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Optimally Rewriting Formulas and Database Queries: A Confluence of Term Rewriting, Structural Decomposition, and Complexity

Authors Hubie Chen, Stefan Mengel

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LIPIcs.ICDT.2024.16.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • Theory of computation → Database query processing and optimization (theory)
Keywords
  • width
  • query rewriting
  • structural decomposition
  • term rewriting

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Abstract

A central computational task in database theory, finite model theory, and computer science at large is the evaluation of a first-order sentence on a finite structure. In the context of this task, the width of a sentence, defined as the maximum number of free variables over all subformulas, has been established as a crucial measure, where minimizing width of a sentence (while retaining logical equivalence) is considered highly desirable. An undecidability result rules out the possibility of an algorithm that, given a first-order sentence, returns a logically equivalent sentence of minimum width; this result motivates the study of width minimization via syntactic rewriting rules, which is this article’s focus. For a number of common rewriting rules (which are known to preserve logical equivalence), including rules that allow for the movement of quantifiers, we present an algorithm that, given a positive first-order sentence ϕ, outputs the minimum-width sentence obtainable from ϕ via application of these rules. We thus obtain a complete algorithmic understanding of width minimization up to the studied rules; this result is the first one - of which we are aware - that establishes this type of understanding in such a general setting. Our result builds on the theory of term rewriting and establishes an interface among this theory, query evaluation, and structural decomposition theory.

Cite As Get BibTex

Hubie Chen and Stefan Mengel. Optimally Rewriting Formulas and Database Queries: A Confluence of Term Rewriting, Structural Decomposition, and Complexity. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICDT.2024.16

Author Details

Hubie Chen
  • Department of Informatics, King’s College London, UK
Stefan Mengel
  • Univ. Artois, CNRS, Centre de Recherche en Informatique de Lens (CRIL), France

Funding

  • Chen, Hubie: acknowledges the support of EPSRC grants EP/V039318/1 and EP/X03190X/1.
  • Mengel, Stefan: partially supported by the ANR project EQUUS ANR-19-CE48-0019.

Acknowledgements

The authors thank Carsten Fuhs, Moritz Müller, and Riccardo Treglia for useful feedback and pointers.

References

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