有没有办法在Haskell中模拟线性类型？

我正在建模一个系统,该系统具有创建资源的操作以及使用该资源的其他操作.但是,给定的资源只能被使用一次 - 有没有一种方法可以保证在编译时？

具体来说,让我们说第一次操作烤蛋糕,还有另外两种操作,一种用于"选择吃"蛋糕,一种用于"选择吃蛋糕",我只能做一种或另一种.

-- This is my current "weakly typed" interface:
bake :: IO Cake
eat  :: Cake -> IO ()
keep :: Cake -> IO ()

-- This is OK
do
  brownie <- bake
  muffin <- bake
  eat brownie
  keep muffin

-- Eating and having the same cake is not OK:
do
  brownie <- bake
  eat brownie
  keep brownie -- oops! already eaten!

通过在我们使用之后在蛋糕上设置标记,很容易强制执行限制,即不在运行时保留已经吃过的蛋糕(反之亦然).但有没有办法在编译时强制执行此操作？

顺便说一句,这个问题是为了概念证明,所以我可以使用任何可以给我静电安全的黑魔法.

1> András Kovác..：

---

在Haskell中,这个的基本版本可以用由蛋糕存储索引的GADT表示(由Nat-s 列表表示):

{-# LANGUAGE
  TypeFamilies, GADTs, TypeOperators, PartialTypeSignatures,
  DataKinds, PolyKinds #-}

import GHC.TypeLits
import Data.Proxy
import GHC.Exts

-- Allocate a new cake
type family New cs where
  New '[]       = 0
  New (c ': cs) = c + 1

-- Constraint satisfiable if "c" is in "cs"
type family Elem c cs :: Constraint where
  Elem c (c ': cs)  = ()
  Elem c (c' ': cs) = Elem c cs

type family Remove c cs where
  Remove c '[]        = '[]  
  Remove c (c ': cs)  = cs
  Remove c (c' ': cs) = c' ': Remove c cs

data Bake :: [Nat] -> [Nat] -> * -> * where
  Pure :: a -> Bake cs cs a
  Bake :: (Proxy (New cs) -> Bake (New cs ': cs) cs' a) -> Bake cs cs' a
  Eat  :: Elem c cs => Proxy c -> Bake (Remove c cs) cs' a -> Bake cs cs' a
  Keep :: Elem c cs => Proxy c -> Bake cs cs' a -> Bake cs cs' a

ok :: Bake '[] _ _
ok =
  Bake $ \cake1 ->
  Bake $ \cake2 ->
  Eat cake1 $
  Keep cake2 $
  Eat cake2 $
  Pure ()

not_ok :: Bake '[] _ _
not_ok =
  Bake $ \cake1 ->
  Bake $ \cake2 ->
  Eat cake1 $
  Keep cake1 $ -- we already ate that
  Eat cake2 $
  Pure ()  

遗憾的是,我们无法从Bake操作中删除类型注释并保留要推断的类型:

foo =
  Bake $ \cake1 ->
  Bake $ \cake2 ->
  Eat cake1 $
  Pure ()

-- Error: Could not deduce (Elem (New cs0) (New cs0 + 1 : New cs0 : cs0))

显然,(Elem (New cs0) (New cs0 + 1 : New cs0 : cs0))对所有人来说都是可以满足的cs0,但GHC无法看到这一点,因为它不能决定是否New cs0不等New cs0 + 1,因为GHC不能假设任何关于灵活cs0变量的东西.

如果我们添加NoMonomorphismRestriction,foo将进行类型检查,但这会使得所有Elem约束都被推到顶部甚至是错误的程序.这仍然会阻止对不正确的术语做任何有用的事情,但这是一个相当丑陋的解决方案.

---

更一般地说,我们可以表达Bake一个索引的自由monad,它可以得到我们的do注释RebindableSyntax,并且允许对它的定义BakeF比我们之前看到的更清晰.它也可以像普通的Freemonad 一样减少样板,尽管我发现在实际代码中人们不太可能在两个不同的场合找到索引的免费monad.

{-# LANGUAGE
  TypeFamilies, GADTs, TypeOperators, PartialTypeSignatures, StandaloneDeriving,
  DataKinds, PolyKinds, NoImplicitPrelude, RebindableSyntax, DeriveFunctor #-}

import Prelude hiding (Monad(..))
import GHC.TypeLits
import Data.Proxy
import GHC.Exts

class IxFunctor f where
  imap :: (a -> b) -> f i j a -> f i j b

class IxFunctor m => IxMonad m where
  return :: a -> m i i a
  (>>=)  :: m i j a -> (a -> m j k b) -> m i k b
  fail   :: String -> m i j a

infixl 1 >>
infixl 1 >>=

(>>) :: IxMonad m => m i j a -> m j k b -> m i k b
ma >> mb = ma >>= const mb

data IxFree f i j a where
  Pure :: a -> IxFree f i i a
  Free :: f i j (IxFree f j k a) -> IxFree f i k a

liftf :: IxFunctor f => f i j a -> IxFree f i j a
liftf = Free . imap Pure

instance IxFunctor f => IxFunctor (IxFree f) where
  imap f (Pure a)  = Pure (f a)
  imap f (Free fa) = Free (imap (imap f) fa)

instance IxFunctor f => IxMonad (IxFree f) where
  return = Pure
  Pure a  >>= f = f a
  Free fa >>= f = Free (imap (>>= f) fa)
  fail = error

-- Old stuff for Bake

type family New cs where
  New '[]       = 0
  New (c ': cs) = c + 1

type family Elem c cs :: Constraint where
  Elem c (c ': cs)  = ()
  Elem c (c' ': cs) = Elem c cs

type family Remove c cs where
  Remove c '[]        = '[]  
  Remove c (c ': cs)  = cs
  Remove c (c' ': cs) = c' ': Remove c cs

-- Now the return type indices of BakeF directly express the change
-- from the old store to the new store.
data BakeF cs cs' k where
  BakeF :: (Proxy (New cs) -> k) -> BakeF cs (New cs ': cs) k
  EatF  :: Elem c cs => Proxy c -> k -> BakeF cs (Remove c cs) k
  KeepF :: Elem c cs => Proxy c -> k -> BakeF cs cs k

deriving instance Functor (BakeF cs cs')
instance IxFunctor BakeF where imap = fmap

type Bake = IxFree BakeF

bake   = liftf (BakeF id)
eat  c = liftf (EatF c ())
keep c = liftf (KeepF c ())

ok :: Bake '[] _ _
ok = do
  cake1 <- bake
  cake2 <- bake
  eat cake1
  keep cake2
  eat cake2

-- not_ok :: Bake '[] _ _
-- not_ok = do
--   cake1 <- bake
--   cake2 <- bake
--   eat cake1
--   keep cake1 -- already ate it
--   eat cake2

---

[here](https://github.com/effectfully/random-stuff/blob/master/FreeKitchen.agda)是相同的,但是带有严格正索引的自由monad而不是索引(只有形状是索引的)容器.我没有包含`keep`构造函数,因为我不满意`keep`既不改变输入也不改变输出索引.在我看来,"keep"应该从像"eat"这样的上下文中排除蛋糕,但在输出中包含蛋糕.我的[原始代码](https://github.com/effectfully/random-stuff/blob/master/Kitchen.agda)就是这样做的,但我想我们需要对'BakeF`仿函数进行双重索引.

2> gallais..：

---

Polakow在他的Haskell Symposium论文中展示了在Haskell(pdf)中嵌入完整的线性lambda演算如何做到这一点.

主要思想是使用输入和输出上下文为每个构造函数编制索引,以跟踪各个子项中消耗的资源.