Source: http://lambda.jimpryor.net/topics/week7_introducing_monads/
Timestamp: 2019-04-21 14:29:53+00:00

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The tradition in the functional programming literature is to introduce monads using a metaphor: monads are spacesuits, monads are monsters, monads are burritos. These metaphors can be helpful, and they can be unhelpful. There's a backlash about the metaphors that tells people to instead just look at the formal definition. We'll give that to you below, but it's sometimes sloganized as A monad is just a monoid in the category of endofunctors, what's the problem?. Without some intuitive guidance, this can also be unhelpful. We'll try to find a good balance.
The closest we will come to metaphorical talk is to suggest that monadic types place values inside of boxes, and that monads wrap and unwrap boxes to expose or enclose the values inside of them. In any case, our emphasis will be on starting with the abstract structure of monads, followed in coming weeks by instances of monads from the philosophical and linguistics literature.
After you've read this once and are coming back to re-read it to try to digest the details further, the "endofunctors" that slogan is talking about are a combination of our boxes and their associated maps. Their "monoidal" character is captured in the Monad Laws, for which see below.
The idea is that whatever type the free type variable α might be instantiated to, we will have a "type box" of a certain sort that "contains" values of type α. For instance, if α list is our box type, and α instantiates to the type int, then in this context, int list is the type of a boxed integer.
Warning: although our initial motivating examples are readily thought of as "containers" (lists, trees, and so on, with αs as their "elements"), with later examples we discuss it will be less natural to describe the boxed types that way. For example, where R is some fixed type, R -> α will be one box type we work extensively with.
for the type of a boxed int.
In the first, P has become int and Q has become bool. (The boxed type Q is bool).
We have to be careful though not to to unthinkingly equivocate between different kinds of boxes.
As semanticists, you are no doubt familiar with the debates between those who insist that propositions are sets of worlds and those who insist they are context change potentials. We hope to show you, in coming weeks, that propositions are (certain sorts of) Kleisli arrows. But this doesn't really compete with the other proposals; it is a generalization of them. Both of the other proposed structures can be construed as specific Kleisli arrow types.
We'll need a family of functions to help us work with box types. As will become clear, these have to be defined differently for each box type.
In Haskell, this is the function fmap from the Prelude and Data.Functor; also called <$> in Data.Functor and Control.Applicative, and also called Control.Applicative.liftA and Control.Monad.liftM.
In Haskell, this is called Control.Applicative.liftA2 and Control.Monad.liftM2.
This notion is exemplified by Just for the box type Maybe α and by the singleton function for the box type List α. It will be a way of boxing values with your box type that plays a distinguished role in the various Laws and interdefinitions we present below.
In Haskell, this is called Control.Monad.return and Control.Applicative.pure. In other theoretical contexts it is sometimes called unit or η. All of these names are somewhat unfortunate. First, it has little to do with η-reduction in the Lambda Calculus. Second, it has little to do with the () : unit value we discussed in earlier classes. Third, it has little to do with the return keyword in C and other languages; that's more closely related to continuations, which we'll discuss in later weeks. Finally, this doesn't perfectly align with other uses of "pure" in the literature. ⇧'d values will generally be "pure" in the other senses, but other boxed values can be too.
For all these reasons, we're thinking it will be clearer in our discussion to use a different name. In the class presentation Jim called it 𝟭; and in an earlier draft of this page we (only) called it mid ("m" plus "identity"); but now we're trying out ⇧ as a symbolic alternative. But in the end, we might switch to just using η.
We'll use ¢ as a left-associative infix operator, reminiscent of (the right-associative) $ which is just ordinary function application (also expressed by mere left-associative juxtaposition). In the class presentation Jim called ¢ ⚫; and in an earlier draft of this page we called it m$. In Haskell, it's called Control.Monad.ap or Control.Applicative.<*>.
In Haskell, this is Control.Monad.<=<.
In Haskell, this is Control.Monad.>=>. We will move freely back and forth between using <=< (aka mcomp) and using >=>, which is just <=< with its arguments flipped. <=< has the virtue that it corresponds more closely to the ordinary mathematical symbol ○. But >=> has the virtue that its types flow more naturally from left to right.
In the class handout, we gave the types for >=> twice, and once was correct but the other was a typo. The above is the correct typing.
Haskell uses the symbol >>= but calls it "bind". This is not well chosen from the perspective of formal semantics, since it's only loosely connected with what we mean by "binding." But the name is too deeply entrenched to change. We've at least preprended an "m" to the front of "bind". In some presentations this operation is called ★.
In Haskell, this is Control.Monad.join. In other theoretical contexts it is sometimes called μ.
The menagerie isn't quite as bewildering as you might suppose. Many of these will be interdefinable. For example, here is how mcomp and mbind are related: k <=< j ≡ \a. (j a >>= k). We'll state some other interdefinitions below.
