X-Git-Url: http://git.megacz.com/?p=ghc-hetmet.git;a=blobdiff_plain;f=docs%2Fusers_guide%2Fglasgow_exts.xml;h=e680d6cc046ad6aa6371cdbf3c11cfdad14944d8;hp=e801b5693d333c1a0217e5899f4d20945660ef71;hb=2f4e21c6f741995e20cc3b53b109ff9edf18eb3c;hpb=56e0d057074e9a22d8fe56392c02e5ad2dede65d diff --git a/docs/users_guide/glasgow_exts.xml b/docs/users_guide/glasgow_exts.xml index e801b56..e680d6c 100644 --- a/docs/users_guide/glasgow_exts.xml +++ b/docs/users_guide/glasgow_exts.xml @@ -106,9 +106,7 @@ documentation describes all the libraries that come with GHC. This option enables the language extension defined in the - Haskell 98 Foreign Function Interface Addendum plus deprecated - syntax of previous versions of the FFI for backwards - compatibility. + Haskell 98 Foreign Function Interface Addendum. New reserved words: foreign. @@ -116,10 +114,10 @@ documentation describes all the libraries that come with GHC. - ,: + ,: - These two flags control how generalisation is done in + These two flags control how generalisation is done. See . @@ -243,6 +241,14 @@ documentation describes all the libraries that come with GHC. + + + Enables overloaded string literals (see ). + + + + Enables lexically-scoped type variables (see -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 @@ -647,7 +653,7 @@ 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 @@ -905,18 +911,46 @@ fromInteger :: Integer -> Bool -> Bool you should be all right. - + +Postfix operators - - -Type system extensions + +GHC allows a small extension to the syntax of left operator sections, which +allows you to define postfix operators. The extension is this: the left section + + (e !) + +is equivalent (from the point of view of both type checking and execution) to the expression + + ((!) e) + +(for any expression e and operator (!). +The strict Haskell 98 interpretation is that the section is equivalent to + + (\y -> (!) e y) + +That is, the operator must be a function of two arguments. GHC allows it to +take only one argument, and that in turn allows you to write the function +postfix. + +Since this extension goes beyond Haskell 98, it should really be enabled +by a flag; but in fact it is enabled all the time. (No Haskell 98 programs +change their behaviour, of course.) + +The extension does not extend to the left-hand side of function +definitions; you must define such a function in prefix form. + + + - -Data types and type synonyms - + + +Extensions to data types and type synonyms + + Data types with no constructors With the flag, GHC lets you declare @@ -934,9 +968,9 @@ 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 @@ -1003,9 +1037,9 @@ to be written infix, very much like expressions. More specifically: - + - + Liberalised type synonyms @@ -1095,10 +1129,10 @@ this will be rejected: because GHC does not allow unboxed tuples on the left of a function arrow. - + - + Existentially quantified data constructors @@ -1192,7 +1226,7 @@ that collection of packages in a uniform manner. You can express quite a bit of object-oriented-like programming this way. - + Why existential? @@ -1215,9 +1249,9 @@ But Haskell programmers can safely think of the ordinary adding a new existential quantification construct. - + - + Type classes @@ -1277,9 +1311,9 @@ Notice the way that the syntax fits smoothly with that used for universal quantification earlier. - + - + Record Constructors @@ -1331,20 +1365,6 @@ 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: @@ -1356,10 +1376,10 @@ setTag obj t = obj{ tag = t } - + - + Restrictions @@ -1485,7 +1505,7 @@ 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 ) @@ -1510,2334 +1530,2476 @@ declarations. Define your own instances! - - + + +Declaring data types with explicit constructor signatures - -Class declarations +GHC allows you to declare an algebraic data type by +giving the type signatures of constructors explicitly. For example: + + data Maybe a where + Nothing :: Maybe a + Just :: a -> Maybe a + +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: + + data Foo = forall a. MkFoo a (a -> Bool) + data Foo' where { MKFoo :: a -> (a->Bool) -> Foo' } + + +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: + + data Set a where + MkSet :: Eq a => [a] -> Set a - -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). + makeSet :: Eq a => [a] -> Set a + makeSet xs = MkSet (nub xs) + + 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. +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 + + data Eq a => Set' a = MkSet' [a] + +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. -All the extensions are enabled by the flag. +For example, a possible application of GHC's behaviour is to reify dictionaries: + + data NumInst a where + MkNumInst :: Num a => NumInst a + + intInst :: NumInst Int + intInst = MkNumInst + + plus :: NumInst a -> a -> a -> a + plus MkNumInst p q = p + q + +Here, a value of type NumInst a is equivalent +to an explicit (Num a) dictionary. - -Multi-parameter type classes -Multi-parameter type classes are permitted. 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 Collection c a where - union :: c a -> c a -> c a - ...etc. + data T a where + T1 :: Eq b => b -> T b + T2 :: (Show c, Ix c) => c -> [c] -> T c + - - - - -The superclasses of a class declaration + +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: + + 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 ... + + - -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: + +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. For example, these two declarations are equivalent - class Functor (m k) => FiniteMap m k where - ... + data Maybe1 a where { + Nothing1 :: Maybe1 a ; + Just1 :: a -> Maybe1 a + } deriving( Eq, Ord ) - class (Monad m, Monad (t m)) => Transform t m where - lift :: m a -> (t m) a + data Maybe2 a = Nothing2 | Just2 a + deriving( Eq, Ord ) + + +You can use record syntax on a GADT-style data type declaration: + + 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). -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: +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 + + + +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: - class C a where { - op :: D b => a -> b -> b - } - - class C a => D a where { ... } +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. + + + + +Generalised Algebraic Data Types (GADTs) -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.) +Generalised Algebraic Data Types generalise ordinary algebraic data types +by allowing constructors to have richer return types. Here is an 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 data types. This generality allows us to +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) + +The key point about GADTs is that 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 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. + - -Class method types + +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: - -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 + num :: Term Int -> Term Int + arg :: Term Bool -> Term Int -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. - + +Deriving clause for classes <literal>Typeable</literal> and <literal>Data</literal> -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: +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. + + - - class Coll s a where - empty :: s - insert :: s -> a -> s - + +Generalised derived instances for newtypes -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 - + +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 -Alternatively Coll might be rewritten + + newtype Dollars = Dollars Int + - - class Coll s a where - empty :: s a - insert :: s a -> a -> s a +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! + -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: + 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 same Num dictionary +for Dollars as for Int. Notionally, the compiler +derives an instance declaration of the form - - class CollE s where - empty :: s + + instance Num Int => Num Dollars + - class CollE s => Coll s a where - insert :: s -> a -> s - +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 - -Background on functional dependencies + + 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 + -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 +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 -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. +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 sidestep this specific problem by removing the empty member from the -class declaration. However, although the remaining members, insert and member, -do not have ambiguous types, we still run into problems when we try to use -them. For example, consider the following two functions: - - f x y = insert x . insert y - g = f True 'a' - -for which GHC infers the following types: - - f :: (Collects a c, Collects b c) => a -> b -> c -> c - g :: (Collects Bool c, Collects Char c) => c -> c + +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]) -Notice that the type for f allows the two parameters x and y to be assigned -different types, even though it attempts to insert each of the two values, one -after the other, into the same collection. If we're trying to model collections -that contain only one type of value, then this is clearly an inaccurate -type. Worse still, the definition for g is accepted, without causing a type -error. As a result, the error in this code will not be flagged at the point -where it appears. Instead, it will show up only when we try to use g, which -might even be in a different module. - -An attempt to use constructor classes +The derived instance is obtained by completing the application of the +class to the new type: - -Faced with the problems described above, some Haskell programmers might be -tempted to use something like the following version of the class declaration: - - class Collects e c where - empty :: c e - insert :: e -> c e -> c e - member :: e -> c e -> Bool + + instance StateMonad [tok] (State [tok] (Failure m)) => + StateMonad [tok] (Parser tok m) -The key difference here is that we abstract over the type constructor c that is -used to form the collection type c e, and not over that collection type itself, -represented by ce in the original class declaration. This avoids the immediate -problems that we mentioned above: empty has type Collects e c => c -e, which is not ambiguous. -The function f from the previous section has a more accurate type: - - f :: (Collects e c) => e -> e -> c e -> c e - -The function g from the previous section is now rejected with a type error as -we would hope because the type of f does not allow the two arguments to have -different types. -This, then, is an example of a multiple parameter class that does actually work -quite well in practice, without ambiguity problems. -There is, however, a catch. This version of the Collects class is nowhere near -as general as the original class seemed to be: only one of the four instances -for Collects -given above can be used with this version of Collects because only one of -them---the instance for lists---has a collection type that can be written in -the form c e, for some type constructor c, and element type e. - - -Adding functional dependencies +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 -To get a more useful version of the Collects class, Hugs provides a mechanism -that allows programmers to specify dependencies between the parameters of a -multiple