On the Tractability of Explaining Decisions of Classifiers

Authors Martin C. Cooper , João Marques-Silva

Thumbnail PDF


  • Filesize: 0.76 MB
  • 18 pages

Document Identifiers

Author Details

Martin C. Cooper
  • IRIT, Université de Toulouse III, France
João Marques-Silva
  • IRIT, CNRS, Toulouse, France

Cite AsGet BibTex

Martin C. Cooper and João Marques-Silva. On the Tractability of Explaining Decisions of Classifiers. In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 210, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Explaining decisions is at the heart of explainable AI. We investigate the computational complexity of providing a formally-correct and minimal explanation of a decision taken by a classifier. In the case of threshold (i.e. score-based) classifiers, we show that a complexity dichotomy follows from the complexity dichotomy for languages of cost functions. In particular, submodular classifiers allow tractable explanation of positive decisions, but not negative decisions (assuming P≠NP). This is an example of the possible asymmetry between the complexity of explaining positive and negative decisions of a particular classifier. Nevertheless, there are large families of classifiers for which explaining both positive and negative decisions is tractable, such as monotone or linear classifiers. We extend tractable cases to constrained classifiers (when there are constraints on the possible input vectors) and to the search for contrastive rather than abductive explanations. Indeed, we show that tractable classes coincide for abductive and contrastive explanations in the constrained or unconstrained settings.

Subject Classification

ACM Subject Classification
  • Theory of computation → Machine learning theory
  • Theory of computation → Problems, reductions and completeness
  • machine learning
  • tractability
  • explanations
  • weighted constraint satisfaction


