The Programming of Algebra

TL;DR

This paper introduces an algebraic framework based on module theory and linear maps to improve the efficiency and expressiveness of relational database query evaluation, especially for cyclic queries.

Contribution

It develops a novel algebraic approach using modules, polysets, and compact maps to optimize relational query processing beyond traditional list-based methods.

Findings

Provides a worst-case optimal evaluation method for cyclic relational queries.

Introduces algebraic joins and intersection algorithms that improve query efficiency.

Demonstrates the framework's ability to handle infinite and complex data structures efficiently.

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.