parameter class (For readers with an interest in theoretical -foundations and previous work: The use of dependency information can be seen -both as a generalization of the proposal for `parametric type classes' that was -put forward by Chen, Hudak, and Odersky, or as a special case of Mark Jones's -later framework for "improvement" of qualified types. The -underlying ideas are also discussed in a more theoretical and abstract setting -in a manuscript [implparam], where they are identified as one point in a -general design space for systems of implicit parameterization.). +Derived instance declarations are constructed as follows. Consider the +declaration (after expansion of any type synonyms) -To start with an abstract example, consider a declaration such as: - - class C a b where ... - -which tells us simply that C can be thought of as a binary relation on types -(or type constructors, depending on the kinds of a and b). Extra clauses can be -included in the definition of classes to add information about dependencies -between parameters, as in the following examples: - - class D a b | a -> b where ... - class E a b | a -> b, b -> a where ... + + 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) -The notation a -> b used here between the | and where -symbols --- not to be -confused with a function type --- indicates that the a parameter uniquely -determines the b parameter, and might be read as "a determines b." Thus D is -not just a relation, but actually a (partial) function. Similarly, from the two -dependencies that are included in the definition of E, we can see that E -represents a (partial) one-one mapping between types. +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. -More generally, dependencies take the form x1 ... xn -> y1 ... ym, -where x1, ..., xn, and y1, ..., yn are type variables with n>0 and -m>=0, meaning that the y parameters are uniquely determined by the x -parameters. Spaces can be used as separators if more than one variable appears -on any single side of a dependency, as in t -> a b. Note that a class may be -annotated with multiple dependencies using commas as separators, as in the -definition of E above. Some dependencies that we can write in this notation are -redundant, and will be rejected because they don't serve any useful -purpose, and may instead indicate an error in the program. Examples of -dependencies like this include a -> a , -a -> a a , -a -> , etc. There can also be -some redundancy if multiple dependencies are given, as in -a->b, - b->c , a->c , and -in which some subset implies the remaining dependencies. Examples like this are -not treated as errors. Note that dependencies appear only in class -declarations, and not in any other part of the language. In particular, the -syntax for instance declarations, class constraints, and types is completely -unchanged. + +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 + -By including dependencies in a class declaration, we provide a mechanism for -the programmer to specify each multiple parameter class more precisely. The -compiler, on the other hand, is responsible for ensuring that the set of -instances that are in scope at any given point in the program is consistent -with any declared dependencies. For example, the following pair of instance -declarations cannot appear together in the same scope because they violate the -dependency for D, even though either one on its own would be acceptable: +GHC now allows stand-alone deriving declarations, enabled by -fglasgow-exts: - instance D Bool Int where ... - instance D Bool Char where ... - -Note also that the following declaration is not allowed, even by itself: - - instance D [a] b where ... - -The problem here is that this instance would allow one particular choice of [a] -to be associated with more than one choice for b, which contradicts the -dependency specified in the definition of D. More generally, this means that, -in any instance of the form: + 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: - instance D t s where ... + newtype Foo a = MkFoo (State Int a) + + derive instance MonadState Int Foo -for some particular types t and s, the only variables that can appear in s are -the ones that appear in t, and hence, if the type t is known, then s will be -uniquely determined. +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 + -The benefit of including dependency information is that it allows us to define -more general multiple parameter classes, without ambiguity problems, and with -the benefit of more accurate types. To illustrate this, we return to the -collection class example, and annotate the original definition of Collects -with a simple dependency: - - class Collects e ce | ce -> e where - empty :: ce - insert :: e -> ce -> ce - member :: e -> ce -> Bool - -The dependency ce -> e here specifies that the type e of elements is uniquely -determined by the type of the collection ce. Note that both parameters of -Collects are of kind *; there are no constructor classes here. Note too that -all of the instances of Collects that we gave earlier can be used -together with this new definition. +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). -What about the ambiguity problems that we encountered with the original -definition? The empty function still has type Collects e ce => ce, but it is no -longer necessary to regard that as an ambiguous type: Although the variable e -does not appear on the right of the => symbol, the dependency for class -Collects tells us that it is uniquely determined by ce, which does appear on -the right of the => symbol. Hence the context in which empty is used can still -give enough information to determine types for both ce and e, without -ambiguity. More generally, we need only regard a type as ambiguous if it -contains a variable on the left of the => that is not uniquely determined -(either directly or indirectly) by the variables on the right. +All the extensions are enabled by the flag. + + +Multi-parameter type classes -Dependencies also help to produce more accurate types for user defined -functions, and hence to provide earlier detection of errors, and less cluttered -types for programmers to work with. Recall the previous definition for a -function f: - - f x y = insert x y = insert x . insert y - -for which we originally obtained a type: +Multi-parameter type classes are permitted. For example: + + - f :: (Collects a c, Collects b c) => a -> b -> c -> c + class Collection c a where + union :: c a -> c a -> c a + ...etc. -Given the dependency information that we have for Collects, however, we can -deduce that a and b must be equal because they both appear as the second -parameter in a Collects constraint with the same first parameter c. Hence we -can infer a shorter and more accurate type for f: + + + + + +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: + + - f :: (Collects a c) => a -> a -> c -> c + class Functor (m k) => FiniteMap m k where + ... + + class (Monad m, Monad (t m)) => Transform t m where + lift :: m a -> (t m) a -In a similar way, the earlier definition of g will now be flagged as a type error. + + -Although we have given only a few examples here, it should be clear that the -addition of dependency information can help to make multiple parameter classes -more useful in practice, avoiding ambiguity problems, and allowing more general -sets of instance declarations. +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.) - - - -Instance declarations - -Relaxed rules for instance declarations -An instance declaration has the form - - instance ( assertion1, ..., assertionn) => class type1 ... typem where ... - -The part before the "=>" is the -context, while the part after the -"=>" is the head of the instance declaration. - - -In Haskell 98 the head of an instance declaration -must be of the form C (T a1 ... an), where -C is the class, T is a type constructor, -and the a1 ... an are distinct type variables. -Furthermore, the assertions in the context of the instance declaration -must be of the form C a where a -is a type variable that occurs in the head. - - -The flag loosens these restrictions -considerably. Firstly, multi-parameter type classes are permitted. Secondly, -the context and head of the instance declaration can each consist of arbitrary -(well-kinded) assertions (C t1 ... tn) subject only to the -following rules: - - -For each assertion in the context: - -No type variable has more occurrences in the assertion than in the head -The assertion has fewer constructors and variables (taken together - and counting repetitions) than the head - - + +Class method types -The coverage condition. For each functional dependency, -tvsleft -> -tvsright, of the class, -every type variable in -S(tvsright) must appear in -S(tvsleft), where S is the -substitution mapping each type variable in the class declaration to the -corresponding type in the instance declaration. - - -These restrictions ensure that context reduction terminates: each reduction -step makes the problem smaller by at least one -constructor. For example, the following would make the type checker -loop if it wasn't excluded: + +Haskell 98 prohibits class method types to mention constraints on the +class type variable, thus: - instance C a => C a where ... + class Seq s a where + fromList :: [a] -> s a + elem :: Eq a => a -> s a -> Bool -For example, these are OK: - - instance C Int [a] -- Multiple parameters - instance Eq (S [a]) -- Structured type in head +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. + - -- Repeated type variable in head - instance C4 a a => C4 [a] [a] - instance Stateful (ST s) (MutVar s) - -- Head can consist of type variables only - instance C a - instance (Eq a, Show b) => C2 a b + + - -- Non-type variables in context - instance Show (s a) => Show (Sized s a) - instance C2 Int a => C3 Bool [a] - instance C2 Int a => C3 [a] b - -But these are not: + +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. - -- Context assertion no smaller than head - instance C a => C a where ... - -- (C b b) has more more occurrences of b than the head - instance C b b => Foo [b] where ... + 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 -The same restrictions apply to instances generated by -deriving clauses. Thus the following is accepted: +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: + - data MinHeap h a = H a (h a) - deriving (Show) + class Coll s a where + empty :: s + insert :: s -> a -> s -because the derived instance + +is not OK, because the type of empty doesn't mention +a. Functional dependencies can make the type variable +reachable: - instance (Show a, Show (h a)) => Show (MinHeap h a) + class Coll s a | s -> a where + empty :: s + insert :: s -> a -> s -conforms to the above rules. - - -A useful idiom permitted by the above rules is as follows. -If one allows overlapping instance declarations then it's quite -convenient to have a "default instance" declaration that applies if -something more specific does not: +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: + + - instance C a where - op = ... -- Default + class CollE s where + empty :: s + + class CollE s => Coll s a where + insert :: s -> a -> s -You can find lots of background material about the reason for these -restrictions in the paper -Understanding functional dependencies via Constraint Handling Rules. - - -Undecidable instances - -Sometimes even the rules of are too onerous. -For