Futhark.IR.Mem.LMAD

This module contains a representation of linear-memory accessor descriptors (LMAD); see work by Zhu, Hoeflinger and David.

Documentation

type Shape num = [num]

The shape of an index function.

data LMAD num

LMAD's representation consists of a general offset and for each dimension a stride, 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 \( \sigma + \{ (n_1, s_1), \ldots, (n_k, s_k) \} \), where \(n\) and \(s\) denote the shape and stride of each dimension, denotes the set of points:

\[ \{ ~ \sigma + i_1 * s_1 + \ldots + i_m * s_m ~ | ~ 0 \leq i_1 < n_1, \ldots, 0 \leq i_m < n_m ~ \} \]

Constructors

LMAD

Fields

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

Instances

Foldable LMAD

Defined in Futhark.IR.Mem.LMAD

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.LMAD

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)

Functor LMAD

Defined in Futhark.IR.Mem.LMAD

Methods

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

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

Show num => Show (LMAD num)

Defined in Futhark.IR.Mem.LMAD

Methods

showsPrec :: Int -> LMAD num -> ShowS

show :: LMAD num -> String

showList :: [LMAD num] -> ShowS

FreeIn num => FreeIn (LMAD num)

Defined in Futhark.IR.Mem.LMAD

Methods

freeIn' :: LMAD num -> FV

Substitute num => Rename (LMAD num)

Defined in Futhark.IR.Mem.LMAD

Methods

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

Substitute num => Substitute (LMAD num)

Defined in Futhark.IR.Mem.LMAD

Methods

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

Eq num => Eq (LMAD num)

Defined in Futhark.IR.Mem.LMAD

Methods

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

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

Ord num => Ord (LMAD num)

Defined in Futhark.IR.Mem.LMAD

Methods

compare :: LMAD num -> LMAD num -> Ordering

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

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

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

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

max :: LMAD num -> LMAD num -> LMAD num

min :: LMAD num -> LMAD num -> LMAD num

Pretty num => Pretty (LMAD num)

Defined in Futhark.IR.Mem.LMAD

Methods

pretty :: LMAD num -> Doc ann

prettyList :: [LMAD num] -> Doc ann

data LMADDim num

A single dimension in an LMAD.

Constructors

LMADDim

Fields

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

Instances

Show num => Show (LMADDim num)

Defined in Futhark.IR.Mem.LMAD

Methods

showsPrec :: Int -> LMADDim num -> ShowS

show :: LMADDim num -> String

showList :: [LMADDim num] -> ShowS

FreeIn num => FreeIn (LMADDim num)

Defined in Futhark.IR.Mem.LMAD

Methods

freeIn' :: LMADDim num -> FV

Eq num => Eq (LMADDim num)

Defined in Futhark.IR.Mem.LMAD

Methods

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

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

Ord num => Ord (LMADDim num)

Defined in Futhark.IR.Mem.LMAD

Methods

compare :: LMADDim num -> LMADDim num -> Ordering

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

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

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

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

max :: LMADDim num -> LMADDim num -> LMADDim num

min :: LMADDim num -> LMADDim num -> LMADDim num

data Monotonicity

The physical element ordering alongside a dimension, i.e. the sign of the stride.

Constructors

Inc

Increasing.

Dec

Decreasing.

Unknown

Unknown.

Instances

Show Monotonicity

Defined in Futhark.IR.Mem.LMAD

Methods

showsPrec :: Int -> Monotonicity -> ShowS

show :: Monotonicity -> String

showList :: [Monotonicity] -> ShowS

Eq Monotonicity

Defined in Futhark.IR.Mem.LMAD

Methods

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

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

Ord Monotonicity

Defined in Futhark.IR.Mem.LMAD

Methods

compare :: Monotonicity -> Monotonicity -> Ordering

(<) :: Monotonicity -> Monotonicity -> Bool

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

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

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

max :: Monotonicity -> Monotonicity -> Monotonicity

min :: Monotonicity -> Monotonicity -> Monotonicity

Pretty Monotonicity

Defined in Futhark.IR.Mem.LMAD

Methods

pretty :: Monotonicity -> Doc ann

prettyList :: [Monotonicity] -> Doc ann

type Permutation = [Int]

lmadShape :: LMAD num -> Shape num

Shape of an LMAD.

lmadShapeBase :: LMAD num -> Shape num

Shape of an LMAD, ignoring permutations.

substituteInLMAD :: Ord a => Map a (TPrimExp t a) -> LMAD (TPrimExp t a) -> LMAD (TPrimExp t a)

Substitute a name with a PrimExp in an LMAD.

permuteInv :: Permutation -> [a] -> [a]

permuteFwd :: Permutation -> [a] -> [a]

conservativeFlatten :: LMAD (TPrimExp Int64 VName) -> Maybe (LMAD (TPrimExp Int64 VName))

Conservatively flatten a list of LMAD dimensions

Since not all LMADs can actually be flattened, we try to overestimate the flattened array instead. This means that any "holes" in betwen dimensions will get filled out. conservativeFlatten :: (IntegralExp e, Ord e, Pretty e) => LMAD e -> LMAD e

disjoint :: [(VName, PrimExp VName)] -> Names -> LMAD (TPrimExp Int64 VName) -> LMAD (TPrimExp Int64 VName) -> Bool

Returns True if the two LMADs could be proven disjoint.

Uses some best-approximation heuristics to determine disjointness. For two 1-dimensional arrays, we can guarantee whether or not they are disjoint, but as soon as more than one dimension is involved, things get more tricky. Currently, we try to conservativelyFlatten any LMAD with more than one dimension.

disjoint2 :: scope -> asserts -> [(VName, PrimExp VName)] -> Names -> LMAD (TPrimExp Int64 VName) -> LMAD (TPrimExp Int64 VName) -> Bool

disjoint3 :: Map VName Type -> [PrimExp VName] -> [(VName, PrimExp VName)] -> [PrimExp VName] -> LMAD (TPrimExp Int64 VName) -> LMAD (TPrimExp Int64 VName) -> Bool

dynamicEqualsLMAD :: Eq num => LMAD (TPrimExp t num) -> LMAD (TPrimExp t num) -> TPrimExp Bool num

Dynamically determine if two LMAD are equal.

True if offset and constituent LMADDim are equal.

lmadPermutation :: LMAD num -> Permutation

makeRotIota

Arguments

:: IntegralExp num

=> Monotonicity

-> num

Offset

-> [num]

Shape

-> LMAD num

Generalised iota with user-specified offset and rotates.

invertMonotonicity :: Monotonicity -> Monotonicity