# Graphical Piecewise-Linear Algebra

arXiv: https://arxiv.org/abs/2111.03956 · HTML: https://mlsanity.com/paper/2111.03956/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Nov 2021.

## Built on by (most cited first, 1 of 1)

- [The Topos of Transformer Networks](https://mlsanity.com/paper/2403.18415/) (2024)

## Abstract (arXiv)

Graphical (Linear) Algebra is a family of diagrammatic languages allowing to reason about different kinds of subsets of vector spaces compositionally. It has been used to model various application domains, from signal-flow graphs to Petri nets and electrical circuits. In this paper, we introduce to the family its most expressive member to date: Graphical Piecewise-Linear Algebra, a new language to specify piecewise-linear subsets of vector spaces. Like the previous members of the family, it comes with a complete axiomatisation, which means it can be used to reason about the corresponding semantic domain purely equationally, forgetting the set-theoretic interpretation. We show completeness using a single axiom on top of Graphical Polyhedral Algebra, and show that this extension is the smallest that can capture a variety of relevant constructs. Finally, we showcase its use by modelling the behaviour of stateless electronic circuits of ideal elements, a domain that had remained outside the remit of previous diagrammatic languages.