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Machine learning and logic: a new frontier in artificial intelligence

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Abstract

Machine learning and logical reasoning have been the two foundational pillars of Artificial Intelligence (AI) since its inception, and yet, until recently the interactions between these two fields have been relatively limited. Despite their individual success and largely independent development, there are new problems on the horizon that seem solvable only via a combination of ideas from these two fields of AI. These problems can be broadly characterized as follows: how can learning be used to make logical reasoning and synthesis/verification engines more efficient and powerful, and in the reverse direction, how can we use reasoning to improve the accuracy, generalizability, and trustworthiness of learning. In this perspective paper, we address the above-mentioned questions with an emphasis on certain paradigmatic trends at the intersection of learning and reasoning. Our intent here is not to be a comprehensive survey of all the ways in which learning and reasoning have been combined in the past. Rather we focus on certain recent paradigms where corrective feedback loops between learning and reasoning seem to play a particularly important role. Specifically, we observe the following three trends: first, the use of learning techniques (especially, reinforcement learning) in sequencing, selecting, and initializing proof rules in solvers/provers; second, combinations of inductive learning and deductive reasoning in the context of program synthesis and verification; and third, the use of solver layers in providing corrective feedback to machine learning models in order to help improve their accuracy, generalizability, and robustness with respect to partial specifications or domain knowledge. We believe that these paradigms are likely to have significant and dramatic impact on AI and its applications for a long time to come.

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Notes

  1. Given that the focus of this paper is to provide a perspective on certain emerging trends in AI, there is no experimental or scientific data associated with it.

  2. Conceptually, the terms abstraction and approximation are very similar in their technical meaning in the broad context of program analysis, synthesis, and verification. Hence, we use them interchangeably.

  3. It is possible to imagine more complex configurations with multiple kinds of abstractors and refiners, but we won’t discuss them for the sake of brevity.

  4. In this paper, we interchangeably use the terms logical reasoners, symbolic reasoners, decision procedures, and solvers (dually, provers), since all such systems fundamentally reason about mathematical formulas and decide whether they are satisfiable (dually, valid).

References

  1. Alur R, Bodik R, Juniwal G, Martin Milo MK, Raghothaman M, Seshia SA, Singh R, Solar-Lezama A, Torlak E, Udupa A (2013) Syntax-guided synthesis. In: Proceedings of the IEEE international conference on formal methods in computer-aided design (FMCAD), pp 1–17

  2. Abate A, David C, Kesseli P, Kroening D, Polgreen E (2018) Counterexample guided inductive synthesis modulo theories. In: Computer aided verification-30th international conference (CAV), volume 10981 of Lecture Notes in Computer Science, pp 270–288. Springer

  3. Atserias A, Fichte JK, Thurley M (2009) Clause-learning algorithms with many restarts and bounded-width resolution. In: Kullmann O (ed) Theory and Applications of Satisfiability Testing - SAT. Springer, Berlin Heidelberg, Berlin, Heidelberg, pp 114–127

    MATH  Google Scholar 

  4. Angluin D (1988) Queries and concept learning. Mach Learn 2(4):319–342

    Article  MathSciNet  MATH  Google Scholar 

  5. Audemard Gilles, Simon Laurent (2013) Glucose 2.3 in the SAT 2013 Competition. In: Proceedings of SAT competition 2013, pp 42–43

  6. Alur R, Singh R, Fisman D, Solar-Lezama A (2018) Search-based program synthesis. Commun ACM 61(12):84–93

    Article  Google Scholar 

  7. Ashok D, Scott J, Wetzel SJ, Panju M, Ganesh V (2021) Logic guided genetic algorithms. In: 35th AAAI conference on artificial intelligence, AAAI 2021, 33rd Conference on innovative applications of artificial intelligence, IAAI 2021, The 11th symposium on educational advances in artificial intelligence, EAAI 2021, virtual event, Feb 2–9, 2021, pp 15753–15754. AAAI Press

  8. Alaa AM, Schaar M van der (2019) Demystifying black-box models with symbolic metamodels. Adv Neural Inf Process Syst 32

