# Kind inference for types and classes

This page summarises several alternative designs for doing kind inference for types and classes, under -XPolyKinds. Though originally written with respect to #9200, there are more issues at play here. Relevant other tickets: #9201, #9427, #14451, #14847, #15142, and (to a lesser extent) #12088, #12643, #14668, #15561, and #15987, which are all about instances.

See also GhcKinds/KindInference/Examples for a growing set of examples to consider when evaluating any new algorithm.

## Simplifying getInitialKinds (#15142)

In terms, we make a drastic distinction between how we check a term with a type signature and how we check a term without. When a term has a type signature, that signature is used for the term, and (for example) polymorphic recursion is possible. Without a type signature, the term is assigned type alpha. During type checking, alpha will unify only with a monotype, and thus polymorphic recursion is impossible. (Of course, after constraint generation and solving, we then generalize any unconstrained variables.)

The goal here is to do the same in type declarations. When a type declaration has a kind signature (in other words, a CUSK, which is a sloppy encoding of a kind signature), we check it with that kind signature. Otherwise, we assign it kind kappa and gather constraints on kappa, allowing kappa only to be a monokind, generalizing later.

For CUSKs, this is pretty close to accurate. But it's nothing like what we do in the non-CUSK case. Currently, non-CUSKs get an "initial kind", formed from whatever we can divine from the type head. (For example, data T a gets kappa -> Type, using T's arity and the fact that it's a data type.) Then we do constraint solving. Simon wonders if we can simplify by just guessing kappa.

The problem is that we will lose some currently-accepted definitions. For example, consider

data Prox k (a :: k)


The kind of Prox is forall k -> k -> Type. Note that this type contains forall k ->, not forall k .. If we had assigned Prox :: kappa, we would be unable to infer the correct kind for Prox. This kind is currently produced right in getInitialKind, with no extra quantification necessary. The k is lexically dependent, and so GHC uses forall k -> for it.

Another interesting case is

data Prox2 k a = MkP2 (Prox k a)


Here, it's "obvious" that Prox2 should have the same kind as Prox, even though k isn't obviously a dependent variable. This is actually rejected today, because GHC requires that all dependent variables be manifestly so, in the LHS.

Here is a good place to note that the term-level scheme has an exception: if the definition is non-recursive and top-level, then we don't guess a type alpha; instead, we just infer the type of the term from the RHS. This allows the term to be assigned a polytype or higher-rank type. If we did the same in types (noting that Prox2 isn't recursive), then perhaps we could accept Prox2.

Note that we want to have a good understanding of what will be accepted and what will not be. And we do not want to accept polymorphic recursion without a signature.

### Simon's suggestion

Here was one idea: assign every tyvar in the LHS a fresh kind variable kappa_n. Then use those to check any kind signatures on the LHS and the definition on the RHS. So, given

data S2 k (a :: k) b = MkS (S2 k b a)


we would start with k :: kappa1, a :: kappa2, and b :: kappa3. We would soon get the following wanteds:

kappa1 ~ Type    -- from the ":: k"
kappa2 ~ k       -- from the ":: k"
kappa3 ~ kappa2  -- because b is used as the second arg to S2 in the RHS


and all will work out.

The problem with this approach is that it also accepts the polymorphic recursive S3:

data S3 k (a :: k) b = MkS (S3 Type b a)


So we think this isn't the right answer.

One other possibility in trying to wrangle this is to not let the kappa_n to unify with other tyvars introduced in the datatype definition. Part of this plan would require checking any LHS kind annotations before assigning the kappa_n. In this scenario, both S2 and S3 would be rejected, because we could never get kappa3 to unify with k.

One big drawback of Adam's idea is that it is very different from what happens in terms. Annotating some tyvar kinds is just like having a partial type signature. Yet, if I say f :: forall a. a -> _, GHC allows the wildcard to be replaced with a. Simon pointed out that partial type signatures are themselves a mess, and that we're not sure we want to duplicate it all "one level up". But we all agreed that they are useful and that uniformity is a Good Thing.

### Comparison to Agda

Incredibly, all the following are accepted in Agda:

data Q (k : Set) (a : k) : Set1 where

data Q2 k a : Set1 where
MkQ2 : Q k a -> Q2 k a

data Q3 k a : Set1 where
MkQ3 : Q3 k a -> Q k a -> Q3 k a

data Q4 k a : Set1 where
MkQ4 : Q4 ℕ 5 -> Q k a -> Q4 k a

data Q5 k a : Set1 where
MkQ5 : Q5 ℕ 3 -> Q5 Bool true -> Q k a -> Q5 k a


I have no clue how they pull that off!

