This article is a re-publication of an article that I wrote some time ago for The Monad.Reader magazine, which is an online magazine about functional programming and Haskell. You can also read the article in the original PDF format as part of the Issue 18 (together with two other interesting articles). The samples from the article can be found on Github.

Monad comprehensions have an interesting history. They were the first syntactic extension for programming with monads. They were implemented in Haskell, but later replaced with plain list comprehensions and monadic do notation. Now, monad comprehensions are back in Haskell, more powerful than ever before!

Redesigned monad comprehensions generalize the syntax for working with lists. Quite interestingly, they also generalize syntax for zipping, grouping and ordering of lists. This article shows how to use some of the new expressive power when working with well-known monads. You'll learn what "parallel composition" means for parsers, a poor man's concurrency monad and an evaluation order monad.

## 1  Introduction

This article is inspired by my earlier work on joinads [1], an extension that adds pattern matching on abstract values to the computation expression syntax in F#. Computation expressions are quite similar to the do notation in Haskell. After implementing the F# version of joinads, I wanted to see how the ideas would look in Haskell. I was quite surprised to find out that a recent extension for GHC adds some of the expressive power of joinads to Haskell.

To add some background: the F# computation expression syntax can be used to work with monads, but also with monoids and a few other abstract notions of computation. It also adds several constructs that generalize imperative features of F#, including while and for loops as well as exception handling. The joinads extension adds support for pattern-matching on "monadic values". For example, you can define a parallel programming monad and use joinads to wait until two parallel computations both complete or wait until the first of the two completes returning a value matching a particular pattern.

How are F# joinads related to Haskell? A recent GHC patch implemented by Nils Schweinsberg [2, 3] brings back support for monad comprehensions to Haskell. The change is now a part of the main branch and will be available in GHC starting with the 7.2 release. The patch doesn't just re-implement original monad comprehensions, but also generalizes recent additions to list comprehensions, allowing parallel monad comprehensions and monadic versions of operations like ordering and grouping [4].

The operation that generalizes parallel comprehensions is closely related to a merge operation that I designed for F# joinads. In the rest of this article, I demonstrate some of the interesting programs that can be written using this operation and the elegant syntax provided by the re-designed monad comprehensions.

### 1.1  Quick review of list comprehensions

List comprehensions are a very powerful mechanism for working with lists in Haskell. I expect that you're already familiar with them, but let me start with a few examples. I will use the examples later to demonstrate how the generalized monad comprehension syntax works in a few interesting cases.

If we have a list animals containing "cat" and "dog" and a list sounds containing animal sounds "meow" and "woof", we can write the following snippets:

> [ a ++ " " ++ s | a <- animals, s <- sounds ]
["cat meow","cat woof","dog meow","dog woof"]

> [ a ++ " " ++ s | a <- animals, s <- sounds, a !! 1 == s !! 1 ]
["dog woof"]

> [ a ++ " " ++ s | a <- animals | s <- sounds ]
["cat meow","dog woof"]


The first example uses just the basic list comprehension syntax. It uses two generators to implement a Cartesian product of the two collections. The second example adds a guard to specify that we want only pairs of strings whose second character is the same. The guard serves as an additional filter for the results.

The last example uses parallel list comprehensions. The syntax is available after enabling the ParallelListComp language extension. It allows us to take elements from multiple lists, so that the nth element of the first list is matched with the nth element of the second list. The same functionality can be easily expressed using the zip function.

### 1.2  Generalizing to monad comprehensions

The three examples we've seen in the previous section are straightforward when working with lists. After installing the latest development snapshot of GHC and turning on the MonadComprehensions language extension, we can use the same syntax for working with further notions of computation. If we instantiate the appropriate type classes, we can even use guards, parallel comprehensions and operations like ordering or grouping. Here are some of the type classes and functions that are used by the desugaring:

class Monad m where
(>>=)  :: m a -> (a -> m b) -> m b
return :: a -> m a

mzero :: m a
mplus :: m a -> m a -> m a

mzip :: m a -> m b -> m (a, b)

guard :: MonadPlus m => Bool -> m ()
guard b = if b then return () else mzero


Aside from Monad, the desugaring also uses the MonadPlus and MonadZip type classes. The former is used only for the guard function, which is also defined above. The latter is a new class which has been added as a generalization of parallel list comprehensions. The name of the function makes it clear that the type class is a generalization of the zip function. The patch also defines a MonadGroup type class that generalizes grouping operations inside list comprehensions, but I will not discuss that feature in this article.

