From ce136f8bc3bfffc60a0c29f42466c309c8cdac63 Mon Sep 17 00:00:00 2001 From: simonpj Date: Mon, 28 Jul 2003 13:31:37 +0000 Subject: [PATCH] [project @ 2003-07-28 13:31:37 by simonpj] Reorganise the type-system-extension part of GlaExts docs --- ghc/docs/users_guide/glasgow_exts.sgml | 2422 ++++++++++++++++---------------- 1 file changed, 1192 insertions(+), 1230 deletions(-) diff --git a/ghc/docs/users_guide/glasgow_exts.sgml b/ghc/docs/users_guide/glasgow_exts.sgml index 9306e08..4602650 100644 --- a/ghc/docs/users_guide/glasgow_exts.sgml +++ b/ghc/docs/users_guide/glasgow_exts.sgml @@ -774,7 +774,11 @@ This name is not supported by GHC. Type system extensions - + + +Data types and type synonyms + + Data types with no constructors With the flag, GHC lets you declare @@ -792,9 +796,9 @@ 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 @@ -845,1746 +849,1703 @@ like expressions. More specifically: - + - -Explicitly-kinded quantification + +Liberalised type synonyms -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: +Type synonmys are like macros at the type level, and +GHC does validity checking on types only after expanding type synonyms. +That means that GHC can be very much more liberal about type synonyms than Haskell 98: -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 - - - + You can write a forall (including overloading) +in a type synonym, thus: + + type Discard a = forall b. Show b => a -> b -> (a, String) - -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. - + f :: Discard a + f x y = (x, show y) - -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. + g :: Discard Int -> (Int,Bool) -- A rank-2 type + g f = f Int True + - + + +You can write an unboxed tuple in a type synonym: + + type Pr = (# Int, Int #) - -Class method types - - -Haskell 98 prohibits class method types to mention constraints on the -class type variable, thus: + h :: Int -> Pr + h x = (# x, x #) + + + + +You can apply a type synonym to a forall type: - class Seq s a where - fromList :: [a] -> s a - elem :: Eq a => a -> s a -> Bool + type Foo a = a -> a -> Bool + + f :: Foo (forall b. b->b) -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). - - -With the GHC lifts this restriction. - +After expanding the synonym, f has the legal (in GHC) type: + + f :: (forall b. b->b) -> (forall b. b->b) -> Bool + + - + +You can apply a type synonym to a partially applied type synonym: + + type Generic i o = forall x. i x -> o x + type Id x = x + + foo :: Generic Id [] + +After epxanding the synonym, foo has the legal (in GHC) type: + + foo :: forall x. x -> [x] + + - -Multi-parameter type classes - + + -This section documents GHC's implementation of multi-parameter type -classes. There's lots of background in the paper Type -classes: exploring the design space (Simon Peyton Jones, Mark -Jones, Erik Meijer). +GHC currently does kind checking before expanding synonyms (though even that +could be changed.) - - - -Types - -GHC imposes the following restrictions on the form of a qualified -type, whether declared in a type signature -or inferred. Consider the type: - +After expanding type synonyms, GHC does validity checking on types, looking for +the following mal-formedness which isn't detected simply by kind checking: + + +Type constructor applied to a type involving for-alls. + + +Unboxed tuple on left of an arrow. + + +Partially-applied type synonym. + + +So, for example, +this will be rejected: - forall tv1..tvn (c1, ...,cn) => type - + type Pr = (# Int, Int #) -(Here, I write the "foralls" explicitly, although the Haskell source -language omits them; in Haskell 1.4, 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 ). + h :: Pr -> Int + h x = ... + +because GHC does not allow unboxed tuples on the left of a function arrow. + - - - + +Existentially quantified data constructors + - Each universally quantified type variable -tvi must be reachable from type. - -A type variable is "reachable" if it it is functionally dependent -(see ) -on the type variables free in type. -The reason for this is that a value with a type that does not obey -this restriction could not be used without introducing -ambiguity. -Here, for example, is an illegal type: +The idea of using existential quantification in data type declarations +was suggested by Laufer (I believe, thought doubtless someone will +correct me), and implemented in Hope+. It's been in Lennart +Augustsson's hbc Haskell compiler for several years, and +proved very useful. Here's the idea. Consider the declaration: + + - forall a. Eq a => Int + data Foo = forall a. MkFoo a (a -> Bool) + | Nil - -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. - - - - 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 data type Foo has two constructors with types: + + - forall a. C a b => burble + MkFoo :: forall a. a -> (a -> Bool) -> Foo + Nil :: Foo + -The next type is illegal because the constraint Eq b does not -mention a: + +Notice that the type variable a in the type of MkFoo +does not appear in the data type itself, which is plain Foo. +For example, the following expression is fine: + + - forall a. Eq b => burble + [MkFoo 3 even, MkFoo 'c' isUpper] :: [Foo] - -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. - - - - + +Here, (MkFoo 3 even) packages an integer with a function +even that maps an integer to Bool; and MkFoo 'c' +isUpper packages a character with a compatible function. These +two things are each of type Foo and can be put in a list. - -Unlike Haskell 1.4, constraints in types do not have to be of -the form (class type-variables). Thus, these type signatures -are perfectly OK +What can we do with a value of type Foo?. In particular, +what happens when we pattern-match on MkFoo? - f :: Eq (m a) => [m a] -> [m a] - g :: Eq [a] => ... + f (MkFoo val fn) = ??? -This choice recovers principal types, a property that Haskell 1.4 does not have. +Since all we know about val and fn is that they +are compatible, the only (useful) thing we can do with them is to +apply fn to val to get a boolean. For example: - - - -Class declarations - - - - - - Multi-parameter type classes are permitted. For example: - - - class Collection c a where - union :: c a -> c a -> c a - ...etc. + f :: Foo -> Bool + f (MkFoo val fn) = fn val - - - - - 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 { ... } - - +What this allows us to do is to package heterogenous values +together with a bunch of functions that manipulate them, and then treat +that collection of packages in a uniform manner. You can express +quite a bit of object-oriented-like programming this way. + -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.) + +Why existential? + + +What has this to do with existential quantification? +Simply that MkFoo has the (nearly) isomorphic type - - - There are no restrictions on the context in a class declaration -(which introduces superclasses), except that the class hierarchy must -be acyclic. So these class declarations are OK: - - class Functor (m k) => FiniteMap m k where - ... - - class (Monad m, Monad (t m)) => Transform t m where - lift :: m a -> (t m) a + MkFoo :: (exists a . (a, a -> Bool)) -> Foo - - - - - All of the class type variables must be reachable (in the sense -mentioned in ) -from the free varibles of each method type -. For example: - - - - class Coll s a where - empty :: s - insert :: s -> a -> s - +But Haskell programmers can safely think of the ordinary +universally quantified type given above, thereby avoiding +adding a new existential quantification construct. + + -is not OK, because the type of empty doesn't mention -a. This rule is a consequence of Rule 1(a), above, for -types, and has the same motivation. + +Type classes -Sometimes, offending class declarations exhibit misunderstandings. For -example, Coll might be rewritten + +An easy extension (implemented in hbc) is to allow +arbitrary contexts before the constructor. For example: + + - class Coll s a where - empty :: s a - insert :: s a -> a -> s a +data Baz = forall a. Eq a => Baz1 a a + | forall b. Show b => Baz2 b (b -> b) + -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: + +The two constructors have the types you'd expect: + + - class CollE s where - empty :: s - - class CollE s => Coll s a where - insert :: s -> a -> s +Baz1 :: forall a. Eq a => a -> a -> Baz +Baz2 :: forall b. Show b => b -> (b -> b) -> Baz - - - - - - - - - - -Instance declarations - - - +But when pattern matching on Baz1 the matched values can be compared +for equality, and when pattern matching on Baz2 the first matched +value can be converted to a string (as well as applying the function to it). +So this program is legal: + - Instance declarations may not overlap. The two instance -declarations - - instance context1 => C type1 where ... - instance context2 => C type2 where ... + f :: Baz -> String + f (Baz1 p q) | p == q = "Yes" + | otherwise = "No" + f (Baz2 v fn) = show (fn v) - -"overlap" if type1 and type2 unify - -However, if you give the command line option --fallow-overlapping-instances -option then overlapping instance declarations are permitted. -However, GHC arranges never to commit to using an instance declaration -if another instance declaration also applies, either now or later. - - - + - EITHER type1 and type2 do not unify +Operationally, in a dictionary-passing implementation, the +constructors Baz1 and Baz2 must store the +dictionaries for Eq and Show respectively, and +extract it on pattern matching. - - - OR type2 is a substitution instance of type1 -(but not identical to type1), or vice versa. +Notice the way that the syntax fits smoothly with that used for +universal quantification earlier. - - -Notice that these rules - - - - make it clear which instance decl to use -(pick the most specific one that matches) + - - - + +Restrictions - do not mention the contexts context1, context2 -Reason: you can pick which instance decl -"matches" based on the type. +There are several restrictions on the ways in which existentially-quantified +constructors can be use. - - -However the rules are over-conservative. Two instance declarations can overlap, -but it can still be clear in particular situations which to use. For example: - - instance C (Int,a) where ... - instance C (a,Bool) where ... - -These are rejected by GHC's rules, but it is clear what to do when trying -to solve the constraint C (Int,Int) because the second instance -cannot apply. Yell if this restriction bites you. - - -GHC is also conservative about committing to an overlapping instance. For example: - - class C a where { op :: a -> a } - instance C [Int] where ... - instance C a => C [a] where ... - - f :: C b => [b] -> [b] - f x = op x - -From the RHS of f we get the constraint C [b]. But -GHC does not commit to the second instance declaration, because in a paricular -call of f, b might be instantiate to Int, so the first instance declaration -would be appropriate. So GHC rejects the program. If you add -GHC will instead silently pick the second instance, without complaining about -the problem of subsequent instantiations. - -Regrettably, GHC doesn't guarantee to detect overlapping instance -declarations if they appear in different modules. GHC can "see" the -instance declarations in the transitive closure of all the modules -imported by the one being compiled, so it can "see" all instance decls -when it is compiling Main. However, it currently chooses not -to look at ones that can't possibly be of use in the module currently -being compiled, in the interests of efficiency. (Perhaps we should -change that decision, at least for Main.) - - + - There are no restrictions on the type in an instance -head, except that at least one must not be a type variable. -The instance "head" is the bit after the "=>" in an instance decl. For -example, these are OK: + When pattern matching, each pattern match introduces a new, +distinct, type for each existential type variable. These types cannot +be unified with any other type, nor can they escape from the scope of +the pattern match. For example, these fragments are incorrect: - instance C Int a where ... - - instance D (Int, Int) where ... - - instance E [[a]] where ... +f1 (MkFoo a f) = a -Note that instance heads may contain repeated type variables. -For example, this is OK: +Here, the type bound by MkFoo "escapes", because a +is the result of f1. One way to see why this is wrong is to +ask what type f1 has: - instance Stateful (ST s) (MutVar s) where ... + f1 :: Foo -> a -- Weird! -See for an experimental -extension to lift this restriction. - - - - - Unlike Haskell 1.4, instance heads may use type -synonyms. As always, using a type synonym is just shorthand for -writing the RHS of the type synonym definition. For example: +What is this "a" in the result type? Clearly we don't mean +this: - type Point = (Int,Int) - instance C Point where ... - instance C [Point] where ... + f1 :: forall a. Foo -> a -- Wrong! -is legal. However, if you added +The original program is just plain wrong. Here's another sort of error - instance C (Int,Int) where ... + f2 (Baz1 a b) (Baz1 p q) = a==q -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: - - - - type P a = [[a]] - instance Monad P where ... - - +It's ok to say a==b or p==q, but +a==q is wrong because it equates the two distinct types arising +from the two Baz1 constructors. -This design decision is independent of all the others, and easily -reversed, but it makes sense to me. -The types in an instance-declaration context must all -be type variables. Thus +You can't pattern-match on an existentially