{"type":"ScholarlyArticle","title":"Certified CNF Translations for Pseudo-Boolean Solving (Extended Abstract).","authors":[{"name":"Stephan Gocht","slug":"stephan-gocht","given":"Stephan","family":"Gocht","affiliation":"Lund University; University of Copenhagen"},{"name":"Ruben Martins","slug":"ruben-martins","given":"Ruben","family":"Martins","affiliation":"Carnegie Mellon University"},{"name":"Jakob Nordström","slug":"jakob-nordstrom","given":"Jakob","family":"Nordström","affiliation":"University of Copenhagen; Lund University"},{"name":"Andy Oertel","slug":"andy-oertel","given":"Andy","family":"Oertel","affiliation":"Lund University; University of Copenhagen"}],"abstract":"The dramatic improvements in Boolean satisfiability (SAT) solving since the turn of the millennium have made it possible to leverage conflict-driven clause learning (CDCL) solvers for many combinatorial problems in academia and industry, and the use of proof logging has played a crucial role in increasing the confidence that the results these solvers produce are correct. However, the fact that SAT proof logging is performed in conjunctive normal form (CNF) clausal format means that it has not been possible to extend guarantees of correctness to the use of SAT solvers for more expressive combinatorial paradigms, where the first step is an unverified translation of the input to CNF.\r\n\r\nIn this work, we show how cutting-planes-based reasoning can provide proof logging for solvers that translate pseudo-Boolean (a.k.a. 0-1 integer linear) decision problems to CNF and then run CDCL. We are hopeful that this is just a first step towards providing a unified proof logging approach that will extend to maximum satisfiability (MaxSAT) solving and pseudo-Boolean optimization in general.","doi":"10.24963/ijcai.2023/716","url":"https://ijcai.org/proceedings/2023/716","pdf":"https://ijcai.org/proceedings/2023/0716.pdf","markdown":"https://ijcai.org/proceedings/2023/716.md","bibtex":"https://ijcai.org/proceedings/2023/bibtex/716","volume":{"slug":"2023","title":"Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2023","url":"https://ijcai.org/proceedings/2023/","year":2023,"location":"Macao, SAR","dates":"19-25 August 2023","editors":"Edith Elkind","isbn":"978-1-956792-03-4","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2023"},"section":"Sister Conferences Best Papers","track":"Sister Conferences Best Papers","number":716,"pages":{"first":6436,"last":6441},"keywords":["Sister Conferences Best Papers: Constraint Satisfaction and Optimization"],"videos":[],"references":[{"position":1,"raw":"[Akgün et al., 2018] Özgür Akgün, Ian P. Gent, Christopher Jefferson, Ian Miguel, and Peter Nightingale. Metamor- phic testing of constraint solvers. In Proc. 24th Inter- national Conference on Principles and Practice of Con- straint Programming (CP '18), pages 727-736, 2018.","title":"Metamorphic testing of constraint solvers","authors":"Ian P Özgür Akgün, Christopher Gent, Ian Jefferson, Peter Miguel, Nightingale","year":"2018","venue":"Proc. 24th International Conference on Principles and Practice of Constraint Programming (CP '18)","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Baek et al., 2021] Seulkee Baek, Mario Carneiro, and Mar- ijn J. H. Heule. A flexible proof format for SAT solver- elaborator communication. In Proc. 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS '21), pages 59-75, 2021.","title":"A flexible proof format for SAT solverelaborator communication","authors":"Baek","year":"2021","venue":"Proc. 