我正在尝试使用dfold定义的here
dfold
:: KnownNat k
=> Proxy (p :: TyFun Nat * -> *)
-> (forall l. SNat l -> a -> (p @@ l) -> p @@ (l + 1))
-> (p @@ 0)
-> Vec k a
-> p @@ k
基本上它是一个折叠,允许您在每个循环后返回一个新类型。
我试图概括在这个项目中定义的bitonicSort:https://github.com/adamwalker/clash-utils/blob/master/src/Clash/Sort.hs
我有两个函数对dfold生成的类型很重要:
bitonicSort
:: forall n a. (KnownNat n, Ord a)
=> (Vec n a -> Vec n a) -- ^ The recursive step
-> (Vec (2 * n) a -> Vec (2 * n) a) -- ^ Merge step
-> Vec (2 * n) a -- ^ Input vector
-> Vec (2 * n) a -- ^ Output vector
bitonicMerge
:: forall n a. (Ord a , KnownNat n)
=> (Vec n a -> Vec n a) -- ^ The recursive step
-> Vec (2 * n) a -- ^ Input vector
-> Vec (2 * n) a -- ^ Output vector
上述项目中使用的示例是:
bitonicSorterExample
:: forall a. (Ord a)
=> Vec 16 a -- ^ Input vector
-> Vec 16 a -- ^ Sorted output vector
bitonicSorterExample = sort16
where
sort16 = bitonicSort sort8 merge16
merge16 = bitonicMerge merge8
sort8 = bitonicSort sort4 merge8
merge8 = bitonicMerge merge4
sort4 = bitonicSort sort2 merge4
merge4 = bitonicMerge merge2
sort2 = bitonicSort id merge2
merge2 = bitonicMerge id
我继续做了一个更通用的版本。
genBitonic :: (Ord a, KnownNat n) =>
(Vec n a -> Vec n a, Vec (2 * n) a -> Vec (2 * n) a)
-> (Vec (2 * n) a -> Vec (2 * n) a, Vec (4 * n) a -> Vec (4 * n) a)
genBitonic (bSort,bMerge) = (bitonicSort bSort bMerge, bitonicMerge bMerge)
bitonicBase :: Ord a => (Vec 1 a -> Vec 1 a, Vec 2 a -> Vec 2 a)
bitonicBase = (id, bitonicMerge id)
有了这个版本,我可以快速制作新的Bitonic Sorts,如下所示:
bSort16 :: Ord a => Vec 16 a -> Vec 16 a
bSort16 = fst $ genBitonic $ genBitonic $ genBitonic $ genBitonic bitonicBase
bSort8 :: Ord a => Vec 8 a -> Vec 8 a
bSort8 = fst $ genBitonic $ genBitonic $ genBitonic bitonicBase
bSort4 :: Ord a => Vec 4 a -> Vec 4 a
bSort4 = fst $ genBitonic $ genBitonic bitonicBase
bSort2 :: Ord a => Vec 2 a -> Vec 2 a
bSort2 = fst $ genBitonic bitonicBase
每个Sort都使用指定大小的向量。
testVec16 :: Num a => Vec 16 a
testVec16 = 9 :> 2 :> 8 :> 6 :> 3 :> 7 :> 0 :> 1 :> 4 :> 5 :> 2 :> 8 :> 6 :> 3 :> 7 :> 0 :> Nil
testVec8 :: Num a => Vec 8 a
testVec8 = 9 :> 2 :> 8 :> 6 :> 3 :> 7 :> 0 :> 1 :> Nil
testVec4 :: Num a => Vec 4 a
testVec4 = 9 :> 2 :> 8 :> 6 :> Nil
testVec2 :: Num a => Vec 2 a
testVec2 = 2 :> 9 :> Nil
快速说明:
我想创建一个为给定Vec生成函数的函数。我相信使用dfold或dtfold,这是正确的路径。
我想用功能genBitonic
做折叠。
然后使用fst
来获得我需要的排序功能。
我有两种可能的设计:
一:使用合成折叠来获得一个基础的函数。
bSort8 :: Ord a => Vec 8 a -> Vec 8 a
bSort8 = fst $ genBitonic.genBitonic.genBitonic $ bitonicBase
在基地被回复之前,它会产生类似的结果
**If composition was performed three times**
foo3 ::
(Ord a, KnownNat n) =>
(Vec n a -> Vec n a, Vec (2 * n) a -> Vec (2 * n) a)
-> (Vec (2 * (2 * (2 * n))) a -> Vec (2 * (2 * (2 * n))) a,
