Comonads in everyday life.

July 9, 2008 at 11:45 am · Filed under Uncategorized ·Tagged Category Theory, Comonad, hvac

This post is a literate haskell file. As is usual with such things, you can go ahead and paste it into a .lhs file and load it right up in ghci. As such, first some boring preliminaries.

> module CoMenu where
> import Control.Applicative; import Data.List; import Data.Tree; import Data.Maybe
> import Network.Frameworks.HVAC; import Network.Frameworks.HVAC.AltController

Anyway, say you’re serving up a website. And say this website has hierarchical menus. Maybe you want to display them with javascript, maybe statically, whatever. But of course, you also need some component of these menus to be generated server-side regardless, so that the structure matches up with the location of the current page in the hierarchy of content. One common way to do this, which isn’t all that bad actually, is to hardcode the whole menu, or to generate it from some list of sections/subsections, etc. that associates them with individual actions, be they .php or .jsp pages, or be they in servlet mappings or, you know, whatever. And in this menu, be it hardcoded or partly generated, there’s some additional code that conditionally displays subhierarchies (i.e. if you’re in the “About” section it will display the subsection selection as well, or etc.) or at a minimum disables the link to the current page, to give some indication of where the user currently is.

There are ways to fancy this up and reduce the overhead, but that’s the general notion — you have a dispatch system, and you have a menu system, and the information in the dispatch and menu systems don’t necessarily correspond. Or if you’re using “configuration by convention” maybe you’ve built something that relies on a correspondence, so that, e.g., a page can “autodiscover” its location in the hierarchy by introspection on its url.

So anyway, you’re describing the structure of your website twice, in perhaps very different formats, and the extent that you can improve on things is really improving the brevity of the formats and throwing some logic to the wind on the assumption that the descriptions really are rendundant.

The natural big little idea that I’m going to discuss then, is how we might use the same structure to describe both the dispatch and the menu, and so cut out all that nonsense. Now doing this is at least slightly trickier than it seems. That’s because in the typical imperative approach we’re thinking in passes from the top down — so even assuming we have a dispatch tree that we walk based on a url parse — and with, e.g., the hvac framework I’m working on, that’s a only a few lines — well, even assuming we have the dispatch down pat, consider the standard imperative approach to generating the menu itself.

For the purposes of this discussion the menu will be a horizontal one, with the top row containing the top level, the next row containing the paths of the selected sublevel, etc. The approach itself will generalize to any sort of menu, however. In any case, we start by rendering the top row. But wait! We have to distinguish the selected section. So, uh, on rendering the link to each section we check if it matches the first portion of the path. And then we react accordingly. And then, using the section we’ve found that matches that first portion we recurse into the next row and soforth.

But wait, that’s not how menus work! Generally, though not always, if you pick the a top section then it won’t show a special “top section” page but will show the page that you get if you “drill down” through first options until you eventually find a leaf. So we’re not talking about a rose tree here, but about a tree with content only at the leaf nodes, which is an irregular structure. On the other hand, maybe there’s content only at the leaf nodes 90% of the time, and 10% of the time there’s content elsewhere. So we’ll actually use a rose tree structure (from Data.Tree) with each node containing the following:

> data MItem a = MItem {
>       miName :: String,
>       miPath :: String,
>       miAction :: Maybe a } deriving Show

Of course with this there’s no guarantee that a leaf node contains a page action either, but it’ll do for the purposes of this discussion.

In any case, you see the issue — the dispatcher “drills down” to an action page, but the rendering of that page depends on the whole context of the menu, not just the initial path we were given. There are various ways to hack around this, but I won’t enumerate them all here. And then of course there’s the insight that even though the menu is “dynamic” on the page that it is rendered in, its static with regards to any given page and as such, a proper pre-traversal of the entire tree can (lazily, mind you) render the menus for every page at once.

