# Generalized algebraic data type

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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)  -- A GADT 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 (1994) and based on pattern matching in ALF.

Generalized algebraic data types were introduced independently by Cheney & Hinze (2003) and prior by Xi, Chen & Chen (2003) 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 & Stuckey (2006) 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. ^
2. ^
3. ^
4. ^ "OCaml 4.00.1". ocaml.org.
5. ^ Cheney & Hinze 2003, p. 25.
6. ^ Cheney & Hinze 2003, pp. 25–26.
7. ^
8. ^ Schrijvers et al. 2009, p. 1.
9. ^ Kmetiuk, Anatolii. "SCALA 3 IS HERE!🎉🎉🎉". scala-lang.org. École Polytechnique Fédérale Lausanne (EPFL) Lausanne, Switzerland. Retrieved 19 May 2021.
10. ^ "SCALA 3 — BOOK ALGEBRAIC DATA TYPES". scala-lang.org. École Polytechnique Fédérale Lausanne (EPFL) Lausanne, Switzerland. Retrieved 19 May 2021.
11. ^ Odersky, Martin. "A Tour of Scala 3 - Martin Odersky". youtube.com. Scala Days Conferences. Archived from the original on 2021-12-19. Retrieved 19 May 2021.
12. ^