## Trouble when fixing #14066

Ticket #14066 requests a more robust way of tracking the scope of type variables. A full discussion is out of scope here, but it required hard thought about kind inference. Here are some notes I (Richard) have drawn up around it:

### Introduction

The central challenge is this:

data Proxy1 a where
Mk :: Proxy1 (a :: k)


To infer the right kind for Proxy1, GHC has to guess a kind for a -- call it kappa -- and then process the type of Mk. What's very bizarre is that the occurrence of Proxy1 in the type of Mk leads GHC to unify kappa with k, even though k has a local scope. In other words, kappa unifies with an out-of-scope skolem. GHC's current (8.4) kind-generalization process ends up quantifying over the k in the kind of Proxy1, and so disaster is averted. But it really works only by accident.

And it doesn't always work.

data Proxy2 a where
Mk1 :: Proxy2 (a :: k)
Mk2 :: Proxy2 (b :: j)


GHC does the same thing here, but it rejects the definition because k doesn't unify with j. This is absurd, because these two variables are skolems that share no scope whatsoever. Interestingly, GHC 7.10 accepts this last definition, so the rejection is a hitherto-unknown regression.

### Simon's Proposed Solution

Datatype declarations are kind-checked in two passes. The first pass looks through all the constructors, accumulating constraints on the type's kind. Then, once the kind is known, all the constructors are checked again with respect to the known kind. Note that we need to look at constructors to determine the kind of the datatype; otherwise, GHC would choke on declarations like

data App a b = MkApp (a b)


The kinds for a and b can be known only by looking at MkApp.

Simon suggested that, in the first pass, we instantiate all the user-written type variables in a constructor's declaration with unification variables and then proceed to infer the kind. This would mean that the j and k in Proxy2's constructors would both be instantiated with unification variables. These would all unify but would remain unconstrained. GHC would then quantify over this kind, as it should. Note that it's possible for GHC to infer something silly here, but it would then be caught on the second pass.

### Aside: Polymorphic Recursion and Type Inference

GHC claims that it infers most general types. But this claim is simply not true in the presence of polymorphic recursion. For example, consider this equation defining f:

f x y = f y x


The most general type for f is forall a b c. a -> b -> c. GHC indeed accepts this type signature for f. However, if f is written without a type signature, GHC infers forall a b. a -> a -> b.

What has happened to GHC's claim of inferring most general types? It is true, but it is true only with respect to the fragment of Haskell that prohibits polymorphic recursion. Note that GHC's inferred type leads to no polymorphic recursion. The lesson here is that a claim of most general types makes sense only with respect to some restriction on the features in the language. (For example, we all know that GHC won't infer a higher-rank type, even if one is more general than another type.)

### Simon's Algorithm Breaks Principal Types

Under Simon's algorithms, some forms of polymorphic recursion are indeed accepted. For example:

data T a where
Mk :: forall k1 k2 (a :: k1) (b :: k2). T b -> T a


Simon's algorithm instantiates k1, k2, a, and b with fresh unification variables. Suppose T is guessed to have kind kappa -> Type. Then, kappa, k1, and k2 all get unified together, with no constraints. GHC will quantify, giving T the kind forall k. k -> Type. During the second pass, this kind works fine, instantiated to k1 and k2. Thus, GHC accept the polymorphic-recursive T.

So: what forms of polymorphic recursion are accepted? Not all of them. Take this example:

data T2 a where
Mk :: forall (a :: k). T2 Maybe -> T2 a


Under Simon's algorithm, the T2 is guessed to have kind kappa -> Type, and then kappa will unify with Type -> Type. T2 is assigned the kind (Type -> Type) -> Type. On the second pass, Mk is rejected, applying T2 to something of kind k. We can see, though, that T2 could be assigned the kind forall k. k -> Type quite profitably. Nevertheless, I agree with Simon's algorithm that rejecting is the correct behavior -- we don't want to infer polymorphic recursion.