Vec (4 * (2 * (2 * n))) a -> Vec (4 * (2 * (2 * n))) a)
二:第二个想法是使用bitonicBase作为值b来开始积累。在我应用fst
之前,这将直接导致我需要它的形式。
编辑vecAcum
只是意味着在dfold
内部建立的价值。
在dfold示例中,他们使用:>
进行折叠,:
只是列表运算符>>> :t (:>)
(:>) :: a -> Vec n a -> Vec (n + 1) a
的向量形式
genBitonic :: (Ord a, KnownNat n) =>
(Vec n a -> Vec n a, Vec (2 * n) a -> Vec (2 * n) a)
-> (Vec (2 * n) a -> Vec (2 * n) a, Vec (4 * n) a -> Vec (4 * n) a)
我想要做的是取两个函数的元组,如:
genBitonic . genBitonic
并撰写它们。所以(Vec n a -> Vec n a, Vec (2 * n) a -> Vec (2 * n) a)
-> (Vec (2 * (2 * n)) a -> Vec (2 * (2 * n)) a, Vec (4 * (2 * n)) a -> Vec (4 * (2 * n)) a)
会有类型:
bitonicBase :: Ord a => (Vec 1 a -> Vec 1 a, Vec 2 a -> Vec 2 a)
bitonicBase = (id, bitonicMerge id)
bSort4 :: Ord a => Vec 4 a -> Vec 4 a
bSort4 = fst $ genBitonic $ genBitonic bitonicBase
那么基本函数就是固化类型的东西。例如
data SplitHalf (a :: *) (f :: TyFun Nat *) :: *
type instance Apply (SplitHalf a) l = (Vec (2^l) a -> Vec (2^l) a, Vec (2 ^ (l + 1)) a -> Vec (2 ^ (l + 1)) a)
generateBitonicSortN2 :: forall k a . (Ord a, KnownNat k) => SNat k -> Vec (2^k) a -> Vec (2^k) a
generateBitonicSortN2 k = fst $ dfold (Proxy :: Proxy (SplitHalf a)) vecAcum base vecMath
where
vecMath = operationList k
vecAcum :: (KnownNat l, KnownNat gl, Ord a) => SNat l
-> (SNat gl -> SplitHalf a @@ gl -> SplitHalf a @@ (gl+1))
-> SplitHalf a @@ l
-> SplitHalf a @@ (l+1)
vecAcum l0 f acc = undefined -- (f l0) acc
base :: (Ord a) => SplitHalf a @@ 0
base = (id,id)
general :: (KnownNat l, Ord a)
=> SNat l
-> SplitHalf a @@ l
-> SplitHalf a @@ (l+1)
general _ (x,y) = (bitonicSort x y, bitonicMerge y )
operationList :: (KnownNat k, KnownNat l, Ord a)
=> SNat k
-> Vec k
(SNat l
-> SplitHalf a @@ l
-> SplitHalf a @@ (l+1))
operationList k0 = replicate k0 general
我正在使用dfold为长度为n的向量构建函数,这相当于在长度为n的向量上进行递归。
我试过了:
我尝试按照dfold下列出的示例进行操作
{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE MagicHash #-}
{-# LANGUAGE PatternSynonyms #-}
{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TemplateHaskell #-}
{-# LANGUAGE TupleSections #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE ViewPatterns #-}
{-# LANGUAGE Trustworthy #-}
我正在使用dfold源代码使用的扩展
Sort.hs:182:71: error:
* Could not deduce (KnownNat l) arising from a use of `vecAcum'
from the context: (Ord a, KnownNat k)
bound by the type signature for:
generateBitonicSortN2 :: (Ord a, KnownNat k) =>
SNat k -> Vec (2 ^ k) a -> Vec (2 ^ k) a
at Sort.hs:181:1-98
Possible fix:
add (KnownNat l) to the context of
a type expected by the context:
SNat l
-> (SNat l0
-> (Vec (2 ^ l0) a -> Vec (2 ^ l0) a,
Vec (2 ^ (l0 + 1)) a -> Vec (2 ^ (l0 + 1)) a)
-> (Vec (2 ^ (l0 + 1)) a -> Vec (2 ^ (l0 + 1)) a,
