{"type":"ScholarlyArticle","title":"Co-Certificate Learning with SAT Modulo Symmetries","authors":[{"name":"Markus Kirchweger","slug":"markus-kirchweger","given":"Markus","family":"Kirchweger","affiliation":"Algorithms and Complexity Group , TU Wien , Austria"},{"name":"Tomáš Peitl","slug":"tomas-peitl","given":"Tomáš","family":"Peitl","affiliation":"Algorithms and Complexity Group , TU Wien , Austria"},{"name":"Stefan Szeider","slug":"stefan-szeider","given":"Stefan","family":"Szeider","affiliation":"Algorithms and Complexity Group , TU Wien , Austria"}],"abstract":"We present a new SAT-based method for generating all graphs up to isomorphism that satisfy a given co-NP property. Our method extends the SAT Modulo Symmetry (SMS) framework with a technique that we call co-certificate learning. If SMS generates a candidate graph that violates the given  co-NP property,\r\nwe obtain a certificate for this violation, i.e., `co-certificate' for the co-NP property. The co-certificate gives rise to a clause that the SAT solver, serving as SMS's backend, learns as part of its CDCL procedure. We demonstrate that SMS plus co-certificate learning is a powerful method that allows us to improve the best-known lower bound on the size of Kochen-Specker vector systems, a problem that is central to the foundations of quantum mechanics and has been studied for over half a century. Our approach is orders of magnitude faster and scales significantly better than a recently proposed SAT-based method.","doi":"10.24963/ijcai.2023/216","url":"https://ijcai.org/proceedings/2023/216","pdf":"https://ijcai.org/proceedings/2023/0216.pdf","markdown":"https://ijcai.org/proceedings/2023/216.md","bibtex":"https://ijcai.org/proceedings/2023/bibtex/216","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":"Constraint Satisfaction and Optimization","track":"Main Track","number":216,"pages":{"first":1944,"last":1953},"keywords":["Constraint Satisfaction and Optimization: CSO: Satisfiabilty","Constraint Satisfaction and Optimization: CSO: Applications","Constraint Satisfaction and Optimization: CSO: Solvers and tools"],"videos":[],"references":[{"position":1,"raw":"[Arends et al., 2011] Felix Arends, Joël Ouaknine, and Charles W. Wampler. On searching for small Kochen- Specker vector systems. In Petr Kolman and Jan Kra- tochvíl, editors, Graph-Theoretic Concepts in Computer Science -37th International Workshop, WG 2011, Teplá Monastery, Czech Republic, June 21-24, 2011. Revised Papers, volume 6986 of Lecture Notes in Computer Sci- ence, pages 23-34. Springer Verlag, 2011. Based on Felix Arends' MSc thesis \"A Lower Bound on the Size of the Smallest Kochen-Specker Vector System in Three Dimen- sions,\" University of Oxford, 2011.","title":"On searching for small Kochen-Specker vector systems","authors":"Felix Arends, Joël Ouaknine, Charles W Wampler","year":"2011","venue":"Based on Felix Arends' MSc thesis \"A Lower Bound on the Size of the Smallest Kochen-Specker Vector System in Three Dimensions","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Barrett et al., 2021] Clark Barrett, Roberto Sebastiani, San- jit A. Seshia, and Cesare Tinelli. Satisfiability modulo the- ories. In A. Biere, M. J. H. Heule, H. van Maaren, and T. Walsh, editors, Handbook of Satisfiability, 2nd Edition, volume 336 of Frontiers in Artificial Intelligence and Ap- plications, chapter 33, pages 1267-1329. IOS Press, 2021.","title":"Satisfiability modulo theories","authors":"Clark Barrett, Roberto Sebastiani, A Sanjit, Cesare Seshia, Tinelli","year":"2021","venue":"Frontiers in Artificial Intelligence and Applications","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Budroni et al., 2022] Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, and Jan øAke Lars- son. Kochen-Specker contextuality. Rev. Mod. Phys., 94:045007, 2022.","title":"Kochen-Specker contextuality","authors":"Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, Jan