I ended up figuring it out. See the video and slides of a talk I gave:
slides/pdf
video
Original question:
In my effort to understand generic recursion schemes (i.e., that use Fix) I have found it useful to write list-only versions of the various schemes. It makes it much easier to understand the actual schemes (without the additional overhead of the Fix stuff).
However, I have not yet figured out how to define list-only versions of zygo and futu.
Here are my specialised definitions so far:
cataL :: (a -> b -> b) -> b -> [a] -> b
cataL f b (a : as) = f a (cataL f b as)
cataL _ b [] = b
paraL :: (a -> [a] -> b -> b) -> b -> [a] -> b
paraL f b (a : as) = f a as (paraL f b as)
paraL _ b [] = b
-- TODO: histo
-- DONE: zygo (see below)
anaL :: (b -> (a, b)) -> b -> [a]
anaL f b = let (a, b') = f b in a : anaL f b'
anaL' :: (b -> Maybe (a, b)) -> b -> [a]
anaL' f b = case f b of
Just (a, b') -> a : anaL' f b'
Nothing -> []
apoL :: ([b] -> Maybe (a, Either [b] [a])) -> [b] -> [a]
apoL f b = case f b of
Nothing -> []
Just (x, Left c) -> x : apoL f c
Just (x, Right e) -> x : e
-- DONE: futu (see below)
hyloL :: (a -> c -> c) -> c -> (b -> Maybe (a, b)) -> b -> c
hyloL f z g = cataL f z . anaL' g
hyloL' :: (a -> c -> c) -> c -> (c -> Maybe (a, c)) -> c
hyloL' f z g = case g z of
Nothing -> z
Just (x,z') -> f x (hyloL' f z' g)
How do you define histo, zygo and futu for lists?
Zygomorphism is the high-falutin' mathsy name we give to folds built from two semi-mutually recursive functions. I'll give an example.
Imagine a function pm :: [Int] -> Int (for plus-minus) which intersperses + and - alternately through a list of numbers, such that pm [v,w,x,y,z] = v - (w + (x - (y + z))). You can write it out using primitive recursion:
lengthEven :: [a] -> Bool
lengthEven = even . length
pm0 [] = 0
pm0 (x:xs) = if lengthEven xs
then x - pm0 xs
else x + pm0 xs
Clearly pm0 is not compositional - you need to inspect the length of the whole list at each position to determine whether you're adding or subtracting. Paramorphism models primitive recursion of this sort, when the folding function needs to traverse the whole subtree at each iteration of the fold. So we can at least rewrite the code to conform to an established pattern.
paraL :: (a -> [a] -> b -> b) -> b -> [a] -> b
paraL f z [] = z
paraL f z (x:xs) = f x xs (paraL f z xs)
pm1 = paraL (\x xs acc -> if lengthEven xs then x - acc else x + acc) 0
But this is inefficient. lengthEven traverses the whole list at each iteration of the paramorphism resulting in an O(n2) algorithm.
We can make progress by noting that both lengthEven and para can be expressed as a catamorphism with foldr...
cataL = foldr
lengthEven' = cataL (\_ p -> not p) True
paraL' f z = snd . cataL (\x (xs, acc) -> (x:xs, f x xs acc)) ([], z)
... which suggests that we may be able to fuse the two operations into a single pass over the list.
pm2 = snd . cataL (\x (isEven, total) -> (not isEven, if isEven
then x - total
else x + total)) (True, 0)
We had a fold which depended on the result of another fold, and we were able to fuse them into one traversal of the list. Zygomorphism captures exactly this pattern.
zygoL :: (a -> b -> b) -> -- a folding function
(a -> b -> c -> c) -> -- a folding function which depends on the result of the other fold
b -> c -> -- zeroes for the two folds
[a] -> c
zygoL f g z e = snd . cataL (\x (p, q) -> (f x p, g x p q)) (z, e)
On each iteration of the fold, f sees its answer from the last iteration as in a catamorphism, but g gets to see both functions' answers. g entangles itself with f.
