A Purely Functional Computer Algebra System Embedded in Haskell

TL;DR

This paper presents a purely functional computer algebra system embedded in Haskell, demonstrating how functional programming techniques can ensure safety, composability, and correctness in algebraic computations.

Contribution

It introduces an embedded domain-specific language in Haskell for computer algebra, implementing advanced algorithms like Gröbner basis methods in a composable and safe manner.

Findings

Successfully implemented Gröbner basis algorithms F4 and F5

Achieved safety, composability, and correctness in algebraic computations

Demonstrated advantages of functional programming in computer algebra systems

Abstract

We demonstrate how methods in Functional Programming can be used to implement a computer algebra system. As a proof-of-concept, we present the computational-algebra package. It is a computer algebra system implemented as an embedded domain-specific language in Haskell, a purely functional programming language. Utilising methods in functional programming and prominent features of Haskell, this library achieves safety, composability, and correctness at the same time. To demonstrate the advantages of our approach, we have implemented advanced Gr\"{o}bner basis algorithms, such as Faug\`{e}re's $F_4$ and $F_5$, in a composable way.