Øake Larsson","year":"2022","venue":"Rev. Mod. Phys","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Chlebíková, 2002] Janka Chlebíková. The structure of ob- structions to treewidth and pathwidth. Discr. Appl. Math., 120(1-3):61-71, 2002.","title":"The structure of obstructions to treewidth and pathwidth","authors":"Janka Chlebíková","year":"2002","venue":"Discr. Appl. Math","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Codish et al., 2016] Michael Codish, Graeme Gange, Avra- ham Itzhakov, and Peter J. Stuckey. Breaking symmetries in graphs: The nauty way. In Principles and Practice of Constraint Programming -22nd International Conference, CP 2016, Toulouse, France, September 5-9, 2016, Pro- ceedings, volume 9892 of Lecture Notes in Computer Sci- ence, pages 157-172. Springer Verlag, 2016.","title":"Breaking symmetries in graphs: The nauty way","authors":"Michael Codish, Graeme Gange, Avraham Itzhakov, Peter J Stuckey","year":"2016","venue":"Principles and Practice of Constraint Programming -22nd International Conference","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Codish et al., 2019] Michael Codish, Alice Miller, Patrick Prosser, and Peter J. Stuckey. Constraints for symmetry breaking in graph representation. Constraints, 24(1):1-24, 2019.","title":"Constraints for symmetry breaking in graph representation","authors":"Michael Codish, Alice Miller, Patrick Prosser, Peter J Stuckey","year":"2019","venue":"Constraints","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[de Moura and Bjørner, 2008] Leonardo Mendonc ¸a de Moura and Nikolaj S. Bjørner. Z3: an efficient SMT solver. In Tools and Algorithms for the Construction and Analysis of Systems, 14th International Conference, TACAS 2008, Held as Part of the Joint European Confer- ences on Theory and Practice of Software, ETAPS 2008, Budapest, Hungary, March 29-April 6, 2008. Proceedings, volume 4963 of Lecture Notes in Computer Science, pages 337-340. Springer Verlag, 2008.","title":"Leonardo Mendonc ¸a de Moura and Nikolaj S. Bjørner. Z3: an efficient SMT solver","authors":"De Moura, Bjørner","year":"2008","venue":"Tools and Algorithms for the Construction and Analysis of Systems, 14th International Conference, TACAS 2008, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2008","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Dolzmann and Sturm, 1997] Andreas Dolzmann and Thomas Sturm. REDLOG: computer algebra meets computer logic. SIGSAM Bull., 31(2):2-9, 1997.","title":"REDLOG: computer algebra meets computer logic","authors":"Sturm Dolzmann, Andreas Dolzmann, Thomas Sturm","year":"1997","venue":"SIGSAM Bull","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Erdős, 1967] P. Erdős. Some remarks on chromatic graphs. Colloq. Math., 16:253-256, 1967.","title":"Some remarks on chromatic graphs","authors":"P Erdős","year":"1967","venue":"Colloq. Math","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Fazekas et al., 2023] Katalin Fazekas, Aina Niemetz, Math- ias Preiner, Markus Kirchweger, Stefan Szeider, and Armin Biere. IPASIR-UP: User propagators for CDCL. In Meena Mahajan and Friedrich Slivovsky, editors, The 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023), July 04-08, 2023, Al- ghero, Italy, LIPIcs. Schloss Dagstuhl -Leibniz-Zentrum für Informatik, 2023. To appear. Fichte et al., 2023]","title":"IPASIR-UP: User propagators for CDCL","authors":"Fazekas","year":"2023","venue":"The 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"Johannes K. Fichte, Markus Hecher, Daniel Le Berre, and Stefan Szeider. The silent (r)evolution of SAT. Communications of the ACM, 66(6):64-72, June 2023.","title":"The silent (r)evolution of SAT","authors":"Johannes K Fichte, Markus Hecher, Daniel Le Berre, Stefan Szeider","year":"2023","venue":"Communications of the ACM","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Goedgebeur, 2020] Jan Goedgebeur. On minimal triangle- free 