quantified +constructor in a let or where group of +bindings. So this is illegal: -instance C a b => Eq (a,b) where ... + f3 x = a==b where { Baz1 a b = x } +Instead, use a case expression: -is OK, but + + f3 x = case x of Baz1 a b -> a==b + + +In general, you can only pattern-match +on an existentially-quantified constructor in a case expression or +in the patterns of a function definition. + +The reason for this restriction is really an implementation one. +Type-checking binding groups is already a nightmare without +existentials complicating the picture. Also an existential pattern +binding at the top level of a module doesn't make sense, because it's +not clear how to prevent the existentially-quantified type "escaping". +So for now, there's a simple-to-state restriction. We'll see how +annoying it is. + + + + + + +You can't use existential quantification for newtype +declarations. So this is illegal: -instance C Int b => Foo b where ... + newtype T = forall a. Ord a => MkT a -is not OK. See for an experimental -extension to lift this restriction. +Reason: a value of type T must be represented as a pair +of a dictionary for Ord t and a value of type t. +That contradicts the idea that newtype should have no +concrete representation. You can get just the same efficiency and effect +by using data instead of newtype. If there is no +overloading involved, then there is more of a case for allowing +an existentially-quantified newtype, because the data +because the data version does carry an implementation cost, +but single-field existentially quantified constructors aren't much +use. So the simple restriction (no existential stuff on newtype) +stands, unless there 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:# + + +data T = forall a. MkT [a] deriving( Eq ) + + +To derive Eq in the standard way we would need to have equality +between the single component of two MkT constructors: + +instance Eq T where + (MkT a) == (MkT b) = ??? + +But a and b have distinct types, and so can't be compared. +It's just about possible to imagine examples in which the derived instance +would make sense, but it seems altogether simpler simply to prohibit such +declarations. Define your own instances! - + + - -Undecidable instances -The rules for instance declarations state that: - -At least one of the types in the head of -an instance declaration must not be a type variable. - -All of the types in the context of -an instance declaration must be type variables. - - -These restrictions ensure that -context reduction terminates: each reduction step removes one type -constructor. For example, the following would make the type checker -loop if it wasn't excluded: + + +Class declarations + + +This section documents GHC's implementation of multi-parameter type +classes. There's lots of background in the paper Type +classes: exploring the design space (Simon Peyton Jones, Mark +Jones, Erik Meijer). + + +There are the following constraints on class declarations: + + + + + Multi-parameter type classes are permitted. For example: + + - instance C a => C a where ... + class Collection c a where + union :: c a -> c a -> c a + ...etc. -There are two situations in which the rule is a bit of a pain. First, -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: + + + + + + + + + The class hierarchy must be acyclic. However, the definition +of "acyclic" involves only the superclass relationships. For example, +this is OK: - instance C a where - op = ... -- Default + class C a where { + op :: D b => a -> b -> b + } + + class C a => D a where { ... } -Second, sometimes you might want to use the following to get the -effect of a "class synonym": +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.) + + + + + + + There are no restrictions on the context in a class declaration +(which introduces superclasses), except that the class hierarchy must +be acyclic. So these class declarations are OK: - class (C1 a, C2 a, C3 a) => C a where { } + class Functor (m k) => FiniteMap m k where + ... - instance (C1 a, C2 a, C3 a) => C a where { } + class (Monad m, Monad (t m)) => Transform t m where + lift :: m a -> (t m) a -This allows you to write shorter signatures: + + + + + + + All of the class type variables must be reachable (in the sense +mentioned in ) +from the free varibles of each method type +. For example: - f :: C a => ... + class Coll s a where + empty :: s + insert :: s -> a -> s -instead of +is not OK, because the type of empty doesn't mention +a. This rule is a consequence of Rule 1(a), above, for +types, and has the same motivation. + +Sometimes, offending class declarations exhibit misunderstandings. For +example, Coll might be rewritten - f :: (C1 a, C2 a, C3 a) => ... + class Coll s a where + empty :: s a + insert :: s a -> a -> s a -Voluminous correspondence on the Haskell mailing list has convinced me -that it's worth experimenting 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. - - -I'm on the lookout for a less brutal solution: a simple rule that preserves decidability while -allowing these idioms interesting idioms. - - +which makes the connection between the type of a collection of +a's (namely (s a)) and the element type a. +Occasionally this really doesn't work, in which case you can split the +class like this: + + + + class CollE s where + empty :: s + + class CollE s => Coll s a where + insert :: s -> a -> s + - -Implicit parameters - - Implicit paramters 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.) - -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. + + + + + +Class method types -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. +Haskell 98 prohibits class method types to mention constraints on the +class type variable, thus: - sort :: (?cmp :: a -> a -> Bool) => [a] -> [a] + class Seq s a where + fromList :: [a] -> s a + elem :: Eq a => a -> s a -> Bool -The dynamic binding constraints are just a new form of predicate in the type class system. +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). -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: - - sortBy :: (a -> a -> Bool) -> [a] -> [a] - - sort :: (?cmp :: a -> a -> Bool) => [a] -> [a] - sort = sortBy ?cmp - +With the GHC lifts this restriction. - -Implicit-parameter type constraints + + + + + +Type signatures + +The context of a type signature -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: +Unlike Haskell 1.4, constraints in types do not have to be of +the form (class type-variables). Thus, these type signatures +are perfectly OK - least :: (?cmp :: a -> a -> Bool) => [a] -> a - least xs = fst (sort xs) + f :: Eq (m a) => [m a] -> [m a] + g :: Eq [a] => ... -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. +This choice recovers principal types, a property that Haskell 1.4 does