Futhark.IR.Mem.IxFun

This module contains a representation for the index function based on linear-memory accessor descriptors; see Zhu, Hoeflinger and David work.

Documentation

data IxFun num

An index function is a mapping from a multidimensional array index space (the domain) to a one-dimensional memory index space. Essentially, it explains where the element at position [i,j,p] of some array is stored inside the flat one-dimensional array that constitutes its memory. For example, we can use this to distinguish row-major and column-major representations.

An index function is represented as a sequence of LMADs.

Constructors

IxFun

Fields

• ixfunLMADs :: NonEmpty (LMAD num)
• base :: Shape num
• ixfunContig :: Bool

  ignoring permutations, is the index function contiguous?

Instances

Functor IxFun

Defined in Futhark.IR.Mem.IxFun

Methods

fmap :: (a -> b) -> IxFun a -> IxFun b

(<$) :: a -> IxFun b -> IxFun a

Foldable IxFun

Defined in Futhark.IR.Mem.IxFun

Methods

fold :: Monoid m => IxFun m -> m

foldMap :: Monoid m => (a -> m) -> IxFun a -> m

foldMap' :: Monoid m => (a -> m) -> IxFun a -> m

foldr :: (a -> b -> b) -> b -> IxFun a -> b

foldr' :: (a -> b -> b) -> b -> IxFun a -> b

foldl :: (b -> a -> b) -> b -> IxFun a -> b

foldl' :: (b -> a -> b) -> b -> IxFun a -> b

foldr1 :: (a -> a -> a) -> IxFun a -> a

foldl1 :: (a -> a -> a) -> IxFun a -> a

toList :: IxFun a -> [a]

null :: IxFun a -> Bool

length :: IxFun a -> Int

elem :: Eq a => a -> IxFun a -> Bool

maximum :: Ord a => IxFun a -> a

minimum :: Ord a => IxFun a -> a

sum :: Num a => IxFun a -> a

product :: Num a => IxFun a -> a

Traversable IxFun

Defined in Futhark.IR.Mem.IxFun

Methods

traverse :: Applicative f => (a -> f b) -> IxFun a -> f (IxFun b)

sequenceA :: Applicative f => IxFun (f a) -> f (IxFun a)

mapM :: Monad m => (a -> m b) -> IxFun a -> m (IxFun b)

sequence :: Monad m => IxFun (m a) -> m (IxFun a)

Eq num => Eq (IxFun num)

Defined in Futhark.IR.Mem.IxFun

Methods

(==) :: IxFun num -> IxFun num -> Bool

(/=) :: IxFun num -> IxFun num -> Bool

Show num => Show (IxFun num)

Defined in Futhark.IR.Mem.IxFun

Methods

showsPrec :: Int -> IxFun num -> ShowS

show :: IxFun num -> String

showList :: [IxFun num] -> ShowS

Pretty num => Pretty (IxFun num)

Defined in Futhark.IR.Mem.IxFun

Methods

ppr :: IxFun num -> Doc

pprPrec :: Int -> IxFun num -> Doc

pprList :: [IxFun num] -> Doc

FreeIn num => FreeIn (IxFun num)

Defined in Futhark.IR.Mem.IxFun

Methods

freeIn' :: IxFun num -> FV

Substitute num => Substitute (IxFun num)

Defined in Futhark.IR.Mem.IxFun

Methods

substituteNames :: Map VName VName -> IxFun num -> IxFun num

Substitute num => Rename (IxFun num)

Defined in Futhark.IR.Mem.IxFun

Methods

rename :: IxFun num -> RenameM (IxFun num)

data LMAD num

LMAD's representation consists of a general offset and for each dimension a stride, rotate factor, number of elements (or shape), permutation, and monotonicity. Note that the permutation is not strictly necessary in that the permutation can be performed directly on LMAD dimensions, but then it is difficult to extract the permutation back from an LMAD.

LMAD algebra is closed under composition w.r.t. operators such as permute, index and slice. However, other operations, such as reshape, cannot always be represented inside the LMAD algebra.

It follows that the general representation of an index function is a list of LMADS, in which each following LMAD in the list implicitly corresponds to an irregular reshaping operation.

However, we expect that the common case is when the index function is one LMAD -- we call this the "nice" representation.