  9. Bouraoui Z, Cornuéjols A, Denoeux T, Destercke S, Dubois D, Guillaume R, Marques-Silva J, Mengin J, Prade H, Schockaert S, Serrurier M, Vrain C (2019) From shallow to deep interactions between knowledge representation, reasoning and machine learning (kay r. amel group). CoRR, abs/1912.06612

  10. Biere A, Heule M, van Maaren H, Walsh T (2009) (eds) Handbook of Satisfiability, vol. 185 of Frontiers in Artificial Intelligence and Applications. IOS Press

  11. Bünz B, Lamm M (2017) Graph neural networks and boolean satisfiability. CoRR, abs/1702.03592

  12. Bengio Y, LeCun Y, Hinton GE (2021) Deep learning for AI. Commun ACM 64(7):58–65

    Article  Google Scholar 

  13. Bak S, Liu C, Johnson TT (2021) The second international verification of neural networks competition (VNN-COMP 2021): Summary and results. CoRR, abs/2109.00498

  14. Ball T, Levin V, Rajamani SK (2011) A decade of software model checking with SLAM. Commun ACM 54(7):68–76

    Article  Google Scholar 

  15. Bansal K, Loos SM, Rabe MN, Szegedy C, Wilcox S (2019) Holist: an environment for machine learning of higher order logic theorem proving. In: Kamalika C and Ruslan S, (eds.) Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine Learning Research, pp 454–463. PMLR

  16. Bader-El-Den MB, Poli R (2007) Generating SAT local-search heuristics using a GP hyper-heuristic framework. In: Nicolas M, El-Ghazali T, Pierre C, Marc S, Evelyne L, (eds), Artificial Evolution, 8th International Conference, Evolution Artificielle, EA 2007, Tours, France, October 29-31, 2007, revised selected papers, volume 4926 of Lecture Notes in Computer Science, Springer, pp 37–49

  17. Clarke E, Biere A, Raimi R, Zhu Y (2001) Bounded model checking using satisfiability solving. Formal Methods Syst Des 19(1):7–34

    Article  MATH  Google Scholar 

  18. Cropper A, Dumancic S, Evans R, Muggleton SH (2022) Inductive logic programming at 30. Mach Learn 111(1):147–172

    Article  MathSciNet  MATH  Google Scholar 

  19. Clarke EM, Emerson EA (1981) Design and synthesis of synchronization skeletons using branching-time temporal logic. In: Logic of programs, pp 52–71

  20. Clarke EM, Fehnker A, Han Z, Krogh BH, Stursberg O, Theobald M (2003) Verification of hybrid systems based on counterexample-guided abstraction refinement. In: TACAS, pp 192–207

  21. Clarke EM, Grumberg O, Jha S, Lu Y, Veith H (2000) Counterexample-guided abstraction refinement. In: 12th international conference on computer aided verification (CAV), vol. 1855 of Lecture Notes in Computer Science, Springer, pp 154–169

  22. Clarke EM, Grumberg O, Jha S, Yuan L, Veith H (2003) Counterexample-guided abstraction refinement for symbolic model checking. J ACM 50(5):752–794

    Article  MathSciNet  MATH  Google Scholar 

  23. Clarke EM, Gupta A, Kukula JH, Strichman O (2002) SAT based abstraction-refinement using ILP and machine learning techniques. In: computer aided verification, 14th international conference (CAV), vol. 2404 of Lecture Notes in Computer Science, Springer, pp 265–279

  24. Cadar C, Ganesh V, Pawlowski PM, Dill DL, Engler DR (2006) EXE: automatically generating inputs of death. In: Proceedings of the 13th ACM conference on computer and communications security, CCS ’06, New York, NY, USA. ACM, pp 322–335

  25. Cook S (1971) The complexity of theorem-proving procedures. In: proceedings of the third annual ACM symposium on theory of computing (STOC), ACM, pp 151–158

  26. Dash T, Chitlangia S, Ahuja A, Srinivasan A (2021) How to tell deep neural networks what we know. CoRR, abs/2107.10295

  27. Davis M, Logemann G, Loveland D (1962) A machine program for theorem-proving. Commun ACM 5(7):394–397

    Article  MathSciNet  MATH  Google Scholar 

  28. De Moura L, Bjørner N (2008) Z3: An efficient smt solver. In: international conference on tools and algorithms for the construction and analysis of systems, Springer, pp 337–340

