X-Git-Url: http://git.megacz.com/?a=blobdiff_plain;f=docs%2Fusers_guide%2Fglasgow_exts.xml;h=6dfda6b84fbeee125d1af01fff7ed64496cfb7b4;hb=586149353a755cf7ba7c0bd499a397c3c8230839;hp=052c9c3dea317662a2e163aa8a1dc30bef852cca;hpb=b55fe864599e77f2ae2a3fbeec899ea7aeeac9f2;p=ghc-hetmet.git
diff --git a/docs/users_guide/glasgow_exts.xml b/docs/users_guide/glasgow_exts.xml
index 052c9c3..6dfda6b 100644
--- a/docs/users_guide/glasgow_exts.xml
+++ b/docs/users_guide/glasgow_exts.xml
@@ -620,7 +620,7 @@ to write clunky would be to use case expressions:
-clunky env var1 var1 = case lookup env var1 of
+clunky env var1 var2 = case lookup env var1 of
Nothing -> fail
Just val1 -> case lookup env var2 of
Nothing -> fail
@@ -645,7 +645,7 @@ Here is how I would write clunky:
-clunky env var1 var1
+clunky env var1 var2
| Just val1 <- lookup env var1
, Just val2 <- lookup env var2
= val1 + val2
@@ -939,14 +939,10 @@ definitions; you must define such a function in prefix form.
-
-Type system extensions
+
+Extensions to data types and type synonyms
-
-
-Data types and type synonyms
-
-
+Data types with no constructorsWith the flag, GHC lets you declare
@@ -964,9 +960,9 @@ not * then an explicit kind annotation must be used
Such data types have only one value, namely bottom.
Nevertheless, they can be useful when defining "phantom types".
-
+
-
+Infix type constructors, classes, and type variables
@@ -1033,9 +1029,9 @@ to be written infix, very much like expressions. More specifically:
-
+
-
+Liberalised type synonyms
@@ -1125,10 +1121,10 @@ this will be rejected:
because GHC does not allow unboxed tuples on the left of a function arrow.
-
+
-
+Existentially quantified data constructors
@@ -1309,7 +1305,7 @@ universal quantification earlier.
-
+Record Constructors
@@ -1361,20 +1357,6 @@ main = do
display (inc (inc counterB)) -- prints "##"
-In GADT declarations (see ), the explicit
-forall may be omitted. For example, we can express
-the same Counter a using GADT:
-
-
-data Counter a where
- NewCounter { _this :: self
- , _inc :: self -> self
- , _display :: self -> IO ()
- , tag :: a
- }
- :: Counter a
-
-
At the moment, record update syntax is only supported for Haskell 98 data types,
so the following function does not work:
@@ -1515,7 +1497,7 @@ are convincing reasons to change it.
You can't use deriving to define instances of a
data type with existentially quantified data constructors.
-Reason: in most cases it would not make sense. For example:#
+Reason: in most cases it would not make sense. For example:;
data T = forall a. MkT [a] deriving( Eq )
@@ -1541,213 +1523,782 @@ declarations. Define your own instances!
-
-
+
+
+Declaring data types with explicit constructor signatures
-
-Class declarations
-
-
-This section, and the next one, documents GHC's type-class extensions.
-There's lots of background in the paper Type
-classes: exploring the design space (Simon Peyton Jones, Mark
-Jones, Erik Meijer).
-
-
-All the extensions are enabled by the flag.
-
-
-
-Multi-parameter type classes
-
-Multi-parameter type classes are permitted. For example:
-
-
+GHC allows you to declare an algebraic data type by
+giving the type signatures of constructors explicitly. For example:
- class Collection c a where
- union :: c a -> c a -> c a
- ...etc.
+ data Maybe a where
+ Nothing :: Maybe a
+ Just :: a -> Maybe a
-
-
-
-
-
-The superclasses of a class declaration
-
-
-There are no restrictions on the context in a class declaration
-(which introduces superclasses), except that the class hierarchy must
-be acyclic. So these class declarations are OK:
-
-
+The form is called a "GADT-style declaration"
+because Generalised Algebraic Data Types, described in ,
+can only be declared using this form.
+Notice that GADT-style syntax generalises existential types ().
+For example, these two declarations are equivalent:
- class Functor (m k) => FiniteMap m k where
- ...
-
- class (Monad m, Monad (t m)) => Transform t m where
- lift :: m a -> (t m) a
+ data Foo = forall a. MkFoo a (a -> Bool)
+ data Foo' where { MKFoo :: a -> (a->Bool) -> Foo' }
-
-
-
-As in Haskell 98, The class hierarchy must be acyclic. However, the definition
-of "acyclic" involves only the superclass relationships. For example,
-this is OK:
-
-
+Any data type that can be declared in standard Haskell-98 syntax
+can also be declared using GADT-style syntax.
+The choice is largely stylistic, but GADT-style declarations differ in one important respect:
+they treat class constraints on the data constructors differently.
+Specifically, if the constructor is given a type-class context, that
+context is made available by pattern matching. For example:
- class C a where {
- op :: D b => a -> b -> b
- }
-
- class C a => D a where { ... }
-
+ data Set a where
+ MkSet :: Eq a => [a] -> Set a
+ makeSet :: Eq a => [a] -> Set a
+ makeSet xs = MkSet (nub xs)
-Here, C is a superclass of D, but it's OK for a
-class operation op of C to mention D. (It
-would not be OK for D to be a superclass of C.)
+ insert :: a -> Set a -> Set a
+ insert a (MkSet as) | a `elem` as = MkSet as
+ | otherwise = MkSet (a:as)
+
+A use of MkSet as a constructor (e.g. in the definition of makeSet)
+gives rise to a (Eq a)
+constraint, as you would expect. The new feature is that pattern-matching on MkSet
+(as in the definition of insert) makes available an (Eq a)
+context. In implementation terms, the MkSet constructor has a hidden field that stores
+the (Eq a) dictionary that is passed to MkSet; so
+when pattern-matching that dictionary becomes available for the right-hand side of the match.
+In the example, the equality dictionary is used to satisfy the equality constraint
+generated by the call to elem, so that the type of
+insert itself has no Eq constraint.
-
-
-
-
-
-
-Class method types
-
-
-Haskell 98 prohibits class method types to mention constraints on the
-class type variable, thus:
+This behaviour contrasts with Haskell 98's peculiar treament of
+contexts on a data type declaration (Section 4.2.1 of the Haskell 98 Report).
+In Haskell 98 the defintion
- class Seq s a where
- fromList :: [a] -> s a
- elem :: Eq a => a -> s a -> Bool
+ data Eq a => Set' a = MkSet' [a]
-The type of elem is illegal in Haskell 98, because it
-contains the constraint Eq a, constrains only the
-class type variable (in this case a).
-GHC lifts this restriction.
-
-
-
-
-
-
-
-Functional dependencies
-
-
- Functional dependencies are implemented as described by Mark Jones
-in “Type Classes with Functional Dependencies”, Mark P. Jones,
-In Proceedings of the 9th European Symposium on Programming,
-ESOP 2000, Berlin, Germany, March 2000, Springer-Verlag LNCS 1782,
-.
-
+gives MkSet' the same type as MkSet above. But instead of
+making available an (Eq a) constraint, pattern-matching
+on MkSet'requires an (Eq a) constraint!
+GHC faithfully implements this behaviour, odd though it is. But for GADT-style declarations,
+GHC's behaviour is much more useful, as well as much more intuitive.
-Functional dependencies are introduced by a vertical bar in the syntax of a
-class declaration; e.g.
+For example, a possible application of GHC's behaviour is to reify dictionaries:
- class (Monad m) => MonadState s m | m -> s where ...
+ data NumInst a where
+ MkNumInst :: Num a => NumInst a
- class Foo a b c | a b -> c where ...
+ intInst :: NumInst Int
+ intInst = MkNumInst
+
+ plus :: NumInst a -> a -> a -> a
+ plus MkNumInst p q = p + q
-There should be more documentation, but there isn't (yet). Yell if you need it.
+Here, a value of type NumInst a is equivalent
+to an explicit (Num a) dictionary.
-Rules for functional dependencies
-In a class declaration, all of the class type variables must be reachable (in the sense
-mentioned in )
-from the free variables of each method type.
-For example:
+The rest of this section gives further details about GADT-style data
+type declarations.
+
+
+
+The result type of each data constructor must begin with the type constructor being defined.
+If the result type of all constructors
+has the form T a1 ... an, where a1 ... an
+are distinct type variables, then the data type is ordinary;
+otherwise is a generalised data type ().
+
+
+The type signature of
+each constructor is independent, and is implicitly universally quantified as usual.
+Different constructors may have different universally-quantified type variables
+and different type-class constraints.
+For example, this is fine:
- class Coll s a where
- empty :: s
- insert :: s -> a -> s
+ data T a where
+ T1 :: Eq b => b -> T b
+ T2 :: (Show c, Ix c) => c -> [c] -> T c
+
-is not OK, because the type of empty doesn't mention
-a. Functional dependencies can make the type variable
-reachable:
+
+Unlike a Haskell-98-style
+data type declaration, the type variable(s) in the "data Set a where" header
+have no scope. Indeed, one can write a kind signature instead:
- class Coll s a | s -> a where
- empty :: s
- insert :: s -> a -> s
+ data Set :: * -> * where ...
+
+or even a mixture of the two:
+
+ data Foo a :: (* -> *) -> * where ...