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS '21)","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Bailleux and Boufkhad, 2003] Olivier Bailleux and Yacine Boufkhad. Efficient CNF encoding of Boolean cardinality constraints. In Proc. 9th International Conference on Prin- ciples and Practice of Constraint Programming (CP '03), pages 108-122, 2003.","title":"Efficient CNF encoding of Boolean cardinality constraints","authors":"Boufkhad Bailleux, Olivier Bailleux, Yacine Boufkhad","year":"2003","venue":"Proc. 9th International Conference on Principles and Practice of Constraint Programming (CP '03)","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Barnett and Biere, 2021] Lee A. Barnett and Armin Biere. Non-clausal redundancy properties. In Proc. 28th Interna- tional Conference on Automated Deduction (CADE-28), pages 252-272, 2021.","title":"Non-clausal redundancy properties","authors":"Lee A Barnett, Armin Biere","year":"2021","venue":"Proc. 28th International Conference on Automated Deduction (CADE-28)","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Barth, 1995] Peter Barth. A Davis-Putnam based enu- meration algorithm for linear pseudo-Boolean optimiza- tion. Technical Report MPI-I-95-2-003, Max-Planck- Institut für Informatik, 1995.","title":"A Davis-Putnam based enumeration algorithm for linear pseudo-Boolean optimization","authors":"Peter Barth","year":"1995","venue":null,"doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Berg et al., 2020] Jeremias Berg, Fahiem Bacchus, and Alex Poole. Abstract cores in implicit hitting set MaxSat solving. In Proc. 23rd International Conference on Theory and Applications of Satisfiability Testing (SAT '20), pages 277-294, 2020.","title":"Abstract cores in implicit hitting set MaxSat solving","authors":"Berg","year":"2020","venue":"Proc. 23rd International Conference on Theory and Applications of Satisfiability Testing (SAT '20)","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Berg et al., 2023] Jeremias Berg, Bart Bogaerts, Jakob Nordström, Andy Oertel, and Dieter Vandesande. Certi- fied core-guided MaxSAT solving. In Proc. 29th Interna- tional Conference on Automated Deduction (CADE-29), 2023. To appear.","title":"Certified core-guided MaxSAT solving","authors":"Jeremias Berg, Bart Bogaerts, Jakob Nordström, Andy Oertel, Dieter Vandesande","year":"2023","venue":"Proc. 29th International Conference on Automated Deduction (CADE-29)","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Biere et al., 2021] Armin Biere, Marijn J. H. Heule, Hans van Maaren, and Toby Walsh, editors. Handbook of Satis- fiability, volume 336 of Frontiers in Artificial Intelligence and Applications. IOS Press, 2nd edition, 2021.","title":"Handbook of Satisfiability","authors":null,"year":"2021","venue":"Frontiers in Artificial Intelligence and Applications","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Bogaerts et al., 2022] Bart Bogaerts, Stephan Gocht, Cia- ran McCreesh, and Jakob Nordström. Certified symmetry and dominance breaking for combinatorial optimisation. In Proc. 36th AAAI Conference on Artificial Intelligence (AAAI '22), pages 3698-3707, 2022.","title":"Certified symmetry and dominance breaking for combinatorial optimisation","authors":"Bogaerts","year":"2022","venue":"Proc. 36th AAAI Conference on Artificial Intelligence (AAAI '22)","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Brummayer et al., 2010] Robert Brummayer, Florian Lons- ing, and Armin Biere. Automated testing and debugging of SAT and QBF solvers. In Proc. 13th International Confer- ence on Theory and Applications of Satisfiability Testing (SAT '10), pages 44-57, 2010.","title":"Automated testing and debugging of SAT and QBF solvers","authors":"Robert Brummayer, Florian Lonsing, Armin Biere","year":"2010","venue":"Proc. 13th International Conference on Theory and Applications of Satisfiability Testing (SAT '10)","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Bryant et al., 2022] Randal E. Bryant, Armin Biere, and Marijn J. H. Heule. Clausal proofs for pseudo-Boolean reasoning. In Proc. 28th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS '22), pages 443-461, 2022.","title":"Clausal proofs for pseudo-Boolean reasoning","authors":"Randal E Bryant, Armin Biere, J H Marijn, Heule","year":"2022","venue":"Proc. 28th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS '22)","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Buss and Nordström, 2021] Samuel R. Buss and Jakob Nordström. Proof complexity and SAT solving. In Biere et al. [2021], chapter 7, pages 233-350.","title":"Proof