We'll write pm as a zygomorphism by using the first folding function to count whether the list is even or odd in length and the second one to calculate the total.
pm3 = zygoL (\_ p -> not p) (\x isEven total -> if isEven
then x - total
else x + total) True 0
This is classic functional programming style. We have a higher order function doing the heavy lifting of consuming the list; all we had to do was plug in the logic to aggregate results. The construction evidently terminates (you need only prove termination for foldr), and it's more efficient than the original hand-written version to boot.
Aside: #AlexR points out in the comments that zygomorphism has a big sister called mutumorphism, which captures mutual recursion in all
its glory. mutu generalises zygo in that both the folding
functions are allowed to inspect the other's result from the previous
iteration.
mutuL :: (a -> b -> c -> b) ->
(a -> b -> c -> c) ->
b -> c ->
[a] -> c
mutuL f g z e = snd . cataL (\x (p, q) -> (f x p q, g x p q)) (z, e)
You recover zygo from mutu simply by ignoring the extra argument.
zygoL f = mutuL (\x p q -> f x p)
Of course, all of these folding patterns generalise from lists to the fixed point of an arbitrary functor:
newtype Fix f = Fix { unFix :: f (Fix f) }
cata :: Functor f => (f a -> a) -> Fix f -> a
cata f = f . fmap (cata f) . unFix
para :: Functor f => (f (Fix f, a) -> a) -> Fix f -> a
para f = snd . cata (\x -> (Fix $ fmap fst x, f x))
zygo :: Functor f => (f b -> b) -> (f (b, a) -> a) -> Fix f -> a
zygo f g = snd . cata (\x -> (f $ fmap fst x, g x))
mutu :: Functor f => (f (b, a) -> b) -> (f (b, a) -> a) -> Fix f -> a
mutu f g = snd . cata (\x -> (f x, g x))
Compare the definition of zygo with that of zygoL. Also note that zygo Fix = para, and that the latter three folds can be implemented in terms of cata. In foldology everything is related to everything else.
You can recover the list version from the generalised version.
data ListF a r = Nil_ | Cons_ a r deriving Functor
type List a = Fix (ListF a)
zygoL' :: (a -> b -> b) -> (a -> b -> c -> c) -> b -> c -> List a -> c
zygoL' f g z e = zygo k l
where k Nil_ = z
k (Cons_ x y) = f x y
l Nil_ = e
l (Cons_ x (y, z)) = g x y z
pm4 = zygoL' (\_ p -> not p) (\x isEven total -> if isEven
then x - total
else x + total) True 0
Histomorphism models dynamic programming, the technique of tabulating the results of previous subcomputations. (It's sometimes called course-of-value induction.) In a histomorphism, the folding function has access to a table of the results of earlier iterations of the fold. Compare this with the catamorphism, where the folding function can only see the result of the last iteration. The histomorphism has the benefit of hindsight - you can see all of history.
Here's the idea. As we consume the input list, the folding algebra will output a sequence of bs. histo will jot down each b as it emerges, attaching it to the table of results. The number of items in the history is equal to the number of list layers you've processed - by the time you've torn down the whole list, the history of your operation will have a length equal to that of the list.
This is what the history of iterating a list(ory) looks like:
data History a b = Ancient b | Age a b (History a b)
History is a list of pairs of things and results, with an extra result at the end corresponding to the []-thing. We'll pair up each layer of the input list with its corresponding result.
cataL = foldr
history :: (a -> History a b -> b) -> b -> [a] -> History a b
history f z = cataL (\x h -> Age x (f x h) h) (Ancient z)
Once you've folded up the whole list from right to left, your final result will be at the top of the stack.
headH :: History a b -> b
headH (Ancient x) = x
headH (Age _ x _) = x
histoL :: (a -> History a b -> b) -> b -> [a] -> b
histoL f z = headH . history f z
(It happens that History a is a comonad, but headH (née extract) is all we need to define histoL.)