+
+The type variables (if given) may be explicitly kinded, so we could also write the header for Foo
+like this:
+
+ data Foo a (b :: * -> *) where ...
+
-Alternatively Coll might be rewritten
+
+You can use strictness annotations, in the obvious places
+in the constructor type:
- class Coll s a where
- empty :: s a
- insert :: s a -> a -> s a
+ data Term a where
+ Lit :: !Int -> Term Int
+ If :: Term Bool -> !(Term a) -> !(Term a) -> Term a
+ Pair :: Term a -> Term b -> Term (a,b)
+
+
+You can use a deriving clause on a GADT-style data type
+declaration. For example, these two declarations are equivalent
+
+ data Maybe1 a where {
+ Nothing1 :: Maybe1 a ;
+ Just1 :: a -> Maybe1 a
+ } deriving( Eq, Ord )
-which makes the connection between the type of a collection of
-a's (namely (s a)) and the element type a.
-Occasionally this really doesn't work, in which case you can split the
-class like this:
+ data Maybe2 a = Nothing2 | Just2 a
+ deriving( Eq, Ord )
+
+
+
+You can use record syntax on a GADT-style data type declaration:
- class CollE s where
- empty :: s
-
- class CollE s => Coll s a where
- insert :: s -> a -> s
+ data Person where
+ Adult { name :: String, children :: [Person] } :: Person
+ Child { name :: String } :: Person
+As usual, for every constructor that has a field f, the type of
+field f must be the same (modulo alpha conversion).
-
+
+At the moment, record updates are not yet possible with GADT-style declarations,
+so support is limited to record construction, selection and pattern matching.
+For exmaple
+
+ aPerson = Adult { name = "Fred", children = [] }
+ shortName :: Person -> Bool
+ hasChildren (Adult { children = kids }) = not (null kids)
+ hasChildren (Child {}) = False
+
+
-
-Background on functional dependencies
+
+As in the case of existentials declared using the Haskell-98-like record syntax
+(),
+record-selector functions are generated only for those fields that have well-typed
+selectors.
+Here is the example of that section, in GADT-style syntax:
+
+data Counter a where
+ NewCounter { _this :: self
+ , _inc :: self -> self
+ , _display :: self -> IO ()
+ , tag :: a
+ }
+ :: Counter a
+
+As before, only one selector function is generated here, that for tag.
+Nevertheless, you can still use all the field names in pattern matching and record construction.
+
+
+
-The following description of the motivation and use of functional dependencies is taken
-from the Hugs user manual, reproduced here (with minor changes) by kind
-permission of Mark Jones.
-
-
-Consider the following class, intended as part of a
-library for collection types:
+
+Generalised Algebraic Data Types (GADTs)
+
+Generalised Algebraic Data Types generalise ordinary algebraic data types
+by allowing constructors to have richer return types. Here is an example:
- class Collects e ce where
- empty :: ce
- insert :: e -> ce -> ce
- member :: e -> ce -> Bool
+ data Term a where
+ Lit :: Int -> Term Int
+ Succ :: Term Int -> Term Int
+ IsZero :: Term Int -> Term Bool
+ If :: Term Bool -> Term a -> Term a -> Term a
+ Pair :: Term a -> Term b -> Term (a,b)
-The type variable e used here represents the element type, while ce is the type
-of the container itself. Within this framework, we might want to define
-instances of this class for lists or characteristic functions (both of which
-can be used to represent collections of any equality type), bit sets (which can
-be used to represent collections of characters), or hash tables (which can be
-used to represent any collection whose elements have a hash function). Omitting
-standard implementation details, this would lead to the following declarations:
+Notice that the return type of the constructors is not always Term a, as is the
+case with ordinary data types. This generality allows us to
+write a well-typed eval function
+for these Terms:
- instance Eq e => Collects e [e] where ...
- instance Eq e => Collects e (e -> Bool) where ...
- instance Collects Char BitSet where ...
- instance (Hashable e, Collects a ce)
- => Collects e (Array Int ce) where ...
+ eval :: Term a -> a
+ eval (Lit i) = i
+ eval (Succ t) = 1 + eval t
+ eval (IsZero t) = eval t == 0
+ eval (If b e1 e2) = if eval b then eval e1 else eval e2
+ eval (Pair e1 e2) = (eval e1, eval e2)
-All this looks quite promising; we have a class and a range of interesting
-implementations. Unfortunately, there are some serious problems with the class
-declaration. First, the empty function has an ambiguous type:
+The key point about GADTs is that pattern matching causes type refinement.
+For example, in the right hand side of the equation
- empty :: Collects e ce => ce
+ eval :: Term a -> a
+ eval (Lit i) = ...
-By "ambiguous" we mean that there is a type variable e that appears on the left
-of the => symbol, but not on the right. The problem with
-this is that, according to the theoretical foundations of Haskell overloading,
+the type a is refined to Int. That's the whole point!
+A precise specification of the type rules is beyond what this user manual aspires to,
+but the design closely follows that described in
+the paper Simple
+unification-based type inference for GADTs,
+(ICFP 2006).
+The general principle is this: type refinement is only carried out
+based on user-supplied type annotations.
+So if no type signature is supplied for eval, no type refinement happens,
+and lots of obscure error messages will
+occur. However, the refinement is quite general. For example, if we had:
+
+ eval :: Term a -> a -> a
+ eval (Lit i) j = i+j
+
+the pattern match causes the type a to be refined to Int (because of the type
+of the constructor Lit), and that refinement also applies to the type of j, and
+the result type of the case expression. Hence the addition i+j is legal.
+
+
+These and many other examples are given in papers by Hongwei Xi, and
+Tim Sheard. There is a longer introduction
+on the wiki,
+and Ralf Hinze's
+Fun with phantom types also has a number of examples. Note that papers
+may use different notation to that implemented in GHC.
+
+
+The rest of this section outlines the extensions to GHC that support GADTs.
+
+
+A GADT can only be declared using GADT-style syntax ();
+the old Haskell-98 syntax for data declarations always declares an ordinary data type.
+The result type of each constructor must begin with the type constructor being defined,
+but for a GADT the arguments to the type constructor can be arbitrary monotypes.
+For example, in the Term data
+type above, the type of each constructor must end with Term ty, but
+the ty may not be a type variable (e.g. the Lit
+constructor).
+
+
+
+You cannot use a deriving clause for a GADT; only for
+an ordianary data type.
+
+
+
+As mentioned in , record syntax is supported.
+For example:
+
+ data Term a where
+ Lit { val :: Int } :: Term Int
+ Succ { num :: Term Int } :: Term Int
+ Pred { num :: Term Int } :: Term Int
+ IsZero { arg :: Term Int } :: Term Bool
+ Pair { arg1 :: Term a
+ , arg2 :: Term b
+ } :: Term (a,b)
+ If { cnd :: Term Bool
+ , tru :: Term a
+ , fls :: Term a
+ } :: Term a
+
+However, for GADTs there is the following additional constraint:
+every constructor that has a field f must have
+the same result type (modulo alpha conversion)
+Hence, in the above example, we cannot merge the num
+and arg fields above into a
+single name. Although their field types are both Term Int,
+their selector functions actually have different types:
+
+
+ num :: Term Int -> Term Int
+ arg :: Term Bool -> Term Int
+
+
+
+
+
+
+
+
+
+
+
+
+Deriving clause for classes Typeable and Data
+
+
+Haskell 98 allows the programmer to add "deriving( Eq, Ord )" to a data type
+declaration, to generate a standard instance declaration for classes specified in the deriving clause.
+In Haskell 98, the only classes that may appear in the deriving clause are the standard
+classes Eq, Ord,
+Enum, Ix, Bounded, Read, and Show.
+
+
+GHC extends this list with two more classes that may be automatically derived
+(provided the flag is specified):
+Typeable, and Data. These classes are defined in the library
+modules Data.Typeable and Data.Generics respectively, and the
+appropriate class must be in scope before it can be mentioned in the deriving clause.
+
+An instance of Typeable can only be derived if the
+data type has seven or fewer type parameters, all of kind *.
+The reason for this is that the Typeable class is derived using the scheme
+described in
+
+Scrap More Boilerplate: Reflection, Zips, and Generalised Casts
+.
+(Section 7.4 of the paper describes the multiple Typeable classes that
+are used, and only Typeable1 up to
+Typeable7 are provided in the library.)
+In other cases, there is nothing to stop the programmer writing a TypableX
+class, whose kind suits that of the data type constructor, and
+then writing the data type instance by hand.
+
+
+
+
+Generalised derived instances for newtypes
+
+
+When you define an abstract type using newtype, you may want
+the new type to inherit some instances from its representation. In
+Haskell 98, you can inherit instances of Eq, Ord,
+Enum and Bounded by deriving them, but for any
+other classes you have to write an explicit instance declaration. For
+example, if you define
+
+
+ newtype Dollars = Dollars Int
+
+
+and you want to use arithmetic on Dollars, you have to
+explicitly define an instance of Num:
+
+
+ instance Num Dollars where
+ Dollars a + Dollars b = Dollars (a+b)
+ ...
+
+All the instance does is apply and remove the newtype
+constructor. It is particularly galling that, since the constructor
+doesn't appear at run-time, this instance declaration defines a
+dictionary which is wholly equivalent to the Int
+dictionary, only slower!
+
+
+
+ Generalising the deriving clause
+
+GHC now permits such instances to be derived instead, so one can write
+
+ newtype Dollars = Dollars Int deriving (Eq,Show,Num)
+
+
+and the implementation uses the sameNum dictionary
+for Dollars as for Int. Notionally, the compiler
+derives an instance declaration of the form
+
+
+ instance Num Int => Num Dollars
+
+
+which just adds or removes the newtype constructor according to the type.
