## Closure conversion without classes

The following scheme approaches the problem of mixing converted and unconverted code from the point of view of GHC's Core representation, avoiding the use of classes as much as possible. In particular, the scheme gracefully handles any declarations that themselves cannot be converted, but occur in a converted module. The two essential ideas are that (1) we move between converted and unconverted values/code using a conversion isomorphism and (2) we treat unconverted declarations differently depending on whether or not they involve arrows; e.g., the definition of `Int`

by way of unboxed values (which we cannot convert) doesn't prevent us from using `Int`

s *as is* in converted code.

#### The closure type

We represent closures by

data a :-> b = forall e. !(e -> a -> b) :$ e

and define closure creation and application as

lam :: (a -> b) -> (a :-> b) lam f = const f :$ () ($:) :: (a :-> b) -> a -> b (f :$ e) $: x = f e x

So, we have `(->)_CC == (:->)`

.

### Overview and invariants

The meta-function "`^`

", written postfix, converts from normal code to closure-converted code.

- For each function definition
f::ty = e

create a closure-converted function definition for`f_cc`

, defining the closure-converted version of`f`

,*and*a definition for`f`

with its original type, that defines it in terms of`f_cc`

:f_cc::ty^ = e^ f::ty = fr iso<ty> f_cc

where`ty^`

is the closure-converted version of`ty`

, and`e^`

is the closure-converted version of`e`

.

- For each data type
`T`

, create a closure-converted data type`T_CC`

, whose constructors use`(:->)`

instead of`(->)`

.

- The value
`iso<ty>`

is a pair of functions, converting to and fro between`ty`

and`ty^`

. Like "`^`

", the`iso<ty>`

thing should be considered as a meta-function that works recursively on`ty`

.

Invariants:

e :: ty implies e^ :: ty^ to iso<ty> :: ty -> ty^ fr iso<ty> :: ty^ -> ty

Provided these invariants are maintained, the transformed program will be type-correct.

### Conversion status

All `TyCon`

s, `DataCon`

s, and `Id`

s have a *conversion status* that determines how occurences of these entities are treated during conversion. For an `Id`

named `v`

, we have two alternatives:

- The binding of
`v`

was compiled without conversion and we have to use`v`

itself in converted code, which requires the use of an in-place conversion function. - Otherwise, we have a converted variant
`v_CC`

, and we use`v_CC`

instead of`v`

in converted code.

For a type constructor `T`

and its data constructors `C`

, we have three alternatives:

- The declaration introducing
`T`

and its constructors was compiled without conversion or we were unable to convert it, as it uses some language feature that prevents conversion. - A converted variant
`T_CC`

exists, but coincides with`T`

(e.g., because`T`

neither directly nor indirectly involves arrows). - A converted variant
`T_CC`

exists and differs from`T`

.

In the last two cases, we also have a *conversion constructor* `isoT`

whose type and meaning is described below.

An example of a feature that prevents conversion are unboxed values. We cannot make a closure from a function that has an unboxed argument, as we can neither instantiate the parametric polymorphic closure type with unboxed types, nor can we put unboxed values into the existentially quantified environment of a closure.

### Converting types

#### Conversion of type terms

We determine the converted type `t^`

of `t`

as follows:

T^ = T_CC , if T_CC exists = T , otherwise a^ = a_CC (t1 -> t2)^ = t1 -> t2 , if kindOf t1 == # or kindOf t2 == # = t1^ :-> t2^, otherwise (t1 t2)^ = t1^ t2^ (forall a.t)^ = forall a_CC.t^

Here some examples,

(Int -> Int)^ = Int :-> Int (forall a. [a] -> [a])^ = [a] :-> [a] ([Int -> Int] -> Int)^ = [Int :-> Int] :-> Int (Int# -> Int# -> Int#)^ = Int# -> Int# -> Int# ((Int -> Int) -> Int#)^ = (Int -> Int) -> Int# (Int -> Int -> Int#)^ = Int :-> (Int -> Int#)

Why do we use `(t1 -> t2)^ = t1 -> t2`

when either argument type is unboxed, instead of producing `t1^ -> t2^`

? Because we want to avoid creating conversion constructors (see below) for such types. After all, the conversion constructor `isoArr`

for function arrows works only for arrows of kind `*->*->*`

.

### Conversion constructors

To move between `t`

and `t^`

we use conversion functions. And to deal with type constructors, we need *conversion constructors*; i.e., functions that map conversion functions for type arguments to conversion functions for compound types.

#### Conversion pairs

Conversion functions come in pairs, which we wrap with the following data type for convenience:

data a :<->: b = (:<->:) {to :: a -> b, fr ::b -> a}

The functions witness the isomorphism between the two representations, as usual.

#### Types of convercion constructors

The type of a conversion constructor depends on the kind of the converted type constructor:

isoTy (t::k1->k2) = forall a a_CC. isoTy (a::k1) -> isoTy (t a::k2) isoTy (t::*) = t :<->: t^

where type conversion `t^`

is defined below.

