Abstract
Recent years have witnessed a renewed interest in Boolean functions in explaining binary classifiers in the field of explainable AI (XAI). The standard approach to Boolean functions is based on propositional logic. We present a modal language of a ceteris paribus nature which supports reasoning about binary classifiers and their properties. We study a family of classifier models, axiomatize it and show completeness of our axiomatics. Moreover, we prove that satisfiability checking for our modal language relative to such a class of models is NP-complete. We leverage the language to formalize counterfactual conditional as well as a variety of notions of explanation including abductive, contrastive and counterfactual explanations, and biases. Finally, we present two extensions of our language: a dynamic extension by the notion of assignment enabling classifier change and an epistemic extension in which the classifier’s uncertainty about the actual input can be represented.
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Notes
- 1.
So the classifier we model here is slightly more expressive than Boolean classifier. Introducing decision atoms \(\mathsf {t}(x), \mathsf {t}(y), \dots \) below allows us to encode more than two decision values (classifications). Sometime we also use binary/Boolean classifier in this more general sense. Notice that we cannot use the term psuedo-Boolean, since in Boolean function it means \( Val = \mathfrak {R}\) [5], but we need our \( Val \) staying finite.
- 2.
In fact it appears to be a prime implicant, when we formally introduce this notion.
- 3.
- 4.
There are other options besides measuring distance by cardinality, e.g., distance in sense of subset relation as [2]. We will consider them in further research.
- 5.
A similar approach of ceteris paribus is [11]. They also refine Lewis’ semantics for counterfactual by selecting the closest worlds according to not only the actual world and antecedent, but also a set of formulas where they note as \(\Gamma \). The main technical difference is that they allow any counterfactual-free formula as a member of \(\Gamma \), while in our setting X only contains atomic formulas.
- 6.
A remarkable fact is that not all \(\Rightarrow _X\) satisfy the strong centering condition, which says that the actual world is the only closest world when the antecedent is already true here. To see it, consider a toy classifier model (C, s) such that \(S = \{s, s',s'',s''' \}\) with \(s = \{p, q\}\), \(s' = \{p \}\), \(s'' = \{q\}\), \(s''' = \emptyset \). We have \( closest _{C}(s{,}p{,}\emptyset ) = \{s,s'\}\), rather than \( closest _{C}(s{,}p{,}\emptyset ) = \{s\}\). All the rest of conditions in \(\mathsf {VC}\) are satisfied regardless of what X is.
- 7.
The symbol \(\triangle \) denotes symmetric difference.
- 8.
For the significance of actionablility in XAI, see e.g. [26].
- 9.
Notice that \(\mathsf {cn}_{Y{,} Atm \setminus Dec }\) is just another expression of \(\widehat{s}\) where \(s = Y\).
References
Biran, O., Cotton, C.: Explanation and justification in machine learning: a survey. In: IJCAI 2017 Workshop on Explainable AI (XAI), vol. 8, no. 1, pp. 8–13 (2017)
Borgida, A.: Language features for flexible handling of exceptions in information systems. ACM Trans. Database Syst. (TODS) 10(4), 565–603 (1985)
Caridroit, T., Lagniez, J.-M., Le Berre, D., de Lima, T., Montmirail, V.: A SAT-based approach for solving the modal logic S5-satisfiability problem. In: Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence (AAAI 2017), pp. 3864–3870. AAAI Press (2017)
Charrier, T., Herzig, A., Lorini, E., Maffre, F., Schwarzentruber, F.: Building epistemic logic from observations and public announcements. In: Proceedings of the Fifteenth International Conference on Principles of Knowledge Representation and Reasoning (KR 2016), pp. 268–277. AAAI Press (2016)
Crama, Y., Hammer, P.L.: Boolean Functions: Theory, Algorithms, and Applications. Cambridge University Press, Cambridge (2011)
Dalal, M.: Investigations into a theory of knowledge base revision: preliminary report. In: Proceedings of the Seventh National Conference on Artificial Intelligence, vol. 2, pp. 475–479. Citeseer (1988)
Darwiche, A., Hirth, A.: On the reasons behind decisions. In: 24th European Conference on Artificial Intelligence, ECAI 2020. Frontiers in Artificial Intelligence and Applications, vol. 325, pp. 712–720. IOS Press (2020)
