Abstract
Modal logics for reasoning about interaction in social networks is an active area of research. In this paper we introduce modalities for quantifying over possible “tweets”, i.e., simultaneous messages sent to all an agent’s “followers”, into an existing basic framework for reasoning about this type of network events. Modalities that quantify over informational events in general, and over agent announcements in particular, is also an active area in the study of the dynamics of knowledge and belief. We combine these two directions by interpreting such modalities in social networks. We study the resulting logic, and provide a sound and strongly complete (infinitary) axiomatisation.
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Notes
- 1.
Finitary axiomatisations for both APAL and GAL have been published but later discovered to be unsound [15].
- 2.
We choose to take \([{a}]\) as primary instead of \(\langle {a}\rangle \) only because it makes some proofs simpler.
- 3.
While the axioms schemata are the same, the instances of the axioms are of course different, but validity can be shown in the same way.
- 4.
Strictly speaking the corresponding lemmas in [21] don’t say exactly the same thing since the notion of an MCS is different. However, they can be proven in exactly the same way.
- 5.
For GAL the even more general variant \(\langle G\rangle \) \([{H}]\varphi \rightarrow [{H}]\) \(\langle G\rangle \) \(\varphi \) holds for any sets of agents G, H.
References
Ågotnes, T., Balbiani, P., van Ditmarsch, H., Seban, P.: Group announcement logic. J. Appl. Logic 8(1), 62–81 (2010)
Ågotnes, T., van Ditmarsch, H.: Coalitions and announcements. In: Padgham, L., Parkes, D., Muller, J., Parsons, S. (eds.) Proceedings of AAMAS, pp. 673–680 (2008)
Ågotnes, T., Goranko, V., Jamroga, W., Wooldridge, M.: Knowledge and ability. In: van Ditmarsch, H., Halpern, J.Y., van der Hoek, W., Kooi, B. (eds.) Handbook of Epistemic Logic, pp. 543–589. College Publications (2015)
Ågotnes, T., Walicki, M.: Strongly complete axiomatizations of “Knowing at Most” in syntactic structures. In: Toni, F., Torroni, P. (eds.) CLIMA 2005. LNCS (LNAI), vol. 3900, pp. 57–76. Springer, Heidelberg (2006). https://doi.org/10.1007/11750734_4
Balbiani, P., Baltag, A., van Ditmarsch, H., Herzig, A., Hoshi, T., de Lima, T.: What can we achieve by arbitrary announcements?: A dynamic take on fitch’s knowability. In: Proceedings of TARK, pp. 42–51. ACM (2007)
Balbiani, P., van Ditmarsch, H.: A simple proof of the completeness of APAL. Stud. Logic 8(1), 65–78 (2015)
Baltag, A., Christoff, Z., Rendsvig, R.K., Smets, S.: Dynamic epistemic logic of diffusion and prediction in threshold models. Studia Logica 107(3), 489–531 (2019)
Bozzelli, L., van Ditmarsch, H., French, T., Hales, J., Pinchinat, S.: Refinement modal logic. Inf. Comput. 239, 303–339 (2014)
van Ditmarsch, H., French, T.: Quantifying over boolean announcements. ArXiv abs/1712.05310 (2017)
van Ditmarsch, H., French, T., Pinchinat, S.: Future event logic - axioms and complexity. In: Beklemishev, L.D., Goranko, V., Shehtman, V.B. (eds.) Proceedings of AiML, pp. 77–99. College Publications (2010)
Galimullin, R., Alechina, N.: Coalition and group announcement logic. In: Lang, J. (ed.) Proceedings of TARK, EPTCS, vol. 251, pp. 207–220 (2017)
Goldblatt, R.: Axiomatising the Logic of Computer Programming. Springer, New York (1982)
Hales, J.: Arbitrary action model logic and action model synthesis. In: Proceedings of LICS, pp. 253–262. IEEE Computer Society (2013)
Kooi, B.: Hybrid logics with infinitary proof systems. J. Logic Comput. 16(2), 161–175 (2006)
Kuijer, L.B.: Unsoundness of \(R(\Box )\) (2015). http://personal.us.es/hvd/APAL_counterexample.pdf
Renardel de Lavalette, G.R., Kooi, B.P., Verbrugge, R.: Strong completeness for propositional dynamic logic. In: AiML2002-Advances in Modal Logic, pp. 377–393 (2002)
Morrison, C., Naumov, P.: Group conformity in social networks. J. Logic Lang. Inf. 29(1), 3–19 (2020)
Seligman, J., Liu, F., Girard, P.: Facebook and the epistemic logic of friendship. In: Schipper, B.C. (ed.) Proceedings TARK, pp. 229–238 (2013)
Studer, T.: On the proof theory of the modal mu-calculus. Studia Logica 89(3), 343–363 (2008)
Xiong, Z., Ågotnes, T.: On the logic of balance in social networks. J. Logic Lang. Inf. 29(1), 53–75 (2020)
Xiong, Z., Ågotnes, T., Seligman, J., Zhu, R.: Towards a logic of tweeting. In: Baltag, A., Seligman, J., Yamada, T. (eds.) LORI 2017. LNCS, vol. 10455, pp. 49–64. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-55665-8_4
Acknowledgments
We are indebted to Jeremy Seligman for extensive discussions about the arbitrary tweeting modalities, and in particular the non-compactness observation. The first author is supported by the Project of MOE Liberal arts and Social Sciences Foundation under research no. 20YJC7204002.
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Xiong, Z., Ågotnes, T. (2020). Arbitrary Propositional Network Announcement Logic. In: Martins, M.A., Sedlár, I. (eds) Dynamic Logic. New Trends and Applications. DaLi 2020. Lecture Notes in Computer Science(), vol 12569. Springer, Cham. https://doi.org/10.1007/978-3-030-65840-3_17
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