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.
|Type variables||alpha, beta, gamma|
|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
* -> * -> Constraint.
- The evidence for
InstanceOf sigma1 sigma2is a function
sigma1 -> sigma2. This accounts for the weird order of parameters in
- The canonical forms associated with the constraint are
InstanceOf sigma1 alpha1and
InstanceOf alpha2 sigma2, where
sigma2is not a type variable.
InstanceOf needs to be defined in
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:
InstanceOf (T sigma1 ... sigman) t1---->
(T sigma1 ... sigman) ~ t1
InstanceOf (forall a. Q2 => tau2) (T sigma1 ... sigman)---->
[a/alpha]tau2 ~ (T sigma1 ... sigman) /\ [a/alpha]Q2
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
Implementation note: add the new types of evidence to
Implementation note: extend the desugarer to convert from evidence to actual functions.
Suppose we want to type check
runST ($) (e :: forall s. ST s Int). Let us denote
alpha the type of
beta the type of
gamma the type of the entire expression. The initial set of constraints which are generated (details on generation below) are:
InstanceOf (forall a b. (a -> b) -> a -> b) (alpha -> beta -> gamma) [from ($)] 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)  InstanceOf (forall a. (forall s. ST s a) -> a) alpha  InstanceOf (forall s. ST s Int) beta  ----> [IOCan2] over  ((delta -> epsilon) -> delta -> epsilon) ~ (alpha -> beta -> gamma)  +  and  ----> type deconstruction in  alpha ~ delta -> epsilon beta ~ delta gamma ~ epsilon +  and  ----> substitution in  and  InstanceOf (forall a. (forall s. ST s a) -> a) (delta -> epsilon)  InstanceOf (forall s. ST s Int) delta  ----> [IOCan2] over  ((forall s. ST s eta) -> eta) ~ (delta -> epsilon)  InstanceOf (forall s. ST s Int) delta ----> type deconstruction in  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)  ----> [IOCan3] over  forall s. (_ => InstanceOf (forall s'. ST s' Int) (ST s eta))  ----> [IOCan2] under (=>) of  forall s. (_ => Instance (ST pi Int) (ST s eta))  ----> canonicalization under (=>) forall s. (_ => s ~ pi /\ eta ~ Int)  ----> 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 :)
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)
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
InstanceOf constraints also need to change to functions. This is done in the zonking phase.
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
- Convert all
InstanceOfinto type equality. In the previous case, the type of
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
InstanceOfconstraints to top-level. In this case, the type given to
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
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
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
:_~ 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,
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
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
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
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.