{"type":"ScholarlyArticle","title":"Learning SMT(LRA) Constraints using SMT Solvers","authors":[{"name":"Samuel Kolb","slug":"samuel-kolb","given":"Samuel","family":"Kolb","affiliation":"KU Leuven , Belgium"},{"name":"Stefano Teso","slug":"stefano-teso","given":"Stefano","family":"Teso","affiliation":"KU Leuven , Belgium"},{"name":"Andrea Passerini","slug":"andrea-passerini","given":"Andrea","family":"Passerini","affiliation":"University of Trento , Italy"},{"name":"Luc De Raedt","slug":"luc-de-raedt","given":"Luc","family":"de Raedt","affiliation":"KU Leuven , Belgium"}],"abstract":"We introduce the problem of learning SMT(LRA) constraints from data. SMT(LRA) extends propositional logic with (in)equalities between numerical variables.\r\nMany relevant formal verification problems can be cast as SMT(LRA) instances and SMT(LRA) has supported recent developments in optimization and counting for hybrid Boolean and numerical domains. We introduce SMT(LRA) learning, the task of learning SMT(LRA) formulas from examples of feasible and infeasible instances, and we contribute INCAL, an exact non-greedy algorithm for this setting. Our approach encodes the learning task itself as an SMT(LRA) satisfiability problem that can be solved directly by SMT solvers. INCAL is an incremental algorithm that achieves exact learning by looking only at a small subset of the data, leading to significant speed-ups. We empirically evaluate our approach on both synthetic instances and benchmark problems taken from the SMT-LIB benchmarks repository.","doi":"10.24963/ijcai.2018/323","url":"https://ijcai.org/proceedings/2018/323","pdf":"https://ijcai.org/proceedings/2018/0323.pdf","markdown":"https://ijcai.org/proceedings/2018/323.md","bibtex":"https://ijcai.org/proceedings/2018/bibtex/323","volume":{"slug":"2018","title":"Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2018","url":"https://ijcai.org/proceedings/2018/","year":2018,"location":"Stockholm","dates":"13-19 July 2018","editors":"Jérôme Lang","isbn":"978-0-9992411-2-7","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2018"},"section":"Machine Learning","track":"Main track","number":323,"pages":{"first":2333,"last":2340},"keywords":["Constraints and SAT: Constraints and Data Mining ; Machine Learning","Machine Learning: Relational Learning","Machine Learning: New Problems"],"videos":[],"references":[{"position":1,"raw":"[Alur et al., 2013] Rajeev Alur, Rastislav Bodik, Garvit Ju- niwal, Milo MK Martin, Mukund Raghothaman, Sanjit A Seshia, Rishabh Singh, Armando Solar-Lezama, Emina Torlak, and Abhishek Udupa. Syntax-guided synthesis. In Formal Methods in Computer-Aided Design (FMCAD), 2013, pages 1-8. IEEE, 2013.","title":"Syntax-guided synthesis","authors":"Rajeev Alur, Rastislav Bodik, Garvit Juniwal, Milo Mk Martin, Mukund Raghothaman, A Sanjit, Rishabh Seshia, Armando Singh, Emina Solar-Lezama, Abhishek Torlak, Udupa","year":"2013","venue":"Formal Methods in Computer-Aided Design (FMCAD)","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Barrett et al., 2009] Clark W Barrett, Roberto Sebastiani, Sanjit A Seshia, and Cesare Tinelli. Satisfiability modulo theories. Handbook of satisfiability, 185:825-885, 2009.","title":"Satisfiability modulo theories","authors":"Clark W Barrett, Roberto Sebastiani, Sanjit A Seshia, Cesare Tinelli","year":"2009","venue":"Handbook of satisfiability","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Barrett et al., 2010] Clark Barrett, Aaron Stump, Cesare Tinelli, et al. The smt-lib standard: Version 2.0. In Pro- ceedings of the 8th International Workshop on Satisfiabil- ity Modulo Theories (Edinburgh, England), volume 13, page 14, 2010.","title":"The smt-lib standard: Version 2.0","authors":"Clark Barrett, Aaron Stump, Cesare Tinelli","year":"2010","venue":"Proceedings of the 8th International Workshop on Satisfiability Modulo Theories","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Belle et al., 2015] Vaishak Belle, Andrea Passerini, and Guy Van den Broeck. Probabilistic inference in hybrid domains by