Vec (2 ^ ((l0 + 1) + 1)) a -> Vec (2 ^ ((l0 + 1) + 1)) a))
-> SplitHalf a @@ l
-> SplitHalf a @@ (l + 1)
* In the second argument of `dfold', namely `vecAcum'
In the second argument of `($)', namely
`dfold (Proxy :: Proxy (SplitHalf a)) vecAcum base vecMath'
In the expression:
fst $ dfold (Proxy :: Proxy (SplitHalf a)) vecAcum base vecMath
Sort.hs:182:84: error:
* Could not deduce (KnownNat l0) arising from a use of `vecMath'
from the context: (Ord a, KnownNat k)
bound by the type signature for:
generateBitonicSortN2 :: (Ord a, KnownNat k) =>
SNat k -> Vec (2 ^ k) a -> Vec (2 ^ k) a
at Sort.hs:181:1-98
The type variable `l0' is ambiguous
* In the fourth argument of `dfold', namely `vecMath'
In the second argument of `($)', namely
`dfold (Proxy :: Proxy (SplitHalf a)) vecAcum base vecMath'
In the expression:
fst $ dfold (Proxy :: Proxy (SplitHalf a)) vecAcum base vecMath
Failed, modules loaded: none.
错误消息:
base
**编辑**添加了更多细节。
你的base :: (Ord a) => SplitHalf a @@ 0
base = (id, bitonicMerge id)
案是错的;它应该是
Prelude
总而言之,这是一个完全可用的版本,在GHC 8.0.2上进行了测试(但它应该在基于GHC 8.0.2的CLaSH上完全相同,以operationList
导入的方式为模)。事实证明Vec n ()
的东西除了它的脊椎之外没有被使用,所以我们可以使用{-# LANGUAGE DataKinds, GADTs, KindSignatures #-}
{-# LANGUAGE Rank2Types, ScopedTypeVariables #-}
{-# LANGUAGE TypeFamilies, TypeOperators, UndecidableInstances #-}
{-# OPTIONS_GHC -fplugin GHC.TypeLits.Normalise -fplugin GHC.TypeLits.KnownNat.Solver #-}
{-# OPTIONS_GHC -fno-warn-incomplete-patterns -fno-warn-redundant-constraints #-}
{-# LANGUAGE NoImplicitPrelude #-}
import Prelude (Integer, (+), Num, ($), undefined, id, fst, Int, otherwise)
import CLaSH.Sized.Vector
import CLaSH.Promoted.Nat
import Data.Singletons
import GHC.TypeLits
import Data.Ord
type ExpVec k a = Vec (2 ^ k) a
data SplitHalf (a :: *) (f :: TyFun Nat *) :: *
type instance Apply (SplitHalf a) k = (ExpVec k a -> ExpVec k a, ExpVec (k + 1) a -> ExpVec (k + 1) a)
generateBitonicSortN2 :: forall k a . (Ord a, KnownNat k) => SNat k -> ExpVec k a -> ExpVec k a
generateBitonicSortN2 k = fst $ dfold (Proxy :: Proxy (SplitHalf a)) step base (replicate k ())
where
step :: SNat l -> () -> SplitHalf a @@ l -> SplitHalf a @@ (l+1)
step SNat _ (sort, merge) = (bitonicSort sort merge, bitonicMerge merge)
base = (id, bitonicMerge id)
代替。
*Main> generateBitonicSortN2 (snatProxy Proxy) testVec2
<9,2>
*Main> generateBitonicSortN2 (snatProxy Proxy) testVec4
<9,8,6,2>
*Main> generateBitonicSortN2 (snatProxy Proxy) testVec8
<9,8,7,6,3,2,1,0>
*Main> generateBitonicSortN2 (snatProxy Proxy) testVec16
<9,8,8,7,7,6,6,5,4,3,3,2,2,1,0,0>
*Main>
这按预期工作,例如:
