Locality Theorems in Semiring Semantics

Authors Clotilde Bizière, Erich Grädel , Matthias Naaf

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Clotilde Bizière
  • ENS Paris, France
Erich Grädel
  • RWTH Aachen University, Germany
Matthias Naaf
  • RWTH Aachen University, Germany

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Clotilde Bizière, Erich Grädel, and Matthias Naaf. Locality Theorems in Semiring Semantics. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Semiring semantics of first-order logic generalises classical Boolean semantics by permitting truth values from a commutative semiring, which can model information such as costs or access restrictions. This raises the question to what extent classical model-theoretic properties still apply, and how this depends on the algebraic properties of the semiring. In this paper, we study this question for the classical locality theorems due to Hanf and Gaifman. We prove that Hanf’s locality theorem generalises to all semirings with idempotent operations, but fails for many non-idempotent semirings. We then consider Gaifman normal forms and show that for formulae with free variables, Gaifman’s theorem does not generalise beyond the Boolean semiring. Also for sentences, it fails in the natural semiring and the tropical semiring. Our main result, however, is a constructive proof of the existence of Gaifman normal forms for min-max and lattice semirings. The proof implies a stronger version of Gaifman’s classical theorem in Boolean semantics: every sentence has a Gaifman normal form which does not add negations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Semiring semantics
  • Locality
  • First-order logic


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