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Jaakko Hintikka on Knowledge and Game-Theoretical Semantics pp 187–206Cite as

Known Unknowns: Time Bounds and Knowledge of Ignorance

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Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 12))

Abstract

This paper studies the role that known bounds on message transmission times in a computer network play on the evolution of the epistemic state over time. A connection to cones of causal influence analogous to, and more general than, light cones is presented. Focusing on lower bounds on message transmission times, an analysis is presented of how knowledge about when others are guaranteed to be ignorant about an event of interest (“knowing that they don’t know”) can arise. This has implications in competitive settings, in which knowing about another’s ignorance can provide an advantage.

The Happiness of Fish

Chuangtse and Hueitse had strolled on to the bridge over the Hao, when the former observed: “See how the small fish are darting about!              That is the happiness of the fish.” “You not being a fish yourself, said Hueitse,       “how can you know the happiness of the fish?” “And you not being I,” retorted Chuangtse,       “how can you know that I do not know?”

Chuangtse, circa 300 B.C.

A shorter conference version of this paper appeared as [4].

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Notes

  1. 1.

    Our setting can be thought of as consisting of a single inertial system, in which there is a single, non-relativistic, notion of time for all sites.

  2. 2.

    For any given protocol P and context \(\gamma \), it is possible to formally define the notion of a run of P in \(\gamma \) (see Fagin et al. [11]). The system \(\mathcal{R}(P,\gamma )\) consisting of all runs of P in \(\gamma \) is thus unique and well-defined.

  3. 3.

    Consider a message sent from i to j at time t with upper bound \(b_{i\!j}\). At time \(t+b_{i\!j}-1\), if the message is not delivered, then the combined (distributed) knowledge of i and j implies that it will be delivered in the next round. The delivery is thus not spontaneous with respect to the agents’ knowledge.

  4. 4.

    Full-information protocols are very convenient for the study of information flow and of achievable coordination in distributed computer systems [1].

  5. 5.

    Note that the future and past cones describe the fashion in which information is disseminated in the particular run. As in space-time diagrams, a node is a point on the timeline of one of the sites. The cones in space-time do not represent or imply a branching model of time.

  6. 6.

    We assume that messages between agents are never instantaneous. In practice, the systems we set up in [3, 8] are such that minimal transmission time per channel is 1, so that \(\mathsf {\Box unaff}(\theta |\mathsf {Net})\) gets a richer structure, as described below for the context \(\gamma ^{\mathsf{b}}\).

  7. 7.

    In analogy to the definition of the \(D_{ij}\) values, \(d_{ij}\) is defined as the shortest distance between i and j in the min-weighted network graph.

  8. 8.

    This observation is made in hindsight, as \(\mathsf {\Box aff}(\theta |\mathsf {Net})\) is defined here for the first time.

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Acknowledgements

Yoram Moses is the Israel Pollak Academic Chair at the Technion. We thank Asa Dan for useful discussions that improved the presentation of the paper. This work was supported in part by ISF grant 1520/11.

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Authors and Affiliations

  1. Haifa, Israel

    Ido Ben-Zvi

  2. Technion, Haifa, Israel

    Yoram Moses

Authors
  1. Ido Ben-Zvi
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  2. Yoram Moses
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Correspondence to Yoram Moses .

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Editors and Affiliations

  1. LORIA, CNRS, Université de Lorraine, Vandoeuvre-lès-Nancy, France

    Hans van Ditmarsch

  2. Department of Philosophy, University of Helsinki, Helsinki, Finland

    Gabriel Sandu

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Ben-Zvi, I., Moses, Y. (2018). Known Unknowns: Time Bounds and Knowledge of Ignorance. In: van Ditmarsch, H., Sandu, G. (eds) Jaakko Hintikka on Knowledge and Game-Theoretical Semantics. Outstanding Contributions to Logic, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-62864-6_7

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