# Generalized algebraic data type

In functional programming, a generalized algebraic data type (GADT, also first-class phantom type,[1] guarded recursive datatype,[2] or equality-qualified type[3]) is a generalization of parametric algebraic data types.

## Overview

In a GADT, the product constructors (called data constructors in Haskell) can provide an explicit instantiation of the ADT as the type instantiation of their return value. This allows defining functions with a more advanced type behaviour. For a data constructor of Haskell 2010, the return value has the type instantiation implied by the instantiation of the ADT parameters at the constructor's application.

-- A parametric ADT that is not a GADT
data List a = Nil | Cons a (List a)

integers :: List Int
integers = Cons 12 (Cons 107 Nil)

strings :: List String
strings = Cons "boat" (Cons "dock" Nil)

data Expr a where
EBool  :: Bool     -> Expr Bool
EInt   :: Int      -> Expr Int
EEqual :: Expr Int -> Expr Int  -> Expr Bool

eval :: Expr a -> a
eval e = case e of
EBool a    -> a
EInt a     -> a
EEqual a b -> (eval a) == (eval b)

expr1 :: Expr Bool
expr1 = EEqual (EInt 2) (EInt 3)

ret = eval expr1 -- False

They are currently implemented in the GHC compiler as a non-standard extension, used by, among others, Pugs and Darcs. OCaml supports GADT natively since version 4.00.[4]

The GHC implementation provides support for existentially quantified type parameters and for local constraints.

## History

An early version of generalized algebraic data types were described by (Augustsson Petersson) and based on pattern matching in ALF.

Generalized algebraic data types were introduced independently by (Cheney Hinze) and prior by (Xi Chen) as extensions to ML's and Haskell's algebraic data types.[5] Both are essentially equivalent to each other. They are similar to the inductive families of data types (or inductive datatypes) found in Coq's Calculus of Inductive Constructions and other dependently typed languages, modulo the dependent types and except that the latter have an additional positivity restriction which is not enforced in GADTs.[6]

(Sulzmann Wazny) introduced extended algebraic data types which combine GADTs together with the existential data types and type class constraints.

Type inference in the absence of any programmer supplied type annotations is undecidable[7] and functions defined over GADTs do not admit principal types in general.[8] Type reconstruction requires several design trade-offs and is an area of active research (Peyton Jones, Washburn & Weirich 2004; Peyton Jones et al. 2006.

In spring 2021, Scala 3.0 is released.[9] This major update of Scala introduce the possibility to write GADTs[10] with the same syntax as ADTs, which is not the case in other programming languages according to Martin Odersky.[11]

## Applications

Applications of GADTs include generic programming, modelling programming languages (higher-order abstract syntax), maintaining invariants in data structures, expressing constraints in embedded domain-specific languages, and modelling objects.[12]

### Higher-order abstract syntax

An important application of GADTs is to embed higher-order abstract syntax in a type safe fashion. Here is an embedding of the simply typed lambda calculus with an arbitrary collection of base types, tuples and a fixed point combinator:

data Lam :: * -> * where
Lift :: a                     -> Lam a        -- ^ lifted value
Pair :: Lam a -> Lam b        -> Lam (a, b)   -- ^ product
Lam  :: (Lam a -> Lam b)      -> Lam (a -> b) -- ^ lambda abstraction
App  :: Lam (a -> b) -> Lam a -> Lam b        -- ^ function application
Fix  :: Lam (a -> a)          -> Lam a        -- ^ fixed point

And a type safe evaluation function:

eval :: Lam t -> t
eval (Lift v)   = v
eval (Pair l r) = (eval l, eval r)
eval (Lam f)    = \x -> eval (f (Lift x))
eval (App f x)  = (eval f) (eval x)
eval (Fix f)    = (eval f) (eval (Fix f))

The factorial function can now be written as:

fact = Fix (Lam (\f -> Lam (\y -> Lift (if eval y == 0 then 1 else eval y * (eval f) (eval y - 1)))))
eval(fact)(10)

We would have run into problems using regular algebraic data types. Dropping the type parameter would have made the lifted base types existentially quantified, making it impossible to write the evaluator. With a type parameter we would still be restricted to a single base type. Furthermore, ill-formed expressions such as App (Lam (\x -> Lam (\y -> App x y))) (Lift True) would have been possible to construct, while they are type incorrect using the GADT. A well-formed analogue is App (Lam (\x -> Lam (\y -> App x y))) (Lift (\z -> True)). This is because the type of x is Lam (a -> b), inferred from the type of the Lam data constructor.

## Notes

1. Cheney & Hinze 2003, p. 25.
2. Cheney & Hinze 2003, pp. 25–26.
3. Schrijvers et al. 2009, p. 1.
4. Kmetiuk, Anatolii. "SCALA 3 IS HERE!🎉🎉🎉". École Polytechnique Fédérale Lausanne (EPFL) Lausanne, Switzerland.
5. "SCALA 3 — BOOK ALGEBRAIC DATA TYPES". École Polytechnique Fédérale Lausanne (EPFL) Lausanne, Switzerland.
6. Odersky, Martin. "A Tour of Scala 3 - Martin Odersky". Scala Days Conferences.

Applications
• Augustsson, Lennart; Petersson, Kent (September 1994). "Silly type families".
• Cheney, James; Hinze, Ralf (2003). "First-Class Phantom Types". Technical Report CUCIS TR2003-1901 (Cornell University).
• Xi, Hongwei; Chen, Chiyan; Chen, Gang (2003). "Guarded recursive datatype constructors". Proceedings of the 30th ACM SIGPLAN-SIGACT symposium on Principles of programming languages. ACM Press. pp. 224–235. doi:10.1145/604131.604150. ISBN 978-1581136289.
• Sheard, Tim; Pasalic, Emir (2004). "Meta-programming with built-in type equality". Proceedings of the Fourth International Workshop on Logical Frameworks and Meta-languages (LFM'04), Cork 199: 49–65. doi:10.1016/j.entcs.2007.11.012.
Semantics
Type reconstruction
• Peyton Jones, Simon; Washburn, Geoffrey; Weirich, Stephanie (2004). "Wobbly types: type inference for generalised algebraic data types". Technical Report MS-CIS-05-25 (University of Pennsylvania).
• Peyton Jones, Simon; Vytiniotis, Dimitrios; Weirich, Stephanie; Washburn, Geoffrey (2006). "Simple Unification-based Type Inference for GADTs". Proceedings of the ACM International Conference on Functional Programming (ICFP'06), Portland.
• Sulzmann, Martin; Wazny, Jeremy; Stuckey, Peter J. (2006). "A Framework for Extended Algebraic Data Types". in Hagiya, M.; Wadler, P.. 3945. pp. 46–64.
• Sulzmann, Martin; Schrijvers, Tom; Stuckey, Peter J. (2006). "Principal Type Inference for GHC-Style Multi-Parameter Type Classes". in Kobayashi, Naoki. 4279. pp. 26–43.
• Schrijvers, Tom; Peyton Jones, Simon; Sulzmann, Martin; Vytiniotis, Dimitrios (2009). "Complete and decidable type inference for GADTs". Proceedings of the 14th ACM SIGPLAN international conference on Functional programming. pp. 341–352. doi:10.1145/1596550.1596599. ISBN 9781605583327.
• Lin, Chuan-kai (2010). Practical Type Inference for the GADT Type System (PDF) (Doctoral Dissertation thesis). Portland State University.
Other