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Probabilistic Causes in Markov Chains

Part of the Lecture Notes in Computer Science book series (LNPSE,volume 12971)

Abstract

The paper studies a probabilistic notion of causes in Markov chains that relies on the counterfactuality principle and the probability-raising property. This notion is motivated by the use of causes for monitoring purposes where the aim is to detect faulty or undesired behaviours before they actually occur. A cause is a set of finite executions of the system after which the probability of the effect exceeds a given threshold. We introduce multiple types of costs that capture the consump-tion of resources from different perspectives, and study the complexity of computing cost-minimal causes.

This work was funded by DFG grant 389792660 as part of TRR 248, the Cluster of Excellence EXC 2050/1 (CeTI, project ID 390696704, as part of Germany’s Excellence Strategy), DFG-projects BA-1679/11-1 and BA-1679/12-1, and the Research Training Group QuantLA (GRK 1763).

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References

  1. CP-logic: A language of causal probabilistic events and its relation to logic programming 9

    Google Scholar 

  2. Baier, C., Katoen, J.P.: Principles of Model Checking (Representation and Mind Series). The MIT Press, Cambridge (2008)

    MATH  Google Scholar 

  3. Bartocci, E., et al.: Adaptive runtime verification. In: Qadeer, S., Tasiran, S. (eds.) RV 2012. LNCS, vol. 7687, pp. 168–182. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-35632-2_18

    CrossRef  Google Scholar 

  4. Beer, I., Ben-David, S., Chockler, H., Orni, A., Trefler, R.: Explaining counterexamples using causality. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 94–108. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02658-4_11

    CrossRef  Google Scholar 

  5. Bertsekas, D.P., Tsitsiklis, J.N.: An analysis of stochastic shortest path problems, 16(3), 580–595 (1991)

    Google Scholar 

  6. Braham, M., van Hees, M.: An anatomy of moral responsibility. Mind 121(483), 601–634 (2012)

    CrossRef  Google Scholar 

  7. Brihaye, T., Geeraerts, G., Haddad, A., Monmege, B.: To reach or not to reach? efficient algorithms for total-payoff games. In: Proceedings of the 26th International Conference on Concurrency Theory (CONCUR’15). LIPIcs, vol. 42, pp. 297–310 (2015)

    Google Scholar 

  8. Chadha, R., Sistla, A.P., Viswanathan, M.: On the expressiveness and complexity of randomization in finite state monitors, 56(5) (2009)

    Google Scholar 

  9. Chatterjee, K., Doyen, L., Henzinger, T.A.: The cost of exactness in quantitative reachability. In: Aceto, L., Bacci, G., Bacci, G., Ingólfsdóttir, A., Legay, A., Mardare, R. (eds.) Models, Algorithms, Logics and Tools. LNCS, vol. 10460, pp. 367–381. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63121-9_18

    CrossRef  Google Scholar 

  10. Chockler, H., Halpern, J.Y.: Responsibility and blame: a structural-model approach. J. Artif. Int. Res. 22(1), 93–115 (2004)

    MathSciNet  MATH  Google Scholar 

  11. Chockler, H., Halpern, J.Y., Kupferman, O.: What causes a system to satisfy a specification? ACM Trans. Comput. Logic 9(3), 20:1–20:26 (2008)

    Google Scholar 

  12. Cini, C., Francalanza, A.: An LTL proof system for runtime verification. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 581–595. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46681-0_54

    CrossRef  MATH  Google Scholar 

  13. Daca, P., Henzinger, T.A., Křetínský, J., Petrov, T.: Faster statistical model checking for unbounded temporal properties. In: Chechik, M., Raskin, J.-F. (eds.) TACAS 2016. LNCS, vol. 9636, pp. 112–129. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49674-9_7

    CrossRef  MATH  Google Scholar 

  14. Dash, D., Voortman, M., De Jongh, M.: Sequences of mechanisms for causal reasoning in artificial intelligence. In: Proceedings of the 23rd International Joint Conference on Artificial Intelligence, IJCAI ’13, pp. 839–845. AAAI Press (2013)

    Google Scholar 

  15. Eells, E.: Probabilistic Causality. Cambridge Studies in Probability, Induction and Decision Theory. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  16. Eiter, T., Lukasiewicz, T.: Complexity results for explanations in the structural-model approach. Artif. Intell. 154(1–2), 145–198 (2004)

    MathSciNet  CrossRef  Google Scholar 

  17. Eiter, T., Lukasiewicz, T.: Causes and explanations in the structural-model approach: tractable cases. Artif. Intell. 170(6–7), 542–580 (2006)

    MathSciNet  CrossRef  Google Scholar 

  18. Esparza, J., Kiefer, S., Kretinsky, J., Weininger, M.: Online monitoring \(\omega \)-regular properties in unknown Markov chains. Arxiv preprint, arXiv:2010.08347 (2020)

  19. Faran, R., Kupferman, O.: Spanning the spectrum from safety to liveness. Acta Informatica 55(8), 703–732 (2018). https://doi.org/10.1007/s00236-017-0307-4

    MathSciNet  CrossRef  MATH  Google Scholar 

  20. Feigenbaum, J., Hendler, J.A., Jaggard, A.D., Weitzner, D.J., Wright, R.N.: Accountability and deterrence in online life. ACM, New York (2011)

    Google Scholar 

  21. Fenton-Glynn, L.: A proposed probabilistic extension of the halpern and pearl definition of ‘actual cause’. Br. J. Philos. Sci. 68(4), 1061–1124 (2016)