weighted model integration. In Proceedings of 24th International Joint Conference on Artificial Intelligence (IJCAI), pages 2770-2776, 2015.","title":"Probabilistic inference in hybrid domains by weighted model integration","authors":"Belle","year":"2015","venue":"Proceedings of 24th International Joint Conference on Artificial Intelligence (IJCAI)","doi":null,"url":null,"internal_url":"/proceedings/2015/392"},{"position":5,"raw":"[Bennett, 1994] Kristin P Bennett. Global tree optimization: A non-greedy decision tree algorithm. Computing Science and Statistics, pages 156-156, 1994.","title":"Global tree optimization: A non-greedy decision tree algorithm","authors":"Kristin P Bennett","year":"1994","venue":"Computing Science and Statistics","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Bertsimas and Dunn, 2017] Dimitris Bertsimas and Jack Dunn. Optimal classification trees. Machine Learning, pages 1-44, 2017.","title":"Optimal classification trees","authors":"Dimitris Bertsimas, Jack Dunn","year":"2017","venue":"Machine Learning","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Bessiere et al., 2005] Christian Bessiere, Remi Coletta, Frédéric Koriche, and Barry O'Sullivan. A sat-based ver- sion space algorithm for acquiring constraint satisfaction problems. Machine Learning: ECML 2005, pages 23-34, 2005.","title":"A sat-based version space algorithm for acquiring constraint satisfaction problems","authors":"Bessiere","year":"2005","venue":"Machine Learning: ECML 2005","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Bessiere et al., 2013] Christian Bessiere, Remi Coletta, Em- manuel Hebrard, George Katsirelos, Nadjib Lazaar, Nina Narodytska, Claude-Guy Quimper, Toby Walsh, et al. Con- straint acquisition via partial queries. In IJCAI, volume 13, pages 475-481, 2013.","title":"Constraint acquisition via partial queries","authors":"Bessiere","year":"2013","venue":"IJCAI","doi":null,"url":null,"internal_url":"/proceedings/2013/78"},{"position":9,"raw":"[Beyer et al., 2009] Dirk Beyer, Alessandro Cimatti, Alberto Griggio, M Erkan Keremoglu, and Roberto Sebastiani. Soft- ware model checking via large-block encoding. In Formal Methods in Computer-Aided Design, 2009. FMCAD 2009, pages 25-32. IEEE, 2009.","title":"Software model checking via large-block encoding","authors":"Dirk Beyer, Alessandro Cimatti, Alberto Griggio, M Erkan Keremoglu, Roberto Sebastiani","year":"2009","venue":"Formal Methods in Computer-Aided Design","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Campigotto et al., 2011] Paolo Campigotto, Andrea Passerini, and Roberto Battiti. Active learning of combina- torial features for interactive optimization. In International Conference on Learning and Intelligent Optimization, pages 336-350. Springer, 2011.","title":"Active learning of combinatorial features for interactive optimization","authors":"Paolo Campigotto, Andrea Passerini, Roberto Battiti","year":"2011","venue":"International Conference on Learning and Intelligent Optimization","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Chavira and Darwiche, 2008] Mark Chavira and Adnan Dar- wiche. On probabilistic inference by weighted model count- ing. Artificial Intelligence, 172(6-7):772-799, 2008.","title":"Mark Chavira and Adnan Darwiche. On probabilistic inference by weighted model counting","authors":"Darwiche Chavira","year":"2008","venue":"Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Cimatti et al., 2013] Alessandro Cimatti, Alberto Griggio, Bastiaan Joost Schaafsma, and Roberto Sebastiani. The mathsat5 smt solver. In TACAS, volume 7795, pages 93- 107. Springer, 2013.","title":"The mathsat5 smt solver","authors":"Alessandro Cimatti, Alberto Griggio, Bastiaan Joost Schaafsma, Roberto Sebastiani","year":"2013","venue":"TACAS","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Cussens, 2008] James Cussens. Bayesian network learning by compiling to weighted MAX-SAT. Proc. UAI, pages 105-112, 2008.","title":"Bayesian network learning by compiling to weighted MAX-SAT","authors":"James