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Gilles Audemard, Steve Bellart, Louenas Bounia, Frédéric Koriche, Jean-Marie Lagniez, and Pierre Marquis. On the computational intelligibility of boolean classifiers. CoRR, abs/2104.06172, 2021. URL: http://arxiv.org/abs/2104.06172.
  2. Gilles Audemard, Frédéric Koriche, and Pierre Marquis. On tractable XAI queries based on compiled representations. In KR, pages 838-849, 2020. URL: https://doi.org/10.24963/kr.2020/86.
  3. Pablo Barceló, Mikaël Monet, Jorge Pérez, and Bernardo Subercaseaux. Model interpretability through the lens of computational complexity. In Hugo Larochelle, Marc'Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin, editors, NeurIPS 2020, 2020. URL: https://proceedings.neurips.cc/paper/2020/hash/b1adda14824f50ef24ff1c05bb66faf3-Abstract.html.
  4. Andrei A. Bulatov. A dichotomy theorem for nonuniform CSPs. In FOCS, pages 319-330, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  5. Rainer E. Burkard, Bettina Klinz, and Rüdiger Rudolf. Perspectives of Monge properties in optimization. Discret. Appl. Math., 70(2):95-161, 1996. Google Scholar
  6. Deeparnab Chakrabarty, Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong. Subquadratic submodular function minimization. In STOC, pages 1220-1231, 2017. Google Scholar
  7. Zhi-Zhong Chen and Seinosuke Toda. The complexity of selecting maximal solutions. Inf. Comput., 119(2):231-239, 1995. URL: https://doi.org/10.1006/inco.1995.1087.
  8. David A. Cohen, Martin C. Cooper, Peter Jeavons, and Andrei A. Krokhin. A maximal tractable class of soft constraints. J. Artif. Intell. Res., 22:1-22, 2004. Google Scholar
  9. David A. Cohen, Martin C. Cooper, Peter Jeavons, and Andrei A. Krokhin. The complexity of soft constraint satisfaction. Artif. Intell., 170(11):983-1016, 2006. Google Scholar
  10. Martin C. Cooper, Simon de Givry, Martí Sánchez-Fibla, Thomas Schiex, Matthias Zytnicki, and Tomás Werner. Soft arc consistency revisited. Artif. Intell., 174(7-8):449-478, 2010. Google Scholar
  11. Martin C. Cooper, Simon de Givry, and Thomas Schiex. Graphical models: Queries, complexity, algorithms (tutorial). In STACS, pages 4:1-4:22, 2020. Google Scholar
  12. Nadia Creignou, Sanjeev Khanna, and Madhu Sudan. Complexity classifications of Boolean constraint satisfaction problems, volume 7 of SIAM monographs on discrete mathematics and applications. SIAM, 2001. Google Scholar
  13. Adnan Darwiche. Three modern roles for logic in AI. In PODS, pages 229-243, 2020. URL: https://doi.org/10.1145/3375395.3389131.
  14. Adnan Darwiche and Auguste Hirth. On the reasons behind decisions. In ECAI, pages 712-720, 2020. URL: https://doi.org/10.3233/FAIA200158.
  15. Adnan Darwiche and Pierre Marquis. A knowledge compilation map. J. Artif. Intell. Res., 17:229-264, 2002. URL: https://doi.org/10.1613/jair.989.
  16. Satoru Fujishige. Submodular Functions and Optimisation, volume 58 of Annals of Discrete Mathematics. Elsevier, 2nd edition, 2005. Google Scholar
  17. Riccardo Guidotti, Anna Monreale, Salvatore Ruggieri, Franco Turini, Fosca Giannotti, and Dino Pedreschi. A survey of methods for explaining black box models. ACM Comput. Surv., 51(5):93:1-93:42, 2019. URL: https://doi.org/10.1145/3236009.
  18. Emmanuel Hebrard, Brahim Hnich, Barry O'Sullivan, and Toby Walsh. Finding diverse and similar solutions in constraint programming. In Manuela M. Veloso and Subbarao Kambhampati, editors, Proceedings, The Twentieth National Conference on Artificial Intelligence, pages 372-377. AAAI Press / The MIT Press, 2005. URL: http://www.aaai.org/Library/AAAI/2005/aaai05-059.php.
  19. John Horan and Barry O'Sullivan. Towards diverse relaxations of over-constrained models. In ICTAI 2009, 21st IEEE International Conference on Tools with Artificial Intelligence, pages 198-205. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/ICTAI.2009.89.
  20. Alexey Ignatiev. Towards trustable explainable AI. In IJCAI, pages 5154-5158, 2020. URL: https://doi.org/10.24963/ijcai.2020/726.
  21. Alexey Ignatiev, Nina Narodytska, Nicholas Asher, and João Marques-Silva. From contrastive to abductive explanations and back again. In Matteo Baldoni and Stefania Bandini, editors, AIxIA 2020, volume 12414 of Lecture Notes in Computer Science, pages 335-355. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-77091-4_21.
  22. Alexey Ignatiev, Nina Narodytska, Nicholas Asher, and João Marques-Silva. On relating `why?' and `why not?' explanations. CoRR, abs/2012.11067, 2020. URL: http://arxiv.org/abs/2012.11067.
  23. Alexey Ignatiev, Nina Narodytska, and João Marques-Silva. Abduction-based explanations for machine learning models. In AAAI, pages 1511-1519, 2019. URL: https://doi.org/10.1609/aaai.v33i01.33011511.
