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IS CAUSAL REASONING HARDER THAN PROBABILISTIC REASONING?

Part of: Philosophical aspects of logic and foundations Theory of computing

Published online by Cambridge University Press:  18 May 2022

MILAN MOSSÉ ,
DULIGUR IBELING  and
THOMAS ICARD
MILAN MOSSÉ
Affiliation:
UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720-2390, USA E-mail: milan_mosse@berkeley.edu
DULIGUR IBELING
Affiliation:
STANFORD UNIVERSITY STANFORD, CA 94305, USA E-mail: duligur@stanford.edu E-mail: icard@stanford.edu
THOMAS ICARD
Affiliation:
STANFORD UNIVERSITY STANFORD, CA 94305, USA E-mail: duligur@stanford.edu E-mail: icard@stanford.edu
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Abstract

Many tasks in statistical and causal inference can be construed as problems of entailment in a suitable formal language. We ask whether those problems are more difficult, from a computational perspective, for causal probabilistic languages than for pure probabilistic (or “associational”) languages. Despite several senses in which causal reasoning is indeed more complex—both expressively and inferentially—we show that causal entailment (or satisfiability) problems can be systematically and robustly reduced to purely probabilistic problems. Thus there is no jump in computational complexity. Along the way we answer several open problems concerning the complexity of well-known probability logics, in particular demonstrating the ${\exists \mathbb {R}}$-completeness of a polynomial probability calculus, as well as a seemingly much simpler system, the logic of comparative conditional probability.

Keywords

probabilitycausationlogiccomplexityETR

MSC classification

Primary: 03A05: Philosophical and critical
Secondary: 68Q25: Analysis of algorithms and problem complexity
Type
Research Article
Information
The Review of Symbolic Logic , Volume 17 , Issue 1 , March 2024 , pp. 106 - 131
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