Essentially these say that map is a homomorphism from the algebra of (universe α -> β, operation ○, elsment id) to that of (α -> β, ○', id'), where ○' and id' are ○ and id restricted to arguments of type _. That might be hard to digest because it's so abstract. Think of the following concrete example: if you take a α list (that's our α), and apply id to each of its elements, that's the same as applying id to the list itself. That's the first law. And if you apply the composition of functions g ○ f to each of the list's elements, that's the same as first applying f to each of the elements, and then going through the elements of the resulting list and applying g to each of those elements. That's the second law. These laws obviously hold for our familiar notion of map in relation to lists.
As mentioned at the top of the page, in Category Theory presentations of monads they usually talk about "endofunctors", which are mappings from a Category to itself. In the uses they make of this notion, the endofunctors combine the role of a box type _ and of the map that goes together with it.
The map2ing of composition onto boxes fs and gs of functions, when ¢'d to a box xs of arguments == the ¢ing of fs to the ¢ing of gs to xs: (⇧(○) ¢ fs ¢ gs) ¢ xs = fs ¢ (gs ¢ xs).
When the arguments (the right-hand operand of ¢) are an ⇧'d value, the order of ¢ing doesn't matter: fs ¢ (⇧x) = ⇧($x) ¢ fs. (Though note that it's ⇧($x), or ⇧(\f. f x) that gets ¢d onto fs, not the original ⇧x.) Here's an example where the order does matter: [succ,pred] ¢ [1,2] == [2,3,0,1], but [($1),($2)] ¢ [succ,pred] == [2,0,3,1]. This Law states a class of cases where the order is guaranteed not to matter.
A consequence of the laws already stated is that when the left-hand operand of ¢ is a ⇧'d value, the order of ¢ing doesn't matter either: ⇧f ¢ xs == map (flip ($)) xs ¢ ⇧f.
If you studied algebra, you'll remember that a monoid is a universe with some associative operation that has an identity. For example, the natural numbers form a monoid with multiplication as the operation and 1 as the identity, or with addition as the operation and 0 as the identity. Strings form a monoid with concatenation as the operation and the empty string as the identity. (This example shows that the operation need not be commutative.) Monads are a kind of generalization of this notion, and that's why they're named as they are. The key difference is that for monads, the values being operated on need not be of the same type. They can be, if they're all Kleisli arrows of a single type P -> P. But they needn't be. Their types only need to "cohere" in the sense that the output type of the one arrow is a boxing of the input type of the next.
If you have any of mcomp, mpmoc, mbind, or join, you can use them to define the others. Also, with these functions you can define ¢ and map2 from MapNables. So with Monads, all you really need to get the whole system of functions are a definition of ⇧, on the one hand, and one of mcomp, mbind, or join, on the other.
The first of these says that if you have a triply-boxed type, and you first merge the inner two boxes (with map join), and then merge the resulting box with the outermost box, that's the same as if you had first merged the outer two boxes, and then merged the resulting box with the innermost box. The second law says that if you take a box type and wrap a second box around it (with ⇧) and then merge them, that's the same as if you had done nothing, or if you had instead wrapped a second box around each element of the original (with map ⇧, leaving the original box on the outside), and then merged them.
A word of advice: if you're doing any work in this conceptual neighborhood and need a Greek letter, don't use μ. In addition to the preceding usage, there's also a use in recursion theory (for the minimization operator), in type theory (as a fixed point operator for types), and in the λμ-calculus, which is a formal system that deals with continuations, which we will focus on later in the course. So μ already exhibits more ambiguity than it can handle. We link to further reading about the Category Theory origins of Monads below.
There isn't any single ⇧ function, or single mbind function, and so on. For each new box type, this has to be worked out in a useful way. And as we hinted, in many cases the choice of box type still leaves some latitude about how they should be defined. We commonly talk about "the List Monad" to mean a combination of the choice of α list for the box type and particular definitions for the various functions listed above. There's also "the ZipList MapNable/Applicative" which combines that same box type with other choices for (some of) the functions. Many of these packages also define special-purpose operations that only make sense for that system, but not for other Monads or Mappables.
As hinted in last week's homework and explained in class, the operations available in a Mappable system exactly preserve the "structure" of the boxed type they're operating on, and moreover are only sensitive to what content is in the corresponding original position. If you say map f [1,2,3], then what ends up in the first position of the result depends only on how f and 1 combine.
For MapNable operations, on the other hand, the structure of the result may instead be a complex function of the structure of the original arguments. But only of their structure, not of their contents. And if you say map2 f [10,20] [1,2,3], what ends up in the first position of the result depends only on how f and 10 and 1 combine.
With map, you can supply an f such that map f [3,2,0,1] == [[3,3,3],[2,2],,]. But you can't transform [3,2,0,1] to [3,3,3,2,2,1], and you can't do that with MapNable operations, either. That would involve the structure of the result (here, the length of the list) being sensitive to the content, and not merely the structure, of the original.
For Monads (Composables), on the other hand, you can perform more radical transformations of that sort. For example, join (map (\x. dup x x) [3,2,0,1]) would give us [3,3,3,2,2,1] (for a suitable definition of dup).