example, sometimes you might want to use the following to get the -effect of a "class synonym": - - class (C1 a, C2 a, C3 a) => C a where { } + +Background on functional dependencies - instance (C1 a, C2 a, C3 a) => C a where { } - -This allows you to write shorter signatures: +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: - f :: C a => ... + class Collects e ce where + empty :: ce + insert :: e -> ce -> ce + member :: e -> ce -> Bool -instead of +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: - f :: (C1 a, C2 a, C3 a) => ... + 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 ... -The restrictions on functional dependencies () are particularly troublesome. -It is tempting to introduce type variables in the context that do not appear in -the head, something that is excluded by the normal rules. For example: +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: - class HasConverter a b | a -> b where - convert :: a -> b - - data Foo a = MkFoo a - - instance (HasConverter a b,Show b) => Show (Foo a) where - show (MkFoo value) = show (convert value) + empty :: Collects e ce => ce -This is dangerous territory, however. Here, for example, is a program that would make the -typechecker loop: - - class D a - class F a b | a->b - instance F [a] [[a]] - instance (D c, F a c) => D [a] -- 'c' is not mentioned in the head - -Similarly, it can be tempting to lift the coverage condition: +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. + + +We can sidestep this specific problem by removing the empty member from the +class declaration. However, although the remaining members, insert and member, +do not have ambiguous types, we still run into problems when we try to use +them. For example, consider the following two functions: - class Mul a b c | a b -> c where - (.*.) :: a -> b -> c - - instance Mul Int Int Int where (.*.) = (*) - instance Mul Int Float Float where x .*. y = fromIntegral x * y - instance Mul a b c => Mul a [b] [c] where x .*. v = map (x.*.) v + f x y = insert x . insert y + g = f True 'a' -The third instance declaration does not obey the coverage condition; -and indeed the (somewhat strange) definition: +for which GHC infers the following types: - f = \ b x y -> if b then x .*. [y] else y + f :: (Collects a c, Collects b c) => a -> b -> c -> c + g :: (Collects Bool c, Collects Char c) => c -> c -makes instance inference go into a loop, because it requires the constraint -(Mul a [b] b). - - -Nevertheless, GHC allows you to experiment with more liberal rules. If you use -the experimental flag --fallow-undecidable-instances -option, you can use arbitrary -types in both an instance context and instance head. Termination is ensured by having a -fixed-depth recursion stack. If you exceed the stack depth you get a -sort of backtrace, and the opportunity to increase the stack depth -with N. +Notice that the type for f allows the two parameters x and y to be assigned +different types, even though it attempts to insert each of the two values, one +after the other, into the same collection. If we're trying to model collections +that contain only one type of value, then this is clearly an inaccurate +type. Worse still, the definition for g is accepted, without causing a type +error. As a result, the error in this code will not be flagged at the point +where it appears. Instead, it will show up only when we try to use g, which +might even be in a different module. - - +An attempt to use constructor classes - -Overlapping instances - -In general, GHC requires that that it be unambiguous which instance -declaration -should be used to resolve a type-class constraint. This behaviour -can be modified by two flags: --fallow-overlapping-instances - -and --fallow-incoherent-instances -, as this section discusses. Both these -flags are dynamic flags, and can be set on a per-module basis, using -an OPTIONS_GHC pragma if desired (). -When GHC tries to resolve, say, the constraint C Int Bool, -it tries to match every instance declaration against the -constraint, -by instantiating the head of the instance declaration. For example, consider -these declarations: +Faced with the problems described above, some Haskell programmers might be +tempted to use something like the following version of the class declaration: - instance context1 => C Int a where ... -- (A) - instance context2 => C a Bool where ... -- (B) - instance context3 => C Int [a] where ... -- (C) - instance context4 => C Int [Int] where ... -- (D) + class Collects e c where + empty :: c e + insert :: e -> c e -> c e + member :: e -> c e -> Bool -The instances (A) and (B) match the constraint C Int Bool, -but (C) and (D) do not. When matching, GHC takes -no account of the context of the instance declaration -(context1 etc). -GHC's default behaviour is that exactly one instance must match the -constraint it is trying to resolve. -It is fine for there to be a potential of overlap (by -including both declarations (A) and (B), say); an error is only reported if a -particular constraint matches more than one. - - - -The flag instructs GHC to allow -more than one instance to match, provided there is a most specific one. For -example, the constraint C Int [Int] matches instances (A), -(C) and (D), but the last is more specific, and hence is chosen. If there is no -most-specific match, the program is rejected. +The key difference here is that we abstract over the type constructor c that is +used to form the collection type c e, and not over that collection type itself, +represented by ce in the original class declaration. This avoids the immediate +problems that we mentioned above: empty has type Collects e c => c +e, which is not ambiguous. -However, GHC is conservative about committing to an overlapping instance. For example: +The function f from the previous section has a more accurate type: - f :: [b] -> [b] - f x = ... + f :: (Collects e c) => e -> e -> c e -> c e -Suppose that from the RHS of f we get the constraint -C Int [b]. But -GHC does not commit to instance (C), because in a particular -call of f, b might be instantiate -to Int, in which case instance (D) would be more specific still. -So GHC rejects the program. If you add the flag , -GHC will instead pick (C), without complaining about -the problem of subsequent instantiations. +The function g from the previous section is now rejected with a type error as +we would hope because the type of f does not allow the two arguments to have +different types. +This, then, is an example of a multiple parameter class that does actually work +quite well in practice, without ambiguity problems. +There is, however, a catch. This version of the Collects class is nowhere near +as general as the original class seemed to be: only one of the four instances +for Collects +given above can be used with this version of Collects because only one of +them---the instance for lists---has a collection type that can be written in +the form c e, for some type constructor c, and element type e. + + +Adding functional dependencies + -The willingness to be overlapped or incoherent is a property of -the instance declaration itself, controlled by the -presence or otherwise of the -and flags when that mdodule is -being defined. Neither flag is required in a module that imports and uses the -instance declaration. Specifically, during the lookup process: - - -An instance declaration is ignored during the lookup process if (a) a more specific -match is found, and (b) the instance declaration was compiled with -. The flag setting for the -more-specific instance does not matter. - - -Suppose an instance declaration does not matche the constraint being looked up, but -does unify with it, so that it might match when the constraint is further -instantiated. Usually GHC will regard this as a reason for not committing to -some other constraint. But if the instance declaration was compiled with -, GHC will skip the "does-it-unify?" -check for that declaration. - - -All this makes it possible for a library author to design a library that relies on -overlapping instances without the library client having to know. +To get a more useful version of the Collects class, Hugs provides a mechanism +that allows programmers to specify dependencies between the parameters of a +multiple parameter class (For readers with an interest in theoretical +foundations and previous work: The use of dependency information can be seen +both as a generalization of the proposal for `parametric type classes' that was +put forward by Chen, Hudak, and Odersky, or as a special case of Mark Jones's +later framework for "improvement" of qualified types. The +underlying ideas are also discussed in a more theoretical and abstract setting +in a manuscript [implparam], where they are identified as one point in a +general design space for systems of implicit parameterization.). + +To start with an abstract example, consider a declaration such as: + + class C a b where ... + +which tells us simply that C can be thought of as a binary relation on types +(or type constructors, depending on the kinds of a and b). Extra clauses can be +included in the definition of classes to add information about dependencies +between parameters, as in the following examples: + + class D a b | a -> b where ... + class E a b | a -> b, b -> a where ... + +The notation a -> b used here between the | and where +symbols --- not to be +confused with a function type --- indicates that the a parameter uniquely +determines the b parameter, and might be read as "a determines b." Thus D is +not just a relation, but actually a (partial) function. Similarly, from the two +dependencies that are included in the definition of E, we can see that E +represents a (partial) one-one mapping between types. -The flag implies the - flag, but not vice versa. + +More generally, dependencies take the form x1 ... xn -> y1 ... ym, +where x1, ..., xn, and y1, ..., yn are type variables with n>0 and +m>=0, meaning that the y parameters are uniquely determined by the x +parameters. Spaces can be used as separators if more than one variable appears +on any single side of a dependency, as in t -> a b. Note that a class may be +annotated with multiple dependencies using commas as separators, as in the +definition of E above. Some dependencies that we can write in this notation are +redundant, and will be rejected because they don't serve any useful +purpose, and may instead indicate an error in the program. Examples of +dependencies like this include a -> a , +a -> a a , +a -> , etc. There can also be +some redundancy if multiple dependencies are given, as in +a->b, + b->c , a->c , and +in which some subset implies the remaining dependencies. Examples like this are +not treated as errors. Note that dependencies appear only in class +declarations, and not in any other part of the language. In particular, the +syntax for instance declarations, class constraints, and types is completely +unchanged. - - - -Type synonyms in the instance head - -Unlike Haskell 98, instance heads may use type -synonyms. (The instance "head" is the bit after the "=>" in an instance decl.) -As always, using a type synonym is just shorthand for -writing the RHS of the type synonym definition. For example: - - +By including dependencies in a class declaration, we provide a mechanism for +the programmer to specify each multiple parameter class more precisely. The +compiler, on the other hand, is responsible for ensuring that the set of +instances that are in scope at any given point in the program is consistent +with any declared dependencies. For example, the following pair of instance +declarations cannot appear together in the same scope because they violate the +dependency for D, even though either one on its own would be acceptable: - type Point = (Int,Int) - instance C Point where ... - instance C [Point] where ... + instance D Bool Int where ... + instance D Bool Char where ... - - -is legal. However, if you added - - +Note also that the following declaration is not allowed, even by itself: - instance C (Int,Int) where ... + instance D [a] b where ... - - -as well, then the compiler will complain about the overlapping -(actually, identical) instance declarations. As always, type synonyms -must be fully applied. You cannot, for example, write: - - +The problem here is that this instance would allow one particular choice of [a] +to be associated with more than one choice for b, which contradicts the +dependency specified in the definition of D. More generally, this means that, +in any instance of the form: - type P a = [[a]] - instance Monad P where ... + instance D t s where ... - - -This design decision is independent of all the others, and easily -reversed, but it makes sense to me. - +for some particular types t and s, the only variables that can appear in s are +the ones that appear in t, and hence, if the type t is known, then s will be +uniquely determined. - - - - - - -Type signatures - -The context of a type signature -Unlike Haskell 98, constraints in types do not have to be of -the form (class type-variable) or -(class (type-variable type-variable ...)). Thus, -these type signatures are perfectly OK +The benefit of including dependency information is that it allows us to define +more general multiple parameter classes, without ambiguity problems, and with +the benefit of more accurate types. To illustrate this, we return to the +collection class example, and annotate the original definition of Collects +with a simple dependency: - g :: Eq [a] => ... - g :: Ord (T a ()) => ... + class Collects e ce | ce -> e where + empty :: ce + insert :: e -> ce -> ce + member :: e -> ce -> Bool +The dependency ce -> e here specifies that the type e of elements is uniquely +determined by the type of the collection ce. Note that both parameters of +Collects are of kind *; there are no constructor classes here. Note too that +all of the instances of Collects that we gave earlier can be used +together with this new definition. -GHC imposes the following restrictions on the constraints in a type signature. -Consider the type: - +What about the ambiguity problems that we encountered with the original +definition? The empty function still has type Collects e ce => ce, but it is no +longer necessary to regard that as an ambiguous type: Although the variable e +does not appear on the right of the => symbol, the dependency for class +Collects tells us that it is uniquely determined by ce, which does appear on +the right of the => symbol. Hence the context in which empty is used can still +give enough information to determine types for both ce and e, without +ambiguity. More generally, we need only regard a type as ambiguous if it +contains a variable on the left of the => that is not uniquely determined +(either directly or indirectly) by the variables on the right. + + +Dependencies also help to produce more accurate types for user defined +functions, and hence to provide earlier detection of errors, and less cluttered +types for programmers to work with. Recall the previous definition for a +function f: - forall tv1..tvn (c1, ...,cn) => type + f x y = insert x y = insert x . insert y - -(Here, we write the "foralls" explicitly, although the Haskell source -language omits them; in Haskell 98, all the free type variables of an -explicit source-language type signature are universally quantified, -except for the class type variables in a class declaration. However, -in GHC, you can give the foralls if you want. See ). +for which we originally obtained a type: + + f :: (Collects a c, Collects b c) => a -> b -> c -> c + +Given the dependency information that we have for Collects, however, we can +deduce that a and b must be equal because they both appear as the second +parameter in a Collects constraint with the same first parameter c. Hence we +can infer a shorter and more accurate type for f: + + f :: (Collects a c) => a -> a -> c -> c + +In a similar way, the earlier definition of g will now be flagged as a type error. - +Although we have given only a few examples here, it should be clear that the +addition of dependency information can help to make multiple parameter classes +more useful in practice, avoiding ambiguity problems, and allowing more general +sets of instance declarations. + + + + - - - - - Each universally quantified type variable -tvi must be reachable from type. - -A type variable a is "reachable" if it it appears -in the same constraint as either a type variable free in in -type, or another reachable type variable. -A value with a type that does not obey -this reachability restriction cannot be used without introducing -ambiguity; that is why the type is rejected. -Here, for example, is an illegal type: - - - - forall a. Eq a => Int - + +Instance declarations + +Relaxed rules for instance declarations -When a value with this type was used, the constraint Eq tv -would be introduced where tv is a fresh type variable, and -(in the dictionary-translation implementation) the value would be -applied to a dictionary for Eq tv. The difficulty is that we -can never know which instance of Eq to use because we never -get any more information about tv. +An instance declaration has the form + + instance ( assertion1, ..., assertionn) => class type1 ... typem where ... + +The part before the "=>" is the +context, while the part after the +"=>" is the head of the instance declaration. + -Note -that the reachability condition is weaker than saying that a is -functionally dependent on a type variable free in -type (see ). The reason for this is there -might be a "hidden" dependency, in a superclass perhaps. So -"reachable" is a conservative approximation to "functionally dependent". -For example, consider: - - class C a b | a -> b where ... - class C a b => D a b where ... - f :: forall a b. D a b => a -> a - -This is fine, because in fact a does functionally determine b -but that is not immediately apparent from f's type. +In Haskell 98 the head of an instance declaration +must be of the form C (T a1 ... an), where +C is the class, T is a type constructor, +and the a1 ... an are distinct type variables. +Furthermore, the assertions in the context of the instance declaration +must be of the form C a where a +is a type variable that occurs in the head. - - - - Every constraint ci must mention at least one of the -universally quantified type variables tvi. - -For example, this type is OK because C a b mentions the -universally quantified type variable b: - +The flag loosens these restrictions +considerably. Firstly, multi-parameter type classes are permitted. Secondly, +the context and head of the instance declaration can each consist of arbitrary +(well-kinded) assertions (C t1 ... tn) subject only to the +following rules: + + +For each assertion in the context: + +No type variable has more occurrences in the assertion than in the head +The assertion has fewer constructors and variables (taken together + and counting repetitions) than the head + + +The coverage condition. For each functional dependency, +tvsleft -> +tvsright, of the class, +every type variable in +S(tvsright) must appear in +S(tvsleft), where S is the +substitution mapping each type variable in the class declaration to the +corresponding type in the instance declaration. + + +These restrictions ensure that context reduction terminates: each reduction +step makes the problem smaller by at least one +constructor. For example, the following would make the type checker +loop if it wasn't excluded: - forall a. C a b => burble + instance C a => C a where ... +For example, these are OK: + + instance C Int [a] -- Multiple parameters + instance Eq (S [a]) -- Structured type in head + -- Repeated type variable in head + instance C4 a a => C4 [a] [a] + instance Stateful (ST s) (MutVar s) -The next type is illegal because the constraint Eq b does not -mention a: - + -- Head can consist of type variables only + instance C a + instance (Eq a, Show b) => C2 a b + -- Non-type variables in context + instance Show (s a) => Show (Sized s a) + instance C2 Int a => C3 Bool [a] + instance C2 Int a => C3 [a] b + +But these are not: - forall a. Eq b => burble + -- Context assertion no smaller than head + instance C a => C a where ... + -- (C b b) has more more occurrences of b than the head + instance C b b => Foo [b] where ... - - -The reason for this restriction is milder than the other one. The -excluded types are never useful or necessary (because the offending -context doesn't need to be witnessed at this point; it can be floated -out). Furthermore, floating them out increases sharing. Lastly, -excluding them is a conservative choice; it leaves a patch of -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: +The same restrictions apply to instances generated by +deriving clauses. Thus the following is accepted: - type1 -> forall a1..an. context2 => type2 -==> - forall a1..an. context2 => type1 -> type2 + data MinHeap h a = H a (h a) + deriving (Show) -(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: +because the derived instance - g :: Int -> Int -> forall b. b -> Int + instance (Show a, Show (h a)) => Show (MinHeap h a) +conforms to the above rules. + -When doing this hoisting operation, GHC eliminates duplicate constraints. For -example: - - type Foo a = (?x::Int) => Bool -> a - g :: Foo (Foo Int) - -means +A useful idiom permitted by the above rules is as follows. +If one allows overlapping instance declarations then it's quite +convenient to have a "default instance" declaration that applies if +something more specific does not: - g :: (?x::Int) => Bool -> Bool -> Int + instance C a where + op = ... -- Default - - - - - - -Implicit parameters - - Implicit parameters are implemented as described in -"Implicit parameters: dynamic scoping with static types", -J Lewis, MB Shields, E Meijer, J Launchbury, -27th ACM Symposium on Principles of Programming Languages (POPL'00), -Boston, Jan 2000. +You can find lots of background material about the reason for these +restrictions in the paper +Understanding functional dependencies via Constraint Handling Rules. + -(Most of the following, stil rather incomplete, documentation is -due to Jeff Lewis.) - -Implicit parameter support is enabled with the option -. + +Undecidable instances -A variable is called dynamically bound when it is bound by the calling -context of a function and statically bound when bound by the callee's -context. In Haskell, all variables are statically bound. Dynamic -binding of variables is a notion that goes back to Lisp, but was later -discarded in more modern incarnations, such as Scheme. Dynamic binding -can be very confusing in an untyped language, and unfortunately, typed -languages, in particular Hindley-Milner typed languages like Haskell, -only support static scoping of variables. - - -However, by a simple extension to the type class system of Haskell, we -can support dynamic binding. Basically, we