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Haskell/Understanding arrows

Arrows, like monads, express computations that happen within a context. However, they are a more general abstraction than monads, and thus allow for contexts beyond what the Monad class makes possible. The essential difference between the abstractions can be summed up thus:

Just as we think of a monadic type m a as representing a 'computation delivering an a'; so we think of an arrow type a b c, (that is, the application of the parameterised type a to the two parameters b and c) as representing 'a computation with input of type b delivering a c'; arrows make the dependence on input explicit.
—John Hughes, Generalising Monads to Arrows [1]

This chapter has two main parts. Firstly, we will consider the main ways in which arrow computations differ from those expressed by the functor classes we are used to, and also briefly present some of the core arrow-related type classes. Secondly, we will study the parser example used by John Hughes in the original presentation of arrows.

Pocket guide to ArrowEdit

Arrows look a lot like functionsEdit

The first step towards understanding arrows is realising how similar they are to functions. Like (->), the type constructor of an Arrow instance has kind * -> * -> *, that is, it takes two type arguments − unlike, say, a Monad, which takes only one. Crucially, Arrow has Category as a superclass. Category is, to put it very roughly, the class for things that can be composed like functions:

class Category y where
    id  :: y a a                   -- identity for composition.
    (.) :: y b c -> y a b -> y a c -- associative composition.

Functions have an instance of Category − in fact, they are Arrows as well.

A practical consequence of this similarity is that you have to think in point-free terms when looking at expressions full of Arrow operators, such as this example from the tutorial:

(total &&& (const 1 ^>> total)) >>> arr (uncurry (/))

Otherwise you will quickly get lost looking for the values to apply things on. In any case, it is easy to get lost even if you look at such expressions in the right way. That's what proc notation is all about: adding extra variable names and whitespace while making some operators implicit, so that Arrow code gets easier to follow.

Before continuing, we should mention that Control.Category also defines (<<<) = (.) and (>>>) = flip (.), which is very commonly used to compose arrows from left to right.

Arrow glides between Applicative and MonadEdit

In spite of the warning we gave just above, arrows can be compared to applicative functors and monads. The trick is making the functors look more like arrows, and not the opposite. That is, you should not compare Arrow y => y a b with Applicative f => f a or Monad m => m a, but rather with:

  • Applicative f => f (a -> b), the type of static morphisms i.e. the values to the left of (<*>); and
  • Monad m => a -> m b, the type of Kleisli morphisms i.e. the functions to the right of (>>=) [2].

Morphisms are the sort of things that can have Category instances, and indeed we could write instances of Category for both static and Kleisli morphisms. This modest twisting is enough for a sensible comparison.

If this argument reminds you of the sliding scale of power discussion, in which we compared Functor, Applicative and Monad, that is a sign you are paying attention, as we are following exactly the same route. Back then, we remarked how the types of the morphisms limit how they can, or cannot, create effects. Monadic binds can induce near-arbitrary changes to the effects of a computation depending on the a values given to the Kleisli morphism, while the isolation between the functorial wrapper and the function arrow in static morphisms mean the effects in an applicative computation do not depend at all on the values within the functor [3].

What sets arrows apart from this point of view is that in Arrow y => y a b there is no such connection between the context y and a function arrow to determine so rigidly the range of possibilities. Both static and Kleisli morphisms can be made into Arrows, and conversely an instance of Arrow can be made as limited as an Applicative one or as powerful as a Monad one [4]. More interestingly, we can use Arrow to take a third option and have both applicative-like static effects and monad-like dynamic effects in a single context, but kept separate from each other. Arrows make it possible to fine tune how effects are to be combined. That is the main thrust of the classic example of the arrow-based parser, which we will have a look at near the end of this chapter.

An Arrow can multitaskEdit

These are the Arrow methods:

class Category y => Arrow y where
    -- Minimal implementation: arr and first
    arr    :: (a -> b) -> y a b                 -- converts function to arrow
    first  :: y a b -> y (a, c) (b, c)          -- maps over first component

    second :: y a b -> y (c, a) (c, b)          -- maps over second component
    (***)  :: y a c -> y b d -> y (a, b) (c, d) -- first and second combined
    (&&&)  :: y a b -> y a c -> y a (b, c)      -- (***) on a duplicated value

With these methods, we can carry out multiple computations at each step of what seems to be a linear chain of composed arrows. That is done by keeping values used in separate computations as elements of pairs in a (possibly nested) pair, and then using the pair-handling functions to reach each value when desired. That allows, for instance, saving intermediate values for later or using functions with multiple arguments conveniently [5].

