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DataParallel/ClosureConversion Up?

## Closure conversion without indexed types

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.

### Conversion status

To all `TyCon`

s, `DataCon`

s, and `Id`

s, we add a value of type

data StatusCC a = NoCC -- Declaration has not been converted | ConvCC a -- Here is the converted version

For example, `Id`

gets a field of type `StatusCC Id`

. A declaration `thisDecl`

can be in one of three categories:

`NoCC`

: We did not convert that declaration, either because it was declared in an unconverted module or because it uses some feature that prevents conversion.`ConvCC thisDecl`

: Original and converted declaration coincide (e.g., type declarations not involving arrows directly or indirectly).`ConvCC convDecl`

: The variant`convDecl`

is the closure-converted form of`thisDecl`

.

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.

### 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.

### Converting type declarations

#### Preliminaries

The alternatives of `TyCon.TyCon`

get a new field `tyConCC :: StatusCC TyCon`

. This field is `NoCC`

for data constructors for which we have no conversion and `ConvCC T_CC`

if we have a conversion, where the converted declaration `T_CC`

may coincide with `T`

. In the latter case, there is also *conversion constructor* `isoT`

between values inhabitating types formed from the original and converted constructor. The type of these functions is as follows:

isoTy (T::k1->..->kn->*) = forall _1 .. _n _1_CC .. _n_CC. isoTy (_1::k1) -> .. -> isoTy (_n::kn) -> (C _1 .. _n :<->: C_CC _1_CC .. _n_CC)

(The type variables beginning with underscores are bound here; we add one underscore for each level of kinding.)

As an example, consider

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

The type of the conversion constructor is as follows (using more meaningful type variable names):

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 (tff tfInt ) x) toT (T2 y) = T2 (to (tff tfBool) y) frT (T1 x) = T1 (fr (tff tfInt ) x) frT (T2 y) = T2 (fr (tff tfBool) y)

where `isoInt`

and `isoBool`

are the conversion constructors for `Int`

s and `Bool`

s.

Moreover, we represent closures - the converted form of function arrows - as follows:

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

So, the function array constructor `(->)::*->*->*`

has a `StatusCC`

value of `ConvCC ((:->)`

with `isoArr`

as the conversion constructor.

Closure application is defined as

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

#### 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`

with`tyConCC S == NoCC`

, we cannot convert`T`

and set its`tyConCC`

field to`NoCC`

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

that are mentioned in`T`

's definiton have`tyConCC S == ConvCC S`

, we do not convert`T`

, but set its`tyConCC`

field to`ConvCC 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, we set its`tyConCC`

field to`NoCC`

- except if Case 2 applies. - Otherwise, we generate a converted type declaration
`T_CC`

together a conversion constructor`isoT`

, and set`tyConCC`

to`ConvCC T_CC`

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

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

`FunTyCon`

: It's`StatusCC`

value was defined above. We handle any occurence of the function type constructor like that of an algabraic type constructor with the`StatusCC`

value given above, but we may not want to explcitly store that value in a field of`FunTyCon`

, as`(:->)`

would then probably need to go into`TyWiredIn`

in.`TupleTyCon`

: The`StatusCC`

value of a tuple constructor`T`

is`ConvCC T`

; i.e., we don't need converted tuple type constructors. However, we need a suitable conversion functions`isoT`

for all supported tuple types somewhere. Unfortunately, there are many tuple types, and hence, many conversion functions. An alternative might be to special case tuples during conversion generation and just inline the needed case construct.`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. We assume their converted and unconverted form are identical, which implies that they never inhibit conversion and that they need no conversion constructors.`CoercionTyCon`

and`SuperKindTyCon`

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

#### Converting data constructors

We convert data constructors by converting their argument types and their representation `DataCon`

gets a new filed `dcCC :: StatusCC DataCon`

. In particular, the signature of the worker is converted. However, in contrast to other functions, we only convert the argument and result types; the arrows tying them together are left intact. For example, if the original wrapper has the type signature

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

the converted wrapper is

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

As a consequence, whenever we convert a *partial* wrapper application in an expression, we need to introduce a closure on the spot.

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

#### Examples

For example, when we convert

data Int = I# Int#

the `tyConCC`

field of `Int`

is set to `ConvCC Int`

and we have

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

As another example, the `tyConCC`

field of

data Maybe a = Nothing | Just a

has a value of `ConvCC Maybe`

and we have

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 processed in the same way as other data types and value bindings, respectively; i.e., they get a `StatusCC`

field and we generate converted versions and conversion constructors as usual.

