Optimally Rewriting Formulas and Database Queries: A Confluence of Term Rewriting, Structural Decomposition, and Complexity
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915947568168960 |
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| author | Chen, Hubie Mengel, Stefan |
| author_facet | Chen, Hubie Mengel, Stefan |
| contents | 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 \emph{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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10229 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimally Rewriting Formulas and Database Queries: A Confluence of Term Rewriting, Structural Decomposition, and Complexity Chen, Hubie Mengel, Stefan Logic in Computer Science Databases 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 \emph{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. |
| title | Optimally Rewriting Formulas and Database Queries: A Confluence of Term Rewriting, Structural Decomposition, and Complexity |
| topic | Logic in Computer Science Databases |
| url | https://arxiv.org/abs/2411.10229 |