{-# LANGUAGE GADTs, DataKinds, TypeFamilies, UndecidableInstances,
FlexibleInstances, FlexibleContexts, ConstraintKinds,
UndecidableSuperClasses, TypeOperators #-}
import GHC.TypeLits
import GHC.Exts (Constraint)
import Data.Proxy
data Peano = Z | S Peano
data SPeano n where
SZ :: SPeano Z
SS :: SPeano n -> SPeano (S n)
type family PowerOfTwo p where
PowerOfTwo Z = 1
PowerOfTwo (S p) = 2 * PowerOfTwo p
type family KnownPowersOfTwo p :: Constraint where
KnownPowersOfTwo Z = ()
KnownPowersOfTwo (S p) = (KnownNat (PowerOfTwo p), KnownPowersOfTwo p)
data Vec (n :: Nat) a -- abstract
type OnVec n a = Vec n a -> Vec n a
type GenBitonic n a = (OnVec n a, OnVec (2 * n) a)
genBitonic :: (Ord a, KnownNat n) => GenBitonic n a -> GenBitonic (2 * n) a
genBitonic = undefined
bitonicBase :: Ord a => GenBitonic 1 a
bitonicBase = undefined
genBitonicN :: (Ord a, KnownPowersOfTwo p) => SPeano p -> GenBitonic (PowerOfTwo p) a
genBitonicN SZ = bitonicBase
genBitonicN (SS p) = genBitonic (genBitonicN p)
我正在使用CLaSH将其合成为VHDL,因此我不能使用递归来应用t次
我不理解这句话,但除此之外:
genBitonicN
KnownNat (PowerOfTwo p)
是通过对代表幂的peano数的递归来定义的。在每个递归步骤中弹出一个新的KnownPowersOfTwo
(通过genBitonicN
类型系列)。在价值水平bSortN
是微不足道的,只是做你想要的。但是我们需要一些额外的机器来定义一个方便的type family Lit n where
Lit 0 = Z
Lit n = S (Lit (n - 1))
class IPeano n where
speano :: SPeano n
instance IPeano Z where
speano = SZ
instance IPeano n => IPeano (S n) where
speano = SS speano
class (n ~ PowerOfTwo (PowerOf n), KnownPowersOfTwo (PowerOf n)) =>
IsPowerOfTwo n where
type PowerOf n :: Peano
getPower :: SPeano (PowerOf n)
instance IsPowerOfTwo 1 where
type PowerOf 1 = Lit 0
getPower = speano
instance IsPowerOfTwo 2 where
type PowerOf 2 = Lit 1
getPower = speano
instance IsPowerOfTwo 4 where
type PowerOf 4 = Lit 2
getPower = speano
instance IsPowerOfTwo 8 where
type PowerOf 8 = Lit 3
getPower = speano
instance IsPowerOfTwo 16 where
type PowerOf 16 = Lit 4
getPower = speano
-- more powers go here
bSortN :: (IsPowerOfTwo n, Ord a) => OnVec n a
bSortN = fst $ genBitonicN getPower
:
bSort1 :: Ord a => OnVec 1 a
bSort1 = bSortN
bSort2 :: Ord a => OnVec 2 a
bSort2 = bSortN
bSort4 :: Ord a => OnVec 4 a
bSort4 = bSortN
bSort8 :: Ord a => OnVec 8 a
bSort8 = bSortN
bSort16 :: Ord a => OnVec 16 a
bSort16 = bSortN
这里有些例子:
IsPowerOfTwo
我不知道我们是否可以避免为每个2的幂定义bSortN
。
更新:这是pnatToSPeano :: IPeano (Lit n) => proxy n -> SPeano (Lit n)
pnatToSPeano _ = speano
bSortNP :: (IPeano (Lit p), KnownPowersOfTwo (Lit p), Ord a)
=> proxy p -> OnVec (PowerOfTwo (Lit p)) a
bSortNP = fst . genBitonicN . pnatToSPeano
的另一个变种:
bSort16 :: Ord a => OnVec 16 a
bSort16 = bSortNP (Proxy :: Proxy 4)
一个例子:
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