impredicative 2015

New impredicativity (June 2015)

The goal is to build a better story for impredicative and higher-rank polymorphism in GHC. For that aim we introduce a new type of constraint, InstanceOf t1 t2, which expresses that type t2 is an instance of t1. This new type of constraint is inspired on ideas from the MLF and HML systems.

This is the result of discussion between Alejandro Serrano Mena <A.SerranoMena@…>, Jurriaan Hage, Dimitrios Vytiniotis, and Simon PJ.

The most up-to-date description is available here: Impredicativity in GHC (PDF)

The rest of the document is kept for historical purposes, and because it contains useful information about how the design is implemented inside GHC.

Notation

Type variables	`alpha`, `beta`, `gamma`
Type constructors	`T`
Type families	`F`
Constraints	`Q`
Monomorphic types	`mu ::= alpha \| a \| mu -> mu \| T mu ... mu \| F mu ... mu`
Types without top-level `forall`	`tau ::= alpha \| a \| sigma -> sigma \| T sigma ... sigma \| F sigma ... sigma`
Polymorphic types	`sigma ::= forall a. Q => tau`

Some basic facts

InstanceOf has kind * -> * -> Constraint.
The evidence for InstanceOf sigma1 sigma2 is a function sigma1 -> sigma2. This accounts for the weird order of parameters in InstanceOf.
The canonical forms associated with the constraint are InstanceOf sigma1 alpha1 and InstanceOf alpha2 sigma2, where sigma2 is not a type variable.

Implementation note: InstanceOf needs to be defined in libraries/ghc-prim/GHC/Types.hs.

Implementation note: new canonical forms need to be defined in compiler/typecheck/TcRnTypes.hs by extending the Ct data type.

Changes to constraint solver

Luckily, in order to work with InstanceOf constraints, we only need to add new rules to the canonicalization step in the solver. These rules are:

[IOCan1] InstanceOf (T sigma1 ... sigman) t1 ----> (T sigma1 ... sigman) ~ t1
[IOCan2] InstanceOf (forall a. Q2 => tau2) (T sigma1 ... sigman) ----> [a/alpha]tau2 ~ (T sigma1 ... sigman) /\ [a/alpha]Q2
[IOCan3] InstanceOf sigma2 (forall a. Q1 => tau1)---->forall a. (Q1 => InstanceOf sigma2 tau1)`

But we also need to generate evidence for each of these steps!

Implementation note: implement these rules in compiler/typecheck/TcCanonical.hs.

Implementation note: add the new types of evidence to EvTerm in compiler/typecheck/TcEvidence.hs.

Implementation note: extend the desugarer to convert from evidence to actual functions.

Example

Suppose we want to type check runST ($) (e :: forall s. ST s Int). Let us denote alpha the type of runST, beta the type of e and gamma the type of the entire expression. The initial set of constraints which are generated (details on generation below) are:

InstanceOf (forall a b. (a -> b) -> a -> b) (alpha -> beta -> gamma)  [from ($)]
InstanceOf (forall a. (forall s. ST s a) -> a) alpha                  [from runST]
InstanceOf (forall s. ST s Int)                beta                   [from e]

The series of solving steps are:

InstanceOf (forall a b. (a -> b) -> a -> b) (alpha -> beta -> gamma)  [1]
InstanceOf (forall a. (forall s. ST s a) -> a) alpha                  [2]
InstanceOf (forall s. ST s Int)                beta                   [3]

----> [IOCan2] over [1]

((delta -> epsilon) -> delta -> epsilon) ~ (alpha -> beta -> gamma)   [4]
+ [2] and [3]

----> type deconstruction in [4]

alpha ~ delta -> epsilon
beta  ~ delta
gamma ~ epsilon
+ [2] and [3]

----> substitution in [2] and [3]

InstanceOf (forall a. (forall s. ST s a) -> a) (delta -> epsilon)     [5]
InstanceOf (forall s. ST s Int) delta                                 [6]

----> [IOCan2] over [5]

((forall s. ST s eta) -> eta) ~ (delta -> epsilon)                    [7]
InstanceOf (forall s. ST s Int) delta

----> type deconstruction in [7]

delta   ~ forall s. ST s eta
epsilon ~ eta
InstanceOf (forall s. ST s Int) delta

----> substitution

InstanceOf (forall s. ST s Int) (forall s. ST s eta)                  [8]

----> [IOCan3] over [8]

forall s. (_ => InstanceOf (forall s'. ST s' Int) (ST s eta))         [9]

----> [IOCan2] under (=>) of [9]

forall s. (_ => Instance (ST pi Int) (ST s eta))                      [10]

----> canonicalization under (=>)

forall s. (_ => s ~ pi /\ eta ~ Int)                                  [11]

----> float constraints out of (=>)

eta ~ Int
forall s. (_ => s ~ pi)

----> FINISHED!

