I'm getting used to Haskell's higher-order functions. Usually I can replace explicit patterns of recursion with functions like map, fold, and scan. However, I often run into the following recursion pattern which I don't understand how to express using higher-order functions:
f (x:[]) = k x
f (x:xs) = g x (f xs)
For instance, suppose I am representing analytic tableaux. Then I create a data type such as:
data Tableau = N Expr | S Expr (Tableau) | B Expr (Tableau) (Tableau)
If I want to convert a list of Exprs into a tableau structure, I want a function part of which might resemble:
f (x:[]) = N x
f (x:xs) = S x (f xs)
Now, I see three options: (1) create a function which decides, given a tableau and a list, whether the next branch in the tableau should be an S or N (or B, but we'll ignore that case); (2) use a higher-order function to encapsulate the recursion pattern of f; (3) use a function like f.
What would the best option be?
I'd most probably use the following:
f xs = foldr g (k (last xs)) (init xs)
It basically means that the end of the list is replaced by k x when folding. Thanks to lazy evaluation present everywhere, it works even for infinite lists.
There are two other solutions - adding empty case and using Maybe.
A) adding empty case:
It would be best if f [] was well-defined. Then, you could write the definition as
f [] = c
f (x:xs) = g x (f xs)
which is f = foldr g c. For example, if you change
data Tableau = N Expr | S Expr Tableau | B Expr Tableau Tableau
to
data Tableau = N | S Expr Tableau | B Expr Tableau Tableau
then you can represent single-element tableaux as S expr N, and the function is defined as one-liner
f = foldr S N
It's the best solution as long the empty case makes sense.
B) use Maybe:
On the other hand, if f [] cannot be sensibly defined, it's worse.
Partial functions are often considered ugly. To make it total, you can use Maybe. Define
f [] = Nothing
f [x] = Just (k x)
f (x:xs) = Just (g x w)
where Just w = f xs
It is a total function - that's better.
But now you can rewrite the function into:
f [] = Nothing
f (x:xs) = case f xs of
Nothing -> Just (k x)
Just w -> Just (g x w)
which is a right fold:
addElement :: Expr -> Maybe Tableaux -> Maybe Tableaux
addElement x Nothing = Just (N x)
addElement x (Just w) = Just (S x w)
f = foldr addElement Nothing
In general, folding is idiomatic and should be used when you fit the recursion pattern. Otherwise use explicit recursion or try to reuse existing combinators. If there's a new pattern, make a combinator, but only if you'll use the pattern a lot - otherwise it's overkill. In this case, the pattern is fold for nonempty lists defined by: data List a = End a | Cons a (List a).
If I've understood the question properly, then here's my evaluation of your options:
It's probably a bit nasty to have to match the (presumably arbitrarily complex?) tableau out from underneath the constructor in order to write that function. This approach seems somewhat brittle, although it would probably work just fine.
I don't see the need to generalise the pattern out, given that it's a recursive function operating on a recursive structure. Introducing a higher-order pattern would (I think) obfuscate the actual logic behind performing a recursive traversal of this data structure.
I think this makes a good deal of sense. As you've observed, it's a reasonably recognised "pattern", but I think it matches the description of the algorithm well to write it down in exactly this way. It may not be as generalisable or reusable, but given that it's essentially the crux of the algorithmic approach, I think it makes sense to write the cases directly as you have done in a function like f. This would be my favoured approach.
Sorry to not provide a particularly concrete answer, but it is a reasonably subjective answer, so given the three options above, I would pick option 3 for reasons of clarity and readability.
Related
Using the following catamorphism for natural numbers I can implement various arithmetic algorithms whithout having to deal with recursion:
cataNat :: b -> (b -> b) -> Natural -> b
cataNat zero succ = go
where
go n = if (n <= 0) then zero else succ (go (n - 1))
fib :: Natural -> Natural
fib = fst . cataNat (0, 1) (\(a, b) -> (b, a + b))
cataNat looks similar to primitive recursion to me. At least each application of it seems garuanteed to terminate, no matter which combination of zero and succ is provided. With each iteration the overall problem is decomposed by the smallest/simplest problem instance. So even if it is technically not primitive recursion it seems to be equally expressive. If this is true it would mean that a catamorphism is not enough to express general recursion. We would probably need a hylomorphism for that. Is my reasoning correct, that is, does the equivalence hold for any type of catamorphism, not just for natural numbers?
Primitive recursion corresponds directly to a paramorphism.
