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monadplus [2014/08/24 02:46]
nikolaj created
monadplus [2014/08/24 02:47] (current)
nikolaj
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 ==== Haskell class ==== ==== Haskell class ====
 <​code/​Haskell> ​ <​code/​Haskell> ​
-    class Monad m where +>
-        return :: a -m a +
-        (>>​=) ​ :: m a -> (a -> m b) -> m b+
 </​code>​ </​code>​
  
 ==== Discussion ==== ==== Discussion ====
 === Postulates === === Postulates ===
-Right unit, left unit and associativity:​ 
- 
-^ m >>= return $\ \leftrightsquigarrow\ $ m ^ 
-^ return x >>= f $\ \leftrightsquigarrow\ $ f x ^ 
-^ (m >>= f) >>= g $\ \leftrightsquigarrow\ $ m >>= (\x -> f x >>= g) ^ 
- 
->In reference to [[Counit-unit adjunction]]:​ 
-> 
->"​F(f)"​ the progrmaming way: If fmap is the function mapping for a functor "​F"​ for which we define >==, note that "(fmap f) mx" will be written as "mx >>= return . f". But >>= is a strong gadget, in that, of course, its second argument doesn'​t have to be of the form "​return 'some function'"​. 
-> 
->In Haskell, functions of several variables are implemented using lambda expressions and currying - here's how this interacts with >==: First note that the above is the same as "mx >>= \x->​(return (f x))". If f is a function of two arguments x,y, then for arguments mx,my, the "​lifted"​ function is "mx >>= \x->(my >>=\y ->​(return (f x y)))". Syntactic sugar: Changing the symbols in between this expression gives Haskells do-noation "do {x <- mx; y <- my; return (f x y)}"​. ​ 
-> 
->​liftM2 ​ f mx my = mx >>= \x->(my >>= \y ->​(return (f x y))) 
->​liftM2 ​ f mg my = mg >>= \g->(my >>= \y ->​(return (f g y))) 
->liftM2 id mg my = mg >>= \g->(my >>= \y ->​(return (g y))) 
-><​*>​ mg my = mg >>= \g->(my >>= \y ->​(return (g y))) 
-> 
->Let $n\in\{0,​1\}$. Among other things, given a function of type $a\to m^n\ b$, a monad enables you to get a function of type $m\ a\to m\ b$.  
->​Consider a function space $A=a\to b$ and fix a term $p : m\ a$. Construct the higher order function $G:A\to m\ b$, which takes any function $g:A$ (for which $n=0$) and, using the monad, applies it to $p$. Next use the monad a second time, now on $G$ (which has $n=1$), to get a function from $m\ A$ to $m\ b$. Lambda abstraction over $p$ and flipping the arguments gives a function $<*> : m\ (a\to b)\to m\ a\to m\ b$. In this way, we've successfully lifted the apply function. 
->Given '​return',​ then '​join'​ (or '>>​='​) induces '<​*>'​. The converse isn't true and so the Applicative concept lies between Functor and Monad. 
-> 
->​Abstraction over $ma$ and flipping arguments gives a map $<*> : m (a->b)$ 
-> 
->Monads sometimes have a "zero element"​ of monadic, acting like the multiplicative zero for bind. "​Nothing"​ and the empty list "​[]"​ are examples. Haskells list comprehension works by returning the empty list for elements x where a predicate returns false. 
  
 === Associated methods === === Associated methods ===
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 ==== Parents ==== ==== Parents ====
 === Subset of === === Subset of ===
-[[Applicative]]+[[Monad . Haskell]]
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