express the use of a -dynamically bound variable as a constraint on the type. These -constraints lead to types of the form (?x::t') => t, which says "this -function uses a dynamically-bound variable ?x -of type t'". For -example, the following expresses the type of a sort function, -implicitly parameterized by a comparison function named cmp. +Sometimes even the rules of are too onerous. +For example, sometimes you might want to use the following to get the +effect of a "class synonym": - sort :: (?cmp :: a -> a -> Bool) => [a] -> [a] + class (C1 a, C2 a, C3 a) => C a where { } + + instance (C1 a, C2 a, C3 a) => C a where { } -The dynamic binding constraints are just a new form of predicate in the type class system. - - -An implicit parameter occurs in an expression using the special form ?x, -where x is -any valid identifier (e.g. ord ?x is a valid expression). -Use of this construct also introduces a new -dynamic-binding constraint in the type of the expression. -For example, the following definition -shows how we can define an implicitly parameterized sort function in -terms of an explicitly parameterized sortBy function: +This allows you to write shorter signatures: - sortBy :: (a -> a -> Bool) -> [a] -> [a] + f :: C a => ... + +instead of + + f :: (C1 a, C2 a, C3 a) => ... + +The restrictions on functional dependencies () are particularly troublesome. +It is tempting to introduce type variables in the context that do not appear in +the head, something that is excluded by the normal rules. For example: + + class HasConverter a b | a -> b where + convert :: a -> b + + data Foo a = MkFoo a - sort :: (?cmp :: a -> a -> Bool) => [a] -> [a] - sort = sortBy ?cmp + instance (HasConverter a b,Show b) => Show (Foo a) where + show (MkFoo value) = show (convert value) - +This is dangerous territory, however. Here, for example, is a program that would make the +typechecker loop: + + class D a + class F a b | a->b + instance F [a] [[a]] + instance (D c, F a c) => D [a] -- 'c' is not mentioned in the head + +Similarly, it can be tempting to lift the coverage condition: + + class Mul a b c | a b -> c where + (.*.) :: a -> b -> c - -Implicit-parameter type constraints - -Dynamic binding constraints behave just like other type class -constraints in that they are automatically propagated. Thus, when a -function is used, its implicit parameters are inherited by the -function that called it. For example, our sort function might be used -to pick out the least value in a list: + instance Mul Int Int Int where (.*.) = (*) + instance Mul Int Float Float where x .*. y = fromIntegral x * y + instance Mul a b c => Mul a [b] [c] where x .*. v = map (x.*.) v + +The third instance declaration does not obey the coverage condition; +and indeed the (somewhat strange) definition: - least :: (?cmp :: a -> a -> Bool) => [a] -> a - least xs = head (sort xs) + f = \ b x y -> if b then x .*. [y] else y -Without lifting a finger, the ?cmp parameter is -propagated to become a parameter of least as well. With explicit -parameters, the default is that parameters must always be explicit -propagated. With implicit parameters, the default is to always -propagate them. +makes instance inference go into a loop, because it requires the constraint +(Mul a [b] b). -An implicit-parameter type constraint differs from other type class constraints in the -following way: All uses of a particular implicit parameter must have -the same type. This means that the type of (?x, ?x) -is (?x::a) => (a,a), and not -(?x::a, ?x::b) => (a, b), as would be the case for type -class constraints. +Nevertheless, GHC allows you to experiment with more liberal rules. If you use +the experimental flag +-fallow-undecidable-instances +option, you can use arbitrary +types in both an instance context and instance head. Termination is ensured by having a +fixed-depth recursion stack. If you exceed the stack depth you get a +sort of backtrace, and the opportunity to increase the stack depth +with N. - You can't have an implicit parameter in the context of a class or instance -declaration. For example, both these declarations are illegal: - - class (?x::Int) => C a where ... - instance (?x::a) => Foo [a] where ... - -Reason: exactly which implicit parameter you pick up depends on exactly where -you invoke a function. But the ``invocation'' of instance declarations is done -behind the scenes by the compiler, so it's hard to figure out exactly where it is done. -Easiest thing is to outlaw the offending types. + + + + +Overlapping instances -Implicit-parameter constraints do not cause ambiguity. For example, consider: +In general, GHC requires that that it be unambiguous which instance +declaration +should be used to resolve a type-class constraint. This behaviour +can be modified by two flags: +-fallow-overlapping-instances + +and +-fallow-incoherent-instances +, as this section discusses. Both these +flags are dynamic flags, and can be set on a per-module basis, using +an OPTIONS_GHC pragma if desired (). + +When GHC tries to resolve, say, the constraint C Int Bool, +it tries to match every instance declaration against the +constraint, +by instantiating the head of the instance declaration. For example, consider +these declarations: - f :: (?x :: [a]) => Int -> Int - f n = n + length ?x - - g :: (Read a, Show a) => String -> String - g s = show (read s) + instance context1 => C Int a where ... -- (A) + instance context2 => C a Bool where ... -- (B) + instance context3 => C Int [a] where ... -- (C) + instance context4 => C Int [Int] where ... -- (D) -Here, g has an ambiguous type, and is rejected, but f -is fine. The binding for ?x at f's call site is -quite unambiguous, and fixes the type a. - - - - -Implicit-parameter bindings +The instances (A) and (B) match the constraint C Int Bool, +but (C) and (D) do not. When matching, GHC takes +no account of the context of the instance declaration +(context1 etc). +GHC's default behaviour is that exactly one instance must match the +constraint it is trying to resolve. +It is fine for there to be a potential of overlap (by +including both declarations (A) and (B), say); an error is only reported if a +particular constraint matches more than one. + -An implicit parameter is bound using the standard -let or where binding forms. -For example, we define the min function by binding -cmp. +The flag instructs GHC to allow +more than one instance to match, provided there is a most specific one. For +example, the constraint C Int [Int] matches instances (A), +(C) and (D), but the last is more specific, and hence is chosen. If there is no +most-specific match, the program is rejected. + + +However, GHC is conservative about committing to an overlapping instance. For example: - min :: [a] -> a - min = let ?cmp = (<=) in least + f :: [b] -> [b] + f x = ... +Suppose that from the RHS of f we get the constraint +C Int [b]. But +GHC does not commit to instance (C), because in a particular +call of f, b might be instantiate +to Int, in which case instance (D) would be more specific still. +So GHC rejects the program. If you add the flag , +GHC will instead pick (C), without complaining about +the problem of subsequent instantiations. -A group of implicit-parameter bindings may occur anywhere a normal group of Haskell -bindings can occur, except at top level. That is, they can occur in a let -(including in a list comprehension, or do-notation, or pattern guards), -or a where clause. -Note the following points: +The willingness to be overlapped or incoherent is a property of +the instance declaration itself, controlled by the +presence or otherwise of the +and flags when that mdodule is +being defined. Neither flag is required in a module that imports and uses the +instance declaration. Specifically, during the lookup process: -An implicit-parameter binding group must be a -collection of simple bindings to implicit-style variables (no -function-style bindings, and no type signatures); these bindings are -neither polymorphic or recursive. - - -You may not mix implicit-parameter bindings with ordinary bindings in a -single let -expression; use two nested lets instead. -(In the case of where you are stuck, since you can't nest where clauses.) +An instance declaration is ignored during the lookup process if (a) a more specific +match is found, and (b) the instance declaration was compiled with +. The flag setting for the +more-specific instance does not matter. - -You may put multiple implicit-parameter bindings in a -single binding group; but they are not treated -as a mutually recursive group (as ordinary let bindings are). -Instead they are treated as a non-recursive group, simultaneously binding all the implicit -parameter. The bindings are not nested, and may be re-ordered without changing -the meaning of the program. -For example, consider: - - f t = let { ?x = t; ?y = ?x+(1::Int) } in ?x + ?y - -The use of ?x in the binding for ?y does not "see" -the binding for ?x, so the type of f is - - f :: (?x::Int) => Int -> Int - +Suppose an instance declaration does not matche the constraint being looked up, but +does unify with it, so that it might match when the constraint is further +instantiated. Usually GHC will regard this as a reason for not committing to +some other constraint. But if the instance declaration was compiled with +, GHC will skip the "does-it-unify?" +check for that declaration. +These rules make it possible for a library author to design a library that relies on +overlapping instances without the library client having to know. + + +If an instance declaration is compiled without +, +then that instance can never be overlapped. This could perhaps be +inconvenient. Perhaps the rule should instead say that the +overlapping instance declaration should be compiled in +this way, rather than the overlapped one. Perhaps overlap +at a usage site should be permitted regardless of how the instance declarations +are compiled, if the flag is +used at the usage site. (Mind you, the exact usage site can occasionally be +hard to pin down.) We are interested to receive feedback on these points. + +The flag implies the + flag, but not vice versa. - -Implicit parameters and polymorphic recursion + +Type synonyms in the instance head -Consider these two definitions: +Unlike Haskell 98, instance heads may use type +synonyms. (The instance "head" is the bit after the "=>" in an instance decl.) +As always, using a type synonym is just shorthand for +writing the RHS of the type synonym definition. For example: + + - len1 :: [a] -> Int - len1 xs = let ?acc = 0 in len_acc1 xs + type Point = (Int,Int) + instance C Point where ... + instance C [Point] where ... + - len_acc1 [] = ?acc - len_acc1 (x:xs) = let ?acc = ?acc + (1::Int) in len_acc1 xs - ------------ +is legal. However, if you added - len2 :: [a] -> Int - len2 xs = let ?acc = 0 in len_acc2 xs - len_acc2 :: (?acc :: Int) => [a] -> Int - len_acc2 [] = ?acc - len_acc2 (x:xs) = let ?acc = ?acc + (1::Int) in len_acc2 xs - -The only difference between the two groups is that in the second group -len_acc is given a type signature. -In the former case, len_acc1 is monomorphic in its own -right-hand side, so the implicit parameter ?acc is not -passed to the recursive call. In the latter case, because len_acc2 -has a type signature, the recursive call is made to the -polymoprhic version, which takes ?acc -as an implicit parameter. So we get the following results in GHCi: - Prog> len1 "hello" - 