6-chromatic graphs. J. Graph Theory, 93(1):34-48, 2020.","title":"Jan Goedgebeur. On minimal trianglefree 6-chromatic graphs","authors":"Goedgebeur","year":"2020","venue":"J. Graph Theory","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Heule et al., 2018] Marijn J. H. Heule, Oliver Kullmann, and Armin Biere. Cube-and-conquer for satisfiability. In Youssef Hamadi and Lakhdar Sais, editors, Handbook of Parallel Constraint Reasoning, pages 31-59. Springer, 2018.","title":"Cube-and-conquer for satisfiability","authors":"J H Marijn, Heule, Armin Oliver Kullmann, Biere","year":"2018","venue":"Handbook of Parallel Constraint Reasoning","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Itzhakov and Codish, 2015] Avraham Itzhakov and Michael Codish. Breaking symmetries in graph search with canon- izing sets. CoRR, abs/1511.08205, 2015.","title":"Breaking symmetries in graph search with canonizing sets","authors":"Codish Itzhakov, Avraham Itzhakov, Michael Codish","year":"2015","venue":null,"doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Janota and Marques-Silva, 2011] Mikolás Janota and João P. Marques-Silva. Abstraction-based algorithm for 2QBF. In Karem A. Sakallah and Laurent Simon, editors, Theory and Applications of Satisfiability Testing -SAT 2011, volume 6695 of Lecture Notes in Computer Science, pages 230-244. Springer Verlag, 2011.","title":"Abstraction-based algorithm for 2QBF","authors":"Marques-Silva ; Mikolás Janota, João P Janota, Marques-Silva","year":"2011","venue":"Theory and Applications of Satisfiability Testing -SAT 2011","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Janota et al., 2012] Mikolás Janota, William Klieber, João Marques-Silva, and Edmund M. Clarke. Solving QBF with counterexample guided refinement. In Theory and Applications of Satisfiability Testing -SAT 2012, volume 7317 of Lecture Notes in Computer Science, pages 114- 128. Springer Verlag, 2012.","title":"Solving QBF with counterexample guided refinement","authors":"Janota","year":"2012","venue":"Theory and Applications of Satisfiability Testing -SAT 2012","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Jensen and Toft, 1995] Tommy R. Jensen and Bjarne Toft. Graph coloring problems. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Ltd, 1995.","title":"Graph coloring problems","authors":"Tommy R Jensen, Bjarne Toft","year":"1995","venue":"Wiley-Interscience Series in Discrete Mathematics and Optimization","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Karp, 1972] Richard M. Karp. Reducibility among combi- natorial problems. In Complexity of computer computa- tions (Proc. Sympos., IBM Thomas J. Watson Res. Cen- ter, Yorktown Heights, N.Y., 1972), pages 85-103. Plenum, New York, 1972.","title":"Reducibility among combinatorial problems","authors":"Richard M Karp","year":"1972","venue":"Complexity of computer computations (Proc. Sympos","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Kirchweger and Szeider, 2021] Markus Kirchweger and Stefan Szeider. SAT modulo symmetries for graph gener- ation. In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021), LIPIcs, page 39:1-39:17. Dagstuhl, 2021.","title":"SAT modulo symmetries for graph generation","authors":"Szeider Kirchweger, Markus Kirchweger, Stefan Szeider","year":"2021","venue":"27th International Conference on Principles and Practice of Constraint Programming (CP 2021)","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Kirchweger et al., 2022] Markus Kirchweger, Manfred Scheucher, and Stefan Szeider. A SAT attack on Rota's Basis Conjecture. In Theory and Applications of Satisfia- bility Testing -SAT 2022 -25th International Conference, Haifa, Israel, August 2-5, 2022, Proceedings, 2022.","title":"A SAT attack on Rota's Basis Conjecture","authors":"Markus Kirchweger, Manfred Scheucher, Stefan Szeider","year":"2022","venue":"Theory and Applications of Satisfiability Testing -SAT 2022 -25th International Conference","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Kirchweger et al., 2023a] Markus Kirchweger, Tomáš Peitl, and Stefan Szeider. A SAT solver's opinion on the Erdős-Faber-Lovász conjecture. In Meena Mahajan and Friedrich Slivovsky, editors, The 26th International Con- ference on Theory and Applications of Satisfiability Test- ing (SAT 2023), July 04-08, 2023, Alghero, Italy, LIPIcs. Schloss Dagstuhl -Leibniz-Zentrum für Informatik, 2023. To appear.","title":"A SAT solver's opinion on the Erdős-Faber-Lovász conjecture","authors":"Kirchweger","year":"2023","venue":"The 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Kirchweger et al., 2023b] Markus Kirchweger, Manfred Scheucher, and Stefan Szeider. SAT-based generation of planar graphs. In Meena Mahajan and Friedrich Slivovsky, editors, The 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023), July 04-08, 2023, Alghero, Italy, LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. To appear.","title":"SAT-based generation of planar graphs","authors":"Kirchweger","year":"2023","venue":"The 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Kochen and Specker, 1967] Simon Kochen and Ernst Specker. The problem of hidden variables in quantum mechanics. J. Math. Mech., 17(1):59-87, 1967.","title":"The problem of hidden variables in quantum mechanics","authors":"Specker Kochen, Simon Kochen, Ernst Specker","year":"1967","venue":"J. Math. Mech","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Li et al., 2022] Zhengyu Li, Curtis Bright, and Vijay Ganesh. A SAT solver + computer algebra attack on the minimum Kochen-Specker problem. Technical report, School of Computer Science at the University of Wind- sor, November 2022. https://cbright.myweb.cs.uwindsor. ca/reports/nmi-ks-preprint.pdf.","title":"A SAT solver + computer algebra attack on the minimum Kochen-Specker problem","authors":"Zhengyu Li, Curtis Bright, Vijay Ganesh","year":"2022","venue":null,"doi":null,"url":"https://cbright.myweb.cs.uwindsor.ca/reports/nmi-ks-preprint.pdf","internal_url":"/proceedings/2024/210"},{"position":25,"raw":"[Marques-Silva et al., 2009] João P. Marques-Silva, Inês Lynce, and Sharad Malik. Conflict-driven clause learn- ing SAT solvers. In Armin Biere, Marijn Heule, Hans van Maaren, and Toby Walsh, editors, Handbook of Satisfia- bility, pages 131-153. IOS Press, 2009.","title":"Conflict-driven clause learning SAT solvers","authors":"P João, Inês Marques-Silva, Sharad Lynce, Malik","year":"2009","venue":"Handbook of Satisfiability","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[McCreesh et al., 2020] Ciaran McCreesh, Patrick Prosser, and James Trimble. The glasgow subgraph solver: Us- ing constraint programming to tackle hard subgraph iso- morphism problem variants. In Fabio Gadducci and Timo Kehrer, editors, Graph Transformation -13th Interna- tional Conference, ICGT 2020, Held as Part of STAF 2020, Bergen, Norway, June 25-26, 2020, Proceedings, volume 12150 of Lecture Notes in Computer Science, pages 316- 324. Springer Verlag, 2020.","title":"The glasgow subgraph solver: Using constraint programming to tackle hard subgraph isomorphism problem variants","authors":"Ciaran Mccreesh, Patrick Prosser, James Trimble","year":"2020","venue":"Graph Transformation -13th International Conference, ICGT 2020, Held as Part of STAF 2020","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[McKay and Piperno, 2014] Brendan D. McKay and Adolfo Piperno. Practical graph isomorphism, II. J. Symbolic Comput., 60:94-112, 2014.","title":"Practical graph isomorphism","authors":"Brendan D Mckay, Adolfo Piperno","year":"2014","venue":"II. J. Symbolic Comput","doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[McKay, 1998] Brendan D. McKay. Isomorph-free exhaus- tive generation. J. Algorithms, 26(2):306-324, 1998.","title":"Isomorph-free exhaustive generation","authors":"Brendan D