Arboreal categories and equi-resource homomorphism preservation theorems

The classical homomorphism preservation theorem, due to Łoś, Lyndon and Tarski, states that a first-order sentence φ is preserved under homomorphisms between structures if, and only if, it is equivalent to an existential positive sentence ψ. Given a notion of (syntactic) complexity of sentences, an...

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Bibliographic Details
Published in:Annals of pure and applied logic 2024-06, Vol.175 (6), p.103423, Article 103423
Main Authors: Abramsky, Samson, Reggio, Luca
Format: Article
Language:English
Subjects:
First-order logic
Game comonads
Guarded logics
Homomorphism preservation theorems
Logical resources
Modal logic
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Summary:The classical homomorphism preservation theorem, due to Łoś, Lyndon and Tarski, states that a first-order sentence φ is preserved under homomorphisms between structures if, and only if, it is equivalent to an existential positive sentence ψ. Given a notion of (syntactic) complexity of sentences, an “equi-resource” homomorphism preservation theorem improves on the classical result by ensuring that ψ can be chosen so that its complexity does not exceed that of φ. We describe an axiomatic approach to equi-resource homomorphism preservation theorems based on the notion of arboreal category. This framework is then employed to establish novel homomorphism preservation results, and improve on known ones, for various logic fragments, including first-order, guarded and modal logics.
ISSN:0168-0072
DOI:10.1016/j.apal.2024.103423