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.
Related
I have been reading up on adjunctions during the last couple of days. While I start to understand their importance from a theoretical point of view, I wonder how and why people use them in Haskell. Data.Functor.Adjunction provides an implementation and among its instances are free functor / forgetful functor and curry / uncurry. Again those are very interesting from the theoretical view point but I can't see how I would use them for more practical programming problems.
Are there examples of programming problems people solved using Data.Functor.Adjunction and why you would prefer this implementation over others?
Preliminary note: This answer is a bit speculative. Much like the question, it was built from studying Data.Functor.Adjunction.
I can think of three reasons why there aren't many use cases for the Adjunction class in the wild.
Firstly, all Hask/Hask adjunctions are ultimately some variation on the currying adjunction, so the spectrum of potential instances isn't all that large to begin with. Many of the adjunctions one might be interested on aren't Hask/Hask.
Secondly, while an Adjunction instance gives you a frankly awesome amount of other instances for free, in many cases those instances already exist somewhere else. To pick the ur-example, we might very easily implement StateT in terms of Control.Monad.Trans.Adjoint:
newtype StateT s m a = StateT { runStateT :: s -> m (s, a) }
deriving (Functor, Applicative, Monad) via AdjointT ((,) s) ((->) s) m
deriving MonadTrans via AdjointT ((,) s) ((->) s)
-- There is also a straightforward, fairly general way to implement MonadState.
However, no one needs to actually do that, because there is a perfectly good StateT in transformers. That said, if you do have an Adjunction instance of your own, you might be in luck. One little thing I have thought of that might make sense (even if I haven't actually seen it out there) are the following functors:
data Dilemma a = Dilemma { fstDil :: a, sndDil a }
data ChoiceF a = Fst a | Snd a
We might write an Adjunction ChoiceF Dilemma instance, which reflects how Dilemma (ChoiceF a) is materialised version of State Bool a. Dilemma (ChoiceF a) can be thought of as a step in a decision tree: choosing one side of the Dilemma tells you, through the ChoiceF constructors, what choice is to be made next. The Adjunction instance would then give us a monad transformer for Dilemma (ChoiceF a) for free.
(Another possibility might be exploiting the Free f/Cofree u adjunction. Cofree Dilemma a is an infinite tree of outcomes, while Free ChoiceF a is a path leading to an outcome. I hazard there is some mileage to get out of that.)
Thirdly, while there are many useful functions for right adjoints in Data.Functor.Adjunction, most of the functionality they provide is also available through Representable and/or Distributive, so most places where they might be used end up sticking with the superclasses instead.
Data.Functor.Adjunction, of course, also offers useful functions for left adjoints. On the one hand, left adjoints (which are isomorphic to pairs i.e. containers that hold a single element) are probably less versatile than right adjoints (which are isomorphic to functions i.e. functors with a single shape); on the other hand, there doesn't seem to be any canonical class for left adjoints (not yet, at least), so that might lead to opportunities for actually using Data.Functor.Adjunction functions. Incidentally, Chris Penner's battleship example you suggested arguably fits the bill, as it does rely on the left adjoint and how it can be used to encode the representation of the right adjoint:
zapWithAdjunction :: Adjunction f u => (a -> b -> c) -> u a -> f b -> c
zapWithAdjunction #CoordF #Board :: (a -> b -> c) -> Board a -> CoordF b -> c
checkHit :: Vessel -> Weapon -> Bool
shoot :: Board Vessel -> CoordF Weapon -> Bool
CoordF, the left adjoint, carries coordinates for the board and a payload. zapWithAdjunction makes it possible to (quite literally, in this case), target the position while using the payload.
I am working through the section on Functors in the Typeclassopedia.
A simple intuition is that a Functor represents a “container” of some sort, along with the ability to apply a function uniformly to every element in the container.
OK. So, functors appear pretty natural for inductive types like lists or trees.
Functors also appear pretty simple if the number of elements is fixed to a low number. For example, with Maybe you just have to be concerned about "Nothing" or "Just a" -- two things.
So, how would you make something like a graph, that could potentially have loops, an instance of Functor? I think a more generalized way to put it is, how do non-inductive types "fit into" Functors?
