6.4.18. Required type arguments


Allow visible dependent quantification forall x -> in types of terms.

This feature is only partially implemented in GHC. In this section we describe the implemented subset, while the full specification can be found in GHC Proposal #281.

The RequiredTypeArguments extension enables the use of visible dependent quantification in types of terms:

id     :: forall a.   a -> a         -- invisible dependent quantification
id_vdq :: forall a -> a -> a         --   visible dependent quantification

The arrow in forall a -> is part of the syntax and not a function arrow, just like the dot in forall a. is not a type operator.

The choice between forall a. and forall a -> does not have any effect on program execution. Both quantifiers introduce type variables, which are erased during compilation. Rather, the main difference is in the syntax used at call sites:

x1 = id       True     -- invisible forall, the type argument is inferred by GHC
x2 = id @Bool True     -- invisible forall, the type argument is supplied by the programmer

x3 = id_vdq _    True  --   visible forall, the type argument is inferred by GHC
x4 = id_vdq Bool True  --   visible forall, the type argument is supplied by the programmer Terminology: Dependent quantifier

Both forall a. and forall a -> are said to be “dependent” because the result type depends on the supplied type argument:

id @Integer :: Integer -> Integer
id @String  :: String  -> String

id_vdq Integer :: Integer -> Integer
id_vdq String  :: String  -> String

Notice how the RHS of the signature is influenced by the LHS.

This is in contrast to the function arrow ->, which is a non-dependent quantifier:

putStrLn "Hello" :: IO ()
putStrLn "World" :: IO ()

The type of putStrLn is String -> IO (). No matter what string we pass as input, the result type IO () does not depend on it.

This notion of dependence is weaker than the one used in dependently-typed languages (see Relation to Π-types). Terminology: Visible quantifier

We say that forall a. is an invisible quantifier and forall a -> is a visible quantifier. This notion of “visibility” is unrelated to implicit quantification, which happens when the quantifier is omitted:

id     ::             a -> a    -- implicit quantification, invisible forall
id     :: forall a.   a -> a    -- explicit quantification, invisible forall
id_vdq :: forall a -> a -> a    -- explicit quantification,   visible forall

The property of “visibility” actually describes whether the corresponding type argument is visible at the definition site and at call sites:

-- Invisible quantification
id :: forall a. a -> a
id x = x                -- defn site: `a` is not mentioned
call_id = id True       -- call site: `a` is invisibly instantiated to `Bool`

-- Visible quantification
id_vdq :: forall a -> a -> a
id_vdq t x = x                  -- defn site: `a` is visibly bound to `t`
call_id_vdq = id_vdq Bool True  -- call site: `a` is visibly instantiated to `Bool`

In the equation for id there is just one binder on the LHS, x, and it corresponds to the value argument, not to the type argument. Compare that with the definition of id_vdq:

id_vdq :: forall a -> a -> a
id_vdq t x = x

This time we have two binders on the LHS:

  • t, corresponding to forall a -> in the signature
  • x, corresponding to a -> in the signature

The bound t can be used in subsequent patterns, as well as on the right-hand side of the equation:

id_vdq :: forall a -> a -> a
id_vdq t (x :: t) = x :: t
--     ↑       ↑         ↑
--   bound    used      used

We use the terms “visible type argument” and “required type argument” interchangeably. Relation to TypeApplications

RequiredTypeArguments are similar to TypeApplications in that we pass a type to a function as an explicit argument. The difference is that type applications are optional: it is up to the caller whether to write id @Bool True or id True. By default, the compiler infers that the type variable is instantiated to Bool. The existence of a type argument is not reflected syntactically in the expression, it is invisible unless we use a visibility override, i.e. @.

Required type arguments are compulsory. They must appear syntactically at call sites:

x1 = id_vdq Bool True    -- OK
x2 = id_vdq      True    -- not OK

You may use an underscore to infer a required type argument:

x3 = id_vdq _ True       -- OK

That is, it is mostly a matter of syntax whether to use forall a. with type applications or forall a ->. One advantage of required type arguments is that they are never ambiguous. Consider the type of Foreign.Storable.sizeOf:

sizeOf :: forall a. Storable a => a -> Int

The value parameter is not actually used, its only purpose is to drive type inference. At call sites, one might write sizeOf (undefined :: Bool) or sizeOf @Bool undefined. Either way, the undefined is entirely superfluous and exists only to avoid an ambiguous type variable.

With RequiredTypeArguments, we can imagine a slightly different API:

sizeOf :: forall a -> Storable a => Int

If sizeOf had this type, we could write sizeOf Bool without passing a dummy value.

Required type arguments are erased during compilation. While the source program appears to bind and pass required type arguments alongside value arguments, the compiled program does not. There is no runtime overhead associated with required type arguments relative to the usual, invisible type arguments. Relation to ExplicitNamespaces

A required type argument is syntactically indistinguishable from a value argument. In a function call f arg1 arg2 arg3, it is impossible to tell, without looking at the type of f, which of the three arguments are required type arguments, if any.

At the same time, one of the design goals of GHC is to be able to perform name resolution (find the binding sites of identifiers) without involving the type system. Consider:

data Ty = Int | Double | String deriving Show
main = print Int

In this example, there are two constructors named Int in scope:

  • The type constructor Int of kind Type (imported from Prelude)
  • The data constructor Int of type Ty (defined locally)

How does the compiler or someone reading the code know that print Int is supposed to refer to the data constructor, not the type constructor? In GHC, this is resolved as follows. Each identifier is said to occur either in type syntax or term syntax, depending on the surrounding syntactic context:

-- Examples of X in type syntax
type T = X      -- RHS of a type synonym
data D = MkD X  -- field of a data constructor declaration
a :: X          -- RHS of a type signature
b = f (c :: X)  -- RHS of a type signature (in expressions)
f (x :: X) = x  -- RHS of a type signature (in patterns)

-- Examples of X in term syntax
c X = a         -- LHS of a function equation
c a = X         -- RHS of a function equation

One could imagine the entire program “zoned” into type syntax and term syntax, each zone having its own rules for name resolution:

  • In type syntax, type constructors take precedence over data constructors.
  • In term syntax, data constructors take precedence over type constructors.

This means that in the print Int example, the data constructor is selected solely based on the fact that the Int occurs in term syntax. This is firmly determined before GHC attempts to type-check the expression, so the type of print does not influence which of the two Ints is passed to it.

This may not be the desired behavior in a required type argument. Consider:

vshow :: forall a -> Show a => a -> String
vshow t x = show (x :: t)

s1 = vshow Int    42      -- "42"
s2 = vshow Double 42      -- "42.0"

The function calls vshow Int 42 and vshow Double 42 are written in term syntax, while the intended referents of Int and Double are the respective type constructors. As long as there are no data constructors named Int or Double in scope, the example works as intended. However, if such clashing constructor names are introduced, they may disrupt name resolution:

data Ty = Int | Double | String

vshow :: forall a -> Show a => a -> String
vshow t x = show (x :: t)

s1 = vshow Int    42      -- error: Expected a type, but ‘Int’ has kind ‘Ty’
s2 = vshow Double 42      -- error: Expected a type, but ‘Double’ has kind ‘Ty’

In this example the intent was to refer to Int and Double as types, but the names were resolved in favor of data constructors, resulting in type errors.

The example can be fixed with the help of ExplicitNamespaces, which allows embedding type syntax into term syntax using the type keyword:

s1 = vshow (type Int)    42
s2 = vshow (type Double) 42

A similar problem occurs with list and tuple syntax. In type syntax, [a] is the type of a list, i.e. Data.List.List a. In term syntax, [a] is a singleton list, i.e. a : []. A naive attempt to use the list type as a required type argument will result in a type error:

s3 = vshow [Int] [1,2,3]  -- error: Expected a type, but ‘[Int]’ has kind ‘[Type]’

The problem is that GHC assumes [Int] to stand for Int : [] instead of the intended Data.List.List Int. This, too, can be solved using the type keyword:

s3 = vshow (type [Int]) [1,2,3]

Since the type keyword is merely a namespace disambiguation mechanism, it need not apply to the entire type argument. Using it to disambiguate only a part of the type argument is also valid:

f :: forall a -> ...   -- `f`` is a function that expects a required type argument

r1 = f (type (Either () Int))           -- `type` applied to the entire type argument
r2 = f (Either (type ()) Int)           -- `type` applied to one part of it
r3 = f (Either (type ()) (type Int))    -- `type` applied to multiple parts

That is, the expression Either (type ()) (type Int) does not indicate that Either is applied to two type arguments; rather, the entire expression is a single type argument and type is used to disambiguate parts of it.

