Difference between revisions of "APL"

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.

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)

Basic operations

-- 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  (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] = 
--  shape    = 

-- rank = shape . shape  -- Rank 1 and shape 
-- 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 x shape i ....
simpleIndex a is = chooseIndex a (disclose (reduce (outer catenate) zilde is))

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  []

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  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]]