But we may sometimes slip.
mzero is a value of type α that is exemplified by Nothing for the box type Maybe α and by  for the box type List α. It has the behavior that anything ¢ mzero == mzero == mzero ¢ anything == mzero >>= anything. In Haskell, this notion is called Control.Applicative.empty or Control.Monad.mzero.
Haskell has a notion >> definable as \u v. map (const id) u ¢ v, or as \u v. u >>= const v. This is often useful, and u >> v won't in general be identical to just v. For example, using the box type List α, [1,2,3] >> [4,5] == [4,5,4,5,4,5]. But in the special case of mzero, it is a consequence of what we said above that anything >> mzero == mzero. Haskell also calls >> Control.Applicative.*>.
Haskell has a correlative notion Control.Applicative.<*, definable as \u v. map const u ¢ v. For example, [1,2,3] <* [4,5] == [1,1,2,2,3,3].
mapconst is definable as map ○ const. For example mapconst 4 [1,2,3] == [4,4,4]. Haskell calls mapconst <$ in Data.Functor and Control.Applicative. They also use $> for flip mapconst, and Control.Monad.void for mapconst ().
The Identity monad is favored by mimes.
In words, mcomp k j a feeds the a (which has type α) to j, which returns a list of βs; each β in that list is fed to k, which returns a list of γs. The final result is the concatenation of those lists of γs.
j 7 produced [49, 14], which after being fed through k gave us [49, 50, 14, 15].
These implementations of <=< and ¢ for lists use the "crossing" strategy for pairing up multiple lists, as opposed to the "zipping" strategy. Nothing forces that choice; you could also define ¢ using the "zipping" strategy instead. (But then you wouldn't be able to build a corresponding Monad; see below.) Haskell talks of the List Monad in the first case, and the ZipList Applicative in the second case.
Then we retain only those αs for which f returns Some b; when f returns None, we just leave out any corresponding element in the result.
Now we can have as many elements in the result for a given α as k cares to return. Another way to write catmap k xs is as (Haskell) concat (map k xs) or (OCaml) List.flatten (List.map k xs). And this is just the definition of mbind or >>= for the List Monad. The definition of mcomp or <=<, that we gave above, differs only in that it's the way to compose two functions j and k, that you'd want to catmap, rather than the way to catmap one of those functions over a value that's already a list.
This example is a good intuitive basis for thinking about the notions of mbind and mcomp more generally. Thus mbind for the option/Maybe type takes an option value, applies k to its element (if there is one), and returns the resulting option value. mbind for a tree with α-labeled leaves would apply k to each of the leaves, and return a tree containing arbitrarily large subtrees in place of all its former leaves, depending on what k returned.
.   3       >>=(α,unit) tree  (\a ->  / \  )  ==>   / \     .
Though as we warned before, only some of the Monads we'll be working with are naturally thought of "containers"; so in other cases the similarity of their mbind operations to what we have here will be more abstract.
The question came up in class of when box types might fail to be Mappable, or Mappables might fail to be MapNables, or MapNables might fail to be Monads.
But if on the other hand, your box type is α -> R, you'll find that there is no way to define a map operation that takes arbitrary functions of type P -> Q and values of the boxed type P, that is P -> R, and returns values of the boxed type Q.
But for some types neither of these will be the case. For function types, as we already mentioned, == is not decidable. If the functions have suitable types, they do form a monoid with ○ as the operation and id as the identity; but many function types won't be such that arbitrary functions of that type are composable. So when R is the type of functions from ints to bools, for example, we won't have any way to write a ¢ that satisfies the constraints stated above.
For the third failure, that is examples of MapNables that aren't Monads, we'll just state that lists where the map2 operation is taken to be zipping rather than taking the Cartesian product (what in Haskell are called ZipLists), these are claimed to exemplify that failure. But we aren't now in a position to demonstrate that to you.
Eugenio Moggi, Notions of Computation and Monads: Information and Computation 93 (1) 1991. This paper is available online, but would be very difficult reading for members of this seminar, so we won't link to it. However, the next two papers should be accessible.
Philip Wadler. The essence of functional programming: invited talk, 19'th Symposium on Principles of Programming Languages, ACM Press, Albuquerque, January 1992.
Philip Wadler. Monads for Functional Programming: in M. Broy, editor, Marktoberdorf Summer School on Program Design Calculi, Springer Verlag, NATO ASI Series F: Computer and systems sciences, Volume 118, August 1992. Also in J. Jeuring and E. Meijer, editors, Advanced Functional Programming, Springer Verlag, LNCS 925, 1995. Some errata fixed August 2001.
There's a long list of monad tutorials linked at the Haskell wiki (we linked to this at the top of the page), and on our own Offsite Reading page. (Skimming the titles is somewhat amusing.) If you are confused by monads, make use of these resources. Read around until you find a tutorial pitched at a level that's helpful for you.

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