Arboretum.hs: Symbolic manipulation for algebras of graphs

2026-04-28

AI Summary

This work addresses the limitations of existing tools in symbolic computation over algebraic graph structures—such as trees and forests—which often lack flexibility, safety, and fidelity to formal mathematical definitions. To overcome these shortcomings, the authors design and implement an extensible symbolic computation framework for graph algebras in Haskell, leveraging its strong static typing and declarative nature to ensure close alignment between code and mathematical formalism while enabling compile-time safety guarantees. Moving beyond the constraints of traditional imperative approaches, the framework also integrates LaTeX-based visualization capabilities to facilitate the exploration of novel algebraic operations. By providing an intuitive, transparent, and composable foundation, this system supports advanced research in algebraic graph theory and numerical analysis—including structures beyond Butcher series—and demonstrates significant practical utility.

Abstract

We design the Arboretum.hs package for symbolic computations with algebras of trees and more general graphs in Haskell. Thanks to the declarative nature of functional programming, the package's implementation closely follows mathematical definitions, making the code intuitive and transparent for users working with algebraic and combinatorial structures. To assist with current mathematical research, Arboretum.hs supports experimentation by facilitating the introduction of new algebraic operations, as well as providing functionality for rendering trees and forests through LaTeX integration. Compared to recent imperative implementations in languages such as Julia or Python, Arboretum.hs offers greater flexibility for manipulating and extending tree-based structures. Its use of Haskell enables safe programming and strong compile-time guarantees, serving both as a practical computational tool and a foundation for further research in algebraic combinatorics, beyond the setting of trees usually considered in the implementation of Butcher series, which are a fundamental tool for the analysis of numerical integrators.

Problem

• symbolic manipulation
• algebras of graphs
• tree-based structures
• algebraic combinatorics
• Butcher series

Innovation

• symbolic computation
• graph algebras
• functional programming
• Haskell
• algebraic combinatorics