6.11.6. Partial Type Signatures


Type checker will allow inferred types for holes.

A partial type signature is a type signature containing special placeholders called wildcards. A wildcard is written as an underscore (e.g. “_”) or, if NamedWildCards is enabled, any identifier with a leading underscore (e.g. “_foo”, “_bar”). Partial type signatures are to type signatures what Typed Holes are to expressions. During compilation these wildcards or holes will generate an error message that describes which type was inferred at the hole’s location, and information about the origin of any free type variables. GHC reports such error messages by default.

Unlike Typed Holes, which make the program incomplete and will generate errors when they are evaluated, this needn’t be the case for holes in type signatures. The type checker is capable (in most cases) of type-checking a binding with or without a type signature. A partial type signature bridges the gap between the two extremes, the programmer can choose which parts of a type to annotate and which to leave over to the type-checker to infer.

By default, the type-checker will report an error message for each hole in a partial type signature, informing the programmer of the inferred type. When the PartialTypeSignatures extension is enabled, the type-checker will accept the inferred type for each hole, generating warnings instead of errors. Additionally, these warnings can be silenced with the -Wno-partial-type-signatures flag.

However, because GHC must infer the type when part of a type is left out, it is unable to use polymorphic recursion. The same restriction takes place when the type signature is omitted completely.

A partial type signature als makes GHC generalise the binding even if MonoLocalBinds is on; see Let-generalisation. Syntax

A (partial) type signature has the following form: forall a b .. . (C1, C2, ..) => tau. It consists of three parts:

  • The type variables: a b ..
  • The constraints: (C1, C2, ..)
  • The (mono)type: tau

We distinguish three kinds of wildcards. Type Wildcards

Wildcards occurring within the monotype (tau) part of the type signature are type wildcards (“type” is often omitted as this is the default kind of wildcard). Type wildcards can be instantiated to any monotype like Bool or Maybe [Bool], including functions and higher-kinded types like (Int -> Bool) or Maybe.

not' :: Bool -> _
not' x = not x
-- Inferred: Bool -> Bool

maybools :: _
maybools = Just [True]
-- Inferred: Maybe [Bool]

just1 :: _ Int
just1 = Just 1
-- Inferred: Maybe Int

filterInt :: _ -> _ -> [Int]
filterInt = filter -- has type forall a. (a -> Bool) -> [a] -> [a]
-- Inferred: (Int -> Bool) -> [Int] -> [Int]

For instance, the first wildcard in the type signature not' would produce the following error message:

Test.hs:4:17: error:
    • Found type wildcard ‘_’ standing for ‘Bool’
      To use the inferred type, enable PartialTypeSignatures
    • In the type signature:
        not' :: Bool -> _
    • Relevant bindings include
        not' :: Bool -> Bool (bound at Test.hs:5:1)

When a wildcard is not instantiated to a monotype, it will be generalised over, i.e. replaced by a fresh type variable, e.g.

foo :: _ -> _
foo x = x
-- Inferred: forall t. t -> t

filter' :: _
filter' = filter -- has type forall a. (a -> Bool) -> [a] -> [a]
-- Inferred: (a -> Bool) -> [a] -> [a] Named Wildcards

Status:Included in GHC2024, GHC2021

Allow naming of wildcards (e.g. _x) in type signatures.

Type wildcards can also be named by giving the underscore an identifier as suffix, i.e. _a. These are called named wildcards. All occurrences of the same named wildcard within one type signature will unify to the same type. For example:

f :: _x -> _x
f ('c', y) = ('d', error "Urk")
-- Inferred: forall t. (Char, t) -> (Char, t)

The named wildcard forces the argument and result types to be the same. Lacking a signature, GHC would have inferred forall a b. (Char, a) -> (Char, b). A named wildcard can be mentioned in constraints, provided it also occurs in the monotype part of the type signature to make sure that it unifies with something:

somethingShowable :: Show _x => _x -> _
somethingShowable x = show x
-- Inferred type: Show a => a -> String

somethingShowable' :: Show _x => _x -> _
somethingShowable' x = show (not x)
-- Inferred type: Bool -> String

Besides an extra-constraints wildcard (see Extra-Constraints Wildcard), only named wildcards can occur in the constraints, e.g. the _x in Show _x.

When ScopedTypeVariables is on, the named wildcards of a function signature scope over the function body just like explicitly-forall’d type variables (Lexically scoped type variables), even though there is no explicit forall. For example:

f :: _a -> _a
f x = let g :: _a -> _a
          g = ...
      in ...

Here the named wildcard _a scopes over the body of f, thereby binding the occurrences of _a in the signature of g. All four occurrences stand for the same type.

Named wildcards should not be confused with type variables. Even though syntactically similar, named wildcards can unify with monotypes as well as be generalised over (and behave as type variables).

In the first example above, _x is generalised over (and is effectively replaced by a fresh type variable a). In the second example, _x is unified with the Bool type, and as Bool implements the Show type class, the constraint Show Bool can be simplified away.

By default, GHC (as the Haskell 2010 standard prescribes) parses identifiers starting with an underscore in a type as type variables. To treat them as named wildcards, the NamedWildCards extension should be enabled. The example below demonstrated the effect.

foo :: _a -> _a
foo _ = False

Compiling this program without enabling NamedWildCards produces the following error message complaining about the type variable _a no matching the actual type Bool.