The more I think about it, the more I realize that inductive / non-inductive doesn't really matter. Inductive types are just easier to define fmap for...
If I wanted to make a graph an instance of Functor, I would have to implement a graph traversal algorithm inside fmap; for example it would probably have to use a helper function that would keep track of the visited nodes. At this point, I am now wondering why bother defining it as a Functor instead of just writing this as a function itself? E.g. map vs fmap for lists...?
I hope someone with experience, war stories, and scars can shed some light. Thanks!
Well let's assume you define a graph like this
data Graph a = Node a [Graph a]
Then fmap is just defined precisely as you would expect
instance Functor Graph where
fmap f (Node a ns) = Node (f a) (map (fmap f) ns)
Now, if there's a loop then we'd have had to do something like
foo = Node 1 [bar]
bar = Node 2 [foo]
Now fmap is sufficiently lazy that you can evaluate part of it's result without forcing the rest of the computation, so it works just as well as any knot-tied graph representation would!
In general this is the trick: fmap is lazy so you can treat it's results just as you would treat any non-inductive values in Haskell (: carefully).
Also, you should define fmap vs the random other functions since
fmap is a good, well known API with rules
Your container now places well with things expecting Functors
You can abstract away other bits of your program so they depend on Functor, not your Graph
In general when I see something is a functor I think "Ah wonderful, I know just how to use that" and when I see
superAwesomeTraversal :: (a -> b) -> Foo a -> Foo b
I get a little worried that this will do unexpected things..
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.
Every time someone promises to "explain monads", my interest is piqued, only to be replaced by frustration when the alleged "explanation" is a long list of examples terminated by some off-hand remark that the "mathematical theory" behind the "esoteric ideas" is "too complicated to explain at this point".
Now I'm asking for the opposite. I have a solid grasp on category theory and I'm not afraid of diagram chasing, Yoneda's lemma or derived functors (and indeed on monads and adjunctions in the categorical sense).
Could someone give me a clear and concise definition of what a monad is in functional programming? The fewer examples the better: sometimes one clear concept says more than a hundred timid examples. Haskell would do nicely as a language for demonstration though I'm not picky.
This question has some good answers: Monads as adjunctions
More to the point, Derek Elkins' "Calculating Monads with Category Theory" article in TMR #13 should have the sort of constructions you're looking for: http://www.haskell.org/wikiupload/8/85/TMR-Issue13.pdf
Finally, and perhaps this is really the closest to what you're looking for, you can go straight to the source and look at Moggi's seminal papers on the topic from 1988-91: http://www.disi.unige.it/person/MoggiE/publications.html
See in particular "Notions of computation and monads".
My own I'm sure too condensed/imprecise take:
Begin with a category Hask whose objects are Haskell types, and whose morphisms are functions. Functions are also objects in Hask, as are products. So Hask is Cartesian closed. Now introduce an arrow mapping every object in Hask to MHask which is a subset of the objects in Hask. Unit!
Next introduce an arrow mapping every arrow on Hask to an arrow on MHask. This gives us map, and makes MHask a covariant endofunctor. Now introduce an arrow mapping every object in MHask which is generated from an object in MHask (via unit) to the object in MHask which generates it. Join! And from the that, MHask is a monad (and a monoidal endofunctor to be more precise).
I'm sure there is a reason why the above is deficient, which is why I'd really direct you, if you're looking for formalism, to the Moggi papers in particular.
As a compliment to Carl's answer, a Monad in Haskell is (theoretically) this:
class Monad m where
join :: m (m a) -> m a
return :: a -> m a
fmap :: (a -> b) -> m a -> m b
Note that "bind" (>>=) can be defined as
x >>= f = join (fmap f x)
According to the Haskell Wiki
A monad in a category C is a triple (F : C → C, η : Id → F, μ : F ∘ F → F)
...with some axioms. For Haskell, fmap, return, and join line up with F, η, and μ, respectively. (fmap in Haskell defines a Functor). If I'm not mistaken, Scala calls these map, pure, and join respectively. (Scala calls bind "flatMap")
Ok, using Haskell terminology and examples...
