GADTs for dummies
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For a long time, I didn't understand what GADTs were or how they could be used. It was sort of a conspiracy of silence — people who understood GADTs thought
that everything was obvious and didn't need further explanation, but I still
couldn't understand them.
Now that I have an idea of how it works, I think that it was really obvious. :) So, I want to share my understanding of GADTs. Maybe the way I realized how GADTs work could help someone else. See also Generalised algebraic datatype
Contents 
1 Type functions
A "data" declaration is a way to declare both type constructor and data constructors. For example,
data Either a b = Left a  Right b
declares type constructor "Either" and two data constructors "Left" and "Right". Ordinary Haskell functions works with data constructors:
isLeft (Left a) = True isLeft (Right b) = False
but there is also an analogous way to work with type constructors!
type X a = Either a a
declares a TYPE FUNCTION named "X". Its parameter "a" must be some type and it returns some type as its result. We can't use "X" on data values, but we can use it on type values. Type constructors declared with "data" statements and type functions declared with "type" statements can be used together to build arbitrarily complex types. In such "computations" type constructors serves as basic "values" and type functions as a way to process them.
Indeed, type functions in Haskell are very limited compared to ordinary functions  they don't support pattern matching, nor multiple statements, nor recursion.
2 Hypothetical Haskell extension  Fullfeatured type functions
Let's build a hypothetical Haskell extension that mimics, for type functions, the wellknown ways to define ordinary functions, including pattern matching:
type F [a] = Set a
multiple statements (this is meaningful only in the presence of pattern matching):
type F Bool = Char F String = Int
and recursion (which again needs pattern matching and multiple statements):
type F [a] = F a F (Map a b) = F b F (Set a) = F a F a = a
As you may already have guessed, this last definition calculates a simple base type of arbitrarilynested collections, e.g.:
F [[[Set Int]]] = F [[Set Int]] = F [Set Int] = F (Set Int) = F Int = Int
Let's not forget about statement guards:
type F a  IsSimple a == TrueType = a
Here we define type function F only for simple datatypes by using a guard type function "IsSimple":
type IsSimple Bool = TrueType IsSimple Int = TrueType .... IsSimple Double = TrueType IsSimple a = FalseType data TrueType = T data FalseType = F
These definitions seem a bit odd, and while we are in imaginary land, let's consider a way to write this shorter:
type F a  IsSimple a = a type IsSimple Bool IsSimple Int .... IsSimple Double
Here, we defined a list of simple types. The implied result of all written statements for "IsSimple" is True, and False for everything else. Essentially, "IsSimple" is a TYPE PREDICATE!
I really love it! :) How about constructing a predicate that traverses a complex type trying to decide whether it contains "Int" anywhere?
type HasInt Int HasInt [a] = HasInt a HasInt (Set a) = HasInt a HasInt (Map a b)  HasInt a HasInt (Map a b)  HasInt b
or a type function that substitutes one type with another inside arbitrarilydeep types:
type Replace t a b  t==a = b Replace [t] a b = [Replace t a b] Replace (Set t) a b = Set (Replace t a b) Replace (Map t1 t2) a b = Map (Replace t1 a b) (Replace t2 a b) Replace t a b = t
3 One more hypothetical extension  multivalue type functions
Let's add more fun! We will introduce one more hypothetical Haskell extension  type functions that may have MULTIPLE VALUES. Say,
type Collection a = [a] Collection a = Set a Collection a = Map b a
So, "Collection Int" has "[Int]", "Set Int" and "Map String Int" as its values, i.e. different collection types with elements of type "Int".
Pay attention to the last statement of the "Collection" definition, where we used the type variable "b" that was not mentioned on the left side, nor defined in any other way. Since it's perfectly possible for the "Collection" function to have multiple values, using some free variable on the right side that can be replaced with any type is not a problem at all. "Map Bool Int", "Map [Int] Int" and "Map Int Int" all are possible values of "Collection Int" along with "[Int]" and "Set Int".
