# Hazelnut: A Bidirectionally Typed Structure Editor Calculus

arXiv: https://arxiv.org/abs/1607.04180 · HTML: https://mlsanity.com/paper/1607.04180/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jul 2016.

## Built on by (most cited first, 8 of 8)

- [Statically Contextualizing Large Language Models with Typed Holes](https://mlsanity.com/paper/2409.00921/) (2024)
- [LACUNA: Safe Agents as Recursive Program Holes](https://mlsanity.com/paper/2605.28617/) (2026)
- [Report of the Workshop on Program Synthesis for Scientific Computing](https://mlsanity.com/paper/2102.01687/) (2021)
- [A generalised editor calculus (Short Paper)](https://mlsanity.com/paper/2505.18778/) (2025)
- [Generative Compilation: On-the-Fly Compiler Feedback as AI Generates Code](https://mlsanity.com/paper/2607.13921/) (2026)
- [Volition Elicitation: Operational Semantics for People and Their Machines](https://mlsanity.com/paper/2607.14138/) (2026)
- [Decode-Time Grammars: Constrained LLM Generation over a Refinement Order of Grammar Fragments](https://mlsanity.com/paper/2607.18357/) (2026)
- [Semantic Prefix Oracles for LLM Decoding: Contracts and Differential Validation](https://mlsanity.com/paper/2609.35425/) (2026)

## Abstract (arXiv)

Structure editors allow programmers to edit the tree structure of a program directly. This can have cognitive benefits, particularly for novice and end-user programmers. It also simplifies matters for tool designers, because they do not need to contend with malformed program text. This paper defines Hazelnut, a structure editor based on a small bidirectionally typed lambda calculus extended with holes and a cursor. Hazelnut goes one step beyond syntactic well-formedness: its edit actions operate over statically meaningful incomplete terms. Naively, this would force the programmer to construct terms in a rigid "outside-in" manner. To avoid this problem, the action semantics automatically places terms assigned a type that is inconsistent with the expected type inside a hole. This safely defers the type consistency check until the term inside the hole is finished. Hazelnut is a foundational type-theoretic account of typed structure editing, rather than an end-user tool itself. To that end, we describe how Hazelnut's rich metatheory, which we have mechanized in Agda, guides the definition of an extension to the calculus. We also discuss various plausible evaluation strategies for terms with holes, and in so doing reveal connections with gradual typing and contextual modal type theory, the Curry-Howard interpretation of contextual modal logic. Finally, we discuss how Hazelnut's semantics lends itself to implementation as a functional reactive program. Our reference implementation is written using js_of_ocaml.