0 - Prog> len2 "hello" - 5 + instance C (Int,Int) where ... -Adding a type signature dramatically changes the result! This is a rather -counter-intuitive phenomenon, worth watching out for. - - -Implicit parameters and monomorphism -GHC applies the dreaded Monomorphism Restriction (section 4.5.5 of the -Haskell Report) to implicit parameters. For example, consider: +as well, then the compiler will complain about the overlapping +(actually, identical) instance declarations. As always, type synonyms +must be fully applied. You cannot, for example, write: + + - f :: Int -> Int - f v = let ?x = 0 in - let y = ?x + v in - let ?x = 5 in - y + type P a = [[a]] + instance Monad P where ... -Since the binding for y falls under the Monomorphism -Restriction it is not generalised, so the type of y is -simply Int, not (?x::Int) => Int. -Hence, (f 9) returns result 9. -If you add a type signature for y, then y -will get type (?x::Int) => Int, so the occurrence of -y in the body of the let will see the -inner binding of ?x, so (f 9) will return -14. - - - - -Linear implicit parameters - -Linear implicit parameters are an idea developed by Koen Claessen, -Mark Shields, and Simon PJ. They address the long-standing -problem that monads seem over-kill for certain sorts of problem, notably: - - - distributing a supply of unique names - distributing a supply of random numbers - distributing an oracle (as in QuickCheck) - - -Linear implicit parameters are just like ordinary implicit parameters, -except that they are "linear" -- that is, they cannot be copied, and -must be explicitly "split" instead. Linear implicit parameters are -written '%x' instead of '?x'. -(The '/' in the '%' suggests the split!) +This design decision is independent of all the others, and easily +reversed, but it makes sense to me. + - -For example: - - import GHC.Exts( Splittable ) + - data NameSupply = ... - - splitNS :: NameSupply -> (NameSupply, NameSupply) - newName :: NameSupply -> Name - instance Splittable NameSupply where - split = splitNS + + +Type signatures - 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' - ...more equations for f... - -Notice that the implicit parameter %ns is consumed - - once by the call to newName - once by the recursive call to f - - +The context of a type signature -So the translation done by the type checker makes -the parameter explicit: - - f :: NameSupply -> Env -> Expr -> Expr - f ns env (Lam x e) = Lam x' (f ns1 env e) - where - (ns1,ns2) = splitNS ns - 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 +Unlike Haskell 98, constraints in types do not have to be of +the form (class type-variable) or +(class (type-variable type-variable ...)). Thus, +these type signatures are perfectly OK - g x = (x, %ns, %ns) + g :: Eq [a] => ... + g :: Ord (T a ()) => ... -and GHC will infer + + +GHC imposes the following restrictions on the constraints in a type signature. +Consider the type: + - g :: (Splittable a, %ns :: a) => b -> (b,a,a) + forall tv1..tvn (c1, ...,cn) => type -The Splittable class is built into GHC. It's exported by module -GHC.Exts. + +(Here, we write the "foralls" explicitly, although the Haskell source +language omits them; in Haskell 98, all the free type variables of an +explicit source-language type signature are universally quantified, +except for the class type variables in a class declaration. However, +in GHC, you can give the foralls if you want. See ). + -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. + + + + + Each universally quantified type variable +tvi must be reachable from type. + +A type variable a is "reachable" if it it appears +in the same constraint as either a type variable free in in +type, or another reachable type variable. +A value with a type that does not obey +this reachability restriction cannot be used without introducing +ambiguity; that is why the type is rejected. +Here, for example, is an illegal type: + - You cannot have implicit parameters (whether linear or not) - in the context of a class or instance declaration. - - + + forall a. Eq a => Int + -Warnings +When a value with this type was used, the constraint Eq tv +would be introduced where tv is a fresh type variable, and +(in the dictionary-translation implementation) the value would be +applied to a dictionary for Eq tv. The difficulty is that we +can never know which instance of Eq to use because we never +get any more information about tv. + -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: +Note +that the reachability condition is weaker than saying that a is +functionally dependent on a type variable free in +type (see ). The reason for this is there +might be a "hidden" dependency, in a superclass perhaps. So +"reachable" is a conservative approximation to "functionally dependent". +For example, consider: - f :: (%ns :: NameSupply) => Env -> Expr -> Expr - f env (Lam x e) = Lam (newName %ns) (f env e) - where - env' = extend env x (newName %ns) + class C a b | a -> b where ... + class C a b => D a b where ... + f :: forall a b. D a b => a -> a -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. +This is fine, because in fact a does functionally determine b +but that is not immediately apparent from f's type. + + + -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. - + Every constraint ci must mention at least one of the +universally quantified type variables tvi. + +For example, this type is OK because C a b mentions the +universally quantified type variable b: - -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) + forall a. C a b => burble -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 + + +The next type is illegal because the constraint Eq b does not +mention a: + + - foo x = let - foom 0 = [] - foom n = x : foom (n-1) - in - foom + forall a. Eq b => burble -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. - - - - -Explicitly-kinded quantification +The reason for this restriction is milder than the other one. The +excluded types are never useful or necessary (because the offending +context doesn't need to be witnessed at this point; it can be floated +out). Furthermore, floating them out increases sharing. Lastly, +excluding them is a conservative choice; it leaves a patch of +territory free in case we need it later. - -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. + + + + + +Implicit parameters - -Arbitrary-rank polymorphism - + Implicit parameters are implemented as described in +"Implicit parameters: dynamic scoping with static types", +J Lewis, MB Shields, E Meijer, J Launchbury, +27th ACM Symposium on Principles of Programming Languages (POPL'00), +Boston, Jan 2000. + + +(Most of the following, stil rather incomplete, documentation is +due to Jeff Lewis.) + +Implicit parameter support is enabled with the option +. -Haskell type signatures are implicitly quantified. The new keyword forall -allows us to say exactly what this means. For example: +A variable is called dynamically bound when it is bound by the calling +context of a function and statically bound when bound by the callee's +context. In Haskell, all variables are statically bound. Dynamic +binding of variables is a notion that goes back to Lisp, but was later +discarded in more modern incarnations, such as Scheme. Dynamic binding +can be very confusing in an untyped language, and unfortunately, typed +languages, in particular Hindley-Milner typed languages like Haskell, +only support static scoping of variables. +However, by a simple extension to the type class system of Haskell, we +can support dynamic binding. Basically, we express the use of a +dynamically bound variable as a constraint on the type. These +constraints lead to types of the form (?x::t') => t, which says "this +function uses a dynamically-bound variable ?x +of type t'". For +example, the following expresses the type of a sort function, +implicitly parameterized by a comparison function named cmp. - g :: b -> b - -means this: - - g :: forall b. (b -> b) + sort :: (?cmp :: a -> a -> Bool) => [a] -> [a] -The two are treated identically. +The dynamic binding constraints are just a new form of predicate in the type class system. - -However, GHC's type system supports arbitrary-rank -explicit universal quantification in -types. -For example, all the following types are legal: +An implicit parameter occurs in an expression using the special form ?x, +where x is +any valid identifier (e.g. ord ?x is a valid expression). +Use of this construct also introduces a new +dynamic-binding constraint in the type of the expression. +For example, the following definition +shows how we can define an implicitly parameterized sort function in +terms of an explicitly parameterized sortBy function: - 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 + sortBy :: (a -> a -> Bool) -> [a] -> [a] - 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. + sort :: (?cmp :: a -> a -> Bool) => [a] -> [a] + sort = sortBy ?cmp + + + +Implicit-parameter type constraints -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: +Dynamic binding constraints behave just like other type class +constraints in that they are automatically propagated. Thus, when a +function is used, its implicit parameters are inherited by the +function that called it. For example, our sort function might be used +to pick out the least value in a list: - x1 :: [forall a. a->a] - x2 :: (forall a. a->a, Int) - x3 :: Maybe (forall a. a->a) + least :: (?cmp :: a -> a -> Bool) => [a] -> a + least xs = head (sort xs) -Of course forall becomes a keyword; you can't use forall as -a type variable any more! +Without lifting a finger, the ?cmp parameter is +propagated to become a parameter of least as well. With explicit +parameters, the default is that parameters must always be explicit +propagated. With implicit parameters, the default is to always +propagate them. - - - -Examples - - -In a data or newtype declaration one can quantify -the types of the constructor arguments. Here are several examples: +An implicit-parameter type constraint differs from other type class constraints in the +following way: All uses of a particular implicit parameter must have +the same type. This means that the type of (?x, ?x) +is (?x::a) => (a,a), and not +(?x::a, ?x::b) => (a, b), as would be the case for type +class constraints. + You can't have an implicit parameter in the context of a class or instance +declaration. For example, both these declarations are illegal: + + class (?x::Int) => C a where ... + instance (?x::a) => Foo [a] where ... + +Reason: exactly which implicit parameter you pick up depends on exactly where +you invoke a function. But the ``invocation'' of instance declarations is done +behind the scenes by the compiler, so it's hard to figure out exactly where it is done. +Easiest thing is to outlaw the offending types. - +Implicit-parameter constraints do not cause ambiguity. For example, consider: -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 - } + f :: (?x :: [a]) => Int -> Int + f n = n + length ?x -newtype Swizzle = MkSwizzle (Ord a => [a] -> [a]) + g :: (Read a, Show a) => String -> String + g s = show (read s) - +Here, g has an ambiguous type, and is rejected, but f +is fine. The binding for ?x at f's call site is +quite unambiguous, and fixes the type a. + - -The constructors have rank-2 types: - + +Implicit-parameter bindings - +An implicit parameter is bound using the standard +let or where binding forms. +For example, we define the min function