We get that the type assigned to the whole expression is gamma ~ epsilon ~ eta ~ Int, as we expected :)

Evidence generation

For [IOCan1] we want to find evidence for W1 :: InstanceOf (T sigma1 ... sigman) t1 from W2 :: (T sigma1 ... sigman) ~ t1. Such an evidence must be a function W1 :: (T sigma1 ... sigman) -> t1. We can get it by applying the coercion resulting from W2. More schematically:

W1 :: InstanceOf (T sigma1 ... sigman) t1

---->

W1 :: T sigma1 ... sigman -> t1
W1 = \x -> x |> W2

W2 :: (T sigma1 ... sigman) ~ t1

W1 :: InstanceOf (forall a. Q1 ... Qn => tau2) (T sigma1 ... sigman)

---->

W1 :: (forall a. Q1 ... Qn => tau2) -> T sigma1 ... sigman
W1 = \x -> (x alpha V1 ... Vn) |> W2

W2 :: [a/alpha]tau2 ~ (T sigma1 ... sigman)
V1 :: [a/alpha]Q1, ..., Vn :: [a/alpha]Qn

The case [IOCan3] is the most complex one: we need to generate a function from the evidence generated by an implication. Such an implication generates a series of bindings, which we denote here using []. Note that we abstract by values, types and constraints, but this is OK, because it is a System FC term.

W1 :: InstanceOf sigma2 (forall a. Q1 => tau1)

---->

W1 :: sigma2 -> (forall a. Q1 => tau1)
W1 = \x -> /\a -> \(d : Q1) -> let [] in (W2 x)

W2 :: forall a. (d : Q1) => (W2 :: InstanceOf sigma2 tau1)

Zonking

While in the solver we want InstanceOf constraints to have their own identity. However, when converted to Core, they must be converted into functions. This means that the types of EvTerms with InstanceOf constraints also need to change to functions. This is done in the zonking phase.

Design choice: `InstanceOf` and `->`

In the designed proposed above, -> is treated as any other type constructor. That means that if we are canonicalizing InstanceOf (sigma3 -> sigma4) (sigma1 -> sigma2), the result is sigma1 ~ sigma3 /\ sigma2 ~ sigma4. That is, -> is treated invariantly in both arguments. Other possible design choices are:

-> treated co- and contravariantly, leading to InstanceOf sigma3 sigma1 /\ InstanceOf sigma2 sigma4.
Treat only the co-domain covariantly, leading to sigma1 ~ sigma3 /\ InstanceOf sigma2 sigma4.

Which are the the benefits of each option?

Changes to approximation

One nasty side-effect of this approach is that the solver may produce non-Haskell 2010 types. For example, when type checking singleton id, where singleton :: forall a. a -> [a] and id :: forall a. a -> a, the result would be forall a. InstanceOf (forall b. b -> b) a => [a]. In short, we want to get rid of the InstanceOf constraints once a binding has been found to type check. This process is part of a larger one which in GHC is known as approximation.

There are two main procedures to move to types without InstanceOf constraints:

Convert all InstanceOf into type equality. In the previous case, the type of singleton id is forall a. a ~ forall b. b -> b => [a], or equivalently, [forall b. b -> b].
Generate a type with the less possible polymorphism, by moving quantifiers out of the InstanceOf constraints to top-level. In this case, the type given to singleton id is forall b. [b -> b].

We aim to implement the second option, since it leads to types which are more similar to those already inferred by GHC. Note that this approximation only applies to unannotated top-level bindings: the user can always ask to give [forall a. a -> a] as a type for singleton id via an annotation.

The procedure works by appling repeatedly the following rules:

InstanceOf (forall b. Q => tau) a  ---->  a ~ [b/beta]tau  /\  [b/beta]Q
InstanceOf a (forall b. Q => tau)  ---->  a ~ (forall b. Q => tau)

The first rule is a version of [IOCon2] which applies to canonical InstanceOf constraints. The second rule ensures that the InstanceOf constraint is satisfied.

Implementation note: change the simplifyInfer function in compiler/typecheck/TcSimplify.hs to generate candidate approximations using the previous two rules.

Changes to constraint generation

Constraint generation is the phase prior to solving, in which constraints reflecting the relations between types in the program are created. We describe constraint generation rules in this section using the same formalism as OutsideIn(X), that is, as a judgement Gamma |- e: tau --> C: under a environment Gamma, the expression e is assigned type tau subject to constraints C.

In principle, the only rule that needs to change is that of variables in the term level, which is the point in which instantiation may happen:

   x : sigma \in \Gamma        alpha fresh
--------------------------------------------- [VAR]
Gamma |- x : alpha --> InstanceOf sigma alpha

Unfortunately, this is not enough. Suppose we have the following piece of code:

(\f -> (f 1, f True)) (if ... then id else not)

We want to typecheck it, and we give the argument f a type variable alpha, and each of its appearances the types variables beta and gamma. The constraints that are generated are:

InstanceOf alpha beta  [usage in (f 1)]
InstanceOf alpha gamma [usage in (f True)]
InstanceOf (forall a. a -> a) alpha
InstanceOf (Bool -> Bool)     alpha

At this point we are stuck, since we have no rule that could be applied. One might think about applying transitivity of InstanceOf, but this is just calling trouble, because it is not clear how to do this without losing information.

