# 7.6. Arrow notation

Arrows are a generalization of monads introduced by John Hughes. For more details, see

• “Generalising Monads to Arrows”, John Hughes, in Science of Computer Programming 37, pp67–111, May 2000.

• A New Notation for Arrows”, Ross Paterson, in ICFP, Sep 2001.

• Arrows and Computation”, Ross Paterson, in The Fun of Programming, Palgrave, 2003.

and the arrows web page at http://www.haskell.org/arrows/. With the -farrows flag, GHC supports the arrow notation described in the second of these papers. What follows is a brief introduction to the notation; it won't make much sense unless you've read Hughes's paper. This notation is translated to ordinary Haskell, using combinators from the Control.Arrow module.

The extension adds a new kind of expression for defining arrows, of the form proc pat -> cmd, where proc is a new keyword. The variables of the pattern are bound in the body of the proc-expression, which is a new sort of thing called a command. The syntax of commands is as follows:
 ```cmd ::= exp1 -< exp2 | exp1 -<< exp2 | do { cstmt1 .. cstmtn ; cmd } | let decls in cmd | if exp then cmd1 else cmd2 | case exp of { calts } | cmd1 qop cmd2 | (| aexp cmd1 .. cmdn |) | \ pat1 .. patn -> cmd | cmd aexp | ( cmd ) cstmt ::= let decls | pat <- cmd | rec { cstmt1 .. cstmtn } | cmd```
Commands produce values, but (like monadic computations) may yield more than one value, or none, and may do other things as well. For the most part, familiarity with monadic notation is a good guide to using commands. However the values of expressions, even monadic ones, are determined by the values of the variables they contain; this is not necessarily the case for commands.

A simple example of the new notation is the expression
 `proc x -> f -< x+1`
We call this a procedure or arrow abstraction. As with a lambda expression, the variable x is a new variable bound within the proc-expression. It refers to the input to the arrow. In the above example, -< is not an identifier but an new reserved symbol used for building commands from an expression of arrow type and an expression to be fed as input to that arrow. (The weird look will make more sense later.) It may be read as analogue of application for arrows. The above example is equivalent to the Haskell expression
 `arr (\ x -> x+1) >>> f`
That would make no sense if the expression to the left of -< involves the bound variable x. More generally, the expression to the left of -< may not involve any local variable, i.e. a variable bound in the current arrow abstraction. For such a situation there is a variant -<<, as in
 `proc x -> f x -<< x+1`
which is equivalent to
 `arr (\ x -> (f, x+1)) >>> app`
so in this case the arrow must belong to the ArrowApply class. Such an arrow is equivalent to a monad, so if you're using this form you may find a monadic formulation more convenient.

## 7.6.1. do-notation for commands

Another form of command is a form of do-notation. For example, you can write
 ```proc x -> do y <- f -< x+1 g -< 2*y let z = x+y t <- h -< x*z returnA -< t+z```
You can read this much like ordinary do-notation, but with commands in place of monadic expressions. The first line sends the value of x+1 as an input to the arrow f, and matches its output against y. In the next line, the output is discarded. The arrow returnA is defined in the Control.Arrow module as arr id. The above example is treated as an abbreviation for
 ```arr (\ x -> (x, x)) >>> first (arr (\ x -> x+1) >>> f) >>> arr (\ (y, x) -> (y, (x, y))) >>> first (arr (\ y -> 2*y) >>> g) >>> arr snd >>> arr (\ (x, y) -> let z = x+y in ((x, z), z)) >>> first (arr (\ (x, z) -> x*z) >>> h) >>> arr (\ (t, z) -> t+z) >>> returnA```
Note that variables not used later in the composition are projected out. After simplification using rewrite rules (see Section 7.9) defined in the Control.Arrow module, this reduces to
 ```arr (\ x -> (x+1, x)) >>> first f >>> arr (\ (y, x) -> (2*y, (x, y))) >>> first g >>> arr (\ (_, (x, y)) -> let z = x+y in (x*z, z)) >>> first h >>> arr (\ (t, z) -> t+z)```
which is what you might have written by hand. With arrow notation, GHC keeps track of all those tuples of variables for you.

Note that although the above translation suggests that let-bound variables like z must be monomorphic, the actual translation produces Core, so polymorphic variables are allowed.

It's also possible to have mutually recursive bindings, using the new rec keyword, as in the following example:
 ```counter :: ArrowCircuit a => a Bool Int counter = proc reset -> do rec output <- returnA -< if reset then 0 else next next <- delay 0 -< output+1 returnA -< output```
The translation of such forms uses the loop combinator, so the arrow concerned must belong to the ArrowLoop class.

## 7.6.2. Conditional commands

In the previous example, we used a conditional expression to construct the input for an arrow. Sometimes we want to conditionally execute different commands, as in
 ```proc (x,y) -> if f x y then g -< x+1 else h -< y+2```
which is translated to
 ```arr (\ (x,y) -> if f x y then Left x else Right y) >>> (arr (\x -> x+1) >>> f) ||| (arr (\y -> y+2) >>> g)```
Since the translation uses |||, the arrow concerned must belong to the ArrowChoice class.

There are also case commands, like
 ```case input of [] -> f -< () [x] -> g -< x+1 x1:x2:xs -> do y <- h -< (x1, x2) ys <- k -< xs returnA -< y:ys```
The syntax is the same as for case expressions, except that the bodies of the alternatives are commands rather than expressions. The translation is similar to that of if commands.

## 7.6.3. Defining your own control structures

As we're seen, arrow notation provides constructs, modelled on those for expressions, for sequencing, value recursion and conditionals. But suitable combinators, which you can define in ordinary Haskell, may also be used to build new commands out of existing ones. The basic idea is that a command defines an arrow from environments to values. These environments assign values to the free local variables of the command. Thus combinators that produce arrows from arrows may also be used to build commands from commands. For example, the ArrowChoice class includes a combinator
 `ArrowChoice a => (<+>) :: a e c -> a e c -> a e c`
so we can use it to build commands:
 ```expr' = proc x -> returnA -< x <+> do symbol Plus -< () y <- term -< () expr' -< x + y <+> do symbol Minus -< () y <- term -< () expr' -< x - y```
This is equivalent to
 ```expr' = (proc x -> returnA -< x) <+> (proc x -> do symbol Plus -< () y <- term -< () expr' -< x + y) <+> (proc x -> do symbol Minus -< () y <- term -< () expr' -< x - y)```
It is essential that this operator be polymorphic in e (representing the environment input to the command and thence to its subcommands) and satisfy the corresponding naturality property
 `arr k >>> (f <+> g) = (arr k >>> f) <+> (arr k >>> g)`
at least for strict k. (This should be automatic if you're not using seq.) This ensures that environments seen by the subcommands are environments of the whole command, and also allows the translation to safely trim these environments. The operator must also not use any variable defined within the current arrow abstraction.

We could define our own operator
 ```untilA :: ArrowChoice a => a e () -> a e Bool -> a e () untilA body cond = proc x -> if cond x then returnA -< () else do body -< x untilA body cond -< x```
and use it in the same way. Of course this infix syntax only makes sense for binary operators; there is also a more general syntax involving special brackets:
 ```proc x -> do y <- f -< x+1 (|untilA (increment -< x+y) (within 0.5 -< x)|)```

## 7.6.4. Primitive constructs

Some operators will need to pass additional inputs to their subcommands. For example, in an arrow type supporting exceptions, the operator that attaches an exception handler will wish to pass the exception that occurred to the handler. Such an operator might have a type
 `handleA :: ... => a e c -> a (e,Ex) c -> a e c`
where Ex is the type of exceptions handled. You could then use this with arrow notation by writing a command
 `body `handleA` \ ex -> handler`
so that if an exception is raised in the command body, the variable ex is bound to the value of the exception and the command handler, which typically refers to ex, is entered. Though the syntax here looks like a functional lambda, we are talking about commands, and something different is going on. The input to the arrow represented by a command consists of values for the free local variables in the command, plus a stack of anonymous values. In all the prior examples, this stack was empty. In the second argument to handleA, this stack consists of one value, the value of the exception. The command form of lambda merely gives this value a name.

More concretely, the values on the stack are paired to the right of the environment. So when designing operators like handleA that pass extra inputs to their subcommands, More precisely, the type of each argument of the operator (and its result) should have the form
 `a (...(e,t1), ... tn) t`
where e is a polymorphic variable (representing the environment) and ti are the types of the values on the stack, with t1 being the "top". The polymorphic variable e must not occur in a, ti or t. However the arrows involved need not be the same. Here are some more examples of suitable operators:
 ```bracketA :: ... => a e b -> a (e,b) c -> a (e,c) d -> a e d runReader :: ... => a e c -> a' (e,State) c runState :: ... => a e c -> a' (e,State) (c,State)```
We can supply the extra input required by commands built with the last two by applying them to ordinary expressions, as in
 ```proc x -> do s <- ... (|runReader (do { ... })|) s```
which adds s to the stack of inputs to the command built using runReader.

The command versions of lambda abstraction and application are analogous to the expression versions. In particular, the beta and eta rules describe equivalences of commands. These three features (operators, lambda abstraction and application) are the core of the notation; everything else can be built using them, though the results would be somewhat clumsy. For example, we could simulate do-notation by defining
 ```bind :: Arrow a => a e b -> a (e,b) c -> a e c u `bind` f = returnA &&& u >>> f bind_ :: Arrow a => a e b -> a e c -> a e c u `bind_` f = u `bind` (arr fst >>> f)```
We could simulate do by defining
 ```cond :: ArrowChoice a => a e b -> a e b -> a (e,Bool) b cond f g = arr (\ (e,b) -> if b then Left e else Right e) >>> f ||| g```

## 7.6.5. Differences with the paper

• Instead of a single form of arrow application (arrow tail) with two translations, the implementation provides two forms "-<" (first-order) and "-<<" (higher-order).

• User-defined operators are flagged with banana brackets instead of a new form keyword.

## 7.6.6. Portability

Although only GHC implements arrow notation directly, there is also a preprocessor (available from the arrows web page>) that translates arrow notation into Haskell 98 for use with other Haskell systems. You would still want to check arrow programs with GHC; tracing type errors in the preprocessor output is not easy. Modules intended for both GHC and the preprocessor must observe some additional restrictions:

• The module must import Control.Arrow.

• The preprocessor cannot cope with other Haskell extensions. These would have to go in separate modules.

• Because the preprocessor targets Haskell (rather than Core), let-bound variables are monomorphic.