So what we want is to traverse the tree in such a way that we preserve the context of the location we’re in with regards to everywhere else in the tree, so that we can render the menu properly, relative urls and all. And we want to do this entire traversal only once, rather than piecewise and repeatedly with each request. So what recursion scheme works? Well, fmap doesn’t, nor are any varients of traverse or mapAccum strong enough — they only tell us where we’ve been, not where we’re going. If only we could associate the context with each node beforehand — then a simple map would do the trick!

Well, the contents of a node plus all the associated context of a node have a common representation — the zipper, which is precisely the “one-hole context” of a datastructure. By definition, the zipper, or derivative, of any data structure, will contain enough information to reconstruct both the structure as well as an indext to a node within it. So here’s a simple and partial implementation of a zipper over a Tree. The included functions actually take it to and from a Forest, which is really just an alias for [Tree a].

> data Zipper a = Zipper {
>       prevLevels :: [Zipper a],
>       leftForest :: Forest a,
>       rightForest :: Forest a } deriving Show

> instance Functor Zipper where
>     fmap f (Zipper p l r) = Zipper (fmap f <$> p) (fmap f <$> l) (fmap f <$> r)

> forest2zip :: Forest a -> Zipper a
> forest2zip ts = Zipper [] [] ts

> zip2forest :: Zipper a -> Forest a
> zip2forest z = case prevLevels z of
>                  [] -> rightForest z
>                  pl -> rightForest . head . reverse $ pl

Now actually we’d typically include a host of functions for “walking around” within the zipper, including generalized depth-first traversal, etc. But it turns out that for our purposes we only ever need to step down, so the rest is ommited for brevity.

> zDown :: Zipper a -> Maybe (Zipper a)
> zDown x = case rightForest x of
>             [] -> Nothing
>             (t:_) -> case subForest t of
>                      (t':ts) -> Just $ Zipper {
>                                         prevLevels = x : prevLevels x,
>                                         leftForest = [],
>                                         rightForest = t':ts }
>                      _ -> Nothing

This gives us a representation of each node and its context. But still, how to associate them with each node, so as to preserve the overall structure? We could concievably turn our entire menu into a zipper, and then step through it, at each location replacing the action by one which includes the proper menu. But this would mean introducing another irregularity — holding

Either (Zipper a, a) a

or some equivalent (c.f. Conor McBride’s Clowns and Jokers). Yeesh! We want to do it all at once. I.e. we want a signature of

Zipper a -> Zipper (Zipper a)

. Wait! Hang on! That’s a specialization of “duplicate” which is to a comonad as “join” is to a monad! And in fact, as any fule kno, The dual of (monadic) substitution is (comonadic) redecoration. Oh, and it gets better! Behind every zipper is a comonad. And better yet! We also know that a rose tree is a cofree comonad over the list functor, which should somewhat help to tie this all together in terms of why and how a zipper is a comonad.

In any case, we provide the basic comonad class, and an instance for our zipper:

> class Functor w => Comonad w where
>         extract :: w a -> a
>         duplicate :: w a -> w (w a)
>         extend :: (w a -> b) -> w a -> w b
>         extend f = fmap f . duplicate
>         duplicate = extend id
>
> instance Comonad Zipper where
>     extract = rootLabel . head . rightForest
>     duplicate z@(Zipper p l r) = Zipper p' l' r'
>        where p' = fmap duplicate p
>              (l',r') = splitAt (length l) fs
>              lrs = l ++ r
>              fs = map (go p) $ zip3 (inits lrs) (tails lrs) lrs
>              go :: [Zipper a] -> ([Tree a],[Tree a],Tree a) -> Tree (Zipper a)
>              go ls (i,t,tr) =
>                         let z = Zipper {prevLevels = ls,
>                                         leftForest = i,
>                                         rightForest = t}
>                             sf' = subForest tr
>                         in tr {rootLabel = z,
>                                subForest = map (go (z:ls)) $
>                                            zip3 (inits sf') (tails sf') sf'}

Ok, so that duplicate code isn’t the easiest to write. Nonetheless, the beautiful thing is you only have to write it once. And it’s not that painful once you realize that we’ve really just got a generalization of the list comonad. But with all that traversal out of the way, a function to render menus almost writes itself. This one is parameterized over the function that draws the actual menu, and includes a little tweak to deal with irregular trees — if there’s no action at the given node, it traverses down until it finds a node that does provide an action.

> renderMenu ::
>     (Zipper (MItem (String -> a)) -> String)
>     -> Forest (MItem (String -> a)) -> Forest (MItem a)
> renderMenu renderRows = zip2forest . extend go . forest2zip
>     where
>       drillToAct :: Zipper (MItem a) -> Zipper (MItem a)
>       drillToAct z = case rightForest z of
>                        [] -> z
>                        (t:ts) -> case miAction . rootLabel $ t of
>                                    Nothing -> case zDown z of
>                                                 Nothing -> z
>                                                 Just z' -> drillToAct z'
>                                    _ -> z
>       go z = (extract z) {miAction = ($ renderRows z) <$>
>                           miAction (extract . drillToAct $ z)}

And here’s one example of a function that turns that zipper into a rendered menu. It makes lots of controversial design choices, has some quirks, and is really just hacked up for the purposes of this discussion, but the main point is that the legwork we did before lets us abstract the rendering from the traversal, so we can swap in whatever we want with no hassle.

> menuRows :: Zipper (MItem a) -> String
> menuRows z = concatMap (renderRow curDepth "") (reverse $ prevLevels z) ++
>              renderRow curDepth "" z ++
>              (case zDown z of
>                         Just z' -> renderRow curDepth ((miPath . extract) z ++ "/") z'
>                         Nothing -> "")
>     where curDepth = (length . prevLevels) z
>           renderRow curDepth prevPath z =
>                     "<div>" ++
>                     intercalate " | " (map mkLink (leftForest z)) ++
>                     (if null prevPath
>                        then " | " ++ (miName . extract) z ++ " | " ++
>                             intercalate " | "
>                                (map mkLink (tail . rightForest $ z))
>                        else " | " ++ intercalate " | "
>                                (map mkLink (rightForest z))) ++
>                     "</div>"
>               where mkLink x = "<a href='" ++ concat (replicate depth "../") ++
>                                prevPath ++
>                                (miPath . rootLabel) x ++ "/'>" ++
>                                (miName . rootLabel) x ++ "</a>"
>                     depth = curDepth - (length . prevLevels $ z)

And to round out the picture, the promised simple dispatch function, and below it, some sample data.

> menu2dispatch :: Forest (MItem (String -> HCGI q s CGIResult))
>                 -> HCGI q s CGIResult
> menu2dispatch mis' = endPath *>
>                        (fromJust . miAction . rootLabel . head) renderedMenu
>                      <|> go renderedMenu
>     where renderedMenu = renderMenu menuRows mis'
>           hasPath t pth = (miPath . rootLabel) t == pth
>           go mis = do
>             p <- takePath
>             case find (`hasPath` p) mis of
>               Nothing -> continue
>               Just mi -> endPath *> (fromJust . miAction . rootLabel) mi
>                          <|> go (subForest mi)
>
> foo = [
>        Node 2 [Node 10 [Node 11 []], Node 23 []],
>        Node 4 [Node 5 [], Node 7 []],
>        Node 8 [Node 9 [], Node 13 []],
>        Node 25 []
>       ]
>
> bar = map (fmap (\x -> MItem {
>                       miName = show x,
>                       miPath = show x,
>                       miAction = if even x
>                                  then Nothing
>                                  else Just $
>                                        \y -> y ++ "\n" ++ "body: " ++ show x}))
>       foo

> {-
> *CoMenu> fromJust . miAction . rootLabel . head $ renderMenu menuRows bar
> "<div> | 2 | <a href='4/'>4</a> | <a href='8/'>8</a> | <a href='25/'>25</a></div><div> | <a href='2/10/'>10</a> | <a href='2/23/'>23</a></div>\nbody: 11"
> -}

So yeah, comonads in practical day-to-day programming. Not as rare as you would suspect, and occasionally just what the doctor ordered.

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