+
+
+
+We can also derive instances of constructor classes in a similar
+way. For example, suppose we have implemented state and failure monad
+transformers, such that
+
+
+ instance Monad m => Monad (State s m)
+ instance Monad m => Monad (Failure m)
+
+In Haskell 98, we can define a parsing monad by
+
+ type Parser tok m a = State [tok] (Failure m) a
+
+
+which is automatically a monad thanks to the instance declarations
+above. With the extension, we can make the parser type abstract,
+without needing to write an instance of class Monad, via
+
+
+ newtype Parser tok m a = Parser (State [tok] (Failure m) a)
+ deriving Monad
+
+In this case the derived instance declaration is of the form
+
+ instance Monad (State [tok] (Failure m)) => Monad (Parser tok m)
+
+
+Notice that, since Monad is a constructor class, the
+instance is a partial application of the new type, not the
+entire left hand side. We can imagine that the type declaration is
+``eta-converted'' to generate the context of the instance
+declaration.
+
+
+
+We can even derive instances of multi-parameter classes, provided the
+newtype is the last class parameter. In this case, a ``partial
+application'' of the class appears in the deriving
+clause. For example, given the class
+
+
+ class StateMonad s m | m -> s where ...
+ instance Monad m => StateMonad s (State s m) where ...
+
+then we can derive an instance of StateMonad for Parsers by
+
+ newtype Parser tok m a = Parser (State [tok] (Failure m) a)
+ deriving (Monad, StateMonad [tok])
+
+
+The derived instance is obtained by completing the application of the
+class to the new type:
+
+
+ instance StateMonad [tok] (State [tok] (Failure m)) =>
+ StateMonad [tok] (Parser tok m)
+
+
+
+
+As a result of this extension, all derived instances in newtype
+ declarations are treated uniformly (and implemented just by reusing
+the dictionary for the representation type), except
+Show and Read, which really behave differently for
+the newtype and its representation.
+
+
+
+ A more precise specification
+
+Derived instance declarations are constructed as follows. Consider the
+declaration (after expansion of any type synonyms)
+
+
+ newtype T v1...vn = T' (t vk+1...vn) deriving (c1...cm)
+
+
+where
+
+
+ The ci are partial applications of
+ classes of the form C t1'...tj', where the arity of C
+ is exactly j+1. That is, C lacks exactly one type argument.
+
+
+ The k is chosen so that ci (T v1...vk) is well-kinded.
+
+
+ The type t is an arbitrary type.
+
+
+ The type variables vk+1...vn do not occur in t,
+ nor in the ci, and
+
+
+ None of the ci is Read, Show,
+ Typeable, or Data. These classes
+ should not "look through" the type or its constructor. You can still
+ derive these classes for a newtype, but it happens in the usual way, not
+ via this new mechanism.
+
+
+Then, for each ci, the derived instance
+declaration is:
+
+ instance ci t => ci (T v1...vk)
+
+As an example which does not work, consider
+
+ newtype NonMonad m s = NonMonad (State s m s) deriving Monad
+
+Here we cannot derive the instance
+
+ instance Monad (State s m) => Monad (NonMonad m)
+
+
+because the type variable s occurs in State s m,
+and so cannot be "eta-converted" away. It is a good thing that this
+deriving clause is rejected, because NonMonad m is
+not, in fact, a monad --- for the same reason. Try defining
+>>= with the correct type: you won't be able to.
+
+
+
+Notice also that the order of class parameters becomes
+important, since we can only derive instances for the last one. If the
+StateMonad class above were instead defined as
+
+
+ class StateMonad m s | m -> s where ...
+
+
+then we would not have been able to derive an instance for the
+Parser type above. We hypothesise that multi-parameter
+classes usually have one "main" parameter for which deriving new
+instances is most interesting.
+
+Lastly, all of this applies only for classes other than
+Read, Show, Typeable,
+and Data, for which the built-in derivation applies (section
+4.3.3. of the Haskell Report).
+(For the standard classes Eq, Ord,
+Ix, and Bounded it is immaterial whether
+the standard method is used or the one described here.)
+
+
+
+
+
+
+Stand-alone deriving declarations
+
+
+GHC now allows stand-alone deriving declarations, enabled by -fglasgow-exts:
+
+ data Foo a = Bar a | Baz String
+
+ derive instance Eq (Foo a)
+
+The token "derive" is a keyword only when followed by "instance";
+you can use it as a variable name elsewhere.
+The stand-alone syntax is generalised for newtypes in exactly the same
+way that ordinary deriving clauses are generalised ().
+For example:
+
+ newtype Foo a = MkFoo (State Int a)
+
+ derive instance MonadState Int Foo
+
+GHC always treats the last parameter of the instance
+(Foo in this exmample) as the type whose instance is being derived.
+
+
+
+
+
+
+
+
+
+Other type system extensions
+
+
+Class declarations
+
+
+This section, and the next one, documents GHC's type-class extensions.
+There's lots of background in the paper Type
+classes: exploring the design space (Simon Peyton Jones, Mark
+Jones, Erik Meijer).
+
+
+All the extensions are enabled by the flag.
+
+
+
+Multi-parameter type classes
+
+Multi-parameter type classes are permitted. For example:
+
+
+
+ class Collection c a where
+ union :: c a -> c a -> c a
+ ...etc.
+
+
+
+
+
+
+The superclasses of a class declaration
+
+
+There are no restrictions on the context in a class declaration
+(which introduces superclasses), except that the class hierarchy must
+be acyclic. So these class declarations are OK:
+
+
+
+ class Functor (m k) => FiniteMap m k where
+ ...
+
+ class (Monad m, Monad (t m)) => Transform t m where
+ lift :: m a -> (t m) a
+
+
+
+
+
+As in Haskell 98, The class hierarchy must be acyclic. However, the definition
+of "acyclic" involves only the superclass relationships. For example,
+this is OK:
+
+
+
+ class C a where {
+ op :: D b => a -> b -> b
+ }
+
+ class C a => D a where { ... }
+
+
+
+Here, C is a superclass of D, but it's OK for a
+class operation op of C to mention D. (It
+would not be OK for D to be a superclass of C.)
+
+
+
+
+
+
+
+Class method types
+
+
+Haskell 98 prohibits class method types to mention constraints on the
+class type variable, thus:
+
+ class Seq s a where
+ fromList :: [a] -> s a
+ elem :: Eq a => a -> s a -> Bool
+
+The type of elem is illegal in Haskell 98, because it
+contains the constraint Eq a, constrains only the
+class type variable (in this case a).
+GHC lifts this restriction.
+
+
+
+
+
+
+
+Functional dependencies
+
+
+ Functional dependencies are implemented as described by Mark Jones
+in “Type Classes with Functional Dependencies”, Mark P. Jones,
+In Proceedings of the 9th European Symposium on Programming,
+ESOP 2000, Berlin, Germany, March 2000, Springer-Verlag LNCS 1782,
+.
+
+
+Functional dependencies are introduced by a vertical bar in the syntax of a
+class declaration; e.g.
+
+ class (Monad m) => MonadState s m | m -> s where ...
+
+ class Foo a b c | a b -> c where ...
+
+There should be more documentation, but there isn't (yet). Yell if you need it.
+
+
+Rules for functional dependencies
+
+In a class declaration, all of the class type variables must be reachable (in the sense
+mentioned in )
+from the free variables of each method type.
+For example:
+
+
+ class Coll s a where
+ empty :: s
+ insert :: s -> a -> s
+
+
+is not OK, because the type of empty doesn't mention
+a. Functional dependencies can make the type variable
+reachable:
+
+ class Coll s a | s -> a where
+ empty :: s
+ insert :: s -> a -> s
+
+
+Alternatively Coll might be rewritten
+
+
+ class Coll s a where
+ empty :: s a
+ insert :: s a -> a -> s a
+
+
+
+which makes the connection between the type of a collection of
+a's (namely (s a)) and the element type a.
+Occasionally this really doesn't work, in which case you can split the
+class like this:
+
+
+
+ class CollE s where
+ empty :: s
+
+ class CollE s => Coll s a where
+ insert :: s -> a -> s
+
+
+
+
+
+
+Background on functional dependencies
+
+The following description of the motivation and use of functional dependencies is taken
+from the Hugs user manual, reproduced here (with minor changes) by kind
+permission of Mark Jones.
+
+
+Consider the following class, intended as part of a
+library for collection types:
+
+ class Collects e ce where
+ empty :: ce
+ insert :: e -> ce -> ce
+ member :: e -> ce -> Bool
+
+The type variable e used here represents the element type, while ce is the type
+of the container itself. Within this framework, we might want to define
+instances of this class for lists or characteristic functions (both of which
+can be used to represent collections of any equality type), bit sets (which can
+be used to represent collections of characters), or hash tables (which can be
+used to represent any collection whose elements have a hash function). Omitting
+standard implementation details, this would lead to the following declarations:
+
+ instance Eq e => Collects e [e] where ...
+ instance Eq e => Collects e (e -> Bool) where ...
+ instance Collects Char BitSet where ...
+ instance (Hashable e, Collects a ce)
+ => Collects e (Array Int ce) where ...
