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 a parametric algebraic data type (ADT).
Overview
editIn 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 Glasgow Haskell Compiler (GHC) 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
editAn 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 the algebraic data types of ML and Haskell.[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 annotation, 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 was released.[9] This major update of Scala introduced the possibility to write GADTs[10] with the same syntax as algebraic data types, which is not the case in other programming languages according to Martin Odersky.[11]
Applications
editApplications 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
editAn 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, product 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)
Problems would have occurred 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, it is still restricted to one base type. Further, 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
editNotes
edit- ^ Cheney & Hinze 2003.
- ^ Xi, Chen & Chen 2003.
- ^ Sheard & Pasalic 2004.
- ^ "OCaml 4.00.1". ocaml.org.
- ^ Cheney & Hinze 2003, p. 25.
- ^ Cheney & Hinze 2003, pp. 25–26.
- ^ Peyton Jones, Washburn & Weirich 2004, p. 7.
- ^ Schrijvers et al. 2009, p. 1.
- ^ Kmetiuk, Anatolii. "Scala 3 Is Here!". scala-lang.org. École Polytechnique Fédérale Lausanne (EPFL) Lausanne, Switzerland. Retrieved 19 May 2021.
- ^ "Scala 3 – Book Algebraic Data Types". scala-lang.org. École Polytechnique Fédérale Lausanne (EPFL) Lausanne, Switzerland. Retrieved 19 May 2021.
- ^ 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.
- ^ Peyton Jones, Washburn & Weirich 2004, p. 3.
Further reading
edit- Applications
- Augustsson, Lennart; Petersson, Kent (September 1994). "Silly type families" (PDF).
- Cheney, James; Hinze, Ralf (2003). "First-Class Phantom Types". Technical Report CUCIS TR2003-1901. Cornell University. hdl:1813/5614.
- 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. CiteSeerX 10.1.1.59.4622. doi:10.1145/604131.604150. ISBN 978-1581136289. S2CID 15095297.
- 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
- 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" (PDF). 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" (PDF). 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. (eds.). 8th International Symposium on Functional and Logic Programming (FLOPS 2006). Lecture Notes in Computer Science. Vol. 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 (ed.). Programming Languages and Systems: 4th Asian Symposium (APLAS 2006). Lecture Notes in Computer Science. Vol. 4279. pp. 26–43.
- Schrijvers, Tom; Peyton Jones, Simon; Sulzmann, Martin; Vytiniotis, Dimitrios (2009). "Complete and decidable type inference for GADTs" (PDF). Proceedings of the 14th ACM SIGPLAN international conference on Functional programming. pp. 341–352. doi:10.1145/1596550.1596599. ISBN 9781605583327. S2CID 11272015.
- Lin, Chuan-kai (2010). Practical Type Inference for the GADT Type System (PDF) (Doctoral Dissertation thesis). Portland State University. Archived from the original (PDF) on 2016-06-11. Retrieved 2011-08-08.
- Other
- 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.
External links
edit- Generalised Algebraic Datatype Page on the Haskell wiki
- Generalised Algebraic Data Types in the GHC Users' Guide
- Generalized Algebraic Data Types and Object-Oriented Programming
- GADTs – Haskell Prime – Trac
- Papers about type inference for GADTs, bibliography by Simon Peyton Jones
- Type inference with constraints, bibliography by Simon Peyton Jones
- Emulating GADTs in Java via the Yoneda lemma