Advertisement

A Dynamic Epistemic Logic Analysis of the Equality Negation Task

  • Éric Goubault
  • Marijana Lazić
  • Jérémy LedentEmail author
  • Sergio Rajsbaum
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12005)

Abstract

In this paper we study the solvability of the equality negation task in a simple wait-free model where processes communicate by reading and writing shared variables or exchanging messages. In this task, two processes start with a private input value in the set \(\left\{ 0,1,2 \right\} \), and after communicating, each one must decide a binary output value, so that the outputs of the processes are the same if and only if the input values of the processes are different. This task is already known to be unsolvable; our goal here is to prove this result using the dynamic epistemic logic (DEL) approach introduced by Goubault, Ledent and Rajsbaum in GandALF 2018. We show that in fact, there is no epistemic logic formula that explains why the task is unsolvable. We fix this issue by extending the language of our DEL framework, which allows us to construct such a formula, and discuss its utility.

Keywords

Dynamic epistemic logic Distributed computing Equality negation 

Notes

Acknowledgements

The authors were supported by DGA project “Validation of Autonomous Drones and Swarms of Drones” and the academic chair “Complex Systems Engineering” of Ecole Polytechnique-ENSTA-Télécom-Thalès-Dassault-Naval Group-DGA-FX-FDO-Fondation ParisTech, by the UNAM-PAPIIT project IN109917 and IN106520, by the France-Mexico Binational SEP-CONACYT-ANUIES-ECOS grant M12M01, by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 787367 (PaVeS), as well as by the Austrian Science Fund (FWF) through Doctoral College LogiCS (W1255-N23).

References

  1. 1.
    Attiya, H., Welch, J.: Distributed Computing: Fundamentals, Simulations and Advanced Topics. Wiley, New York (2004)CrossRefGoogle Scholar
  2. 2.
    Baltag, A., Moss, L., Solecki, S.: The logic of common knowledge, public announcements, and private suspicions. In: TARK VII, pp. 43–56 (1998).  https://doi.org/10.1007/978-3-319-20451-2_38Google Scholar
  3. 3.
    Baltag, A., Renne, B.: Dynamic epistemic logic. In: The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University (2016). https://plato.stanford.edu/archives/win2016/entries/dynamic-epistemic/
  4. 4.
    Biran, O., Moran, S., Zaks, S.: A combinatorial characterization of the distributed 1-solvable tasks. J. Algorithms 11(3), 420–440 (1990).  https://doi.org/10.1016/0196-6774(90)90020-FMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chor, B., Israeli, A., Li, M.: On processor coordination using asynchronous hardware. In: Proceedings of the Sixth Annual ACM Symposium on Principles of Distributed Computing, PODC 1987, pp. 86–97. ACM, New York (1987).  https://doi.org/10.1145/41840.41848
  6. 6.
    Ditmarsch, H.V., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-1-4020-5839-4CrossRefzbMATHGoogle Scholar
  7. 7.
    Fischer, M.J., Lynch, N.A., Paterson, M.: Impossibility of distributed consensus with one faulty process. J. ACM 32(2), 374–382 (1985).  https://doi.org/10.1145/3149.214121MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gafni, E., Koutsoupias, E.: Three-processor tasks are undecidable. SIAM J. Comput. 28(3), 970–983 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Goubault, E., Lazić, M., Ledent, J., Rajsbaum, S.: A dynamic epistemic logic analysis of the equality negation task. CoRR abs/1909.03263 (2019). http://arxiv.org/abs/1909.03263
  10. 10.
    Goubault, E., Lazić, M., Ledent, J., Rajsbaum, S.: Wait-free solvability of equality negation tasks. In: 33rd International Symposium on Distributed Computing, DISC 2019, 14–18 October 2019, Budapest, Hungary (2019). https://drops.dagstuhl.de/opus/volltexte/2019/11328/
  11. 11.
    Goubault, É., Ledent, J., Rajsbaum, S.: A simplicial complex model for dynamic epistemic logic to study distributed task computability. In: Proceedings Ninth International Symposium on Games, Automata, Logics, and Formal Verification, GandALF 2018, Saarbrücken, Germany, 26–28th September 2018, pp. 73–87 (2018).  https://doi.org/10.4204/EPTCS.277.6MathSciNetCrossRefGoogle Scholar
  12. 12.
    Herlihy, M., Kozlov, D., Rajsbaum, S.: Distributed Computing Through Combinatorial Topology. Elsevier-Morgan Kaufmann, Amsterdam (2013).  https://doi.org/10.1016/C2011-0-07032-1CrossRefzbMATHGoogle Scholar
  13. 13.
    Herlihy, M.: Wait-free synchronization. ACM Trans. Program. Lang. Syst. 13(1), 124–149 (1991).  https://doi.org/10.1145/114005.102808CrossRefGoogle Scholar
  14. 14.
    Herlihy, M., Rajsbaum, S.: The decidability of distributed decision tasks (extended abstract). In: Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing (STOC), El Paso, Texas, USA, 4–6 May 1997, pp. 589–598 (1997).  https://doi.org/10.1145/258533.258652
  15. 15.
    Herlihy, M., Shavit, N.: The topological structure of asynchronous computability. J. ACM 46(6), 858–923 (1999).  https://doi.org/10.1145/331524.331529MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jayanti, P.: On the robustness of herlihy’s hierarchy. In: Proceedings of the Twelfth Annual ACM Symposium on Principles of Distributed Computing, PODC 1993, pp. 145–157. ACM, New York (1993).  https://doi.org/10.1145/164051.164070
  17. 17.
    Kozlov, D.: Combinatorial Algebraic Topology. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-71962-5CrossRefzbMATHGoogle Scholar
  18. 18.
    Lo, W., Hadzilacos, V.: All of us are smarter than any of us: nondeterministic wait-free hierarchies are not robust. SIAM J. Comput. 30(3), 689–728 (2000).  https://doi.org/10.1137/S0097539798335766MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Loui, M.C., Abu-Amara, H.H.: Memory requirements for agreement among unreliable asynchronous processes. In: Advances in Computing research, pp. 163–183. JAI Press, Greenwich (1987)Google Scholar
  20. 20.
    Lynch, N.A.: Distributed Algorithms. Morgan Kaufmann Publishers Inc., San Francisco (1996)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Éric Goubault
    • 1
  • Marijana Lazić
    • 2
  • Jérémy Ledent
    • 4
    Email author
  • Sergio Rajsbaum
    • 3
  1. 1.LIX, CNRS, École Polytechnique, Institut Polytechnique de ParisPalaiseauFrance
  2. 2.TU MünchenMunichGermany
  3. 3.Instituto de MatemáticasUNAMMexico CityMexico
  4. 4.University of StrathclydeGlasgowUK

Personalised recommendations