Our solution is to make this situation impossible by generating beta ~ alpha and gamma ~ alpha instead of their InstanceOf counterparts. We do this by changing the [VAR] rule in such a way that ~ is generated when the variable comes from an unannotated abstraction or unannotated let. The environment is responsible for keeping track of this fact for each binding, by a small tag.

    x :_~ sigma \in \Gamma
------------------------------ [VAR~]
Gamma |- x : sigma --> nothing

Notice the change from : to :_~ in the rule. As stated above, some other rules need to be changed in order to generate this tag for their enclosed variables:

alpha fresh    Gamma, (x :_~ alpha) |- e : tau --> C
----------------------------------------------------
       Gamma |- \x -> e : alpha -> tau --> C

Gamma, (x :_~ alpha) |- e : tau1 --> C1    Gamma, (x :_~ alpha) |- b : tau2 --> C2
----------------------------------------------------------------------------------
           Gamma |- let x = e in b : tau2 --> C1 /\ C2 /\ alpha ~ tau1

With this change, our initial example leads to an error (f cannot be applied to both Bool and Int), from which one can recover by adding an extra annotation. This is a better situation, though, that getting stuck in the middle of the solving process.

Summary: a PDF with the entire set of rules is available as an attachment. only-gen.pdf

Implementation note: the type of local environments, TcLclEnv in compiler/typecheck/TcExpr.hs, needs to be upgraded to take into account whether a variable is tagged as generating ~. Maybe just change type TcTypeEnv = NameEnv (TcTyThing, Bool)?

Implementation note: constraint generation appears in GHC source code as tcExpr in compiler/typecheck/TcExpr.hs.

Adding propagation

Still, this is not enough! Suppose you write the following code:

f :: (forall a. a -> a) -> (Int, Bool)
f x = (x 1, x True)

g = f (\x -> x)

None of them will work! The problem is that, in the first case, we do not use the information in the signature when generating constraints for the function. Thus, x will be added to the environment with the ~ tag, effectively forbidding to be applied to both Ints and Bools.

In the second case the solver does not know that it should generalize at the point of the \x -> x expression. Thus, we will come to a point where we have tau -> tau ~ forall a. a -> a, which leads to an error, since quantified and not quantified types cannot be equated.

However, we expect both cases to work. After all, the information is there, we only have to make it flow to the right place. This is exactly the goal of adding propagation to the constraint generation phase. Lucikly, GHC already does some propagation now, as reflected in the type of the function tcExpr. The main change is that, whereas the current implementation pushes down and infers shapes of functions, the new one is simpler, and only pushes information down. A PDF with the rules is available as an attachment. only-prop.pdf

The most surprising rule is the one named [AppFun], which applies when we have a block of known expressions f1 ... fm whose type can be recovered from the environment followed by some other freely-shaped expressions. For example, the case of f (\x -> x) above, where f is in the environment of g. In that case, we compute the type that the first block ought to have, and propagate it to the rest of arguments.

The reason for including a block of fis is to cover cases such as runST $ do ..., or more clearly ($) runST (do ...), where some combinators are used between functions. Should the rule [AppFun] only include the case f e1 ... fm, the common runST $ do ... could not be typed without an annotation.

Type classes and families

There are some unwanted interactions between type classes and families and the InstanceOf constraint. For example, if we want to type [] == [], we obtain as canonical constraints:

Eq a /\ InstanceOf (forall b. [b]) a

At this point we are stuck. We need to instantiate b before Eq can scrutinize its argument to check whether an instance is available. One possibility is to instantiate by default every type linked to a variable appearing in a type class or type family.

That solution poses its own problems. Consider the following type family:

type family F a b
type instance F [a] b = b -> b

Using the rule of always instantiating, the result of gamma ~ F [Int] b, InstanceOf (forall a. a -> a) b is gamma ~ (delta -> delta) -> (delta -> delta). We have lost polymorphism in a way which was not expected. What we hoped is to get gamma ~ (forall a. a -> a) -> (forall a. a -> a).

Thus, we need to have a way to instantiate variables appearing in type classes and families, but only as necessary. We do this by temporarily instantiating variables when checking for axiom application, and returning extra constraints which make this instantiation possible if the match is successful. For example, in the previous case we want to apply the axiom forall e. Eq e => Eq [e], and thus we need to instantiate a. We return as residual constraints Eq e /\ Eq a ~ Eq [e], and the solver takes care of the rest, that is, InstanceOf (forall b. [b]) [e].

Implementation note: the changes need to be done in the lookupInstEnv' function in compiler/types/InstEnv.hs. The solver needs to be changed at compiler/typecheck/TcInteract.hs to use the new information.