A Simplicial Model for KB4_n: Epistemic Logic with Agents That May Die

Authors Éric Goubault , Jérémy Ledent , Sergio Rajsbaum

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Author Details

Éric Goubault
  • LIX, CNRS, École Polytechnique, Institut Polytechnique de Paris, France
Jérémy Ledent
  • MSP Group, University of Strathclyde, Glasgow, Scotland
Sergio Rajsbaum
  • National Autonomous University of Mexico, Mexico City, Mexico

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Éric Goubault, Jérémy Ledent, and Sergio Rajsbaum. A Simplicial Model for KB4_n: Epistemic Logic with Agents That May Die. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


The standard semantics of multi-agent epistemic logic S5_n is based on Kripke models whose accessibility relations are reflexive, symmetric and transitive. This one dimensional structure contains implicit higher-dimensional information beyond pairwise interactions, that we formalized as pure simplicial models in a previous work in Information and Computation 2021 [Éric Goubault et al., 2021]. Here we extend the theory to encompass simplicial models that are not necessarily pure. The corresponding class of Kripke models are those where the accessibility relation is symmetric and transitive, but might not be reflexive. Such models correspond to the epistemic logic KB4_n. Impure simplicial models arise in situations where two possible worlds may not have the same set of agents. We illustrate it with distributed computing examples of synchronous systems where processes may crash.

Subject Classification

ACM Subject Classification
  • Theory of computation → Modal and temporal logics
  • Epistemic logic
  • Simplicial complexes
  • Distributed computing