Dhurandhar, A., et al.: Explanations based on the missing: towards contrastive explanations with pertinent negatives. In: Advances in Neural Information Processing Systems, pp. 592–603 (2018)
Dretske, F: Meaningful perception. An Invitation to Cognitive Science: Visual Cognition, pp. 331–352 (1995)
Fagin, R., Moses, Y., Halpern, J.Y., Vardi, M.Y.: Reasoning about Knowledge. MIT Press, Cambridge (1995)
Girard, P., Triplett, M.A.: Ceteris paribus logic in counterfactual reasoning. In: TARK 2015, pp. 176–193 (2016)
Grossi, D., Lorini, E., Schwarzentruber, F.: The ceteris paribus structure of logics of game forms. J. Artif. Intell. Res. 53, 91–126 (2015)
Hempel, C.G., Oppenheim, P.: Studies in the logic of explanation. Philos. Sci. 15(2), 135–175 (1948)
Herzig, A., Lorini, E.: A modal logic of perceptual belief. In: Lihoreau, F., Rebuschi, M. (eds.) Epistemology, Context, and Formalism. SL, vol. 369, pp. 197–211. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-02943-6_12
Herzig, A., Lorini, E., Maffre, F.: A poor man’s epistemic logic based on propositional assignment and higher-order observation. In: van der Hoek, W., Holliday, W.H., Wang, W. (eds.) LORI 2015. LNCS, vol. 9394, pp. 156–168. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48561-3_13
Ignatiev, A., Cooper, M.C., Siala, M., Hebrard, E., Marques-Silva, J.: Towards formal fairness in machine learning. In: Simonis, H. (ed.) CP 2020. LNCS, vol. 12333, pp. 846–867. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58475-7_49
Ignatiev, A., Narodytska, N., Asher, N., Marques-Silva, J.: From contrastive to abductive explanations and back again. In: Baldoni, M., Bandini, S. (eds.) AIxIA 2020. LNCS (LNAI), vol. 12414, pp. 335–355. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77091-4_21
Ignatiev, A., Narodytska, N., Marques-Silva, J.: Abduction-based explanations for machine learning models. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 33, pp. 1511–1519 (2019)
Kment, B.: Counterfactuals and explanation. Mind 115(458), 261–310 (2006)
Lewis, D.: Counterfactuals. Harvard University Press, Cambridge (1973)
Martens, D., Provost, F.: Explaining data-driven document classifications. MIS Q. 38(1), 73–100 (2014)
Mittelstadt, B., Russell, C., Wachter, S.: Explaining explanations in AI. In: Proceedings of the Conference on Fairness, Accountability, and Transparency, pp. 279–288 (2019)
Mothilal, R.K., Sharma, A., Tan, C.: Explaining machine learning classifiers through diverse counterfactual explanations. In: Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency, pp. 607–617 (2020)
Quine, W.V.: A way to simplify truth functions. Am. Math. Mon. 62(9), 627–631 (1955)
Shi, W., Shih, A., Darwiche, A., Choi, A.: On tractable representations of binary neural networks. arXiv preprint arXiv:2004.02082 (2020)
Sokol, K., Flach, P.A.: Counterfactual explanations of machine learning predictions: opportunities and challenges for AI safety. In: SafeAI@ AAAI (2019)
Van Benthem, J., Van Eijck, J., Kooi, B.: Logics of communication and change. Inf. Comput. 204(11), 1620–1662 (2006)
van der Hoek, W., Iliev, P., Wooldridge, M.J.: A logic of revelation and concealment. In: Proceedings of the International Conference on Autonomous Agents and Multiagent Systems, (AAMAS 2012), pp. 1115–1122. IFAAMAS (2012)
Van Der Hoek, W., Troquard, N., Wooldridge, M.J.: Knowledge and control. In: Proceedings of the 10th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2021), pp. 719–726. IFAAMAS (2011)
van Ditmarsch, H., van Der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Synthese Library, vol. 337. Springer, Heidelberg (2007)
van Ditmarsch, H.P., van der Hoek, W., Kooi, B.P.: Dynamic epistemic logic with assignment. In: Proceedings of the 4th International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS 2005), pp. 141–148. ACM (2005)
Verma, S., Dickerson, J., Hines, K.: Counterfactual explanations for machine learning: a review. arXiv preprint arXiv:2010.10596 (2020)
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Support from the ANR-3IA Artificial and Natural Intelligence Toulouse Institute is gratefully acknowledged.
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Liu, X., Lorini, E. (2021). A Logic for Binary Classifiers and Their Explanation. In: Baroni, P., Benzmüller, C., Wáng, Y.N. (eds) Logic and Argumentation. CLAR 2021. Lecture Notes in Computer Science(), vol 13040. Springer, Cham. https://doi.org/10.1007/978-3-030-89391-0_17
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