## Desugaring of array comprehensions

Wadler's desugaring for list comprehensions is not suitable for arrays, as we need to use collective operations to get good parallel code. The build/foldr desugaring, although using collective operations, isn't a good match for how the array operations are implemented. In fact, the *naive* desugaring from the H98 report is a much better fit:

(1) [: e | :] = [:e:] (2) [: e | b, qs :] = if b then [: e | qs :] else [::] (3) [: e | p <- a, qs :] = let ok p = [: e | qs :] ok _ = [::] in concatMapP ok a (4) [: e | let ds, qs :] = let ds in [: e | qs :] (5) [: e | qs | qss :] = [: e | (XS, XSS) <- zip [: XS | qs :] [: XSS | qss :] :] where XS & XSS are the bound variables in qs & qss

In particular, `concatMapP f a` essentially implies to apply the lifted version of `f` directly to `a` and then the concat strips of one level of segment descriptors; i.e., both the `concatP` and the `mapP` vanish due to vectorisation.

## Problem with the naive rules

Nevertheless, these rules are not entirely satisfactory. For example, `[:e | x <- a, b:]` turns into

concatMap (\x -> if b then [:e:] else [::]) a

which is a fairly complicated way to perform

mapP (\x -> e) . filterP (\x -> b) $ a

even when taking vectorisation into account. Under vectorisation, the conditional implies `filterP (\x -> b)`, but adds an expensive, and here useless, merge operation. Maybe these overheads can be optimised away. However, for the moment, we use a desugaring that is based on the above rules, but generates code that should be better suited to array processing.

## Modified rules

The idea is to flatten out the processing of comprehensions to some degree by defining a transformation function `<< . >>` that gets two arguments: a pattern `pa` and a desugared expression `ea`, where we are guaranteed that `ea` is array valued and all its elements match `pa`. The semantics of the transformation function is given by

<<[: e | qs :]>> pa ea = [: e | pa <- ea, qs :] = concatMap (\pa -> [: e | qs :]) ea

We have the second line by applying Rule (3).

Using this definition of `<< . >>`, we can derive a new set of desugaring rules. The derivation proceeds by unfold/fold transformations and some properties of the involved combinators. The resulting rules are the following:

(1') <<[: e | :]>> pa ea = mapP (\pa -> e) ea (2') <<[: e | b, qs :]>> pa ea = <<[: e | qs :]>> pa (filterP (\pa -> b) ea) (3') <<[: e | p <- a, qs :]>> pa ea = let ok p = True ok p = False in <<[: q | qs :]>> (pa, p) (crossMapP ea (\pa -> filterP ok a)) (4') <<[: e | let ds, qs :]>> pa ea = <<[: e | qs :]>> (pa, XS) (mapP (\v@pa -> let ds in (v, XS)) ea) where XS are the variables bound by ds (5') <<[: e | qs | qss :]>> pa ea = <<[: e | qss :]>> (pa, XS) (zipP ea [: XS | qs :]) where XS are the variables bound by qs

The typical array processing comprehensions containing only generators, guards, and parallel comprehensions (but not cross-products and lets) are translated into a straight combination of `mapP`, `filterP`, and `zipP` by these rules, which is exactly what we want.