After some thinking, Adam, Csongor and I came up with this rule: Simon's algorithm accepts polymorphic recursive definitions when the recursive occurrences are at types that instantiate kind variables with kind variables, but never concrete kinds. Note that in T, the recursive occurrence replaces k1 with k2, but T2 replaces k with Type -> Type. Let's call a polymorphic recursive definition where recursive occurrences replace variables with variables "pseudopolymorphic recursive".

However, Simon's algorithm does not always infer the most general type with respect to the fragment of the language containing pseudopolymorphic recursion (but not general polymorphic recursion). For example:

data T3 a b where
Mk :: T3 b a -> T3 a b


Simon's algorithm infers the kind forall k. k -> k -> Type for T3, even though it could have the kind forall k1 k2. k1 -> k2 -> Type and remain pseudopolymorphic recursive. Thus, I claim that Simon's algorithm does not infer most general types.

## Inferring dependency

A related problem to the above is inferring dependency between binders, some notes follow.

In GHC 8.4, the following declaration for Proxy2 is rejected:

data Proxy k (a :: k)
data Proxy2 k a = MkP (Proxy k a)


Proxy's kind is forall k -> k -> Type. According to the rule that dependent variables must be manifestly so (implemented by getInitialKind), the declaration for Proxy2 is rejected, because only after checking the RHS do we find out about the dependency. (This rule is documented in Section 10.11.13 of the user manual).

However, arguably inference should be able to determine that Proxy2 :: forall k -> k -> Type. A possible way would be to skip getInitialKind and look at the RHS immediately (according to Richard, implementing this is not that simple).

Consider the following, GADT-style definition of Proxy2:

data Proxy2 k a where
MkP :: Proxy k a -> Proxy2 k a


This definition is accepted today (GHC 8.4), but why?

With -fprint-explicit-foralls, Proxy2's signature is actually the following:

data Proxy2 k k1 (a :: k) where
MkP :: forall k (a :: k). (Proxy k a) -> Proxy2 k k a


It is a kind-indexed GADT, with the MkP constructor holding k ~ k1, and its kind forall {k}. Type -> k -> Type is not manifestly dependent. This is not wrong, but there are certainly cases where (short of having a CUSK) it is desired that the inferred signature of a GADT is not kind-indexed.

Consider the singleton type for Bool:

data SingBool b where
SingTrue :: SingBool 'True
SingFalse :: SingBool 'False


The inferred kind is Bool -> Type, but forall {k}. k -> Type would not be invalid either.

A related problem, where it's much more apparent that the more specialised kind is better is type families:

type family F a where
F Int = Bool


Here, F is inferred to be Type -> Type. Indeed, if instead it was generalised to k -> j, then F Int could only be reduced to Bool if its return kind was specified.

So the question is: if the non-GADT Proxy2 will be inferred to have the dependent kind, should the GADT Proxy2 be too?

## New Proposal

### Preprocessing: Translation from GADT syntax to H98 syntax

1. The scope of the type/kind variables are explicit;
2. Avoid the appearance of recursion in return types;
3. Make sure the different representations of the same data type behave the same.

Example:

data Proxy2 a where
Mk1 :: Proxy2 (a :: k)
Mk2 :: Proxy2 (b :: j)

-- is translated to

data Proxy2 c
= forall k a. ((a :: k) ~ c) => Mk1
| forall j b. ((b :: j) ~ c) => Mk2


After the translation step, we can focus on the kind inference algorithm for only H98 syntax.

### Kind Inference Strategy for H98 syntax

CUSK. If the data type has CUSK, we use the provided kind to do type-checking.

Non-CUSK. If there is no CUSK:

1. Depending on whether the datatype is defined recursively, we use different strategies.
1. If the definition is recursive, we guess a monokind kappa as the initial kind of the data type. Reject any dependency. E.g.
 -- P :: kappa
data P a b = ...

2. If the definition is non-recursive, for all variables, we guess kappa kind for them. Allow dependency. There is no need to get an initial kind, as the data type is not used in RHS. E.g.
 --  a::kappa1, b::kappa2, and kappa2 might be unified with a
data P a b = ...