Cussens","year":"2008","venue":"Proc. UAI","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[De Moura and Bjørner, 2008] Leonardo De Moura and Nikolaj Bjørner. Z3: An efficient smt solver. Tools and Algorithms for the Construction and Analysis of Systems, pages 337-340, 2008.","title":"Leonardo De Moura and Nikolaj Bjørner. Z3: An efficient smt solver","authors":"De Moura, Bjørner","year":"2008","venue":"Tools and Algorithms for the Construction and Analysis of Systems","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Fagerberg et al., 2012] Rolf Fagerberg, Christoph Flamm, Daniel Merkle, and Philipp Peters. Exploring chemistry using smt. In Principles and Practice of Constraint Pro- gramming, pages 900-915. Springer, 2012.","title":"Exploring chemistry using smt","authors":"Rolf Fagerberg, Christoph Flamm, Daniel Merkle, Philipp Peters","year":"2012","venue":"Principles and Practice of Constraint Programming","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Gulwani et al., 2011] Sumit Gulwani, Vijay Anand Kor- thikanti, and Ashish Tiwari. Synthesizing geometry con- structions. In ACM SIGPLAN Notices, volume 46, pages 50-61. ACM, 2011.","title":"Synthesizing geometry constructions","authors":"Sumit Gulwani, Vijay Anand Korthikanti, Ashish Tiwari","year":"2011","venue":"ACM SIGPLAN Notices","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Gulwani et al., 2017] Sumit Gulwani, Oleksandr Polozov, Rishabh Singh, et al. Program synthesis. Foundations and Trends R in Programming Languages, 4(1-2):1-119, 2017.","title":"Program synthesis","authors":"Sumit Gulwani, Oleksandr Polozov, Rishabh Singh","year":"2017","venue":"Foundations and Trends R in Programming Languages","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Harada et al., 1995] Mikako Harada, Andrew Witkin, and David Baraff. Interactive physically-based manipulation of discrete/continuous models. In Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, pages 199-208. ACM, 1995.","title":"Interactive physically-based manipulation of discrete/continuous models","authors":"Mikako Harada, Andrew Witkin, David Baraff","year":"1995","venue":"Proceedings of the 22nd annual conference on Computer graphics and interactive techniques","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Jaakkola et al., 2010] Tommi Jaakkola, David Sontag, Amir Globerson, and Marina Meila. Learning Bayesian network structure using LP relaxations. In Proc. of the Thirteenth In- ternational Conference on Artificial Intelligence and Statis- tics, pages 358-365, 2010.","title":"Learning Bayesian network structure using LP relaxations","authors":"Tommi Jaakkola, David Sontag, Amir Globerson, Marina Meila","year":"2010","venue":"Proc. of the Thirteenth International Conference on Artificial Intelligence and Statistics","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Jabbari et al., 2016] Shahin Jabbari, Ryan M Rogers, Aaron Roth, and Steven Z Wu. Learning from rational behavior: Predicting solutions to unknown linear programs. In Ad- vances in Neural Information Processing Systems, pages 1570-1578, 2016.","title":"Learning from rational behavior: Predicting solutions to unknown linear programs","authors":"Shahin Jabbari, Ryan M Rogers, Aaron Roth, Steven Z Wu","year":"2016","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Kantchelian et al., 2014] Alex Kantchelian, Michael C Tschantz, Ling Huang, Peter L Bartlett, Anthony D Joseph, and J Doug Tygar. Large-margin convex polytope machine. In Advances in Neural Information Processing Systems, pages 3248-3256, 2014.","title":"Large-margin convex polytope machine","authors":"Alex Kantchelian, C Michael, Ling Tschantz, Peter L Huang, Anthony D Bartlett, Doug Joseph, Tygar","year":"2014","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Lee, 2008] Edward A Lee. Cyber physical systems: Design challenges. In Object oriented real-time distributed com- puting (isorc), 2008 11th ieee international symposium on, pages 363-369. IEEE, 2008.","title":"Cyber physical systems: Design challenges","authors":"Lee