You're correct that a catamorphism has equivalent theoretical power to a paramorphism, but they can be different in important ways in operational terms. For an example, let's go to lists instead of Nats.
cata :: b -> (a -> b -> b) -> [a] -> b
cata = flip foldr -- I'm lazy, but this argument order makes a bit more sense for this example
para :: b -> (a -> [a] -> b -> b) -> [a] -> b
para z _ [] = z
para z f (x:xs) = f x xs (para z f xs)
-- Removes the first element from the list which is equal to the other argument
delete1 :: Eq a => a -> [a] -> [a]
delete1 x xs = cata (const []) (\el k found -> if not found && el == x then k True else el : k found) xs False
-- Removes the first element from the list which is equal to the other argument
delete2 :: Eq a => a -> [a] -> [a]
delete2 x xs = para [] (\el raw processed -> if el == x then raw else el : processed) xs
Look at how awkward delete1 is, compared to delete2. Not only do you have to contort your logic by making the result of cata a function, but there's a very real operational cost, too. You have to traverse everything in the list after finding a matching element, and re-create all the (:) constructors. That can have a noticeable cost in efficiency. In comparison, delete2, when it finds the target element, can just use the existing tail of the list for the remainder, without even looking at it. Of course, most uses of foldr (real world, not this example) don't produce a function and don't want access to the unprocessed tail of the list. For them, the catamorphism is going to be slightly more efficient simply because of passing around less data.
So in terms of theoretical power, they're equivalent. In operational terms, each has a use, though catamorphisms are a lot more common.
For some expansion of the idea in more general terms, see the recursion-schemes library. It uses a rather different-looking formulation of the idea so that it can abstract over data types with different shapes, instead of needing a different type for cata/para for each data type they can be applied to. But it really is just an alternate way of packing up the same ideas, and other kinds of morphisms are covered as well, including much more niche (or even possibly useless) ones.
In the ``Kan Extensions for Program Optimisation'' by Ralf Hinze there is the definition of List type based on right Kan extension of the forgetful functor from the category of monoids along itself (section 7.4). The paper gives Haskell implementation as follows:
newtype List a = Abstr {
apply :: forall z . (Monoid z) => (a -> z) -> z
}
I was able to define usual nil and cons constructors:
nil :: List a
nil = Abstr (\f -> mempty)
cons :: a -> List a -> List a
cons x (Abstr app) = Abstr (\f -> mappend (f x) (app f))
With the following instance of Monoid class for Maybe functor, I managed to define head function:
instance Monoid (Maybe a) where
mempty = Nothing
mappend Nothing m = m
mappend (Just a) m = Just a
head :: List a -> Maybe a
head (Abstr app) = app Just
Question: How can one define tail function?
Here is a rather principled solution to implementing head and tail in one go (full gist):
First of all, we know how to append lists (it will be useful later on):
append :: List a -> List a -> List a
append (Abstr xs) (Abstr ys) = Abstr (\ f -> xs f <> ys f)
Then we introduce a new type Split which we will use to detect whether a List is empty or not (and get, in the case it's non empty, a head and a tail):
newtype Split a = Split { outSplit :: Maybe (a, List a) }
This new type forms a monoid: indeed we know how to append two lists.
instance Monoid (Split a) where
mempty = Split Nothing
mappend (Split Nothing) (Split nns) = Split nns
mappend (Split mms) (Split Nothing) = Split mms
mappend (Split (Just (m, ms))) (Split (Just (n, ns))) =
Split $ Just (m, append ms (cons n ns))
Which means that we can get a function from List a to Split a using List a's apply:
split :: List a -> Split a
split xs = apply xs $ \ a -> Split $ Just (a, nil)
head and tail can finally be trivially derived from split:
head :: List a -> Maybe a
head = fmap fst . outSplit . split
tail :: List a -> Maybe (List a)
tail = fmap snd . outSplit . split
This implementation of lists as free monoids is provided in the package fmlist, which notes some interesting properties of it (unlike most implementations of lists, which are right-biased, this one is truly unbiased; you can make an arbitrary tree, and although of course the monoid laws force you to see it as flattened, you can still observe some differences in the infinite case. This is almost a Haskell quirk -- usually, free monoids). It also has an implementation of tail, so that's sort of an answer to your question (but see below).
With these sorts of representations (not just this particular one one, but also e.g. forall r. (a -> r -> r) -> r -> r lists), there are usually some operations (e.g. appending) that become easier, and some (e.g. zip and tail) that become more difficult. This is discussed a bit in various places, e.g. How to take the tail of a functional stream.