At first glance, it seems that multiplevalue functions are meaningless  they can't be used to define datatypes, because we need concrete types here. But if we take another look, they can be useful to define type constraints and type families.
We can also represent a multiplevalue function as a predicate:
type Collection a [a] Collection a (Set a) Collection a (Map b a)
If you're familiar with Prolog, you should know that a predicate, in contrast to a function, is a multidirectional thing  it can be used to deduce any parameter from the other ones. For example, in this hypothetical definition:
head  Collection Int a :: a > Int
we define a 'head' function for any Collection containing Ints.
And in this, again, hypothetical definition:
data Safe c  Collection c a = Safe c a
we deduced element type 'a' from collection type 'c' passed as the parameter to the type constructor.
4 Back to real Haskell  type classes
After reading about all of these glorious examples, you may be wondering "Why doesn't Haskell support fullfeatured type functions?" Hold your breath... Haskell already contains them, and GHC has implemented all of the capabilities mentioned above for more than 10 years! They were just named... TYPE CLASSES! Let's translate all of our examples to their language:
class IsSimple a instance IsSimple Bool instance IsSimple Int .... instance IsSimple Double
The Haskell'98 standard supports type classes with only one parameter. That limits us to only defining type predicates like this one. But GHC and Hugs support multiparameter type classes that allow us to define arbitrarilycomplex type functions
class Collection a c instance Collection a [a] instance Collection a (Set a) instance Collection a (Map b a)
All of the "hypothetical" Haskell extensions we investigated earlier are actually implemented at the type class level!
Pattern matching:
instance Collection a [a]
Multiple statements:
instance Collection a [a] instance Collection a (Set a)
Recursion:
instance (Collection a c) => Collection a [c]
Pattern guards:
instance (IsSimple a) => Collection a (UArray a)
Let's define a type class which contains any collection which uses Int in its elements or indexes:
class HasInt a instance HasInt Int instance (HasInt a) => HasInt [a] instance (HasInt a) => HasInt (Map a b) instance (HasInt b) => HasInt (Map a b)
Another example is a class that replaces all occurrences of 'a' with
'b' in type 't' and return the result as 'res':
class Replace t a b res instance Replace t a a t instance Replace [t] a b [Replace t a b] instance (Replace t a b res) => Replace (Set t) a b (Set res) instance (Replace t1 a b res1, Replace t2 a b res2) => Replace (Map t1 t2) a b (Map res1 res2) instance Replace t a b t
You can compare it to the hypothetical definition we gave earlier. It's important to note that type class instances, as opposed to function statements, are not checked in order. Instead, the most _specific_ instance is automatically selected. So, in the Replace case, the last instance, which is the most general instance, will be selected only if all the others fail to match, which is what we want.
In many other cases this automatic selection is not powerful enough and we are forced to use some artificial tricks or complain to the language developers. The two most wellknown language extensions proposed to solve such problems are instance priorities, which allow us to explicitly specify instance selection order, and '/=' constraints, which can be used to explicitly prohibit unwanted matches:
instance Replace t a a t instance (a/=b) => Replace [t] a b [Replace t a b] instance (a/=b, t/=[_]) => Replace t a b t
You can check that these instances no longer overlap.
In practice, typelevel arithmetic by itself is not very useful. It becomes a
fantastic tool when combined with another feature that type classes provide 
member functions. For example:
class Collection a c where foldr1 :: (a > a > a) > c > a class Num a where (+) :: a > a > a sum :: (Num a, Collection a c) => c > a sum = foldr1 (+)
I'll also be glad to see the possibility of using type classes in data
declarations, like this:
data Safe c = (Collection c a) => Safe c a
but as far as I know, this is not yet implemented.
UNIFICATION
...
5 Back to GADTs
If you are wondering how all of these interesting type manipulations relate to GADTs, here is the answer. As you know, Haskell contains highly developed ways to express datatodata functions. We also know that Haskell contains rich facilities to write typetotype functions in the form of "type" statements and type classes. But how do "data" statements fit into this infrastructure?