2. Collect constraints on kappa from LHS and RHS
3. Solve constraints
4. Do generalization

### Examples of non-CUSKs under the new strategy

-- Accept. Proxy :: forall k -> k -> Type
data Proxy k (a::k)

-- Reject. As it is recursive, it cannot have dependency
data T1 k (a::k) = MkT1 (T1 k a)

-- Accept. Proxy2 :: forall k -> k -> Type
data Proxy2 k a = MkP2 (Proxy k a)

-- Reject. As it is recursive, it cannot have dependency
data T2 k a = MkT2 (Proxy k a) | MkT2' (T2 * Int)


## Current strategy (GHC 7.8)

Running example:

  data T a f = T1 Int (T a f) | T2 (f a)

data S a f = S1 (S Maybe (S Int))


Here is GHC 7.8's strategy:

1. Sort data/type/class declarations into strongly connected components. In our example, there are two recursive SCCs, one for T and one for S.

1. Then, for each SCC in turn:
• Bind the type constructor to a fresh meta-kind variable:
        T :: kappa0

• Kind-check all the declarations of the SCC in this environment. This will generate some unifications, so in the end we get
        kappa0 ~ kappa1 -> (kappa1 -> *) -> *

• Generalise. So we get
        T :: forall k. k -> (k -> *) -> *


Following the same three steps for S fails, because we need polymorphic recursion. Srarting with S :: kappa0, we get

  kappa0 ~ kappa1 -> kappa2 -> *    -- S has two arguments
kappa1 ~ *->*                     -- From (S Maybe)
kappa1 ~ *                        -- From (S Int)
...and some more...


So in GHC 7.8 you can supply a "complete user-supplied kind" (CUSK) for S, thus:

  data S (a :: k) (f :: k -> *) where
S1 :: S Maybe (S Int) -> S a f


Then step 2 works thus:

• Bind the each non-CUSK type constructor to a fresh meta-kind variable, and each CUSK type constructor to its polymorphic kind.
        S :: forall k. k -> (k->*) -> *

• Kind-check all the declarations of the SCC in this environment. With this new kind for S, kind checking succeeds.
• Generalise any non-CUSK type constructors.

## Proposed new strategy

Note: This is implemented. See the commit messages in #9200.

The main proposed change is to the definition of a "complete user-supplied kind" (CUSK). The current story is in Section 7.8.3 of the user manual. Alas, it does not allow CUSKs for class declarations. The new proposal is this:

• A class or datatype is said to have a CUSK if and only if all of its type variables are annotated. Its result kind is, by construction, Constraint or * respectively.
• A type synonym has a CUSK if and only if all of its type variables and its RHS are annotated with kinds.
• A closed type family is said to have a CUSK if and only if all of its type variables and its return type are annotated.
• An open type family always has a CUSK -- unannotated type variables (and return type) default to *. (This is not a change.)
• Update for TypeInType: If -XTypeInType is in effect, any top-level kind given to a datatype must introduce all of its kind variables explicitly, allowing data X :: forall k. k -> * but saying that data X :: k -> * does not have a CUSK.

This is somewhat simpler, it covers classes. See comment:19:ticket:9200 for more exposition. This change alone is enough to satisfy #9200.

Examples:

  class C (a::k) (f::k->*) where ...   -- Has CUSK
class D a (f :: k -> *)  where ...   -- No CUSK (a is un-annotated)
data T (f :: k -> *) where ...       -- Has CUSK

type T (a::k) = a :: k               -- Has CUSK
type S (a::k) = a                    -- No CUSK (RHS is un-annotated)

type family F a :: *                 -- Has CUSK (a defaults to *)

type family F (a::*) :: k where ...  -- Has CUSK
type family F a where ...            -- No CUSK (neither arg nor result annotated)


If a datatype has a CUSK and its kind has any unsolved metavariables after inference (possible with -XTypeInType), an error is reported.

Reasons for Update: It's nice to be able to say data X :: Proxy k -> * and get inference to be able to deduce that you mean X :: forall j (k :: j). Proxy k -> *. Of course, this is possible only with -XTypeInType, so we make it a tad harder to specify a CUSK. Ideally, we'll one day move to a separate type X :: <X's kind> syntax that will obviate this delicate syntactic definition.

## A possible variation

RAE: This variation is partially implemented in tag re-sort-non-cusk-decls at my GitHub fork. It is tracked in ticket #9427.

This algorithm is not quite as expressive as it could be. Consider

   data SS f a b = MkSS (TT a f) (SS f a b)
data TT (a::k) (f::k -> *) :: * where
MkTT :: f a -> SS f a Maybe -> SS f a Int -> TT a f


Here TT has a CUSK, but SS does not. Hence in step 1 we'd bind SS :: kappa0. But in the RHS of TT we use SS at two different kinds, so infernece will fail. We could solve this by giving a CUSK to SS as well.