complexity and SAT solving","authors":"R Samuel, Jakob Buss, Nordström","year":"2021","venue":null,"doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Cheung et al., 2017] Kevin K. H. Cheung, Ambros M. Gleixner, and Daniel E. Steffy. Verifying integer pro- gramming results. In Proc. 19th International Conference on Integer Programming and Combinatorial Optimization (IPCO '17), pages 148-160, 2017.","title":"Verifying integer programming results","authors":"K H Kevin, Ambros M Cheung, Daniel E Gleixner, Steffy","year":"2017","venue":"Proc. 19th International Conference on Integer Programming and Combinatorial Optimization (IPCO '17)","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Cook et al., 1987] William Cook, Collette Rene Coullard, and György Turán. On the complexity of cutting-plane proofs. Discrete Applied Mathematics, 18(1):25-38, 1987.","title":"On the complexity of cutting-plane proofs","authors":"William Cook, Collette, Rene Coullard, György Turán","year":"1987","venue":"Discrete Applied Mathematics","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Cook et al., 2013] William Cook, Thorsten Koch, Daniel E. Steffy, and Kati Wolter. A hybrid branch-and-bound approach for exact rational mixed-integer programming. Mathematical Programming Computation, 5(3):305-344, 2013.","title":"A hybrid branch-and-bound approach for exact rational mixed-integer programming","authors":"William Cook, Thorsten Koch, Daniel E Steffy, Kati Wolter","year":"2013","venue":"Mathematical Programming Computation","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Davies and Bacchus, 2011] Jessica Davies and Fahiem Bac- chus. Solving MAXSAT by solving a sequence of sim- pler SAT instances. In Proc. 17th International Confer- ence on Principles and Practice of Constraint Program- ming (CP '11), pages 225-239, 2011.","title":"Jessica Davies and Fahiem Bacchus. Solving MAXSAT by solving a sequence of simpler SAT instances","authors":"Bacchus Davies","year":"2011","venue":"Proc. 17th International Conference on Principles and Practice of Constraint Programming (CP '11)","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Devriendt et al., 2021a] Jo Devriendt, Ambros Gleixner, and Jakob Nordström. Learn to relax: Integrating 0-1 in- teger linear programming with pseudo-Boolean conflict- driven search. Constraints, 26(1-4):26-55, 2021.","title":"Learn to relax: Integrating 0-1 integer linear programming with pseudo-Boolean conflictdriven search","authors":"Devriendt","year":"2021","venue":"Constraints","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Devriendt et al., 2021b] Jo Devriendt, Stephan Gocht, Emir Demirović, Jakob Nordström, and Peter Stuckey. Cut- ting to the core of pseudo-Boolean optimization: Com- bining core-guided search with cutting planes reasoning. In Proc. 35th AAAI Conference on Artificial Intelligence (AAAI '21), pages 3750-3758, 2021. [Eén and Sörensson, 2006] Niklas Eén and Niklas Sörensson. Translating pseudo-Boolean constraints into SAT. Journal on Satisfiability, Boolean Modeling and Computation, 2(1-4):1-26, 2006.","title":"Cutting to the core of pseudo-Boolean optimization: Combining core-guided search with cutting planes reasoning","authors":"Devriendt","year":"2006","venue":"Proc. 35th AAAI Conference on Artificial Intelligence (AAAI '21)","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Eifler and Gleixner, 2021] Leon Eifler and Ambros Gleixner. A computational status update for exact rational mixed integer programming. In Proc. 22nd International Conference on Integer Programming and Combinatorial Optimization (IPCO '21), pages 163-177, 2021.","title":"A computational status update for exact rational mixed integer programming","authors":"Gleixner Eifler, Leon Eifler, Ambros Gleixner","year":"2021","venue":"Proc. 22nd International Conference on Integer Programming and Combinatorial Optimization (IPCO '21)","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Elffers and Nordström, 2018] Jan Elffers and Jakob Nord- ström. Divide and conquer: Towards faster pseudo- Boolean