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The library provides a layer on top of DPH unlifted arrays to support multi-dimensional arrays, and shape polymorphic operations for fast sequential and parallel execution. The interface for delayed arrays is similar, but in contrast to operations on the former, any operation on a delayed array is not evaluated. To force evaluation, the programmer has to explicitly convert them to a strict array.

The current implementation of the library exposes some implementation details the user of the library shouldn't have to worry about. Once the design of the library is finalised, most of these will be hidden by distinguishing between internal types and representation types.

## Strict Arrays, Delayed Array and Shape Data Type

Both strict and delayed arrays are parametrised with their shape - that is, their dimensionality and size of each dimension. The shape type class has the following interface:

### DArrays

DArrays are collections of values of `primitive' type, which are member of the class Data.Parallel.Array.Unlifted.Elt, which includes all basic types like Float, Double, Bool, Char, Integer, and pairs constructed with Data.Parallel.Array.Unlifted.(:*:), including ().

### The Shape class

Values of class Shape serve two purposes: they can describe the dimensionality and size of an array (in which case we refer to them as 'shape'), and they can also refer to the position of a particular element in the array (in which case we refer to them as an 'index'). It provides the following methods:

class (Show sh, U.Elt sh, Eq sh, Rebox sh) => Shape sh where dim :: sh -> Int -- ^number of dimensions (>= 0) size :: sh -> Int -- ^for a *shape* yield the total number of elements in that array toIndex :: sh -> sh -> Int -- ^corresponding index into a linear representation of -- the array (first argument is the shape) fromIndex:: sh -> Int -> sh -- ^given index into linear row major representation, calculates -- index into array range :: sh -> U.Array sh -- ^all the valid indices in a shape. The following equality should hold: -- map (toIndex sh) (range sh) = [:0..(size sh)-1:] inRange :: sh -> sh -> Bool -- ^Checks if a given index is in the range of an array shape. I.e., -- inRange sh ind == elem ind (range sh) zeroDim :: sh -- ^shape of an array of size zero of the particular dimensionality intersectDim :: sh -> sh -> sh -- ^shape of an array of size zero of the particular dimensionality next:: sh -> sh -> Maybe sh -- ^shape of an array of size zero of the particular dimensionality

Note that a `Shape` has to be in the type class `Elt` imported from `Data.Parallel.Array.Unboxed` so
that it can be an element of `Data.Parallel.Array.Unboxed.Array`.

The following instances are defined

instance Shape () instance Shape sh => Shape (sh :*: Int)

so we have inductively defined n-tuples of `Int` values to represent shapes. This somewhat unusual representation
is necessary to be able to inductively define operations on `Shape`. It should, however, be hidden from the library
user in favour of the common tuple representation.

The multiparameter type class `Subshape sh sh'` contains all pairs of shapes `sh` and `sh'`, for which the dimensionality of `sh'` is
less than that of `sh`.

class (Shape sh, Shape sh') => Subshape sh sh' where addDim :: sh -> sh' -> sh modDim :: sh -> sh' -> sh inject :: sh -> sh' -> sh

The method `addDim` adds the sizes of two shapes (or positions of two indices). If `sh'` is a strict subshape of
`sh`, the fields of `sh` are copied when no corresponding fields of `sh'` exist, accordingly for `modDim`

### Representation, Order of Elements, and Lifted Values

As mentioned when introducing the functions `toIndex` and `range`, the following relationship should hold:

map (toIndex sh) (range sh) = [:0..(size sh)-1:]

this means that, for example

range (() :*: 2 :*: 3) = [() :*: 0 :*: 0, () :*: 0 :*: 1, .....

A scalar value `c`is isomorphic to a zero-dimensional array

DArray () (\_ -> c)

Lifting a scalar value over a shape `dim':

Shape dim => DArray dim (\_ -> c)

## Operations on Arrays and Delayed Arrays

### Array Creation and Conversion

Strict arrays are simply defined as record containing a flat data array and shape information:

data Array dim e where Array:: { arrayShape :: dim -- ^extend of dimensions , arrayData :: U.Array e -- flat parallel array } -> Array dim e deriving Show toArray :: (U.Elt e, Shape dim) => dim -> U.Array e -> Array dim e fromArray:: (U.Elt e, Shape dim) => Array dim e -> U.Array e

Delayed arrays, in contrast, in addition to the shape, only contain a function which, given an index, yields the corresponding element.

data DArray dim e where DArray :: {dArrayShape::dim -> dArrayFn:: (dim -> e)} -> DArray dim e

Delayed arrays can be converted to and from strict arrays:

toDArray:: (U.Elt e, Array.Shape dim) => Array.Array dim e -> DArray dim e fromDArray:: (U.Elt e, Array.Shape dim) => DArray dim e -> Array dim e

the result of `toDArray` is a DArray which contains an indexing function into
an array. In general, the function `dArrayFn` can be much more complex. The function
`forceDArray` (should this be called `normalizeDArray`?) forces the evaluation `dArrayFn` on
every index of the range, and replaces `dArrayFn` by a simple indexing function into an array
of the result values.

forceDArray:: (U.Elt e, A.Shape dim) => DArray dim e -> DArray dim e

## Collection Oriented Operations

### Elementwise Application of Functions

The `map` operation takes a function over element types and applies it to every
data element of the DArray, which can have arbitrary dimensionality. Note that
it is not possible to use this function to apply an operation for example to every
row or column of a matrix. We will discuss how this can be done later on.

