The Programming of Algebra

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Abstract

We present module theory and linear maps as a powerful generalised and computationally efficient framework for the relational data model, which underpins today's relational database systems. Based on universal constructions of modules we obtain compact and computationally efficient data structures for data collections corresponding to union and deletion, repeated union, Cartesian product and key-indexed data. Free modules naturally give rise to polysets, which generalise multisets and facilitate expressing database queries as multilinear maps with asymptotically efficient evaluation on polyset constructors. We introduce compact maps as a way of representing infinite (poly)sets constructible from an infinite base set and its elements by addition and subtraction. We show how natural joins generalise to algebraic joins, while intersection is implemented by a novel algorithm on nested compact maps that carefully avoids visiting parts of the input that do not contribute to the eventual output. Our algebraic framework leads to a worst-case optimal evaluation of cyclic relational queries, which is known to be impossible using textbook query optimisers that operate on lists of records only.

OriginalsprogEngelsk
TidsskriftElectronic Proceedings in Theoretical Computer Science, EPTCS
Vol/bind360
Sider (fra-til)71-92
Antal sider22
ISSN2075-2180
DOI
StatusUdgivet - jun. 2022
Begivenhed9th Workshop on Mathematically Structured Functional Programming, MSFP 2022 - Munich, Tyskland
Varighed: 2 apr. 2022 → …

Konference

Konference9th Workshop on Mathematically Structured Functional Programming, MSFP 2022
Land/OmrådeTyskland
ByMunich
Periode02/04/2022 → …

Bibliografisk note

Funding Information:
Acknowledgements. This work was made possible by Independent Research Fund Denmark grant FUTHARK: Functional Technology for High-performance Architectures and DFF–International Postdoc 0131-00025B. We greatly appreciate and thank the three anonymous referees for their comments and recommendations.

Funding Information:
Thiswork wasmade possible by Independent Research Fund Denmark grant FUTHARK: Functional Technology for High-performance Architectures and DFF-International Postdoc 0131-00025B. We greatly appreciate and thank the three anonymous referees for their comments and recommendations.

Publisher Copyright:
© 2022 Henglein, Kaarsgaard, & Mathiesen.

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