You can find the general desugaring rules in the patch description [2]. In this article, we'll just go through the examples from the previous section and examine what the translation looks like. The following declaration shows how to implement the Monad, MondPlus, and MonadZip type classes for lists:

instance Monad [] where
source >>= f = concat $map f source return a = [a] instance MonadPlus [] where mzero = [] mplus = (++) instance MonadZip [] where mzip = zip  The >>= operation, called bind, applies the provided function to each element of the input list and then concatenates the generated lists. The return function creates a singleton list containing the specified value. The mzero value from MonadPlus type class is an empty list, which means that guard returns [()] when the argument is True and the empty list otherwise. Finally, the mzip function for lists is just zip. Now we have everything we need to look at the desugaring of monad comprehensions. The first example from the previous section used multiple generators and can be translated purely in terms of Monad: animals >>= (\a -> sounds >>= (\s -> return$ a ++ " " ++ b))


Every generator is translated to a binding using >>=. The operations are nested, and the innermost operation always returns the result of the output function. The next snippet shows what happens when we add a predicate to filter the results:

animals >>= (\a -> sounds >>= (\s ->
guard (a !! 1 == s !! 1) >>= (\_ ->
return $a ++ " " ++ s) ))  A predicate is translated into a call to the guard function in the innermost part of the desugared expression. When the function returns mzero value (an empty list), the result of the binding will also be mzero, so the element for which the predicate doesn't hold will be filtered out. Finally, let's look at the translation of the last example: (animals mzip sounds) >>= (\(a, s) -> return$ a ++ " " ++ s)


When we use parallel comprehensions, the inputs of the generators are combined using the mzip function. The result is passed to the bind operation, which applies the output function to values of the combined computation. If we also specified filtering, the guard function would be added to the innermost expression, as in the previous example.

As you can see, the translation of monad comprehensions is quite simple, but it adds expressivity to the syntax for working with monads. In particular, the do notation doesn't provide an equivalent syntactic extension for writing parallel comprehensions. (Constructs like generalized ordering, using functions of type m a -> m a, and generalized grouping, using functions of type m a -> m (m a), add even more expressivity, but that's a topic for another article.) In the next three sections, I show how we could implement the mzip operation for several interesting monads, representing parsers, resumable computations, and parallel computations. At the end of the article, I also briefly consider laws that should hold about the mzip operation.

## 2  Composing parsers in parallel

What does a parallel composition of two parsers mean? Probably the best thing we can do is to run both parsers on the input string and return a tuple with the two results. That sounds quite simple, but what is this construct good for? Let's first implement it and then look at some examples.

### 2.1  Introducing parsers

A parser is a function that takes an input string and returns a list of possible results. It may be empty (if the parser fails) or contain several items (if there are multiple ways to parse the input). The implementation I use in this article mostly follows the one by Hutton and Meijer [5].

newtype Parser a
= Parser (String -> [(a, Int, String)])


The result of parsing is a tuple containing a value of type a produced by the parser, the number of characters consumed by the parser, and the remaining unparsed part of the string. The Int value represents the number of characters consumed by the parser. It is not usually included in the definition, but we'll need it in the implementation of mzip.

Now that we have a definition of parsers, we can create our first primitive parser and a function that runs a parser on an input string and returns the results:

item :: Parser Char
item = Parser (\input -> case input of
""   -> []
c:cs -> [(c, 1, cs)])

run :: Parser a -> [a]
run (Parser p) input =
[ result | (result, _, tail) <- p input, tail == [] ]


The item parser returns the first character of the input string. When it succeeds, it consumes a single character, so it returns 1 as the second element of the tuple. The run function applies the underlying function of the parser to a specified input. As specified by the condition tail == [], the function returns the results of those parsers which parsed the entire input. The next step is to make the parser monadic.

