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An Extension of the Ehrenfeucht-Fraïssé Game for First Order Logics Augmented with Lindström Quantifiers

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Fields of Logic and Computation II

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9300))

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Abstract

We propose an extension of the Ehrenfeucht-Fraïssé game able to deal with logics augmented with Lindström quantifiers. We describe three different games with varying balance between simplicity and ease of use.

Dedicated to Yuri Gurevich on the occasion of his \( {75}{th}\) birthday.

Research partially supported by National Science Foundation grant no. DMS 1101597. Research partially supported by European Research Council grant 338821. Number 1059 on author’s list.

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Notes

  1. 1.

    An undirected graph with no loops and no double edges is called a simple graph.

  2. 2.

    Of course, one may encode zero using the relation.

  3. 3.

    Notice that by our definition in 1. item (5) above, \(Q_0\) is always the existential quantifier, and so our definition coincides with the standard definition when relevant.

  4. 4.

    The atomic sentences appearing in \(\varphi (x)\) are \(x=y\) and \(x\sim y\).

  5. 5.

    \(\varphi \) expresses: “the complete graph \(K_{d(x)}\) is Hamiltonian” which is true when \(d(x)>2\) and false when \(d(x)=2\) (we may treat \(K_0\) and \(K_1\) separately, if needed).

  6. 6.

    We will describe a few variants, hence the subscript.

  7. 7.

    In this case, \(\mathbf {A}_\exists = P(V)\setminus \{\emptyset \}\), so \(\mathrm {Spoiler}\) may choose any non-empty subset \(S_\ell \) of \(V_\ell \).

  8. 8.

    We will consider only logics stronger than first-order, hence the existential quantifier is always assumed to be at \(\mathrm {Spoiler}\) ’ disposal and he will never lose in this manner.

  9. 9.

    We say that \(E_1\), an \((a,k,G_1)\)-equivalence class of a-tuples in \(G_1\) corresponds to \(E_2\) — a set of a-tuples in \(G_2\) if for any \(\bar{x}_1 \in E_1\) and \(\bar{x}_2 \in E_2\) one has

    $$\begin{aligned} G_1 \models \varphi (\bar{x}_1) \Leftrightarrow G_2 \models \varphi (\bar{x}_2) \end{aligned}$$

    for any \(\varphi \in \mathcal {L}\) of quantifier depth at most k.

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Acknowledgements

We would like to thanks the anonymous referees whose comments helped significantly in improving the presentation of this paper and in putting it in the right frame.

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Authors and Affiliations

  1. Bar-Ilan University, 5290002, Ramat-Gan, Israel

    Simi Haber

  2. The Hebrew University of Jerusalem, 91904, Jerusalem, Israel

    Saharon Shelah

  3. Rutgers, Piscataway, NJ, 08854-8019, USA

    Saharon Shelah

Authors
  1. Simi Haber
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  2. Saharon Shelah
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Correspondence to Simi Haber .

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Editors and Affiliations

  1. Steklov Mathematical Institute, Moscow, Russia

    Lev D. Beklemishev

  2. University of Michigan, Ann Arbor, Michigan, USA

    Andreas Blass

  3. Tel Aviv University, Tel Aviv, Israel

    Nachum Dershowitz

  4. Universität des Saarlandes, Saarbrücken, Germany

    Bernd Finkbeiner

  5. Microsoft Research, REDMOND, Washington, USA

    Wolfram Schulte

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Haber, S., Shelah, S. (2015). An Extension of the Ehrenfeucht-Fraïssé Game for First Order Logics Augmented with Lindström Quantifiers. In: Beklemishev, L., Blass, A., Dershowitz, N., Finkbeiner, B., Schulte, W. (eds) Fields of Logic and Computation II. Lecture Notes in Computer Science(), vol 9300. Springer, Cham. https://doi.org/10.1007/978-3-319-23534-9_13

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  • DOI: https://doi.org/10.1007/978-3-319-23534-9_13

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