History labels each layer of the input list with its corresponding result. The cofree comonad captures the pattern of labelling each layer of an arbitrary structure.
data Cofree f a = Cofree { headC :: a, tailC :: f (Cofree f a) }
(I came up with History by plugging ListF into Cofree and simplifying.)
Compare this with the free monad,
data Free f a = Free (f (Free f a))
| Return a
Free is a coproduct type; Cofree is a product type. Free layers up a lasagne of fs, with values a at the bottom of the lasagne. Cofree layers up the lasagne with values a at each layer. Free monads are generalised externally-labelled trees; cofree comonads are generalised internally-labelled trees.
With Cofree in hand, we can generalise from lists to the fixpoint of an arbitrary functor,
newtype Fix f = Fix { unFix :: f (Fix f) }
cata :: Functor f => (f b -> b) -> Fix f -> b
cata f = f . fmap (cata f) . unFix
histo :: Functor f => (f (Cofree f b) -> b) -> Fix f -> b
histo f = headC . cata (\x -> Cofree (f x) x)
and once more recover the list version.
data ListF a r = Nil_ | Cons_ a r deriving Functor
type List a = Fix (ListF a)
type History' a b = Cofree (ListF a) b
histoL' :: (a -> History' a b -> b) -> b -> List a -> b
histoL' f z = histo g
where g Nil_ = z
g (Cons_ x h) = f x h
Aside: histo is the dual of futu. Look at their types.
histo :: Functor f => (f (Cofree f a) -> a) -> (Fix f -> a)
futu :: Functor f => (a -> f (Free f a)) -> (a -> Fix f)
futu is histo with the arrows flipped and with Free replaced by
Cofree. Histomorphisms see the past; futumorphisms predict the future.
And much like cata f . ana g can be fused into a hylomorphism,
histo f . futu g can be fused into a
chronomorphism.
Even if you skip the mathsy parts, this paper by Hinze and Wu features a good, example-driven tutorial on histomorphisms and their usage.
Since no one else has answered for futu yet, I'll try to stumble my way through. I'm going to use ListF a b = Base [a] = ConsF a b | NilF
Taking the type in recursion-schemes: futu :: Unfoldable t => (a -> Base t (Free (Base t) a)) -> a -> t.
I'm going to ignore the Unfoldable constraint and substitute [b] in for t.
(a -> Base [b] (Free (Base [b]) a)) -> a -> [b]
(a -> ListF b (Free (ListF b) a)) -> a -> [b]
Free (ListF b) a) is a list, possibly with an a-typed hole at the end. This means that it's isomorphic to ([b], Maybe a). So now we have:
(a -> ListF b ([b], Maybe a)) -> a -> [b]
Eliminating the last ListF, noticing that ListF a b is isomorphic to Maybe (a, b):
(a -> Maybe (b, ([b], Maybe a))) -> a -> [b]
Now, I'm pretty sure that playing type-tetris leads to the only sensible implementation:
futuL f x = case f x of
Nothing -> []
Just (y, (ys, mz)) -> y : (ys ++ fz)
where fz = case mz of
Nothing -> []
Just z -> futuL f z
Summarizing the resulting function, futuL takes a seed value and a function which may produce at least one result, and possibly a new seed value if it produced a result.
At first I thought this was equivalent to
notFutuL :: (a -> ([b], Maybe a)) -> a -> [b]
notFutuL f x = case f x of
(ys, mx) -> ys ++ case mx of
Nothing -> []
Just x' -> notFutuL f x'
And in practice, perhaps it is, more or less, but the one significant difference is that the real futu guarantees productivity (i.e. if f always returns, you will never be stuck waiting forever for the next list element).
Related
I'm currently doing a course in Haskell, and I have a lot of difficulty understanding the types of functions, particularly when there's function application or lambda expressions. Say for instance the following:
f = (\x -> \y -> \z -> [x (y z), y z])
or
g = \x -> \y -> \z -> x.y.z
I can sort of make some assumptions about the fact that x and y are functions, but I don't have a concrete method for figuring out the types of these functions.