Visualising may help understanding the data flow in an arrow computation. Here are illustrations of (>>>) and the five Arrow methods:

 
Data flow for the mean1 arrow from the tutorial. Rectangles are arrows, rounded rectangles are arrows made with arr, circles are other data flow split/merge points. Other combinators are left implicit. Corresponding code:
(total &&& (const 1 ^>> total))
    >>> arr (uncurry (/))

It is worth mentioning that Control.Arrow defines returnA = arr id as a do-nothing arrow. One of the arrow laws says returnA must be equivalent to the id from the Category instance [6].

An ArrowChoice can be resoluteEdit

If Arrow makes multitasking possible, ArrowChoice forces a decision on what task to do.

class Arrow y => ArrowChoice y where
    -- Minimal implementation: left
    left  :: y a b -> y (Either a c) (Either b c)          -- maps over left choice

    right :: y a b -> y (Either c a) (Either c b)          -- maps over right choice
    (+++) :: y a c -> y b d -> y (Either a b) (Either c d) -- left and right combined
    (|||) :: y a c -> y b c -> y (Either a b) c            -- (+++), then merge results

Either provides a way to tag the values, so that different arrows can handle them depending on whether they are tagged with Left or Right. Note that these methods involving Either are entirely analogous to those involving pairs offered by Arrow.

 
Data flow in a fragment of the getWord example of the tutorial. Blue indicates a Left tag and red indicates Right. Note that the if construct of proc notation sends True to Left and False to Right. Corresponding code:
proc () -> do
  firstTime <- oneShot -< ()
  mPicked <- if firstTime
    then do
      picked <- pickWord rng -< ()
      returnA -< Just picked
    else returnA -< Nothing
  accum' Nothing mplus -< mPicked

An ArrowApply is just boringEdit

As the name suggests, ArrowApply makes it possible to apply arrows to values directly midway through an arrow computation. Ordinary Arrows do not allow that − we can just compose them on and on and on. Application only happens right at the end, once a run-arrow function of some sort is used to get a plain function from the arrow.

class Arrow y => ArrowApply y where
    app :: y (y a b, a) b -- applies first component to second

(For instance, app for functions is uncurry ($) = \(f, x) -> f x .)

app, however, comes at a steep price. Building an arrow as a value within an arrow computation and then eliminating it through application implies allowing the values within the computation to affect the context. That sounds a lot like what monadic binds do. It turns out that an ArrowApply is exactly equivalent to some Monad as long as the ArrowApply laws are followed. The ultimate consequence is that ArrowApply arrows cannot realise any of the interesting possibilities Arrow allows but Monad doesn't, such as having a partly static context.

The real flexibility with arrows comes with the ones that aren't monads, otherwise it's just a clunkier syntax.
—Philippa Cowderoy

Arrow combinators crop up in unexpected placesEdit

Functions are the trivial example of arrows, and so all of the Control.Arrow functions shown above can be used with them. For that reason, it is quite common to see arrow combinators being used in code that otherwise has nothing to do with arrows. Here is a summary of what they do with plain functions, alongside with combinators in other modules that can be used in the same way (in case you prefer the alternative names, or just prefer using simple modules for simple tasks).

Combinator What it does
(specialised to (->))
Alternatives
(>>>) flip (.)
first \f (x, y) -> (f x, y) first (Data.Bifunctor)
second \f (x, y) -> (x, f y) fmap; second (Data.Bifunctor)
(***) \f g (x, y) -> (f x, g y) bimap (Data.Bifunctor)
(&&&) \f g x -> (f x, g x) liftA2 (,) (Control.Applicative)
left Maps over Left case. first (Data.Bifunctor)
right Maps over Right case. fmap; second (Data.Bifunctor)
(+++) Maps over both cases. bimap (Data.Bifunctor)
(|||) Eliminates Either. either (Data.Either)
app \(f, x) -> f x uncurry ($)

The Data.Bifunctor module provides the Bifunctor class, of which pairs and Either are instances. A Bifunctor is very much like a Functor, except that there are two independent ways of mapping functions over it, corresponding to the first and second methods [7].