  29. Duan H, Nejati S, Trimponias G, Poupart P, Ganesh V (2020) Online bayesian moment matching based SAT solver heuristics. In: Proceedings of the 37th international conference on machine learning, ICML 2020, 13–18 July 2020, Virtual Event, vol. 119 of proceedings of machine learning research, PMLR, pp 2710–2719

  30. Devlin D, O’Sullivan B (2008) Satisfiability as a classification problem. In: Proceedings of the 19th Irish Conference on artificial intelligence and cognitive science

  31. Dubois D, Prade H (2019) Towards a reconciliation between reasoning and learning-a position paper. In: Nahla Ben A, Benjamin Q, Martin T, (eds) scalable uncertainty management-13th international conference, SUM 2019, Compiègne, France, Dec 16–18, 2019, Proceedings, vol. 11940 of lecture notes in computer science, Springer, pp 153–168

  32. Eén N, Sörensson N (2004) Theory and applications of satisfiability testing: 6th international conference, SAT 2003, santa margherita ligure, Italy, May 5–8, 2003, selected revised papers, chapter an extensible SAT-solver, Springer Berlin Heidelberg, Berlin, Heidelberg, pp 502–518

  33. Ertel W, Schumann J, Suttner CB (1989) Learning heuristics for a theorem prover using back propagation. In: Johannes R, Karl L, (eds) 5. Österreichische Artificial Intelligence-Tagung, Igls, Tirol, 28. bis 30. Sept 1989, Proceedings, vol. 208 of Informatik-Fachberichte, Springer, pp 87–95

  34. Flint A, Blaschko MB (2012) Perceptron learning of SAT. In: Bartlett PL, Pereira FNC, Burges CJC, Léon B, Weinberger KQ, (eds) Advances in neural information processing systems 25: 26th annual conference on neural information processing systems 2012. Proceedings of a meeting held Dec 3–6, 2012, Lake Tahoe, Nevada, United States, pp 2780–2788

  35. Froleyks N, Heule M, Iser M, Järvisalo M, Suda M (2021) SAT competition 2020. Artif Intell 301:103572

    Article  MathSciNet  MATH  Google Scholar 

  36. Feldman YMY, Immerman N, Sagiv M, Shoham S (2020) Complexity and information in invariant inference. Proc ACM Program Lang 4(POPL):5:1-5:29

    Article  Google Scholar 

  37. First E, Rabe MN, Ringer T, Brun Y (2023) Baldur: whole-proof generation and repair with large language models. CoRR, abs/2303.04910

  38. Fukunaga AS (2002) Automated discovery of composite SAT variable-selection heuristics. In: Dechter R, Kearns MJ, Sutton RS, (eds) proceedings of the eighteenth national conference on artificial intelligence and fourteenth conference on innovative applications of artificial intelligence, July 28 - August 1, 2002, Edmonton, Alberta, Canada, AAAI Press / The MIT Press, pp 641–648

  39. Fukunaga AS (2004) Evolving local search heuristics for SAT using genetic programming. In: Kalyanmoy D, Riccardo P, Wolfgang B, Hans-Georg B, Burke EK, Darwen PJ, Dasgupta D, Floreano D, Foster JA, Mark H, Owen H, Pier Luca L, Lee S, Andrea T, Dirk T, Tyrrell AM, (eds) Genetic and evolutionary computation-GECCO 2004, genetic and evolutionary computation conference, Seattle, WA, USA, June 26–30, 2004, Proceedings, Part II, volume 3103 of Lecture Notes in Computer Science, Springer, pp 483–494

  40. Fukunaga AS (2008) Automated discovery of local search heuristics for satisfiability testing. Evol Comput 16(1):31–61

    Article  Google Scholar 

  41. Ian JG, Yoshua B, Courville AC (2016) Deep Learning. Adaptive computation and machine learning. MIT Press, Cambridge

    MATH  Google Scholar 

  42. Goldman SA, Kearns MJ (1992) On the complexity of teaching. J Comput Syst Sci 50:303–314

    MathSciNet  Google Scholar 

  43. Gupta A (2006) Learning Abstractions for Model Checking. PhD thesis, Carnegie Mellon University, June

  44. Hutter F, Babic D, Hoos HH, Hu AJ (2007) Boosting verification by automatic tuning of decision procedures. In: 7th international conference on formal methods in computer-aided design (FMCAD), IEEE Computer Society, pp 27–34