Similarly for the following:
h = foldr (&&)
I try to guess and then check via :t in the interpreter, but I'm usually off by quite a bit.
Is there any particular method I can use to find the types of such functions?
You start by assigning type variables to the inputs and the result
f = (\x -> \y -> \z -> [x (y z), y z])
and conclude
f :: a -> b -> c -> d -- (A0)
-- or even (f is not needed)
\x -> \y -> \z -> [x (y z), y z] :: a -> b -> c -> d
that is
x :: a -- (1)
y :: b -- (2)
z :: c -- (3)
[x (y z), y z] :: d -- (4)
You can continue with (4) and conclude
that the type d is a list of d1s, i.e. d ~ [d1] (5)
f :: a -> b -> c -> [d1] -- (A1)
and that the values of the list are of type d1, i.e.
x (y z) :: d1 -- (6)
y z :: d1 -- (7)
From (6) you learn that
x :: e -> d1 -- (8)
y z :: e -- (9)
(1) and (8) unify, i.e. a ~ (e -> d1) and
f :: (e -> d1) -> b -> c -> [d1] -- (A2)
You play this game until you get bored and use GHCi to arrive at
f :: (d1 -> d1) -> (f -> d1) -> f -> [d1] -- (A3)
-- and renaming
f :: (a -> a) -> (b -> a) -> b -> [a] -- (A4)
If you want to learn more and read a paper you can start with Principal type-schemes for functional programs.
Prelude> :t h
h :: Foldable t => Bool -> t Bool -> Bool
Prelude> :t foldr
foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b
Prelude> :t (&&)
(&&) :: Bool -> Bool -> Bool
Prelude>
By "plugging in" (&&) you have removed (a -> b -> b)
so you need to provide the rest to the function
b -> t a -> b
That is restricted by (&&) to be a bool as second param to it, and the second parameter is the t a which is also restricted to being a bool. since a and b needs to be the same type as in the (a->b->b) function.
In recursion schemes, how can I construct something with type definition like (Recursive t, CoRecursive t) -> t -> ? -> t
I try to use recursion-schemes to update nodes. Taking list as an example, I can come up with two methods like:
update :: [a] -> Natural -> a -> [a]
update = para palg where
palg Nil _ _ = []
palg (Cons a (u, _)) 0 b = b : u
palg (Cons a (u, f)) n b = a : f (n-1) b
update' :: [a] -> Natural -> a -> [a]
update' = c2 (apo acoalg) where
c2 f a b c = f (a,b,c)
acoalg ([], _, _) = Nil
acoalg (_:as , 0, b) = Cons b $ Left as
acoalg (a:as , n, b) = Cons a $ Right (as, n-1, b)
However, these two implementations are good. In these two implementations, the constructor of ListF and [] appears in both sides of the equation. And the definition does not appear to be unique. Is there a better way to perform List update with recursion schemes?
Recursion schemes is flexible approach. You can also implement your own variant.
(Reuse cata)
zipo :: (Recursive g, Recursive h) => (Base g (h -> c) -> Base h h -> c) -> g -> h -> c
zipo alg = cata zalg
where
zalg x = alg x <<< project
update :: forall a. [a] -> Natural -> a -> [a]
update xs n a = zipo alg n xs
where
alg :: Maybe ([a] -> [a]) -> ListF a [a] -> [a]
alg _ Nil = []
alg Nothing (Cons y ys) = a:ys
alg (Just n') (Cons y ys) = y:(n' ys)
Also u can implement some parallel version like
zipCata :: (Recursive g, Recursive h) => ((g -> h -> r) -> Base g g -> Base h h -> r) -> g -> h -> r
zipCata phi x y = phi (zipCata phi) (project x) (project y)
update' :: forall a. [a] -> Natural -> a -> [a]
update' xs n a = zipCata alg n xs
where
alg :: (Natural -> [a] -> [a]) -> Maybe Natural -> ListF a [a] -> [a]
alg _ _ Nil = []
alg _ Nothing (Cons _ ys) = a:ys
alg f (Just n) (Cons y ys) = y:(f n ys)
Both variants (also as your) will be get the same result
PS. I hate approach for code sample on SO
Say I have functions which accept the same parameters and I want to test if their outputs are equivalent for the same input.