However, we can also solve it using a plan due to Mark Jones, and one that GHC 7.8 already follows for ordinary, recursive term-level functions. As usual, divide the declarations into SCCs, and then for each SCC do the following:

• Identify which type constructors have Complete User Type Signatures (CUSK). In this example, TT does. Extend the environment with these, fixed, kinds:
       TT :: forall k. k -> (k->*) -> *

• Perform a new strongly-connected component (SCC) analysis on the non-CUSK decls in the SCC, ignoring dependencies on a type constructor with a CUSK. In our example, we get a single recursive SCC, containing SS.
• For each SCC in turn:
• Bind the type constructor to a fresh meta-kind variable:
        SS :: kappa0

• Kind-check all the declarations of the SCC in this environment. This will generate some unifications, so in the end we get
        kappa0 ~ (kappa1 -> *) -> kappa1 -> kappa2 -> *

The kappa1 arises from instantiating TT at its call site in SS
• Generalise. So we get
        SS :: forall k1 k2. (k1->*) -> k1 -> k2 -> *

• Extend the environment with these generalised kind bindings, and kind-check the CUSK declarations.

The Key Point is that we can kind-check SS without looking at TT's definition at all, because we completely know TT's kind. That in turn means that we can exploit inferred polymorphism for SS when kind-checking TT. As we do here: TT uses SS in two different ways (SS f a Maybe) and (SS f a Int).

This strategy is more complicated than the initial proposal, but allows fewer annotations. It's not clear whether it is wroth the bother.

### Typing rule for the new proposal

An algorithm is all very well, but what about the typing judgements? We will pretend that data, type, type family, class declarations look something like this:

  T :: forall k1 k2. (k1 -> *) -> k1 -> _ -> *
data/type/class T a b c = rhs


That is, there is a possibly-partial kind signature, with holes denoted by "_", and a definition "rhs" (eg the consructors of a data type, or equations of a closed type family). In reality there isn't a separate kind signature; instead, it is integrated in the definition; e.g.

  data T (a::k1 -> *) (b::k1) c = MkT (a b) c


The kind variables mentioned in the partial kind signature scope over the "rhs".

Then we have two typing rules, one for CUSK, and one for non-CUSK. We give the rule for data types but modulo keyword it works for classes, closed type families, etc. We ignore arity/saturation issues for type families.

k has at least one missing bit (non-CUSK)
k2 = k[k'1 .. k'n/_]     -- k'1 .. k'n are magically known
kvs2 = fkv(k2)
G, kvs2, T : k2 |- (data T tvs = rhs) ok    -- Monomorphic recursion
----------------------------------------------------- NoCUSK
G |- (T :: forall kvs. k; data T tvs = rhs) :: {T :: forall kvs2. k2}

k has no missing bits (CUSK)
G, T : forall kvs. k |- (T tvs = rhs) ok  -- Polymorphic recursion
----------------------------------------------------- CUSK
G |- (T :: forall kvs. k; data T tvs = rhs) :: {T :: forall kvs. k}


We need two rules, depending on whether or not a CUSK is detected.

The first rule requires the equations to be fully parametric in its kinds, whereas the second allows polymorphic recursion.

For closed type families, these rules are different than the implementation today, because kind inference for closed type families today is ill-specified. See comment:18:ticket:9200.

# OUT-DATED MATERIAL

Now here are some notes and design alternatives that we rejected. It's just here because I hate deleting stuff!

## Notes about CUSKs on type synonyms

Simon What about type synonym declarations? Don't we need a kind signature on the RHS? Also what about specifying the return kind of a type family (open or closed)? Does it default to *, or must you specify it to get a CUSK?

Richard Type synonym declarations can never be recursive, right? So, this issue doesn't affect them. I've answered the other questions above.

Simon Wrong: type synonyms can be recursive through a data type:

  data S (a :: k) (f :: k -> *) = S1 (SSyn (S Int) Maybe)
type SSyn f a = S a f


This will fail despite the CUSK for S because SSyn lacks one. (The variation below would fix this particular example.) I think #9151 is another example.