solving. In Proc. 27th International Joint Conference on Artificial Intelligence (IJCAI '18), pages 1291-1299, 2018.","title":"Jan Elffers and Jakob Nordström. Divide and conquer: Towards faster pseudo-Boolean solving","authors":"Nordström Elffers","year":"2018","venue":"Proc. 27th International Joint Conference on Artificial Intelligence (IJCAI '18)","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Elffers et al., 2020] Jan Elffers, Stephan Gocht, Ciaran Mc- Creesh, and Jakob Nordström. Justifying all differences using pseudo-Boolean reasoning. In Proc. 34th AAAI Conference on Artificial Intelligence (AAAI '20), pages 1486-1494, 2020.","title":"Justifying all differences using pseudo-Boolean reasoning","authors":"Elffers","year":"2020","venue":"Proc. 34th AAAI Conference on Artificial Intelligence (AAAI '20)","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Fu and Malik, 2006] Zhaohui Fu and Sharad Malik. On solving the partial MAX-SAT problem. In Proc. 9th Inter- national Conference on Theory and Applications of Satis- fiability Testing (SAT '06), pages 252-265, 2006.","title":"On solving the partial MAX-SAT problem","authors":"Zhaohui Fu, Sharad Malik","year":"2006","venue":"Proc. 9th International Conference on Theory and Applications of Satisfiability Testing (SAT '06)","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Gillard et al., 2019] Xavier Gillard, Pierre Schaus, and Yves Deville. SolverCheck: Declarative testing of con- straints. In Proc. 25th International Conference on Prin- ciples and Practice of Constraint Programming (CP '19), pages 565-582, 2019.","title":"SolverCheck: Declarative testing of constraints","authors":"Xavier Gillard, Pierre Schaus, Yves Deville","year":"2019","venue":"Proc. 25th International Conference on Principles and Practice of Constraint Programming (CP '19)","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Gocht and Nordström, 2021] Stephan Gocht and Jakob Nordström. Certifying parity reasoning efficiently using pseudo-Boolean proofs. In Proc. 35th AAAI Conference on Artificial Intelligence (AAAI '21), pages 3768-3777, 2021.","title":"Certifying parity reasoning efficiently using pseudo-Boolean proofs","authors":"Nordström Gocht, Stephan Gocht, Jakob Nordström","year":"2021","venue":"Proc. 35th AAAI Conference on Artificial Intelligence (AAAI '21)","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Gocht et al., 2020] Stephan Gocht, Ciaran McCreesh, and Jakob Nordström. Subgraph isomorphism meets cutting planes: Solving with certified solutions. In Proc. 29th In- ternational Joint Conference on Artificial Intelligence (IJ- CAI '20), pages 1134-1140, 2020.","title":"Subgraph isomorphism meets cutting planes: Solving with certified solutions","authors":"Stephan Gocht, Ciaran Mccreesh, Jakob Nordström","year":"2020","venue":"Proc. 29th International Joint Conference on Artificial Intelligence (IJ-CAI '20)","doi":null,"url":null,"internal_url":"/proceedings/2020/158"},{"position":26,"raw":"[Gocht et al., 2022] Stephan Gocht, Ruben Martins, Jakob Nordström, and Andy Oertel. Certified CNF translations for pseudo-Boolean solving. In Proc. 25th International Conference on Theory and Applications of Satisfiability Testing (SAT '22), pages 16:1-16:25, 2022.","title":"Certified CNF translations for pseudo-Boolean solving","authors":"Stephan Gocht, Ruben Martins, Jakob Nordström, Andy Oertel","year":"2022","venue":"Proc. 25th International Conference on Theory and Applications of Satisfiability Testing (SAT '22)","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Goldberg and Novikov, 2003] Evgueni Goldberg and Yakov Novikov. Verification of proofs of unsatisfiability for CNF formulas. In Proc. Conference on Design, Au- tomation and Test in Europe (DATE '03), pages 886-891, 2003.","title":"Verification of proofs of unsatisfiability for CNF formulas","authors":"Evgueni Goldberg, Yakov Novikov","year":"2003","venue":"Proc. Conference on Design, Automation and Test in Europe (DATE '03)","doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Joshi et al., 