not have. -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. - +GHC imposes the following restrictions on the constraints in a type signature. +Consider the type: - 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 ... + forall tv1..tvn (c1, ...,cn) => type -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: - - f :: (?x :: [a]) => Int -> Int - f n = n + length ?x - 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. +(Here, we write the "foralls" explicitly, although the Haskell source +language omits them; in Haskell 1.4, 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 ). - - - -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. - - min :: [a] -> a - min = let ?cmp = (<=) in least - - + + + + -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.) - + Each universally quantified type variable +tvi must be reachable from type. + +A type variable is "reachable" if it it is functionally dependent +(see ) +on the type variables free in type. +The reason for this is that a value with a type that does not obey +this restriction could not be used without introducing +ambiguity. +Here, for example, is an illegal type: + - -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 + forall a. Eq a => Int - - - - - - -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 suppply of random numbers - distributing an oracle (as in QuickCheck) - +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. - -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!) - -For example: - - import GHC.Exts( Splittable ) + + - data NameSupply = ... - - splitNS :: NameSupply -> (NameSupply, NameSupply) - newName :: NameSupply -> Name + + Every constraint ci must mention at least one of the +universally quantified type variables tvi. - instance Splittable NameSupply where - split = splitNS +For example, this type is OK because C a b mentions the +universally quantified type variable b: - f :: (%ns :: NameSupply) => Env -> Expr -> Expr - f env (Lam x e) = Lam x' (f env e) - where - x' = newName %ns - env' = extend env x x' - ...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 - - - -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 - g x = (x, %ns, %ns) + forall a. C a b => burble -and GHC will infer + + +The next type is illegal because the constraint Eq b does not +mention a: + + - g :: (Splittable a, %ns :: a) => b -> (b,a,a) + forall a. Eq b => burble -The Splittable class is built into GHC. It's exported by module -GHC.Exts. + + +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. + - -Other points: - - '?x' and '%x' -are entirely distinct implicit parameters: you - can use them together and they won't intefere with each other. - You can bind linear implicit parameters in 'with' clauses. + - You cannot have implicit parameters (whether linear or not) - in the context of a class or instance declaration. - + -Warnings - + +For-all hoisting -The monomorphism restriction is even more important than usual. -Consider the example above: +It is often convenient to use generalised type synonyms (see ) at the right hand +end of an arrow, thus: - 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' + type Discard a = forall b. a -> b -> a + + g :: Int -> Discard Int + g x y z = x+y -If we replaced the two occurrences of x' by (newName %ns), which is -usually a harmless thing to do, we get: +Simply expanding the type synonym would give - f :: (%ns :: NameSupply) => Env -> Expr -> Expr - f env (Lam x e) = Lam (newName %ns) (f env e) - where - env' = extend env x (newName %ns) + g :: Int -> (forall b. Int -> b -> Int) -But now the name supply is consumed in three places -(the two calls to newName,and the recursive call to f), so -the result is utterly different. Urk! We don't even have -the beta rule. - - -Well, this is an experimental change. With implicit -parameters we have already lost beta reduction anyway, and -(as John Launchbury puts it) we can't sensibly reason about -Haskell programs without knowing their typing. - - - - -Recursive functions -Linear implicit parameters can be particularly tricky when you have a recursive function -Consider +but GHC "hoists" the forall to give the isomorphic type - foo :: %x::T => Int -> [Int] - foo 0 = [] - foo n = %x : foo (n-1) + g :: forall b. Int -> Int -> b -> Int -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: +In general, the rule is this: to determine the type specified by any explicit +user-written type (e.g. in a type signature), GHC expands type synonyms and then repeatedly +performs the transformation: + + type1 -> forall a1..an. context2 => type2 +==> + forall a1..an. context2 => type1 -> type2 + +(In fact, GHC tries to retain as much synonym information as possible for use in +error messages, but that is a usability issue.) This rule applies, of course, whether +or not the forall comes from a synonym. For example, here is another +valid way to write g's type signature: + + g :: Int -> Int -> forall b. b -> Int + + + +When doing this hoisting operation, GHC eliminates duplicate constraints. For +example: - foo x 0 = [] - foo x n = let (x1,x2) = split x - in x1 : foo x2 (n-1) + type Foo a = (?x::Int) => Bool -> a + g :: Foo (Foo Int) -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 +means - foo x = let - foom 0 = [] - foom n = x : foom (n-1) - in - foom + g :: (?x::Int) => Bool -> Bool -> Int -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. + - -Functional dependencies - + +Instance declarations - 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, -. - + +Overlapping instances -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 ... +In general, instance declarations may not overlap. The two instance +declarations - class Foo a b c | a b -> c where ... + + + instance context1 => C type1 where ... + instance context2 => C type2 where ... -There should be more documentation, but there isn't (yet). Yell if you need it. - - - -Arbitrary-rank polymorphism - +"overlap" if type1 and type2 unify + +However, if you give the command line option +-fallow-overlapping-instances +option then overlapping instance declarations are permitted. +However, GHC arranges never to commit to using an instance declaration +if another instance declaration also applies, either now or later. + + + -Haskell type signatures are implicitly quantified. The new keyword forall -allows us to say exactly what this means. For example: + EITHER type1 and type2 do not unify + + + - - g :: b -> b - -means this: - - g :: forall b. (b -> b) - -The two are treated identically. + OR type2 is a substitution instance of type1 +(but not identical to type1), or vice versa. + + +Notice that these rules + + -However, GHC's type system supports arbitrary-rank -explicit universal quantification in -types. -For example, all the following types are