Richard: Good example. But, wouldn't this be fixed with the more-careful dependency checking described immediately below? Then, S, with its CUSK, would be added to the environment and generalized before looking at SSyn, and all is well. I don't like the idea of saying that a type synonym has a CUSK when its tyvars and its RHS are annotated, because the RHS is just a plain type -- there's not really a place in the syntax to put the RHS's kind annotation. Looking at the code, something approaching this is done already, where all non-synonyms are first added to the environment (with getInitialKinds) and then all the synonyms are fully kind-checked, and then all the other decls are kind-checked. The more-careful dependency checking below could be implemented simply by having getInitialKinds notice the CUSK and generalize, I think. Indeed, if I give S a CUSK using the current CUSK syntax, the example above compiles today.

Simon: I agree that, since type syononyms can't be recursive except via a data type, if you use "A possible variation" below, then you can get away with saying that type synonyms cannot have CUSKs. It seems a bit non-orthogonal; and could occasionally be tiresome. Imagine

data T1 a b c = ...S...
data T2 p q r = ...S...
data T3 x y q = ...S...
type S f g h = ...T1...T2...T3...


Then, since you can't decorate S with a CUSK, you might need to annotate all of T1, T2 and T3 with CUSKs.

It would not be hard to say what a CUSK for a type synonym was: just wrap the whole RHS in a kind signature:

type T (a::k1) (b::k1->k2) = b a :: k2


I think I'd argue mildy for that, just to make the design orthogonal.

## Partial kind signature strategy (PARTIAL)

The key idea is that all polymorphism is declared, so nothing gets to be kind-polymorphic unless you say so. But the payoff is that you can give partial kind signatures. Here's the strategy.

1. Sort the declarations into SCCs. No special treatment for CUSKs.
1. For each declaration, extend the environment with a kind binding that has a forall for each explicit user-written kind variable, but meta-kind variables otherwise. These kind annotations amount to partial kind signatures. For example
      data Foo (a :: k1 -> k1) b c = ...

would get a kind binding
      Foo :: forall k1. (k1->k1) -> kappa1 -> kappa2 -> *

Our earlier example would give
      T :: forall k. k -> (k->*) -> *
S :: kappa3 -> kappa4 -> kappa5 -> *

1. Kind-check the declartions in this environment. At a call of Foo, say, we'd instantiate the forall k1 with a fresh meta-kind variable, but would share kappa1, kappa2 among all calls to Foo.
1. Default any unconstrained meta kind variables to *

That's it! No generalisation step. The only polymorphism is that declared by the user.

So our earlier SS/TT example would be rejected because it relies on S being polymorphic in its third parameter. If you want the SS/TT example to work you could write

   data SS (f::k1->*) (a::k1) (b::k2) = MkSS (TT a f) (SS f a b)
data TT (a::k) (f::k->*) where
MkTT :: f a -> SS f a Maybe -> SS f a Int -> TT a f


### Declarative typing rules for (PARTIAL)

I think that (PARTIAL) has a nice declarative typing rule.

Here is what the conventional declarative typing rule, in the absence of polymorphism for a single self-recursive function looks like. (I'm using the term language but the same thing happens at the type level.)

        G, f:t |- e:t
G, f:t |- b:t'
---------------------------
G |- letrec f = e in b : t'


Here the "t" is a monotype (no foralls) that the declarative typing rules clairvoyantly conjures up out of thin air.

Once you add Hindley-Milner style polymorphism, the rule gets a bit more complicated

        G, f:t |- e:t
G, f:gen(G,t) |- b:t'
---------------------------
G |- letrec f = e in b : t'


where 'gen' is generalising.

The (PARTIAL) rule might look like this:

        t = forall vs. sig[t1..tn/_]
vs \not\in ti
G, f : t |- e : forall vs.t
G, f : t |- b:t'
--------------------------- (T-PARTIAL)
G |- letrec f :: forall vs. sig; f = e in b : t'


Here I'm expressing the user-specified knowledge as a signature forall vs.sig, with '_' for bits you don't want to specify.

       f :: forall a. _ -> a -> _


Then the rule intantiates each '_' independently with a clairvoyantly guessed monotype (provided it does not mention the 'vs', or 'a' in this example), and off you go.

