# Getting to the Point. Index Sets and Parallelism-Preserving Autodiff for Pointful Array Programming

arXiv: https://arxiv.org/abs/2104.05372 · HTML: https://mlsanity.com/paper/2104.05372/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Apr 2021.

## Built on by (most cited first, 10 of 10)

- [PartIR: Composing SPMD Partitioning Strategies for Machine Learning](https://mlsanity.com/paper/2401.11202/) (2024)
- [Decomposing reverse-mode automatic differentiation](https://mlsanity.com/paper/2105.09469/) (2021)
- [Machines of finite depth: towards a formalization of neural networks](https://mlsanity.com/paper/2204.12786/) (2022)
- [λA: A Typed Lambda Calculus for LLM Agent Composition](https://mlsanity.com/paper/2604.11767/) (2026)
- [Combinatory Adjoints and Differentiation](https://mlsanity.com/paper/2207.00847/) (2022)
- [nablaSD: Differentiable Programming for Sparse Tensors](https://mlsanity.com/paper/2303.07030/) (2023)
- [Operon: Incremental Construction of Ragged Data via Named Dimensions](https://mlsanity.com/paper/2511.16080/) (2025)
- [A Typed Tensor Language for Shared-State Federated Computation](https://mlsanity.com/paper/2605.21103/) (2026)
- [Differentiate the Evaluator, Not the Program: An Efficient Runtime Representation for Neuro-Symbolic Learning](https://mlsanity.com/paper/2607.03574/) (2026)
- [It's All Just Vectorization: einx, a Universal Notation for Tensor Operations](https://mlsanity.com/paper/2607.27987/) (2026)

## Abstract (arXiv)

We present a novel programming language design that attempts to combine the clarity and safety of high-level functional languages with the efficiency and parallelism of low-level numerical languages. We treat arrays as eagerly-memoized functions on typed index sets, allowing abstract function manipulations, such as currying, to work on arrays. In contrast to composing primitive bulk-array operations, we argue for an explicit nested indexing style that mirrors application of functions to arguments. We also introduce a fine-grained typed effects system which affords concise and automatically-parallelized in-place updates. Specifically, an associative accumulation effect allows reverse-mode automatic differentiation of in-place updates in a way that preserves parallelism. Empirically, we benchmark against the Futhark array programming language, and demonstrate that aggressive inlining and type-driven compilation allows array programs to be written in an expressive, "pointful" style with little performance penalty.