BIBLIOGRAPHY

Abadi, M., & Halpern, J. Y. (1994). Decidability and expressiveness for first-order logics of probability. Information and Computation, 112, 1–36.CrossRefGoogle Scholar
Abrahamsen, M., Adamaszek, A., & Miltzow, T. (2018). The art gallery problem is ∃ℝ-complete. In Diakonikolas, I., and Kempe, D., editors. Proceedings of the 50 th Annual ACM SIGACT Symposium on Theory of Computing. New York: Association for Computing Machinery, pp. 65–73.Google Scholar
Abrahamsen, M., Kleist, L., & Miltzow, T. (2021). Training neural networks is ∃ℝ-complete. In Ranzato, M., editor. Proceedings of the Thirty-Fifth Conference on Neural Information Processing Systems (NeurIPS). Vancouver: Curran Associates, Inc.Google Scholar
Aleksandrowicz, G., Chockler, H., Halpern, J. Y., & Ivrii, A. (2017). The computational complexity of structure-based causality. Journal of Artificial Intelligence Research, 58, 431–451.CrossRefGoogle Scholar
Bareinboim, E., Correa, J., Ibeling, D., & Icard, T. (2022). On Pearl’s hierarchy and the foundations of causal inference. In Geffner, H., Dechter, R., and Halpern, J. Y., editors. Probabilistic and Causal Inference: The Works of Judea Pearl. New York: ACM Books, pp. 509–556.Google Scholar
Belot, G. (2020). Absolutely no free lunches! Theoretical Computer Science, 845, 159–180.CrossRefGoogle Scholar
Bilò, V., & Mavronicolas, M. (2017). ∃ℝ-complete decision problems about symmetric Nash equilibria in symmetric multi-player games. In Vallée, B., and Vollmer, H., editors. 34 th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Dagstuhl: Schloss Dagstuhl-Leibniz-Zentrum für Informatik.Google Scholar
Board, O. (2004). Dynamic interactive epistemology. Games and Economic Behavior, 49(1), 49–80.CrossRefGoogle Scholar
Canny, J. (1988). Some algebraic and geometric computations in PSPACE. In Simon, J., editor. Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing. New York: Association for Computing Machinery, pp. 460–467.CrossRefGoogle Scholar
Cardinal, J. (2015). Computational geometry column 62. ACM SIGACT News, 46(4), 69–78.CrossRefGoogle Scholar
ten Cate, B., Kolaitis, P. G., & Othman, W. (2013). Data exchange with arithmetic operations. In Guerrini, G., editor. Proceedings of the 16 th International Conference on Extending Database Technology. New York: Association for Computing Machinery, pp. 537–548.Google Scholar
Cook, S. A. (1971). The complexity of theorem-proving procedures. In Harrison, M., Banerji, R., and Ullman, J., editors. Proceedings of the Third Annual ACM Symposium on Theory of Computing. New York: Association for Computing Machinery, pp. 151–158.CrossRefGoogle Scholar
Darwiche, A. (2021). Causal inference using tractable circuits. In Ranzato, M., editor. Proceedings of the Thirty-Fifth Conference on Neural Information Processing Systems (NeurIPS). Vancouver: Curran Associates, Inc.Google Scholar
Duarte, G., Finkelstein, N., Knox, D., Mummolo, J., & Shpitser, I. (2021). An automated approach to causal inference in discrete settings. Preprint, arXiv:2109.13471v1.Google Scholar
Efron, B. (1978). Controversies in the foundations of statistics. The American Mathematical Monthly, 85(4), 231–246.CrossRefGoogle Scholar
Eiter, T., & Lukasiewicz, T. (2002). Complexity results for structure-based causality. Artificial Intelligence, 142, 53–89.CrossRefGoogle Scholar
Erickson, J., van der Hoog, I., & Miltzow, T. (2020). Smoothing the gap between NP and ∃ℝ . In Irani, S., editor. IEEE 61 st Annual Symposium on Foundations of Computer Science (FOCS). Durham, NC: IEEE, pp. 1022–1033.Google Scholar
Fagin, R., Halpern, J. Y., & Megiddo, N. (1990). A logic for reasoning about probabilities. Information and Computation, 87(1–2), 78–128.CrossRefGoogle Scholar
de Finetti, B. (1937). La prévision: ses lois logiques, ses sources subjectives. Annales de l’Institut Henri Poincaré, 7, 1–68.Google Scholar
Gelman, A., & Shalizi, C. R. (2013). Philosophy and the practice of Bayesian statistics. British Journal of Mathematical and Statistical Psychology, 66, 8–38.CrossRefGoogle ScholarPubMed
Halpern, J., & Vardi, M. (1989). The complexity of reasoning about knowledge and time. Journal of Computer and System Sciences, 38, 195–237.CrossRefGoogle Scholar
Halpern, J. Y. (2000). Axiomatizing causal reasoning. Journal of Artificial Intelligence Research, 12, 317–337.CrossRefGoogle Scholar
Heifetz, A., & Mongin, P. (2001). Probability logic for type spaces. Games and Economic Behavior, 35, 31–53.CrossRefGoogle Scholar
Ibeling, D. (2018). Causal modeling with probabilistic simulation models. In Bellodi E., and Shrijvers, T., editors. Proceedings of the 5 th International Workshop on Probabilistic Logic Programming (PLP). Ferrara: Italian Association for Artificial Intelligence, pp. 36–48.Google Scholar
Ibeling, D., & Icard, T. (2020). Probabilistic reasoning across the causal hierarchy. In Rossi, F., editor. Proceedings of the 34th AAAI Conference on Artificial Intelligence; revised as arXiv:2001.02889v5. Palo Alto, CA: Association for the Advancement of Artificial Intelligence.Google Scholar
Ibeling, D., & Icard, T. (2021). A topological perspective on causal inference. In Ranzato, M., editor. Proceedings of the Thirty-Fifth Conference on Neural Information Processing Systems (NeurIPS). Vancouver: Curran Associates, Inc.Google Scholar
Ibeling, D., Icard, T., Mierzewski, K., & Mossé, M. (2022). Probing the quantitative–qualitative divide in probabilistic reasoning. Unpublished manuscript.Google Scholar
Kurucz, A. (2007). Combining modal logics. In van Benthem, J., Blackburn, P., and Wolter, F., editors. Handbook of Modal Logic. Amsterdam: Elsevier, pp. 869–924.CrossRefGoogle Scholar
Luce, R. D. (1968). On the numerical representation of qualitative conditional probability. The Annals of Mathematical Statistics, 39(2), 481–491.CrossRefGoogle Scholar
Macintyre, A. & Wilkie, A. J. (1995). On the decidability of the real exponential field. In Odifreddi, P., editor. Kreiseliana. About and Around Georg Kreisel. Wellesley, MA: A. K. Peters, pp. 441–467.Google Scholar
Neyman, J. (1977). Frequentist probability and frequentist statistics. Synthese, 36(1), 97–131.CrossRefGoogle Scholar
Ognjanović, Z., Rašković, M., & Marković, Z. (2016). Probability Logics: Probability-Based Formalization of Uncertain Reasoning. Cham: Springer.CrossRefGoogle Scholar
Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82(4), 669–710.CrossRefGoogle Scholar
Pearl, J. (2009). Causality. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Roth, D. (1996). On the hardness of approximate reasoning. Artificial Intelligence, 82(1), 273–302.CrossRefGoogle Scholar
Ruiz-Vanoye, J. A., Pérez-Ortega, J., Pazos R., R. A., Díaz-Parra, O., Frausto-Solís, J., Huacuja, H. J. F., Cruz-Reyes, L., & Martínez F., J. A. (2011). Survey of polynomial transformations between NP-complete problems. Journal of Computational and Applied Mathematics, 235(16), 4851–4865.CrossRefGoogle Scholar
Schachter, R. D. (1988). Probabilistic inference and influence diagrams. Operations Research, 36, 589–605.CrossRefGoogle Scholar
Schaefer, M. (2009). Complexity of some geometric and topological problems. In Eppstein, D., and Gansner, E. R., editors. International Symposium on Graph Drawing. Berlin: Springer, pp. 334–344.Google Scholar
Schaefer, M. (2013). Realizability of graphs and linkages. In Pach, J., editor. Thirty Essays on Geometric Graph Theory. New York: Springer, pp. 461–482.CrossRefGoogle Scholar
Scott, D., & Krauss, P. (1966). Assigning probabilities to logical formulas. Studies in Logic and the Foundations of Mathematics, 43, 219–264.CrossRefGoogle Scholar
Shalev-Shwartz, S., & Ben-David, S. (2014). Understanding Machine Learning: From Theory to Algorithms. New York: Cambridge University Press.CrossRefGoogle Scholar
Speranski, S. O. (2017). Quantifying over events in probability logic: An introduction. Mathematical Structures in Computer Science, 27(8), 1581–1600.CrossRefGoogle Scholar
Spirtes, P., Glymour, C. N., & Scheines, R. (2000). Causation, Prediction, and Search. Cambridge, MA: MIT Press.Google Scholar
Stalnaker, R. C. (1996). Knowledge, belief and counterfactual reasoning in games. Economics and Philosophy, 12, 133–163.CrossRefGoogle Scholar
Suppes, P., & Zanotti, M. (1981). When are probabilistic explanations possible? Synthese, 48, 191–199.CrossRefGoogle Scholar