  24. Alexey Ignatiev, Nina Narodytska, and João Marques-Silva. On relating explanations and adversarial examples. In NeurIPS, pages 15857-15867, 2019. URL: http://papers.nips.cc/paper/9717-on-relating-explanations-and-adversarial-examples.
  25. Linnea Ingmar, Maria Garcia de la Banda, Peter J. Stuckey, and Guido Tack. Modelling diversity of solutions. In AAAI 2020, pages 1528-1535. AAAI Press, 2020. URL: https://aaai.org/ojs/index.php/AAAI/article/view/5512.
  26. Peter Jeavons and Martin C. Cooper. Tractable constraints on ordered domains. Artif. Intell., 79(2):327-339, 1995. URL: https://doi.org/10.1016/0004-3702(95)00107-7.
  27. Matthew Joseph, Michael J. Kearns, Jamie Morgenstern, and Aaron Roth. Fairness in learning: Classic and contextual bandits. In Daniel D. Lee, Masashi Sugiyama, Ulrike von Luxburg, Isabelle Guyon, and Roman Garnett, editors, NIPS 2016, pages 325-333, 2016. URL: https://proceedings.neurips.cc/paper/2016/hash/eb163727917cbba1eea208541a643e74-Abstract.html.
  28. Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and James W. Thatcher, editors, Proceedings of a symposium on the Complexity of Computer Computations, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  29. Vladimir Kolmogorov, Andrei A. Krokhin, and Michal Rolínek. The complexity of general-valued CSPs. SIAM J. Comput., 46(3):1087-1110, 2017. URL: https://doi.org/10.1137/16M1091836.
  30. Andrei A. Krokhin and Stanislav Zivný. The complexity of valued CSPs. In Andrei A. Krokhin and Stanislav Zivný, editors, The Constraint Satisfaction Problem: Complexity and Approximability, volume 7 of Dagstuhl Follow-Ups, pages 233-266. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/DFU.Vol7.15301.9.
  31. Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong. A faster cutting plane method and its implications for combinatorial and convex optimization. In FOCS, pages 1049-1065, 2015. Google Scholar
  32. Xingchao Liu, Xing Han, Na Zhang, and Qiang Liu. Certified monotonic neural networks. In Hugo Larochelle, Marc'Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin, editors, NeurIPS 2020, 2020. URL: https://proceedings.neurips.cc/paper/2020/hash/b139aeda1c2914e3b579aafd3ceeb1bd-Abstract.html.
  33. João Marques-Silva, Thomas Gerspacher, Martin C. Cooper, Alexey Ignatiev, and Nina Narodytska. Explaining naive Bayes and other linear classifiers with polynomial time and delay. In Hugo Larochelle, Marc'Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin, editors, NeurIPS 2020, 2020. URL: https://proceedings.neurips.cc/paper/2020/hash/eccd2a86bae4728b38627162ba297828-Abstract.html.
  34. João Marques-Silva, Thomas Gerspacher, Martin C. Cooper, Alexey Ignatiev, and Nina Narodytska. Explanations for monotonic classifiers. In Marina Meila and Tong Zhang, editors, ICML 2021, volume 139 of Proceedings of Machine Learning Research, pages 7469-7479. PMLR, 2021. URL: http://proceedings.mlr.press/v139/marques-silva21a.html.
  35. Tim Miller. Explanation in artificial intelligence: Insights from the social sciences. Artif. Intell., 267:1-38, 2019. Google Scholar
  36. James B. Orlin. A faster strongly polynomial time algorithm for submodular function minimization. Math. Program., 118(2):237-251, 2009. URL: https://doi.org/10.1007/s10107-007-0189-2.
  37. Manon Ruffini, Jelena Vucinic, Simon de Givry, George Katsirelos, Sophie Barbe, and Thomas Schiex. Guaranteed diversity & quality for the weighted CSP. In ICTAI 2019, pages 18-25. IEEE, 2019. URL: https://doi.org/10.1109/ICTAI.2019.00012.
  38. Ethan L. Schreiber, Richard E. Korf, and Michael D. Moffitt. Optimal multi-way number partitioning. J. ACM, 65(4):24:1-24:61, 2018. URL: https://doi.org/10.1145/3184400.
  39. Andy Shih, Arthur Choi, and Adnan Darwiche. A symbolic approach to explaining bayesian network classifiers. In IJCAI, pages 5103-5111, 2018. URL: https://doi.org/10.24963/ijcai.2018/708.
  40. Andy Shih, Arthur Choi, and Adnan Darwiche. Compiling bayesian network classifiers into decision graphs. In AAAI, pages 7966-7974, 2019. URL: https://doi.org/10.1609/aaai.v33i01.33017966.
  41. Johan Thapper and Stanislav Zivny. The complexity of finite-valued CSPs. J. ACM, 63(4):37:1-37:33, 2016. Google Scholar
  42. Donald M. Topkis. Minimizing a submodular function on a lattice. Oper. Res., 26(2):305-321, 1978. URL: https://doi.org/10.1287/opre.26.2.305.
  43. Christopher Umans. The minimum equivalent DNF problem and shortest implicants. J. Comput. Syst. Sci., 63(4):597-611, 2001. URL: https://doi.org/10.1006/jcss.2001.1775.
  44. Dmitriy Zhuk. A proof of CSP dichotomy conjecture. In FOCS, pages 331-342, 2017. URL: https://doi.org/10.1109/FOCS.2017.38.
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail