Partial Type Signatures for Haskell

Thomas Winant, Dominique Devriese, Frank Piessens, Tom Schrijvers

Relevant links

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.


Use Keyword = PartialTypeSignatures to ensure that a ticket ends up on these lists.

Open Tickets:

Unexpected defaulting of partial type signatures and inconsistent behaviour when -fdefer-typed-holes is set.
Allow wildcards for parameters functionally determined (also type synonyms)
Typed holes regression in GHC 8.0.2: No skolem info: z_a1sY[sk:2]
Partial type signatures + mutual recursion = confusion
Hole-y partial type signatures greatly slow down compile times
Internal error with PartialTypeSignatures and TH
Core lint error from PartialTypeSignatures
Unhelpful names for wildcard type variables

Closed Tickets:

Multiple constraint wildcards allowed with PartialTypeSignatures
kinded holes
Partial type signatures in class constraints behave unexpectedly
'Fix two more bugs in partial signatures' broke T10846
GHC 8.4.1-alpha regression with PartialTypeSignatures
Bizarre pretty-printing of inferred Coercible constraint in partial type signature
Named wildcards in kinds

Background and motivation


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]

Named Wildcards

Type wildcards can also be named by giving the underscore an indentifier 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")
 -- Inferrred: forall a. (Char, a) -> (Char, a)

The named wildcard forces the arugment 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 unify with something:

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

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

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.

Extra-constraints Wildcard

Another kind of wildcard we propose 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. (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.

Extra-constraints wildcards cannot be named.

NOTE In spite of SLPJ's reasonable proposal to simplify things by not taking the annotated constraints into account when an extra-constraints wildcard is present and just inferring everything from scratch, we still do.


Previously, underscores in types were disallowed by GHC and Haskell 2010, so to remain backwards compatible, wildcards or 'holes in types' should still result in errors. However, the generated error messages can now be much more informative, i.e. they should inform the user of the type each wildcard/hole was instantiated to. As this does not change the set of accepted programs nor the behaviour of accepted programs, this doesn't have to be an extension (similar to TypedHoles).

Furthermore, when the user enables the PartialTypeSignatures extension, the errors are not reported anymore, the inferred type is simply used.

However, named wildcards (_a) are currently parsed as type variables. To also remain compatible on this front, we propose to introduce a separate extension, NamedWildcards. When this extension is enabled, a type variable like _a will be parsed as a named wildcard.

To summarise, the four different cases, depending on the enabled extensions:

PartialTypeSignatures OFF PartialTypeSignatures ON
NamedWildcards OFF Informative errors are reported for hole instantiations, but only for unnamed wildcards. Named wildcards are still parsed as type variables, as before. The types of unnamed wildcards are inferred and used. Named wildcards are still parsed as type variables.
NamedWildcards ON Informative errors are reported for hole instantiations, both for unnamed and named wildcards. The types of both unnamed and named wildcards are inferred and used.

Along with informative errors, we also suggest the user to turn on the PartialTypeSignatures extension.

Let's demonstrate the described behaviour with an example. An example program:

module Example where

foo :: (Show _a, _) => _a -> _
foo x = show (succ x)

Compiled with a prior version of GHC gives:

Example.hs:3:18: parse error on input ‘_’

When compiled with a version of GHC that implements the proposal:

    Found hole ‘_’ with inferred constraints: Enum _a
    Where: ‘_a’ is a rigid type variable bound by
                the inferred type of foo :: Enum _a => _a -> String at Example.hs:3:8
    To use the inferred type, enable PartialTypeSignatures
    In the type signature for ‘foo’: foo :: (Show _a, _) => _a -> _

    Found hole ‘_’ with type: String
    To use the inferred type, enable PartialTypeSignatures
    In the type signature for ‘foo’: foo :: (Show _a, _) => _a -> _

Now the types the wildcards were instantiated to are reported. Note that _a is still treated as a type variable, as prescribed in Haskell 2010. To treat it as a named wildcard, enable the NamedWildcards extension to get:

    Found hole ‘_a’ with type: w_a
    Where: ‘w_a’ is a rigid type variable bound by
                 the inferred type of foo :: Enum w_a => w_a -> String
                 at Example.hs:4:1
    To use the inferred type, enable PartialTypeSignatures
    In the type signature for ‘foo’: foo :: (Show _a, _) => _a -> _

    Found hole ‘_a’ with type: w_a
    Where: ‘w_a’ is a rigid type variable bound by
                 the inferred type of foo :: Enum w_a => w_a -> String
                 at Example.hs:4:1
    To use the inferred type, enable PartialTypeSignatures
    In the type signature for ‘foo’: foo :: (Show _a, _) => _a -> _


An extra error message appears, reporting that _a was instantiated to a new type variable (w_a).

Finally, when the PartialTypeSignatures extension is enabled, the type checker just uses the inferred types for the wildcards and compiles the program without generating any messages.

Alejandro Serrano Mena has added Emacs support for interacting with TypedHoles during his GSOC project. He told me this should work fine with PartialTypeSignatures as it is, but we would like to add the ability to let Emacs fill in the hole's inferred type at the press of a key. We will look into this.

Defer type errors

GHC's -fdefer-type-errors flag defers compile-time type errors to run-time. This flag overlaps partially with our -XPartialTypeSignatures flag.

Holes in type signatures will cause compile-time type errors, but deferring them to run-time with -fdefer-type-errors will make them disappear, as they will never be generated at run-time, unlike real type errors. To be clear, holes in type signatures will not cause run-time type errors with -fdefer-type-errors enabled.

The -XPartialTypeSignatures flag will also make the compiler swallow type errors caused by holes in type signatures, but other type errors will still be generated at compile time.

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.

Partial Expression and Pattern Signatures

Wildcards are allowed in expression and pattern signatures, e.g.

bar1 :: _a -> Bool
bar1 x = (x :: _a)
-- Inferred: Bool -> Bool

bar2 :: Bool -> _a
bar2 (x :: _a) = x
-- Inferred: Bool -> Bool

These wildcards are quantified in the expression or pattern signature they appear in.

We don't support extra-constraints wildcards in such signatures, as the implementation difficulties they pose don't outweigh their usefulness.


We worked out a rigorous formalisation of partial type signatures 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 our proposal as a branch of GHC (master), available at github. This version can be compiled and should type check the majority of the examples in this proposal.

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

  • The parser (Parser.y.pp and RdrHsSyn.lhs) was extended to parse wildcards in types when the appropriate extensions are enabled, and also to check that no wildcards occur in undesired places, e.g. typeclass definitions.
  • HsType has two new constructors: HsWildcardTy for unnamed type wildcards (_) and HsNamedWildcardTy name for named wildcards (_a).
  • The HsForAllTy constructor of HsType has an extra field of type Maybe SrcSpan that stores the extra-constraints wildcard's position, if present. A Maybe SrcSpan was used instead of a boolean for pretty-printing purposes and error messages.
  • The renamer: when renaming TypeSigs, we do a pass over the annotated type signature. In this pass, unnamed wildcards (_) are given a generated name and are converted to named wildcards. The named wildcards are collected and stored in the TypeSig, as if they were existentially quantified in that TypeSig. Duplicates and wildcards that are already into scope are filtered out. A similar thing is done for expression signatures (ExprWithTySig) and pattern signatures (HsWithBndrs).
  • In the type checker, in tcTySig, we create new meta-variables for the wildcards and add them to the type variable environment and store them in the returned TcSigInfo to get them brought in scope when checking the RHSs. A Maybe TyVar is also stored in the TcSigInfo, which will eventually be unified with the inferred extra constraints.
  • Analogously to TypedHoles, when a wildcard is encountered during type checking, an insoluble CHoleCan constraint which contains information about the expected type, is emitted. When the type checker and the constraint solver have finished, these insoluble constraints will be leftover, and are converted to error messages mentioning the inferred type for each wildcard. A similar thing is done for the extra-constraints wildcard which contains a meta-variable that is unified with the inferred extra constraints after the constraints of one group of bindings are solved. If the PartialTypeSignatures extension is enabled, no constraints are emitted as no error messages have to be produced and reported.
  • To typecheck a binding (actually a group of bindings) with a partial type signature, tcPolyInfer, the function used for typechecking bindings without type signatures is reused and modified. The partial type signature is taken into account, as well as the presence of an extra-constraints wildcard.
  • The constraint solver function simplifyInfer is extended such that it also accepts an optional set of annotated constraints as input.
  • The error reporter for Holes is extended to deal with different holes, e.g. a ConstraintsHole originating from an extra-constraints wildcard is treated differently from an ExprHole or TypeHole (all are constructors of the new HoleSort data type).
  • We extended Haddock as well to handle partial type signatures. A single commit updating the GHC API and adding printing capabilities can be found in our fork of Haddock on github.


In this section we'll quickly mention some inelegant solutions, or explain some things that may be non-obvious.

  • Analogously to TypedHoles, we generate insoluble constraints (CHoleCan) to report the types wildcards are instantiated to, however, we might have reused a bit too much of the existing infrastructure for TypedHoles.

    • Errors after the holes reporter (see reportAllUnsolved in TcErrors.lhs) are no longer suppressed, as, contrarily to TypedHoles, wildcards in types don't prevent a program from being complete. In the case of TypedHoles, the holes should be filled before continuing type checking, but this is not needed for wildcards in types, thus other type errors may be reported.

    • There is some code duplication in the two branches of mkHoleError in TcErrors.lhs.

    • We differentiate Holes (i.e. values of CHoleCan) with the HoleSort type, which in case of an extra-constraints wildcard hole contains an TcTyVar field that will unify with the extra inferred constraints. We do this in order to be able to reported the inferred constraints when reporting the hole error.

    • In order to capture the CHoleCan for the extra-constraints wildcard, we wrap a call to emitInsoluble in captureConstraints in the tcTySig function in TcBinds.lhs. This captured constraint, which we'll call wantedSig, is stored in the signature (TcSigInfo) and later added to the wantedBody constraints captured while type checking the body of the function. Both sets of wanted constraints are later merged together in the local definition makeWanted of tcPolyInfer in TcBinds.lhs. This feels kind of dirty.

    • The CHoleCan constraint needs evidence (CtEvidence). For TypedHoles and holes of type wildcards, we can use the inferred type. For the CHoleCan generated by an extra-constraints wildcard, we currently use unitTy.
  • Constraints are solved (i.e. calls to simplifyInfer) per recursive group of bindings, thus, the extra constraints are inferred per recursive group as well. Therefore, they are added to each binding of the group with an extra-constraints wildcards.


The TODOs are sorted by severity, i.e. the first TODO is the most important one, the last one is only of minor importance.

  1. TODO A partial type signature for a local definition is not generalised when the MonoLocalBinds flag is enabled (this flag is implied by GADTs and TypeFamilies!), as intended. However, for functions affected by this restriction (i.e. not all free variables are closed), partial type signatures aren't properly handled.
    {-# LANGUAGE MonoLocalBinds, ScopedTypeVariables #-}
    monoLoc :: forall a. a -> ((a, String), (a, String))
    monoLoc x = (g True , g 'v')
      -- g :: b -> (a, String) -- This signature is ok
      g :: _                   -- This signature is not
      g y = (x, "foo")
    The example above will produce a number of errors of which the following is the vital one:
    Couldn't match expected type ‘Char -> (a, String)’
                with actual type ‘_’
      because type variable ‘a’ would escape its scope

  1. TODO There is a bug in the handling of pattern bindings with partial type signatures, i.e. the partial type signatures are ignored, e.g.
    foo :: Bool -> _
    Just foo = Just id
    -- Inferred: forall a. a -> a
    -- Instead of: Bool -> Bool
    See test case PatBind2.

There is a TODO for this in the code.

  1. TODO One of the examples described in the section about the monomorphism restriction doesn't work:
    {-# LANGUAGE MonomorphismRestriction, PartialTypeSignatures #-}
    charlie :: _ => a
    charlie = 3
    We think that, as the type variable is annotated and thus unrestricted, the constraint Num a should be inferred as an extra constraint. Unfortunately, this doesn't happen:
    No instance for (Num a) arising from the literal ‘3’
    In the expression: 3
    In an equation for ‘charlie’: charlie = 3
    See test case ExtraNumAMROn. Either this needs to be fixed, or our expectation of how the type checker should behave is wrong.
  1. TODO When reporting the types wildcards were instantiated to in error messages, we mention the inferred type of the binding, but this doesn't include the annotated constraints that weren't inferred during typechecking, e.g.
    bar :: (Num a, _) => a -> a -> _
    bar x y = x == y
    Produces the following error message:
        Found hole ‘_’ with inferred constraints: Eq a
        Where: ‘a’ is a rigid type variable bound by
                   the inferred type of bar :: Eq a => a -> a -> Bool at Example.hs:3:8
        To use the inferred type, enable PartialTypeSignatures
        In the type signature for ‘bar’: bar :: (Num a, _) => a -> a -> _
        Found hole ‘_’ with type: Bool
        To use the inferred type, enable PartialTypeSignatures
        In the type signature for ‘bar’: bar :: (Num a, _) => a -> a -> _
    In the 4th line of the output shown above, the inferred type is displayed, but the Num a constraint has disappeared, as it wasn't inferred. Two lines below, it is displayed, but that is because the annotated partial type signature is shown. The actual type of bar, (Num a, Eq a) => a -> a -> Bool, is not shown to the user, as it isn't easily accessible when reporting the error.

Related feature request for TypedHoles: #9091.

  1. TODO The order in which type variables are tidied for printing is incorrect. Wildcards which are generalised over become type variables with names starting with w_. If a type variable with such a name already exists, the existing tidying infrastructure should make sure that the suffixes 1, 2, ... are added in order to differentiate the similarly named type variables. However, the order in which the suffixes are added is unsatisfactory, e.g.
    barry :: _ -> w_
    barry _ = undefined
    -- Inferred (modulo alpha-equivalence): a -> b
    Which causes the following hole error message:
    Found hole ‘_’ with type: w_
    Where: ‘w_’ is a rigid type variable bound by
                the inferred type of barry :: w_ -> w_1 at Example.hs:3:1
    To use the inferred type, enable PartialTypeSignatures
    In the type signature for ‘barry’: barry :: _ -> w_
    The problem is that the existing type variable w_ is renamed to w_1 and the type variable generated by generalising over the wildcard gets w_ as name, instead of the inverse. See test case TidyClash2.

Monomorphism restriction

Regarding the monomorphism restriction, the existing rules apply. Declarations with a partial type signature are treated as declarations without a type signature. Therefore, these declarations will be restricted, and constrained type variables of such declarations may not be generalised over, and thus defaulting will occur when possible.

We illustrate this behaviour with some examples:

-- Remember: the principal type of 3 is forall a. Num a => a
alpha :: _
alpha = 3
-- with MR on, the type variable is restricted and defaults to Integer
-- with MR off, generalise over _, error: no instance for (Num _)

bravo :: _ => _
bravo = 3
-- with MR on, the type variable is restricted and defaults to Integer
-- with MR off, type: forall a. Num a => a

charlie :: _ => a
charlie = 3
-- with MR on, type variable is annotated and thus unrestricted, type: forall a. Num a => a
-- with MR off, type: forall a. Num a => a

The last example (with the monomorphism restriction) on, charlie, might be surprising. As a is a generalised type variable that is already annotated, we don't have to generalise over it anymore. The monomorphism restriction simply doesn't apply.

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?

    Update: we now follow the ScopedTypeVariables extension.
  • 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 w_c. a -> (forall b. (b, w_c) -> b) -> Int
    We will never infer a quantification in the nested quantification (where b is quantified):
    forall a. a -> (forall b w_c. (b, w_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


Not supported: 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.

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.

Last modified 21 months ago Last modified on Apr 3, 2018 7:59:06 AM