2015] Saurabh Joshi, Ruben Martins, and Vasco M. Manquinho. Generalized totalizer encoding for pseudo-Boolean constraints. In Proc. 21st International Conference on Principles and Practice of Constraint Pro- gramming (CP '15), pages 200-209, 2015.","title":"Generalized totalizer encoding for pseudo-Boolean constraints","authors":"Saurabh Joshi, Ruben Martins, Vasco M Manquinho","year":"2015","venue":"Proc. 21st International Conference on Principles and Practice of Constraint Programming (CP '15)","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Kaufmann and Biere, 2021] Daniela Kaufmann and Armin Biere. AMulet 2.0 for verifying multiplier circuits. In Proc. 27th International Conference on Tools and Al- gorithms for the Construction and Analysis of Systems (TACAS '21), pages 357-364, 2021.","title":"AMulet 2.0 for verifying multiplier circuits","authors":"Daniela Kaufmann, Armin Biere","year":"2021","venue":"Proc. 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS '21)","doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Kaufmann et al., 2022] Daniela Kaufmann, Paul Beame, Armin Biere, and Jakob Nordström. Adding dual variables to algebraic reasoning for circuit verification. In Proc. 25th Design, Automation and Test in Europe Conference (DATE '22), pages 1435-1440, 2022.","title":"Adding dual variables to algebraic reasoning for circuit verification","authors":"Daniela Kaufmann, Paul Beame, Armin Biere, Jakob Nordström","year":"2022","venue":"Proc. 25th Design, Automation and Test in Europe Conference (DATE '22)","doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Le Berre and Parrain, 2010] Daniel Le Berre and Anne Par- rain. The Sat4j library, release 2.2. Journal on Satisfiabil- ity, Boolean Modeling and Computation, 7:59-64, 2010.","title":"Daniel Le Berre and Anne Parrain. The Sat4j library, release 2.2","authors":"Le Berre, Parrain","year":"2010","venue":"Journal on Satisfiability, Boolean Modeling and Computation","doi":null,"url":null,"internal_url":null},{"position":32,"raw":"[Marques-Silva and Sakallah, 1999] João P. Marques-Silva and Karem A. Sakallah. GRASP: A search algorithm for propositional satisfiability. IEEE Transactions on Comput- ers, 48(5):506-521, 1999.","title":"GRASP: A search algorithm for propositional satisfiability","authors":"P João, Karem A Marques-Silva, Sakallah","year":"1999","venue":"IEEE Transactions on Computers","doi":null,"url":null,"internal_url":null},{"position":33,"raw":"[McConnell et al., 2011] Ross M. McConnell, Kurt Mehlhorn, Stefan Näher, and Pascal Schweitzer. Certify- ing algorithms. Computer Science Review, 5(2):119-161, 2011.","title":"Certifying algorithms","authors":"Ross M Mcconnell, Kurt Mehlhorn, Stefan Näher, Pascal Schweitzer","year":"2011","venue":"Computer Science Review","doi":null,"url":null,"internal_url":null},{"position":34,"raw":"[Moskewicz et al., 2001] Matthew W. Moskewicz, Conor F. Madigan, Ying Zhao, Lintao Zhang, and Sharad Malik. Chaff: Engineering an efficient SAT solver. In Proc. 38th Design Automation Conference (DAC '01), pages 530-535, 2001.","title":"Chaff: Engineering an efficient SAT solver","authors":"Matthew W Moskewicz, F Conor, Ying Madigan, Lintao Zhao, Sharad Zhang, Malik","year":"2001","venue":"Proc. 38th Design Automation Conference (DAC '01)","doi":null,"url":null,"internal_url":null},{"position":35,"raw":"[Philipp and Steinke, 2015] Tobias Philipp and Peter Steinke. PBLib -a library for encoding pseudo-Boolean constraints into CNF. In Proc. 18th International Confer- ence on Theory and Applications of Satisfiability Testing (SAT '15), pages 9-16, 2015.","title":"PBLib -a library for encoding pseudo-Boolean constraints into CNF","authors":"Steinke Philipp, Tobias Philipp, Peter Steinke","year":"2015","venue":"Proc. 18th International Conference on Theory and Applications of Satisfiability Testing (SAT '15)","doi":null,"url":null,"internal_url":null},{"position":36,"raw":"[Roussel and Manquinho, 2016] Olivier Roussel and Vasco M. Manquinho. Input/output format and solver requirements for the competitions