:conf/icse/AgarwalK13</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ovesyueluva5fms4mpfd2vcqbi">fatcat:ovesyueluva5fms4mpfd2vcqbi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809031741/https://www.cse.iitk.ac.in/users/karkare/pubs/icsews13formalise-id1-p-16138-preprint.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/14/3c/143cbd3c44bce08e9a5a6b880557c4b7538213df.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/formalise.2013.6612272"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

2009 CAV award announcement

Randal E. Bryant, Orna Grumberg, Joseph Sifakis, Moshe Y. Vardi
<span title="2010-05-26">2010</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fyhr5xuigbfmdabfoqp2f7lbum" style="color: black;">Formal methods in system design</a> </i> &nbsp;
This annual award recognizes a specific fundamental contribution or series of outstanding contributions to the CAV field.  ...  The 2009 CAV (Computer-Aided Verification) award was presented to seven individuals who made major advances in creating high-performance Boolean satisfiability solvers.  ...  For example, the rules for a 9 × 9 Sudoku puzzle can be expressed as a number of constraints: that each square is assigned a number between 1 and 9, that no row or column contain any repeated digits, that  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10703-010-0094-9">doi:10.1007/s10703-010-0094-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pdvhglgtnvg4ji3cyljeed3rla">fatcat:pdvhglgtnvg4ji3cyljeed3rla</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20110708182600/http://www.csi.ucd.ie/staff/jpms/docs/cav-award-fmsd10.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/21/6d/216d25b8b7fe88b1b333df03b8977f99fa72bab1.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10703-010-0094-9"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Deductive search for logic puzzles

Cameron Browne
<span title="">2013</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/rhd7ssklwncwpd7fotjzyiag4e" style="color: black;">2013 IEEE Conference on Computational Inteligence in Games (CIG)</a> </i> &nbsp;
Further, it provides an estimate of the deducibility of a given problem for human solvers and offers new ways of understanding deduction puzzles.  ...  Deductive search (DS) is a breadth-first, depthlimited propagation scheme for the constraint-based solution of deduction puzzles, using simple logic operations found in standard constraint satisfaction  ...  INTRODUCTION Sudoku is currently rivalling the crossword as the world's most popular "pencil and paper puzzle", largely because it is language-neutral and can be solved using pure logic without the need  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/cig.2013.6633649">doi:10.1109/cig.2013.6633649</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/cig/Browne13a.html">dblp:conf/cig/Browne13a</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fth3usfmvjbgflgguevnv6doci">fatcat:fth3usfmvjbgflgguevnv6doci</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808055811/http://eldar.mathstat.uoguelph.ca/dashlock/CIG2013/papers/paper_74.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b1/49/b1495eea4c7e0e96894bd954d91dc3bdc01d867a.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/cig.2013.6633649"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Savile Row Manual [article]

Peter Nightingale
<span title="2021-11-12">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Savile Row translates a solver-independent constraint modelling language to the input languages for various solvers including constraint, SAT, and SMT solvers.  ...  Then we describe the functions of the tool, its main features and options and how to install and use it.  ...  Figure 1 : On the left is a Sudoku puzzle.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2201.03472v1">arXiv:2201.03472v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lztbtkbiljfbdkssn34t565afi">fatcat:lztbtkbiljfbdkssn34t565afi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220112011603/https://arxiv.org/pdf/2201.03472v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/22/18/22184e7f1a3807d82ec5d3b68c6e2e24157147ca.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2201.03472v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Satune: synthesizing efficient SAT encoders

Hamed Gorjiara, Guoqing Harry Xu, Brian Demsky
<span title="2020-11-13">2020</span> <i title="Association for Computing Machinery (ACM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/dqy7qc7jkzal5bz3gueys3siz4" style="color: black;">Proceedings of the ACM on Programming Languages (PACMPL)</a> </i> &nbsp;
Modern SAT solvers are extremely efficient at solving boolean satisfiability problems, enabling a wide spectrum of techniques for checking, verifying, and validating real-world programs.  ...  results that are orders-of-magnitude apart, regardless of the underlying solvers used.  ...  ACKNOWLEDGMENTS We thank the anonymous reviewers for their thorough and insightful comments that helped us substantially improve the paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3428214">doi:10.1145/3428214</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fye4birrnzehfo4hbcrrhukzki">fatcat:fye4birrnzehfo4hbcrrhukzki</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220101175816/https://dl.acm.org/doi/pdf/10.1145/3428214" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/33/26/3326432a84944fc640772d3430fb9b0db4542450.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3428214"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>
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