  45. Hart PE, Nilsson NJ, Raphael B (1968) A formal basis for the heuristic determination of minimum cost paths. IEEE Trans Syst Sci Cybern 4(2):100–107

    Article  Google Scholar 

  46. Holden SB (2021) Machine learning for automated theorem proving: learning to solve SAT and QSAT. Found Trends Mach Learn 14(6):807–989

    Article  MATH  Google Scholar 

  47. Hopfield JJ, Tank DW (1985) Neural computation of decisions in optimization problems. Biol Cybern 52:141–152

    Article  MATH  Google Scholar 

  48. Jha S, Gulwani S, Seshia SA, Tiwari A (2010) Oracle-guided component-based program synthesis. In: Proceedings of the 32nd international conference on software engineering (ICSE), pp 215–224

  49. Jha S, Gulwani S, Seshia SA, Tiwari A (2010) Synthesizing switching logic for safety and dwell-time requirements. In: proceedings of the international conference on cyber-physical systems (ICCPS), pp 22–31

  50. Johnson JL (1989) A neural network approach to the 3-satisfiability problem. J Parallel Distributed Comput 6:435–449

    Article  Google Scholar 

  51. Jha S, Seshia SA (2017) A theory of formal synthesis via inductive learning. Acta Inform 54(7):693–726

    Article  MathSciNet  MATH  Google Scholar 

  52. Kim S, Lu PY, Mukherjee S, Gilbert M, Jing L, Čeperić V, Soljačić M (2021) Integration of neural network-based symbolic regression in deep learning for scientific discovery. IEEE Trans Neural Netw Learn Syst 32(9):4166–4177

    Article  Google Scholar 

  53. Kautz HA, Selman B (1992) Planning as satisfiability. In: Bernd N, (ed) 10th European conference on artificial intelligence, ECAI 92, Vienna, Austria, August 3–7, 1992. Proceedings, John Wiley and Sons, pp 359–363

  54. Kurshan R (1994) Automata-theoretic verification of coordinating processes. In: 11th international conference on analysis and optimization of systems–discrete event systems, vol. 199 of LNCS, Springer, pp 16–28

  55. Lewkowycz A, Andreassen A, Dohan D, Dyer E, Michalewski H, Ramasesh VV, Slone A, Anil C, Schlag I, Gutman-Solo T, Wu Y, Neyshabur B, Gur-Ari G, Misra V (2022) Solving quantitative reasoning problems with language models. CoRR, abs/2206.14858

  56. Liang JH, Ganesh V, Poupart P, Czarnecki K (2016) Learning rate based branching heuristic for SAT solvers. In: Nadia C, Daniel LB, (eds) Theory and applications of satisfiability testing–SAT 2016, Cham. Springer International Publishing, pp 123–140

  57. Liang JH (2018) Machine learning for SAT solvers. PhD thesis, University of Waterloo, Canada

  58. Lagoudakis Michail G, Littman Michael L (2001) Learning to select branching rules in the DPLL procedure for satisfiability. Electron Notes Discrete Math 9:344–359

    Article  MATH  Google Scholar 

  59. Liang JH, Oh C, Mathew M, Thomas C, Li C, Ganesh V (2018) Machine learning-based restart policy for CDCL SAT solvers. In: theory and applications of satisfiability testing-SAT 2018 - 21st international conference, SAT 2018, held as part of the federated logic conference, FloC 2018, Oxford, UK, July 9–12, 2018, Proceedings, pp 94–110

  60. Lederman G, Rabe MN, Seshia S, Lee EA (2020) Learning heuristics for quantified boolean formulas through reinforcement learning. In: 8th international conference on learning representations (ICLR), April

  61. Lorenz JH, Wörz F (2020) On the effect of learned clauses on stochastic local search. In: Luca P, Martina S, (eds) Theory and applications of satisfiability testing-SAT 2020–23rd international conference, Alghero, Italy, July 3–10, 2020, Proceedings, vol. 12178 of lecture notes in computer science, Springer, pp 89–106

