A new relational division operation and its implementation

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master_Tlekbay_Dias_2026.pdf (1.23 MB)

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School of Science | Master's thesis
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Date

2025-12-31

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Machine Learning, Data Science and Artificial Intelligence

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Degree programme

Master's Programme in Computer, Communication and Information Sciences
Master's Programme in Computer, Communication and Information Sciences
Master's Programme in Computer, Communication and Information Sciences

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en

Pages

50

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Abstract

Universal quantification — that is, requirements which must be met by all related entities — is a foundational yet challenging concept for SQL to capture efficiently. Whereas such conditions are natural in logical formalisms (such as relational calculus), SQL offers no direct “FOR ALL” quantifier nor a relational division operator. Thus, universal queries are typically expressed through nested negation, set difference, or aggregation, which can obscure intent and lead to varying performance across database systems. This thesis uses relational division as the algebraic foundation for representing universal quantification and looks into how the classical and extended forms can be implemented effectively in SQL. This study contrasts several relational division operators, such as standard division, generalized division, the small and great divide, and the grouped generalized division. A comparative framework is developed for these operators in respect of semantics, expressiveness, realizability, rewriteability, and computational complexity. These results indicate that the grouped generalized division operator is the most expressive and SQL-compatible manner of expressing restricted and grouped universal queries. This naturally matches up with newer features in SQL such as grouping and aggregation. It enables efficient evaluation strategies supported by contemporary query optimizers. This results in the finding that relational division remains a central concept in bridging logical expressiveness and practical query optimization in the relational model.

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Supervisor

Rintanen, Jussi

Keywords

relational division, universal quantification, relational algebra, query optimization, restricted universal queries, grouped generalized division (GGD)

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https://urn.fi/URN:NBN:fi:aalto-202601221881

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[dipl] Perustieteiden korkeakoulu / SCI