Using a histomorphism (histo) from recursion-schemes I can get the a list containing only the odd indexes from an initial list:
import Data.Functor.Foldable
odds :: [a] -> [a]
odds = histo $ \case
Nil -> []
Cons h (_ :< Nil) -> [h]
Cons h (_ :< Cons _ (t :< _)) -> h:t
How can get the same thing using mhisto?
nil = Fix Nil
cons a b = Fix $ Cons a b
list = cons 1 $ cons 2 $ cons 3 $ nil
modds :: Fix (ListF a) -> [a]
modds = mhisto alg where
alg _ _ Nil = []
alg f g (Cons a b) = ?
This is it:
modds :: Fix (ListF a) -> [a]
modds = mhisto alg
where
alg _ _ Nil = []
alg odd pre (Cons a b) = a : case pre b of
Nil -> []
Cons _ b' -> odd b'
GHCi> list = cata embed [1..10] :: Fix (ListF Int)
GHCi> odds (cata embed list)
[1,3,5,7,9]
GHCi> modds list
[1,3,5,7,9]
odd folds the rest of the list, while pre digs the predecessor. Note how the availability of an y -> f y function in the Mendler algebra mirrors the introduction of Cofree in the ordinary histomorphism algebra (in which digging back can be done by reaching for the tail of the Cofree stream):
cata :: Functor f => (f c -> c) -> Fix f -> c
histo :: Functor f => (f (Cofree f c) -> c) -> Fix f -> c
mcata :: (forall y. (y -> c) -> f y -> c) -> Fix f -> c
mhisto :: (forall y. (y -> c) -> (y -> f y) -> f y -> c) -> Fix f -> c
For further reading on mcata and mhisto, see chapters 5 and 6 of Categorical programming with inductive and coinductive types, by Varmo Vene.
I need a function that does this:
>>> func (+1) [1,2,3]
[[2,2,3],[2,3,3],[2,3,4]]
My real case is more complex, but this example shows the gist of the problem. The main difference is that in reality using indexes would be infeasible. The List should be a Traversable or Foldable.
EDIT: This should be the signature of the function:
func :: Traversable t => (a -> a) -> t a -> [t a]
And closer to what I really want is the same signature to traverse but can't figure out the function I have to use, to get the desired result.
func :: (Traversable t, Applicative f) :: (a -> f a) -> t a -> f (t a)
It looks like #Benjamin Hodgson misread your question and thought you wanted f applied to a single element in each partial result. Because of this, you've ended up thinking his approach doesn't apply to your problem, but I think it does. Consider the following variation:
import Control.Monad.State
indexed :: (Traversable t) => t a -> (t (Int, a), Int)
indexed t = runState (traverse addIndex t) 0
where addIndex x = state (\k -> ((k, x), k+1))
scanMap :: (Traversable t) => (a -> a) -> t a -> [t a]
scanMap f t =
let (ti, n) = indexed (fmap (\x -> (x, f x)) t)
partial i = fmap (\(k, (x, y)) -> if k < i then y else x) ti
in map partial [1..n]
Here, indexed operates in the state monad to add an incrementing index to elements of a traversable object (and gets the length "for free", whatever that means):
> indexed ['a','b','c']
([(0,'a'),(1,'b'),(2,'c')],3)
and, again, as Ben pointed out, it could also be written using mapAccumL:
indexed = swap . mapAccumL (\k x -> (k+1, (k, x))) 0
Then, scanMap takes the traversable object, fmaps it to a similar structure of before/after pairs, uses indexed to index it, and applies a sequence of partial functions, where partial i selects "afters" for the first i elements and "befores" for the rest.
> scanMap (*2) [1,2,3]
[[2,2,3],[2,4,3],[2,4,6]]
As for generalizing this from lists to something else, I can't figure out exactly what you're trying to do with your second signature:
func :: (Traversable t, Applicative f) => (a -> f a) -> t a -> f (t a)
because if you specialize this to a list you get:
func' :: (Traversable t) => (a -> [a]) -> t a -> [t a]
and it's not at all clear what you'd want this to do here.