legal: - - f1 :: forall a b. a -> b -> a - g1 :: forall a b. (Ord a, Eq b) => a -> b -> a - - f2 :: (forall a. a->a) -> Int -> Int - g2 :: (forall a. Eq a => [a] -> a -> Bool) -> Int -> Int + make it clear which instance decl to use +(pick the most specific one that matches) - 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 arrrow. As g2 -shows, the polymorphic type on the left of the function arrow can be overloaded. + do not mention the contexts context1, context2 +Reason: you can pick which instance decl +"matches" based on the type. - -The functions f3 and g3 have rank-3 types; -they have rank-2 types on the left of a function arrow. + + + +However the rules are over-conservative. Two instance declarations can overlap, +but it can still be clear in particular situations which to use. For example: + + instance C (Int,a) where ... + instance C (a,Bool) where ... + +These are rejected by GHC's rules, but it is clear what to do when trying +to solve the constraint C (Int,Int) because the second instance +cannot apply. Yell if this restriction bites you. -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,g3 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: +GHC is also conservative about committing to an overlapping instance. For example: - x1 :: [forall a. a->a] - x2 :: (forall a. a->a, Int) - x3 :: Maybe (forall a. a->a) + class C a where { op :: a -> a } + instance C [Int] where ... + instance C a => C [a] where ... + + f :: C b => [b] -> [b] + f x = op x -Of course forall becomes a keyword; you can't use forall as -a type variable any more! +From the RHS of f we get the constraint C [b]. But +GHC does not commit to the second instance declaration, because in a paricular +call of f, b might be instantiate to Int, so the first instance declaration +would be appropriate. So GHC rejects the program. If you add +GHC will instead silently pick the second instance, without complaining about +the problem of subsequent instantiations. - - - -Examples - - -In a data or newtype declaration one can quantify -the types of the constructor arguments. Here are several examples: +Regrettably, GHC doesn't guarantee to detect overlapping instance +declarations if they appear in different modules. GHC can "see" the +instance declarations in the transitive closure of all the modules +imported by the one being compiled, so it can "see" all instance decls +when it is compiling Main. However, it currently chooses not +to look at ones that can't possibly be of use in the module currently +being compiled, in the interests of efficiency. (Perhaps we should +change that decision, at least for Main.) + + + +Type synonyms in the instance head +Unlike Haskell 1.4, 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: + -data T a = T1 (forall b. b -> b -> b) a + type Point = (Int,Int) + instance C Point where ... + instance C [Point] where ... + -data MonadT m = MkMonad { return :: forall a. a -> m a, - bind :: forall a b. m a -> (a -> m b) -> m b - } -newtype Swizzle = MkSwizzle (Ord a => [a] -> [a]) +is legal. However, if you added + + + + instance C (Int,Int) where ... - - -The constructors have rank-2 types: - +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: - -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 + type P a = [[a]] + instance Monad P where ... + +This design decision is independent of all the others, and easily +reversed, but it makes sense to me. + + - -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. + +Undecidable instances + +An instance declaration must normally obey the following rules: + +At least one of the types in the head of +an instance declaration must not be a type variable. +For example, these are OK: + + + instance C Int a where ... + + instance D (Int, Int) where ... + + instance E [[a]] where ... + +but this is not: + + instance F a where ... + +Note that instance heads may contain repeated type variables. +For example, this is OK: + + instance Stateful (ST s) (MutVar s) where ... + + - -As for type signatures, implicit quantification happens for non-overloaded -types too. So if you write this: + +All of the types in the context of +an instance declaration must be type variables. +Thus + +instance C a b => Eq (a,b) where ... + +is OK, but + +instance C Int b => Foo b where ... + +is not OK. + + + +These restrictions ensure that +context reduction terminates: each reduction step removes one type +constructor. For example, the following would make the type checker +loop if it wasn't excluded: - data T a = MkT (Either a b) (b -> b) + instance C a => C a where ... +There are two situations in which the rule is a bit of a pain. First, +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: -it's just as if you had written this: - data T a = MkT (forall b. Either a b) (forall b. b -> b) + instance C a where + op = ... -- Default -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, - +Second, sometimes you might want to use the following to get the +effect of a "class synonym": - - 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 + class (C1 a, C2 a, C3 a) => C a where { } - mkTs :: (forall b. b -> b -> b) -> a -> [T a] - mkTs f x y = [T1 f x, T1 f y] + instance (C1 a, C2 a, C3 a) => C a where { } - - - -The type of the argument can, as usual, be more general than the type -required, as (MkSwizzle reverse) shows. (reverse -does not need the Ord constraint.) - - -When you use pattern matching, the bound variables may now have -polymorphic types. For example: - +This allows you to write shorter signatures: - - f :: T a -> a -> (a, Char) - f (T1 w k) x = (w k x, w 'c' 'd') + f :: C a => ... + - 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) +instead of + + + + f :: (C1 a, C2 a, C3 a) => ... - +Voluminous correspondence on the Haskell mailing list has convinced me +that it's worth experimenting 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. + -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. +I'm on the lookout for a less brutal solution: a simple rule that preserves decidability while +allowing these idioms interesting idioms. - -Type inference - -In general, type inference for arbitrary-rank types is undecideable. -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: + + + +Implicit parameters + + Implicit paramters 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.) -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. +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. -What does it mean to "provide" an explicit type for x? You can do that by -giving a type signature for x directly, using a pattern type signature -(), thus: - - \ f :: (forall a. a->a) -> (f True, f 'c') - -Alternatively, you can give a type signature to the enclosing -context, which GHC can "push down" to find the type for the variable: - - (\ f -> (f True, f 'c')) :: (forall a. a->a) -> (Bool,Char) - -Here the type signature on the expression can be pushed inwards -to give a type signature for f. Similarly, and more commonly, -one can give a type signature for the function itself: - - h :: (forall a. a->a) -> (Bool,Char) - h f = (f True, f 'c') - -You don't need to give a type signature if the lambda bound variable -is a constructor argument. Here is an example we saw earlier: +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. - f :: T a -> a -> (a, Char) - f (T1 w k) x = (w k x, w 'c' 'd') + sort :: (?cmp :: a -> a -> Bool) => [a] -> [a] -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 dynamic binding constraints are just a new form of predicate in the type class system. - - - - - -Implicit quantification - -GHC performs implicit quantification as follows. At the top level (only) of -user-written types, if and only if there is no explicit forall, -GHC finds all the type variables mentioned in the type that are not already -in scope, and universally quantifies them. For example, the following pairs are -equivalent: +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: - f :: a -> a - f :: forall a. a -> a + sortBy :: (a -> a -> Bool) -> [a] -> [a] - g (x::a) = let - h :: a -> b -> b - h x y = y - in ... - g (x::a) = let - h :: forall b. a -> b -> b - h x y = y - in ... + sort :: (?cmp :: a -> a -> Bool) => [a] -> [a] + sort = sortBy ?cmp + + +Implicit-parameter type constraints -Notice that GHC does not find the innermost possible quantification -point. For example: +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: - f :: (a -> a) -> Int - -- MEANS - f :: forall a. (a -> a) -> Int - -- NOT - f :: (forall a. a -> a) -> Int - - - g :: (Ord a => a -> a) -> Int - -- MEANS the illegal type - g :: forall a. (Ord a => a -> a) -> Int - -- NOT - g :: (forall a. Ord a => a -> a) -> Int + least :: (?cmp :: a -> a -> Bool) => [a] -> a + least xs = fst (sort xs) -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. +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. + + +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. - - - - -Liberalised type synonyms - + 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. -Type synonmys are like macros at the type level, and -GHC does validity checking on types only after expanding type synonyms. -That means that GHC can be very much more liberal about type synonyms than Haskell 98: - - You can write a forall (including overloading) -in a type synonym, thus: +Implicit-parameter constraints do not cause ambiguity. For example, consider: - type Discard a = forall b. Show b => a -> b -> (a, String) - - f :: Discard a - f x y = (x, show y) + f :: (?x :: [a]) => Int -> Int + f n = n + length ?x - g :: Discard Int -> (Int,Bool) -- A rank-2 type - g f = f Int True + 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. - + - -You can write an unboxed tuple in a type synonym: - - type Pr = (# Int, Int #) + +Implicit-parameter bindings - h :: Int -> Pr - h x = (# x, x #) + +An implicit parameter is bound using the standard +let or where binding forms. +For example, we define the min function by binding +cmp. + + min :: [a] -> a + min = let ?cmp = (<=) in least + + +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 can apply a type synonym to a forall type: - - type Foo a = a -> a -> Bool - - f :: Foo (forall b. b->b) - -After expanding the synonym, f has the legal (in GHC) type: - - f :: (forall b. b->b) -> (forall b. b->b) -> Bool - +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 can apply a type synonym to a partially applied type synonym: +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: - type Generic i o = forall x. i x -> o x - type Id x = x - - foo :: Generic Id [] + f t = let { ?x = t; ?y = ?x+(1::Int) } in ?x + ?y -After epxanding the synonym, foo has the legal (in GHC) type: +The use of ?x in the binding for ?y does not "see" +the binding for ?x, so the type of f is - foo :: forall x. x -> [x] + f :: (?x::Int) => Int -> Int - + + + + +Linear implicit parameters -GHC currently does kind checking before expanding synonyms (though even that -could be changed.) +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: - -After expanding type synonyms, GHC does validity checking on types, looking for -the following mal-formedness which isn't detected simply by kind checking: - -Type constructor applied to a type involving for-alls. - - -Unboxed tuple on left of an arrow. - - -Partially-applied type synonym. - + distributing a supply of unique names + distributing a suppply of random numbers + distributing an oracle (as in QuickCheck) -So, for example, -this will be rejected: + + +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!) + + +For example: - type Pr = (# Int, Int #) + import GHC.Exts( Splittable ) - h :: Pr -> Int - h x = ... + data NameSupply = ... + + splitNS :: NameSupply -> (NameSupply, NameSupply) + newName :: NameSupply -> Name + + instance Splittable NameSupply where + split = splitNS + + + 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... -because GHC does not allow unboxed tuples on the left of a function arrow. +Notice that the implicit parameter %ns is consumed + + once by the call to newName + once by the recursive call to f + - - - -For-all hoisting -It is often convenient to use generalised type synonyms at the right hand -end of an arrow, thus: +So the translation done by the type checker makes +the parameter explicit: - type Discard a = forall b. a -> b -> a - - g :: Int -> Discard Int - g x y z = x+y + 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' -Simply expanding the type synonym would give +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: - g :: Int -> (forall b. Int -> b -> Int) + class Splittable a where + split :: a -> (a,a) -but GHC "hoists" the forall to give the isomorphic type +The instance for Splittable NameSupply tells GHC how to implement +split for name supplies. But we can simply write - g :: forall b. Int -> Int -> b -> Int + g x = (x, %ns, %ns) -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: +and GHC will infer - type1 -> forall a1..an. context2 => type2 -==> - forall a1..an. context2 => type1 -> type2 + g :: (Splittable a, %ns :: a) => b -> (b,a,a) -(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: +The Splittable class is built into GHC. It's exported by module +GHC.Exts. + + +Other points: + + '?x' and '%x' +are entirely distinct implicit parameters: you + can use them together and they won't intefere with each other. + + + You can bind linear implicit parameters in 'with' clauses. + + You cannot have implicit parameters (whether linear or not) + in the context of a class or instance declaration. + + + +Warnings + + +The monomorphism restriction is