f :: a -> b -> c
g :: a -> b -> c
f a b == g a b
How can I package the parameters a and b in x so I can write the following instead.
f x == g x
What are the best ways to accomplish this without needing to wrap the functions themselves?
The only way to do exactly what you’re asking is to use uncurry:
let
x = (a, b)
in uncurry f x == uncurry g x
(Or uncurryN for N arguments.)
However, instead of packaging the arguments in a tuple, you could use the (->) x instance of Applicative (i.e., functions taking x as input) to implicitly “spread” the arguments to the parameters of both functions, so at least you only have to mention them once. This instance is commonly used in point-free code.
For example, using liftA2 specialised to this instance:
-- General type:
liftA2 :: Applicative f => (a -> b -> c) -> f a -> f b -> f c
-- Specialised to ‘(->) x’ (using TypeApplications syntax):
liftA2 #((->) _) :: (a -> b -> c) -> (x -> a) -> (x -> b) -> (x -> c)
You get this pattern:
liftA2 h f g x
-- =
(h <$> f <*> g) x
-- =
h (f x) (g x)
To lift more arguments, you add another liftA2 or … <$> … <*> …:
liftA2 (liftA2 h) f g x y
-- =
(liftA2 h <$> f <*> g) x y
-- =
h (f x y) (g x y)
So in a case like yours:
f, g :: Int -> Char -> Bool
f i c = chr i == c
g i c = i == ord c
(liftA2 . liftA2) (==) f g :: Int -> Char -> Bool
-- =
liftA2 (liftA2 (==)) f g
-- =
(\ x y -> f x y == g x y)
The N in liftAN corresponds to the number of functions; the number of liftAN calls corresponds to the number of arguments.
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.
While musing what more useful standard class to suggest to this one
class Coordinate c where
createCoordinate :: x -> y -> c x y
getFirst :: c x y -> x
getSecond :: c x y -> y
addCoordinates :: (Num x, Num y) => c x y -> c x y -> c x y
it occured me that instead of something VectorSpace-y or R2, a rather more general beast might lurk here: a Type -> Type -> Type whose two contained types can both be extracted. Hm, perhaps they can be extracted?
Turns out neither the comonad nor bifunctors package contains something called Bicomonad. Question is, would such a class even make sense, category-theoretically? Unlike Bimonad (which also isn't defined, and I couldn't really see how might look), a naïve definition seems plausible:
class Bifunctor c => Bicomonad c where
fst :: c x y -> x
snd :: c x y -> y
bidup :: c x y -> c (c x y) (c x y)
probably with the laws
fst . bidup ≡ id
snd . bidup ≡ id
bimap fst snd . bidup ≡ id
bimap bidup bidup . bidup ≡ bidup . bidup
but I find it disquieting that both fields of the result of bidup contain the same type, and there are quite a number of other, perhaps “better” conceivable signatures.
Any thoughts?
This is not an answer, but for Bimonad, how about this?
class Biapplicative p => Bimonad p where
(>>==) :: p a b -> (a -> b -> p c d) -> p c d
biap :: Bimonad p => p (a -> b) (c -> d) -> p a c -> p b d
biap p q = p >>== \ab cd -> q >>== \a c -> bipure (ab a) (cd c)
instance Bimonad (,) where
(a,b) >>== f = f a b
I don't know if this is categorically right/interesting, or even remotely useful, but it smells right from a Haskell perspective. Does it match your Bicomonad or something similar?