As an example, consider

data T (f::*->*) = T1 (f Int) | T2 (f Bool)

The type of the conversion constructor is as follows :

isoTy (T::(*->*)->*) = forall f f_CC. (forall a a_CC. (a :<->: a_CC) -> (f a :<->: f_CC a_CC)) -> T f :<->: T_CC f_CC

The conversion constructor might be implemented as

isoT isof = toT :<->: frT where toT (T1 x) = T1 (to (isof isoInt ) x) toT (T2 y) = T2 (to (isof isoBool) y) frT (T1 x) = T1 (fr (isof isoInt ) x) frT (T2 y) = T2 (fr (isof isoBool) y)

where `isoInt`

and `isoBool`

are the conversion constructors for `Int`

s and `Bool`

s.

Moreover, the conversion constructor for function arrows is

isoArr :: a :<->: a_CC -- argument conversion -> b :<->: b_CC -- result conversion -> (a -> b) :<->: (a_CC :-> b_CC) isoArr (toa :<->: fra) (tob :<->: frb) = toArr :<->: frArr where toArr f = const (tob . f . fra) :$ () frArr (f :$ e) = frb . f e . toa

### Conversions

#### Rules

To perform the actual conversion of values of a type `t::*`

, we generate a conversion `iso<t>`

of type `t :<->: t^`

as follows:

iso<T> = isoT , if T_CC exists = idIso<*> , otherwise iso<a::k> = idIso<k> iso<t1 -> t2> = idIso<*> , if kindOf t1 == # or kindOf t2 == # = isoArr , otherwise iso<t1> iso<t2> iso<t1 t2> = iso<t1> iso<t2> iso<forall a.t> = toIso :<->: frIso where toIso (x::forall a.t) = /\a. to (iso<t>)@a x@a frIso (x::forall a.t) = /\a. fr (iso<t>)@a x@a

where

idIso<*> = id :<->: id

**OPEN PROBLEM:** What should `idIso<k>`

do for `k`

other than `*`

? We might think

idIso<k1->k2> = \_ -> (idIso<k2>)

would work, but it doesn't always. Take this example

data T a = MkT a unit :: T a -> a unit (MkT x) = x -- converted foo :: f (Int -> Int) -> (forall a. f a -> a) -> Int foo t u = u t 1 -- not converted bar = foo unit

Here, we will have to convert the first argument to `foo`

and that conversion needs to convert the embedded `Int -> Int`

to `Int :-> Int`

, which is hard to do in `foo = fr iso<...> foo_CC`

as we don't know anything about `f::*->*`

.

#### Examples

Here some example conversions:

iso<Int -> Int> = isoArr isoInt isoInt iso<Int -> Int#> = id :<->: id iso<[a -> a]> = isoList (isoArr (id :<->: id) (id :<->: id)) iso<f (Int -> Int)> = ???

### Converting type declarations

#### Conversion rules

If a type declaration for constructor `T`

occurs in a converted module, we need to decide whether to convert the declaration of `T`

. We decide this as follows:

- If the declaration of
`T`

mentions another algebraic type constructor`S`

for which there is**no**`S_CC`

, then we cannot convert`T`

. - If
**all**algebraic type constructors`S`

mentioned in`T`

's definiton have a conversion`S_CC == S`

, we do not convert`T`

, but set`T_CC == T`

and generate a suitable conversion constructor`isoT`

. (NB: The condition implies that`T`

does not mention any function arrows.) - If the declaration of
`T`

uses any features that we cannot (or for the moment, don't want to) convert, simply don't convert it. - Otherwise, we generate a converted type declaration
`T_CC`

together with a conversion constructor`isoT`

. Conversion proceeds by converting all data constructors (see below).

Moreover, we handle other forms of type constructors as follows:

`FunTyCon`

: We have`(->)_CC = (:->)`

.`TupleTyCon`

: We have`(,..,)_CC = (,..,)`

. We may either have a (long) list of conversion constructors`iso(,..,)`

pre-defined or need to generate them inline by generating a suitable case expression where needed.`SynTyCon`

: Closure conversion operates on`coreView`

; hence, we will see no synonyms. (Well, we may see synonym families, but will treat them as not convertible for the moment.)`PrimTyCon`

: We essentially ignore primitive types during conversion, assuming that their converted and unconverted forms coincide. As they cannot contain values of other types, we need no conversion constructor.`CoercionTyCon`

and`SuperKindTyCon`

: They don't categorise values and are ignored during conversion.

#### Conversion constructor

Whenever we have a converted type constructor `T_CC`

, we also need to generate a conversion constructor `isoT`

. If `T`

has one or more arguments, the conversion is non-trivial, even for `T_CC == T`

.

#### Converting data constructors

We convert a data constructor `C :: t1 -> ... -> tn`

by generating a converted constructor `C_CC :: t1^ -> .. -> tn^`

. This includes the generation of a corresponding new worker `Id`

. For example, if the original worker has the type signature

MkT :: (Int -> Int) -> Int

the converted worker is

MkT_CC :: (Int :-> Int) -> Int

As a consequence, whenever we convert a *partial* worker application in an expression, we need to introduce a closure on the spot. (Simon pointed out that this is a rare case anyway.)