+
+All this looks quite promising; we have a class and a range of interesting
+implementations. Unfortunately, there are some serious problems with the class
+declaration. First, the empty function has an ambiguous type:
+
+ empty :: Collects e ce => ce
+
+By "ambiguous" we mean that there is a type variable e that appears on the left
+of the => symbol, but not on the right. The problem with
+this is that, according to the theoretical foundations of Haskell overloading,
we cannot guarantee a well-defined semantics for any term with an ambiguous
type.
@@ -2405,55 +2956,7 @@ territory free in case we need it later.
-
-For-all hoisting
-
-It is often convenient to use generalised type synonyms (see ) at the right hand
-end of an arrow, thus:
-
- type Discard a = forall b. a -> b -> a
-
- g :: Int -> Discard Int
- g x y z = x+y
-
-Simply expanding the type synonym would give
-
- g :: Int -> (forall b. Int -> b -> Int)
-
-but GHC "hoists" the forall to give the isomorphic type
-
- g :: forall b. Int -> Int -> b -> Int
-
-In general, the rule is this: to determine the type specified by any explicit
-user-written type (e.g. in a type signature), GHC expands type synonyms and then repeatedly
-performs the transformation:
-
- type1 -> forall a1..an. context2 => type2
-==>
- forall a1..an. context2 => type1 -> type2
-
-(In fact, GHC tries to retain as much synonym information as possible for use in
-error messages, but that is a usability issue.) This rule applies, of course, whether
-or not the forall comes from a synonym. For example, here is another
-valid way to write g's type signature:
-
- g :: Int -> Int -> forall b. b -> Int
-
-
-
-When doing this hoisting operation, GHC eliminates duplicate constraints. For
-example:
-
- type Foo a = (?x::Int) => Bool -> a
- g :: Foo (Foo Int)
-
-means
-
- g :: (?x::Int) => Bool -> Bool -> Int
-
-
-
-
+
@@ -2747,1011 +3250,749 @@ the parameter explicit:
x' = newName ns2
env = extend env x x'
-Notice the call to 'split' introduced by the type checker.
-How did it know to use 'splitNS'? Because what it really did
-was to introduce a call to the overloaded function 'split',
-defined by the class Splittable:
-
- class Splittable a where
- split :: a -> (a,a)
-
-The instance for Splittable NameSupply tells GHC how to implement
-split for name supplies. But we can simply write
-
- g x = (x, %ns, %ns)
-
-and GHC will infer
-
- g :: (Splittable a, %ns :: a) => b -> (b,a,a)
-
-The Splittable class is built into GHC. It's exported by module
-GHC.Exts.
-
-
-Other points:
-
- '?x' and '%x'
-are entirely distinct implicit parameters: you
- can use them together and they won't intefere with each other.
-
-
- You can bind linear implicit parameters in 'with' clauses.
-
-You cannot have implicit parameters (whether linear or not)
- in the context of a class or instance declaration.
-
-
-
-Warnings
-
-
-The monomorphism restriction is even more important than usual.
-Consider the example above:
-
- f :: (%ns :: NameSupply) => Env -> Expr -> Expr
- f env (Lam x e) = Lam x' (f env e)
- where
- x' = newName %ns
- env' = extend env x x'
-
-If we replaced the two occurrences of x' by (newName %ns), which is
-usually a harmless thing to do, we get:
-
- f :: (%ns :: NameSupply) => Env -> Expr -> Expr
- f env (Lam x e) = Lam (newName %ns) (f env e)
- where
- env' = extend env x (newName %ns)
-
-But now the name supply is consumed in three places
-(the two calls to newName,and the recursive call to f), so
-the result is utterly different. Urk! We don't even have
-the beta rule.
-
-
-Well, this is an experimental change. With implicit
-parameters we have already lost beta reduction anyway, and
-(as John Launchbury puts it) we can't sensibly reason about
-Haskell programs without knowing their typing.
-
-
-
-
-Recursive functions
-Linear implicit parameters can be particularly tricky when you have a recursive function
-Consider
-
- foo :: %x::T => Int -> [Int]
- foo 0 = []
- foo n = %x : foo (n-1)
-
-where T is some type in class Splittable.
-
-Do you get a list of all the same T's or all different T's
-(assuming that split gives two distinct T's back)?
-
-If you supply the type signature, taking advantage of polymorphic
-recursion, you get what you'd probably expect. Here's the
-translated term, where the implicit param is made explicit:
-
- foo x 0 = []
- foo x n = let (x1,x2) = split x
- in x1 : foo x2 (n-1)
-
-But if you don't supply a type signature, GHC uses the Hindley
-Milner trick of using a single monomorphic instance of the function
-for the recursive calls. That is what makes Hindley Milner type inference
-work. So the translation becomes
-
- foo x = let
- foom 0 = []
- foom n = x : foom (n-1)
- in
- foom
-
-Result: 'x' is not split, and you get a list of identical T's. So the
-semantics of the program depends on whether or not foo has a type signature.
-Yikes!
-
-You may say that this is a good reason to dislike linear implicit parameters
-and you'd be right. That is why they are an experimental feature.
-
-
-
-
-
-================ END OF Linear Implicit Parameters commented out -->
-
-
-Explicitly-kinded quantification
-
-
-Haskell infers the kind of each type variable. Sometimes it is nice to be able
-to give the kind explicitly as (machine-checked) documentation,
-just as it is nice to give a type signature for a function. On some occasions,
-it is essential to do so. For example, in his paper "Restricted Data Types in Haskell" (Haskell Workshop 1999)
-John Hughes had to define the data type:
-
- data Set cxt a = Set [a]
- | Unused (cxt a -> ())
-
-The only use for the Unused constructor was to force the correct
-kind for the type variable cxt.
-
-
-GHC now instead allows you to specify the kind of a type variable directly, wherever
-a type variable is explicitly bound. Namely:
-
-data declarations:
-
- data Set (cxt :: * -> *) a = Set [a]
-
-type declarations:
-
- type T (f :: * -> *) = f Int
-
-class declarations:
-
- class (Eq a) => C (f :: * -> *) a where ...
-
-forall's in type signatures:
-
- f :: forall (cxt :: * -> *). Set cxt Int
-
-
-
-
-
-The parentheses are required. Some of the spaces are required too, to
-separate the lexemes. If you write (f::*->*) you
-will get a parse error, because "::*->*" is a
-single lexeme in Haskell.
-
-
-
-As part of the same extension, you can put kind annotations in types
-as well. Thus:
-
- f :: (Int :: *) -> Int
- g :: forall a. a -> (a :: *)
-
-The syntax is
-
- atype ::= '(' ctype '::' kind ')
-
-The parentheses are required.
-
-
-
-
-
-Arbitrary-rank polymorphism
-
-
-
-Haskell type signatures are implicitly quantified. The new keyword forall
-allows us to say exactly what this means. For example:
-
-
-
- g :: b -> b
-
-means this:
-
- g :: forall b. (b -> b)
-
-The two are treated identically.
-
-
-
-However, GHC's type system supports arbitrary-rank
-explicit universal quantification in
-types.
-For example, all the following types are legal:
-
- f1 :: forall a b. a -> b -> a
- g1 :: forall a b. (Ord a, Eq b) => a -> b -> a
-
- f2 :: (forall a. a->a) -> Int -> Int
- g2 :: (forall a. Eq a => [a] -> a -> Bool) -> Int -> Int
-
- f3 :: ((forall a. a->a) -> Int) -> Bool -> Bool
-
-Here, f1 and g1 are rank-1 types, and
-can be written in standard Haskell (e.g. f1 :: a->b->a).
-The forall makes explicit the universal quantification that
-is implicitly added by Haskell.
-
-
-The functions f2 and g2 have rank-2 types;
-the forall is on the left of a function arrow. As g2
-shows, the polymorphic type on the left of the function arrow can be overloaded.
-
-
-The function f3 has a rank-3 type;
-it has rank-2 types on the left of a function arrow.
-
-
-GHC allows types of arbitrary rank; you can nest foralls
-arbitrarily deep in function arrows. (GHC used to be restricted to rank 2, but
-that restriction has now been lifted.)
-In particular, a forall-type (also called a "type scheme"),
-including an operational type class context, is legal:
-
- On the left of a function arrow
- On the right of a function arrow (see )
- As the argument of a constructor, or type of a field, in a data type declaration. For
-example, any of the f1,f2,f3,g1,g2 above would be valid
-field type signatures.
- As the type of an implicit parameter
- In a pattern type signature (see )
-
-There is one place you cannot put a forall:
-you cannot instantiate a type variable with a forall-type. So you cannot
-make a forall-type the argument of a type constructor. So these types are illegal:
-
- x1 :: [forall a. a->a]
- x2 :: (forall a. a->a, Int)
- x3 :: Maybe (forall a. a->a)
-
-Of course forall becomes a keyword; you can't use forall as
-a type variable any more!
-
-
-
-
-Examples
-
-
-
-In a data or newtype declaration one can quantify
-the types of the constructor arguments. Here are several examples:
-
-
-
-
+Notice the call to 'split' introduced by the type checker.
+How did it know to use 'splitNS'? Because what it really did
+was to introduce a call to the overloaded function 'split',
+defined by the class Splittable:
-data T a = T1 (forall b. b -> b -> b) a
-
-data MonadT m = MkMonad { return :: forall a. a -> m a,
- bind :: forall a b. m a -> (a -> m b) -> m b
- }
-
-newtype Swizzle = MkSwizzle (Ord a => [a] -> [a])
+ class Splittable a where
+ split :: a -> (a,a)
-
-
-
-
-The constructors have rank-2 types:
+The instance for Splittable NameSupply tells GHC how to implement
+split for name supplies. But we can simply write
+
+ g x = (x, %ns, %ns)
+
+and GHC will infer
+
+ g :: (Splittable a, %ns :: a) => b -> (b,a,a)
+
+The Splittable class is built into GHC. It's exported by module
+GHC.Exts.
