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Equi-Rank Homomorphism Preservation Theorem on Finite Structures

Author Benjamin Rossman

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ACM Subject Classification
  • Theory of computation → Finite Model Theory
Keywords
  • finite model theory
  • preservation theorems
  • quantifier rank

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Abstract

The Homomorphism Preservation Theorem (HPT) of classical model theory states that a first-order sentence is preserved under homomorphisms if, and only if, it is equivalent to an existential-positive sentence. This theorem remains valid when restricted to finite structures, as demonstrated by the author in [Rossman, 2008; Rossman, 2017] via distinct model-theoretic and circuit-complexity based proofs. In this paper, we present a third (and significantly simpler) proof of the finitary HPT based on a generalized Cai-Fürer-Immerman construction. This method establishes a tight correspondence between syntactic parameters of a homomorphism-preserved sentence (quantifier rank, variable width, alternation height) and structural parameters of its minimal models (tree-width, tree-depth, decomposition height). Consequently, we prove a conjectured "equi-rank" version of the finitary HPT. In contrast, previous versions of the finitary HPT possess additional properties, but incur blow-ups in the quantifier rank of the equivalent existential-positive sentence.

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Benjamin Rossman. Equi-Rank Homomorphism Preservation Theorem on Finite Structures. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CSL.2025.6

Author Details

Benjamin Rossman
  • Duke University, Durham, NC, USA

Acknowledgements

I thank Deepanshu Kush, William He and Ken-ichi Kawarabayashi for stimulating conversions that prompted me to revisit the topic of this paper. Anuj Dawar provided helpful comments on a preliminary draft of this paper. I am grateful to the anonymous referees for numerous suggestions that improved the clarity of this paper. This paper was partially written during a visit to the National Institute of Informations in Tokyo, Japan.

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