Looking more closely at fmlist, though, its solution is pretty unsatisfactory: It just converts the nice balanced tree that you give it to a right-biased list using foldr, which allows it to do regular list operations, but loses the monoidal structure. The tail of a "middle-infinite" list is no longer "middle-infinite", it's just right-infinite like a regular list.
It should be possible to come up with a clever Monoid instance to compute the tail while disturbing the rest of the structure as little as possible, but an obvious one doesn't come to mind off-hand. I can think of a non-clever "brute force" solution, though: Cheat and reify the "list" into a tree using an invalid Monoid instance, inspect the tree, and then fold it back up so the end result is valid. Here's what it would look like with my nonfree package and fmlist:
nail :: FM.FMList a -> FM.FMList a
nail (FM.FM k) = FM.FM $ \f -> foldMap f (nail' (k N))
nail' :: N a -> N a
nail' NEmpty = error "nail' NEmpty"
nail' (N x) = NEmpty
nail' (NAppend l r) =
case normalize l of
NEmpty -> nail' r
N x -> r
l' -> NAppend (nail' l') r
-- Normalize a tree so that the left side of a root NAppend isn't an empty
-- subtree of any shape. If the tree is infinite in a particular way, this
-- won't terminate, so in that sense taking the tail of a list can make it
-- slightly worse (but you were already in pretty bad shape as far as
-- operations on the left side are concerned, and this is a pathological case
-- anyway).
normalize :: N a -> N a
normalize (NAppend l r) =
case normalize l of
NEmpty -> normalize r
l' -> NAppend l' r
normalize n = n
In explaining foldr to Haskell newbies, the canonical definition is
foldr :: (a -> b -> b) -> b -> [a] -> b
foldr _ z [] = z
foldr f z (x:xs) = f x (foldr f z xs)
But in GHC.Base, foldr is defined as
foldr k z = go
where
go [] = z
go (y:ys) = y `k` go ys
It seems this definition is an optimization for speed, but I don't see why using the helper function go would make it faster. The source comments (see here) mention inlining, but I also don't see how this definition would improve inlining.
I can add some important details about GHC's optimization system.
The naive definition of foldr passes around a function. There's an inherent overhead in calling a function - especially when the function isn't known at compile time. It'd be really nice to able to inline the definition of the function if it's known at compile time.
There are tricks available to perform that inlining in GHC - and this is an example of them. First, foldr needs to be inlined (I'll get to why later). foldr's naive implementation is recursive, so cannot be inlined. So a worker/wrapper transformation is applied to the definition. The worker is recursive, but the wrapper is not. This allows foldr to be inlined, despite the recursion over the structure of the list.
When foldr is inlined, it creates a copy of all of its local bindings, too. It's more or less a direct textual inlining (modulo some renaming, and happening after the desugaring pass). This is where things get interesting. go is a local binding, and the optimizer gets to look inside it. It notices that it calls a function in the local scope, which it names k. GHC will often remove the k variable entirely, and will just replace it with the expression k reduces to. And then afterwards, if the function application is amenable to inlining, it can be inlined at this time - removing the overhead of calling a first-class function entirely.
Let's look at a simple, concrete example. This program will echo a line of input with all trailing 'x' characters removed:
dropR :: Char -> String -> String
dropR x r = if x == 'x' && null r then "" else x : r
main :: IO ()
main = do
s <- getLine
putStrLn $ foldr dropR "" s
First, the optimizer will inline foldr's definition and simplify, resulting in code that looks something like this:
main :: IO ()
main = do
s <- getLine
-- I'm changing the where clause to a let expression for the sake of readability
putStrLn $ let { go [] = ""; go (x:xs) = dropR x (go xs) } in go s
And that's the thing the worker-wrapper transformation allows.. I'm going to skip the remaining steps, but it should be obvious that GHC can now inline the definition of dropR, eliminating the function call overhead. This is where the big performance win comes from.
GHC cannot inline recursive functions, so
foldr :: (a -> b -> b) -> b -> [a] -> b
foldr _ z [] = z
foldr f z (x:xs) = f x (foldr f z xs)
cannot be inlined. But
foldr k z = go
where
go [] = z
go (y:ys) = y `k` go ys
is not a recursive function. It is a non-recursive function with a local recursive definition!