-
+Other points:
+
+ '?x' and '%x'
+are entirely distinct implicit parameters: you
+ can use them together and they won't intefere with each other.
+
-
-T1 :: forall a. (forall b. b -> b -> b) -> a -> T a
-MkMonad :: forall m. (forall a. a -> m a)
- -> (forall a b. m a -> (a -> m b) -> m b)
- -> MonadT m
-MkSwizzle :: (Ord a => [a] -> [a]) -> Swizzle
-
+ You can bind linear implicit parameters in 'with' clauses.
+You cannot have implicit parameters (whether linear or not)
+ in the context of a class or instance declaration.
+
-
-Notice that you don't need to use a forall if there's an
-explicit context. For example in the first argument of the
-constructor MkSwizzle, an implicit "forall a." is
-prefixed to the argument type. The implicit forall
-quantifies all type variables that are not already in scope, and are
-mentioned in the type quantified over.
-
+Warnings
-As for type signatures, implicit quantification happens for non-overloaded
-types too. So if you write this:
-
+The monomorphism restriction is even more important than usual.
+Consider the example above:
- data T a = MkT (Either a b) (b -> b)
+ f :: (%ns :: NameSupply) => Env -> Expr -> Expr
+ f env (Lam x e) = Lam x' (f env e)
+ where
+ x' = newName %ns
+ env' = extend env x x'
-
-it's just as if you had written this:
-
+If we replaced the two occurrences of x' by (newName %ns), which is
+usually a harmless thing to do, we get:
- data T a = MkT (forall b. Either a b) (forall b. b -> b)
+ f :: (%ns :: NameSupply) => Env -> Expr -> Expr
+ f env (Lam x e) = Lam (newName %ns) (f env e)
+ where
+ env' = extend env x (newName %ns)
-
-That is, since the type variable b isn't in scope, it's
-implicitly universally quantified. (Arguably, it would be better
-to require explicit quantification on constructor arguments
-where that is what is wanted. Feedback welcomed.)
+But now the name supply is consumed in three places
+(the two calls to newName,and the recursive call to f), so
+the result is utterly different. Urk! We don't even have
+the beta rule.
-
-You construct values of types T1, MonadT, Swizzle by applying
-the constructor to suitable values, just as usual. For example,
+Well, this is an experimental change. With implicit
+parameters we have already lost beta reduction anyway, and
+(as John Launchbury puts it) we can't sensibly reason about
+Haskell programs without knowing their typing.
-
+
+Recursive functions
+Linear implicit parameters can be particularly tricky when you have a recursive function
+Consider
- a1 :: T Int
- a1 = T1 (\xy->x) 3
-
- a2, a3 :: Swizzle
- a2 = MkSwizzle sort
- a3 = MkSwizzle reverse
-
- a4 :: MonadT Maybe
- a4 = let r x = Just x
- b m k = case m of
- Just y -> k y
- Nothing -> Nothing
- in
- MkMonad r b
-
- mkTs :: (forall b. b -> b -> b) -> a -> [T a]
- mkTs f x y = [T1 f x, T1 f y]
+ foo :: %x::T => Int -> [Int]
+ foo 0 = []
+ foo n = %x : foo (n-1)
-
-
-
-
-The type of the argument can, as usual, be more general than the type
-required, as (MkSwizzle reverse) shows. (reverse
-does not need the Ord constraint.)
-
-
-
-When you use pattern matching, the bound variables may now have
-polymorphic types. For example:
-
-
+where T is some type in class Splittable.
-
+Do you get a list of all the same T's or all different T's
+(assuming that split gives two distinct T's back)?
+
+If you supply the type signature, taking advantage of polymorphic
+recursion, you get what you'd probably expect. Here's the
+translated term, where the implicit param is made explicit:
- f :: T a -> a -> (a, Char)
- f (T1 w k) x = (w k x, w 'c' 'd')
-
- g :: (Ord a, Ord b) => Swizzle -> [a] -> (a -> b) -> [b]
- g (MkSwizzle s) xs f = s (map f (s xs))
-
- h :: MonadT m -> [m a] -> m [a]
- h m [] = return m []
- h m (x:xs) = bind m x $ \y ->
- bind m (h m xs) $ \ys ->
- return m (y:ys)
+ foo x 0 = []
+ foo x n = let (x1,x2) = split x
+ in x1 : foo x2 (n-1)
-
-
-
-
-In the function h we use the record selectors return
-and bind to extract the polymorphic bind and return functions
-from the MonadT data structure, rather than using pattern
-matching.
+But if you don't supply a type signature, GHC uses the Hindley
+Milner trick of using a single monomorphic instance of the function
+for the recursive calls. That is what makes Hindley Milner type inference
+work. So the translation becomes
+
+ foo x = let
+ foom 0 = []
+ foom n = x : foom (n-1)
+ in
+ foom
+
+Result: 'x' is not split, and you get a list of identical T's. So the
+semantics of the program depends on whether or not foo has a type signature.
+Yikes!
+
+You may say that this is a good reason to dislike linear implicit parameters
+and you'd be right. That is why they are an experimental feature.
-
-Type inference
+
+
+================ END OF Linear Implicit Parameters commented out -->
+
+
+Explicitly-kinded quantification
-In general, type inference for arbitrary-rank types is undecidable.
-GHC uses an algorithm proposed by Odersky and Laufer ("Putting type annotations to work", POPL'96)
-to get a decidable algorithm by requiring some help from the programmer.
-We do not yet have a formal specification of "some help" but the rule is this:
+Haskell infers the kind of each type variable. Sometimes it is nice to be able
+to give the kind explicitly as (machine-checked) documentation,
+just as it is nice to give a type signature for a function. On some occasions,
+it is essential to do so. For example, in his paper "Restricted Data Types in Haskell" (Haskell Workshop 1999)
+John Hughes had to define the data type:
+
+ data Set cxt a = Set [a]
+ | Unused (cxt a -> ())
+
+The only use for the Unused constructor was to force the correct
+kind for the type variable cxt.
-For a lambda-bound or case-bound variable, x, either the programmer
-provides an explicit polymorphic type for x, or GHC's type inference will assume
-that x's type has no foralls in it.
+GHC now instead allows you to specify the kind of a type variable directly, wherever
+a type variable is explicitly bound. Namely:
+
+data declarations:
+
+ data Set (cxt :: * -> *) a = Set [a]
+
+type declarations:
+
+ type T (f :: * -> *) = f Int
+
+class declarations:
+
+ class (Eq a) => C (f :: * -> *) a where ...
+
+forall's in type signatures:
+
+ f :: forall (cxt :: * -> *). Set cxt Int
+
+
+
-What does it mean to "provide" an explicit type for x? You can do that by
-giving a type signature for x directly, using a pattern type signature
-(), thus:
-
- \ f :: (forall a. a->a) -> (f True, f 'c')
-
-Alternatively, you can give a type signature to the enclosing
-context, which GHC can "push down" to find the type for the variable:
-
- (\ f -> (f True, f 'c')) :: (forall a. a->a) -> (Bool,Char)
-
-Here the type signature on the expression can be pushed inwards
-to give a type signature for f. Similarly, and more commonly,
-one can give a type signature for the function itself:
-
- h :: (forall a. a->a) -> (Bool,Char)
- h f = (f True, f 'c')
-
-You don't need to give a type signature if the lambda bound variable
-is a constructor argument. Here is an example we saw earlier:
-
- f :: T a -> a -> (a, Char)
- f (T1 w k) x = (w k x, w 'c' 'd')
-
-Here we do not need to give a type signature to w, because
-it is an argument of constructor T1 and that tells GHC all
-it needs to know.
+The parentheses are required. Some of the spaces are required too, to
+separate the lexemes. If you write (f::*->*) you
+will get a parse error, because "::*->*" is a
+single lexeme in Haskell.
-
+
+As part of the same extension, you can put kind annotations in types
+as well. Thus:
+
+ f :: (Int :: *) -> Int
+ g :: forall a. a -> (a :: *)
+
+The syntax is
+
+ atype ::= '(' ctype '::' kind ')
+
+The parentheses are required.
+
+
-
-Implicit quantification
+
+Arbitrary-rank polymorphism
+
-GHC performs implicit quantification as follows. At the top level (only) of
-user-written types, if and only if there is no explicit forall,
-GHC finds all the type variables mentioned in the type that are not already
-in scope, and universally quantifies them. For example, the following pairs are
-equivalent:
-
- f :: a -> a
- f :: forall a. a -> a
-
- g (x::a) = let
- h :: a -> b -> b
- h x y = y
- in ...
- g (x::a) = let
- h :: forall b. a -> b -> b
- h x y = y
- in ...
-
+Haskell type signatures are implicitly quantified. The new keyword forall
+allows us to say exactly what this means. For example:
-Notice that GHC does not find the innermost possible quantification
-point. For example:
- f :: (a -> a) -> Int
- -- MEANS
- f :: forall a. (a -> a) -> Int
- -- NOT
- f :: (forall a. a -> a) -> Int
-
-
- g :: (Ord a => a -> a) -> Int
- -- MEANS the illegal type
- g :: forall a. (Ord a => a -> a) -> Int
- -- NOT
- g :: (forall a. Ord a => a -> a) -> Int
+ g :: b -> b
-The latter produces an illegal type, which you might think is silly,
-but at least the rule is simple. If you want the latter type, you
-can write your for-alls explicitly. Indeed, doing so is strongly advised
-for rank-2 types.