### A tricky point about (PARTIAL)

Notice that in this typing rule I say vs \not\in ti. If you don't have that side condition I think complete inference becomes very hard. Suppose MT :: (*->*) -> *, and consider

   data Foo f (a::k) = MkFoo (Foo Maybe Int) (Foo MT Maybe)


Because of the partial kind signature we'll kind-check Foo's RHS with this kind signature for Foo:

   Foo :: forall k. kappa1 -> k -> *


using the unification variable kapp1 for f. Now, if we clairvoyantly decide kappa1 := k->*, as would be allowed by (T-PARTIAL), then indeed the definition if well-kinded. So we'd better infer that, if we are to be complete wrt (T-PARTIAL). But the algorithm will share kappa1 among both calls to Foo, and will therefore unify Maybe with MT and fail.

To gain completeness we need to be less ambitious; hence the side condition in (T-PARTIAL) vs \not\in ti.

But that side condition, in turn, means that this will fail:

  data Foo f (a::k) = MkFoo (f a) (Foo f a)


because here kappa1 must be unified with k->*, which isn't allowed by (T-PARTIAL). Maybe that is acceptable; you can always decorate both of Foo's arguments.

## Generalised partial kind signature strategy (PARGEN)

The (PARGEN) strategy is exactly like (PARTIAL) except that step 4 is different:

1. Generalise over any unconstrained meta kind variable, rather than defaulting to *. Since we are operating at top level, there are no kind variables mentioned in the environment, so no need for the ususal "not free in the environment" check.

So we use the partial kind signatures to express any polymorphism necessary for recursion inside the SCC, but perhaps infer yet more polymorphism that can be used after the SCC. Thus:

data T1 f a = MkT1 (f a) (T f a)
-- Success:  T1 :: forall k. (k->*) -> k -> *

data T1a f (a::k) = MkT1a (f a) (T f a)
-- Failure:  f's kind is unified with skolem k
-- See "tricky point" above

data T2 f a = MkT2 (f a) (T2 Maybe Int) (T2 Monad Maybe)
-- Failure: needs polymorphic recursion

data T3 (f::k->*) (a::k) = MkT3 (f a) (T3 Maybe Int) (T3 Monad Maybe)
-- Success: polymorphic recursion declared

data T4 (f::k->*) a = MkT4 (f a) (T4 Maybe Int) (T4 Monad Maybe)
-- Failure: not all polymorphism in k is declared
-- See "tricky point" above


### Declarative typing rules for (PARGEN)

The declarative rule for (PARGEN) is a combination of the one for (PARTIAL) with the standard generalisation:

        t = forall vs. sig[t1..tn/_]
vs \not\in ti
G, f : t |- e : forall vs.t
G, f : gen(G,t) |- b:t'
--------------------------- (T-PARGEN)
G |- letrec f :: forall vs. sig; f = e in b : t'


The difference from (PARTIAL) is that before type-checking b we generalise t.

Here is the declarative rule for closed type families:

k2 = forall fkv(k1). k1[k'1 .. k'n/_]
fkv(k1) \not\in k'j
forall i: G, F:k2 |- (F ps_i = t_i) ok
k3 = gen(G, k2)
---------------------------------------------------
G |- type family F :: k1 where { F ps_i = t_i } : k3


This is very similar to the declarative typing rule for letrec above. Here, I am ignoring issues with arity/saturation and using a syntax where the kind signature of a type family is given as k1 with blanks, instead of the tyvarbndr syntax used in source code.

This is in fact an improvement over the current state of affairs, which is essentially this rule without the side condition. Because of the omitted side condition, we don't have principal types! For example,

type family X (a :: k) where
X True = False


X could be X :: k -> k or X :: k -> Bool. Neither is more general than the other. GHC chooses X :: k -> Bool, but it's not principled. This is what I get for implementing kind inference for closed type families without writing out declarative rules! In any case, the solution to this problem (closed type families) seems to be the same as the solution for datatypes and classes, quite happily: add the side condition.

### How does (PARGEN) differ from (BASELINE)?

(PARGEN) and (BASELINE) are incomparable.

• The SS/TT example under (BASELINE) will be rejected by (PARGEN) becuase SS will get kind kappa1 -> kappa2 -> kappa3 -> * when kind-checking the SS/TT strongly connected component. But (BASELINE) accepts it by breaking the SCC into two.
• There are obviously examples that are accepted by (PARGEN) but not (BASELINE).

So moving from (BASELINE) to (PARGEN) would be a breaking change, but only in rather obscure circumstances. I am intensely relaxed about that particular backward-compatibility problem!