Finally, the list of LMADs is kept in an IxFun together with the shape of the original array, and a bit to indicate whether the index function is contiguous, i.e., if we instantiate all the points of the current index function, do we get a contiguous memory interval?

By definition, the LMAD denotes the set of points (simplified):

{ o + Sigma_{j=0}^{k} ((i_j+r_j) mod n_j)*s_j, forall i_j such that 0<=i_j<n_j, j=1..k }

Constructors

LMAD

Fields

• lmadOffset :: num
• lmadDims :: [LMADDim num]

Instances

Functor LMAD

Defined in Futhark.IR.Mem.IxFun

Methods

fmap :: (a -> b) -> LMAD a -> LMAD b

(<$) :: a -> LMAD b -> LMAD a

Foldable LMAD

Defined in Futhark.IR.Mem.IxFun

Methods

fold :: Monoid m => LMAD m -> m

foldMap :: Monoid m => (a -> m) -> LMAD a -> m

foldMap' :: Monoid m => (a -> m) -> LMAD a -> m

foldr :: (a -> b -> b) -> b -> LMAD a -> b

foldr' :: (a -> b -> b) -> b -> LMAD a -> b

foldl :: (b -> a -> b) -> b -> LMAD a -> b

foldl' :: (b -> a -> b) -> b -> LMAD a -> b

foldr1 :: (a -> a -> a) -> LMAD a -> a

foldl1 :: (a -> a -> a) -> LMAD a -> a

toList :: LMAD a -> [a]

null :: LMAD a -> Bool

length :: LMAD a -> Int

elem :: Eq a => a -> LMAD a -> Bool

maximum :: Ord a => LMAD a -> a

minimum :: Ord a => LMAD a -> a

sum :: Num a => LMAD a -> a

product :: Num a => LMAD a -> a

Traversable LMAD

Defined in Futhark.IR.Mem.IxFun

Methods

traverse :: Applicative f => (a -> f b) -> LMAD a -> f (LMAD b)

sequenceA :: Applicative f => LMAD (f a) -> f (LMAD a)

mapM :: Monad m => (a -> m b) -> LMAD a -> m (LMAD b)

sequence :: Monad m => LMAD (m a) -> m (LMAD a)

Eq num => Eq (LMAD num)

Defined in Futhark.IR.Mem.IxFun

Methods

(==) :: LMAD num -> LMAD num -> Bool

(/=) :: LMAD num -> LMAD num -> Bool

Show num => Show (LMAD num)

Defined in Futhark.IR.Mem.IxFun

Methods

showsPrec :: Int -> LMAD num -> ShowS

show :: LMAD num -> String

showList :: [LMAD num] -> ShowS

Pretty num => Pretty (LMAD num)

Defined in Futhark.IR.Mem.IxFun

Methods

ppr :: LMAD num -> Doc

pprPrec :: Int -> LMAD num -> Doc

pprList :: [LMAD num] -> Doc

FreeIn num => FreeIn (LMAD num)

Defined in Futhark.IR.Mem.IxFun

Methods

freeIn' :: LMAD num -> FV

Substitute num => Substitute (LMAD num)

Defined in Futhark.IR.Mem.IxFun

Methods

substituteNames :: Map VName VName -> LMAD num -> LMAD num

Substitute num => Rename (LMAD num)

Defined in Futhark.IR.Mem.IxFun

Methods

rename :: LMAD num -> RenameM (LMAD num)

data LMADDim num

Constructors

LMADDim

Fields

• ldStride :: num
• ldRotate :: num
• ldShape :: num
• ldPerm :: Int
• ldMon :: Monotonicity

Instances

Eq num => Eq (LMADDim num)

Defined in Futhark.IR.Mem.IxFun

Methods

(==) :: LMADDim num -> LMADDim num -> Bool

(/=) :: LMADDim num -> LMADDim num -> Bool

Show num => Show (LMADDim num)