### 2.2  Implementing the parser monad

Parsers are well known examples of monads and of monoids. This means that we can implement both the Monad and the MonadPlus type classes for our Parser type. The implementation looks as follows:

instance Monad Parser where
return a = Parser (\input -> [(a, 0, input)])
(Parser p1) >>= f = Parser (\input ->
[ (result, n1 + n2, tail)
| (a, n1, input') <- p1 input
, let (Parser p2) = f a
, (result, n2, tail) <- p2 input' ])

mzero = Parser (\_ -> [])
mplus (Parser p1) (Parser p2) = Parser (\input ->
p1 input ++ p2 input)


The return operation returns a single result containing the specified value that doesn't consume any input. The >>= operation can be implemented using ordinary list comprehensions. It runs the parsers in sequence, returns the result of the second parser and consumes the sum of characters consumed by the first and the second parser. The mzero operation creates a parser that always fails, and mplus represents a nondeterministic choice between two parsers.

The two type class instances allow us to use some of the monad comprehension syntax. We can now use the item primitive to write a few simple parsers:

sat :: (Char -> Bool) -> Parser Char
sat pred   = [ ch | ch <- item, pred ch ]

char, notChar :: Char -> Parser Char
char ch    = sat (ch ==)
notChar ch = sat (ch /=)

some p = [ a:as | a <- p, as <- many p ]
many p = some p mplus return []


The sat function creates a parser that parses a character matching the specified predicate. The generator syntax ch <- item corresponds to monadic binding and is desugared into an application of the >>= operation. Because the Parser type is an instance of MonadPlus, we can use the predicate pred ch as a guard. The desugared version of the function is:

sat pred = item >>= (\ch ->
guard (pred ch) >>= (\_ -> return ch))


| _ <- brackets 'a' 'b' unit, _ <- many $char 'c' ] where unit = return ()  The example uses both parallel and sequential binding, but the sequential composition doesn't involve dependencies on previously parsed results. It uses brackets to parse two groups of the same length, followed (or preceded) by many to consume the remaining group of arbitrary length. ### 2.5 Applicative and parallel parsers Before moving to the next example, let's take a look how mzip relates to applicative functors. As already mentioned, an alternative definition of applicative functors ([8] section 7) uses an operation with exactly the same type signature as mzip. You can find the alternative definition in the following snippet: class Functor f => Monoidal f where unit :: f () (⋆) :: f a -> f b -> f (a, b) pure :: Monoidal f => a -> f a pure a = fmap (const a) unit (<*>) :: Monoidal f => f (a -> b) -> f a -> f b f <*> a = fmap (uncurry ($)) (f ⋆ a)


Applicative functors are more general than monads, which means that every monad is also an applicative functor. When introducing the _mzip_ operation, we attempted to define it in terms of _return_ and _>>=_. That code was actually a definition of ⋆. However, as already explained, we cannot use that definition (or the ⋆ operation) for _mzip_, because that definition isn't always intuitively right. Referring to intuition is always tricky, so I'll express my expectations more formally in terms of laws at the end of the article. However, if we always just used ⋆ as the mzip operation, we wouldn't get any additional expressive power, so there would be no point in using parallel monad comprehensions. The expression [ e | a <- e1 | b <- e2] would mean exactly the same thing as [ e | a <- e1, b <- e2].

For example, the definition of ⋆ for the usual List monad gives us the Cartesian product of lists. This isn't very useful, because we can get that behavior using multiple generators. Instead, parallel list comprehensions use zipping of lists, which comes from a different applicative functor, namely ZipList.

The example with parsers is similar. The implementation of ⋆ would give us sequential composition of parsers. This wouldn't be very useful, so I defined mzip as the intersection of parsers. This is an interesting operation that adds expressive power to the language. The mzip operation also defines an instance of applicative functor for parsers, but a different one.

For the fans of category theory, the ⋆ operation is a natural transformation of a monoidal functor that can be defined by the monad we're working with. The mzip operation can be also viewed as a natural transformation of some monoidal functor, but it may be a different one. One of the additional laws that we can require about mzip is commutativity (more about that later), which means that mzip should be defined by a symmetric monoidal functor.