-
-
-
-
-
-
-Impredicative polymorphism
-
-GHC supports impredicative polymorphism. This means
-that you can call a polymorphic function at a polymorphic type, and
-parameterise data structures over polymorphic types. For example:
+means this:
- f :: Maybe (forall a. [a] -> [a]) -> Maybe ([Int], [Char])
- f (Just g) = Just (g [3], g "hello")
- f Nothing = Nothing
+ g :: forall b. (b -> b)
-Notice here that the Maybe type is parameterised by the
-polymorphic type (forall a. [a] ->
-[a]).
-
-The technical details of this extension are described in the paper
-Boxy types:
-type inference for higher-rank types and impredicativity,
-which appeared at ICFP 2006.
+The two are treated identically.
-
-
-
-Lexically scoped type variables
-
-GHC supports lexically scoped type variables, without
-which some type signatures are simply impossible to write. For example:
+However, GHC's type system supports arbitrary-rank
+explicit universal quantification in
+types.
+For example, all the following types are legal:
-f :: forall a. [a] -> [a]
-f xs = ys ++ ys
- where
- ys :: [a]
- ys = reverse xs
+ f1 :: forall a b. a -> b -> a
+ g1 :: forall a b. (Ord a, Eq b) => a -> b -> a
+
+ f2 :: (forall a. a->a) -> Int -> Int
+ g2 :: (forall a. Eq a => [a] -> a -> Bool) -> Int -> Int
+
+ f3 :: ((forall a. a->a) -> Int) -> Bool -> Bool
+
+ f4 :: Int -> (forall a. a -> a)
-The type signature for f brings the type variable a into scope; it scopes over
-the entire definition of f.
-In particular, it is in scope at the type signature for ys.
-In Haskell 98 it is not possible to declare
-a type for ys; a major benefit of scoped type variables is that
-it becomes possible to do so.
+Here, f1 and g1 are rank-1 types, and
+can be written in standard Haskell (e.g. f1 :: a->b->a).
+The forall makes explicit the universal quantification that
+is implicitly added by Haskell.
-Lexically-scoped type variables are enabled by
-.
+
+The functions f2 and g2 have rank-2 types;
+the forall is on the left of a function arrow. As g2
+shows, the polymorphic type on the left of the function arrow can be overloaded.
-Note: GHC 6.6 contains substantial changes to the way that scoped type
-variables work, compared to earlier releases. Read this section
-carefully!
-
-
-Overview
-
-The design follows the following principles
-
-A scoped type variable stands for a type variable, and not for
-a type. (This is a change from GHC's earlier
-design.)
-Furthermore, distinct lexical type variables stand for distinct
-type variables. This means that every programmer-written type signature
-(includin one that contains free scoped type variables) denotes a
-rigid type; that is, the type is fully known to the type
-checker, and no inference is involved.
-Lexical type variables may be alpha-renamed freely, without
-changing the program.
-
+
+The function f3 has a rank-3 type;
+it has rank-2 types on the left of a function arrow.
-A lexically scoped type variable can be bound by:
+GHC allows types of arbitrary rank; you can nest foralls
+arbitrarily deep in function arrows. (GHC used to be restricted to rank 2, but
+that restriction has now been lifted.)
+In particular, a forall-type (also called a "type scheme"),
+including an operational type class context, is legal:
-A declaration type signature ()
-An expression type signature ()
-A pattern type signature ()
-Class and instance declarations ()
+ On the left or right (see f4, for example)
+of a function arrow
+ On the right of a function arrow (see )
+ As the argument of a constructor, or type of a field, in a data type declaration. For
+example, any of the f1,f2,f3,g1,g2 above would be valid
+field type signatures.
+ As the type of an implicit parameter
+ In a pattern type signature (see )
+Of course forall becomes a keyword; you can't use forall as
+a type variable any more!
+
+
+
+Examples
+
+
-In Haskell, a programmer-written type signature is implicitly quantifed over
-its free type variables (Section
-4.1.2
-of the Haskel Report).
-Lexically scoped type variables affect this implicit quantification rules
-as follows: any type variable that is in scope is not universally
-quantified. For example, if type variable a is in scope,
-then
-
- (e :: a -> a) means (e :: a -> a)
- (e :: b -> b) means (e :: forall b. b->b)
- (e :: a -> b) means (e :: forall b. a->b)
-
+In a data or newtype declaration one can quantify
+the types of the constructor arguments. Here are several examples:
+
-
+
+data T a = T1 (forall b. b -> b -> b) a
+data MonadT m = MkMonad { return :: forall a. a -> m a,
+ bind :: forall a b. m a -> (a -> m b) -> m b
+ }
-
-Declaration type signatures
-A declaration type signature that has explicit
-quantification (using forall) brings into scope the
-explicitly-quantified
-type variables, in the definition of the named function(s). For example:
-
- f :: forall a. [a] -> [a]
- f (x:xs) = xs ++ [ x :: a ]
+newtype Swizzle = MkSwizzle (Ord a => [a] -> [a])
-The "forall a" brings "a" into scope in
-the definition of "f".
+
-This only happens if the quantification in f's type
-signature is explicit. For example:
-
- g :: [a] -> [a]
- g (x:xs) = xs ++ [ x :: a ]
-
-This program will be rejected, because "a" does not scope
-over the definition of "f", so "x::a"
-means "x::forall a. a" by Haskell's usual implicit
-quantification rules.
+
+
+The constructors have rank-2 types:
-
-
-Expression type signatures
+
-An expression type signature that has explicit
-quantification (using forall) brings into scope the
-explicitly-quantified
-type variables, in the annotated expression. For example:
- f = runST ( (op >>= \(x :: STRef s Int) -> g x) :: forall s. ST s Bool )
+T1 :: forall a. (forall b. b -> b -> b) -> a -> T a
+MkMonad :: forall m. (forall a. a -> m a)
+ -> (forall a b. m a -> (a -> m b) -> m b)
+ -> MonadT m
+MkSwizzle :: (Ord a => [a] -> [a]) -> Swizzle
-Here, the type signature forall a. ST s Bool brings the
-type variable s into scope, in the annotated expression
-(op >>= \(x :: STRef s Int) -> g x).
-
-
+
-
-Pattern type signatures
-A type signature may occur in any pattern; this is a pattern type
-signature.
-For example:
-
- -- f and g assume that 'a' is already in scope
- f = \(x::Int, y::a) -> x
- g (x::a) = x
- h ((x,y) :: (Int,Bool)) = (y,x)
-
-In the case where all the type variables in the pattern type sigature are
-already in scope (i.e. bound by the enclosing context), matters are simple: the
-signature simply constrains the type of the pattern in the obvious way.
+Notice that you don't need to use a forall if there's an
+explicit context. For example in the first argument of the
+constructor MkSwizzle, an implicit "forall a." is
+prefixed to the argument type. The implicit forall
+quantifies all type variables that are not already in scope, and are
+mentioned in the type quantified over.
+
-There is only one situation in which you can write a pattern type signature that
-mentions a type variable that is not already in scope, namely in pattern match
-of an existential data constructor. For example:
+As for type signatures, implicit quantification happens for non-overloaded
+types too. So if you write this:
+
- data T = forall a. MkT [a]
+ data T a = MkT (Either a b) (b -> b)
+
- k :: T -> T
- k (MkT [t::a]) = MkT t3
- where
- t3::[a] = [t,t,t]
+it's just as if you had written this:
+
+
+ data T a = MkT (forall b. Either a b) (forall b. b -> b)
-Here, the pattern type signature (t::a) mentions a lexical type
-variable that is not already in scope. Indeed, it cannot already be in scope,
-because it is bound by the pattern match. GHC's rule is that in this situation
-(and only then), a pattern type signature can mention a type variable that is
-not already in scope; the effect is to bring it into scope, standing for the
-existentially-bound type variable.
+
+That is, since the type variable b isn't in scope, it's
+implicitly universally quantified. (Arguably, it would be better
+to require explicit quantification on constructor arguments
+where that is what is wanted. Feedback welcomed.)
+
-If this seems a little odd, we think so too. But we must have
-some way to bring such type variables into scope, else we
-could not name existentially-bound type variables in subequent type signatures.
+You construct values of types T1, MonadT, Swizzle by applying
+the constructor to suitable values, just as usual. For example,
+
-This is (now) the only situation in which a pattern type
-signature is allowed to mention a lexical variable that is not already in
-scope.
-For example, both f and g would be
-illegal if a was not already in scope.
-
+
+ a1 :: T Int
+ a1 = T1 (\xy->x) 3
+
+ a2, a3 :: Swizzle
+ a2 = MkSwizzle sort
+ a3 = MkSwizzle reverse
+
+ a4 :: MonadT Maybe
+ a4 = let r x = Just x
+ b m k = case m of
+ Just y -> k y
+ Nothing -> Nothing
+ in
+ MkMonad r b
-
+ mkTs :: (forall b. b -> b -> b) -> a -> [T a]
+ mkTs f x y = [T1 f x, T1 f y]
+
-
+
-
-Class and instance declarations
-
-The type variables in the head of a class or instance declaration
-scope over the methods defined in the where part. For example:
+
+Implicit quantification
+
+GHC performs implicit quantification as follows. At the top level (only) of
+user-written types, if and only if there is no explicit forall,
+GHC finds all the type variables mentioned in the type that are not already
+in scope, and universally quantifies them. For example, the following pairs are
+equivalent:
+
+ f :: a -> a
+ f :: forall a. a -> a
+ g (x::a) = let
+ h :: a -> b -> b
+ h x y = y
+ in ...