On lists, I'd use the following. Feel free to discard the first element, if not wanted.
> let mymap f [] = [[]] ; mymap f ys#(x:xs) = ys : map (f x:) (mymap f xs)
> mymap (+1) [1,2,3]
[[1,2,3],[2,2,3],[2,3,3],[2,3,4]]
This can also work on Foldable, of course, after one uses toList to convert the foldable to a list. One might still want a better implementation that would avoid that step, though, especially if we want to preserve the original foldable type, and not just obtain a list.
I just called it func, per your question, because I couldn't think of a better name.
import Control.Monad.State
func f t = [evalState (traverse update t) n | n <- [0..length t - 1]]
where update x = do
n <- get
let y = if n == 0 then f x else x
put (n-1)
return y
The idea is that update counts down from n, and when it reaches 0 we apply f. We keep n in the state monad so that traverse can plumb n through as you walk across the traversable.
ghci> func (+1) [1,1,1]
[[2,1,1],[1,2,1],[1,1,2]]
You could probably save a few keystrokes using mapAccumL, a HOF which captures the pattern of traversing in the state monad.
This sounds a little like a zipper without a focus; maybe something like this:
data Zippy a b = Zippy { accum :: [b] -> [b], rest :: [a] }
mapZippy :: (a -> b) -> [a] -> [Zippy a b]
mapZippy f = go id where
go a [] = []
go a (x:xs) = Zippy b xs : go b xs where
b = a . (f x :)
instance (Show a, Show b) => Show (Zippy a b) where
show (Zippy xs ys) = show (xs [], ys)
mapZippy succ [1,2,3]
-- [([2],[2,3]),([2,3],[3]),([2,3,4],[])]
(using difference lists here for efficiency's sake)
To convert to a fold looks a little like a paramorphism:
para :: (a -> [a] -> b -> b) -> b -> [a] -> b
para f b [] = b
para f b (x:xs) = f x xs (para f b xs)
mapZippy :: (a -> b) -> [a] -> [Zippy a b]
mapZippy f xs = para g (const []) xs id where
g e zs r d = Zippy nd zs : r nd where
nd = d . (f e:)
For arbitrary traversals, there's a cool time-travelling state transformer called Tardis that lets you pass state forwards and backwards:
mapZippy :: Traversable t => (a -> b) -> t a -> t (Zippy a b)
mapZippy f = flip evalTardis ([],id) . traverse g where
g x = do
modifyBackwards (x:)
modifyForwards (. (f x:))
Zippy <$> getPast <*> getFuture
I want to be able to use cata from recursion-schemes package for lists in Church encoding.
type ListC a = forall b. (a -> b -> b) -> b -> b
I used a second rank type for convenience, but I don't care. Feel free to add a newtype, use GADTs, etc. if you feel it is necessary.
The idea of Church encoding is widely known and simple:
three :: a -> a -> a -> List1 a
three a b c = \cons nil -> cons a $ cons b $ cons c nil
Basically "abstract unspecified" cons and nil are used instead of "normal" constructors. I believe everything can be encoded this way (Maybe, trees, etc.).
It's easy to show that List1 is indeed isomorphic to normal lists:
toList :: List1 a -> [a]
toList f = f (:) []
fromList :: [a] -> List1 a
fromList l = \cons nil -> foldr cons nil l
So its base functor is the same as of lists, and it should be possible to implement project for it and use the machinery from recursion-schemes.
But I couldn't, so my question is "how do I do that?". For normal lists, I can just pattern match:
decons :: [a] -> ListF a [a]
decons [] = Nil
decons (x:xs) = Cons x xs
Since I cannot pattern-match on functions, I have to use a fold to deconstruct the list. I could write a fold-based project for normal lists:
decons2 :: [a] -> ListF a [a]
decons2 = foldr f Nil
where f h Nil = Cons h []
f h (Cons hh t) = Cons h $ hh : t
However I failed to adapt it for Church-encoded lists:
-- decons3 :: ListC a -> ListF a (ListC a)
decons3 ff = ff f Nil
where f h Nil = Cons h $ \cons nil -> nil
f h (Cons hh t) = Cons h $ \cons nil -> cons hh (t cons nil)
cata has the following signature:
cata :: Recursive t => (Base t a -> a) -> t -> a
To use it with my lists, I need:
To declare the base functor type for the list using type family instance Base (ListC a) = ListF a
To implement instance Recursive (List a) where project = ...
