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Partial Type Signatures for Haskell

Thomas Winant, Dominique Devriese, Frank Piessens, Tom Schrijvers

In Haskell, programmers have a binary choice between omitting the type signature (and relying on type inference) or explicitly providing the type entirely; there are no intermediate options. Partial type signatures have been often proposed as an intermediate option.

In a partial type signature, annotated types can be mixed with inferred types. A type signature is written like before, but can now contain wildcards, written as underscores. The types of these wildcards or unknown types will be inferred by the type checker, e.g.

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

We have been working on a design and an implementation of them for Haskell and GHC. We have worked out the interaction with OutsideIn(X) type inference and generalisation in a paper presented at PADL'14 and a technical report with proofs. This document attempts to describe our design and our current implementation from a more practical point of view in the hope to get some feedback from the GHC developers community.

Motivation and discussion


Before we start, we want to say something about the pragmatics of when and how to use type signatures. Many people have an opinion about when it is appropriate to write type signatures, e.g. we personally tend to always write type signatures for top-level functions but sometimes omit them for local bindings. Sometimes we are a bit more lax in throw-away code or during development.

We emphasise that partial type signatures do not alter such pragmatics. The point is that Haskell already allows a choice between a full signature and no signature at all and all we add is an intermediate technical option: partial signatures. It is still up to the user to choose how much and what kind of type signatures to write for different kinds of definitions. He/she may still want to use partial signatures only for local definitions and/or non-exported (auxiliary) top-level definitions and/or top-level definitions in throw-away code.


So why would you want to use a partial type signatures instead of a normal type signature?

  • Readability Partial type signature can be useful to improve the readability or understandability of your types. Some types are so verbose that they might confuse the programmer. By replacing distracting boilerplate type information with wildcards, you can put the focus on the crucial bits of your type and guide users through your complex types by hiding the less relevant bits.

    In the example below, the constraints make the type signature bloated.
    replaceLoopsRuleP :: (ProductionRule p,
      EpsProductionRule p,
      RecProductionRule p qphi r,
      TokenProductionRule p t,
      PenaltyProductionRule p) =>
      PenaltyExtendedContextFreeRule qphi r t v ->
      (forall ix. qphi ix -> p [r ix]) ->
      (forall ix. qphi ix -> p [r ix]) -> p v
    With the following partial type signature, the type becomes less daunting:
    replaceLoopsRuleP :: _ =>
      PenaltyExtendedContextFreeRule qphi r t v ->
      (forall ix. qphi ix -> p [r ix]) ->
      (forall ix. qphi ix -> p [r ix]) -> p v
  • Development During development, some parts of the type may still be uncertain, unknown or prone to change, and/or you just don't care to annotate them. With a partial type signature, you can annotate exactly the parts of the type signature you know or want to annotate and replace the (unknown) bits you want the type inferencer to figure out for you with underscores (_).

    Type signatures are very helpful during development, they let the type checker verify that the type of the program you wrote matches the type you wanted it to have. They also provide a machine-checked form of documentation. After a quick glance at the type signature, you instantly know quite a lot about the function you're planning to use and/or forgot the type of, as you wrote it more than an hour ago.

    But during development, your functions, and by consequence your types, will often change. Thus, the type signatures have to be updated too, but programmers tend to be lazy, and will just omit the signatures and add them back in the end. Unfortunately, by omitting the type signatures, you lose the valuable advantages described in the previous paragraph. A solution could be to annotate the fixed parts of your signatures and replace the ever-changing parts by wildcards.
  • Uninferrable types Some types cannot be inferred, see the following program (example adapted from Practical type inference for arbitrary-rank types).
    foo x = (x [True, False], x ['a', 'b'])
    test = foo reverse
    -- reverse :: forall a. [a] -> [a]
    The argument of foo, x, is a polymorphic function, making the type of foo (forall a. [a] -> [a]) -> ([Bool], [Char]), which is a higher-rank type. Inferring higher-rank types is isomorphic to higher-order unification, which is known to be undecidable. So generally, higher-rank types cannot be inferred. Therefore, the program above will not typecheck.

    The program above can typecheck when the higher-rank type is annotated, e.g.
    foo :: (forall a. [a] -> [a]) -> ([Bool], [Char])
    foo x = (x [True, False], x ['a', 'b'])
    Unfortunately, we are forced to write a whole type signature, even though only the type of the argument is strictly required to be annotated, the rest could easily be inferred. With a partial type signature, you can annotate just the required bits and leave out the rest to be inferred for you, e.g.
    foo :: (forall a. [a] -> [a]) -> _
    foo x = (x [True, False], x ['a', 'b'])
  • Community Feature requests or wiki pages for partial type signatures in some form or other already exist: #5248, PartialTypeSigs, PartialTypeAnnotations. More recently, a question popped up on Stack Overflow asking if there is such a thing as partial type signatures for Haskell. Judging from the comments on our answer, people seemed to be interested. After presenting the paper at PADL'14, people told us they would like to see it in GHC. All this evidence combined makes us believe there is a real community demand for partial type signatures in GHC.

Existing Workarounds

In many situations, the effect of partial signatures can be achieved using other means. Consider for example the following partial signature

foo :: (forall a. [a] -> [a]) -> _

A similar thing can already be done with the ScopedTypeVariables extension, which allows pattern signatures, e.g.

foo (x :: forall a. [a] -> [a]) =
  (x [True, False], x ['a', 'b'])

A downside to this approach is that the programmer is sprinkling type hints throughout his or her programs, instead of nicely separating types from terms with a type signature. It's also not directly clear how this can be applied to arbitrary parts of a type, e.g. how would you say something like monadicComputation :: _ Int etc.

Other workarounds exist as well (see the paper for a discussion). They are typically based on adding computationally useless functions or dead code and we find them all less natural and elegant than our partial signatures.

Our Design

A good way to think of a partial type signature is as follows. If a function has a type signature, GHC checks that it has exactly that type. But if it has a partial type signature GHC proceeds exactly as if it were inferring the type for the function (especially including generalisation), except that it additionally forces the function's type to have the shape given by the partial type signature.

We now describe the syntax and the semantics of our partial type signatures.

Type Wildcards

A (partial) type signature has the following form:

forall a b .. . (C1, C2, ..) => tau

It consists of three parts:

  1. The type variables: a b ..
  2. The constraints: (C1, C2, ..)
  3. The (mono)type: tau

We call wildcards occurring within the monotype (tau) part of the type signature type wildcards. Type wildcards can be instantiated to any monotype like Bool or Maybe [Bool], e.g.

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

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

Wildcards can unify with function types, e.g.

qux :: Int -> _
qux i b = i == i && b
-- Inferred: Int -> Bool -> Bool

Additionally, when they are not instantiated to a monotype, they will be generalised over, e.g.

bar :: _ -> _
bar x = x
-- Inferred: forall a. a -> a

bar2 :: _ -> _ -> _
bar2 x f = f x
-- Inferred: forall a b. a -> (a -> b) -> b

Each wildcard will be independently instantiated (see Named wildcards for dependent instantiation), e.g. the three wildcards in bar2 are each instantiated to a different type.

As type wildcards can be generalised over, additional type variables can be universally quantified. One should expect an implicit 'wildcard' in the forall part of the type signature, e.g.

bar3 :: forall a. a -> (a -> _) -> _
bar3 x f = f x
-- Inferred: forall a b. a -> (a -> b) -> b

In addition to the explicitly quantified type variable a, the inferred type now contains a new type variable b. As type variables are implicitly universally quantified in Haskell, we chose not to make this kind of forall 'wildcard' explicit.

Wildcards can also unify with annotated type variables, e.g.

filter' :: _ -> [a] -> [a]
filter' = filter
-- Inferred: forall a. (a -> Bool) -> [a] -> [a]
-- Same as the type of filter

As wildcards can unify with functions, just one type wildcard is enough to infer the whole type of a function of any arity (albeit without constraints).

filter'' :: _
filter'' = filter
-- Inferred: forall a. (a -> Bool) -> [a] -> [a]
-- Same as the type of filter and filter'

And type-constructors, e.g.

justify :: a -> _ a
justify = Just
-- Inferred: forall a. a -> Maybe a

tupleIt :: a -> _ a a
tupleIt x = (x, x)
-- Inferred: forall a. a -> (a, a)

nestedTCs :: a -> _ (_ (_ _ _))
nestedTCs = Just . (: []) . Left
-- Inferred: forall a b. a -> Maybe [Either a b]

Constraint Wildcards

NOTE: we no longer intend to support constraint wildcards as described below. Only named wildcards also occurring in monotype and an extra-constraints wildcard will be allowed.

SLPJ I'm honestly not sure it's worth the complexity you have here. The power to weight ratio seems poor. I'd drop constraint wildcards, and implement only "extra constraint" wildcards. End SLPJ

thomasw Seems reasonable to us. This certainly makes things easier. End thomasw

We call wildcards occurring within a constraint (inside a C in (C1, C2, ..)) constraint wildcards, e.g.

fstIsBool :: (Bool, _) ~ a => a -> Bool
fstIsBool (b1, b2) = not b1 && b2
-- Inferred: (Bool, Bool) -> Bool

class Two a b | a -> b where
  one :: a -> a
  two :: b -> b

-- Ignore the second parameter of the typeclass
secondParam :: Two a _ => a -> a
secondParam x = one x
-- Inferred type: forall a b. Two a b -> a -> a

GHC's constraint solver doesn't unify constraints with each other. E.g. Eq _ or _ a will never be unified with Eq a. The problem the constraint solver is faced with is "deduce Eq a from Eq _, figuring out what the _ should be instantiated to". Or, worse, "deduce Eq a from _ a" or something even less constrained. The constraint solver is absolutely not set up to figure out how to fill in existential variables in the "givens".

So the following program will not work:

-- Neither partial type signature will work
impossible :: Eq _ => a -> a -> Bool
impossible :: _  a => a -> a -> Bool
impossible x y = x == y
-- Actual type: forall a. Eq a => a -> a -> Bool

Note that constraints are not unified for good reasons. One immediate problem is already that it could lead to ambiguities, consider for instance the following program.

-- Incorrect partial type signature
ambi :: _ a => a -> String
ambi x = show (succ x) ++ show (x == x)
-- Actual type:
--   forall a. (Enum a, Eq a, Show a) => a -> String

As constraints are unordered, the constraint solver wouldn't know which one of the inferred constraints (Enum a, Eq a, Show a) the partially annotated constraint (_ a) should be unified with, it would have to guess. Regardless of whether constraints are unified, this program would have been rejected anyway, as only one constraint is partially annotated instead of all three.

However, as demonstrated in the first two examples, we think constraint wildcards can be of (limited) use, provided that certain restrictions are imposed on their usage. More on this later.

Extra-constraints Wildcard

A third kind of wildcard we propose is the extra-constraints wildcard, not to be confused with a constraint wildcard. Whereas constraint wildcards occur inside the Cs in the constraints part (C1, C2, ..) of a partial type signature, an extra-constraints wildcard occurs as a C inside the constraints part.

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. (Show a, Enum a, Eq a) => a -> String

An extra-constraints wildcard shouldn't prevent the programmer from already listing the constraints he knows or wants to annotate, e.g.

-- Also a correct partial type signature:
arbitCs' :: (Enum a, _) => a -> String
arbitCs' = arbitCs
-- Inferred:
--   forall a. (Enum a, Show a, Eq 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

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.

Named Wildcards

A fourth and final kind of wildcard we propose is the named wildcard. A named wildcard is a wildcard suffixed with an identifier. All occurrences of the same named wildcard within one type signature will unify to the same type. They are particularly useful to express constraints on unknown types, e.g.

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

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

Named wildcards should not be confused with type variables. Even though syntactically similar, named wildcards can unify with concrete types 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). In the second example, _x is unified with the Bool type, and as Bool implements the Show typeclass, the constraint Show Bool can be simplified away.

Currently, a named wildcard is in scope in the type signature where it appears, but also in signatures in the right-hand side of the implementation. See the issues section for more discussion.

Extension Flags

Previously, underscores in types were disallowed and caused parse errors. With this extension, underscores will be accepted by the parser. To remain backwards compatible (also with Haskell 2010), we require the user to enable our PartialTypeSignatures extension. Using wildcards without this flag will trigger a parse error with a suggestion to enable the flag.

Currently, GHC parses identifiers prefixed with an underscore in type signatures, i.e. what we consider named wildcards, as type variables, as prescribed in Haskell 2010. To also remain backwards compatible on this front, we propose to only parse them as named wildcards when the extension NamedWildcards is enabled. This extension will be turned on automatically when the PartialTypeSignatures extension is turned on.

We opted for a separate flag because unnamed wildcards cause programs previously rejected to be accepted, whereas named wildcards change the behaviour of previously accepted programs. With this extra flag, the user can still use (unnamed) wildcards with the (No)NamedWildcards extension explicitly turned off, to maintain backwards compatibility.

Local Definitions

With the introduction of OutsideIn(X), the GADTs extension and the TypeFamilies extension, trigger the MonoLocalBinds flag. When it is enabled, types of local bindings without a signature no longer get generalised as described in OutsideIn(X).

For the same reasons, we intend to perform no generalisation over wildcards in partial type signatures for local bindings, e.g.

monoLoc :: forall a. a -> ((a, Bool), (a, Char))
monoLoc x =  let g :: _ -> _
                 g y = (x, y)
             in (g True, g 'v')

When MonoLocalBinds is disabled, the program should work with or without the partial type signature. But when the MonoLocalBinds extension is enabled, it should no longer typecheck, as the type of g is no longer generalised, again with or without the partial type signature.

GHC is slightly more liberal than the strict rule described in the OutsideIn(X) paper though. As explained in Let generalisation in GHC 7.0, even with the MonoLocalBinds extension enabled, a local binding without a type signature will still be generalised when all its free variables are closed. Translating this to partial type signatures: the following example should type check, as it does without a type signature.

safeLoc :: (Bool, Char)
safeLoc = let f :: _ -> _
              f x = x
          in (f True, f 'v')

Furthermore, we believe it necessary to disallow extra-constraints wildcards in partial type signatures for local bindings, as the generalisation over constraints is exactly what led to let should not be generalised.

All of this is currently not yet fully implemented. We have also not yet worked out what we should do precisely if MonoLocalBinds is not enabled, although our intention is to try and align as closely as possible to the generalisation that happens in the absence of a type signature.


We worked out a rigorous formalisation of partial type signature including typing rules, extending the existing formalisation of GHC's type inference system, OutsideIn(X) (as described in OutsideIn(X): Modular type inference with local assumptions). Additionally, we proved soundness of our new typing rules.

We mentioned before that a single type wildcard is enough to infer any type, albeit without constraints. When we combine this with the fact that the presence of a single extra-constraints wildcard is enough to infer any number of constraints, one can see that the partial type signature _ => _ is the same as omitting the type signature and relying completely on type inference. We formulated this, i.e. that _ => _ is equivalent to omitting the type signature, as a theorem and proved it. We also showed that on non-partial type signatures, our new rules have the same effect as the old rules.

See the technical report for more details.


We implemented a subset of our proposal as a branch of GHC (master), available at github. This version can be compiled and should type check a large number of examples in this proposal.

Here is a summary of the changes we have made in the implementation:

  • We have extended the HsSyn representation of Haskell code with two new forms of types: HsWildcardTy for anonymous _ types and HsNamedWildcardTy for named wildcards _a.
  • The parser has been modified to parse them if the appropriate extensions are enabled.
  • The renamer: when renaming TypeSigs, we do a pass over the annotated type signature. We replace named wildcard types with normal type variables but collect and store them as being existentially quantified at the level of TypeSig. In the process, we do the same for anonymous wildcards, replacing them with existentially quantified anonymous type variables.
  • In the type checker, in tcTySig, we create new flexible meta-variables for the wildcards and add them to the type variable environment and store them in the returned sig_fun to get them brought in scope when checking the RHSs.
  • In tcPolyBinds, we introduce a new "Generalisation Plan" called WcGen which is used when there is only one funbind in a group of bindings and its signature contains wildcards. This binding is treated by a new function called tcPolyCombi, which applies our generalisation rules.
  • The changes in tcPolyCombi are accompanied by corresponding changes in mkExport for quantifying over the correct set of type variables.

The two last steps are not yet satisfactory. Most importantly, we do not yet distinguish between local binds and top-level binds. What we think is necessary is to make two versions of tcPolyCombi: one for local binds and one for top-level binds.

