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How to Share Knowledge by Gossiping

  • Andreas Herzig
  • Faustine Maffre
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9571)

Abstract

Given n agents each of which has a secret (a fact not known to anybody else), the classical version of the gossip problem is to achieve shared knowledge of all secrets in a minimal number of phone calls. There exist protocols achieving shared knowledge in \(2(n{-}2)\) calls: when the protocol terminates everybody knows all the secrets. We generalize that problem and focus on higher-order shared knowledge: how many calls does it take to obtain that everybody knows that everybody knows all secrets? More generally, how many calls does it take to obtain shared knowledge of order k? This requires not only the communication of secrets, but also the communication of knowledge about secrets. We give a protocol that works in \((k{+}1)(n{-}2)\) steps and prove that it is correct: it achieves shared knowledge of level k. The proof is presented in a dynamic epistemic logic that is based on the observability of propositional variables by agents.

Keywords

Gossip Epistemic logic Shared knowledge Common knowledge 

Notes

Acknowledgements

We would like to acknowledge several discussions about the gossip problem at the inspiring August 2015 workshop “To be announced” in Leiden, in particular with Hans van Ditmarsch, Jan van Eijck, Malvin Gattinger, Louwe Kuijer, Christian Muise, Pere Pardo, Rahim Ramezanian and Francois Schwarzentruber. We are also grateful to Davide Grossi, Emiliano Lorini and Martin Cooper.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Andreas Herzig
    • 1
  • Faustine Maffre
    • 1
  1. 1.IRIT, Université de ToulouseToulouse Cedex 9France

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