A monad, in functional programming, is a composition pattern for data types with the kind * -> *.
class Monad (m :: * -> *) where
return :: a -> m a
(>>=) :: m a -> (a -> m b) -> m b
(There's more to the class than that in Haskell, but those are the important parts.)
A data type is a monad if it can implement that interface while satisfying three conditions in the implementation. These are the "monad laws", and I'll leave it to those long-winded explanations for the full explanation. I summarize the laws as "(>>= return) is an identity function, and (>>=) is associative." It's really not more than that, even if it can be expressed more precisely.
And that's all a monad is. If you can implement that interface while preserving those behavioral properties, you have a monad.
That explanation is probably shorter than you expected. That's because the monad interface really is very abstract. The incredible level of abstraction is part of why so many different things can be modeled as monads.
What's less obvious is that as abstract as the interface is, it allows generically modeling any control-flow pattern, regardless of the actual monad implementation. This is why the Control.Monad package in GHC's base library has combinators like when, forever, etc. And this is why the ability to explicitly abstract over any monad implementation is powerful, especially with support from a type system.
You should read the paper by Eugenio Moggi "Notions of computations and monads" which explain the then proposed role of monads to structure denotational semantic of effectful languages.
Also there is a related question:
References for learning the theory behind pure functional languages such as Haskell?
As you don't want hand-waving, you have to read scientific papers, not forum answers or tutorials.
A monad is a monoid in the category of endofunctors, whats the problem?.
Humor aside, I personally believe that monads, as they are used in Haskell and functional programming, are better understood from the monads-as-an-interface point of view (as in Carl's and Dan's answers) instead of from the monads-as-the-term-from-category-theory point of view. I have to confess that I only really internalized the whole monad thing when I had to use a monadic library from another language in a real project.
You mention that you didn't like all the "lots of examples" tutorials. Has anyone ever pointed you to the Awkward squad paper? It focuses manly in the IO monad but the introduction gives a good technical and historical explanation of why the monad concept was embraced by Haskell in the first place.
I don't really know what I'm talking about, but here's my take:
Monads are used to represent computations. You can think of a normal procedural program, which is basically a list of statements, as a bunch of composed computations. Monads are a generalization of this concept, allowing you to define how the statements get composed. Each computation has a value (it could just be ()); the monad just determines how the value strung through a series of computations behaves.
Do notation is really what makes this clear: it's basically a special sort of statement-based language that lets you define what happens between statements. It's as if you could define how ";" worked in C-like languages.
In this light all of the monads I've used so far makes sense: State doesn't affect the value but updates a second value which is passed along from computation to computation in the background; Maybe short-circuits the value if it ever encounters a Nothing; List lets you have a variable number of values passed through; IO lets you have impure values passed through in a safe way. The more specialized monads I've used like Gen and Parsec parsers are also similar.
Hopefully this is a clear explanation which isn't completely off-base.
Since you understand monads in the category-theoretic sense I am interpreting your question as being about the presentation of monads in functional programming.
Thus my answer avoids any explanation of what a monad is, or any intuition about its meaning or use.
Answer: In Haskell a monad is presented, in an internal language for some category, as the (internalised) maps of a Kleisli triple.
Explanation:
It is hard to be precise about the properties of the "Hask category", and these properties are largely irrelevant for understanding Haskell's presentation of monads.
Instead, for this discussion, it is more useful to understand Haskell as an internal language for some category C. Haskell functions define morphisms in C and Haskell types are objects in C, but the particular category in which these definitions are made is unimportant.
Parameteric data types, e.g. data F a = ..., are object mappings, e.g. F : |C| -> |C|.
The usual description of a monad in Haskell is in Kleisli triple (or Kleisli extension) form:
class Monad m where
return :: a -> m a
(>>=) :: m a -> (a -> m b) -> m b
where:
m is the object mapping m :|C| -> |C|
return is the unit operation on objects
>>= (pronounced "bind" by Haskellers) is the extension operation on morphisms but with its first two parameters swapped (cf. usual signature of extension (-)* : (a -> m b) -> m a -> m b)
(These maps are themselves internalised as families of morphisms in C, which is possible since m :|C| -> |C|).
Haskell's do-notation (if you have come across this) is therefore an internal language for Kleisli categories.
The Haskell wikibook page has a good basic explanation.