Abstract
We present a proof-theoretical and model-theoretical approach to reasoning about knowledge and conditional probability. We extend both the language of epistemic logic and the language of linear weight formulas, allowing statements like “Agent Ag knows that the probability of A given B is at least a half”. We axiomatize this logic, provide corresponding semantics and prove that the axiomatization is sound and strongly complete. We also show that the logic is decidable.
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References
Alechina, N.: Logic with probabilistic operators. In: Proceedings of the ACCOLADE 1994, pp. 121–138 (1995)
Doder, D., Marinković, B., Maksimović, P., Perović, A.: A logic with conditional probability operators. Publications de L’Institut Mathematique Ns 87(101), 85–96 (2010)
Doder, D., Ognjanović, Z.: Probabilistic logics with independence and confirmation. Studia Logica 105(5), 943–969 (2017)
Doder, D., Ognjanovic, Z.: A probabilistic logic for reasoning about uncertain temporal information. In: Meila, M., Heskes, T. (eds.) Proceedings of the Thirty-First Conference on Uncertainty in Artificial Intelligence, UAI 2015, Amsterdam, The Netherlands, pp. 248–257. AUAI Press (2015)
Fagin, R., Halpern, J.Y.: Reasoning about knowledge and probability. J. ACM 41(2), 340–367 (1994)
Fagin, R., Halpern, J.Y., Megiddo, N.: A logic for reasoning about probabilities. Inf. Comput. 87, 78–128 (1990)
Fagin, R., Geanakoplos, J., Halpern, J.Y., Vardi, M.Y.: The hierarchical approach to modeling knowledge and common knowledge. Int. J. Game Theory 28(3), 331–365 (1999)
Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. MIT Press, Cambridge, MA (2003)
Frisch, A., Haddawy, P.: Anytime deduction for probabilistic logic. Artif. Intell. 69, 93–122 (1994)
Halpern, J.Y., Pucella, R.: A logic for reasoning about evidence. J. Artif. Intell. Res. 26, 1–34 (2006)
van der Hoek, W.: Some considerations on the logic pfd. J. Appl. Non-Classical Logics 7(3) (1997)
Hughes, G.E., Cresswell, M.J.: A Companion to Modal Logic. Methuen London, New York (1984)
de Lavalette, G.R.R., Kooi, B., Verbrugge, R.: A strongly complete proof system for propositional dynamic logic. In: AiML2002–Advances in Modal Logic (Conference Proceedings), pp. 377–393 (2002)
Marchioni, E., Godo, L.: A logic for reasoning about coherent conditional probability: a modal fuzzy logic approach. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 213–225. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30227-8_20
Marinkovic, B., Glavan, P., Ognjanovic, Z., Studer, T.: A temporal epistemic logic with a non-rigid set of agents for analyzing the blockchain protocol. J. Logic Comput. (2019)
Marinkovic, B., Ognjanovic, Z., Doder, D., Perovic, A.: A propositional linear time logic with time flow isomorphic to \(\omega \)\({}^{\text{2 }}\). J. Appl. Log. 12(2), 208–229 (2014)
Milošević, M., Ognjanović, Z.: A first-order conditional probability logic. Logic J. IGPL 20(1), 235–253 (2012)
Ognjanovic, Z., Markovic, Z., Raskovic, M., Doder, D., Perovic, A.: A propositional probabilistic logic with discrete linear time for reasoning about evidence. Ann. Math. Artif. Intell. 65(2–3), 217–243 (2012)
Ognjanović, Z., Rašković, M., Marković, Z.: Probability Logics: Probability-based Formalization of Uncertain Reasoning. Springer, New York (2016). https://doi.org/10.1007/978-3-319-47012-2
Rašković, M., Ognjanović, Z., Marković, Z.: A logic with conditional probabilities. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 226–238. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30227-8_21
Savic, N., Doder, D., Ognjanovic, Z.: Logics with lower and upper probability operators. Int. J. Approx. Reason. 88, 148–168 (2017)
Tomović, S., Ognjanović, Z., Doder, D.: Probabilistic common knowledge among infinite number of agents. In: Destercke, S., Denoeux, T. (eds.) ECSQARU 2015. LNCS (LNAI), vol. 9161, pp. 496–505. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-20807-7_45
Tomovic, S., Ognjanovic, Z., Doder, D.: A first-order logic for reasoning about knowledge and probability. ACM Trans. Comput. Log. 21(2), 16:1–16:30 (2020)
Wolter, F.: First order common knowledge logics. Studia Logica 65(2), 249–271 (2000)
Acknowledgments
This work has been partially funded by the Science Fund of the Republic of Serbia through the project Advanced artificial intelligence techniques for analysis and design of system components based on trustworthy BlockChain technology - AI4TrustBC (the first and the third author).
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Dautović, Š., Doder, D., Ognjanović, Z. (2021). An Epistemic Probabilistic Logic with Conditional Probabilities. In: Faber, W., Friedrich, G., Gebser, M., Morak, M. (eds) Logics in Artificial Intelligence. JELIA 2021. Lecture Notes in Computer Science(), vol 12678. Springer, Cham. https://doi.org/10.1007/978-3-030-75775-5_19
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