APL

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An APL library for Haskell

APL is an array language with a highly-functional flavour, and a rich set of carefully-thought-out array operations. It would be interesting to build a Haskell library that offered a Haskell rendering of APL's array algebra, possibly (though not definitely) bound to a mature implementation.

This page collects thoughts loosely based around that idea.

APL arrays in Haskell

data Array elt -- Forces uniform element types

  • Need rank-zero arrays
  • In a rank-3 arrary, any of the axes can have zero length

   e.g. (3 x 0 x 4) array  /= (0 x 0 x 4) array

  • Prototypical item? Leave open for now. A minefield.

Major question: should the rank be visible in the type at all? Plan A: data Array elt Plan (Succ A): data Array rank elt

The main API, using Plan A

data Array a  -- Rectangular!

type Scalar a = Array a   -- Rank 0, length 1
type Vector a = Array a   -- Rank 1!
type Matrix a = Array a   -- Rank 2!
type Axis = Nat

-- Arrays can be added, multiplied, etc
class Num a where
  (+) :: Num a => a -> a -> 

instance Num a => Num (Array a) where ..

-- All indexing is 0-based (Dijkstra)
</hasekll>

Basic operations
<haskell>
-- A rank-0 array contains one item
zilde   :: Vector a       -- Empty vector, rank 1
enclose :: a -> Scalar a  -- Returns a rank 0 array, of
                          -- depth one greater than input

encloseA :: Nat             -- r
         -> Array a         -- Rank n
         -> Array (Array a) -- Outer rank r, inner rank n-r
-- encloseA 2 (Array [3,1,4]) : Array [3] (Array [1,4])
-- enclose a = encloseA 0  (when a is an array)

disclose :: Scalar a -> Array a
discloseA :: Array (Array a) -> Array a

iota :: Nat -> Array Nat  -- iota n = [0 1 2 3 ... n-1]

ravel :: Array a -> Vector a  -- Flattens an array of arbitrary rank
                              -- (including zero!) in row-major order
                              -- Does not change depth

reshape :: Vector Nat    -- s: the shape
        -> Array a       -- Arbitrary shape    
        -> Array a       -- Shape of result = s
-- NB: ravel a = product (shape a) `reshape` a

product  :: Num a => a -> Array a -> Array a
productA :: Num a => Axis -> a -> Array a -> Array a
  -- Result has rank one smaller than input

shape :: Array a -> Vector Nat
--  shape [[1 3] [5 2] [3 9]] = [3 2]
--  shape [3 2] = [2] 
--  shape [2]   = [1] 

-- rank = shape . shape  -- Rank 1 and shape [1] 
-- Or we could have rank :: Array a -> Nat

Swapping rank and depth

rankOperator :: 
-- This is in J and it is somehow lovely in a way 
-- that ordinary mortals cannot understand
-- rankOp 1 (+) A B

-- Swapping rank and depth
-- Array [2,4,5,3] Float 
--  --> Array [2,5] (Array [4,3] Float)
-- And the reverse!

rankOp rank-spec f = reshape rank-spec . f . unreshape rank-spec


transpose :: Vector Nat    -- Permutation of (iota (rank arg))
          -> Array a       -- Arg
          -> Array a
-- Permutes the axes

More operations

class Item a wher
  depth :: a -> Nat

instance Item Float where
  depth _ = 0
instance Item a => Item (Array a) where
  depth _ = 1 + depth (undefined :: a)


catenate :: Array a -> Array a -> Array a
-- Concatenates on last axis (or first in J)
-- Checks for shape compatibility
catenateA :: Axis -> Array a -> Array a -> Array a
--  You can specify the axis