This means that, as #bheklilr writes, in map (foldr (+) 0) the foldr can be inlined and hence f and z replaced by (+) and 0 in the new go, and great things can happen, such as unboxing of the intermediate value.
As the comments say:
-- Inline only in the final stage, after the foldr/cons rule has had a chance
-- Also note that we inline it when it has *two* parameters, which are the
-- ones we are keen about specialising!
In particular, note the "we inline it when it has two parameters, which are the ones we are keen about specialising!"
What this is saying is that when foldr gets inlined, it's getting inlined only for the specific choice of f and z, not for the choice of the list getting folded. I'm not expert, but it would seem it would make it possible to inline it in situations like
map (foldr (+) 0) some_list
so that the inline happens in this line and not after map has been applied. This makes it optimizable in more situations and more easily. All the helper function does is mask the 3rd argument so {-# INLINE #-} can do its thing.
One tiny important detail not mentioned in other answers is that GHC, given a function definition like
f x y z w q = ...
cannot inline f until all of the arguments x, y, z, w, and q are applied. This means that it's often advantageous to use the worker/wrapper transformation to expose a minimal set of function arguments which must be applied before inlining can occur.
What is the GHC equivalent of OCaml's -rectypes for allowing recursive types? I don't see one in the documentation. Is it a hidden feature?
There isn't one unfortunately, all recursion must go through a data type. However, if you're willing to put up with a bit of headache you can still write recursive types pretty easily.
newtype RecArr b a = RecArr {unArr :: RecArr b a -> b}
unfold = unArr
fold = RecArr
Now we can fold and unfold our RecArr to unfold our recursion to our hearts content. This is a little painful because it's manual, but completely workable. As a demonstration, here's the y combinator written using fold and unfold.
y f = (\x -> f (unfold x x)) $ fold (\x -> f (unfold x x))
factorial f n = if n == 0 then 1 else n * f (n-1)
main = print (y factorial 5) -- prints 120
There is none. All recursion has to go through nominal types. That is, you have to define a data type.
I am reading a tutorial that uses the following example (that I'll generalize somewhat):
f :: Foo -> (Int, Foo)
...
fList :: Foo -> [Int]
fList foo = x : fList bar
where
(x, bar) = f foo
My question lies in the fact that it seems you can refer to x and bar, by name, outside of the tuple where they are obtained. This would seem to act like destructuring parameter lists in other languages, if my guess is correct. (In other words, I didn't have to do the following:)
fList foo = (fst tuple) : fList (snd tuple)
where
tuple = f foo
Am I right about this behavior? I've never seen it mentioned yet in the tutorials/books I've been reading. Can someone point me to more info on the subject?
Edit: Can anything (lists, arrays, etc.) be destructured in a similar way, or can you only do this with tuples?
Seeing your edit, I think what your asking about is Pattern matching.
And to answer your question: Yes, anything you can construct, you can also 'deconstruct' using the constructors. For example, you're probably familiar with this form of pattern matching:
head :: [a] -> a
head (x:xs) = x
head [] = error "Can't take head of empty list"
However, there are more places where you can use pattern matching, other valid notations are:
head xs = case xs of
(y:ys) -> y
[] -> error "Can't take head of empty list"
head xs = let (y:ys) = xs
in y
head xs = y
where
(y:ys) = xs
Note that the last two examples are a bit different from the first to because they give different error messages when you call them with an empty list.
Although these examples are specific to lists, you can do the same with other data types, like so:
first :: (a, b) -> a
first tuple = x
where
(x, y) = tuple
second :: (a, b) -> b
second tuple = let (x, y) = tuple
in y
fromJust :: Maybe a -> a
fromJust ma = x
where
(Just x) = ma
Again, the last function will also crash if you call it with Nothing.
To sum up; if you can create something using constructors (like (:) and [] for lists, or (,) for tuples, or Nothing and Just for Maybe), you can use those same constructors to do pattern matching in a variety of ways.
Am I right about this behavior?
Yes. The names exist only in the block where you have defined them, though. In your case, this means the logical unit that your where clause is applied to, i.e. the expression inside fList.
Another way to look at it is that code like this
x where x = 3
is roughly equivalent to
let x = 3 in x
Yes, you're right. Names bound in a where clause are visible to the full declaration preceding the where clause. In your case those names are f and bar.
(One of the hard things about learning Haskell is that it is not just permitted but common to use variables in the source code in locations that precede the locations where those variables are defined.)
The place to read more about where clauses is in the Haskell 98 Report or in one of the many fine tutorials to be found at haskell.org.