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  1. Federico Battiston, Giulia Cencetti, Iacopo Iacopini, Vito Latora, Maxime Lucas, Alice Patania, Jean-Gabriel Young, and Giovanni Petri. Networks beyond pairwise interactions: Structure and dynamics. Physics Reports, 874:1-92, 2020. Networks beyond pairwise interactions: Structure and dynamics. URL: https://doi.org/10.1016/j.physrep.2020.05.004.
  2. Elizabeth Borowsky and Eli Gafni. Generalized FLP impossibility result for t-resilient asynchronous computations. In S. Rao Kosaraju, David S. Johnson, and Alok Aggarwal, editors, Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, May 16-18, 1993, San Diego, CA, USA, pages 91-100. ACM, 1993. URL: https://doi.org/10.1145/167088.167119.
  3. Armando Castañeda, Pierre Fraigniaud, Ami Paz, Sergio Rajsbaum, Matthieu Roy, and Corentin Travers. Synchronous t-resilient consensus in arbitrary graphs. In Mohsen Ghaffari, Mikhail Nesterenko, Sébastien Tixeuil, Sara Tucci, and Yukiko Yamauchi, editors, Stabilization, Safety, and Security of Distributed Systems - 21st International Symposium, SSS 2019, Pisa, Italy, October 22-25, 2019, Proceedings, volume 11914 of Lecture Notes in Computer Science, pages 53-68. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-34992-9_5.
  4. Hans van Ditmarsch, Éric Goubault, Jérémy Ledent, and Sergio Rajsbaum. Knowledge and simplicial complexes. CoRR, abs/2002.08863, 2020. To appear in Philosophy of Computing- Themes from IACAP 2019, Eds.: Björn Lundgren, and Nancy Abigail Nuñez Hernández. URL: http://arxiv.org/abs/2002.08863.
  5. Cynthia Dwork and Yoram Moses. Knowledge and common knowledge in a byzantine environment: Crash failures. Inf. Comput., 88(2):156-186, 1990. URL: https://doi.org/10.1016/0890-5401(90)90014-9.
  6. Ronald Fagin, Joseph Y. Halpern, Yoram Moses, and Moshe Y. Vardi. Reasoning About Knowledge. MIT Press, Cambridge, MA, USA, 2003. Google Scholar
  7. Michael J. Fischer and Nancy A. Lynch. A lower bound for the time to assure interactive consistency. Information Processing Letters, 14(4):183-186, 1982. URL: https://doi.org/10.1016/0020-0190(82)90033-3.
  8. James Garson. Modal Logic. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Summer 2021 edition, 2021. Google Scholar
  9. Éric Goubault, Marijana Lazic, Jérémy Ledent, and Sergio Rajsbaum. Wait-free solvability of equality negation tasks. In Jukka Suomela, editor, 33rd International Symposium on Distributed Computing, DISC 2019, October 14-18, 2019, Budapest, Hungary, volume 146 of LIPIcs, pages 21:1-21:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.DISC.2019.21.
  10. Éric Goubault, Jérémy Ledent, and Sergio Rajsbaum. A simplicial complex model for dynamic epistemic logic to study distributed task computability. Inf. Comput., 278:104597, 2021. URL: https://doi.org/10.1016/j.ic.2020.104597.
  11. Joseph Y. Halpern and Yoram Moses. Knowledge and common knowledge in a distributed environment. J. ACM, 37(3):549-587, 1990. URL: https://doi.org/10.1145/79147.79161.
  12. M. Herlihy, D. Kozlov, and S. Rajsbaum. Distributed Computing Through Combinatorial Topology. Morgan Kaufmann, San Francisco, CA, USA, 2013. Google Scholar
  13. M. Herlihy and N. Shavit. The topological structure of asynchronous computability. J. ACM, 46(6):858-923, November 1999. URL: https://doi.org/10.1145/331524.331529.
  14. Maurice Herlihy. Wait-free synchronization. ACM Trans. Program. Lang. Syst., 13(1):124-149, January 1991. URL: https://doi.org/10.1145/114005.102808.
  15. Maurice Herlihy, Sergio Rajsbaum, and Mark R. Tuttle. An overview of synchronous message-passing and topology. Electronic Notes in Theoretical Computer Science, 39(2):1-17, 2000. URL: https://doi.org/10.1016/S1571-0661(05)01148-5.
  16. Dmitry N. Kozlov. Combinatorial Algebraic Topology, volume 21 of Algorithms and computation in mathematics. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-71962-5.
  17. Nancy A. Lynch. Distributed Algorithms. Morgan Kaufmann, 1996. Google Scholar
  18. John C. Mitchell and Eugenio Moggi. Kripke-style models for typed lambda calculus. Annals of Pure and Applied Logic, 51:99-124, 1996. Google Scholar
  19. Andrea Mock and Ismar Volic. Political structures and the topology of simplicial complexes, 2021. Google Scholar
  20. Yoram Moses. Knowledge in Distributed Systems, pages 1051-1055. Springer New York, New York, NY, 2016. URL: https://doi.org/10.1007/978-1-4939-2864-4_606.
  21. Michael Okun. Strong order-preserving renaming in the synchronous message passing model. Theor. Comput. Sci., 411(40–42):3787-3794, September 2010. URL: https://doi.org/10.1016/j.tcs.2010.06.001.
  22. Y. Moses R. Fagin, J. Halpern and M. Vardi. Reasoning About Knowledge. MIT Press, 1995. Google Scholar
  23. Michael E. Saks and Fotios Zaharoglou. Wait-free k-set agreement is impossible: The topology of public knowledge. SIAM J. Comput., 29(5):1449-1483, 2000. URL: https://doi.org/10.1137/S0097539796307698.
  24. Hans van Ditmarsch. Wanted dead or alive: Epistemic logic for impure simplicial complexes. In Alexandra Silva, Renata Wassermann, and Ruy J. G. B. de Queiroz, editors, Logic, Language, Information, and Computation - 27th International Workshop, WoLLIC 2021, Virtual Event, October 5-8, 2021, Proceedings, volume 13038 of Lecture Notes in Computer Science, pages 31-46. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-88853-4_3.
  25. Hans van Ditmarsch, Éric Goubault, Marijana Lazic, Jérémy Ledent, and Sergio Rajsbaum. A dynamic epistemic logic analysis of equality negation and other epistemic covering tasks. J. Log. Algebraic Methods Program., 121:100662, 2021. URL: https://doi.org/10.1016/j.jlamp.2021.100662.
  26. Frans Voorbraak. Generalized kripke models for epistemic logic. In Proceedings of the Fourth Conference on Theoretical Aspects of Reasoning about Knowledge, TARK '92, pages 214-228, San Francisco, CA, USA, 1992. Morgan Kaufmann Publishers Inc. Google Scholar
  27. Koki Yagi and Susumu Nishimura. Logical obstruction to set agreement tasks for superset-closed adversaries, 2021. Google Scholar
  28. Xiong Zheng and Vijay K. Garg. Byzantine lattice agreement in synchronous message passing systems. In Hagit Attiya, editor, 34th International Symposium on Distributed Computing, DISC 2020, October 12-16, 2020, Virtual Conference, volume 179 of LIPIcs, pages 32:1-32:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.DISC.2020.32.
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