Defined in Futhark.IR.Mem.IxFun

Methods

showsPrec :: Int -> LMADDim num -> ShowS

show :: LMADDim num -> String

showList :: [LMADDim num] -> ShowS

data Monotonicity

Constructors

Inc

Dec

Unknown

monotonously increasing, decreasing or unknown

Instances

Eq Monotonicity

Defined in Futhark.IR.Mem.IxFun

Methods

(==) :: Monotonicity -> Monotonicity -> Bool

(/=) :: Monotonicity -> Monotonicity -> Bool

Show Monotonicity

Defined in Futhark.IR.Mem.IxFun

Methods

showsPrec :: Int -> Monotonicity -> ShowS

show :: Monotonicity -> String

showList :: [Monotonicity] -> ShowS

Pretty Monotonicity

Defined in Futhark.IR.Mem.IxFun

Methods

ppr :: Monotonicity -> Doc

pprPrec :: Int -> Monotonicity -> Doc

pprList :: [Monotonicity] -> Doc

index :: (IntegralExp num, Eq num) => IxFun num -> Indices num -> num

Compute the flat memory index for a complete set inds of array indices and a certain element size elem_size.

iota :: IntegralExp num => Shape num -> IxFun num

iota.

iotaOffset :: IntegralExp num => num -> Shape num -> IxFun num

iota with offset.

permute :: IntegralExp num => IxFun num -> Permutation -> IxFun num

Permute dimensions.

rotate :: (Eq num, IntegralExp num) => IxFun num -> Indices num -> IxFun num

Rotate an index function.

reshape :: (Eq num, IntegralExp num) => IxFun num -> ShapeChange num -> IxFun num

Reshape an index function.

slice :: (Eq num, IntegralExp num) => IxFun num -> Slice num -> IxFun num

Slice an index function.

rebase :: (Eq num, IntegralExp num) => IxFun num -> IxFun num -> IxFun num

Rebase an index function on top of a new base.

shape :: (Eq num, IntegralExp num) => IxFun num -> Shape num

Shape of an index function.

rank :: IntegralExp num => IxFun num -> Int

The number of dimensions in the domain of the input function.

linearWithOffset :: (Eq num, IntegralExp num) => IxFun num -> num -> Maybe num

If the memory support of the index function is contiguous and row-major (i.e., no transpositions, repetitions, rotates, etc.), then this should return the offset from which the memory-support of this index function starts.

rearrangeWithOffset :: (Eq num, IntegralExp num) => IxFun num -> num -> Maybe (num, [(Int, num)])

Similar restrictions to linearWithOffset except for transpositions, which are returned together with the offset.

isDirect :: (Eq num, IntegralExp num) => IxFun num -> Bool

Is this is a row-major array?

isLinear :: (Eq num, IntegralExp num) => IxFun num -> Bool

Is this a row-major array starting at offset zero?

substituteInIxFun :: Ord a => Map a (TPrimExp t a) -> IxFun (TPrimExp t a) -> IxFun (TPrimExp t a)

Substitute a name with a PrimExp in an index function.

leastGeneralGeneralization :: Eq v => IxFun (PrimExp v) -> IxFun (PrimExp v) -> Maybe (IxFun (PrimExp (Ext v)), [(PrimExp v, PrimExp v)])

Generalization (anti-unification)

Anti-unification of two index functions is supported under the following conditions: 0. Both index functions are represented by ONE lmad (assumed common case!) 1. The support array of the two indexfuns have the same dimensionality (we can relax this condition if we use a 1D support, as we probably should!) 2. The contiguous property and the per-dimension monotonicity are the same (otherwise we might loose important information; this can be relaxed!) 3. Most importantly, both index functions correspond to the same permutation (since the permutation is represented by INTs, this restriction cannot be relaxed, unless we move to a gated-LMAD representation!)

existentialize :: (IntExp t, Eq v, Pretty v) => IxFun (TPrimExp t v) -> State [TPrimExp t v] (Maybe (IxFun (TPrimExp t (Ext v))))

closeEnough :: IxFun num -> IxFun num -> Bool

When comparing index functions as part of the type check in KernelsMem, we may run into problems caused by the simplifier. As index functions can be generalized over if-then-else expressions, the simplifier might hoist some of the code from inside the if-then-else (computing the offset of an array, for instance), but now the type checker cannot verify that the generalized index function is valid, because some of the existentials are computed somewhere else. To Work around this, we've had to relax the KernelsMem type-checker a bit, specifically, we've introduced this function to verify whether two index functions are "close enough" that we can assume that they match. We use this instead of `ixfun1 == ixfun2` and hope that it's good enough.