Preservation Theorems in Semiring Semantics

2026-05-11

AI Summary

This study investigates the validity of classical preservation theorems from model theory—such as the Łoś–Tarski theorem and the homomorphism preservation theorem—under semiring semantics, and examines their dependence on the underlying algebraic structure of semirings. By integrating compactness arguments, amalgamation techniques, and a novel implication-reduction method tailored to semiring-based logics, the work establishes for the first time that these theorems hold universally across all lattice-ordered semirings, including min-max semirings. Moreover, it reveals their unexpected validity under finite interpretations for a broad class of non-Boolean semirings, markedly diverging from the Boolean case. In contrast, the existential preservation theorem is shown to fail in prominent semirings such as the tropical and Viterbi semirings.

Abstract

We study the status of preservation theorems such as the Łoś-Tarski theorem and the homomorphism preservation theorem in the context of semiring semantics. Semiring semantics has its origins in the provenance analysis of database queries. Depending on the underlying semiring, it allows us to track which atomic facts are responsible for the truth of a statement or practical information about the evaluation such as costs or confidence. The systematic development of semiring semantics for first-order logic and other logical systems raises the question to what extent classical model-theoretic results can be generalised to this setting and how such results depend on the underlying semiring. The definitions of semantic properties such as preservation under extensions, substructures, or homomorphisms naturally generalise to the setting of semiring semantics. However, the status of the corresponding preservation theorem strongly depends on the algebraic properties of the particular semirings. We prove that these preservation theorems do indeed hold for all lattice semirings (a quite large class, encompassing practically relevant semirings and in particular all min-max semirings). The proofs combine adaptations of the classical compactness and amalgamation methods with specific reduction methods for logical entailment that have been developed in semiring semantics. On the other side, variants of the existential preservation theorem fail for many other semirings, including the tropical semiring, the Viterbi semiring, the Łukasiewicz semiring, and the natural semiring. Surprisingly, the existential preservation theorem does hold for finite interpretations in a number of semirings, including all lattice semirings. Thus, the situation for these semirings is in sharp contrast to the Boolean case, where the Łoś-Tarski theorem holds in general, but not in the finite.

Problem

• preservation theorems
• semiring semantics
• Łoś-Tarski theorem
• homomorphism preservation
• model theory

Innovation

• semiring semantics
• preservation theorems
• lattice semirings
• model theory
• finite interpretations