The Nonexistence of Unicorns and Many-Sorted Löwenheim–Skolem Theorems

Benjamin Przybocki; Guilherme Toledo; Yoni Zohar; Clark Barrett

doi:10.1007/978-3-031-71162-6_34

The Nonexistence of Unicorns and Many-Sorted Löwenheim–Skolem Theorems

Benjamin Przybocki, Guilherme Toledo, Yoni Zohar, Clark Barrett

Department of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

Stable infiniteness, strong finite witnessability, and smoothness are model-theoretic properties relevant to theory combination in satisfiability modulo theories. Theories that are strongly finitely witnessable and smooth are called strongly polite and can be effectively combined with other theories. Toledo, Zohar, and Barrett conjectured that stably infinite and strongly finitely witnessable theories are smooth and therefore strongly polite. They called counterexamples to this conjecture unicorn theories, as their existence seemed unlikely. We prove that, indeed, unicorns do not exist. We also prove versions of the Löwenheim–Skolem theorem and the Łoś–Vaught test for many-sorted logic.

Original language	English
Title of host publication	Formal Methods - 26th International Symposium, FM 2024, Proceedings
Editors	André Platzer, Kristin Yvonne Rozier, Matteo Pradella, Matteo Rossi
Publisher	Springer Science and Business Media Deutschland GmbH
Pages	658-675
Number of pages	18
ISBN (Print)	9783031711619
DOIs	https://doi.org/10.1007/978-3-031-71162-6_34
State	Published - 2025
Event	26th International Symposium on Formal Methods, FM 2024 - Milan, Italy Duration: 9 Sep 2024 → 13 Sep 2024

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	14933 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	26th International Symposium on Formal Methods, FM 2024
Country/Territory	Italy
City	Milan
Period	9/09/24 → 13/09/24

Bibliographical note

Publisher Copyright:
© The Author(s) 2025.

Access to Document

10.1007/978-3-031-71162-6_34

Cite this

Przybocki, B., Toledo, G., Zohar, Y., & Barrett, C. (2025). The Nonexistence of Unicorns and Many-Sorted Löwenheim–Skolem Theorems. In A. Platzer, K. Y. Rozier, M. Pradella, & M. Rossi (Eds.), Formal Methods - 26th International Symposium, FM 2024, Proceedings (pp. 658-675). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 14933 LNCS). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/978-3-031-71162-6_34

Przybocki, Benjamin ; Toledo, Guilherme ; Zohar, Yoni et al. / The Nonexistence of Unicorns and Many-Sorted Löwenheim–Skolem Theorems. Formal Methods - 26th International Symposium, FM 2024, Proceedings. editor / André Platzer ; Kristin Yvonne Rozier ; Matteo Pradella ; Matteo Rossi. Springer Science and Business Media Deutschland GmbH, 2025. pp. 658-675 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "Stable infiniteness, strong finite witnessability, and smoothness are model-theoretic properties relevant to theory combination in satisfiability modulo theories. Theories that are strongly finitely witnessable and smooth are called strongly polite and can be effectively combined with other theories. Toledo, Zohar, and Barrett conjectured that stably infinite and strongly finitely witnessable theories are smooth and therefore strongly polite. They called counterexamples to this conjecture unicorn theories, as their existence seemed unlikely. We prove that, indeed, unicorns do not exist. We also prove versions of the L{\"o}wenheim–Skolem theorem and the {\L}o{\'s}–Vaught test for many-sorted logic.",

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Przybocki, B, Toledo, G, Zohar, Y & Barrett, C 2025, The Nonexistence of Unicorns and Many-Sorted Löwenheim–Skolem Theorems. in A Platzer, KY Rozier, M Pradella & M Rossi (eds), Formal Methods - 26th International Symposium, FM 2024, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 14933 LNCS, Springer Science and Business Media Deutschland GmbH, pp. 658-675, 26th International Symposium on Formal Methods, FM 2024, Milan, Italy, 9/09/24. https://doi.org/10.1007/978-3-031-71162-6_34

The Nonexistence of Unicorns and Many-Sorted Löwenheim–Skolem Theorems. / Przybocki, Benjamin; Toledo, Guilherme; Zohar, Yoni et al.
Formal Methods - 26th International Symposium, FM 2024, Proceedings. ed. / André Platzer; Kristin Yvonne Rozier; Matteo Pradella; Matteo Rossi. Springer Science and Business Media Deutschland GmbH, 2025. p. 658-675 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 14933 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Przybocki B, Toledo G, Zohar Y, Barrett C. The Nonexistence of Unicorns and Many-Sorted Löwenheim–Skolem Theorems. In Platzer A, Rozier KY, Pradella M, Rossi M, editors, Formal Methods - 26th International Symposium, FM 2024, Proceedings. Springer Science and Business Media Deutschland GmbH. 2025. p. 658-675. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-031-71162-6_34