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DOI: 10.4230/LIPIcs.CSL.2025.12

First-Order Logic with Equicardinality in Random Graphs

Authors: Simi Haber, Tal Hershko, Mostafa Mirabi, and Saharon Shelah

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract

We answer a question of Blass and Harary about the validity of the zero-one law in random graphs for extensions of first-order logic (FOL). For a given graph property P, the Lindström extension of FOL by P is defined as the minimal (regular) extension of FOL able to express P. For several graph properties P (e.g. Hamiltonicity), it is known that the Lindström extension by P is also able to interpret a segment of arithmetic, and thus strongly disobeys the zero-one law. Common to all these properties is the ability to express the Härtig quantifier, a natural extension of FOL testing if two definable sets are of the same size. We prove that the Härtig quantifier is sufficient for the interpretation of arithmetic, thus providing a general result which implies all known cases of Lindström extensions which are able to interpret a segment of arithmetic.

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Simi Haber, Tal Hershko, Mostafa Mirabi, and Saharon Shelah. First-Order Logic with Equicardinality in Random Graphs. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{haber_et_al:LIPIcs.CSL.2025.12,
  author =	{Haber, Simi and Hershko, Tal and Mirabi, Mostafa and Shelah, Saharon},
  title =	{{First-Order Logic with Equicardinality in Random Graphs}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{12:1--12:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.12},
  URN =		{urn:nbn:de:0030-drops-227694},
  doi =		{10.4230/LIPIcs.CSL.2025.12},
  annote =	{Keywords: finite model theory, first-order logic, monadic second-order logic, random graphs, zero-one laws, generalized quantifiers, equicardinality}
}

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First-Order Logic with Equicardinality in Random Graphs

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