I fail at both steps.
The newtype wrapper turned out to be the crucial step I missed. Here is the code along with a sample catamorphism from recursion-schemes.
{-# LANGUAGE LambdaCase, Rank2Types, TypeFamilies #-}
import Data.Functor.Foldable
newtype ListC a = ListC { foldListC :: forall b. (a -> b -> b) -> b -> b }
type instance Base (ListC a) = ListF a
cons :: a -> ListC a -> ListC a
cons x (ListC xs) = ListC $ \cons' nil' -> x `cons'` xs cons' nil'
nil :: ListC a
nil = ListC $ \cons' nil' -> nil'
toList :: ListC a -> [a]
toList f = foldListC f (:) []
fromList :: [a] -> ListC a
fromList l = foldr cons nil l
instance Recursive (ListC a) where
project xs = foldListC xs f Nil
where f x Nil = Cons x nil
f x (Cons tx xs) = Cons x $ tx `cons` xs
len = cata $ \case Nil -> 0
Cons _ l -> l + 1
In Haskell, I recently found the following function useful:
listCase :: (a -> [a] -> b) -> [a] -> [b]
listCase f [] = []
listCase f (x:xs) = f x xs : listCase f xs
I used it to generate sliding windows of size 3 from a list, like this:
*Main> listCase (\_ -> take 3) [1..5]
[[2,3,4],[3,4,5],[4,5],[5],[]]
Is there a more general recursion scheme which captures this pattern? More specifically, that allows you to generate a some structure of results by repeatedly breaking data into a "head" and "tail"?
What you are asking for is a comonad. This may sound scarier than monad, but is a simpler concept (YMMV).
Comonads are Functors with additional structure:
class Functor w => Comonad w where
extract :: w a -> a
duplicate :: w a -> w (w a)
extend :: (w a -> b) -> w a -> w b
(extendand duplicate can be defined in terms of each other)
and laws similar to the monad laws:
duplicate . extract = id
duplicate . fmap extract = id
duplicate . duplicate = fmap duplicate . duplicate
Specifically, the signature (a -> [a] -> b) takes non-empty Lists of type a. The usual type [a] is not an instance of a comonad, but the non-empty lists are:
data NE a = T a | a :. NE a deriving Functor
instance Comonad NE where
extract (T x) = x
extract (x :. _) = x
duplicate z#(T _) = T z
duplicate z#(_ :. xs) = z :. duplicate xs
The comonad laws allow only this instance for non-empty lists (actually a second one).
Your function then becomes
extend (take 3 . drop 1 . toList)
Where toList :: NE a -> [a] is obvious.
This is worse than the original, but extend can be written as =>> which is simpler if applied repeatedly.
For further information, you may start at What is the Comonad typeclass in Haskell?.
This looks like a special case of a (jargon here but it can help with googling) paramorphism, a generalisation of primitive recursion to all initial algebras.
Reimplementing ListCase
Let's have a look at how to reimplement your function using such a combinator. First we define the notion of paramorphism: a recursion principle where not only the result of the recursive call is available but also the entire substructure this call was performed on:
The type of paraList tells me that in the (:) case, I will have access to the head, the tail and the value of the recursive call on the tail and that I need to provide a value for the base case.