Questions and issues

  • Constraint wildcards: NOTE: we no longer intend to support constraint wildcards. Only named wildcards also occurring in monotype and an extra-constraints wildcard will be allowed. The examples below demonstrating the named wildcard in the constraints look useful to us (and already work in the implementation).
    somethingShowable :: Show _x => _x -> _
    somethingShowable x = show x
    -- Inferred type: Show x => x -> String
    somethingShowable' :: Show _x => _x -> _
    somethingShowable' x = show (not x)
    -- Inferred type: Bool -> String
  • The scope of named wildcards: We currently treat all named wildcards as scoped type variables, i.e. if we mention a named wildcard in a top-level partial type signature, then it can also be mentioned in local bindings and it will refer to the same type variable. This means, for example, that the following example won't work:
    test2 :: _a -> _a
    test2 x = let y :: _a
                  y = (x,x)
              in fst y
    The problem is that we will get a constraint _a ~ (_a, _a), and we will get an error about not being able to construct the infinite type _a. If _a were not scoped, then the inner _a could be unified with (_a', _a') where _a' is the outer _a and everything would work.

    A remaining question here is if we should treat named wildcards as scoped only if the ScopedTypeVariables extension is enabled. Our understanding is that not treating type variables as scoped is only done for backwards compatibility, so there seems little point in doing it for named wildcards?
  • Local definitions: no generalisation and no extra constraints wildcards in partial type signatures for local definitions, are we too strict? What when NoMonoLocalBinds is turned on in conjunction with GADTs or TypeFamilies?
  • Generally, we are interested in any problems you may find in this design: major problems or small corner cases that we didn't think of or look at yet, etc.

Some detailed notes

  • Generalising over equality constraints? Consider the following example:
    anyTypeThatIsBool :: _ => a
    anyTypeThatIsBool = True
    A question that we have considered is whether one would like this to type-check. Note that it can be typed as
    anyTypeThatIsBool :: a ~ Bool => a
    anyTypeThatIsBool = True
    We currently do not infer this kind of equality constraints (even when there is an extra constraints wildcard): technically, a will be instantiated to a skolem type variable and an equality for such a variable will be considered absurd, similarly to a constraint like Int ~ Bool. As a result, it will not be generalised over. Note that if we were to allow inferring this kind of constraints, then any universally quantified type variable would behave as a wildcard in the presence of an extra constraints wildcard.

    Note however that this is only the case for equality constraints that directly apply to universally quantified type variables. For example, the following does work:
    type family F a
    data Proxy a = Proxy a
    anyTypeWhoseFIsBool :: _ => Proxy a -> F a
    anyTypeWhoseFIsBool _ = True
    This will type-check fine and the constraint F a ~ Bool will be happily inferred. Unlike a ~ Bool, it is not considered absurd since it does not directly apply to a universally quantified type variable.
  • Disallow other uses: partial type signatures should not be allowed in every case a type (signature) is needed, for instance in type class definitions, data type definitions, foreign declarations, type synonyms, associated type instantiations, ... Our implementation currently checks for these cases, but we might have missed one. We believe that in nearly every case but value bindings, partial type signatures (or types containing wildcards) should be disallowed. However, we think they should be admissible in InstanceSigs (not yet implemented).
  • Extra-constraints wildcard position: We only allow one extra-constraints wildcard in a signature: at the outer quantification of the signature. Consider for example the function multiCs:
    multiCs x = show (succ x) ++ show (x == x)
    GHC infers the type (Show a, Enum a, Eq a) => a -> String for it, but the type signature provided below is also valid (with the RankNTypes extension enabled).
    multiCs :: (Show a) => a -> (Enum a, Eq a) => String
    We currently only allow one extra-constraints wildcard, in the outermost set of constraints. Otherwise, we might get ambiguous situations like this:
    multiCs :: (Show a, _) => a -> (Enum a, _) => String
  • Higher-rank types: Consider the following partial type signature:
    forall a. a -> (forall b. (b, _c) -> b) -> Int
    We believe that generalising over the _c named wildcard should lead to a top-level quantification (where a is quantified) of the resulting type variable:
    forall a tw_c. a -> (forall b. (b, tw_c) -> b) -> Int
    We will never infer a quantification in the nested quantification (where b is quantified):
    forall a. a -> (forall b tw_c. (b, tw_c) -> b) -> Int
    The latter is equivalent to inferring higher-rank types, which, as we mentioned before, is not something we can do.

    Additionally, we do not allow an extra-constraints wildcard in a nested 'forall', e.g.
    f :: (forall a. _ => a -> a) -> b -> b