-- Swapping rank with depth
flatten :: Vector (Vector a) -> Array a

vector :: [a] -> Vector a

each :: (a -> b) -> Array a -> Array b

simpleIndex   -- m [a;b]
  :: Array a            -- a: The array to index
  -> Vector (Array Nat) -- i: Outer array is a tuple
                        --    of length = rank a
  -> Array a            -- shape result = shape i[0] x shape i[1] ....
simpleIndex a is = chooseIndex a (disclose (reduce (outer catenate) zilde is)) 
    -- We aren't quite sure about this definition

reduce :: (Array a -> Array a -> Array a) -> Array a
                    -- All rank one smaller than input
       -> Array a   -- Input
       -> Array a   -- Rank one smaller than input
                    --   except that rank 0 input gives identity

reduceA :: Axis -> (a -> a -> a) -> a
        -> Array a
        -> Array a   -- Rank one smaller than ieput


outer :: (a -> b -> c)        -- Function argument
      -> Array a -> Array b   -- Two array arguments, arg1, arg2
      -> Array c              -- Shape result = shape arg1 ++ shape arg2

-- simpleIndex a (vector [enclose 3, enclose 4]) :: Array a  (rank 0)
--  mat [3 ; 4]

chooseIndex   -- m [b]
   :: Array a            -- a: The array to index
   -> Array (Vector Nat) -- i: Inner arrays are the index tuples
                         --    of length = rank a
   -> Array a            -- shape result = shape i

Various monomorphic pick operators

pick1 :: Vector (Matrix (Vector a))
     -> Array (Nat, (Nat,Nat), Nat)
     -> Array a
 
pick2 :: Vector (Matrix a)
      -> Array (Nat, (Nat,Nat))
      -> Array a
 
-- Just an instance of pick2
pick2 :: Vector (Matrix (Vector a))
      -> Array (Nat, (Nat,Nat))
      -> Array (Vector a)

Reference implementation

This implementation of the above API is intended to give its semantics. It is not intended to run fast!

data Array a = Arr Shape [a]

type Shape = [Nat]
-- Invariant: Arr s xs: product s = length xs

ravel :: Array a -> Vector a
ravel (Arr s a) = Arr [product s] a

zilde :: Vector a       -- Empty vector, rank 1
zilde = Arr [0] []

enclose :: a -> Scalar a  -- Returns a rank 0 array, of
                          -- depth one greater than input
enclose x = Arr [] [x]

disclose :: Scalar a -> a   
disclose (Arr [] [x]) = x

discloseA (Arr outer items)
  = Arr (outer ++ inner) [ i | Arr _ is <- items, i <- is ] 
  where
    (Arr inner _ : _) = items


transpose :: Vector Nat    -- Permutation of (iota (rank arg))
          -> Array a       -- Arg
          -> Array a
transpose (Arr [n] perm) (Arr shape items)
  = assert (n == length shape ) $
    Arr (permute perm shape) 
        (scramble ... items)

-- Property: disclose (enclose x) == x

shape :: Array a -> Vector Nat
shape (Arr s a) = Arr [1] s

reshape :: Vector Nat    -- s: the shape
        -> Array a       -- Arbitrary shape    
        -> Array a       -- Shape of result = s
reshape (Arr [n] s) (Arr s' elts)
  | null elts = error "Reshape on empty array"
  | otherwise
  = Arr s (take (product s) (cycle elts))

each :: (a -> b) -> Array a -> Array b
each f (Arr s xs) = Arr s (map f xs)

reduce :: (Array a -> Array a -> Array a) -> Array a
                      -- All rank one smaller than input
         -> Array a   -- Input
         -> Array a   -- Rank one smaller than input
                      --   except that rank 0 input gives identity
reduce k z (Arr [] [item])
  = Arr [] [item]   -- Identity on rank 0
reduce k z (Arr (s:ss) items)
  = foldr k z (chop ss items)

encloseA :: Nat -> Array a -> Array a
encloseA n (Arr shape items)
  = Arr outer_shape (chop inner_shape items)
  where
    (outer_shape, inner_shape) = splitAt n shape

chop :: Shape -> [item] -> [Array item]
chop s [] = []
chop s is = Arr s i : chop s is'
          where
            (i,is') = splitAt (product s) is

A couple of examples

 x = 0 1 2  : Array Float   = Arr [2,3] [0,1,2,3,4,5]
     3 4 5   
 enclose x :: Array (Array Float)
   = Arr [] [Arr [2,3] [0,1,2,3,4,5]]