  62. Tom M (1997) Mitchell. McGraw-Hill, Machine Learning

    Google Scholar 

  63. Marques-Silva JP, Sakallah KA (1996) GRASP-A new search algorithm for satisfiability. In: proceedings of the 1996 IEEE/ACM international conference on computer-aided design, ICCAD ’96, Washington, DC, USA. IEEE Computer Society, pp 220–227

  64. Manna Z, Waldinger R (1980) A deductive approach to program synthesis. ACM Trans Program Lang Syst (TOPLAS) 2(1):90–121

    Article  MATH  Google Scholar 

  65. Nejati S, Frioux LL, Ganesh V (2020) A machine learning based splitting heuristic for divide-and-conquer solvers. In: Helmut S, (ed) Principles and practice of constraint programming-26th international conference, CP 2020, Louvain-la-Neuve, Belgium, Sept 7–11, 2020, Proceedings, vol. 12333 of lecture notes in computer science, Springer, pp 899–916

  66. OpenAI. GPT-4 technical report. CoRR, abs/2303.08774, 2023

  67. Panju M (2021) Automated Knowledge Discovery using Neural Networks. PhD thesis, University of Waterloo, Ontario, Canada

  68. Polgreen E, Cheang K, Gaddamadugu P, Godbole A, Laeufer K, Lin S, Manerkar YA, Mora F, Seshia SA (2022) UCLID5: multi-modal formal modeling, verification, and synthesis. In: computer aided verification-34th international conference (CAV), vol. 13371 of lecture notes in computer science, Springer, pp 538–551

  69. Pipatsrisawat K, Darwiche A (2011) On the power of clause-learning SAT solvers as resolution engines. Artif Intell 175(2):512–525

    Article  MathSciNet  MATH  Google Scholar 

  70. Andrei P, Polat ES, Alexander F, Mathias U, Müslüm A, Viet-Man L, Klaus P, Martin E, Trang TTN (2022) An overview of machine learning techniques in constraint solving. J Intell Inf Syst 58(1):91–118

    Article  Google Scholar 

  71. Pimpalkhare N, Mora F, Polgreen E, Seshia SA (2021) MedleySolver: online SMT algorithm selection. In: 24th international conference on theory and applications of satisfiability testing (SAT), vol. 12831 of lecture notes in computer science, Springer, pp 453–470

  72. Pogancic MV, Paulus A, Musil V, Martius G, Rolínek M (2020) Differentiation of blackbox combinatorial solvers. In: 8th international conference on learning representations, ICLR 2020, Addis Ababa, Ethiopia, April 26–30, 2020. OpenReview.net

  73. Polgreen E, Reynolds A, Seshia SA (2022) Satisfiability and synthesis modulo oracles. In: Proceedings of the 23rd international conference on verification, model checking, and abstract interpretation (VMCAI)

  74. De Raedt L (2008) Logical and relational learning. Cognitive technologies. Springer, Berlin

    Google Scholar 

  75. Russell S, Norvig P (2020) Artificial intelligence: a modern approach, 4th edn. Pearson, London

    MATH  Google Scholar 

  76. Rossi F, van Beek P, Walsh T (2006) editors. Handbook of Constraint Programming, vol. 2 of Foundations of Artificial Intelligence. Elsevier

  77. Richard SS, Andrew GB (1998) Reinforcement learning-an introduction. Adaptive computation and machine learning. MIT Press, Cambridge

    Google Scholar 

  78. Seshia SA (2005) Adaptive eager boolean encoding for arithmetic reasoning in verification. PhD thesis, Carnegie Mellon University

  79. Seshia SA (2012) Sciduction: combining induction, deduction, and structure for verification and synthesis. In: proceedings of the design automation conference (DAC), pp 356–365

  80. Seshia SA (2015) Combining induction, deduction, and structure for verification and synthesis. Proc IEEE 103(11):2036–2051

    Article  Google Scholar 

  81. Selsam D, Lamm M, Bünz B, Liang P, de Moura L, Dill DL (2019) Learning a SAT solver from single-bit supervision. In: 7th international conference on learning representations, ICLR 2019, New Orleans, LA, USA, May 6–9, 2019. OpenReview.net