+ g (x::a) = let
+ h :: forall b. a -> b -> b
+ h x y = y
+ in ...
+
+
+
+Notice that GHC does not find the innermost possible quantification
+point. For example:
- class C a where
- op :: [a] -> a
+ f :: (a -> a) -> Int
+ -- MEANS
+ f :: forall a. (a -> a) -> Int
+ -- NOT
+ f :: (forall a. a -> a) -> Int
- op xs = let ys::[a]
- ys = reverse xs
- in
- head ys
+
+ g :: (Ord a => a -> a) -> Int
+ -- MEANS the illegal type
+ g :: forall a. (Ord a => a -> a) -> Int
+ -- NOT
+ g :: (forall a. Ord a => a -> a) -> Int
+The latter produces an illegal type, which you might think is silly,
+but at least the rule is simple. If you want the latter type, you
+can write your for-alls explicitly. Indeed, doing so is strongly advised
+for rank-2 types.
-
-
-Deriving clause for classes Typeable and Data
-
-Haskell 98 allows the programmer to add "deriving( Eq, Ord )" to a data type
-declaration, to generate a standard instance declaration for classes specified in the deriving clause.
-In Haskell 98, the only classes that may appear in the deriving clause are the standard
-classes Eq, Ord,
-Enum, Ix, Bounded, Read, and Show.
-
-
-GHC extends this list with two more classes that may be automatically derived
-(provided the flag is specified):
-Typeable, and Data. These classes are defined in the library
-modules Data.Typeable and Data.Generics respectively, and the
-appropriate class must be in scope before it can be mentioned in the deriving clause.
-
-An instance of Typeable can only be derived if the
-data type has seven or fewer type parameters, all of kind *.
-The reason for this is that the Typeable class is derived using the scheme
-described in
-
-Scrap More Boilerplate: Reflection, Zips, and Generalised Casts
-.
-(Section 7.4 of the paper describes the multiple Typeable classes that
-are used, and only Typeable1 up to
-Typeable7 are provided in the library.)
-In other cases, there is nothing to stop the programmer writing a TypableX
-class, whose kind suits that of the data type constructor, and
-then writing the data type instance by hand.
+
+Impredicative polymorphism
+
+GHC supports impredicative polymorphism. This means
+that you can call a polymorphic function at a polymorphic type, and
+parameterise data structures over polymorphic types. For example:
+
+ f :: Maybe (forall a. [a] -> [a]) -> Maybe ([Int], [Char])
+ f (Just g) = Just (g [3], g "hello")
+ f Nothing = Nothing
+
+Notice here that the Maybe type is parameterised by the
+polymorphic type (forall a. [a] ->
+[a]).
+
+The technical details of this extension are described in the paper
+Boxy types:
+type inference for higher-rank types and impredicativity,
+which appeared at ICFP 2006.
-
-Generalised derived instances for newtypes
+
+Lexically scoped type variables
+
-When you define an abstract type using newtype, you may want
-the new type to inherit some instances from its representation. In
-Haskell 98, you can inherit instances of Eq, Ord,
-Enum and Bounded by deriving them, but for any
-other classes you have to write an explicit instance declaration. For
-example, if you define
-
-
- newtype Dollars = Dollars Int
-
-
-and you want to use arithmetic on Dollars, you have to
-explicitly define an instance of Num:
-
-
- instance Num Dollars where
- Dollars a + Dollars b = Dollars (a+b)
- ...
+GHC supports lexically scoped type variables, without
+which some type signatures are simply impossible to write. For example:
+
+f :: forall a. [a] -> [a]
+f xs = ys ++ ys
+ where
+ ys :: [a]
+ ys = reverse xs
-All the instance does is apply and remove the newtype
-constructor. It is particularly galling that, since the constructor
-doesn't appear at run-time, this instance declaration defines a
-dictionary which is wholly equivalent to the Int
-dictionary, only slower!
+The type signature for f brings the type variable a into scope; it scopes over
+the entire definition of f.
+In particular, it is in scope at the type signature for ys.
+In Haskell 98 it is not possible to declare
+a type for ys; a major benefit of scoped type variables is that
+it becomes possible to do so.
+
+Lexically-scoped type variables are enabled by
+.
+Note: GHC 6.6 contains substantial changes to the way that scoped type
+variables work, compared to earlier releases. Read this section
+carefully!
+
+Overview
- Generalising the deriving clause
+The design follows the following principles
+
+A scoped type variable stands for a type variable, and not for
+a type. (This is a change from GHC's earlier
+design.)
+Furthermore, distinct lexical type variables stand for distinct
+type variables. This means that every programmer-written type signature
+(includin one that contains free scoped type variables) denotes a
+rigid type; that is, the type is fully known to the type
+checker, and no inference is involved.
+Lexical type variables may be alpha-renamed freely, without
+changing the program.
+
+
-GHC now permits such instances to be derived instead, so one can write
-
- newtype Dollars = Dollars Int deriving (Eq,Show,Num)
-
-
-and the implementation uses the sameNum dictionary
-for Dollars as for Int. Notionally, the compiler
-derives an instance declaration of the form
-
-
- instance Num Int => Num Dollars
-
-
-which just adds or removes the newtype constructor according to the type.
+A lexically scoped type variable can be bound by:
+
+A declaration type signature ()
+An expression type signature ()
+A pattern type signature ()
+Class and instance declarations ()
+
+In Haskell, a programmer-written type signature is implicitly quantifed over
+its free type variables (Section
+4.1.2
+of the Haskel Report).
+Lexically scoped type variables affect this implicit quantification rules
+as follows: any type variable that is in scope is not universally
+quantified. For example, if type variable a is in scope,
+then
+
+ (e :: a -> a) means (e :: a -> a)
+ (e :: b -> b) means (e :: forall b. b->b)
+ (e :: a -> b) means (e :: forall b. a->b)
+
+
-We can also derive instances of constructor classes in a similar
-way. For example, suppose we have implemented state and failure monad
-transformers, such that
-
- instance Monad m => Monad (State s m)
- instance Monad m => Monad (Failure m)
-
-In Haskell 98, we can define a parsing monad by
-
- type Parser tok m a = State [tok] (Failure m) a
-
+
-which is automatically a monad thanks to the instance declarations
-above. With the extension, we can make the parser type abstract,
-without needing to write an instance of class Monad, via
-
- newtype Parser tok m a = Parser (State [tok] (Failure m) a)
- deriving Monad
+
+Declaration type signatures
+A declaration type signature that has explicit
+quantification (using forall) brings into scope the
+explicitly-quantified
+type variables, in the definition of the named function(s). For example:
+
+ f :: forall a. [a] -> [a]
+ f (x:xs) = xs ++ [ x :: a ]
-In this case the derived instance declaration is of the form
-
- instance Monad (State [tok] (Failure m)) => Monad (Parser tok m)
-
-
-Notice that, since Monad is a constructor class, the
-instance is a partial application of the new type, not the
-entire left hand side. We can imagine that the type declaration is
-``eta-converted'' to generate the context of the instance
-declaration.
+The "forall a" brings "a" into scope in
+the definition of "f".
-
-
-We can even derive instances of multi-parameter classes, provided the
-newtype is the last class parameter. In this case, a ``partial
-application'' of the class appears in the deriving
-clause. For example, given the class
-
-
- class StateMonad s m | m -> s where ...
- instance Monad m => StateMonad s (State s m) where ...
-
-then we can derive an instance of StateMonad for Parsers by
-
- newtype Parser tok m a = Parser (State [tok] (Failure m) a)
- deriving (Monad, StateMonad [tok])
+This only happens if the quantification in f's type
+signature is explicit. For example:
+
+ g :: [a] -> [a]
+ g (x:xs) = xs ++ [ x :: a ]
+This program will be rejected, because "a" does not scope
+over the definition of "f", so "x::a"
+means "x::forall a. a" by Haskell's usual implicit
+quantification rules.
+
+
-The derived instance is obtained by completing the application of the
-class to the new type:
+
+Expression type signatures
-
- instance StateMonad [tok] (State [tok] (Failure m)) =>
- StateMonad [tok] (Parser tok m)
+An expression type signature that has explicit
+quantification (using forall) brings into scope the
+explicitly-quantified
+type variables, in the annotated expression. For example:
+
+ f = runST ( (op >>= \(x :: STRef s Int) -> g x) :: forall s. ST s Bool )
+Here, the type signature forall a. ST s Bool brings the
+type variable s into scope, in the annotated expression
+(op >>= \(x :: STRef s Int) -> g x).
-
-As a result of this extension, all derived instances in newtype
- declarations are treated uniformly (and implemented just by reusing
-the dictionary for the representation type), except
-Show and Read, which really behave differently for
-the newtype and its representation.
-
- A more precise specification
+
+Pattern type signatures
-Derived instance declarations are constructed as follows. Consider the
-declaration (after expansion of any type synonyms)
-
-
- newtype T v1...vn = T' (t vk+1...vn) deriving (c1...cm)
-
-
-where
-
-
- The ci are partial applications of
- classes of the form C t1'...tj', where the arity of C
- is exactly j+1. That is, C lacks exactly one type argument.