## 3  Parallelizing cooperative computations

As the name parallel monad comprehensions suggests, we can use the syntax for running computations in parallel. Unlike comprehensions with multiple generators, parallel comprehensions cannot have any dependencies between the composed bindings. This means that the computations can be evaluated independently.

In this section, I demonstrate the idea using a poor man's concurrency monad inspired by Claessen [10]. The monad can be used to implement a lightweight cooperative concurrency. When running two computations in parallel, we can allow interleaving of atomic actions from the two threads.

### 3.1  Modelling resumable computations

The example I demonstrate here models computations using resumptions. This concept is slightly simpler than the original poor man's concurrency monad (which is based on continuations). A resumption is a computation that has either finished and produced some value or has not finished, in which case it can run one atomic step and produce a new resumption:

data Monad m => Resumption m r
= Step (m (Resumption m r))
| Done r


The type is parametrized over a monad and a return type. When evaluating a resumption, we repeatedly run the computation step-by-step. While evaluating, we perform the effects allowed by the monad m until we eventually get a result of type r. If you're interested in more details about the Resumption type, you can find similar definitions in Harrison's cheap threads [11] and Papaspyrou's resumption transformer [12].

Now that we have a type, we can write a function to create a Resumption that runs a single atomic action and then completes, and a function that runs a Resumption:

run :: Monad m => Resumption m r -> m r
run (Done r) = return r
run (Step m) = m >>= run

action :: Monad m => m r -> Resumption m r
action a = Step [ Done r | r <- a ]


The run function takes a resumption that may perform effects specified by the monad m and runs the resumption inside the monad until it reaches Done. The function runs the whole computation sequentially (and it cannot be implemented differently). The cooperative concurrency can be added later by creating a combinator that interleaves the steps of two Resumption computations.

The action function is quite simple. It returns a Step that runs the specified action inside the monad m and then wraps the result using the Done constructor. I implemented the function using monad comprehensions to demonstrate the notation again, but it could be equally written using combinators or the do notation.

### 3.2  Implementing the resumption monad

You can see the implementation of Monad instance for Resumption m below. The return operation creates a new resumption in the "done" state which contains the specified value. The >>= operation constructs a resumption that gets the result of the first resumption and then calls the function f. When the left parameter is Done, we apply the function to the result and wrap the application inside return (because the function is pure) and Step. When the left parameter is Step, we create a resumption that runs the step and then uses >>= recursively to continue running steps from the left argument until it finishes.

instance Monad m => Monad (Resumption m) where
return a = Done a
(Done r) >>= f = Step $return (f r) (Step s) >>= f = Step$ do
next <- s
return $next >>= f instance MonadTrans Resumption where lift = action  The listing also defines an instance of MonadTrans to make Resumption a monad transformer. The lift function takes a computation in the monad m and turns it into a computation in the Resumption monad. This is exactly what our function for wrapping atomic actions does. Equipped with the two type class instances and the run function, we can write some interesting computations. The following function creates a computation that runs for the specified number of steps, prints some string in each step and then returns a specified value: printLoop :: String -> Int -> a -> Resumption IO a printLoop str count result = do lift$ putStrLn str
if count == 1 then return result
else printLoop str (count - 1) result

cats = run $printLoop "meow" 3 "cat"  The function is written using the do notation. It first prints the specified string, using lift to turn the IO () action into a single-step Resumption IO (). When the counter reaches one, it returns the result; otherwise it continues looping. The snippet defines a simple computation cats that prints "meow" three times and then returns a string "cat". The computations created by printLoop are not fully opaque. If we have two computations like cats, we can treat them as sequences of steps and interleave them. This is what the mzip operation does. ### 3.3 Parallel composition of resumptions If we have two resumptions, we can compose them to run in sequence either using the do notation or using a monad comprehension with two generators. To run them in parallel, we need to implement interleaving of the steps as shown in the next snippet: instance Monad m => MonadZip (Resumption m) where mzip (Done a) (Done b) = Done (a, b) mzip sa sb = Step [ mzip a b | a <- step sa, b <- step sb ] where step (Done r) = return$ Done r
step (Step sa) = sa


The result of mzip is a resumption that consists of multiple steps. In each step, it performs one step of both of the resumptions given as arguments. When it reaches a state when both of the resumptions complete and produce results, it returns a tuple containing the results using Done. A step is performed using an effectful step function. To keep the implementation simple, we keep applying step to both of the resumptions and then recursively combine the results. Applying step to a resumption that has already completed isn't a mistake. This operation doesn't do anything and just returns the original resumption (without performing any effects).

Once we define mzip, we can start using the parallel comprehension syntax for working with resumptions. The next snippet demonstrates two ways of composing resumptions. In both examples, we compose a computation that prints "meow" two times and then returns "cat" with a computation that prints "woof" three times and then returns "dog":

animalsSeq =
[ c ++ " and " ++ d
| c <- printLoop "meow" 2 "cat"
, d <- printLoop "woof" 3 "dog" ]

animalsPar =
[ c ++ " and " ++ d
| c <- printLoop "meow" 2 "cat"
| d <- printLoop "woof" 3 "dog" ]


The only difference between the two examples is that the first one composes the operations using multiple generators (separated by comma) and the second one uses parallel comprehensions (separated by bar).

When you run the first example, the program prints meow, meow, woof, woof, woof and then returns a string "cat and dog". The second program interleaves the steps of the two computations and prints meow, woof, meow, woof, woof and then returns the same string.

## 4  Composing computations in parallel

In the previous section, we used the parallel comprehension syntax to create computations that model parallelism using resumptions. Resumptions can be viewed as lightweight cooperative threads. They are useful abstraction, but the simple implementation in the previous section does not give us any speed-up on multi-core CPU. This section will follow a similar approach, but we look at how to implement actual parallelism based on evaluation strategies.

Marlow et al. [13] introduced an Eval monad for explicitly specifying evaluation order. I will start by briefly introducing the monad, so don't worry if you're not familiar with it already. I'll then demonstrate how to define a MonadZip instance for this monad. This way, we can use the parallel comprehension syntax for actually running computations in parallel.

### 4.1  Introducing the evaluation-order monad

The evaluation-order monad is represented by a type Eval a. When writing code inside the monad, we can use several functions of type a -> Eval a that are called strategies. These functions take a value of type a, which may be unevaluated, and wrap it inside the monad. A strategy can specify how to evaluate the (unevaluated) value. The two most common strategies are rpar and rseq (both are functions of type a -> Eval a). The rpar strategy starts evaluating the value in background and rseq evaluates the value eagerly before returning.

A typical pattern is to use the do notation to spawn one computation in parallel and then run another computation sequentially. This way we can easily parallelize two function calls:

fib38 = runEval $do a <- rpar$ fib 36
b <- rseq $fib 37 return$ a + b


The example shows how to calculate the 38th Fibonacci number. It starts calculating fib 36 in parallel with the rest of the computation and then calculates fib 37 sequentially. The do block creates a value of type Eval Integer. We then pass this value to runEval, which returns the wrapped value. Because we used the rpar and rseq combinators when constructing the computation, the returned value will be already evaluated.

However, it is worth noting that the return operation of the monad doesn't specify any evaluation order. The function runEval . return is just an identity function that doesn't force evaluation of the argument. The evaluation order is specified by additional combinators such as rpar.

The Eval monad is implemented in the parallel package [14] and very well explained in the paper by Marlow et al. [13]. You can find the definition of Eval and its monad instance in the snippet below. We don't need to know how rpar and rseq work, so we omit them from the listing.

data Eval a = Done a

runEval :: Eval a -> a
runEval (Done x) = x

return x = Done x
Done x >>= k = k x


The Eval a type is simple. It just wraps a value of type a. The runEval function unwraps the value and the Monad instance implements composition of computations in the usual way. The power of the monad comes from the evaluation annotations we can add. We transform values of type a into values of type Eval a; while doing so, we can specify the evaluation strategy. The strategy is usually given using combinators, but if we add an instance of the MonadZip class, we can also specify the evaluation order using the monad comprehension syntax.

### 4.2  Specifying parallel evaluation order

The mzip operator for the evaluation order monad encapsulates a common pattern that runs two computations in parallel. We've seen an example in the previous section - the first computation is started using rpar and the second one is evaluated eagerly in parallel using rseq. You can find the implementation in the next snippet.

instance MonadZip Eval where
mzip ea eb = do
a <- rpar $runEval ea b <- rseq$ runEval eb
return (a, b)


A tricky aspect of Eval is that it may represent computations with explicitly specified evaluation order (created, for example, using rpar). We can also create computations without specifying evaluation order using return. The fact that return doesn't evaluate the values makes it possible to implement the mzip function, because the required type signature is Eval a -> Eval b -> Eval (a, b).

The two arguments already have to be values of type Eval, but we want to specify the evaluation order using mzip after creating them. If all Eval values were already evaluated, then the mzip operation couldn't have any effect. However, if we create values using return, we can then apply mzip and specify how they should be evaluated later. This means that mzip only works for Eval computations created using return.

Once we understand this, implementing mzip is quite simple. It extracts the underlying (unevaluated) values using runEval, specifies the evaluation order using rpar and rseq and then returns a result of type Eval (a, b), which now carries the evaluation order specification. Let's look how we can write a sequential and parallel version of a snippet that calculates the 38th Fibonacci number:

fibTask n = return $fib n fib38seq = runEval [ a + b | a <- fibTask 36 , b <- fibTask 37 ] fib38par = runEval [ a + b | a <- fibTask 36 | b <- fibTask 37 ]  The snippet first declares a helper function fibTask that creates a delayed value using the sequential fib function and wraps it inside the Eval monad without specifying evaluation strategy. Then we can use the function as a source for generators in the monad comprehension syntax. The first example runs the entire computation sequentially - aside from some wrapping and unwrapping, there are no evaluation order specifications. The second example runs the two sub-computations in parallel. The evaluation order annotations are added by the mzip function from the desugared parallel comprehension syntax. To run the program using multiple threads, you need to compile it using GHC with the -threaded option. Then you can run the resulting application with command line arguments +RTS -N2 -RTS, which specifies that the runtime should use two threads. I measured the performance on a dual-core Intel Core 2 Duo CPU (2.26GHz). The time needed to run the first version was approximately 13 seconds while the second version completes in 9 seconds. ### 4.3 Writing parallel algorithms The +/- 1.4x speedup is less than the maximal 2x speedup, because the example parallelizes two calculations that do not take equally long. To generate a better potential for parallelism, we can implement a recursive pfib function that splits the computation into two parallel branches recursively until it reaches some threshold: pfib :: Integer -> Eval Integer pfib n | n <= 35 = return$ fib n
pfib n = [ a + b | a <- pfib $n - 1 | b <- pfib$ n - 2 ]


I hope you'll agree that the declaration looks quite neat. A nice consequence of using parallel comprehensions is that we can see which parts of the computation will run in parallel without any syntactic noise. We just replace a comma with a bar to get a parallel version! The compiler also prevents us from trying to parallelize code that cannot run in parallel, because of data dependencies. For example, let's look at the Ackermann function:

ack :: Integer -> Integer -> Eval Integer
ack 0 n = return $n + 1 ack m 0 = ack (m - 1) 1 ack m n = [ a | na <- ack m (n - 1) , a <- ack (m - 1) na ]  The Ackermann function is a well-known function from computability theory. It is interesting because it grows very fast (as a result, it cannot be expressed using primitive recursion). For example, the value of ack 4 2 is 265536 - 3. We're probably not going to be able to finish the calculation, no matter how many cores our CPU has. However, we can still try to parallelize the function by replacing the two sequential generators with a parallel comprehension: ack m n = [ a | na <- ack m (n - 1) | a <- ack (m - 1) na ]  If you try compiling this snippet, you get an error message saying Not in scope: na. We can easily see what went wrong if we look at the desugared version: ((ack m (n - 1)) mzip (ack (m - 1) na)) >>= (\(na, a) -> a)  The problem is that the second argument to mzip attempts to access the value na which is defined later. The value is the result of the first expression, so we can access it only after both of the two parallelized operations complete. In other words, there is a data dependency between the computations that we're trying to parallelize. If we were not using parallel monad comprehensions, we could mistakenly think that we can parallelize the function and write the following: ack m n = runEval$ do
na <- rpar $ack m (n - 1) a <- rseq$ ack (m - 1) na
return a


This would compile, but it wouldn't run in parallel! The value na needs to be evaluated before the second call, so the second call to ack will block until the first one completes. This demonstrates a nice aspect of writing parallel computations using the comprehension syntax. Not only that the syntax is elegant, but the desugaring also performs a simple sanity check on our code.

## 5  Parallel comprehension laws

I have intentionally postponed the discussion about laws to the end of the article. So far, we have looked at three different implementations of mzip. I discussed some of the expectations informally to aid intuition. The type of mzip partially specifies how the operation should behave, but not all well-typed implementations are intuitively right.

In my understanding, the laws about mzip are still subject to discussion, although some were already proposed in the discussion about the GHC patch [2]. I hope to contribute to the discussion in this section. We first look at the laws that can be motivated by the category theory behind the operation, and then discuss additional laws inspired by the work on F# joinads.

### 5.1  Basic laws of parallel bindings

As already briefly mentioned, the mzip operation can be viewed as a natural transformation defined by some monoidal functor [15]. In practice, this means that the mzip operation should obey two laws. The first one is usually called naturality and it specifies the behavior of mzip with respect to the map function of a monad (the function can be implemented in terms of bind and return and corresponds to liftM from the Haskell base library). The second law is associativity, and we can express it using a helper function assoc ((a,b),c) = (a,(b,c)):

map (f × g) (mzip a b) ≡ mzip (map f a) (map g b) (naturality)
mzip a (mzip b c) ≡ map assoc (mzip (mzip a b) c) (associativity)

The naturality law specifies that we can change the order of applying mzip and map. The equation has already been identified as a law in the discussion about the patch [2]. The law is also required by applicative functors [8] - this is not surprising as applicative functors are also monoidal.

The associativity law is also very desirable. When we write a comprehension such as [e | a <- m1 | b <- m2 | c <- m3], the desugaring first needs to zip two of the three inputs and then zip the third with the result, because mzip is a binary operation. The order of zipping feels like an implementation detail, but if we don't require associativity, it may affect the meaning of our code.

### 5.2  Parallel binding as monoidal functor

A monoidal functor in category theory defines a natural transformation (corresponding to our mzip), but also a special value called unit. A Haskell representation of monoidal functor is the Monoidal type class. For a monoidal functor f, the type of unit is f (). Every applicative functor in Haskell has a unit. For example, the unit value for ZipList is an infinite list of () values. We do not necessarily need to add unit to the definition of MonadZip, because it is not needed by the desugaring. However, it is interesting to explore how a special munit value would behave.

The laws for monoidal functors specify that if we combine munit with any other value using mzip, we can recover the original value using map snd. In the language of monad comprehensions, the law says that the expression [ e | a <- m ] should be equivalent to [ e | a <- m | () <- munit].

It is important to realize that the computation created using monadic return isn't the same thing as unit of the monoidal functor associated with mzip. For lists, return () creates a singleton list. The return operation is unit of a monoidal functor defined by the monad, but the unit associated with mzip belongs to a different monoidal functor!

### 5.3  Symmetry of parallel binding

Another sensible requirement for mzip (which exists in F# joinads) is that reordering of the arguments only changes the order of elements in the resulting tuple. In theory, this means that the monoidal functor defining mzip is symmetric. We can specify the law using a helper function swap (a,b) = (b,a):

mzip a b ≡ map swap (mzip a b) (symmetry)

The symmetry law specifies that [(x, y) | x <- a | y <- b] should mean the same thing as [(x, y) | y <- b | x <- a]. This may look like a very strong requirement, but it fits quite well with the usual intuition about the zip operation and parallel monad comprehensions. The symmetry law holds for lists, parsers and the evaluation order monad. For the poor man's concurrency monad, it holds if we treat effects that occur within a single step of the evaluation as unordered (which may be a reasonable interpretation). For some monads, such as the State monad, it is not possible to define a symmetric mzip operation.

The three laws that we've seen so far are motivated by the category theory laws for the (symmetric) monoidal functors that define our mzip operation. However, we also need to relate the functors in some way to the monads with which we are combining them.

### 5.4  Relations with additional operations

The discussion about the patch [2] suggests one more law that relates the mzip operation with the map operation of the monad, called information preservation:

map fst (mzip a b) ≡ a ≡ map snd (mzip b a) (information preservation)

The law specifies that combining a computation with some other computation using mzip and then recovering the original form of the value using map doesn't lose information. However, this law is a bit tricky. For example, it doesn't hold for lists if a and b are lists of different length, since the zip function restricts the length of the result to the length of the shorter list. Similarly, it doesn't hold for parsers (from the first section) that consume a different number of characters.

However, the law expresses an important requirement: when we combine certain computations, it should be possible to recover the original components. A law that is similar to (\ref{law:ipreserve}) holds for applicative functors, but with a slight twist:

map fst (mzip a munit) ≡ a ≡ map snd (mzip munit a) (applicative information preservation)

Instead of zipping two arbitrary monadic values, the law for applicative functors zips an arbitrary value with unit. In case of lists, unit is an infinite list, so the law holds. Intuitively, this holds because unit has a maximal structure (in this case, the structure is the length of the list).

Ideally, we'd like to say that combining two values with the same structure creates a new value which also has the same structure. There is no way to refer to the "structure" of a monadic value directly, but we can create values with a given structure using map. This weaker law holds for both lists and parsers. Additionally, we can describe the case when one of the two computations is mzero and when both of them are created using return:

map fst (mzip a (map f a)) ≡ a ≡ map snd (mzip (map g a) a) (weak information preservation)
map fst (mzip a mzero) ≡ mzero ≡ map snd (mzip mzero a) (zero)
mzip (return a) (return b) ≡ return (a, b)} (unit merge)

The weak information preservation law is quite similar to the original information preservation law. The only difference is that instead of zipping with an arbitrary monadic value b, we're zipping a with a value constructed using map (for any total functions f and g). This means that the actual value(s) that the monadic value contains (or produces) can be different, but the structure will be the same, because map is required to preserve the structure.

The zero law specifies that mzero is the zero element with respect to mzip. It almost seems that it holds of necessity, because mzero with type m a doesn't contain any value of type a, so there is no way we could construct a value of type (a,b). This second law complements the first, but it is not difficult to see that it contradicts the original information preservation.

Finally, the unit merge law describes how mzip works with respect to the monadic return operation. This is interesting, because we're relating mzip of one applicative functor with the unit of another applicative functor (defined by the monad). The law isn't completely arbitrary; an equation with a similar structure is required for causal commutative arrows [16]. In terms of monad comprehension syntax, the law says that [ e | a <- return e1 | b <- return e2 ] is equivalent to [ e | (a, b) <- return (e1, e2)].

I believe that the three laws I proposed in this section partly answer the question of how to relate the two structures combined by parallel monad comprehensions - the MonadPlus type class with the symmetric monoidal functor that defines mzip.

## Conclusions

After some time, monad comprehensions are back in Haskell! The recent GHC patch makes them even more useful by generalizing additional features of list comprehensions including parallel binding and support for operations like ordering and grouping. In this article, I focused on the first generalization, although the remaining two are equally interesting.

We looked at three examples: parallel composition of parsers (which applies parsers to the same input), parallel composition of resumptions (which interleaves the steps of computations) and parallel composition of an evaluation order monad (which runs two computations in parallel). Some of the examples are inspired by my previous work on joinads that add a similar language extension to F#. The F# version includes an operation very similar to mzip from the newly included MonadZip type class. I also proposed several laws - some of them inspired by category theory and some by my work on F# joinads - hoping that this article may contribute to the discussion about the laws required by MonadZip.

### Acknowledgements

Thanks to Alan Mycroft for inspiring discussion about some of the monads demonstrated in this article and to Dominic Orchard for many useful comments on a draft of the article as well as discussion about category theory and the mzip` laws. I'm also grateful to Brent Yorgey for proofreading the article and suggesting numerous improvements.