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• Higher-order type constructors and functors in Ocaml

  - by sdcvvc

  Can the following polymorphic functions let id x = x;; let compose f g x = f (g x);; let rec fix f = f (fix f);; (*laziness aside*) be written for types/type constructors or modules/functors? I tried type 'x id = Id of 'x;; type 'f 'g 'x compose = Compose of ('f ('g 'x));; type 'f fix = Fix of ('f (Fix 'f));; for types but it doesn't work. Here's a Haskell version for types: data Id x = Id x data Compose f g x = Compose (f (g x)) data Fix f = Fix (f (Fix f)) -- examples: l = Compose [Just 'a'] :: Compose [] Maybe Char type Natural = Fix Maybe -- natural numbers are fixpoint of Maybe n = Fix (Just (Fix (Just (Fix Nothing)))) :: Natural -- n is 2 -- up to isomorphism composition of identity and f is f: iso :: Compose Id f x -> f x iso (Compose (Id a)) = a

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• Continuation monad "interface"

  - by sdcvvc

  The state monad "interface" class MonadState s m where get :: m s put :: s -> m () (+ return and bind) allows to construct any possible computation with State monad without using State constructor. For example, State $ \s -> (s+1, s-1) can be written as do s <- get put (s-1) return (s+1) Similarily, I never have to use Reader constructor, because I can create that computation using ask, return and (>>=). Precisely: Reader f == ask >>= return . f. Is it the same true for continuations - is it possible to write all instances of Cont r a using callCC (the only function in MonadCont), return and bind, and never type something like Cont (\c -> ...)?

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• What is your favourite cleverly written functional code?

  - by sdcvvc

  What are your favourite short, mind-blowing snippets in functional languages? My two favourite ones are (Haskell): powerset = filterM (const [True, False]) foldl f v xs = foldr (\x g a -> g (f a x)) id xs v -- from Hutton's tutorial (I tagged the question as Haskell, but examples in all languages - including non-FP ones - are welcome as long as they are in functional spirit.)

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• Are monads Writer m and Either e categorically dual?

  - by sdcvvc

  I noticed there is a dual relation between Writer m and Either e monads. If m is a monoid, then unit :: () -> m join :: (m,m) -> m can be used to form a monad: return is composition: a -> ((),a) -> (m,a) join is composition: (m,(m,a)) -> ((m,m),a) -> (m,a) The dual of () is Void (empty type), the dual of product is coproduct. Every type e can be given "comonoid" structure: unit :: Void -> e join :: Either e e -> e in the obvious way. Now, return is composition: a -> Either Void a -> Either e a join is composition: Either e (Either e a) -> Either (Either e e) a -> Either e a and this is the Either e monad. The arrows follow exactly the same pattern. Question: Is it possible to write a single generic code that will be able to perform both as Either e and as Writer m depending on the monoid given?

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• What are uses of polymorphic kinds?

  - by sdcvvc

  Polymorphic kinds are an extension to Haskell's type system, supported by UHC, allowing data A x y = A (x y) to be typed (kinded?) as a -> (a -> *) -> *. What are they useful for?

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