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Compactness in Semiring Semantics

Authors Sophie Brinke , Anuj Dawar , Erich Grädel , Lovro Mrkonjić , Matthias Naaf

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LIPIcs.CSL.2026.13.pdf
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ACM Subject Classification
  • Theory of computation → Logic
Keywords
  • Semiring semantics
  • compactness

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Abstract

Semiring provenance was originally introduced in database theory with the aim of explaining why certain tuples are (not) contained in the answer of a query. To this end, logical statements are not just evaluated to true or false but to values in a commutative semiring. Depending on the underlying semiring, this allows us to track descriptions of the atomic facts that are responsible for the truth of a statement or practical information about the evaluation such as costs or confidence. Recently, this approach has been expanded to a systematic study of semiring semantics for first-order logic and other logical systems. This raises the question to what extent model-theoretic results can be generalised to semiring semantics and how this relates to the algebraic properties of the underlying semiring. 
Here we investigate the availability of compactness in semiring semantics. The appropriate setting for this is based on absorptive semirings with well-defined infinitary products. Compactness can be stated either in terms of satisfiability or in terms of entailment, and these two variants are trivially equivalent in Boolean semantics. However, this is no longer the case in semiring semantics. Compactness in terms of satisfiability, defined as the existence of non-zero valuations, indeed generalises to every infinitary absorptive semiring. For compactness in terms of entailment the situation is different. The entailment relation naturally extends to semiring semantics (via the natural order on the semiring) but this yields a stronger variant of compactness, which fails for certain important semirings, including the tropical semiring and the Łukasiewicz semiring. Our main positive results show that strong compactness does indeed hold for all finite semirings and all lattice semirings.

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Sophie Brinke, Anuj Dawar, Erich Grädel, Lovro Mrkonjić, and Matthias Naaf. Compactness in Semiring Semantics. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 13:1-13:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CSL.2026.13

Author Details

Sophie Brinke
  • RWTH Aachen University, Germany
Anuj Dawar
  • University of Cambridge, UK
Erich Grädel
  • RWTH Aachen University, Germany
Lovro Mrkonjić
  • RWTH Aachen University, Germany
Matthias Naaf
  • RWTH Aachen University, Germany

References

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