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Algebraic structures for untyped Racket
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Algebraic Racket

Racket Package Documentation Build Status Coverage Status

A Racket extension for untyped algebraic data structures.

What is Algebraic Racket?

Algebraic structures provide the operational under-pinnings for algebraic data types. What's missing is the static typing constraints.

The current release provides a #lang algebraic/racket/base that extends #lang racket/base with:

  1. First class, lexically scoped, naturally ordered data constructors, and

  2. A consistent destructuring syntax for functions and macros.

Algebraic Data

As the name implies, Algebraic Racket works with two kinds of structure: sums and products.

(data Maybe (Nothing Just))

This defines a sum named Maybe and two constituent products named Nothing and Just. It also defines three membership predicates named Maybe?, Nothing?, and Just? for recognizing products and their instances.

The elements of a sum are ordered by position so they can be compared and sorted.

> (data A-Z (A B C D E F G H I J K L M N O P Q R S T U V W X Y Z))
> (let ([Πs (shuffle (data->list (sum A-Z)))])
    (values Πs (sort Πs data-less-than?)))
'(G O M W V C Q H T K F S Y U Z A B R J N E P X I L D)
'(A B C D E F G H I J K L M N O P Q R S T U V W X Y Z)
> (data Z-A (Z Y X W V U T S R Q P O N M L K J I H G F E D C B A))
> (let ([Πs (shuffle (data->list (sum Z-A)))])
    (values Πs (sort Πs data-less-than?)))
'(R C U N Y Z X L A K D H B J S V E G I W O M P Q F T)
'(Z Y X W V U T S R Q P O N M L K J I H G F E D C B A)

The products Nothing and Just can be matched against directly or they can be used to construct instances of the products.

An instance is a transparent data structure that resembles a tagged tuple. Algebraic Racket imposes no restrictions on the type or number of arguments accepted by a product.

> (values (Just) (Just 1) (Just 1 2))
(Just 1)
(Just 1 2)

Destructuring Syntax

Algebraic Racket provides φ/phi and function forms (and multi-arg variants) which extend the variable binding sites of the λ/lambda form with support for pattern-based destructuring.

> (define maybe
      [(n _ Nothing) n]
      [(_ f (Just x)) (f x)]))
> (values
   (maybe #f values (Just 123))
   (maybe #f values Nothing))

It also provides μ/mu and macro forms for syntax transformers.

> (define-syntax unroll-pow
    (μ* (b:number p:number) '(* #,@(make-list (var p) #'b))))
> (unroll-pow 2 5)
'(* 2 2 2 2 2)

Destructuring Patterns

Patterns for literal data are designed to look and feel as similar as possible to the terms they match, including:

  • Plain old Racket structs

  • Unquoted literal values: boolean, number, string, bytes, char

  • Unquoted containers that look like literals: pair, list, vector, hash

  • Quoted data, including symbols

  • Quasiquoted data with escapes

Other notable features include:

  • Pattern guards, aliases, rest args, and wildcard/variable naming conventions all have a consistent look and feel across all function and macro forms.

  • Macros also support ellipsis patterns and the :syntax-class naming convention. Use syntax/parse to define your own classes.

  • Regular expression patterns for functional string processing.

Installation and Use

Algebraic Racket is distributed in the algebraic package in the official Racket package repository. It can be installed from DrRacket's package manager, or with raco pkg from the command line.

raco pkg install algebraic

To start using Algebraic Racket, set the initial line of your Racket source file to:

#lang algebraic/racket/base

The package is fully documented, including a tutorial series of interpreters developed for and with Algebraic Racket.

Related Work

Plain old Racket structs

Prefab struct types are globally scoped, which has been a problem for me in the past. Non-prefab structs are better in that regard, except basic features like type hierarchies and functional updaters have a tendency to interfere with each other.

Algebraic data constructors are lexically scoped and have no fixed arity. Instances of algebraic data are always transparent.

Despite their differences, plain old Racket structs work pretty well with Algebraic Racket's destructuring syntax. They're always there when you need them.


Hackett is an excellent platform for pure, lazy functional programming in the Racket software ecosystem. If I needed pure and lazy evaluation semantics today, I'd try Hackett before considering Haskell seriously again. Having said that, I don't usually want pure and lazy evaluation semantics.

Wouldn't it be nice if I could import just the features I wanted from Hackett and ignore the rest? That is the promise of language-oriented programming, after all. For all I know, it's possible now.

Unfortunately, it isn't yet possible to mix and match off-the-shelf features to comprise a tool as robust as Hackett (at least, that isn't Hackett itself). Algebraic Racket is a step in that direction.

Typed Racket

Typed Racket is a robust platform with a broad scope, and it keeps getting better. The type system offers the static analyses I care about and the documentation is excellent. After spending some time building interpreters in Typed Racket, I decided I still wanted something else for two reasons:

  1. Fumbling with types for common Racket idioms (like apply) became a distraction, and

  2. Uncompiled code loads very slowly.

This may say more about my ignorance of Typed Racket than anything else. Nonetheless, Algebraic Racket takes advantage of my intuition for Haskell syntax while staying responsive enough for rapid iterative exploratory programming.


Redex is a domain-specific language for semantics engineering. It does a lot more for the semantics engineer than Algebraic Racket ever will, like typesetting and testing automation.

On the other hand, developing full applications with Redex is probably not for the faint of heart. With Algebraic Racket, you get all the bells and whistles of #lang racket/base and then some.

Project Goals

Algebraic Racket is the first major milestone in a larger effort to produce a complete language-oriented toolkit for integrating algebraic structure into more sophisticated designs.

The project aims to:

  1. Implement and document the forms, functions, and syntax classes comprising Algebraic Racket for maximum potential reuse.

  2. Support the development of modular type systems and other language-facing components as libraries.


Pull requests of any size are welcome. For help creating one, please start a thread on racket-users. For major changes, please open an issue first to discuss what you would like to change.

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