## All of the above (ALL)

Combine (BASELINE) and (NEWCUSK), for the CUSK stuff, with (PARGEN) for type with partial kind signatures. This would type the most programs, but is the most complicated.

## Type signatures

Another place that we currently (i.e. using (BASELINE)) do kind generalisation is in type signatures. If you write

f :: m a -> m a
f = ...


then the type signature is kind-generalised thus:

This user-written signature
f :: m a -> m a
means this (BASELINE)
f :: forall k (a:k) (m:k->*). m a -> m a


And f's RHS had better be that polymorphic.

However (PARTIAL) does no kind generalisation, and it would be consistent to cease doing so for type signatures too. so:

This user-written signature
f :: m a -> m a
means this (PARTIAL)
f :: forall (a:*) (m:k->*). m a -> m a


If you want the kind-polymorphic one, you'd have to write thus

This user-written signature
f :: forall k (a:k) (m:k->*). m a -> m a
means this (PARTIAL)
f :: forall k (a:k) (m:k->*). m a -> m a


## Reflection

I think we could reasonably switch to (PARTIAL) throughout.

As Richard's comments in TcHsType point out, we don't want maximal polymorphism. His example is:

    type family F a where
F Int = Bool
F Bool = Char


We could generate

   F :: forall k1 k2. k1 -> k2


so that (F Maybe) is well-kinded, but stuck. But that's probably not what we want. It would be better to get F :: * -> *

    type family G a f b where
G Int  f b = f b
G Bool f b = Char -> f b


You could just about argue that the programmer intends

   F :: forall k. * -> (k->*) -> k -> *


It's quite similar to this:

  data PT f a = MkPT (f a)


which today, using (BASELINE), we infer to have kind

  PT :: forall k. (k->*) -> k -> *


But I'd be perfectly happy if PT got a monomorphic inferred kind, which is what (PARTIAL) would do:

  PT :: (*->*) -> * -> *


If you want the poly-kinded PT, use a signature:

  -- Any of these would do
data PT f             (a :: k) = MkPT (f a)
data PT (f :: k -> *) a        = MkPT (f a)
data PT (f :: k -> *) (a :: k) = MkPT (f a)


One oddity is that we'd do (BASELINE) for terms and (PARTIAL) for types. But perhaps that's ok. They are different.

• Terms ought to be as polymorphic as possible but arguably not types. Examples above. Also, since kind polymorphism is still in its infancy, maybe it's no bad thing if all kind polymorphism is explicitly signalled every time a kind-polymorphic binder is introduced.
• Terms have well-established separate type signatures, but we don't have a syntax for separate kind signatures of types and classes.

If we moved from (BASELINE) to (PARTIAL), some programs that work now would fail:

• the original S/T example above
• a data type like PT where the user did actually want the kind-polymorphic version.

But that might be a price worth paying for the simplicity, uniformity, and predictability you'd get in exchange.

Richard: I think changing to (PARTIAL) throughout would be a mistake, as lots of code would fail to compile. Kind polymorphism by default in datatypes and classes has been around since 7.4, and I suspect there is quite a bit of code that such a change would disrupt.

On the other hand, I think changing to (PARGEN) throughout would work nicely. I believe that it would allow all current code to type-check (except for the weird example that probably should be rejected in #9201). If we were to choose (PARGEN) over (ALL), it's possible that some code would become more polymorphic, as (PARGEN) is more polymorphic than (BASELINE) in the presence of a CUSK. However, I don't believe that this could be a breaking change, and I would prefer going with (PARGEN) over (ALL) for the sake of simplicity -- no need to have two systems around.

I can't figure out a way that (BASELINE) and (PARGEN) are different in type signatures for terms. This version doesn't have quite as nice a declarative typing rule because the type is generalized over kind variables that go completely unmentioned in the type -- a straightforward forall ftv(t). t doesn't quite do it. We need to generalize over seen variables, infer kinds, and then generalize over meta-kind variables. But, this is what is done today.

(Because open type families do not have a body, they would still be considered to have a CUSK, where un-annotated type variables default to have kind *.)

In comment:5:ticket:9200, I discuss "good" polymorphism and "bad" polymorphism. This discussion, in retrospect, seems tangential at this point. It really only makes sense when discussing closed type families, which aren't at the heart of the problems here. End Richard