even more important than usual. +Consider the example above: - g :: Int -> Int -> forall b. b -> Int + f :: (%ns :: NameSupply) => Env -> Expr -> Expr + f env (Lam x e) = Lam x' (f env e) + where + x' = newName %ns + env' = extend env x x' + +If we replaced the two occurrences of x' by (newName %ns), which is +usually a harmless thing to do, we get: + + f :: (%ns :: NameSupply) => Env -> Expr -> Expr + f env (Lam x e) = Lam (newName %ns) (f env e) + where + env' = extend env x (newName %ns) +But now the name supply is consumed in three places +(the two calls to newName,and the recursive call to f), so +the result is utterly different. Urk! We don't even have +the beta rule. -When doing this hoisting operation, GHC eliminates duplicate constraints. For -example: +Well, this is an experimental change. With implicit +parameters we have already lost beta reduction anyway, and +(as John Launchbury puts it) we can't sensibly reason about +Haskell programs without knowing their typing. + + + + +Recursive functions +Linear implicit parameters can be particularly tricky when you have a recursive function +Consider - type Foo a = (?x::Int) => Bool -> a - g :: Foo (Foo Int) + foo :: %x::T => Int -> [Int] + foo 0 = [] + foo n = %x : foo (n-1) -means +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: - g :: (?x::Int) => Bool -> Bool -> Int + foo x 0 = [] + foo x n = let (x1,x2) = split x + in x1 : foo x2 (n-1) + +But if you don't supply a type signature, GHC uses the Hindley +Milner trick of using a single monomorphic instance of the function +for the recursive calls. That is what makes Hindley Milner type inference +work. So the translation becomes + + foo x = let + foom 0 = [] + foom n = x : foom (n-1) + in + foom +Result: 'x' is not split, and you get a list of identical T's. So the +semantics of the program depends on whether or not foo has a type signature. +Yikes! + +You may say that this is a good reason to dislike linear implicit parameters +and you'd be right. That is why they are an experimental feature. - + + - -Existentially quantified data constructors +<sect2 id="functional-dependencies"> +<title>Functional dependencies - -The idea of using existential quantification in data type declarations -was suggested by Laufer (I believe, thought doubtless someone will -correct me), and implemented in Hope+. It's been in Lennart -Augustsson's hbc Haskell compiler for several years, and -proved very useful. Here's the idea. Consider the declaration: + 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. - data Foo = forall a. MkFoo a (a -> Bool) - | Nil - - - - - -The data type Foo has two constructors with types: - - - + class (Monad m) => MonadState s m | m -> s where ... - - MkFoo :: forall a. a -> (a -> Bool) -> Foo - Nil :: Foo + class Foo a b c | a b -> c where ... - - - - -Notice that the type variable a in the type of MkFoo -does not appear in the data type itself, which is plain Foo. -For example, the following expression is fine: +There should be more documentation, but there isn't (yet). Yell if you need it. + - - - - [MkFoo 3 even, MkFoo 'c' isUpper] :: [Foo] - - - -Here, (MkFoo 3 even) packages an integer with a function -even that maps an integer to Bool; and MkFoo 'c' -isUpper packages a character with a compatible function. These -two things are each of type Foo and can be put in a list. - + +Explicitly-kinded quantification -What can we do with a value of type Foo?. In particular, -what happens when we pattern-match on MkFoo? +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. - - - - f (MkFoo val fn) = ??? - - +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 + + -Since all we know about val and fn is that they -are compatible, the only (useful) thing we can do with them is to -apply fn to val to get a boolean. For example: +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. - - - f :: Foo -> Bool - f (MkFoo val fn) = fn val - - +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. + - -What this allows us to do is to package heterogenous values -together with a bunch of functions that manipulate them, and then treat -that collection of packages in a uniform manner. You can express -quite a bit of object-oriented-like programming this way. - - -Why existential? +<sect2 id="universal-quantification"> +<title>Arbitrary-rank polymorphism -What has this to do with existential quantification? -Simply that MkFoo has the (nearly) isomorphic type +Haskell type signatures are implicitly quantified. The new keyword forall +allows us to say exactly what this means. For example: - - - MkFoo :: (exists a . (a, a -> Bool)) -> Foo + g :: b -> b - - - - -But Haskell programmers can safely think of the ordinary -universally quantified type given above, thereby avoiding -adding a new existential quantification construct. - - - - - -Type classes - - -An easy extension (implemented in hbc) is to allow -arbitrary contexts before the constructor. For example: - - - - +means this: -data Baz = forall a. Eq a => Baz1 a a - | forall b. Show b => Baz2 b (b -> b) + g :: forall b. (b -> b) - - - - -The two constructors have the types you'd expect: +The two are treated identically. - +However, GHC's type system supports arbitrary-rank +explicit universal quantification in +types. +For example, all the following types are legal: -Baz1 :: forall a. Eq a => a -> a -> Baz -Baz2 :: forall b. Show b => b -> (b -> b) -> Baz - + 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 - -But when pattern matching on Baz1 the matched values can be compared -for equality, and when pattern matching on Baz2 the first matched -value can be converted to a string (as well as applying the function to it). -So this program is legal: + 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. - - - - f :: Baz -> String - f (Baz1 p q) | p == q = "Yes" - | otherwise = "No" - f (Baz2 v fn) = show (fn v) - - +The functions f2 and g2 have rank-2 types; +the forall is on the left of a function arrrow. As g2 +shows, the polymorphic type on the left of the function arrow can be overloaded. - -Operationally, in a dictionary-passing implementation, the -constructors Baz1 and Baz2 must store the -dictionaries for Eq and Show respectively, and -extract it on pattern matching. +The functions f3 and g3 have rank-3 types; +they have rank-2 types on the left of a function arrow. - -Notice the way that the syntax fits smoothly with that used for -universal quantification earlier. +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,g3 above would be valid +field type signatures. + As the type of an implicit parameter + In a pattern type signature (see ) + +There is one place you cannot put a forall: +you cannot instantiate a type variable with a forall-type. So you cannot +make a forall-type the argument of a type