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 { (+_CC) :: a :-> a :-> a, negate_CC :: a :-> a } dNumInt_CC :: Num_CC Int -- Int \equiv Int_CC dNumInt_CC = Num_CC (to isoIntToIntToInt primAddInt) (to isoIntToInt primNegateInt) where isoIntToIntToInt = isoArr isoInt isoIntToInt isoIntToInt = isoArr isoInt isoInt

chak: revision front

### Converting type terms

We determine the converted type `t^`

of `t`

as follows:

T^ = T_CC , if available T , otherwise a^ = a (t1 t2)^ = t1^ t2^ (t1 -> t2)^ = Clo t1 t2 (forall a.t)^ = forall a.t^ (C t1 => t2)^ = C_CC t1^ => t2^ , if available C t1^ => t2^ , otherwise

### Converting value bindings

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

, we generate `f_CC :: t^`

. The alternatives `GlobalId`

and `LocalId`

of `Var.Var`

get a new field `idCC :: StatusCC Id`

whose values, for a declaration `f`

, we determine as follows:

- If
`Id`

's declaration uses any features that we cannot (or currently, don't want to) convert, set`idCC`

to`NoCC`

. - If all type constructors involved in
`f`

's type are marked`NoCC`

or`AsIsCC`

, we set`f`

's`idCC`

field to`AsIsCC`

. - Otherwise, convert
`f`

and set its`ifCC`

field to`ConvCC f_CC`

.

### Converting core terms

Apart from the standard rules, we need to handle the following special cases:

- We come across a value variable
`v`

where`idCC v == NoCC`

whose type is`t`

: we generate`convert t v`

(see below). - We come across a case expression where the scrutinised type
`T`

has`tyConCC T == NoCC`

: we leave the case expression as is (i.e., unconverted), but make sure that the`idCC`

field of all variables bound by patterns in the alternatives have their`idCC`

field as`NoCC`

. (This implies that the previous case will kick in and convert the (unconverted) values obtained after decomposition.) - Whenever we have an FC
`cast`

from or to a newtype`T`

, where`tyConCC T == NoCC`

, we need to add a`convert tau`

or`trevnoc tau`

, respectively. We can spot these casts by inspecting the kind of every coercion used in a cast. One side of the equality will have the newtype constructor. - We come across a dfun: If its
`idCC`

field is`NoCC`

, we keep the selection as is, but apply`convert t e`

from it, where`t`

is the type of the selected method and`e`

the selection expression. If`idCC`

is`ConvCC d_CC`

, and the dfun's class is converted,`d_CC`

is fully converted. If it's class is not converted, we also keep the selection unconverted, but have a bit less to do in`convert t e`

.**TODO**This needs to be fully worked out.

### Generating conversions

Whenever we had `convert t e`

above, where `t`

is an unconverted type and `e`

a converted expression, we need to generate some conversion code. This works roughly as follows in a type directed manner:

convert T = id , if tyConCC T == NoCC or AsIsCC = to_T , otherwise convert a = id convert (t1 t2) = convert t1 (convert t2) convert (t1 -> t2) = createClosure using (trevnoc t1) and (convert t2) on argument and result resp.

where `trevnoc`

is the same as `convert`

, but using `from_T`

instead of `to_T`

.

The idea is that conversions for parametrised types are parametrised over conversions of their parameter types. Wherever we call a function using parametrised types, we will know these type parameters (and hence can use `convert`

) to compute their conversions. This fits well, because it is at occurences of `Id`

s that have `idCC == NoCC`

where we have to perform conversion.

The only remaining problem is that a type parameter to a function may itself be a type parameter got from a calling function; so similar to classes, we need to pass conversion functions with every type parameter. So, maybe we want to stick `fr`

and `to`

into a class after all and requires that all functions used in converted contexts have the appropriate contexts in their signatures.

### 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`

.

#### Conversion functions

Similar to `HasGenerics`

and instead of storing `Id`

of conversion constructors, we can derive from the name of the `TyCon`

.

#### Data constructors

How to exactly handle the worker and wrapper? Can we replace arrows by closure types in the worker? Or do we always have to add a wrapper?

**Simpler''' Don't try to make a complete cloned data constructor. By the time of CC, its all just Core and so wrappers are just like any other global function.
**

#### Original functions

The previous story was that when vectorising `f`

and generating `f_CC`

, we now define

f :: tau f = trevnoc tau f_CC

Now, with the approximate conversion scheme above, we may not have `trevnoc tau`

. In this case, we still generate `f_CC`

, but also leave the rhs of `f`

alone (i.e., compile the original functions).

When we give up on converting a complete right-hand side, we still want to convert all subexpressions that we can convert.