Outside a required type argument, it is illegal to use type:

r4 = type Int  -- illegal use of ‘type’

Finally, there are types that require the type keyword only due to limitations of the current implementation:

a1 = f (type (Int -> Bool))                       -- function type
a2 = f (type (Read T => T))                       -- constrained type
a3 = f (type (forall a. a))                       -- universally quantified type
a4 = f (type (forall a. Read a => String -> a))   -- a combination of the above

This restriction will be relaxed in a future release of GHC. Effect on implicit quantification

Implicit quantification is said to occur when GHC inserts an implicit forall to bind type variables:

const :: a -> b -> a               -- implicit quantification
const :: forall a b. a -> b -> a   -- explicit quantification

Normally, implicit quantification is unaffected by term variables in scope:

f a = ...  -- the LHS binds `a`
  where const :: a -> b -> a
           -- implicit quantification over `a` takes place
           -- despite the `a` bound on the LHS of `f`

When RequiredTypeArguments is in effect, names bound in term syntax are not implicitly quantified. This allows us to accept the following example:

readshow :: forall a -> (Read a, Show a) => String -> String
readshow t s = show (read s :: t)

s1 = readshow Int    "42"      -- "42"
s2 = readshow Double "42"      -- "42.0"

Note how t is bound on the LHS of a function equation (term syntax), and then used in a type annotation (type syntax). Under the usual rules for implicit quantification, the t would have been implicitly quantified:

-- RequiredTypeArguments
readshow t s = show (read s :: t)   -- the `t` is captured
--       ↑                     ↑
--      bound                 used

-- NoRequiredTypeArguments
readshow t s = show (read s :: t)   -- the `t` is implicitly quantified as follows:
readshow t s = show (read s :: forall t. t)
--       ↑                            ↑  ↑
--      bound                      bound used

On the one hand, taking the current scope into account allows us to accept programs like the one above. On the other hand, some existing programs will no longer compile:

a = 42
f :: a -> a    -- RequiredTypeArguments: the top-level `a` is captured

Because of that, merely enabling RequiredTypeArguments might lead to type errors of this form:

Term variable ‘a’ cannot be used here
  (term variables cannot be promoted)

There are two possible ways to fix this error:

a = 42
f1 :: b -> b              -- (1) use a different variable name
f2 :: forall a. a -> a    -- (2) use an explicit forall

If you are converting a large codebase to be compatible with RequiredTypeArguments, consider using -Wterm-variable-capture during the migration. It is a warning that detects instances of implicit quantification incompatible with RequiredTypeArguments:

The type variable ‘a’ is implicitly quantified,
even though another variable of the same name is in scope:
  ‘a’ defined at ... Relation to Π-types

Both forall a. and forall a -> are dependent quantifiers in the narrow sense defined in Terminology: Dependent quantifier. However, neither of them constitutes a dependent function type (Π-type) that might be familiar to users coming from dependently-typed languages or proof assistants.

  • Haskell has always had functions whose result value depends on the argument value:

    not True  = False   -- argument value: True;  result value: False
    (*2) 5    = 10      -- argument value: 5;     result value: 10

    This captures the usual idea of a function, denoted a -> b.

  • Haskell also has functions whose result type depends on the argument type:

    id    @Int  :: Int  -> Int    -- argument type: Int;  result type: Int  -> Int
    id_vdq Bool :: Bool -> Bool   -- argument type: Bool; result type: Bool -> Bool

    This captures the idea of parametric polymorphism, denoted forall a. b or forall a -> b.

  • Furthermore, Haskell has functions whose result value depends on the argument type:

    maxBound @Int8   = 127    -- argument type: Int8;  result value: 127
    maxBound @Int16  = 32767  -- argument type: Int16; result value: 32767

    This captures the idea of ad-hoc (class-based) polymorphism, denoted C a => b.

  • However, Haskell does not have direct support for functions whose result type depends on the argument value. In the literature, these are often called “dependent functions”, or “Π-types”.


    type F :: Bool -> Bool
    type family F b where
      F True  = ...
      F False = ...
    f :: Bool -> Bool
    f True  = ...
    f False = ...

    In this example, we define a type family F to pattern-match on b at the type level; and a function f to pattern-match on b at the term level. However, it is impossible to quantify over b in such a way that both F and f could be applied to it:

    depfun :: forall (b :: Bool) -> F b  -- Allowed
    depfun b = ... (f b) ...             -- Not allowed

    It is illegal to pass b to f because b does not exist at runtime. Types and type arguments are erased before runtime.

The RequiredTypeArguments extension does not add dependent functions, which would be a much bigger step. Rather RequiredTypeArguments just makes it possible for the type arguments of a function to be compulsory.