My answer: they just define a typetodata constructor translation. Moreover, this translation may give multiple results. Say, the following definition:
data Maybe a = Just a  Nothing
defines typetodata constructors function "Maybe" that has a parameter "a" and for each "a" has two possible results  "Just a" and "Nothing". We can rewrite it in the same hypothetical syntax that was used above for multivalue type functions:
data Maybe a = Just a Maybe a = Nothing
Or how about this:
data List a = Cons a (List a) List a = Nil
and this:
data Either a b = Left a Either a b = Right b
But how flexible are "data" definitions? As you should remember, "type" definitions were very limited in their features, while type classes, on the other hand, were more developed than ordinary Haskell functions facilities. What about features of "data" definitions examined as sort of functions?
On the one hand, they supports multiple statements and multiple results and can be recursive, like the "List" definition above. On the other, that's all  no pattern matching or even type constants on the left side and no guards.
Lack of pattern matching means that the left side can contain only free type variables. That in turn means that the left sides of all "data" statements for a type will be essentially the same. Therefore, repeated left sides in multistatement "data" definitions are omitted and instead of
data Either a b = Left a Either a b = Right b
we write just
data Either a b = Left a  Right b
And here we finally come to GADTs! It's just a way to define data types using
pattern matching and constants on the left side of "data" statements!
Let's say we want to do this:
data T String = D1 Int T Bool = D2 T [a] = D3 (a,a)
We cannot do this using a standard data definition. So, now we must use a GADT definition:
data T a where D1 :: Int > T String D2 :: T Bool D3 :: (a,a) > T [a]
Amazed? After all, GADTs seem to be a really simple and obvious extension to data type definition facilities.
The idea here is to allow a data constructor's return type to be specified directly:
data Term a where Lit :: Int > Term Int Pair :: Term a > Term b > Term (a,b) ...
In a function that performs pattern matching on Term, the pattern match gives type as well as value information. For example, consider this function:
eval :: Term a > a eval (Lit i) = i eval (Pair a b) = (eval a, eval b) ...
If the argument matches Lit, it must have been built with a Lit constructor, so type 'a' must be Int, and hence we can return 'i' (an Int) in the right hand side. The same thing applies to the Pair constructor.
6 Further reading
The best paper on type level arithmetic using type classes I've seen is "Faking it: simulating dependent types in Haskell" ( http://www.cs.nott.ac.uk/~ctm/faking.ps.gz ). Most of this article comes from his work.
A great demonstration of typelevel arithmetic is in the TypeNats package, which "defines typelevel natural numbers and arithmetic operations on them including addition, subtraction, multiplication, division and GCD" ( darcs get partial tag '0.1' http://www.eecs.tufts.edu/~rdocki01/typenats/ )
I should also mention here Oleg Kiselyov's page on typelevel programming in Haskell: http://okmij.org/ftp/Haskell/types.html
There are plenty of GADTrelated papers, but the best one for beginners
is "Fun with phantom types"
(http://www.informatik.unibonn.de/~ralf/publications/With.pdf).
Phantom types is another name of GADT. You should also know that this
paper uses old GADT syntax. This paper is a mustread because it
contains numerous examples of practical GADT usage  a theme completely
omitted from my article.
Other GADTrelated papers:
"Dynamic Optimization for Functional Reactive Programming using Generalized Algebraic Data Types" http://www.cs.nott.ac.uk/~nhn/Publications/icfp2005.pdf
"Phantom types" (actually more scientific version of "Fun with phantom types") http://citeseer.ist.psu.edu/rd/0,606209,1,0.25,Download/http:qSqqSqwww.informatik.unibonn.deqSq~ralfqSqpublicationsqSqPhantom.ps.gz
"Phantom types and subtyping" http://arxiv.org/ps/cs.PL/0403034
"Existentially quantified type classes" by Stuckey, Sulzmann and Wazny (URL?)