module ListCase where
paraList :: (a -> [a] -> b -> b) -- cons
-> b -- nil
-> [a] -> b -- resulting function on lists
paraList c n [] = n
paraList c n (x : xs) = c x xs $ paraList c n xs
We can now give an alternative definition of listCase:
listCase' :: (a -> [a] -> b) -> [a] -> [b]
listCase' c = paraList (\ x xs tl -> c x xs : tl) []
Considering the general case
In the general case, we are interested in building a definition of paramorphism for all data structures defined as the fixpoint of a (strictly positive) functor. We use the traditional fixpoint operator:
newtype Fix f = Fix { unFix :: f (Fix f) }
This builds an inductive structure layer by layer. The layers have an f shape which maybe better grasped by recalling the definition of List using this formalism. A layer is either Nothing (we're done!) or Just (head, tail):
newtype ListF a as = ListF { unListF :: Maybe (a, as) }
type List a = Fix (ListF a)
nil :: List a
nil = Fix $ ListF $ Nothing
cons :: a -> List a -> List a
cons = curry $ Fix . ListF .Just
Now that we have this general framework, we can define para generically for all Fix f where f is a functor:
para :: Functor f => (f (Fix f, b) -> b) -> Fix f -> b
para alg = alg . fmap (\ rec -> (rec, para alg rec)) . unFix
Of course, ListF a is a functor. Meaning we could use para to reimplement paraList and listCase.
instance Functor (ListF a) where fmap f = ListF . fmap (fmap f) . unListF
paraList' :: (a -> List a -> b -> b) -> b -> List a -> b
paraList' c n = para $ maybe n (\ (a, (as, b)) -> c a as b) . unListF
listCase'' :: (a -> List a -> b) -> List a -> List b
listCase'' c = paraList' (\ x xs tl -> cons (c x xs) tl) nil
You can implement a simple bijection toList, fromList to test it if you want. I could not be bothered to reimplement take so it's pretty ugly:
toList :: [a] -> List a
toList = foldr cons nil
fromList :: List a -> [a]
fromList = paraList' (\ x _ tl -> x : tl) []
*ListCase> fmap fromList . fromList . listCase'' (\ _ as -> toList $ take 3 $ fromList as). toList $ [1..5]
[[2,3,4],[3,4,5],[4,5],[5],[]]
I need binary combinators of the type
(a -> Bool) -> (a -> Bool) -> a -> Bool
or maybe
[a -> Bool] -> a -> Bool
(though this would just be the foldr1 of the first, and I usually only need to combine two boolean functions.)
Are these built-in?
If not, the implementation is simple:
both f g x = f x && g x
either f g x = f x || g x
or perhaps
allF fs x = foldr (\ f b -> b && f x) True fs
anyF fs x = foldr (\ f b -> b || f x) False fs
Hoogle turns up nothing, but sometimes its search doesn't generalise properly. Any idea if these are built-in? Can they be built from pieces of an existing library?
If these aren't built-in, you might suggest new names, because these names are pretty bad. In fact that's the main reason I hope that they are built-in.
Control.Monad defines an instance Monad ((->) r), so
ghci> :m Control.Monad
ghci> :t liftM2 (&&)
liftM2 (&&) :: (Monad m) => m Bool -> m Bool -> m Bool
ghci> liftM2 (&&) (5 <) (< 10) 8
True
You could do the same with Control.Applicative.liftA2.
Not to seriously suggest it, but...
ghci> :t (. flip ($)) . flip all
(. flip ($)) . flip all :: [a -> Bool] -> a -> Bool
ghci> :t (. flip ($)) . flip any
(. flip ($)) . flip any :: [a -> Bool] -> a -> Bool
It's not a builtin, but the alternative I prefer is to use type classes to generalize
the Boolean operations to predicates of any arity:
module Pred2 where
class Predicate a where
complement :: a -> a
disjoin :: a -> a -> a
conjoin :: a -> a -> a
instance Predicate Bool where
complement = not
disjoin = (||)
conjoin = (&&)
instance (Predicate b) => Predicate (a -> b) where
complement = (complement .)
disjoin f g x = f x `disjoin` g x
conjoin f g x = f x `conjoin` g x
-- examples:
ge :: Ord a => a -> a -> Bool
ge = complement (<)
pos = (>0)
nonzero = pos `disjoin` (pos . negate)
zero = complement pos `conjoin` complement (pos . negate)
I love Haskell!
I don't know builtins, but I like the names you propose.
getCoolNumbers = filter $ either even (< 42)
Alternately, one could think of an operator symbol in addition to typeclasses for alternatives.
getCoolNumbers = filter $ even <|> (< 42)