  82. Solar-Lezama A, Tancau L, Bodík R, Seshia SA, Saraswat VA (2006) Combinatorial sketching for finite programs. In: proceedings of the 12th international conference on architectural support for programming languages and operating systems (ASPLOS), ACM Press, pp 404–415

  83. Scott J, Niemetz A, Preiner M, Nejati S, Ganesh V (2021) Machsmt: a machine learning-based algorithm selector for SMT solvers. In: Jan Friso G, Kim Guldstrand L, (eds) Tools and algorithms for the construction and analysis of systems-27th international conference, TACAS 2021, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021, Luxembourg City, Luxembourg, March 27–April 1, 2021, Proceedings, Part II, vol. 12652 of Lecture Notes in Computer Science, Springer, pp 303–325

  84. Scott J, Panju M, Ganesh V (2020) LGML: logic guided machine learning. In: the thirty-fourth AAAI conference on artificial intelligence, AAAI 2020, the thirty-second innovative applications of artificial intelligence conference, IAAI 2020, the tenth AAAI symposium on educational advances in artificial intelligence, EAAI 2020, New York, NY, USA, Feb 7–12, 2020, AAAI Press, pp 13909–13910

  85. Marques SJP, Sakallah KA (1996) GRASP-a new search algorithm for satisfiability. In: Rutenbar RA, Otten RHJM, (eds) proceedings of the 1996 IEEE/ACM international conference on computer-aided design, ICCAD 1996, San Jose, CA, USA, Nov 10–14, 1996, IEEE Computer Society / ACM, pp 220–227

  86. Seshia SA, Subramanyan P (2018) UCLID5: integrating modeling, verification, synthesis, and learning. In: proceedings of the 15th ACM/IEEE international conference on formal methods and models for codesign (MEMOCODE)

  87. Seshia SA, Sadigh D, Sastry SS (2022) Toward verified artificial intelligence. Commun ACM 65(7):46–55

    Article  Google Scholar 

  88. Seshia SA, Sharygina N, Tripakis S (2018) Modeling for verification. In: Clarke EM, Thomas H, Helmut V, (eds) Handbook of Model Checking, chapter 3. Springer

  89. Sarker Md, Kamruzzaman ZL, Aaron E, Pascal H (2021) Neuro-symbolic artificial intelligence. AI Commun 34(3):197–209

    Article  MathSciNet  MATH  Google Scholar 

  90. van Harmelen F, Lifschitz V, Porter BW (2008) (eds) Handbook of Knowledge Representation, vol.  3 of Foundations of Artificial Intelligence. Elsevier

  91. The Verification of Neural Networks Library (VNN-LIB). www.vnnlib.org, 2019

  92. Wang PW, Donti PL, Wilder B, Zico KJ (2019) Satnet: bridging deep learning and logical reasoning using a differentiable satisfiability solver. In: Kamalika C, Ruslan S, (eds) proceedings of the 36th international conference on machine learning, ICML 2019, 9–15 June 2019, Long Beach, California, USA, vol. 97 of Proceedings of Machine Learning Research, PMLR, pp 6545–6554

  93. Jeannette MW (2021) Trustworthy AI. Commun ACM 64(10):64–71

    Article  Google Scholar 

  94. Lin X, Hutter F, Hoos HH, Leyton-Brown K (2008) SATzilla: portfolio-based algorithm selection for SAT. J Artif Intell Res 32(1):565–606

    MATH  Google Scholar 

Download references

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  1. Vijay Ganesh, Sanjit A. Seshia, Somesh Jha have contributed equally to this work.

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  1. University of Waterloo, Ontario, Canada

    Vijay Ganesh

  2. University of California, Berkeley, USA

    Sanjit A. Seshia

  3. University of Wisconsin, Madison, USA

    Somesh Jha

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  1. Vijay Ganesh
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Ganesh, V., Seshia, S.A. & Jha, S. Machine learning and logic: a new frontier in artificial intelligence. Form Methods Syst Des 60, 426–451 (2022). https://doi.org/10.1007/s10703-023-00430-1

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