Mckay","year":"1998","venue":"J. Algorithms","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Ohrimenko et al., 2009] Olga Ohrimenko, Peter J. Stuckey, and Michael Codish. Propagation via lazy clause genera- tion. Constraints, 14(3):357-391, September 2009.","title":"Propagation via lazy clause generation","authors":"Olga Ohrimenko, Peter J Stuckey, Michael Codish","year":"2009","venue":"Constraints","doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Östergård, 2014] Patric R. J. Östergård. On the minimum size of 4-uniform hypergraphs without property B. Discr. Appl. Math., 163:199-204, 2014.","title":"On the minimum size of 4-uniform hypergraphs without property","authors":"R J Patric, Östergård","year":"2014","venue":"B. Discr. Appl. Math","doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Pavičić et al., 2005] Mladen Pavičić, Jean-Pierre Merlet, Brendan McKay, and Norman D Megill. Kochen-Specker vectors. J. Phys. A Math. Theor., 38(7):1577, 2005.","title":"Kochen-Specker vectors","authors":"Mladen Pavičić, Jean-Pierre Merlet, Brendan Mckay, Norman D Megill","year":"2005","venue":"J. Phys. A Math. Theor","doi":null,"url":null,"internal_url":null},{"position":32,"raw":"[Peres, 1991] A Peres. Two simple proofs of the Kochen- Specker theorem. J. Phys. A Math. Theor., 24(4):L175, feb 1991.","title":"Two simple proofs of the Kochen-Specker theorem","authors":"Peres","year":"1991","venue":"J. Phys. A Math. Theor","doi":null,"url":null,"internal_url":null},{"position":33,"raw":"[Prestwich, 2009] Steven David Prestwich. Cnf encodings. In Armin Biere, Marijn Heule, Hans van Maaren, and Toby Walsh, editors, Handbook of Satisfiability, pages 75-97. IOS Press, 2009.","title":"Cnf encodings","authors":"Steven David, Prestwich","year":"2009","venue":"Handbook of Satisfiability","doi":null,"url":null,"internal_url":null},{"position":34,"raw":"[Sinz, 2005] Carsten Sinz. Towards an optimal cnf encoding of boolean cardinality constraints. In Peter van Beek, ed- itor, Principles and Practice of Constraint Programming -CP 2005, 11th International Conference, CP 2005, Sit- ges, Spain, October 1-5, 2005, Proceedings, volume 3709 of Lecture Notes in Computer Science, pages 827-831. Springer Verlag, 2005.","title":"Towards an optimal cnf encoding of boolean cardinality constraints","authors":"Carsten Sinz","year":"2005","venue":"Principles and Practice of Constraint Programming -CP 2005, 11th International Conference, CP 2005","doi":null,"url":null,"internal_url":null},{"position":35,"raw":"[Uijlen and Westerbaan, 2016] Sander Uijlen and Bas West- erbaan. A Kochen-Specker system has at least 22 vectors. New Gener. Comput., 34(1-2):3-23, 2016.","title":"Sander Uijlen and Bas Westerbaan. A Kochen-Specker system has at least 22 vectors","authors":"Westerbaan Uijlen","year":"2016","venue":"New Gener. Comput","doi":null,"url":null,"internal_url":null},{"position":36,"raw":"[Wetzler et al., 2014] Nathan Wetzler, Marijn J. H. Heule, and Warren A. Hunt. DRAT-trim: Efficient checking and trimming using expressive clausal proofs. In Theory and Applications of Satisfiability Testing -SAT 2014, volume 8561 of Lecture Notes in Computer Science, pages 422- 429. Springer Verlag, 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":"Theory and Applications of Satisfiability Testing -SAT 2014","doi":null,"url":null,"internal_url":null},{"position":37,"raw":"[Zulkoski et al., 2017] Edward Zulkoski, Curtis Bright, Al- bert Heinle, Ilias S. Kotsireas, Krzysztof Czarnecki, and Vijay Ganesh. Combining SAT solvers with computer al- gebra systems to verify combinatorial conjectures. J. Au- tom. Reason., 58(3):313-339, 2017.","title":"Combining SAT solvers with computer algebra systems to verify combinatorial conjectures","authors":"Edward Zulkoski, Curtis Bright, Albert Heinle, Ilias S Kotsireas, Krzysztof Czarnecki, Vijay Ganesh","year":"2017","venue":"J. Autom. Reason","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}