by binding +cmp. -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 + min :: [a] -> a + min = let ?cmp = (<=) in least - - - - -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. - -As for type signatures, implicit quantification happens for non-overloaded -types too. So if you write this: +A group of implicit-parameter bindings may occur anywhere a normal group of Haskell +bindings can occur, except at top level. That is, they can occur in a let +(including in a list comprehension, or do-notation, or pattern guards), +or a where clause. +Note the following points: + + +An implicit-parameter binding group must be a +collection of simple bindings to implicit-style variables (no +function-style bindings, and no type signatures); these bindings are +neither polymorphic or recursive. + + +You may not mix implicit-parameter bindings with ordinary bindings in a +single let +expression; use two nested lets instead. +(In the case of where you are stuck, since you can't nest where clauses.) + + +You may put multiple implicit-parameter bindings in a +single binding group; but they are not treated +as a mutually recursive group (as ordinary let bindings are). +Instead they are treated as a non-recursive group, simultaneously binding all the implicit +parameter. The bindings are not nested, and may be re-ordered without changing +the meaning of the program. +For example, consider: - data T a = MkT (Either a b) (b -> b) + f t = let { ?x = t; ?y = ?x+(1::Int) } in ?x + ?y - -it's just as if you had written this: - +The use of ?x in the binding for ?y does not "see" +the binding for ?x, so the type of f is - data T a = MkT (forall b. Either a b) (forall b. b -> b) + f :: (?x::Int) => Int -> Int - -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.) + + - -You construct values of types T1, MonadT, Swizzle by applying -the constructor to suitable values, just as usual. For example, - + - +Implicit parameters and polymorphic recursion + +Consider these two definitions: - 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 + len1 :: [a] -> Int + len1 xs = let ?acc = 0 in len_acc1 xs - mkTs :: (forall b. b -> b -> b) -> a -> [T a] - mkTs f x y = [T1 f x, T1 f y] - + len_acc1 [] = ?acc + len_acc1 (x:xs) = let ?acc = ?acc + (1::Int) in len_acc1 xs - + ------------ - -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.) - + len2 :: [a] -> Int + len2 xs = let ?acc = 0 in len_acc2 xs - -When you use pattern matching, the bound variables may now have -polymorphic types. For example: + len_acc2 :: (?acc :: Int) => [a] -> Int + len_acc2 [] = ?acc + len_acc2 (x:xs) = let ?acc = ?acc + (1::Int) in len_acc2 xs + +The only difference between the two groups is that in the second group +len_acc is given a type signature. +In the former case, len_acc1 is monomorphic in its own +right-hand side, so the implicit parameter ?acc is not +passed to the recursive call. In the latter case, because len_acc2 +has a type signature, the recursive call is made to the +polymoprhic version, which takes ?acc +as an implicit parameter. So we get the following results in GHCi: + + Prog> len1 "hello" + 0 + Prog> len2 "hello" + 5 + +Adding a type signature dramatically changes the result! This is a rather +counter-intuitive phenomenon, worth watching out for. + - +Implicit parameters and monomorphism +GHC applies the dreaded Monomorphism Restriction (section 4.5.5 of the +Haskell Report) to implicit parameters. For example, consider: - 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) + f :: Int -> Int + f v = let ?x = 0 in + let y = ?x + v in + let ?x = 5 in + y - +Since the binding for y falls under the Monomorphism +Restriction it is not generalised, so the type of y is +simply Int, not (?x::Int) => Int. +Hence, (f 9) returns result 9. +If you add a type signature for y, then y +will get type (?x::Int) => Int, so the occurrence of +y in the body of the let will see the +inner binding of ?x, so (f 9) will return +14. + + + + - -Scoped type variables - + +Explicitly-kinded quantification -A lexically scoped type variable can be bound by: +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: -A declaration type signature () -A pattern type signature () -A result type signature () +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 + -For example: - -f (xs::[a]) = ys ++ ys - where - ys :: [a] - ys = reverse xs - -The pattern (xs::[a]) includes a type signature for xs. -This brings the type variable a into scope; it scopes over -all the patterns and right hand sides for this equation for f. -In particular, it is in scope at the type signature for y. -At ordinary type signatures, such as that for ys, any type variables -mentioned in the type signature that are not in scope are -implicitly universally quantified. (If there are no type variables in -scope, all type variables mentioned in the signature are universally -quantified, which is just as in Haskell 98.) In this case, since a -is in scope, it is not universally quantified, so the type of ys is -the same as that of xs. 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. +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. - -Scoped type variables are implemented in both GHC and Hugs. Where the -implementations differ from the specification below, those differences -are noted. - + +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 + -So much for the basic idea. Here are the details. +Haskell type signatures are implicitly quantified. The new keyword forall +allows us to say exactly what this means. For example: - - -What a scoped type variable means -A lexically-scoped type variable is simply -the name for a type. The restriction it expresses is that all occurrences -of the same name mean the same type. For example: - f :: [Int] -> Int -> Int - f (xs::[a]) (y::a) = (head xs + y) :: a + g :: b -> b -The pattern type signatures on the left hand side of -f express the fact that xs -must be a list of things of some type a; and that y -must have this same type. The type signature on the expression (head xs) -specifies that this expression must have the same type a. -There is no requirement that the type named by "a" is -in fact a type variable. Indeed, in this case, the type named by "a" is -Int. (This is a slight liberalisation from the original rather complex -rules, which specified that a pattern-bound type variable should be universally quantified.) -For example, all of these are legal: - +means this: - t (x::a) (y::a) = x+y*2 - - f (x::a) (y::b) = [x,y] -- a unifies with b + g :: forall b. (b -> b) + +The two are treated identically. + - g (x::a) = x + 1::Int -- a unifies with Int + +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 - h x = let k (y::a) = [x,y] -- a is free in the - in k x -- environment + f2 :: (forall a. a->a) -> Int -> Int + g2 :: (forall a. Eq a => [a] -> a -> Bool) -> Int -> Int - k (x::a) True = ... -- a unifies with Int - k (x::Int) False = ... + f3 :: ((forall a. a->a) -> Int) -> Bool -> Bool - w :: [b] -> [b] - w (x::a) = x -- a unifies with [b] + f4 :: Int -> (forall a. a -> a) - - - - -Scope and implicit quantification - +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. + -All the type variables mentioned in a pattern, -that are not already in scope, -are brought into scope by the pattern. We describe this set as -the type variables bound by the pattern. -For example: - - f (x::a) = let g (y::(a,b)) = fst y - in - g (x,True) - -The pattern (x::a) brings the type variable -a into scope, as well as the term -variable x. The pattern (y::(a,b)) -contains an occurrence of the already-in-scope type variable a, -and brings into scope the type variable b. +The function f3 has a rank-3 type; +it has rank-2 types on the left of a function arrow. - - - -The type variable(s) bound by the pattern have the same scope -as the term variable(s) bound by the pattern. For example: - - let - f (x::a) = <...rhs of f...> - (p::b, q::b) = (1,2) - in <...body of let...> - -Here, the type variable a scopes over the right hand side of f, -just like x does; while the type variable b scopes over the -body of the let, and all the other definitions in the let, -just like p and q do. -Indeed, the newly bound type variables also scope over any ordinary, separate -type signatures in the let group. +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 or right (see f4, for example) +of a function arrow + 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! - - - - -The type variables bound by the pattern may be -mentioned in ordinary type signatures or pattern -type signatures anywhere within their scope. - - + +Examples + - - In ordinary type signatures, any type variable mentioned in the -signature that is in scope is not universally quantified. - +In a data or newtype declaration one can quantify +the types of the constructor arguments. Here are several examples: - - - - Ordinary type signatures do not bring any new type variables -into scope (except in the type signature itself!). So this is illegal: - f :: a -> a - f x = x::a - +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 + } -It's illegal because a is not in scope in the body of f, -so the ordinary signature x::a is equivalent to x::forall a.a; -and that is an incorrect typing. +newtype Swizzle = MkSwizzle (Ord a => [a] -> [a]) + - - -The pattern type signature is a monotype: +The constructors have rank-2 types: - - -A pattern type signature cannot contain any explicit forall quantification. - - - -The type variables bound by a pattern type signature can only be instantiated to monotypes, -not to type schemes. - - - -There is no implicit universal quantification on pattern type signatures (in contrast to -ordinary type signatures). - - - - - - - -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 +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 - -(Not implemented in Hugs yet, Dec 98). - - - + +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. - + +As for type signatures, implicit quantification happens for non-overloaded +types too. So if you write this: - -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 ] - -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 ] + data T a = MkT (Either a b) (b -> b) -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. - - - - -Where a pattern type signature can occur - - -A pattern type signature can occur in any pattern. For example: - - - - -A pattern type signature can be on an arbitrary sub-pattern, not -just on a variable: +it's just as if you had written this: - f ((x,y)::(a,b)) = (y,x) :: (b,a) + data T a = MkT (forall b. Either a b) (forall b. b -> b) - +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.) - - - Pattern type signatures, including the result part, can be used -in lambda abstractions: - - - (\ (x::a, y) :: a -> x) - +You construct values of types T1, MonadT, Swizzle by applying +the constructor to suitable values, just as usual. For example, - - - Pattern type signatures, including the result part, can be used -in case expressions: - case e of { ((x::a, y) :: (a,b)) -> x } - + 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 -Note that the -> symbol in a case alternative -leads to difficulties when parsing a type signature in the pattern: in -the absence of the extra parentheses in the example above, the parser -would try to interpret the -> as a function -arrow and give a parse error later. + mkTs :: (forall b. b -> b -> b) -> a -> [T a] + mkTs f x y = [T1 f x, T1 f