96 lines
2.1 KiB
Markdown
96 lines
2.1 KiB
Markdown
# Applicative Functors
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Types: `Bool`, `Int`, `Char`, `[Char] = String`
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Type Constructors: `Maybe`, `[]`
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(type) classes: `Eq`, `Show`, `Functor`
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```haskell
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newtype Parser a = P ( String -> [a, String])
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parse :: Parser a -> String -> [(a, String)]
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parse (P p) s = p s -- s's can be cancelled from both sides
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instance Functor Parser where
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-- fmap :: (a -> b) -> Parser a -> Parser b
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fmap g pa = P (\s -> [(g x, s1) |
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(x,s1) <- parse pa s])
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```
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Applicative - motivation
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```haskell
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Functor f
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fmap0 :: a -> f a
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fmap1 :: (a -> b) -> f a -> f b
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-- cannot do this with functors ie cannot deal with multiple parameters
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fmap2 :: (a -> b -> c) -> f a -> f b -> f c
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fmap3 :: (a -> ... n) -> f a -> ... f n
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```
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`Functor f` can do `fmap1` however cannot do `fmap0` or `fmap2` etc.
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**Remember**: `a -> b -> c == a -> (b -> c)`
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For `fmap2` we can use `fmap2 :: (a -> (b -> c)) -> f a -> f (a -> b)`
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would need: `f(b -> c) -> f b -> f c`
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```haskell
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class Functor f => Applicative f where
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pure :: a -> f a
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(<*>) :: f (a -> b) -> f a -> f b
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-- <*> infix operator
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-- NOTE its f (a -> b) and not (a -> b) in fmap1
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-- fmap1 not part of the applicative class
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```
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Writing `fmap3` in an applicative functor
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```haskell
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fmap3 :: g x y z = (pure g) <*> x <*> y <*> z
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```
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##### Example Maybe
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```haskell
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instance Applicative Maybe where
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-- pure :: a -> Maybe a
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pure x = Just x
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-- (<*>) :: Maybe (a -> b) -> Maybe a -> Maybe b
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Just g <*> (Just x) = Just (g x)
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_ <*> _ = Nothing
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```
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##### Example Lists
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```haskell
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instance Applicative [] where
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-- pure :: a -> [a]
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pure x = [x]
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-- (<*>) :: [a -> b] -> [a] -> [b]
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gs <*> xs = [g x | g <- gs, x <- xs]
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```
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##### Example Parser
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```haskell
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instance Applicative Parser where
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-- pure :: a -> Parser a
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-- newtype Parser a = P ( String -> [(a, String)] )
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pure x = P (\s -> [(x,s)])
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-- <*> :: Parser (a -> b) -> Parser a -> Parser b
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pf <*> pa = P (\s -> [ (f x, s2) |
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(f, s1) <- parse pf s,
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(x, s2) <- parse pa s1)])
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```
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All parse does is apply a parser
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`parse :: Parser a -> String -> [(a, String)]`
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Where `P` is the constructor
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`parse ( P p ) = p`
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