Exercises
  1. Write implementations for second, (***) and (&&&). Use just (>>>), arr, and first (plus any plain functions) to implement second; after that, you can use the other combinators once you have implemented them.
  2. Write an implementation for right in terms of left.
  3. Implement liftY2 :: Arrow y =>
    (a -> b -> c) -> y r a -> y r b -> y r c

Using arrowsEdit

Avoiding leaksEdit

Arrows were originally motivated by an efficient parser design found by Swierstra and Duponcheel[8]. In order to have a reference point for discussing the benefits of their design, let's begin with a very quick look at how monadic parsers work.

Here is a really bare-bones illustration of a monadic parser type:

newtype Parser s a = Parser { runParser :: [s] -> Maybe (a, [s]) }

A Parser is a function that takes a stream of input (here, a list of type [s]), and, depending on what it finds in the input, returns either a result (of type [a]) and a stream (often, the input stream minus some of the input that is consumed by the parser), or Nothing. When we say a parser succeeded or failed, we refer to whether it produced a result. While real world parser types, such as ParsecT from Parsec [9], can be a lot more complex in order to provide various other features (notably, informative error messages upon failure), this simple Parser is good enough for our current purposes.

For instance, this is a parser for a single character in a string:

char :: Char -> Parser Char Char
char c = Parser $ \input -> case input of
    [] -> Nothing
    x : xs -> if x == c then Just (c, xs) else Nothing

If the first character in the input string is c, char c succeeds, consuming the first character and giving c as a result; otherwise, it fails:

GHCi> runParser (char 'H') "Hello"
Just ('H',"ello")
GHCi> runParser (char 'G') "Hello"
Nothing

A second look at the Parser type shows it is essentially State [s] over Maybe. As such, it is not surprising we can make it an instance of Applicative, Monad, Alternative, and so forth (you might want to give it a try). That gives us a very flexible set of combinators for building more complex parsers out of simpler ones. Parsers can be ran in sequence...

isHel :: Parser Char ()
isHel = char 'H' *> char 'e' *> char 'l' *> pure ()

... have their results combined...

string :: String -> Parser Char String
string = traverse char

... or be tried as alternatives to each other:

solfege :: Parser Char String
solfege = string "Do" <|> string "Re" <|> string "Mi"

Through this last example, we can indicate which issue Swierstra and Duponcheel were trying to tackle. When solfege is ran on the string "Fa", we can't detect the parser will fail until all of the three alternatives have failed. If we had more complex parsers in which one of the alternatives might fail only after attempting to consume a lot of the input stream, we would have to descend down the chain of parsers in the very same way. All of the input that can possibly be consumed by later parsers must be retained in memory in case one of them happens to be able to consume it. That can lead to much more space usage than you would naively expect − a situation often called a space leak.

Can it be done better?Edit

Swierstra and Duponcheel (1996) noticed that, when dealing with some kinds of parsing tasks, a smarter parser could immediately fail upon seeing the very first character. For example, in the solfege parser above, the choice of first letter parsers was limited to letters 'D', 'R' and 'M', for "Do", "Re" and "Mi" respectively. This smarter parser would also be able to garbage collect input sooner because it could look ahead to see if any other parsers might be able to consume the input, and drop input that could not be consumed. This new parser is a lot like the monadic parsers with the major difference that it exports static information. It's like a monad, but it also tells you what it can parse.

There's one major problem. This doesn't fit into the Monad interface. Monadic composition works with (a -> m b) functions, and functions alone. There's no way to attach static information. You have only one choice, throw in some input, and see if it passes or fails.

Back when this issue first arose, the monadic interface was being touted as a completely general purpose tool in the functional programming community, so finding that there was some particularly useful code that just couldn't fit into that interface was something of a setback. This is where arrows come in. John Hughes's Generalising monads to arrows proposed the arrows abstraction as new, more flexible tool.

Static and dynamic parsersEdit

Let us examine Swierstra and Duponcheel's parser in greater detail, from the perspective of arrows as presented by Hughes. The parser has two components: a fast, static parser which tells us if the input is worth trying to parse; and a slow, dynamic parser which does the actual parsing work.

import Control.Arrow
import qualified Control.Category as Cat
import Data.List (union)

data Parser s a b = P (StaticParser s) (DynamicParser s a b)
data StaticParser s = SP Bool [s]
newtype DynamicParser s a b = DP ((a, [s]) -> (b, [s]))

The static parser consists of a flag, which tells us if the parser can accept the empty input, and a list of possible starting characters. For example, the static parser for a single character would be as follows:

spCharA :: Char -> StaticParser Char
spCharA c = SP False [c]

It does not accept the empty string (False) and the list of possible starting characters consists only of c.

