如何使用免费monads实现Reader？ [英] How do I implement Reader using free monads?

问题描述

好的，我已经知道如何使用 Reader （和 ReaderT ，未显示） a href =http://hackage.haskell.org/package/operational =noreferrer> 运营 包裹：

  { - ＃LANGUAGE GADTs，ScopedTypeVariables＃ - } 
 
导入Control.Monad.Operational 
 
数据ReaderI ra其中
 Ask :: ReaderI rr 
 
类型阅读器ra =程序（ReaderI r）a 
 
询问::读者rr 
询问= singleton询问
 
 runReader :: forall r a。阅读器r a  - > r  - > a 
 runReader = interpretWithMonad evalI 
其中evalI :: forall b。 ReaderI r b  - > （r  - > b）
 evalI Ask = id 
  

但我无法计算我的生活如何做到这一点与免费monads（我使用爱德华Kmett的 免费 包）。我得到的最接近的是这个，我的理解是作弊（（（ - >）r）已经是单子）：

  import Control.Monad.Free 
 
类型阅读器ra =免费（（ - >）r）a 
 
 ask :: Reader rr 
 ask = Free Pure 
 
 runReader :: Reader ra  - > r  - > a 
 runReader（Pure a）_ = a 
 runReader（Free k）r = runReader（kr）r 
 
  - 或者更简单明了：
 -  
  - > runReader =缩进
  

即使这不像我猜想的那样愚蠢，这不是我因为我想要的基本上是能够检查 Reader 作为数据...

解决方案

我不认为这是可以做的，除非他们有你的方式。但是，我不认为这对读者来说是独一无二的。考虑免费的monad版本的作家

  data WriterF ma = WriterF ma派生（Functor）
 
类型Writer m = Free（WriterF m）
  

显然， WriterF 与作者是同构的，但这确实表现出我们期望的简单代数的方式

  algebraWriter :: Monoid m => WriterF m（m，a） - > （m，a）
 algebraWriter（WriterF m1（m2，a））=（m1≠m2，a）
  

因此

  runWriter :: Monoid m =>作家m a  - > （m，a）
 runWriter（Pure a）=（mempty，a）
 runWriter（Free x）= algebraWriter。 fmap runWriter $ x 
  

同样，我认为免费阅读器为

 类型ReaderF r =（ - >）r 
 
类型Reader r = Free（ReaderF r）
  

我喜欢这个，因为添加它们会给你状态monad

<$ p $ （（ReaderF x）：+：（WriterF x））

runState :: State xa - > x - > （a，x）
runState（Pure a）x =（a，x）
runState（Free（Inl f））x = runState（fx）x
runState（Free（Inr WriterF xf）））_ = runState fx

请注意，您的操作解决方案可以与通过使用免费函子可以免费，就像运行时可以使用的任何东西一样



  data FreeFunctor fx = forall a。 FreeFunctor（fa）（a  - > x）
  

但是， FreeFunctor ReaderI 也同构于（ - >）。

Ok, so I have figured out how to implement Reader (and ReaderT, not shown) using the operational package:

{-# LANGUAGE GADTs, ScopedTypeVariables #-}

import Control.Monad.Operational

data ReaderI r a where
    Ask :: ReaderI r r

type Reader r a = Program (ReaderI r) a

ask :: Reader r r
ask = singleton Ask

runReader :: forall r a. Reader r a -> r -> a
runReader = interpretWithMonad evalI
    where evalI :: forall b. ReaderI r b -> (r -> b)
          evalI Ask = id

But I can't figure out for my life how to do this with free monads (I'm using Edward Kmett's free package). The closest I've gotten is this, which I understand is cheating (something about how ((->) r) is already a monad):

import Control.Monad.Free

type Reader r a = Free ((->) r) a

ask :: Reader r r
ask = Free Pure

runReader :: Reader r a -> r -> a
runReader (Pure a) _ = a
runReader (Free k) r = runReader (k r) r

-- Or, more simply and tellingly:
--
-- > runReader = retract

Even if this wasn't as dumb as I suspect it is, it's not what I want because what I want, basically, is to be able to inspect a Reader as data...

解决方案

I don't think it can be done except they way you have. But, I don't think this is unique to reader. Consider the free monad version of writer

data WriterF m a = WriterF m a deriving (Functor)

type Writer m = Free (WriterF m)

obviously, WriterF is isomorphic to writer, but this does behave the way we would expect with the simple algebra

algebraWriter :: Monoid m => WriterF m (m,a) -> (m,a)
algebraWriter (WriterF m1 (m2,a)) = (m1 <> m2,a)

thus

runWriter :: Monoid m => Writer m a -> (m,a)
runWriter (Pure a) = (mempty,a)
runWriter (Free x) = algebraWriter . fmap runWriter $ x

Similarly, I think of the Free reader as

type ReaderF r = (->) r

type Reader r = Free (ReaderF r)

I like this, because adding them gives you the state monad

type State x = Free ((ReaderF x) :+: (WriterF x))

runState :: State x a -> x -> (a,x)
runState (Pure a) x                    = (a,x)
runState (Free (Inl f)) x              = runState (f x) x
runState (Free (Inr (WriterF x f))) _  = runState f x

Note, that your operational solution could be made to work with Free by using the "free functor", as can anything that works with operational

data FreeFunctor f x = forall a. FreeFunctor (f a) (a -> x)

but, that FreeFunctor ReaderI is also isomorphic to (->).

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