Test.hs:5:9: error:
    • Couldn't match expected type ‘_a’ with actual type ‘Bool’
      ‘_a’ is a rigid type variable bound by
        the type signature for:
          foo :: forall _a. _a -> _a
        at Test.hs:4:8
    • In the expression: False
      In an equation for ‘foo’: foo _ = False
    • Relevant bindings include foo :: _a -> _a (bound at Test.hs:5:1)

Compiling this program with NamedWildCards (as well as PartialTypeSignatures) enabled produces the following error message reporting the inferred type of the named wildcard _a.

Test.hs:4:8: warning: [-Wpartial-type-signatures]
    • Found type wildcard ‘_a’ standing for ‘Bool’
    • In the type signature:
        foo :: _a -> _a
    • Relevant bindings include
        foo :: Bool -> Bool (bound at Test.hs:5:1) Extra-Constraints Wildcard

The third kind of wildcard is the extra-constraints wildcard. The presence of an extra-constraints wildcard indicates that an arbitrary number of extra constraints may be inferred during type checking and will be added to the type signature. In the example below, the extra-constraints wildcard is used to infer three extra constraints.

arbitCs :: _ => a -> String
arbitCs x = show (succ x) ++ show (x == x)
-- Inferred:
--   forall a. (Enum a, Eq a, Show a) => a -> String
-- Error:
Test.hs:5:12: error:
    Found constraint wildcard ‘_’ standing for ‘(Show a, Eq a, Enum a)’
    To use the inferred type, enable PartialTypeSignatures
    In the type signature:
      arbitCs :: _ => a -> String

An extra-constraints wildcard shouldn’t prevent the programmer from already listing the constraints they know or want to annotate, e.g.

-- Also a correct partial type signature:
arbitCs' :: (Enum a, _) => a -> String
arbitCs' x = arbitCs x
-- Inferred:
--   forall a. (Enum a, Show a, Eq a) => a -> String
-- Error:
Test.hs:9:22: error:
    Found constraint wildcard ‘_’ standing for ‘()’
    To use the inferred type, enable PartialTypeSignatures
    In the type signature:
      arbitCs' :: (Enum a, _) => a -> String

An extra-constraints wildcard can also lead to zero extra constraints to be inferred, e.g.

noCs :: _ => String
noCs = "noCs"
-- Inferred: String
-- Error:
Test.hs:13:9: error:
    Found constraint wildcard ‘_’ standing for ‘()’
    To use the inferred type, enable PartialTypeSignatures
    In the type signature:
      noCs :: _ => String

As a single extra-constraints wildcard is enough to infer any number of constraints, only one is allowed in a type signature and it should come last in the list of constraints.

Extra-constraints wildcards cannot be named. Where can they occur?

Partial type signatures are allowed for bindings, pattern and expression signatures, except that extra-constraints wildcards are not supported in pattern or expression signatures. In the following example a wildcard is used in each of the three possible contexts.

{-# LANGUAGE ScopedTypeVariables #-}
foo :: _
foo (x :: _) = (x :: _)
-- Inferred: forall w_. w_ -> w_

Anonymous and named wildcards can occur on the left hand side of a type or data instance declaration; see Wildcards on the LHS of data and type family instances.

Anonymous wildcards are also allowed in visible type applications/ visible kind applications (Visible type application). If you want to specify only the second type argument to wurble, then you can say wurble @_ @Int where the first argument is a wildcard.

Standalone deriving declarations permit the use of a single, extra-constraints wildcard, like so:

deriving instance _ => Eq (Foo a)

This denotes a derived Eq (Foo a) instance where the context is inferred, in much the same way that ordinary deriving clauses do. Any other use of wildcards in a standalone deriving declaration is prohibited.

In all other contexts, type wildcards are disallowed, and a named wildcard is treated as an ordinary type variable. For example:

class C _ where ...          -- Illegal
instance Eq (T _)            -- Illegal (currently; would actually make sense)
instance Eq _a => Eq (T _a)  -- Perfectly fine, same as  Eq a => Eq (T a)

Partial type signatures can also be used in Template Haskell splices.

  • Declaration splices: partial type signature are fully supported.

    {-# LANGUAGE TemplateHaskell, NamedWildCards #-}
    $( [d| foo :: _ => _a -> _a -> _
           foo x y = x == y|] )
  • Expression splices: anonymous and named wildcards can be used in expression signatures. Extra-constraints wildcards are not supported, just like in regular expression signatures.

    {-# LANGUAGE TemplateHaskell, NamedWildCards #-}
    $( [e| foo = (Just True :: _m _) |] )
  • Typed expression splices: the same wildcards as in (untyped) expression splices are supported.

  • Pattern splices: anonymous and named wildcards can be used in pattern signatures. Note that ScopedTypeVariables has to be enabled to allow pattern signatures. Extra-constraints wildcards are not supported, just like in regular pattern signatures.

    {-# LANGUAGE TemplateHaskell, ScopedTypeVariables #-}
    foo $( [p| (x :: _) |] ) = x
  • Type splices: only anonymous wildcards are supported in type splices. Named and extra-constraints wildcards are not.

    {-# LANGUAGE TemplateHaskell #-}
    foo :: $( [t| _ |] ) -> a
    foo x = x