The dynamic parser needs a little more dissecting. What we see is a function that goes from (a, [s]) to (b, [s]). It is useful to think in terms of sequencing two parsers: each parser consumes the result of the previous parser (a), along with the remaining bits of input stream ([s]), it does something with a to produce its own result b, consumes a bit of string and returns that. So, as an example of this in action, consider a dynamic parser (Int, String) -> (Int, String), where the Int represents a count of the characters parsed so far. The table below shows what would happen if we sequence a few of them together and set them loose on the string "cake" :


result remaining
before 0 cake
after first parser 1 ake
after second parser 2 ke
after third parser 3 e


So the point here is that a dynamic parser has two jobs : it does something to the output of the previous parser (informally, a -> b), and it consumes a bit of the input string, (informally, [s] -> [s]), hence the type DP ((a,[s]) -> (b,[s])). Now, in the case of a dynamic parser for a single character (type (Char, String) -> (Char, String)), the first job is trivial. We ignore the output of the previous parser, return the character we have parsed and consume one character off the stream:

dpCharA :: Char -> DynamicParser Char a Char
dpCharA c = DP (\(_,_:xs) -> (c,xs))

This might lead you to ask a few questions. For instance, what's the point of accepting the output of the previous parser if we're just going to ignore it? And shouldn't the dynamic parser be making sure that the current character off the stream matches the character to be parsed by testing x == c? The answer to the second question is no − and in fact, this is part of the point: the work is not necessary because the check would already have been performed by the static parser. Naturally, things are only so simple because we are testing just one character. If we were writing a parser for several characters in sequence we would need dynamic parsers that actually tested the second and further characters; and if we wanted to build an output string by chaining several parsers of characters then we would need the output of previous parsers.

Time to put both parsers together. Here is our S+D style parser for a single character:

charA :: Char -> Parser Char a Char
charA c = P (SP False [c]) (DP (\(_,_:xs) -> (c,xs)))

In order to actually use the parser, we need a runParser function that runs the static tests and applies the dynamic parser to the input:

-- The Eq constraint on s is needed so that we can use elem.
runParser :: Eq s => Parser s a b -> a -> [s] -> Maybe (b, [s])
runParser (P (SP emp _) (DP p)) a []
    | emp = Just (p (a, []))
    | otherwise = Nothing
runParser (P (SP _ start) (DP p)) a input@(x:_)
    | x `elem` start = Just (p (a, input))
    | otherwise = Nothing

Note how runParser gives out a function with type [s] -> Maybe (b, [s]) function, which is essentially the same as the monadic parser we demonstrated earlier on. Now we can use charA much in the same way we used char back then. (The main difference is the () which we pass as a dummy argument. That makes sense here because charA doesn't actually use the initial value of type a. With other parsers, though, that need not be the case.)

GHCi> runParser (charA 'D') () "Do"
Just ('D',"o")
GHCi> runParser (charA 'D') () "Re"
Nothing

Bringing the arrow combinators inEdit

With the preliminary bit of exposition done, we are now going to implement the Arrow class for Parser s, and by doing so, give you a glimpse of what makes arrows useful. So let's get started:

instance Eq s => Arrow (Parser s) where

arr should convert an arbitrary function into a parsing arrow. In this case, we have to use "parse" in a very loose sense: the resulting arrow accepts the empty string, and only the empty string (its set of first characters is []). Its sole job is to take the output of the previous parsing arrow and do something with it. That being so, it does not consume any input.

    arr f = P (SP True []) (DP (\(b,s) -> (f b,s)))

Likewise, the first combinator is relatively straightforward. Given a parser, we want to produce a new parser that accepts a pair of inputs (b,d). The first component of the input b, is what we actually want to parse. The second part passes through untouched:

    first (P sp (DP p)) = P sp (DP (\((b,d),s) ->
        let (c, s') = p (b,s)
        in  ((c,d),s')))

We also have to supply the Category instance. id is entirely obvious, as id = arr id must hold:

instance Eq s => Cat.Category (Parser s) where
    id = P (SP True []) (DP (\(b,s) -> (b,s)))
    -- Or simply: id = P (SP True []) (DP id)

On the other hand, the implementation of (.) requires a little more thought. We want to take two parsers, and return a combined parser incorporating the static and dynamic parsers of both arguments:

