Generalized algebraic data type

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In functional programming, a generalized algebraic data type (GADT, also first-class phantom type,(Cheney Hinze) guarded recursive datatype,(Xi Chen) or equality-qualified type(Sheard Pasalic)) is a generalization of parametric algebraic data types.


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 = Cons 12 (Cons 107 Nil)       -- the type of integers is List Int
strings = Cons "boat" (Cons "dock" Nil) -- the type of strings is List String

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 = EEqual (EInt 2) (EInt 3)        -- the type of expr1 is Expr Bool
ret = eval expr1                        -- ret is 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.[1]

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


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.(Cheney Hinze) 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.(Cheney Hinze)

(Sulzmann Wazny) introduced extended algebraic data types which combine GADTs together with the existential data types and type class constraints introduced by (Perry 1991), (Läufer Odersky) and (Läufer 1996).

Type inference in the absence of any programmer supplied type annotations is undecidable(Peyton Jones Washburn) and functions defined over GADTs do not admit principal types in general.(Schrijvers Peyton Jones) Type reconstruction requires several design trade-offs and is an area of active research (Peyton Jones, Washburn & Weirich 2004; Peyton Jones et al. 2006; Pottier & Régis-Gianas 2006; Sulzmann, Schrijvers & Stuckey 2006; Simonet & Pottier 2007; Schrijvers et al. 2009; Lin & Sheard 2010a; Lin & Sheard 2010b; Vytiniotis, Peyton Jones & Schrijvers 2010; Vytiniotis et al. 2011).


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.(Peyton Jones Washburn)

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)))))

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.

See also


Further reading

  • Patricia Johann and Neil Ghani (2008). "Foundations for Structured Programming with GADTs".
  • Arie Middelkoop, Atze Dijkstra and S. Doaitse Swierstra (2011). "A lean specification for GADTs: system F with first-class equality proofs". Higher-Order and Symbolic Computation.
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.. 8th International Symposium on Functional and Logic Programming (FLOPS 2006). 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. Programming Languages and Systems: 4th Asian Symposium (APLAS 2006). 4279. pp. 26–43. 
  • Schrijvers, Tom; Peyton Jones, Simon; Sulzmann, Martin; Vytiniotis, Dimitrios (2009). "Complete and Decidable Type Inference for GADTs". Proceedings of the ACM International Conference on Functional Programming (ICFP'09), Edinburgh. 
  • Lin, Chuan-kai (2010). Practical Type Inference for the GADT Type System (PDF) (Doctoral Dissertation thesis). Portland State University.CS1 maint: ref=harv (link)
  • Andrew Kennedy and Claudio V. Russo. "Generalized algebraic data types and object-oriented programming". In Proceedings of the 20th annual ACM SIGPLAN conference on Object oriented programming, systems, languages, and applications. ACM Press, 2005.

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