We do not specially handle wrappers of data constructors or field selectors. They are converted just like any other toplevel function.

#### Examples

For example, when we convert

data Int = I# Int#

we get `Int_CC = Int`

and we have

isoInt :: Int :<->: Int isoInt = toInt :<->: frInt where toInt (I# i#) = I# i# frInt (I# i#) = I# i#

As another example,

data Maybe a = Nothing | Just a

implies `Maybe_CC = Maybe`

and

isoMaybe :: (a :<->: a_CC) -> (Maybe a :<->: Maybe a_CC) isoMaybe isoa = toMaybe :<->: frMaybe where toMaybe isoa Nothing = Nothing toMaybe isoa (Just x) = Just (to isoa x) frMaybe isoa Nothing = Nothing frMaybe isoa (Just x) = Just (fr isoa x)

### Converting classes and instances

We don't alter class and instance declarations in any way. However, the dictionary type constructors and dfuns are converted in the same way as other data types and value bindings, respectively.

As an example, assume `Num Int`

were defined as

class Num a where (+) :: a -> a -> a negate :: a -> a instance Num Int where (+) = primAddInt negate = primNegateInt

with the Core code being

data Num a = Num { (+) :: a -> a -> a, negate :: a -> a } dNumInt = Num Int dNumInt = Num primAddInt primNegateInt

Then, closure conversion gives us

data Num_CC a = Num_CC (a :-> a :-> a) (a :-> a) (+_CC) :: Num_CC a :-> a :-> a :-> a negate_CC :: Num_CC a :-> a :-> a dNumInt_CC :: Num_CC Int -- as Int_CC = Int dNumInt_CC = Num_CC (to isoIntToIntToInt primAddInt) (to isoIntToInt primNegateInt) where isoIntToIntToInt = isoArr isoInt isoIntToInt isoIntToInt = isoArr isoInt isoInt

### Converting value bindings

#### Bindings

For every binding

f :: t = e

we generate

f_CC :: t^ = e^

#### Toplevel

When converting a toplevel binding for `f :: t`

, we generate `f_CC :: t^`

and redefine `f`

as

f :: t = fr iso<t> f_CC

#### Examples

Given

add :: Num a -> a -> a add = \dNum x -> (+) dNum x 1

we generate

add :: Num a -> a -> a add = fr isoFun add_CC where isoFun = isoNum (id :<->: id) `isoArr` (id :<->: id) `isoArr` (id :<->: id) add_CC :: Num_CC a :-> a :-> a add_CC = lam $ \dNum -> (\dNum x -> (+_CC) $: dNum $: x $: 1) :$ dNum

If `add`

is used in unconverted code it will still refer to the converted computation `add_CC`

; i.e., we can use converted (and subsequently vectorised) code from unconverted/unvectorised code just by importing a converted/vectorised module as normal into an unconverted module.

### Converting terms

To keep the translation rules simpler, we assume that for every converted data constructor `C_CC :: forall a1..an. t1 -> .. -> tm -> t`

, there is a conversion wrapper

$WC_CC :: forall a1..an. t1 :-> .. :-> tm :-> t $WC_CC = /\a1..an -> lam_m (C_CC@a1@..@an)

where the family of functions

lam_1 :: (a1 -> b) -> (a1 :-> b) lam_1 = lam lam_n :: (a1 -> .. -> an -> b) -> (a1 :-> .. :-> an :-> b) lam_n f = lam_(n-1) $ \x1 .. x(n-1) -> uncurry_(n-1) f :$ (x1, .., x(n-1)

turns an `n`

-ary function into an `n`

-ary closure. (NB: This is not the same as `(to iso<..>)`

for that type, as we do not convert the types of the arguments of the function.)

We translate terms as follows:

cc[[C]] | if C_CC exists = $WC_CC cc[[x::t]] | if x_CC exists = x_CC | otherwise = to iso<t> x cc[[lit]] = lit cc[[e1 e2]] = cc[[e1]] $: cc[e2] cc[[e@t]] = cc[[e]]@t^ cc[[\x -> e]] = (\(y1, .., yn) x_CC -> cc[e]]) :$ (y1, .., yn) where y1 .. yn = FV e cc[[/\a -> e]] = /\a -> cc[e]] cc[let x = e1 in e2] = let x_CC = cc[[e1]] in cc[[e2]] cc[[case e x::t of alts]]= case cc[[e]] x_CC::t^ of cc[[alts]] cc[[e `cast` t]] = cc[[e]] `cast` t^ cc[[alt1; ..; altn]] = cc[[alt1]]; ..; cc[altn]] cc[[default -> e]] = default -> cc[[e]] cc[[C x1 .. xn -> e]] | if C_CC exists = C_CC x1_CC .. xn_CC -> cc[[e]] | otherwise = C x1 .. xn -> cc[[e]]

### TODO

#### Examples

Have an example with two modules one unconverted, where the converted imports the unconverted.

Also have an example that motivates why we have to vectorise/CC declarations such as `Int`

.