y] + - + +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.) + - -To avoid ambiguity, the type after the “::” in a result -pattern signature on a lambda or case must be atomic (i.e. a single -token or a parenthesised type of some sort). To see why, -consider how one would parse this: +When you use pattern matching, the bound variables may now have +polymorphic types. For example: + + - \ x :: a -> b -> x - + 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) + - - - - Pattern type signatures can bind existential type variables. -For example: - - - - data T = forall a. MkT [a] - - f :: T -> T - f (MkT [t::a]) = MkT t3 - where - t3::[a] = [t,t,t] - +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. + + + +Type inference + +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: - - - - - -Pattern type signatures -can be used in pattern bindings: - +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. + + +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 x = let (y, z::a) = x in ... - f1 x = let (y, z::Int) = x in ... - f2 (x::(Int,a)) = let (y, z::a) = x in ... - f3 :: (b->b) = \x -> x + \ f :: (forall a. a->a) -> (f True, f 'c') - -In all such cases, the binding is not generalised over the pattern-bound -type variables. Thus f3 is monomorphic; f3 -has type b -> b for some type b, -and not forall b. b -> b. -In contrast, the binding +Alternatively, you can give a type signature to the enclosing +context, which GHC can "push down" to find the type for the variable: - f4 :: b->b - f4 = \x -> x + (\ f -> (f True, f 'c')) :: (forall a. a->a) -> (Bool,Char) -makes a polymorphic function, but b is not in scope anywhere -in f4's scope. - - - - - -Pattern type signatures are completely orthogonal to ordinary, separate -type signatures. The two can be used independently or together. - - - - -Result type signatures - - -The result type of a function can be given a signature, thus: - - +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: - f (x::a) :: [a] = [x,x,x] + 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 final :: [a] after all the patterns gives a signature to the -result type. Sometimes this is the only way of naming the type variable -you want: + - - f :: Int -> [a] -> [a] - f n :: ([a] -> [a]) = let g (x::a, y::a) = (y,x) - in \xs -> map g (reverse xs `zip` xs) - + +Implicit quantification - -The type variables bound in a result type signature scope over the right hand side -of the definition. However, consider this corner-case: +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: - rev1 :: [a] -> [a] = \xs -> reverse xs + f :: a -> a + f :: forall a. a -> a - foo ys = rev (ys::[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 ... -The signature on rev1 is considered a pattern type signature, not a result -type signature, and the type variables it binds have the same scope as rev1 -itself (i.e. the right-hand side of rev1 and the rest of the module too). -In particular, the expression (ys::[a]) is OK, because the type variable a -is in scope (otherwise it would mean (ys::forall a.[a]), which would be rejected). -As mentioned above, rev1 is made monomorphic by this scoping rule. -For example, the following program would be rejected, because it claims that rev1 -is polymorphic: +Notice that GHC does not find the innermost possible quantification +point. For example: - rev1 :: [b] -> [b] - rev1 :: [a] -> [a] = \xs -> reverse xs - - + f :: (a -> a) -> Int + -- MEANS + f :: forall a. (a -> a) -> Int + -- NOT + f :: (forall a. a -> a) -> Int - -Result type signatures are not yet implemented in Hugs. - + 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 <literal>Typeable</literal> and <literal>Data</literal> - -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. + +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]). -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. +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 same Num 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". - +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. + + -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 + +Expression type signatures - - 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]) +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). + -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) + +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. +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] -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. + k :: T -> T + k (MkT [t::a]) = MkT t3 + where + t3::[a] = [t,t,t] + +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. + + +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. + + + - A more precise specification + + + +Class and instance declarations -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 +The type variables in the head of a class or instance declaration +scope over the methods defined in the where part. For example: - - 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.) + + class C a where + op :: [a] -> a + + op xs = let ys::[a] + ys = reverse xs + in + head ys + + Generalised typing of mutually recursive bindings @@ -3901,193 +4063,85 @@ pattern binding must have the same context. For example, this is fine: - - - - - - -Generalised Algebraic Data Types + +Overloaded string literals + -Generalised Algebraic Data Types (GADTs) 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. + +GHC supports overloaded string literals. Normally a +string literal has type String, but with overloaded string +literals enabled (with -foverloaded-strings) + a string literal has type (IsString a) => a. - The extensions to GHC are these: - - - 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: - + +This means that the usual string syntax can be used, e.g., for packed strings +and other variations of string like types. String literals behave very much +like integer literals, i.e., they can be used in both expressions and patterns. +If used in a pattern the literal with be replaced by an equality test, in the same +way as an integer literal is. + + +The class IsString is defined as: - 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) +class IsString a where + fromString :: String -> a - - - - -You can use strictness annotations, in the obvious places -in the constructor type: +And the only predefined instance is the obvious one to make strings work as usual: - 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) +instance IsString [Char] where + fromString cs = cs - - - -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 + + +A small example: - data Maybe1 a where { - Nothing1 :: Maybe a ; - Just1 :: a -> Maybe 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. - +newtype MyString = MyString String deriving (Eq, Show) +instance IsString MyString where + fromString = MyString - -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. +greet :: MyString -> MyString +greet "hello" = "world" +greet other = other - 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 +main = do + print $ greet "hello" + print $ greet "fool" -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. - - + +Note that deriving Eq is necessary for the pattern matching +to work since it gets translated into an equality comparison. + -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 } - - - - - + + Template Haskell -Template Haskell allows you to do compile-time meta-programming in Haskell. There is a "home page" for -Template Haskell at -http://www.haskell.org/th/, while -the background to +Template Haskell allows you to do compile-time meta-programming in +Haskell. +The background to the main technical innovations is discussed in " Template Meta-programming for Haskell" (Proc Haskell Workshop 2002). -The details of the Template Haskell design are still in flux. Make sure you -consult the online library reference material + + +There is a Wiki page about +Template Haskell at +http://www.haskell.org/th/, and that is the best place to look for +further details. +You may also +consult the online +Haskell library reference material (search for the type ExpQ). [Temporary: many changes to the original design are described in "http://research.microsoft.com/~simonpj/tmp/notes2.ps". -Not all of these changes are in GHC 6.2.] +Not all of these changes are in GHC 6.6.] The first example from that paper is set out below as a worked example to help get you started. @@ -4171,6 +4225,14 @@ Tim Sheard is going to expand it.) (It would make sense to do so, but it's hard to implement.) + + Furthermore, you can only run a function at compile time if it is imported + from another module that is not part of a mutually-recursive group of modules + that includes the module currently being compiled. For example, when compiling module A, + you can only run Template Haskell functions imported from B if B does not import A (directly or indirectly). + The reason should be clear: to run B we must compile and run A, but we are currently type-checking A. + + The flag -ddump-splices shows the expansion of all top-level splices as they happen. @@ -4797,8 +4859,8 @@ Because the preprocessor targets Haskell (rather than Core), GHC supports an extension of pattern matching called bang patterns. Bang patterns are under consideration for Haskell Prime. The the -Haskell prime feature description contains more discussion and examples +url="http://hackage.haskell.org/trac/haskell-prime/wiki/BangPatterns">Haskell +prime feature description contains more discussion and examples than the material below. @@ -4875,7 +4937,7 @@ 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: @@ -5903,12 +5965,6 @@ The following are good consumers: - length - - - - - ++ (on its first argument) @@ -6176,7 +6232,7 @@ r) GHCziBase.ZMZN GHCziBase.Char -> GHCziBase.ZMZN GHCziBase.Cha r) -> tpl2}) - (%note "foo" + (%note "bar" eta); @@ -6198,6 +6254,22 @@ r) -> described in this section. All are exported by GHC.Exts. + The <literal>seq</literal> function + +The function seq is as described in the Haskell98 Report. + + seq :: a -> b -> b + +It evaluates its first argument to head normal form, and then returns its +second argument as the result. The reason that it is documented here is +that, despite seq's polymorphism, its +second argument can have an unboxed type, or +can be an unboxed tuple; for example (seq x 4#) +or (seq x (# p,q #)). This requires b +to be instantiated to an unboxed type, which is not usually allowed. + + + The <literal>inline</literal> function The inline function is somewhat experimental. @@ -6256,6 +6328,11 @@ If lazy were not lazy, par would look strict in y which would defeat the whole purpose of par. + +Like seq, the argument of lazy can have +an unboxed type. + + The <literal>unsafeCoerce#</literal> function @@ -6271,16 +6348,20 @@ It is generally used when you want to write a program that you know is well-typed, but where Haskell's type system is not expressive enough to prove that it is well typed. + +The argument to unsafeCoerce# can have unboxed types, +although extremely bad things will happen if you coerce a boxed type +to an unboxed type. + + + Generic classes - (Note: support for generic classes is currently broken in - GHC 5.02). - The ideas behind this extension are described in detail in "Derivable type classes", Ralf Hinze and Simon Peyton Jones, Haskell Workshop, Montreal Sept 2000, pp94-105. @@ -6552,7 +6633,7 @@ can be completely switched off by -Monomorphic patteern bindings +Monomorphic pattern bindings