There are quite a few of questions here about whether or not certain transformations of types that involve Monads are possible.
For instance, it's possible to make a function of type f :: Monad m => [m a] -> m [a], but impossible to make a function of type g :: Monad m => m [a] -> [m a] as a proper antifunction to the former. (IE: f . g = id)
I want to understand what rules one can use to determine if a function of that type can or cannot be constructed, and why these types cannot be constructed if they disobey these rules.
The way that I've always thought about monads is that a value of type Monad m => m a is some program of type m that executes and produces an a. The monad laws reinforce this notion by thinking of composition of these programs as "do thing one then do thing two", and produce some sort of combination of the results.
Right unit Taking a program and just returning its value should
be the same as just running the original program.
m >>= return = m
Left unit If you create a simple program that just returns a value,
and then pass that value to a function that creates a new program, then
the resulting program should just be as if you called the function on the
value.
return x >>= f = f x
Associativity If you execute a program m, feed its result into a function f that produces another program, and then feed that result into a third function g that also produces a program, then this is identical to creating a new function that returns a program based on feeding the result of f into g, and feeding the result of m into it.
(m >>= f) >>= g = m >>= (\x -> f x >>= g)
Using this intuition about a "program that creates a value" can come to some conclusions about what it means for the functions that you've provided in your examples.
Monad m => [m a] -> m [a] Deviating from the intuitive definition of what this function should do is hard: Execute each program in sequence and collect the results. This produces another program that produces a list of results.
Monad m => m [a] -> [m a] This doesn't really have a clear intuitive definition, since it's a program that produces a list. You can't create a list without getting access to the resulting values which in this case means executing a program. Certain monads, that have a clear way to extract a value from a program, and provide some variant of m a -> a, like the State monad, can have sane implementations of some function like this. It would have to be application specific though. Other monads, like IO, you cannot escape from.
Monad m => (a -> m b) -> m (a -> b) This also doesn't really have a clear intuitive definition. Here you have a function f that produces a program of type m b, but you're trying to return a function of type m (a -> b). Based on the a, f creates completely different programs with different executing semantics. You cannot encompass these variations in a single program of type m (a -> b), even if you can provide a proper mapping of a -> b in the programs resulting value.
This intuition doesn't really encompass the idea behind monads completely. For example, the monadic context of a list doesn't really behave like a program.
Something easy to remember is : "you can't escape from a Monad" (it's kind of design for it). Transforming m [a] to [m a] is a form of escape, so you can't.
On the other hand you can easily create a Monad from something (using return) so traversing ([m a] -> m [a]) is usually possible.
If you take a look at "Monad laws", monad only constrain you to define a composition function but not reverse function.
In the first example you can compose the list elements.
In the second one Monad m => m [a] -> [m a], you cannot split an action into multiple actions ( action composition is not reversible).
Example:
Let's say you have to read 2 values.
s1 <- action
s2 <- action
Doing so, action result s2 depends by the side effect made by s1.
You can bind these 2 actions in 1 action to be executed in the same order, but you cannot split them and execute action from s2 without s1 made the side effect needed by the second one.
Not really an answer, and much too informal for my linking, but nevertheless I have a few interesting observations that won't fit into a comment. First, let's consider this function you refer to:
f :: Monad m => [m a] -> m [a]
This signature is in fact stronger than it needs to be. The current generalization of this is the sequenceA function from Data.Traversable:
sequenceA :: (Traversable t, Applicative f) -> t (f a) -> f (t a)
...which doesn't need the full power of Monad, and can work with any Traversable and not just lists.
Second: the fact that Traversable only requires Applicative is I think really significant to this question, because applicative computations have a "list-like" structure. Every applicative computation can be rewritten to have the form f <$> a1 <*> ... <*> an for some f. Or, informally, every applicative computation can be seen as a list of actions a1, ... an (heterogeneous on the result type, homogeneous in the functor type), plus an n-place function to combine their results.
If we look at sequenceA through this lens, all it does is choose an f built out of the appropriate nested number of list constructors:
sequenceA [a1, ..., an] == f <$> a1 <*> ... <*> an
where f v1 ... vn = v1 : ... : vn : []
Now, I haven't had the chance to try and prove this yet, but my conjectures would be the following:
Mathematically speaking at least, sequenceA has a left inverse in free applicative functors. If you have a Functor f => [FreeA f a] and you sequenceA it, what you get is a list-like structure that contains those computations and a combining function that makes a list out of their results. I suspect however that it's not possible to write such a function in Haskell (unSequenceFreeA :: (Traversable t, Functor f) => FreeA f (t a) -> Maybe (t (Free f a))), because you can't pattern match on the structure of the combining function in the FreeA to tell that it's of the form f v1 ... vn = v1 : ... : vn : [].
sequenceA doesn't have a right inverse in a free applicative, however, because the combining function that produces a list out of the results from the a1, ... an actions may do anything; for example, return a constant list of arbitrary length (unrelated to the computations that the free applicative value performs).
Once you move to non-free applicative functors, there will no longer be a left inverse for sequenceA, because the non-free applicative functor's equations translate into cases where you can no longer tell apart which of two t (f a) "action lists" was the source for a given f (t a) "list-producing action."
While hacking something up earlier, I created the following code:
newtype Callback a = Callback { unCallback :: a -> IO (Callback a) }
liftCallback :: (a -> IO ()) -> Callback a
liftCallback f = let cb = Callback $ \x -> (f x >> return cb) in cb
runCallback :: Callback a -> IO (a -> IO ())
runCallback cb =
do ref <- newIORef cb
return $ \x -> readIORef ref >>= ($ x) . unCallback >>= writeIORef ref
Callback a represents a function that handles some data and returns a new callback that should be used for the next notification. A callback which can basically replace itself, so to speak. liftCallback just lifts a normal function to my type, while runCallback uses an IORef to convert a Callback to a simple function.
The general structure of the type is:
data T m a = T (a -> m (T m a))
It looks much like this could be isomorphic to some well-known mathematical structure from category theory.
But what is it? Is it a monad or something? An applicative functor? A transformed monad? An arrow, even? Is there a search engine similar Hoogle that lets me search for general patterns like this?
The term you are looking for is free monad transformer. The best place to learn how these work is to read the "Coroutine Pipelines" article in issue 19 of The Monad Reader. Mario Blazevic gives a very lucid description of how this type works, except he calls it the "Coroutine" type.
I wrote up his type in the transformers-free package and then it got merged into the free package, which is its new official home.
Your Callback type is isomorphic to:
type Callback a = forall r . FreeT ((->) a) IO r
To understand free monad transformers, you need to first understand free monads, which are just abstract syntax trees. You give the free monad a functor which defines a single step in the syntax tree, and then it creates a Monad from that Functor that is basically a list of those types of steps. So if you had:
Free ((->) a) r
That would be a syntax tree that accepts zero or more as as input and then returns a value r.
However, usually we want to embed effects or make the next step of the syntax tree dependent on some effect. To do that, we simply promote our free monad to a free monad transformer, which interleaves the base monad between syntax tree steps. In the case of your Callback type, you are interleaving IO in between each input step, so your base monad is IO:
FreeT ((->) a) IO r
The nice thing about free monads is that they are automatically monads for any functor, so we can take advantage of this to use do notation to assemble our syntax tree. For example, I can define an await command that will bind the input within the monad:
import Control.Monad.Trans.Free
await :: (Monad m) => FreeT ((->) a) m a
await = liftF id
Now I have a DSL for writing Callbacks:
import Control.Monad
import Control.Monad.Trans.Free
printer :: (Show a) => FreeT ((->) a) IO r
printer = forever $ do
a <- await
lift $ print a
Notice that I never had to define the necessary Monad instance. Both FreeT f and Free f are automatically Monads for any functor f, and in this case ((->) a) is our functor, so it automatically does the right thing. That's the magic of category theory!
Also, we never had to define a MonadTrans instance in order to use lift. FreeT f is automatically a monad transformer, given any functor f, so it took care of that for us, too.
Our printer is a suitable Callback, so we can feed it values just by deconstructing the free monad transformer:
feed :: [a] -> FreeT ((->) a) IO r -> IO ()
feed as callback = do
x <- runFreeT callback
case x of
Pure _ -> return ()
Free k -> case as of
[] -> return ()
b:bs -> feed bs (k b)
The actual printing occurs when we bind runFreeT callback, which then gives us the next step in the syntax tree, which we feed the next element of the list.
Let's try it:
>>> feed [1..5] printer
1
2
3
4
5
However, you don't even need to write all this up yourself. As Petr pointed out, my pipes library abstracts common streaming patterns like this for you. Your callback is just:
forall r . Consumer a IO r
The way we'd define printer using pipes is:
printer = forever $ do
a <- await
lift $ print a
... and we can feed it a list of values like so:
>>> runEffect $ each [1..5] >-> printer
1
2
3
4
5
I designed pipes to encompass a very large range of streaming abstractions like these in such a way that you can always use do notation to build each streaming component. pipes also comes with a wide variety of elegant solutions for things like state and error handling, and bidirectional flow of information, so if you formulate your Callback abstraction in terms of pipes, you tap into a ton of useful machinery for free.
If you want to learn more about pipes, I recommend you read the tutorial.
The general structure of the type looks to me like
data T (~>) a = T (a ~> T (~>) a)
where (~>) = Kleisli m in your terms (an arrow).
Callback itself doesn't look like an instance of any standard Haskell typeclass I can think of, but it is a Contravariant Functor (also known as Cofunctor, misleadingly as it turns out). As it is not included in any of the libraries that come with GHC, there exist several definitions of it on Hackage (use this one), but they all look something like this:
class Contravariant f where
contramap :: (b -> a) -> f a -> f b
-- c.f. fmap :: (a -> b) -> f a -> f b
Then
instance Contravariant Callback where
contramap f (Callback k) = Callback ((fmap . liftM . contramap) f (f . k))
Is there some more exotic structure from category theory that Callback possesses? I don't know.
I think that this type is very close to what I have heard called a 'Circuit', which is a type of arrow. Ignoring for a moment the IO part (as we can have this just by transforming a Kliesli arrow) the circuit transformer is:
newtype CircuitT a b c = CircuitT { unCircuitT :: a b (c, CircuitT a b c) }
This is basicall an arrow that returns a new arrow to use for the next input each time. All of the common arrow classes (including loop) can be implemented for this arrow transformer as long as the base arrow supports them. Now, all we have to do to make it notionally the same as the type you mention is to get rid of that extra output. This is easily done, and so we find:
Callback a ~=~ CircuitT (Kleisli IO) a ()
As if we look at the right hand side:
CircuitT (Kleisli IO) a () ~=~
(Kliesli IO) a ((), CircuitT (Kleisli IO) a ()) ~=~
a -> IO ((), CircuitT (Kliesli IO) a ())
And from here, you can see how this is similar to Callback a, except we also output a unit value. As the unit value is in a tuple with something else anyway, this really doesn't tell us much, so I would say they're basically the same.
N.B. I used ~=~ for similar but not entirely equivalent to, for some reason. They are very closely similar though, in particular note that we could convert a Callback a into a CircuitT (Kleisli IO) a () and vice-versa.
EDIT: I would also fully agree with the ideas that this is A) a monadic costream (monadic operation expecitng an infinite number of values, I think this means) and B) a consume-only pipe (which is in many ways very similar to the circuit type with no output, or rather output set to (), as such a pipe could also have had output).
Just an observation, your type seems quite related to Consumer p a m appearing in the pipes library (and probably other similar librarties as well):
type Consumer p a = p () a () C
-- A Pipe that consumes values
-- Consumers never respond.
where C is an empty data type and p is an instance of Proxy type class. It consumes values of type a and never produces any (because its output type is empty).
For example, we could convert a Callback into a Consumer:
import Control.Proxy
import Control.Proxy.Synonym
newtype Callback m a = Callback { unCallback :: a -> m (Callback m a) }
-- No values produced, hence the polymorphic return type `r`.
-- We could replace `r` with `C` as well.
consumer :: (Proxy p, Monad m) => Callback m a -> () -> Consumer p a m r
consumer c () = runIdentityP (run c)
where
run (Callback c) = request () >>= lift . c >>= run
See the tutorial.
(This should have been rather a comment, but it's a bit too long.)
I am learning Haskell and trying to understand Monads. I have two questions:
From what I understand, Monad is just another typeclass that declares ways to interact with data inside "containers", including Maybe, List, and IO. It seems clever and clean to implement these 3 things with one concept, but really, the point is so there can be clean error handling in a chain of functions, containers, and side effects. Is this a correct interpretation?
How exactly is the problem of side-effects solved? With this concept of containers, the language essentially says anything inside the containers is non-deterministic (such as i/o). Because lists and IOs are both containers, lists are equivalence-classed with IO, even though values inside lists seem pretty deterministic to me. So what is deterministic and what has side-effects? I can't wrap my head around the idea that a basic value is deterministic, until you stick it in a container (which is no special than the same value with some other values next to it, e.g. Nothing) and it can now be random.
Can someone explain how, intuitively, Haskell gets away with changing state with inputs and output? I'm not seeing the magic here.
The point is so there can be clean error handling in a chain of functions, containers, and side effects. Is this a correct interpretation?
Not really. You've mentioned a lot of concepts that people cite when trying to explain monads, including side effects, error handling and non-determinism, but it sounds like you've gotten the incorrect sense that all of these concepts apply to all monads. But there's one concept you mentioned that does: chaining.
There are two different flavors of this, so I'll explain it two different ways: one without side effects, and one with side effects.
No Side Effects:
Take the following example:
addM :: (Monad m, Num a) => m a -> m a -> m a
addM ma mb = do
a <- ma
b <- mb
return (a + b)
This function adds two numbers, with the twist that they are wrapped in some monad. Which monad? Doesn't matter! In all cases, that special do syntax de-sugars to the following:
addM ma mb =
ma >>= \a ->
mb >>= \b ->
return (a + b)
... or, with operator precedence made explicit:
ma >>= (\a -> mb >>= (\b -> return (a + b)))
Now you can really see that this is a chain of little functions, all composed together, and its behavior will depend on how >>= and return are defined for each monad. If you're familiar with polymorphism in object-oriented languages, this is essentially the same thing: one common interface with multiple implementations. It's slightly more mind-bending than your average OOP interface, since the interface represents a computation policy rather than, say, an animal or a shape or something.
Okay, let's see some examples of how addM behaves across different monads. The Identity monad is a decent place to start, since its definition is trivial:
instance Monad Identity where
return a = Identity a -- create an Identity value
(Identity a) >>= f = f a -- apply f to a
So what happens when we say:
addM (Identity 1) (Identity 2)
Expanding this, step by step:
(Identity 1) >>= (\a -> (Identity 2) >>= (\b -> return (a + b)))
(\a -> (Identity 2) >>= (\b -> return (a + b)) 1
(Identity 2) >>= (\b -> return (1 + b))
(\b -> return (1 + b)) 2
return (1 + 2)
Identity 3
Great. Now, since you mentioned clean error handling, let's look at the Maybe monad. Its definition is only slightly trickier than Identity:
instance Monad Maybe where
return a = Just a -- same as Identity monad!
(Just a) >>= f = f a -- same as Identity monad again!
Nothing >>= _ = Nothing -- the only real difference from Identity
So you can imagine that if we say addM (Just 1) (Just 2) we'll get Just 3. But for grins, let's expand addM Nothing (Just 1) instead:
Nothing >>= (\a -> (Just 1) >>= (\b -> return (a + b)))
Nothing
Or the other way around, addM (Just 1) Nothing:
(Just 1) >>= (\a -> Nothing >>= (\b -> return (a + b)))
(\a -> Nothing >>= (\b -> return (a + b)) 1
Nothing >>= (\b -> return (1 + b))
Nothing
So the Maybe monad's definition of >>= was tweaked to account for failure. When a function is applied to a Maybe value using >>=, you get what you'd expect.
Okay, so you mentioned non-determinism. Yes, the list monad can be thought of as modeling non-determinism in a sense... It's a little weird, but think of the list as representing alternative possible values: [1, 2, 3] is not a collection, it's a single non-deterministic number that could be either one, two or three. That sounds dumb, but it starts to make some sense when you think about how >>= is defined for lists: it applies the given function to each possible value. So addM [1, 2] [3, 4] is actually going to compute all possible sums of those two non-deterministic values: [4, 5, 5, 6].
Okay, now to address your second question...
Side Effects:
Let's say you apply addM to two values in the IO monad, like:
addM (return 1 :: IO Int) (return 2 :: IO Int)
You don't get anything special, just 3 in the IO monad. addM does not read or write any mutable state, so it's kind of no fun. Same goes for the State or ST monads. No fun. So let's use a different function:
fireTheMissiles :: IO Int -- returns the number of casualties
Clearly the world will be different each time missiles are fired. Clearly. Now let's say you're trying to write some totally innocuous, side effect free, non-missile-firing code. Perhaps you're trying once again to add two numbers, but this time without any monads flying around:
add :: Num a => a -> a -> a
add a b = a + b
and all of a sudden your hand slips, and you accidentally typo:
add a b = a + b + fireTheMissiles
An honest mistake, really. The keys were so close together. Fortunately, because fireTheMissiles was of type IO Int rather than simply Int, the compiler is able to avert disaster.
Okay, totally contrived example, but the point is that in the case of IO, ST and friends, the type system keeps effects isolated to some specific context. It doesn't magically eliminate side effects, making code referentially transparent that shouldn't be, but it does make it clear at compile time what scope the effects are limited to.
So getting back to the original point: what does this have to do with chaining or composition of functions? Well, in this case, it's just a handy way of expressing a sequence of effects:
fireTheMissilesTwice :: IO ()
fireTheMissilesTwice = do
a <- fireTheMissiles
print a
b <- fireTheMissiles
print b
Summary:
A monad represents some policy for chaining computations. Identity's policy is pure function composition, Maybe's policy is function composition with failure propogation, IO's policy is impure function composition and so on.
Let me start by pointing at the excellent "You could have invented monads" article. It illustrates how the Monad structure can naturally manifest while you are writing programs. But the tutorial doesn't mention IO, so I will have a stab here at extending the approach.
Let us start with what you probably have already seen - the container monad. Let's say we have:
f, g :: Int -> [Int]
One way of looking at this is that it gives us a number of possible outputs for every possible input. What if we want all possible outputs for the composition of both functions? Giving all possibilities we could get by applying the functions one after the other?
Well, there's a function for that:
fg x = concatMap g $ f x
If we put this more general, we get
fg x = f x >>= g
xs >>= f = concatMap f xs
return x = [x]
Why would we want to wrap it like this? Well, writing our programs primarily using >>= and return gives us some nice properties - for example, we can be sure that it's relatively hard to "forget" solutions. We'd explicitly have to reintroduce it, say by adding another function skip. And also we now have a monad and can use all combinators from the monad library!
Now, let us jump to your trickier example. Let's say the two functions are "side-effecting". That's not non-deterministic, it just means that in theory the whole world is both their input (as it can influence them) as well as their output (as the function can influence it). So we get something like:
f, g :: Int -> RealWorld# -> (Int, RealWorld#)
If we now want f to get the world that g left behind, we'd write:
fg x rw = let (y, rw') = f x rw
(r, rw'') = g y rw'
in (r, rw'')
Or generalized:
fg x = f x >>= g
x >>= f = \rw -> let (y, rw') = x rw
(r, rw'') = f y rw'
in (r, rw'')
return x = \rw -> (x, rw)
Now if the user can only use >>=, return and a few pre-defined IO values we get a nice property again: The user will never actually see the RealWorld# getting passed around! And that is a very good thing, as you aren't really interested in the details of where getLine gets its data from. And again we get all the nice high-level functions from the monad libraries.
So the important things to take away:
The monad captures common patterns in your code, like "always pass all elements of container A to container B" or "pass this real-world-tag through". Often, once you realize that there is a monad in your program, complicated things become simply applications of the right monad combinator.
The monad allows you to completely hide the implementation from the user. It is an excellent encapsulation mechanism, be it for your own internal state or for how IO manages to squeeze non-purity into a pure program in a relatively safe way.
Appendix
In case someone is still scratching his head over RealWorld# as much as I did when I started: There's obviously more magic going on after all the monad abstraction has been removed. Then the compiler will make use of the fact that there can only ever be one "real world". That's good news and bad news:
It follows that the compiler must guarantuee execution ordering between functions (which is what we were after!)
But it also means that actually passing the real world isn't necessary as there is only one we could possibly mean: The one that is current when the function gets executed!
Bottom line is that once execution order is fixed, RealWorld# simply gets optimized out. Therefore programs using the IO monad actually have zero runtime overhead. Also note that using RealWorld# is obviously only one possible way to put IO - but it happens to be the one GHC uses internally. The good thing about monads is that, again, the user really doesn't need to know.
You could see a given monad m as a set/family (or realm, domain, etc.) of actions (think of a C statement). The monad m defines the kind of (side-)effects that its actions may have:
with [] you can define actions which can fork their executions in different "independent parallel worlds";
with Either Foo you can define actions which can fail with errors of type Foo;
with IO you can define actions which can have side-effects on the "outside world" (access files, network, launch processes, do a HTTP GET ...);
you can have a monad whose effect is "randomness" (see package MonadRandom);
you can define a monad whose actions can make a move in a game (say chess, Go…) and receive move from an opponent but are not able to write to your filesystem or anything else.
Summary
If m is a monad, m a is an action which produces a result/output of type a.
The >> and >>= operators are used to create more complex actions out of simpler ones:
a >> b is a macro-action which does action a and then action b;
a >> a does action a and then action a again;
with >>= the second action can depend on the output of the first one.
The exact meaning of what an action is and what doing an action and then another one is depends on the monad: each monad defines an imperative sublanguage with some features/effects.
Simple sequencing (>>)
Let's say with have a given monad M and some actions incrementCounter, decrementCounter, readCounter:
instance M Monad where ...
-- Modify the counter and do not produce any result:
incrementCounter :: M ()
decrementCounter :: M ()
-- Get the current value of the counter
readCounter :: M Integer
Now we would like to do something interesting with those actions. The first thing we would like to do with those actions is to sequence them. As in say C, we would like to be able to do:
// This is C:
counter++;
counter++;
We define an "sequencing operator" >>. Using this operator we can write:
incrementCounter >> incrementCounter
What is the type of "incrementCounter >> incrementCounter"?
It is an action made of two smaller actions like in C you can write composed-statements from atomic statements :
// This is a macro statement made of several statements
{
counter++;
counter++;
}
// and we can use it anywhere we may use a statement:
if (condition) {
counter++;
counter++;
}
it can have the same kind of effects as its subactions;
it does not produce any output/result.
So we would like incrementCounter >> incrementCounter to be of type M (): an (macro-)action with the same kind of possible effects but without any output.
More generally, given two actions:
action1 :: M a
action2 :: M b
we define a a >> b as the macro-action which is obtained by doing (whatever that means in our domain of action) a then b and produces as output the result of the execution of the second action. The type of >> is:
(>>) :: M a -> M b -> M b
or more generally:
(>>) :: (Monad m) => m a -> m b -> m b
We can define bigger sequence of actions from simpler ones:
action1 >> action2 >> action3 >> action4
Input and outputs (>>=)
We would like to be able to increment by something else that 1 at a time:
incrementBy 5
We want to provide some input in our actions, in order to do this we define a function incrementBy taking an Int and producing an action:
incrementBy :: Int -> M ()
Now we can write things like:
incrementCounter >> readCounter >> incrementBy 5
But we have no way to feed the output of readCounter into incrementBy. In order to do this, a slightly more powerful version of our sequencing operator is needed. The >>= operator can feed the output of a given action as input to the next action. We can write:
readCounter >>= incrementBy
It is an action which executes the readCounter action, feeds its output in the incrementBy function and then execute the resulting action.
The type of >>= is:
(>>=) :: Monad m => m a -> (a -> m b) -> m b
A (partial) example
Let's say I have a Prompt monad which can only display informations (text) to the user and ask informations to the user:
-- We don't have access to the internal structure of the Prompt monad
module Prompt (Prompt(), echo, prompt) where
-- Opaque
data Prompt a = ...
instance Monad Prompt where ...
-- Display a line to the CLI:
echo :: String -> Prompt ()
-- Ask a question to the user:
prompt :: String -> Prompt String
Let's try to define a promptBoolean message actions which asks for a question and produces a boolean value.
We use the prompt (message ++ "[y/n]") action and feed its output to a function f:
f "y" should be an action which does nothing but produce True as output;
f "n" should be an action which does nothing but produce False as output;
anything else should restart the action (do the action again);
promptBoolean would look like this:
-- Incomplete version, some bits are missing:
promptBoolean :: String -> M Boolean
promptBoolean message = prompt (message ++ "[y/n]") >>= f
where f result = if result == "y"
then ???? -- We need here an action which does nothing but produce `True` as output
else if result=="n"
then ???? -- We need here an action which does nothing but produce `False` as output
else echo "Input not recognised, try again." >> promptBoolean
Producing a value without effect (return)
In order to fill the missing bits in our promptBoolean function, we need a way to represent dummy actions without any side effect but which only outputs a given value:
-- "return 5" is an action which does nothing but outputs 5
return :: (Monad m) => a -> m a
and we can now write out promptBoolean function:
promptBoolean :: String -> Prompt Boolean
promptBoolean message :: prompt (message ++ "[y/n]") >>= f
where f result = if result=="y"
then return True
else if result=="n"
then return False
else echo "Input not recognised, try again." >> promptBoolean message
By composing those two simple actions (promptBoolean, echo) we can define any kind of dialogue between the user and your program (the actions of the program are deterministic as our monad does not have a "randomness effect").
promptInt :: String -> M Int
promptInt = ... -- similar
-- Classic "guess a number game/dialogue"
guess :: Int -> m()
guess n = promptInt "Guess:" m -> f
where f m = if m == n
then echo "Found"
else (if m > n
then echo "Too big"
then echo "Too small") >> guess n
The operations of a monad
A Monad is a set of actions which can be composed with the return and >>= operators:
>>= for action composition;
return for producing a value without any (side-)effect.
These two operators are the minimal operators needed to define a Monad.
In Haskell, the >> operator is needed as well but it can in fact be derived from >>=:
(>>): Monad m => m a -> m b -> m b
a >> b = a >>= f
where f x = b
In Haskell, an extra fail operator is need as well but this is really a hack (and it might be removed from Monad in the future).
This is the Haskell definition of a Monad:
class Monad m where
return :: m a
(>>=) :: m a -> (a -> m b) -> m b
(>>) :: m a -> m b -> m b -- can be derived from (>>=)
fail :: String -> m a -- mostly a hack
Actions are first-class
One great thing about monads is that actions are first-class. You can take them in a variable, you can define function which take actions as input and produce some other actions as output. For example, we can define a while operator:
-- while x y : does action y while action x output True
while :: (Monad m) => m Boolean -> m a -> m ()
while x y = x >>= f
where f True = y >> while x y
f False = return ()
Summary
A Monad is a set of actions in some domain. The monad/domain define the kind of "effects" which are possible. The >> and >>= operators represent sequencing of actions and monadic expression may be used to represent any kind of "imperative (sub)program" in your (functional) Haskell program.
The great things are that:
you can design your own Monad which supports the features and effects that you want
see Prompt for an example of a "dialogue only subprogram",
see Rand for an example of "sampling only subprogram";
you can write your own control structures (while, throw, catch or more exotic ones) as functions taking actions and composing them in some way to produce a bigger macro-actions.
MonadRandom
A good way of understanding monads, is the MonadRandom package. The Rand monad is made of actions whose output can be random (the effect is randomness). An action in this monad is some kind of random variable (or more exactly a sampling process):
-- Sample an Int from some distribution
action :: Rand Int
Using Rand to do some sampling/random algorithms is quite interesting because you have random variables as first class values:
-- Estimate mean by sampling nsamples times the random variable x
sampleMean :: Real a => Int -> m a -> m a
sampleMean n x = ...
In this setting, the sequence function from Prelude,
sequence :: Monad m => [m a] -> m [a]
becomes
sequence :: [Rand a] -> Rand [a]
It creates a random variable obtained by sampling independently from a list of random variables.
There are three main observations concerning the IO monad:
1) You can't get values out of it. Other types like Maybe might allow to extract values, but neither the monad class interface itself nor the IO data type allow it.
2) "Inside" IO is not only the real value but also that "RealWorld" thing. This dummy value is used to enforce the chaining of actions by the type system: If you have two independent calculations, the use of >>= makes the second calculation dependent on the first.
3) Assume a non-deterministic thing like random :: () -> Int, which isn't allowed in Haskell. If you change the signature to random :: Blubb -> (Blubb, Int), it is allowed, if you make sure that nobody ever can use a Blubb twice: Because in that case all inputs are "different", it is no problem that the outputs are different as well.
Now we can use the fact 1): Nobody can get something out of IO, so we can use the RealWord dummy hidden in IO to serve as a Blubb. There is only one IOin the whole application (the one we get from main), and it takes care of proper sequentiation, as we have seen in 2). Problem solved.
One thing that often helps me to understand the nature of something is to examine it in the most trivial way possible. That way, I'm not getting distracted by potentially unrelated concepts. With that in mind, I think it may be helpful to understand the nature of the Identity Monad, as it's the most trivial implementation of a Monad possible (I think).
What is interesting about the Identity Monad? I think it is that it allows me to express the idea of evaluating expressions in a context defined by other expressions. And to me, that is the essence of every Monad I've encountered (so far).
If you already had a lot of exposure to 'mainstream' programming languages before learning Haskell (like I did), then this doesn't seem very interesting at all. After all, in a mainstream programming language, statements are executed in sequence, one after the other (excepting control-flow constructs, of course). And naturally, we can assume that every statement is evaluated in the context of all previously executed statements and that those previously executed statements may alter the environment and the behavior of the currently executing statement.
All of that is pretty much a foreign concept in a functional, lazy language like Haskell. The order in which computations are evaluated in Haskell is well-defined, but sometimes hard to predict, and even harder to control. And for many kinds of problems, that's just fine. But other sorts of problems (e.g. IO) are hard to solve without some convenient way to establish an implicit order and context between the computations in your program.
As far as side-effects go, specifically, often they can be transformed (via a Monad) in to simple state-passing, which is perfectly legal in a pure functional language. Some Monads don't seem to be of that nature, however. Monads such as the IO Monad or the ST monad literally perform side-effecting actions. There are many ways to think about this, but one way that I think about it is that just because my computations must exist in a world without side-effects, the Monad may not. As such, the Monad is free to establish a context for my computation to execute that is based on side-effects defined by other computations.
Finally, I must disclaim that I am definitely not a Haskell expert. As such, please understand that everything I've said is pretty much my own thoughts on this subject and I may very well disown them later when I understand Monads more fully.
the point is so there can be clean error handling in a chain of functions, containers, and side effects
More or less.
how exactly is the problem of side-effects solved?
A value in the I/O monad, i.e. one of type IO a, should be interpreted as a program. p >> q on IO values can then be interpreted as the operator that combines two programs into one that first executes p, then q. The other monad operators have similar interpretations. By assigning a program to the name main, you declare to the compiler that that is the program that has to be executed by its output object code.
As for the list monad, it's not really related to the I/O monad except in a very abstract mathematical sense. The IO monad gives deterministic computation with side effects, while the list monad gives non-deterministic (but not random!) backtracking search, somewhat similar to Prolog's modus operandi.
With this concept of containers, the language essentially says anything inside the containers is non-deterministic
No. Haskell is deterministic. If you ask for integer addition 2+2 you will always get 4.
"Nondeterministic" is only a metaphor, a way of thinking. Everything is deterministic under the hood. If you have this code:
do x <- [4,5]
y <- [0,1]
return (x+y)
it is roughly equivalent to Python code
l = []
for x in [4,5]:
for y in [0,1]:
l.append(x+y)
You see nondeterminism here? No, it's deterministic construction of a list. Run it twice, you'll get the same numbers in the same order.
You can describe it this way: Choose arbitrary x from [4,5]. Choose arbitrary y from [0,1]. Return x+y. Collect all possible results.
That way seems to involve nondeterminism, but it's only a nested loop (list comprehension). There is no "real" nondeterminism here, it's simulated by checking all possibilities. Nondeterminism is an illusion. The code only appears to be nondeterministic.
This code using State monad:
do put 0
x <- get
put (x+2)
y <- get
return (y+3)
gives 5 and seems to involve changing state. As with lists it's an illusion. There are no "variables" that change (as in imperative languages). Everything is nonmutable under the hood.
You can describe the code this way: put 0 to a variable. Read the value of a variable to x. Put (x+2) to the variable. Read the variable to y, and return y+3.
That way seems to involve state, but it's only composing functions passing additional parameter. There is no "real" mutability here, it's simulated by composition. Mutability is an illusion. The code only appears to be using it.
Haskell does it this way: you've got functions
a -> s -> (b,s)
This function takes and old value of state and returns new value. It does not involve mutability or change variables. It's a function in mathematical sense.
For example the function "put" takes new value of state, ignores current state and returns new state:
put x _ = ((), x)
Just like you can compose two normal functions
a -> b
b -> c
into
a -> c
using (.) operator you can compose "state" transformers
a -> s -> (b,s)
b -> s -> (c,s)
into a single function
a -> s -> (c,s)
Try writing the composition operator yourself. This is what really happens, there are no "side effects" only passing arguments to functions.
From what I understand, Monad is just another typeclass that declares ways to interact with data [...]
...providing an interface common to all those types which have an instance. This can then be used to provide generic definitions which work across all monadic types.
It seems clever and clean to implement these 3 things with one concept [...]
...the only three things that are implemented are the instances for those three types (list, Maybe and IO) - the types themselves are defined independently elsewhere.
[...] but really, the point is so there can be clean error handling in a chain of functions, containers, and side effects.
Not just error handling e.g. consider ST - without the monadic interface, you would have to pass the encapsulated-state directly and correctly...a tiresome task.
How exactly is the problem of side-effects solved?
Short answer: Haskell solves manages them by using types to indicate their presence.
Can someone explain how, intuitively, Haskell gets away with changing state with inputs and output?
"Intuitively"...like what's available over here? Let's try a simple direct comparison instead:
From How to Declare an Imperative by Philip Wadler:
(* page 26 *)
type 'a io = unit -> 'a
infix >>=
val >>= : 'a io * ('a -> 'b io) -> 'b io
fun m >>= k = fn () => let
val x = m ()
val y = k x ()
in
y
end
val return : 'a -> 'a io
fun return x = fn () => x
val putc : char -> unit io
fun putc c = fn () => putcML c
val getc : char io
val getc = fn () => getcML ()
fun getcML () =
valOf(TextIO.input1(TextIO.stdIn))
(* page 25 *)
fun putcML c =
TextIO.output1(TextIO.stdOut,c)
Based on these two answers of mine, this is my Haskell translation:
type IO a = OI -> a
(>>=) :: IO a -> (a -> IO b) -> IO b
m >>= k = \ u -> let !(u1, u2) = part u in
let !x = m u1 in
let !y = k x u2 in
y
return :: a -> IO a
return x = \ u -> let !_ = part u in x
putc :: Char -> IO ()
putc c = \ u -> putcOI c u
getc :: IO Char
getc = \ u -> getcOI u
-- primitives
data OI
partOI :: OI -> (OI, OI)
putcOI :: Char -> OI -> ()
getcOI :: OI -> Char
Now remember that short answer about side-effects?
Haskell manages them by using types to indicate their presence.
Data.Char.chr :: Int -> Char -- no side effects
getChar :: IO Char -- side effects at
{- :: OI -> Char -} -- work: beware!
I'm trying to wrap my head around the enumerator library and ran into a situation where I want to build a new Enumeratee in terms of two existing Enumeratees. Let's say I have the enumeratees:
e1 :: Enumeratee x y m b
e2 :: Enumeratee y z m b
I feel that I should be able to combine them into one enumeratee
e3 :: Enumeratee x z m b
but I couldn't find an existing function to do this in the package. I tried to write such a function myself, but my understanding of iteratees is still so limited that I couldn't figure out a way to get all the complex types to match.
Did I just miss some basic combinator, or are Enumeratees even supposed to be composable with each other?
In theory they are composable, but the types are a bit tricky. The difficulty is that the final parameter b of the first enumeratee isn't actually b; it's another iteratee!. Here's the type of the ><> operator from iteratee, which composes enumeratees:
Prelude Data.Iteratee> :t (><>)
(><>)
:: (Monad m, Nullable s1) =>
(forall x. Enumeratee s1 s2 m x)
-> Enumeratee s2 s3 m a -> Enumeratee s1 s3 m a
Note the extra forall in the first enumeratee; this indicates that a Rank-2 type is at work. If the enumerator author wants to maintain H98 compatibility (I believe this was one of the original goals), this approach is unavailable.
It is possible to write this type signature in a form which doesn't require Rank-2 types, but it's either longer, not clear from the type that it's actually two enumeratee's that are being composed, or both. For example, this is ghc's inferred type for (><>):
Prelude Data.Iteratee> :t (><>>)
(><>>)
:: (Monad m, Nullable s) =>
(b -> Iteratee s m (Iteratee s' m a1))
-> (a -> b) -> a -> Iteratee s m a1
Although these types are for iteratee combinators, hopefully it's enough information you'll be able to apply them to enumerator.
I ran with this problem a while ago, you need to have an Iteratee first (or an Enumerator) in order to make the composition of Enumeratees.
You can either start by doing this:
module Main where
import Data.Enumerator
import qualified Data.Enumerator.List as EL
main :: IO ()
main = run_ (enum $$ EL.consume) >>= print
where
enum = (enumList 5 [1..] $= EL.isolate 100) $= EL.filter pairs
pairs = (==0) . (`mod` 2)
The previous code composes a list of enumeratees together to create a new enumerator, and then it is applied to the consume Iteratee.
The ($=) serves to compose an Enumerator and an Enumeratee to create a new enumerator, while the (=$) can be used to compose an Iteratee with an Enumeratee to create a new Iteratee. I recommend the latter given that types won't bust your balls when composing a list of Enumeratees using (=$):
module Main where
import Data.Enumerator
import qualified Data.Enumerator.List as EL
main :: IO ()
main = run_ (enumList 5 [1..] $$ it) >>= print
where
it = foldr (=$)
EL.consume
[ EL.isolate 100
, EL.filter ((==0) . (`mod` 2))
]
If you would try to implement the same function above by creating an Enumerator instead of an Iteratee, you will get an infinite recursive type error when using foldl' ($=) (enumList 5 [1..]) [list-of-enumeratees].
Hope this helps.