Edward","year":"2008","venue":"Object oriented real-time distributed computing (isorc)","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Monostori et al., 1996] L Monostori, A Márkus, Hendrik Van Brussel, and E Westkämpfer. Machine learning ap- proaches to manufacturing. CIRP Annals-Manufacturing Technology, 45(2), 1996.","title":"Machine learning approaches to manufacturing","authors":"Monostori, Hendrik Márkus, Van Brussel, Westkämpfer","year":"1996","venue":"CIRP Annals-Manufacturing Technology","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Mooney, 1995] Raymond J Mooney. Encouraging experi- mental results on learning cnf. Machine Learning, 19(1):79- 92, 1995.","title":"Encouraging experimental results on learning cnf","authors":"J Raymond, Mooney","year":"1995","venue":"Machine Learning","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Muggleton and De Raedt, 1994] Stephen Muggleton and Luc De Raedt. Inductive logic programming: Theory and methods. The Journal of Logic Programming, 19:629-679, 1994.","title":"Inductive logic programming: Theory and methods","authors":"De Muggleton, ; Raedt, Luc Stephen Muggleton, Raedt De","year":"1994","venue":"The Journal of Logic Programming","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Norouzi et al., 2015] Mohammad Norouzi, Maxwell Collins, Matthew A Johnson, David J Fleet, and Pushmeet Kohli. Efficient non-greedy optimization of decision trees. In Advances in Neural Information Processing Systems, pages 1729-1737, 2015.","title":"Efficient non-greedy optimization of decision trees","authors":"Mohammad Norouzi, Maxwell Collins, Matthew A Johnson, David J Fleet, Pushmeet Kohli","year":"2015","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Pawlak and Krawiec, 2017] Tomasz P Pawlak and Krzysztof Krawiec. Automatic synthesis of constraints from examples using mixed integer linear programming. European Journal of Operational Research, 261(3):1141-1157, 2017.","title":"Automatic synthesis of constraints from examples using mixed integer linear programming","authors":"P Tomasz, Krzysztof Pawlak, Krawiec","year":"2017","venue":"European Journal of Operational Research","doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Quinlan, 1990] J Ross Quinlan. Learning logical definitions from relations. Machine learning, 5(3):239-266, 1990.","title":"Learning logical definitions from relations","authors":"Quinlan Ross","year":"1990","venue":"Machine learning","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Sebastiani and Tomasi, 2015] Roberto Sebastiani and Silvia Tomasi. Optimization modulo theories with linear rational costs. ACM Transactions on Computational Logic (TOCL), 16(2):12, 2015.","title":"Optimization modulo theories with linear rational costs","authors":"Tomasi Sebastiani, Roberto Sebastiani, Silvia Tomasi","year":"2015","venue":"ACM Transactions on Computational Logic (TOCL)","doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Teso et al., 2015] Stefano Teso, Roberto Sebastiani, and An- drea Passerini. Structured learning modulo theories. Artifi- cial Intelligence, 2015.","title":"Structured learning modulo theories","authors":"Stefano Teso, Roberto Sebastiani, Andrea Passerini","year":"2015","venue":null,"doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Valiant, 1984] Leslie G Valiant. A theory of the learnable. Communications of the ACM, 27(11):1134-1142, 1984.","title":"A theory of the learnable","authors":"G Leslie, Valiant","year":"1984","venue":"Communications of the ACM","doi":null,"url":null,"internal_url":null},{"position":32,"raw":"[Yordanov et al., 2013] Boyan Yordanov, Christoph M Win- tersteiger, Youssef Hamadi, and Hillel Kugler. Smt-based analysis of biological computation. In NASA formal meth- ods symposium, pages 78-92. Springer, 2013.","title":"Smt-based analysis of biological computation","authors":"Boyan Yordanov, Christoph M Wintersteiger, Youssef Hamadi, Hillel Kugler","year":"2013","venue":"NASA formal methods symposium","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}