of pseudo- Boolean solvers. Revision 2324. Available at http://www.cril.univ-artois.fr/PB16/format.pdf, 2016.","title":"Input/output format and solver requirements for the competitions of pseudo-Boolean solvers","authors":"Olivier Roussel, M Vasco, Manquinho","year":"2016","venue":"Revision 2324","doi":null,"url":"http://www.cril.univ-artois.fr/PB16/format.pdf","internal_url":null},{"position":37,"raw":"[Sinz, 2005] Carsten Sinz. Towards an optimal CNF encod- ing of Boolean cardinality constraints. In Proc. 11th Inter- national Conference on Principles and Practice of Con- straint Programming (CP '05), pages 827-831, 2005.","title":"Towards an optimal CNF encoding of Boolean cardinality constraints","authors":"Carsten Sinz, Sinz","year":"2005","venue":"Proc. 11th International Conference on Principles and Practice of Constraint Programming (CP '05)","doi":null,"url":null,"internal_url":null},{"position":38,"raw":"[Smirnov et al., 2021] Pavel Smirnov, Jeremias Berg, and Matti Järvisalo. Pseudo-Boolean optimization by im- plicit hitting sets. In Proc. 27th International Conference on Principles and Practice of Constraint Programming (CP '21), pages 51:1-51:20, 2021.","title":"Pseudo-Boolean optimization by implicit hitting sets","authors":"Pavel Smirnov, Jeremias Berg, Matti Järvisalo","year":"2021","venue":"Proc. 27th International Conference on Principles and Practice of Constraint Programming (CP '21)","doi":null,"url":null,"internal_url":null},{"position":39,"raw":"[Smirnov et al., 2022] Pavel Smirnov, Jeremias Berg, and Matti Järvisalo. Improvements to the implicit hitting set approach to pseudo-Boolean optimization. In Proc. 25th International Conference on Theory and Applications of Satisfiability Testing (SAT '22), pages 13:1-13:18, 2022.","title":"Improvements to the implicit hitting set approach to pseudo-Boolean optimization","authors":"Smirnov","year":"2022","venue":"Proc. 25th International Conference on Theory and Applications of Satisfiability Testing (SAT '22)","doi":null,"url":null,"internal_url":null},{"position":40,"raw":"[Van Gelder, 2008] Allen Van Gelder. Verifying RUP proofs of propositional unsatisfiability. In 10th International Symposium on Artificial Intelligence and Mathematics (ISAIM '08), 2008.","title":"Verifying RUP proofs of propositional unsatisfiability","authors":"Allen Van Gelder","year":"2008","venue":"10th International Symposium on Artificial Intelligence and Mathematics (ISAIM '08)","doi":null,"url":null,"internal_url":null},{"position":41,"raw":"[Vandesande et al., 2022] Dieter Vandesande, Wolf De Wulf, and Bart Bogaerts. QMaxSATpb: A certified MaxSAT solver. In Proc. 16th International Conference on Logic Programming and Non-monotonic Reasoning (LP- NMR '22), pages 429-442, 2022.","title":"QMaxSATpb: A certified MaxSAT solver","authors":"Dieter Vandesande, Wolf De Wulf, Bart Bogaerts","year":"2022","venue":"Proc. 16th International Conference on Logic Programming and Non-monotonic Reasoning (LP-NMR '22)","doi":null,"url":null,"internal_url":null},{"position":42,"raw":"[Warners, 1998] Joost P. Warners. A linear-time transforma- tion of linear inequalities into conjunctive normal form. Information Processing Letters, 68(2):63-69, 1998.","title":"A linear-time transformation of linear inequalities into conjunctive normal form","authors":"P Joost, Warners","year":"1998","venue":"Information Processing Letters","doi":null,"url":null,"internal_url":null},{"position":43,"raw":"[Wetzler et al., 2014] Nathan Wetzler, Marijn J. H. Heule, and Warren A. Hunt Jr. DRAT-trim: Efficient checking and trimming using expressive clausal proofs. In Proc. 17th International Conference on Theory and Applications of Satisfiability Testing (SAT '14), pages 422-429, 2014.","title":"DRAT-trim: Efficient checking and trimming using expressive clausal proofs","authors":"Nathan Wetzler, J H Marijn, Warren A Heule, Hunt","year":"2014","venue":"Proc. 17th International Conference on Theory and Applications of Satisfiability Testing (SAT '14)","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}