`map:: (U.Elt a, U.Elt b, A.Shape dim) => (a -> b) -> DArray dim a -> DArray dim b`

Similarily, `zip` and `zipWith` apply to every data element in the array as well. Both arguments
have to have the same dimensionality (which is enforced by the type system). If they have a different
shape, the result will have the intersection of both shapes. For example, zipping an array of shape
`(() :*: 4 :*: 6)` and `(() :*: 2 :*: 8)` results in an array of shape `(() :*: 2 :*: 6)`.

zipWith:: (U.Elt a, U.Elt b, U.Elt c, A.Shape dim) => (a -> b -> c) -> DArray dim a -> DArray dim b-> DArray dim c zip:: (U.Elt a, U.Elt b, A.Shape dim) => DArray dim a -> DArray dim b-> DArray dim (a :*: b)

The function `fold` collapses the values of the innermost rows of an array of at least dimensionality 1.

fold :: (U.Elt e, A.Shape dim) => (e -> e-> e) -> e -> DArray (dim :*: Int) e -> DArray dim e

Again, it's not possible to use `fold` directly to collapse an array along any other axis, but, as
we will see shortly, this can be easily done using other functions in combination with `fold`.

TODO: MISSING: description of mapStencil

### Shape Polymorphic Computations on Arrays

The library provides a range of operation where the dimensionality of the result depends on the dimensionality of the argument in a non-trivial manner, which we want to be reflected in the type system. Examples of such functions are generalised selection, which allows for extraction of subarrays of arbitrary dimension, and generalised replicate, which allows replication of an array in any dimension (or dimensions). For example, given a three dimensional matrix, we can use select to extract scalar element values, rows, columns, as well as two dimensional matrices along any of the three axes.

For selection, we can informally state the relationship between dimensionality of the argument, the selector, and the result as follows:

select:: Array dim e -> <select dim' of dim array> -> Array dim' e

Another example for such a generalised function would be a generalised
`map`, which can apply a function to all elements, all rows, all
columns, or submatrices of different orientation of a multidimensional
array.

For the former example, we need a way to express the relationship between the
shape of the argument and the shape and orientation of the result, as well as
the numerical position of the structure (i.e., first, second, third element).
In case of the generalised `map`, we don't need the numerical information, since
the operation will be applied to all elements, rows, columns etc.

To express this dependency between input and output shape and orientation,
as well as possibly a concrete position, the library provides the `Index` GADT,
which expresses a relationship between the source and the projected dimension.
It is defined as follows:

data Index a initialDim projectedDim where IndexNil :: Index a initialDim initialDim IndexAll :: (Shape init, Shape proj) => Index a init proj -> Index a (init :*: Int) (proj :*: Int) IndexFixed :: (Shape init, Shape proj) => a -> Index a init proj -> Index a (init :*: Int) proj

To refer to a specific element, the type parameter `a` is instantiated with the type `Int`, otherwise
with the unit type:

type SelectIndex = Index Int type MapIndex = Index ()

Given this definition, the type of `select` now becomes

select:: (U.Elt e, Shape dim, Shape dim') => Array dim e -> SelectIndex dim dim' -> Array dim' e

Example:

arr:: Array (() :*: Int :*: Int :*: Int) Double arr' :: () :*: Int :*: Int arr' = select arr (IndexFixed 3 (IndexAll (IndexAll IndexNil)))

We could generalise this further, to extract from any array `arr` which is at least one dimensional
the third element:

arr:: Shape dim => Array (dim :*: Int) Double arr' :: Array dim Double arr' = select arr (IndexFixed 3 IndexNil)

The index type is also used to express the type of generalised replicate

replicate:: Array dim' e -> SelectIndex dim dim' -> Array dim e

which, given an array, can be used to expand it along any dimension. For example,

simpleReplicate:: (U.Elt e, Shape dim) => Array dim e -> Int -> Array (dim :*: Int) e simpleReplicate arr n = replicate arr (IndexFixed n IndexNil)

replicates the argument array (which can of any dimensionality) `n` times and behaves
thus similarly to list replicate, whereas

elementwiseReplicate:: (U.Elt e, Shape dim) => Array (dim :*: Int) e -> Int -> Array (dim :*: Int :*: Int) e elementwiseReplicate arr n = replicate arr (IndexAll (IndexFixed n IndexNil))

replicates each element of an array `n` times (similarly to `map (replicate n)` on lists).

Even though the index type is well suited to express the relationship between the selector/multiplicator and the dimensionality of the argument and the result array, it is inconvenient to use, as the examples demonstrate. We therefore do need to add another layer to improve the usability of the library.

Note that the library provides no way to statically check the pre- and postconditions on the actual size of arguments and results. This has to be done at run time using assertions.

## Open Questions

### Do we need array comprehension on DArrays?

### Wave computations

#### Generalised Stencil Operation ===

MapStencil is currently not lifted, and doesn't run in parallel. Need to come up with a generalised version.