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.
| Original language | English |
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| Title of host publication | 33rd EACSL Annual Conference on Computer Science Logic, CSL 2025 |
| Editors | Jorg Endrullis, Sylvain Schmitz |
| Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |
| ISBN (Electronic) | 9783959773621 |
| DOIs | |
| State | Published - 3 Feb 2025 |
| Event | 33rd EACSL Annual Conference on Computer Science Logic, CSL 2025 - Amsterdam, Netherlands Duration: 10 Feb 2025 → 14 Feb 2025 |
Publication series
| Name | Leibniz International Proceedings in Informatics, LIPIcs |
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| Volume | 326 |
| ISSN (Print) | 1868-8969 |
Conference
| Conference | 33rd EACSL Annual Conference on Computer Science Logic, CSL 2025 |
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| Country/Territory | Netherlands |
| City | Amsterdam |
| Period | 10/02/25 → 14/02/25 |
Bibliographical note
Publisher Copyright:© Simi Haber, Tal Hershko, Mostafa Mirabi, and Saharon Shelah.
Keywords
- equicardinality
- finite model theory
- first-order logic
- generalized quantifiers
- monadic second-order logic
- random graphs
- zero-one laws