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Dynamic Logics of Imperfect Information: From Teams and Games to Transitions

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Jaakko Hintikka on Knowledge and Game-Theoretical Semantics

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

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Abstract

We introduce a new semantical formalism for logics of imperfect information, based on Game Logic (and, in particular, on van Benthem, Ghosh and Lu’s Concurrent Dynamic Game Logic). This new kind of semantics combines aspects from game theoretic semantics and from team semantics, and demonstrates how logics of imperfect information can be seen as languages for reasoning about games. Finally we show that, for a very expressive fragment of our language, a simpler semantics is available.

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Notes

  1. 1.

    In this work, we will not describe in much detail the definitions of the logics of imperfect information nor the motivations which led to their development. The interested reader who is not familiar with this field of research is referred to [28, 22] for a thorough introduction to the area.

  2. 2.

    There are exceptions, however: for example, [25, 6] present two, not entirely equivalent, varieties of game-theoretic semantics based on the concept of Nash Equilibria for undetermined games.

  3. 3.

    Team semantics can also be found mentioned under the names of Hodges semantics and of trump semantics.

  4. 4.

    Strictly speaking, this is the case only with respect to sentences. With respect to open formulas, this is true only if the domain of the team is presumed finite and fixed.

  5. 5.

    We will not attempt to give here a summary of these results; apart from [27], the interested reader could refer for example to [17,18,19,20, 4]. But this brief list far from complete.

  6. 6.

    [8] contains a few basic results along these lines, as does [7].

  7. 7.

    Or, simply, a variable.

  8. 8.

    That \(R(G) = (G^d)^d\) would then follow at once from the fact that, in our formalism, the second player — representing the environment — has no knowledge restrictions.

  9. 9.

    Or, to be more formal, if and only if there exists an unused constant symbol c and an element \(m \in \texttt {Dom}(M)\) such that \(X \in \Vert \phi [c/p]\Vert _{M(c \mapsto m)}\).

  10. 10.

    In other words, in \(\langle \gamma \rangle \phi \) the belief formula \(\phi \) specifies a winning condition for the game formula \(\gamma \).

  11. 11.

    It is not difficult to see that the transition semantics for this connective would be: \(M \models _{X \rightarrow Y} \gamma _1 \cap \gamma _2\) if and only if there exist \(Y_1\) and \(Y_2\) such that \(Y_1 \cup Y_2 = Y\), \(M \models _{X \rightarrow Y_1} \gamma _1\) and \(M \models _{X \rightarrow Y_2} \gamma _2\).

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Acknowledgements

The author wishes to thank Jouko Väänänen for a number of useful suggestions and comments about previous versions of this work. Furthermore, the author thankfully acknowledges the support of the EUROCORES LogICCC LINT programme.

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Authors and Affiliations

  1. Free University of Bozen-Bolzano, Bolzano, Italy

    Pietro Galliani

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  1. Pietro Galliani
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Correspondence to Pietro Galliani .

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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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Galliani, P. (2018). Dynamic Logics of Imperfect Information: From Teams and Games to Transitions. 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_12

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