    -- The Eq s constraint is needed for using union here.
    (P (SP empty1 start1) (DP p2)) .
      (P (SP empty2 start2) (DP p1)) =
        P (SP (empty1 && empty2)
              (if not empty1 then start1 else start1 `union` start2))
          (DP (p2.p1))

Combining the dynamic parsers is easy enough; we just do function composition. Putting the static parsers together requires a little bit of thought. First of all, the combined parser can only accept the empty string if both parsers do. Fair enough, now how about the starting symbols? Well, the parsers are supposed to be in a sequence, so the starting symbols of the second parser shouldn't really matter. If life were simple, the starting symbols of the combined parser would only be start1. Alas, life is not simple, because parsers could very well accept the empty input. If the first parser accepts the empty input, then we have to account for this possibility by accepting the starting symbols from both the first and the second parsers

So what do arrows buy us?Edit

If you look back at our Parser type and blank out the static parser section, you might notice that this looks a lot like the arrow instances for functions.

arr f = \(b, s) -> (f b, s)
first p = \((b, d), s) ->
            let (c, s') = p (b, s)
            in  ((c, d), s'))

id = id
p2 . p1 = p2 . p1

There's the odd s variable out for the ride, which makes the definitions look a little strange, but the outline of e.g. the simple first functions is there. Actually, what you see here is roughly the arrow instance for the State monad/Kleisli morphism (let f :: b -> c, p :: b -> State s c and (.) actually be (<=<) = flip (>=>)).

That's fine, but we could have easily done that with bind in classic monadic style, with first becoming just an odd helper function that could be easily written with a bit of pattern matching. But remember, our Parser type is not just the dynamic parser − it also contains the static parser.

arr f = SP True []
first sp = sp
(SP empty1 start1) >>> (SP empty2 start2) = (SP (empty1 && empty2)
        (if not empty1 then start1 else start1 `union` start2))

This is not at all a function, it's just pushing around some data types, and it cannot be expressed in a monadic way. But the Arrow interface can deal with just as well. And when we combine the two types, we get a two-for-one deal: the static parser data structure goes along for the ride along with the dynamic parser. The Arrow interface lets us transparently compose and manipulate the two parsers, static and dynamic, as a unit, which we can then run as a traditional, unified function.

Arrows in practiceEdit

Some examples of libraries using arrows:

  • Opaleye (library documentation), a library for SQL generation.
  • The Haskell XML Toolbox (project page and library documentation) uses arrows for processing XML. There is a Wiki page in the Haskell Wiki with a somewhat Gentle Introduction to HXT.
  • Netwire (library documentation) is a library for functional reactive programming (FRP). FRP is a functional paradigm for handling events and time-varying values, with use cases including user interfaces, simulations and games. Netwire has an arrow interface as well as an applicative one.
  • Yampa (Haskell Wiki page library documentation) is another arrow-based FRP library, and a predecessor to Netwire.
  • Hughes' arrow-style parsers were first described in his 2000 paper, but a usable implementation wasn't available until May 2005, when Einar Karttunen released PArrows.

See alsoEdit

Notes

  1. The paper that introduced arrows. It is freely accessible through its publisher.
  2. Those two concepts are usually known as static arrows and Kleisli arrows respectively. Since using the word "arrow" with two subtly different meanings would make this text horribly confusing, we opted for "morphism", which is a synonym for this alternative meaning.
  3. Incidentally, that is why they are called static: the effects are set in stone by the sequencing of computations; the generated values cannot affect them.
  4. For details, see Idioms are oblivious, arrows are meticulous, monads are promiscuous, by Sam Lindley, Philip Wadler and Jeremy Yallop.
  5. "Conveniently" is arguably too strong a word, though, given how confusing handling nested tuples can get. Ergo, proc notation.
  6. Arrow has laws, and so do the other arrow classes we are discussing in these two chapters. We won't pause to pore over the laws here, but you can check them in the Control.Arrow documentation.
  7. Data.Bifunctor was only added to the core GHC libraries in version 7.10, so it might not be installed if you are using an older version. In that case, you can install the bifunctors package, which also includes several other bifunctor-related modules
  8. Swierstra, Duponcheel. Deterministic, error correcting parser combinators. [1]
  9. Parsec is a popular and powerful parsing library. See the parsec documentation on Hackage for more information.

AcknowledgementsEdit