    MathSciNet  CrossRef  Google Scholar 

  22. Gill, J.: Computational complexity of probabilistic turing machines. SIAM J. Comput. 6(4), 675–695 (1977)

    MathSciNet  CrossRef  Google Scholar 

  23. Gondi, K., Patel, Y., Sistla, A.P.: Monitoring the full range of w-regular properties of stochastic systems. In: Jones, N.D., Müller-Olm, M. (eds.) VMCAI 2009. LNCS, vol. 5403, pp. 105–119. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-93900-9_12

    CrossRef  Google Scholar 

  24. Haase, C., Kiefer, S.: The odds of staying on budget. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 234–246. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47666-6_19

    CrossRef  Google Scholar 

  25. Haase, C., Kiefer, S.: The complexity of the kth largest subset problem and related problems, 116(2) (2016)

    Google Scholar 

  26. Halpern, J.Y.: A modification of the Halpern-Pearl definition of causality. In: Proceedings of IJCAI’15, pp. 3022–3033. AAAI Press (2015)

    Google Scholar 

  27. Halpern, J.Y., Pearl, J.: Causes and explanations: a structural-model approach: part 1: causes. In: Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence (UAI), pp. 194–202 (2001)

    Google Scholar 

  28. Huang, Y., Kleinberg, S.: Fast and accurate causal inference from time series data. In: Proceedings of FLAIRS 2015, pp. 49–54. AAAI Press (2015)

    Google Scholar 

  29. Ibrahim, A., Pretschner, A.: From checking to inference: actual causality computations as optimization problems. In: Hung, D.V., Sokolsky, O. (eds.) ATVA 2020. LNCS, vol. 12302, pp. 343–359. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-59152-6_19

    CrossRef  Google Scholar 

  30. Ibrahim, A., Pretschner, A., Klesel, T., Zibaei, E., Kacianka, S., Pretschner, A.: Actual causality canvas: a general framework for explanation-based socio-technical constructs. In: Proceedings of ECAI’20, pp. 2978–2985. IOS Press Ebooks (2020)

    Google Scholar 

  31. Kalajdzic, K., Bartocci, E., Smolka, S.A., Stoller, S.D., Grosu, R.: Runtime verification with particle filtering. In: Legay, A., Bensalem, S. (eds.) RV 2013. LNCS, vol. 8174, pp. 149–166. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40787-1_9

    CrossRef  Google Scholar 

  32. Kleinberg, S.: A logic for causal inference in time series with discrete and continuous variables. In: Proceedings of IJCAI’11, pp. 943–950 (2011)

    Google Scholar 

  33. Kleinberg, S., Hripcsak, G.: A review of causal inference for biomedical informatics. J. Biomed. Inform. 44(6), 1102–12 (2011)

    CrossRef  Google Scholar 

  34. Kleinberg, S., Mishra, B.: The temporal logic of causal structures. In: Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence (UAI), pp. 303–312 (2009)

    Google Scholar 

  35. Kleinberg, S., Mishra, B.: The temporal logic of token causes. In: Proceedings of KR’10, pp. 575–577. AAAI Press (2010)

    Google Scholar 

  36. Miller, T.: Explanation in artificial intelligence: insights from the social sciences. Artif. Intell. 267, 1–38 (2017)

    MathSciNet  CrossRef  Google Scholar 

  37. Pearl, J.: Causality, 2nd edn. Cambridge University Press, Cambridge (2009)

    CrossRef  Google Scholar 

  38. Piribauer, J., Baier, C.: Partial and conditional expectations in Markov decision processes with integer weights. In: Bojańczyk, M., Simpson, A. (eds.) FoSSaCS 2019. LNCS, vol. 11425, pp. 436–452. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17127-8_25

    CrossRef  MATH  Google Scholar 

  39. Reichenbach, H.: The Direction of Time. Dover Publications, Mineola (1956)

    CrossRef  Google Scholar 

  40. Sistla, A.P., Srinivas, A.R.: Monitoring temporal properties of stochastic systems. In: Logozzo, F., Peled, D.A., Zuck, L.D. (eds.) VMCAI 2008. LNCS, vol. 4905, pp. 294–308. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78163-9_25

    CrossRef  Google Scholar 

  41. Stoller, S.D.: Runtime verification with state estimation. In: Khurshid, S., Sen, K. (eds.) RV 2011. LNCS, vol. 7186, pp. 193–207. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29860-8_15

    CrossRef  Google Scholar 

  42. Toda, S.: PP is as hard as the polynomial-time hierarchy, 20, 865–877 (1991)

    Google Scholar 

  43. Vennekens, J., Bruynooghe, M., Denecker, M.: Embracing events in causal modelling: interventions and counterfactuals in CP-logic. In: Janhunen, T., Niemelä, I. (eds.) JELIA 2010. LNCS (LNAI), vol. 6341, pp. 313–325. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15675-5_27

    CrossRef  MATH  Google Scholar 

  44. Zheng, M., Kleinberg, S.: A method for automating token causal explanation and discovery. In: Proceedings of FLAIRS’17 (2017)

    Google Scholar 

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Baier, C., Funke, F., Jantsch, S., Piribauer, J., Ziemek, R. (2021). Probabilistic Causes in Markov Chains. In: Hou, Z., Ganesh, V. (eds) Automated Technology for Verification and Analysis. ATVA 2021. Lecture Notes in Computer Science(), vol 12971. Springer, Cham. https://doi.org/10.1007/978-3-030-88885-5_14

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  • DOI: https://doi.org/10.1007/978-3-030-88885-5_14

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