constructor. So these types are illegal: + + x1 :: [forall a. a->a] + x2 :: (forall a. a->a, Int) + x3 :: Maybe (forall a. a->a) + +Of course forall becomes a keyword; you can't use forall as +a type variable any more! - - -Restrictions + +Examples + -There are several restrictions on the ways in which existentially-quantified -constructors can be use. +In a data or newtype declaration one can quantify +the types of the constructor arguments. Here are several examples: - - - - - When pattern matching, each pattern match introduces a new, -distinct, type for each existential type variable. These types cannot -be unified with any other type, nor can they escape from the scope of -the pattern match. For example, these fragments are incorrect: + +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 + } - -f1 (MkFoo a f) = a +newtype Swizzle = MkSwizzle (Ord a => [a] -> [a]) + -Here, the type bound by MkFoo "escapes", because a -is the result of f1. One way to see why this is wrong is to -ask what type f1 has: + +The constructors have rank-2 types: + + - f1 :: Foo -> a -- Weird! +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 + -What is this "a" in the result type? Clearly we don't mean -this: + +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: - f1 :: forall a. Foo -> a -- Wrong! + data T a = MkT (Either a b) (b -> b) - -The original program is just plain wrong. Here's another sort of error - +it's just as if you had written this: - f2 (Baz1 a b) (Baz1 p q) = a==q + data T a = MkT (forall b. Either a b) (forall b. b -> b) - -It's ok to say a==b or p==q, but -a==q is wrong because it equates the two distinct types arising -from the two Baz1 constructors. - - +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 can't pattern-match on an existentially quantified -constructor in a let or where group of -bindings. So this is illegal: +You construct values of types T1, MonadT, Swizzle by applying +the constructor to suitable values, just as usual. For example, + + - f3 x = a==b where { Baz1 a b = x } - - -Instead, use a case expression: + 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 - - f3 x = case x of Baz1 a b -> a==b + mkTs :: (forall b. b -> b -> b) -> a -> [T a] + mkTs f x y = [T1 f x, T1 f y] -In general, you can only pattern-match -on an existentially-quantified constructor in a case expression or -in the patterns of a function definition. - -The reason for this restriction is really an implementation one. -Type-checking binding groups is already a nightmare without -existentials complicating the picture. Also an existential pattern -binding at the top level of a module doesn't make sense, because it's -not clear how to prevent the existentially-quantified type "escaping". -So for now, there's a simple-to-state restriction. We'll see how -annoying it is. + + +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.) - - -You can't use existential quantification for newtype -declarations. So this is illegal: +When you use pattern matching, the bound variables may now have +polymorphic types. For example: + + - newtype T = forall a. Ord a => MkT a - - + f :: T a -> a -> (a, Char) + f (T1 w k) x = (w k x, w 'c' 'd') -Reason: a value of type T must be represented as a pair -of a dictionary for Ord t and a value of type t. -That contradicts the idea that newtype should have no -concrete representation. You can get just the same efficiency and effect -by using data instead of newtype. If there is no -overloading involved, then there is more of a case for allowing -an existentially-quantified newtype, because the data -because the data version does carry an implementation cost, -but single-field existentially quantified constructors aren't much -use. So the simple restriction (no existential stuff on newtype) -stands, unless there are convincing reasons to change it. + 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) + - - - You can't use deriving to define instances of a -data type with existentially quantified data constructors. +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. + + -Reason: in most cases it would not make sense. For example:# + +Type inference + +In general, type inference for arbitrary-rank types is undecideable. +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: + + +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: -data T = forall a. MkT [a] deriving( Eq ) + \ f :: (forall a. a->a) -> (f True, f 'c') + +Alternatively, you can give a type signature to the enclosing +context, which GHC can "push down" to find the type for the variable: + + (\ f -> (f True, f 'c')) :: (forall a. a->a) -> (Bool,Char) + +Here the type signature on the expression can be pushed inwards +to give a type signature for f. Similarly, and more commonly, +one can give a type signature for the function itself: + + h :: (forall a. a->a) -> (Bool,Char) + h f = (f True, f 'c') + +You don't need to give a type signature if the lambda bound variable +is a constructor argument. Here is an example we saw earlier: + + f :: T a -> a -> (a, Char) + f (T1 w k) x = (w k x, w 'c' 'd') +Here we do not need to give a type signature to w, because +it is an argument of constructor T1 and that tells GHC all +it needs to know. + -To derive Eq in the standard way we would need to have equality -between the single component of two MkT constructors: + + + + +Implicit quantification + +GHC performs implicit quantification as follows. At the top level (only) of +user-written types, if and only if there is no explicit forall, +GHC finds all the type variables mentioned in the type that are not already +in scope, and universally quantifies them. For example, the following pairs are +equivalent: -instance Eq T where - (MkT a) == (MkT b) = ??? - + f :: a -> a + f :: forall a. a -> a -But a and b have distinct types, and so can't be compared. -It's just about possible to imagine examples in which the derived instance -would make sense, but it seems altogether simpler simply to prohibit such -declarations. Define your own instances! + g (x::a) = let + h :: a -> b -> b + h x y = y + in ... + g (x::a) = let + h :: forall b. a -> b -> b + h x y = y + in ... + - + +Notice that GHC does not find the innermost possible quantification +point. For example: + + f :: (a -> a) -> Int + -- MEANS + f :: forall a. (a -> a) -> Int + -- NOT + f :: (forall a. a -> a) -> Int - + g :: (Ord a => a -> a) -> Int + -- MEANS the illegal type + g :: forall a. (Ord a => a -> a) -> Int + -- NOT + g :: (forall a. Ord a => a -> a) -> Int + +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. - - + + + Scoped type variables @@ -5182,3 +5143,4 @@ Just to finish with, here's another example I rather like: ;;; sgml-parent-document: ("users_guide.sgml" "book" "chapter" "sect1") *** ;;; End: *** --> + -- 1.7.10.4