-
-
- The k is chosen so that ci (T v1...vk) is well-kinded.
-
-
- The type t is an arbitrary type.
-
-
- The type variables vk+1...vn do not occur in t,
- nor in the ci, and
-
-
- None of the ci is Read, Show,
- Typeable, or Data. These classes
- should not "look through" the type or its constructor. You can still
- derive these classes for a newtype, but it happens in the usual way, not
- via this new mechanism.
-
-
-Then, for each ci, the derived instance
-declaration is:
-
- instance ci t => ci (T v1...vk)
-
-As an example which does not work, consider
-
- newtype NonMonad m s = NonMonad (State s m s) deriving Monad
-
-Here we cannot derive the instance
-
- instance Monad (State s m) => Monad (NonMonad m)
-
-
-because the type variable s occurs in State s m,
-and so cannot be "eta-converted" away. It is a good thing that this
-deriving clause is rejected, because NonMonad m is
-not, in fact, a monad --- for the same reason. Try defining
->>= with the correct type: you won't be able to.
+A type signature may occur in any pattern; this is a pattern type
+signature.
+For example:
+
+ -- f and g assume that 'a' is already in scope
+ f = \(x::Int, y::a) -> x
+ g (x::a) = x
+ h ((x,y) :: (Int,Bool)) = (y,x)
+
+In the case where all the type variables in the pattern type sigature are
+already in scope (i.e. bound by the enclosing context), matters are simple: the
+signature simply constrains the type of the pattern in the obvious way.
+There is only one situation in which you can write a pattern type signature that
+mentions a type variable that is not already in scope, namely in pattern match
+of an existential data constructor. For example:
+
+ data T = forall a. MkT [a]
-Notice also that the order of class parameters becomes
-important, since we can only derive instances for the last one. If the
-StateMonad class above were instead defined as
-
-
- class StateMonad m s | m -> s where ...
+ k :: T -> T
+ k (MkT [t::a]) = MkT t3
+ where
+ t3::[a] = [t,t,t]
-
-then we would not have been able to derive an instance for the
-Parser type above. We hypothesise that multi-parameter
-classes usually have one "main" parameter for which deriving new
-instances is most interesting.
+Here, the pattern type signature (t::a) mentions a lexical type
+variable that is not already in scope. Indeed, it cannot already be in scope,
+because it is bound by the pattern match. GHC's rule is that in this situation
+(and only then), a pattern type signature can mention a type variable that is
+not already in scope; the effect is to bring it into scope, standing for the
+existentially-bound type variable.
-Lastly, all of this applies only for classes other than
-Read, Show, Typeable,
-and Data, for which the built-in derivation applies (section
-4.3.3. of the Haskell Report).
-(For the standard classes Eq, Ord,
-Ix, and Bounded it is immaterial whether
-the standard method is used or the one described here.)
+
+If this seems a little odd, we think so too. But we must have
+some way to bring such type variables into scope, else we
+could not name existentially-bound type variables in subequent type signatures.
+
+
+This is (now) the only situation in which a pattern type
+signature is allowed to mention a lexical variable that is not already in
+scope.
+For example, both f and g would be
+illegal if a was not already in scope.
+
+
-
+
+
+Class and instance declarations
+
+The type variables in the head of a class or instance declaration
+scope over the methods defined in the where part. For example:
+
+
+
+ class C a where
+ op :: [a] -> a
+
+ op xs = let ys::[a]
+ ys = reverse xs
+ in
+ head ys
+
+
+
Generalised typing of mutually recursive bindings
@@ -3818,183 +4059,6 @@ pattern binding must have the same context. For example, this is fine:
-
-
-
-Generalised Algebraic Data Types (GADTs)
-
-Generalised Algebraic Data Types generalise ordinary algebraic data types by allowing you
-to give the type signatures of constructors explicitly. For example:
-
- data Term a where
- Lit :: Int -> Term Int
- Succ :: Term Int -> Term Int
- IsZero :: Term Int -> Term Bool
- If :: Term Bool -> Term a -> Term a -> Term a
- Pair :: Term a -> Term b -> Term (a,b)
-
-Notice that the return type of the constructors is not always Term a, as is the
-case with ordinary vanilla data types. Now we can write a well-typed eval function
-for these Terms:
-
- eval :: Term a -> a
- eval (Lit i) = i
- eval (Succ t) = 1 + eval t
- eval (IsZero t) = eval t == 0
- eval (If b e1 e2) = if eval b then eval e1 else eval e2
- eval (Pair e1 e2) = (eval e1, eval e2)
-
-These and many other examples are given in papers by Hongwei Xi, and
-Tim Sheard. There is a longer introduction
-on the wiki,
-and Ralf Hinze's
-Fun with phantom types also has a number of examples. Note that papers
-may use different notation to that implemented in GHC.
-
-
-The rest of this section outlines the extensions to GHC that support GADTs.
-It is far from comprehensive, but the design closely follows that described in
-the paper Simple
-unification-based type inference for GADTs,
-which appeared in ICFP 2006.
-
-
- Data type declarations have a 'where' form, as exemplified above. The type signature of
-each constructor is independent, and is implicitly universally quantified as usual. Unlike a normal
-Haskell data type declaration, the type variable(s) in the "data Term a where" header
-have no scope. Indeed, one can write a kind signature instead:
-
- data Term :: * -> * where ...
-
-or even a mixture of the two:
-
- data Foo a :: (* -> *) -> * where ...
-
-The type variables (if given) may be explicitly kinded, so we could also write the header for Foo
-like this:
-
- data Foo a (b :: * -> *) where ...
-
-
-
-
-There are no restrictions on the type of the data constructor, except that the result
-type must begin with the type constructor being defined. For example, in the Term data
-type above, the type of each constructor must end with ... -> Term ....
-
-
-
-You can use record syntax on a GADT-style data type declaration:
-
-
- data Term a where
- Lit { val :: Int } :: Term Int
- Succ { num :: Term Int } :: Term Int
- Pred { num :: Term Int } :: Term Int
- IsZero { arg :: Term Int } :: Term Bool
- Pair { arg1 :: Term a
- , arg2 :: Term b
- } :: Term (a,b)
- If { cnd :: Term Bool
- , tru :: Term a
- , fls :: Term a
- } :: Term a
-
-For every constructor that has a field f, (a) the type of
-field f must be the same; and (b) the
-result type of the constructor must be the same; both modulo alpha conversion.
-Hence, in our example, we cannot merge the num and arg
-fields above into a
-single name. Although their field types are both Term Int,
-their selector functions actually have different types:
-
-
- num :: Term Int -> Term Int
- arg :: Term Bool -> Term Int
-
-
-At the moment, record updates are not yet possible with GADT, so support is
-limited to record construction, selection and pattern matching:
-
-
- someTerm :: Term Bool
- someTerm = IsZero { arg = Succ { num = Lit { val = 0 } } }
-
- eval :: Term a -> a
- eval Lit { val = i } = i
- eval Succ { num = t } = eval t + 1
- eval Pred { num = t } = eval t - 1
- eval IsZero { arg = t } = eval t == 0
- eval Pair { arg1 = t1, arg2 = t2 } = (eval t1, eval t2)
- eval t@If{} = if eval (cnd t) then eval (tru t) else eval (fls t)
-
-
-
-
-
-You can use strictness annotations, in the obvious places
-in the constructor type:
-
- data Term a where
- Lit :: !Int -> Term Int
- If :: Term Bool -> !(Term a) -> !(Term a) -> Term a
- Pair :: Term a -> Term b -> Term (a,b)
-
-
-
-
-You can use a deriving clause on a GADT-style data type
-declaration, but only if the data type could also have been declared in
-Haskell-98 syntax. For example, these two declarations are equivalent
-
- data Maybe1 a where {
- Nothing1 :: Maybe1 a ;
- Just1 :: a -> Maybe1 a
- } deriving( Eq, Ord )
-
- data Maybe2 a = Nothing2 | Just2 a
- deriving( Eq, Ord )
-
-This simply allows you to declare a vanilla Haskell-98 data type using the
-where form without losing the deriving clause.
-
-
-
-Pattern matching causes type refinement. For example, in the right hand side of the equation
-
- eval :: Term a -> a
- eval (Lit i) = ...
-
-the type a is refined to Int. (That's the whole point!)
-A precise specification of the type rules is beyond what this user manual aspires to, but there is a paper
-about the ideas: "Wobbly types: practical type inference for generalised algebraic data types", on Simon PJ's home page.
-
- The general principle is this: type refinement is only carried out based on user-supplied type annotations.
-So if no type signature is supplied for eval, no type refinement happens, and lots of obscure error messages will
-occur. However, the refinement is quite general. For example, if we had:
-
- eval :: Term a -> a -> a
- eval (Lit i) j = i+j
-
-the pattern match causes the type a to be refined to Int (because of the type
-of the constructor Lit, and that refinement also applies to the type of j, and
-the result type of the case expression. Hence the addition i+j is legal.
-
-
-
-
-
-Notice that GADTs generalise existential types. For example, these two declarations are equivalent:
-
- data T a = forall b. MkT b (b->a)
- data T' a where { MKT :: b -> (b->a) -> T' a }
-
-
-
-
-
-
@@ -4807,7 +4871,7 @@ f !x = 3
Is this a definition of the infix function "(!)",
or of the "f" with a bang pattern? GHC resolves this
-ambiguity inf favour of the latter. If you want to define
+ambiguity in favour of the latter. If you want to define
(!) with bang-patterns enabled, you have to do so using
prefix notation: