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For well over a year, I have been intensely using lift, return, and constructors such as EitherT, ReaderT, and so forth. I've read Real World Haskell, Learn You a Haskell, almost every monad tutorial out there, and tried writing my own. Yet, I constantly remain confused about these three operations. Any time I am writing new code I try to figure out which of the three to use, and it almost always takes me an hour or more on the first function in a particular block of code.
What is an intuitive understanding of the three? Simple types are insufficient, as in all three cases I can instantly recite the types to you. What is a meaning for what these do that is consistent across all of the standard monad transformers?
(Unfortunately, if you respond in math terms, I'm still not going to understand you. While I can write code to solve math problems and can set up time complexity based on the code I see, I cannot after many years of trying to work in Haskell relate math terms to programming terms.)
return takes a pure computation and turns it into a computation which claims to have some monad-y side-effects, but doesn't.
lift takes a computation that has some side-effects, and adds more.
EitherT, ReaderT, and so on take a computation that already has all the side-effects you're interested in and "spells them differently" -- for example, where before your state was spelled as a function that returns an updated value, it is now spelled as a State(T)-ful computation.
So let's say you have a computation. In a lazy language like Haskell you'd write
comp1 :: a
and know that this computation will be performed upon request and result in a value of type a.
Let's say you have a similar computation, but in addition to computing a value of type a, it might "fail" for some reason or another. For example, a might be Integer and this computation will "fail" if its a division by zero. We're write this now as
comp2 :: Maybe a
where the Maybe constructor "tags" the a to indicate failure.
Let's say we have a similar computation as before, but now we are allowed to fail, but also collect a log during the computation. "Log collecting" is called Writer so we'd like to tag our type with Writer as well as Maybe. Unfortunately
comp3_bad :: (Writer String) Maybe a
doesn't make any sense. The definition of writer allows for a single parameter, not two. We can consider a bit of what the underlying mechanics of this combined effect would be, though—it needs to return a Maybe paired with the log... or perhaps if the computation fails, the log is discarded. There are two options
comp3_1 :: (String, Maybe a)
comp3_2 :: Maybe (String, a)
If we unpack the Writer, we can see that these are equivalent to
comp3_1' :: Writer String (Maybe a)
comp3_2' :: Maybe (Writer String a)
This pattern of nesting is called composition. If you want to combine the effects of two monads then you'd like to compose them. For some monads this works directly, though it's a little cumbersome.
Unfortunately, some monads start to break the monad laws once they are composed. They can still be "stacked" but not in the normal way. So, we allow each type to determine its stacking method by creating the transformer version <monad>T.
newtype WriterT w m a = WriterT { runWriterT :: m (w, a) }
newtype MaybeT m a = MaybeT { runMaybeT :: m (Maybe a) }
-- note that
WriterT String Maybe a == Maybe (String, a)
MaybeT (Writer String) a == (String, Maybe a)
These composed stacks of monads are called monad transformer stacks and they allow you to assemble side effects in layers.
So what happens if we have two different, but similar stacks that we'd like to use together. For instance, we can consider Maybe to be a monad... or a monad transformer stack of a single layer. Compare that to WriterT String Maybe which is a monad transformer stack of two layers, the bottom of which is Maybe.
These two stacks are very similar, but we cannot transport computations from one to the other. Or rather, we can, but it's fairly annoying
transport :: Maybe a -> WriterT String Maybe a
transport Nothing = WriterT Nothing
transport (Just a) = WriterT (Just ("", a))
this transport forms a general pattern where we "add another layer" onto a stack. This general pattern is called lift
lift :: Maybe a -> WriterT String Maybe a
Or, written polymorphically we see the extra layer t being prepended.
lift :: MonadTrans t => m a -> t m a
Finally, we've come a long way from our pure computation at the beginning
comp1 :: a
and demonstrated that we can lift simple transformer stacks into more complex ones. Can we consider comp1 to be living in the very simplest of transformer stacks—the empty stack?
It turns out that this is actually a really valid point of view. We can even "lift" comp1 into a more sophisticated transformer stack... but the terminology changes slightly.
return :: Monad m => a -> m a
So, it's valid to think of return as lifting a pure computation into a basic monad. This is a foundational principle of monads even—that they can embed pure computations within them.
Comonoids are mentioned, for example, in Haskell's distributive library docs:
Due to the lack of non-trivial comonoids in Haskell, we can restrict ourselves to requiring a Functor rather than some Coapplicative class.
After a little searching I found a StackOverflow answer that explains this a bit more with the laws that comonoids would have to satisfy. So I think I understand why there's only one possible instance for a hypothetical Comonoid typeclass in Haskell.
Thus, to find a nontrivial comonoid, I suppose we'd have to look in some other category. Surely, if category theorists have a name for comonoids, then there are some interesting ones. The other answers on that page seem to hint at an example involving Supply, but I couldn't figure one out that still satisfies the laws.
I also turned to Wikipedia: there's a page for monoids that doesn't reference category theory, which seems to me as an adequate description of Haskell's Monoid typeclass, but "comonoid" redirects to a category-theoretic description of monoids and comonoids together that I can't understand, and there still don't seem to be any interesting examples.
So my questions are:
Can comonoids be explained in non-category-theoretic terms like monoids?
What is a simple example of an interesting comonoid, even if it's not a Haskell type? (Could one be found in a Kleisli category over a familiar Haskell monad?)
edit: I am not sure if this is actually category-theoretically correct, but what I was imagining in the parenthetical of question 2 was nontrivial definitions of delete :: a -> m () and split :: a -> m (a, a) for some specific Haskell type a and Haskell monad m that satisfy Kleisli-arrow versions of the comonoid laws in the linked answer. Other examples of comonoids are still welcome.
As Phillip JF mentioned, comonoids are interesting to talk about in substructural logics. Let's talk about linear lambda calculus. This is much like your normal typed lambda calculus except that every variable must be used exactly once.
To get a feel, let's count linear functions of given types, i.e.
a -> a
has exactly one inhabitant, id. While
(a,a) -> (a,a)
has two, id and flip. Note that in regular lambda calculus (a,a) -> (a,a) has four inhabitants
(a, b) ↦ (a, a)
(a, b) ↦ (b, b)
(a, b) ↦ (a, b)
(a, b) ↦ (b, a)
but the first two require that we use one of the arguments twice while discarding the other. This is exactly the essence of linear lambda calculus—disallowing those kinds of functions.
As a quick aside, what's the point of linear LC? Well, we can use it to model linear effects or resource usage. If, for instance, we have a file type and a few transformers it might look like
data File
open :: String -> File
close :: File -> () -- consumes a file, but we're ignoring purity right now
t1 :: File -> File
t2 :: File -> File
and then the following are valid pipelines:
close . t1 . t2 . open
close . t2 . t1 . open
close . t1 . open
close . t2 . open
but this "branching" computation isn't
let f1 = open "foo"
f2 = t1 f1
f3 = t2 f1
in close f3
since we used f1 twice.
Now, you might be wondering something at this point about what things must follow the linear rules. For instance, I decided that some pipelines don't have to include both t1 and t2 (compare the enumeration exercise from before). Further, I introduced the open and close functions which happily create and destroy the File type despite that being a violation of linearity.
Indeed, we might posit the existence of functions which violate linearity—but not all clients may. It's much like the IO monad—all of the secrets live inside the implementation of IO so that users work in a "pure" world.
And this is where Comonoid comes in.
class Comonoid m where
destroy :: m -> ()
split :: m -> (m, m)
A type that instantiates Comonoid in a linear lambda calculus is a type which has carry-along destruction and duplication rules. In other words, it's a type which isn't very much bound by linear lambda calculus at all.
Since Haskell doesn't implement the linear lambda calculus rules at all, we can always instantiate Comonoid
instance Comonoid a where
destroy a = ()
split a = (a, a)
Or, perhaps the other way to think of it is that Haskell is a linear LC system that just happens to instantiate Comonoid for every type and applies destroy and split for you automatically.
A monoid in the usual sense is the same as a categorical monoid in the category of sets. One would expect that a comonoid in the usual sense is the same as a categorical comonoid in the category of sets. But every set in the category of sets is a comonoid in a trivial way, so apparently there is no non-categorical description of comonoids which would be parallel to that of monoids.
Just like a monad is a monoid in the category of endofunctors (what's the problem?), a comonad is a comonoid in the category of endofunctors (what's the coproblem?) So yes, any comonad in Haskell would be an example of a comonoid.
Well one way we can think of a monoid is as hooked to any particular product construction that we're using, so in Set we'd take this signature:
mul : A * A -> A
one : A
to this one:
dup : A -> A * A
one : A
but the idea of duality is that the logical statements that you can make all have duals which can be applied to the dual objects, and there is another way of stating what a monoid is, and that's being agnostic to the choice of product construction and then when we take the costructure we can take the coproduct in the output, like:
div : A -> A + A
one : A
where + is a tagged sum. Here we essentially have that every single term which is in this type is always ready to produce a new bit, which is implicitly derived from the tag used to denote the left or the right instance of A. I personally think this is really damn cool. I think the cool version of the things that people were talking about above is when you don't particularly construct that for monoids, but for monoid actions.
A monoid M is said to act on a set A if there's a function
act : M * A -> A
where we have the following rules
act identity a = a
act f (act g a) = act (f * g) a
If we want a co-action, what exactly do we want?
act : A -> M * A
this generates us a stream of the type of our comonoid! I'm having a lot of trouble coming up with the laws for these systems, but I think they must be around somewhere so I'm gonna keep looking tonight. If somebody can tell me them or that I'm wrong about these things in some way or another, also interested in that.
As a physicist, the most common example I deal with is coalgebras, which are comonoid objects in the category of vector spaces, with the monoidal structure usually given by the tensor product.
In that case, there is a bijection between monoid and comonoid objects, since you can just take the adjoint or transpose of the product and unit maps to get a coproduct and a counit that satisfy the comonoid axioms.
In some branches of physics, it is very common to see objects that have both an algebra and a coalgebra structure with some compatibility axioms. The two most common cases are Hopf algebras and Frobenius algebras. They are very convenient for constructing states or solution that are entangled or correlated.
In programming, the simplest nontrivial example I can think of would be reference counted pointers such as shared_ptr in C++ and Rc in Rust, along with their weak equivalents. You can copy them, which is a nontrivial operation that bumps up the refcount (so the two copies are distinct from the initial state). You can drop (call the destructor) on one, which is nontrivial because it bumps down the refcount of any other refcounted pointer that points to the same piece of data.
Furthermore, weak pointers are a great example of a comonoid action. You can use the co-action to generate a weak pointer from a shared pointer. This can be easily checked by noting that creating one from a shared pointer and immediately dropping it is a unit operation, and creating one & cloning it is equivalent to creating two from the shared pointer.
This is a general thing you see with nontrivial coproducts and their co-actions: when they don't reduce to a copying operation, they intuitively imply some form of action at a distance between the two halves, while also adding an operation that erases one half to leave the other independent.
To clarify the question: it is about the merits of the monad type class (as opposed to just its instances without the unifying class).
After having read many references (see below),
I came to the conclusion that, actually, the monad class is there to solve only one, but big and crucial, problem: the 'chaining' of functions on types with context. Hence, the famous sentence "monads are programmable semicolons".
In fact, a monad can be viewed as an array of functions with helper operations.
I insist on the difference between the monad class, understood as a general interface for other types; and these other types instantiating the class (thus, "monadic types").
I understand that the monad class by itself, only solves the chaining of operators because mainly, it only mandates its type instances
to have bind >>= and return, and tell us how they must behave. And as a bonus, the compiler greatyly helps the coding providing do notation for monadic types.
On the other hand,
it is each individual type instantiating the monad class which solves each concrete problem, but not merely for being a instance of Monad. For instance Maybe solves "how a function returns a value or an error", State solves "how to have functions with global state", IO solves "how to interact with the outside world", and so on. All theses classes encapsulate a value within a context.
But soon or later, we will need to chain operations on such context-types. I.e., we will need to organize calls to functions on these types in a particular sequence (for an example of such a problem, please read the example about multivalued functions in You could have invented monads).
And you get solved the problem of chaining, if you have each type be an instance of the monad class.
For the chaining to work you need >>= just with the exact signature it has, no other. (See this question).
Therefore, I guess that the next time you define a context data type T for solving something, if you need to sequence calls of functions (on values of T) consider making T an instance of Monad (if you need "chaining with choice" and if you can benefit from the do notation). And to make sure you are doing it right, check that T satisfies the monad laws
Then, I ask two questions to the Haskell experts:
A concrete question: is there any other problem that the monad class solves by ifself (leaving apart monadic classes)? If any, then, how it compares in relevance to the problem of chaining operations?
An optional general question: are my conclusions right, am I misunderstanding something?
References
Tutorials
Monads in pictures Definitely worth it; read this one first.
Fistful of monads
You could have invented monads
Monads are trees (pdf)
StackOverflow Questions & Answers
How to detect a monad
On the signature of >>= monad operator
You're definitely on to something in the way that you're stating this—there are many things that Monad means and you've separated them out well.
That said, I would definitely say that chaining operations is not the primary thing solved by Monads. That can be solved using plain Functors (with some trouble) or easily with Applicatives. You need to use the full monad spec when "chaining with choice". In particular, the tension between Applicative and Monad comes from Applicative needing to know the entire structure of the side-effecting computation statically. Monad can change that structure at runtime and thus sacrifices some analyzability for power.
To make the point more clear, the only place you deal with a Monad but not any specific monad is if you're defining something with polymorphism constrained to be a Monad. This shows up repeatedly in the Control.Monad module, so we can examine some examples from there.
sequence :: [m a] -> m [a]
forever :: m a -> m b
foldM :: (a -> b -> m a) -> a -> [b] -> m a
Immediately, we can throw out sequence as being particular to Monad since there's a corresponding function in Data.Traversable, sequenceA which has a type slightly more general than Applicative f => [f a] -> f [a]. This ought to be a clear indicator that Monad isn't the only way to sequence things.
Similarly, we can define foreverA as follows
foreverA :: Applicative f => f a -> f b
foreverA f = flip const <$> f <*> foreverA f
So more ways to sequence non-Monad types. But we run into trouble with foldM
foldM :: (Monad m) => (a -> b -> m a) -> a -> [b] -> m a
foldM _ a [] = return a
foldM f a (x:xs) = f a x >>= \fax -> foldM f fax xs
If we try to translate this definition to Applicative style we might write
foldA :: (Applicative f) => (a -> b -> f a) -> a -> [b] -> f a
foldA _ a [] = pure a
foldA f a (x:xs) = foldA f <$> f a x <*> xs
But Haskell will rightfully complain that this doesn't typecheck--each recursive call to foldA tries to put another "layer" of f on the result. With Monad we could join those layers down, but Applicative is too weak.
So how does this translate to Applicatives restricting us from runtime choices? Well, that's exactly what we express with foldM, a monadic computation (a -> b -> m a) which depends upon its a argument, a result from a prior monadic computation. That kind of thing simply doesn't have any meaning in the more purely sequential world of Applicative.
To solve the problem of chaining operations on an individual monadic type, it's not at all necessary to make it an instance of Monad and be sure the monad laws are satisfied. You could just implement a chaining operation directly on your type.
It would probably be very similar to the monadic bind, but not necessarily exactly the same (recall that bind for lists is concatMap, a function that exists anyway, but with the arguments in a different order). And you wouldn't have to worry about the monad laws, because you would have a slightly different interface for each type, so they wouldn't have any common requirements.
To ask what problem the Monad type class itself solves, look at all the functions (in Control.Monad and else where) that work on values in any monadic type. The problem solved is code reuse! Monad is exactly the part of all the monadic types that is common to each and every one of them. That part is sufficient on its own to write useful computations. All of these functions could be implemented for any individual monadic type (often more directly), but they've already been implemented for all monadic types, even the ones that don't exist yet.
You don't write a Monad instance so that you can chain operations on your type (often you already have a way of chaining, in fact). You write a Monad instance for all the code that automatically comes along with the Monad instance. Monad isn't about solving any problem for any single type, it's about a way of viewing many disparate types as instances of a single unifying concept.
I my way to learn Haskell I'm starting to grasp the monad concept and starting to use the known monads in my code but I'm still having difficulties approaching monads from a designer point of view. In OO there are several rules like, "identify nouns" for objects, watch for some kind of state and interface... but I'm not able to find equivalent resources for monads.
So how do you identify a problem as monadic in nature? What are good design patterns for monadic design? What's your approach when you realize that some code would be better refactored into a monad?
A helpful rule of thumb is when you see values in a context; monads can be seen as layering "effects" on:
Maybe: partiality (uses: computations that can fail)
Either: short-circuiting errors (uses: error/exception handling)
[] (the list monad): nondeterminism (uses: list generation, filtering, ...)
State: a single mutable reference (uses: state)
Reader: a shared environment (uses: variable bindings, common information, ...)
Writer: a "side-channel" output or accumulation (uses: logging, maintaining a write-only counter, ...)
Cont: non-local control-flow (uses: too numerous to list)
Usually, you should generally design your monad by layering on the monad transformers from the standard Monad Transformer Library, which let you combine the above effects into a single monad. Together, these handle the majority of monads you might want to use. There are some additional monads not included in the MTL, such as the probability and supply monads.
As far as developing an intuition for whether a newly-defined type is a monad, and how it behaves as one, you can think of it by going up from Functor to Monad:
Functor lets you transform values with pure functions.
Applicative lets you embed pure values and express application — (<*>) lets you go from an embedded function and its embedded argument to an embedded result.
Monad lets the structure of embedded computations depend on the values of previous computations.
The easiest way to understand this is to look at the type of join:
join :: (Monad m) => m (m a) -> m a
This means that if you have an embedded computation whose result is a new embedded computation, you can create a computation that executes the result of that computation. So you can use monadic effects to create a new computation based on values of previous computations, and transfer control flow to that computation.
Interestingly, this can be a weakness of structuring things monadically: with Applicative, the structure of the computation is static (i.e. a given Applicative computation has a certain structure of effects that cannot change based on intermediate values), whereas with Monad it is dynamic. This can restrict the optimisation you can do; for instance, applicative parsers are less powerful than monadic ones (well, this isn't strictly true, but it effectively is), but they can be optimised better.
Note that (>>=) can be defined as
m >>= f = join (fmap f m)
and so a monad can be defined simply with return and join (assuming it's a Functor; all monads are applicative functors, but Haskell's typeclass hierarchy unfortunately doesn't require this for historical reasons).
As an additional note, you probably shouldn't focus too heavily on monads, no matter what kind of buzz they get from misguided non-Haskellers. There are many typeclasses that represent meaningful and powerful patterns, and not everything is best expressed as a monad. Applicative, Monoid, Foldable... which abstraction to use depends entirely on your situation. And, of course, just because something is a monad doesn't mean it can't be other things too; being a monad is just another property of a type.
So, you shouldn't think too much about "identifying monads"; the questions are more like:
Can this code be expressed in a simpler monadic form? With which monad?
Is this type I've just defined a monad? What generic patterns encoded by the standard functions on monads can I take advantage of?
Follow the types.
If you find you have written functions with all of these types
(a -> b) -> YourType a -> YourType b
a -> YourType a
YourType (YourType a) -> YourType a
or all of these types
a -> YourType a
YourType a -> (a -> YourType b) -> YourType b
then YourType may be a monad. (I say “may” because the functions must obey the monad laws as well.)
(Remember you can reorder arguments, so e.g. YourType a -> (a -> b) -> YourType b is just (a -> b) -> YourType a -> YourType b in disguise.)
Don't look out only for monads! If you have functions of all of these types
YourType
YourType -> YourType -> YourType
and they obey the monoid laws, you have a monoid! That can be valuable too. Similarly for other typeclasses, most importantly Functor.
There's the effect view of monads:
Maybe - partiality / failure short-circuiting
Either - error reporting / short-circuiting (like Maybe with more information)
Writer - write only "state", commonly logging
Reader - read-only state, commonly environment passing
State - read / write state
Resumption - pausable computation
List - multiple successes
Once you are familiar with these effects its easy to build monads combining them with monad transformers. Note that combining some monads needs special care (particularly Cont and any monads with backtracking).
One thing important to note is there aren't many monads. There are some exotic ones that aren't in the standard libraries e.g the probability monad and variations of the Cont monad like Codensity. But unless you are doing something mathematical its unlikely you will invent (or discover) a new monad, however if you use Haskell long enough you'll build many monads that are different combinations of the standard ones.
Edit - Also note that the order you stack monad transformers results in different monads:
If you add ErrorT (transformer) to a Writer monad, you get this monad Either err (log,a) - you can only access the log if you have no error.
If you add WriterT (transfomer) to an Error monad, you get this monad (log, Either err a) which always gives access to the log.
This is sort of a non-answer, but I feel it is important to say anyways. Just ask! StackOverflow, /r/haskell, and the #haskell irc channel are all great places to get quick feedback from smart people. If you are working on a problem, and you suspect that there's some monadic magic that could make it easier, just ask! The Haskell community loves to solve problems, and is ridiculously friendly.
Don't misunderstand, I'm not encouraging you to never learn for yourself. Quite the contrary, interacting with the Haskell community is one of the best ways to learn. LYAH and RWH, 2 Haskell books that are freely available online, come highly recommended as well.
Oh, and don't forget to play, play, play! As you play around with monadic code, you'll start to get the feel of what "shape" monads have, and when monadic combinators can be useful. If you're rolling your own monad, then usually the type system will guide you to an obvious, simple solution. But to be honest, you should rarely need to roll your own instance of Monad, since Haskell libraries provide tons of useful things as mentioned by other answerers.
There's a common notion that one sees in many programming languages of an "infectious function tag" -- some special behavior for a function that must extend to its callers as well.
Rust functions can be unsafe, meaning they perform operations that can potentially violate memory unsafety. unsafe functions can call normal functions, but any function that calls an unsafe function must be unsafe as well.
Python functions can be async, meaning they return a promise rather than an actual value. async functions can call normal functions, but invocation of an async function (via await) can only be done by another async function.
Haskell functions can be impure, meaning they return an IO a rather than an a. Impure functions can call pure functions, but impure functions can only be called by other impure functions.
Mathematical functions can be partial, meaning they don't map every value in their domain to an output. The definitions of partial functions can reference total functions, but if a total function maps some of its domain to a partial function, it becomes partial as well.
While there may be ways to invoke a tagged function from an untagged function, there is no general way, and doing so can often be dangerous and threatens to break the abstraction the language tries to provide.
The benefit, then, of having tags is that you can expose a set of special primitives that are given this tag and have any function that uses these primitives make that clear in its signature.
Say you're a language designer and you recognize this pattern, and you decide that you want to allow user-defined tags. Let's say the user defined a tag Err, representing computations that may throw an error. A function using Err might look like this:
function div <Err> (n: Int, d: Int): Int
if d == 0
throwError("division by 0")
else
return (n / d)
If we wanted to simplify things, we might observe that there's nothing erroneous about taking arguments - it's computing the return value where problems might arise. So we can restrict tags to functions that take no arguments, and have div return a closure rather than the actual value:
function div(n: Int, d: Int): <Err> () -> Int
() =>
if d == 0
throwError("division by 0")
else
return (n / d)
In a lazy language such as Haskell, we don't need the closure, and can just return a lazy value directly:
div :: Int -> Int -> Err Int
div _ 0 = throwError "division by 0"
div n d = return $ n / d
It is now apparent that, in Haskell, tags need no special language support - they are ordinary type constructors. Let's make a typeclass for them!
class Tag m where
We want to be able to call an untagged function from a tagged function, which is equivalent to turning an untagged value (a) into a tagged value (m a).
addTag :: a -> m a
We also want to be able to take a tagged value (m a) and apply a tagged function (a -> m b) to get a tagged result (m b):
embed :: m a -> (a -> m b) -> m b
This, of course, is precisely the definition of a monad! addTag corresponds to return, and embed corresponds to (>>=).
It is now clear that "tagged functions" are merely a type of monad. As such, whenever you spot a place where a "function tag" could apply, chances are you've got a place suitable for a monad.
P.S. Regarding the tags I've mentioned in this answer: Haskell models impurity with the IO monad and partiality with the Maybe monad. Most languages implement async/promises fairly transparently, and there seems to be a Haskell package called promise that mimics this functionality. The Err monad is equivalent to the Either String monad. I'm not aware of any language that models memory unsafety monadically, it could be done.
In "Learn You a Haskell for Great Good!" author claims that Applicative IO instance is implemented like this:
instance Applicative IO where
pure = return
a <*> b = do
f <- a
x <- b
return (f x)
I might be wrong, but it seems that both return, and do-specific constructs (some sugared binds (>>=) ) comes from Monad IO. Assuming that's correct, my actual question is:
Why Applicative IO implementation depends on Monad IO functions/combinators?
Isn't Applicative less powerfull concept than Monad?
Edit (some clarifications):
This implementation is against my intuition, because according to Typeclassopedia article it's required for a given type to be Applicative before it can be made Monad (or it should be in theory).
(...) according to Typeclassopedia article it's required for a given type to be Applicative before it can be made Monad (or it should be in theory).
Yes, your parenthetical aside is exactly the issue here. In theory, any Monad should also be an Applicative, but this is not actually required, for historical reasons (i.e., because Monad has been around longer). This is not the only peculiarity of Monad, either.
Consider the actual definitions of the relevant type classes, taken from the base package's source on Hackage.
Here's Applicative:
class Functor f => Applicative f where
pure :: a -> f a
(<*>) :: f (a -> b) -> f a -> f b
(*>) :: f a -> f b -> f b
(<*) :: f a -> f b -> f a
...about which we can observe the following:
The context is correct given currently existing type classes, i.e., it requires Functor.
It's defined in terms of function application, rather than in (possibly more natural from a mathematical standpoint) terms of lifting tuples.
It includes technically superfluous operators equivalent to lifting constant functions.
Meanwhile, here's Monad:
class Monad m where
(>>=) :: m a -> (a -> m b) -> m b
(>>) :: m a -> m b -> m b
return :: a -> m a
fail :: String -> m a
...about which we can observe the following:
The context not only ignores Applicative, but also Functor, both of which are logically implied by Monad but not explicitly required.
It's also defined in terms of function application, rather than the more mathematically natural definition using return and join.
It includes a technically superfluous operator equivalent to lifting a constant function.
It also includes fail which doesn't really fit in at all.
In general, the ways that the Monad type class differs from the mathematical concept it's based on can be traced back through its history as an abstraction for programming. Some, like the function application bias it shares with Applicative, are a reflection of existing in a functional language; others, like fail or the lack of an appropriate class context, are historical accidents more than anything else.
What it all comes down to is that having an instance of Monad implies an instance for Applicative, which in turn implies an instance for Functor. A class context merely formalizes this explicitly; it remains true regardless. As it stands, given a Monad instance, both Functor and Applicative can be defined in a completely generic way. Applicative is "less powerful" than Monad in exactly the same sense that it is more general: Any Monad is automatically Applicative if you copy+paste the generalized instance, but there exist Applicative instances which cannot be defined as a Monad.
A class context, like Functor f => Applicative f says two things: That the latter implies the former, and that a definition must exist to fulfill that implication. In many cases, defining the latter implicitly defines the former anyway, but the compiler cannot deduce that in general, and thus requires both instances to be written out explicitly. The same thing can be observed with Eq and Ord--the latter obviously implies the former, but you still need to define an Eq instance in order to define one for Ord.
The IO type is abstract in Haskell, so if you want to implement a general Applicative for IO you have to do it with the operations that are supported by IO. Since you can implement Applicative in terms of the Monad operations that seems like a good choice. Can you think of another way to implement it?
And yes, Applicative is in some sense less powerful than Monad.
Isn't Applicative a less powerful concept than Monad?
Yes, and therefore whenever you have a Monad you can always make it an Applicative. You could replace IO with any other monad in your example and it would be a valid Applicative instance.
As an analogy, while a color printer may be considered more powerful than a grayscale printer, you can still use one to print a grayscale image.
Of course, one could also base a Monad instance on an Applicative and set return = pure, but you won't be able to define >>= generally. This is what Monad being more powerful means.
In a perfect world every Monad would be an Applicative (so we had class Applicative a => Monad a where ...), but for historical reasons both type classes are independend. So your observation that this definition is kind of "backwards" (using the more powerful abstaction to implement the less powerful one) is correct.
You already have perfectly good answers for older versions of GHC, but in the latest version you actually do have class Applicative m => Monad m so your question needs another answer.
In terms of GHC implementation: GHC just checks what instances are defined for a given type before it tries to compile any of them.
In terms of code semantics: class Applicative m => Monad m doesn't mean the Applicative instance has to be defined "first", just that if it hasn't been defined by the end of your program then the compiler will abort.