1 <!-- λ Λ ∀ ≡ α β γ ρ ω Ω ○ μ η δ ζ ξ ⋆ ★ • ∙ ● 𝟎 𝟏 𝟐 𝟘 𝟙 𝟚 𝟬 𝟭 𝟮 ¢ ⇧ -->
4 The [[tradition in the functional programming
5 literature|https://wiki.haskell.org/Monad_tutorials_timeline]] is to
6 introduce monads using a metaphor: monads are spacesuits, monads are
7 monsters, monads are burritos. These metaphors can be helpful, and they
8 can be unhelpful. There's a backlash about the metaphors that tells people
9 to instead just look at the formal definition. We'll give that to you below, but it's
10 sometimes sloganized as
11 [A monad is just a monoid in the category of endofunctors, what's the problem?](http://stackoverflow.com/questions/3870088).
12 Without some intuitive guidance, this can also be unhelpful. We'll try to find a good balance.
15 The closest we will come to metaphorical talk is to suggest that
16 monadic types place values inside of *boxes*, and that monads wrap
17 and unwrap boxes to expose or enclose the values inside of them. In
18 any case, our emphasis will be on starting with the abstract structure
19 of monads, followed in coming weeks by instances of monads from the philosophical and
20 linguistics literature.
22 > <small>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 `map`s. Their "monoidal" character is captured in the Monad Laws, for which see below.</small>
28 ## Box types: type expressions with one free type variable ##
30 Recall that we've been using lower-case Greek letters
31 <code>α, β, γ, ...</code> as type variables. We'll
32 use `P`, `Q`, `R`, and `S` as schematic metavariables over type expressions, that may or may not contain unbound
33 type variables. For instance, we might have
42 A *box type* will be a type expression that contains exactly one free
43 type variable. (You could extend this to expressions with more free variables; then you'd have
44 to specify which one of them the box is capturing. But let's keep it simple.) Some examples (using OCaml's type conventions):
48 (α, R) tree (assuming R contains no free type variables)
51 The idea is that whatever type the free type variable `α` might be instantiated to,
52 we will have a "type box" of a certain sort that "contains" values of type `α`. For instance,
53 if `α list` is our box type, and `α` instantiates to the type `int`, then in this context, `int list`
54 is the type of a boxed integer.
56 Warning: although our initial motivating examples are readily thought of as "containers" (lists, trees, and so on, with `α`s 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.
58 Also, for clarity: the *box type* is the type `α list` (or as we might just say, the `list` type operator); the *boxed type* is some specific instantiation of the free type variable `α`. We'll often write boxed types as a box containing what the free
59 type variable instantiates to. So if our box type is `α list`, and `α` instantiates to the specific type `int`, we write:
61 <code><u>int</u></code>
63 for the type of a boxed `int`.
69 A lot of what we'll be doing concerns types that are called *Kleisli arrows*. Don't worry about why they're called that, or if you like go read some Category Theory. All we need to know is that these are functions whose type has the form:
71 <code>P -> <u>Q</u></code>
73 That is, they are functions from values of one type `P` to a boxed type `Q`, for some choice of box and of type expressions `P` and `Q`.
74 For instance, the following are Kleisli arrow types:
76 <code>int -> <u>bool</u></code>
78 <code>int list -> <u>int list</u></code>
80 In the first, `P` has become `int` and `Q` has become `bool`. (The boxed type <code><u>Q</u></code> is <code><u>bool</u></code>).
82 Note that either of the schemas `P` or `Q` are permitted to themselves be boxed
83 types. That is, if `α list` is our box type, we can write the second type as:
85 <code><u>int</u> -> <u>int list</u></code>
87 And also what the rhs there is a boxing of is itself a boxed type (with the same kind of box):, so we can write it as:
89 <code><u>int</u> -> <span class="box2">int</span></code>
91 We have to be careful though not to to unthinkingly equivocate between different kinds of boxes.
93 Here are some examples of values of these Kleisli arrow types, where the box type is `α list`, and the Kleisli arrow types are <code>int -> <u>int</u></code> (that is, `int -> int list`) or <code>int -> <u>bool</u></code>:
97 \x. prime_factors_of x
100 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](/images/faye_dunaway.jpg) with the other proposals; it is a generalization of them. Both of the other proposed structures can be construed as specific Kleisli arrow types.
103 ## A family of functions for each box type ##
105 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.
107 Here are the types of our crucial functions, together with our pronunciation, and some other names the functions go by. (Usually the type doesn't fix their behavior, which will be discussed below.)
109 <code>map (/mæp/): (P -> Q) -> <u>P</u> -> <u>Q</u></code>
111 > In Haskell, this is the function `fmap` from the `Prelude` and `Data.Functor`; also called `<$>` in `Data.Functor` and `Control.Applicative`, and also called `Control.Applicative.liftA` and `Control.Monad.liftM`.
113 <code>map2 (/mæptu/): (P -> Q -> R) -> <u>P</u> -> <u>Q</u> -> <u>R</u></code>
115 > In Haskell, this is called `Control.Applicative.liftA2` and `Control.Monad.liftM2`.
117 <code>⇧ or mid (/εmaidεnt@tI/): P -> <u>P</u></code>
119 > This notion is exemplified by `Just` for the box type `Maybe α` and by the singleton function for the box type `List α`. It will be a way of boxing values with your box type that plays a distinguished role in the various Laws and interdefinitions we present below.
121 > In Haskell, this is called `Control.Monad.return` and `Control.Applicative.pure`. In other theoretical contexts it is sometimes called `unit` or `η`. All of these names are somewhat unfortunate. First, it has little to do with `η`-reduction in the Lambda Calculus. Second, it has little to do with the `() : unit` value we discussed in earlier classes. Third, it has little to do with the `return` keyword in C and other languages; that's more closely related to continuations, which we'll discuss in later weeks. Finally, this doesn't perfectly align with other uses of "pure" in the literature. `⇧`'d values _will_ generally be "pure" in the other senses, but other boxed values can be too.
123 > For all these reasons, we're thinking it will be clearer in our discussion to use a different name. In the class presentation Jim called it `𝟭`; and in an earlier draft of this page we (only) called it `mid` ("m" plus "identity"); but now we're trying out `⇧` as a symbolic alternative. But in the end, we might switch to just using `η`.
125 <code>¢ or mapply (/εm@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>
127 > We'll use `¢` as a left-associative infix operator, reminiscent of (the right-associative) `$` 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.<*>`.
129 <code><=< or mcomp : (Q -> <u>R</u>) -> (P -> <u>Q</u>) -> (P -> <u>R</u>)</code>
131 > In Haskell, this is `Control.Monad.<=<`.
133 <code>>=> or flip mcomp : (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
135 > In Haskell, this is `Control.Monad.>=>`. We will move freely back and forth between using `<=<` (aka `mcomp`) and using `>=>`, which
136 is just `<=<` with its arguments flipped. `<=<` has the virtue that it corresponds more
137 closely to the ordinary mathematical symbol `○`. But `>=>` has the virtue
138 that its types flow more naturally from left to right.
140 > 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.
142 <code>>>= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
144 > 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 `★`.
146 <code>=<< or flip mbind : (Q -> <u>R</u>) -> (<u>Q</u>) -> (<u>R</u>)</code>
148 <code>join: <span class="box2">P</span> -> <u>P</u></code>
150 > In Haskell, this is `Control.Monad.join`. In other theoretical contexts it is sometimes called `μ`.
152 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: <code>k <=< j ≡ \a. (j a >>= k)</code>. We'll state some other interdefinitions below.
154 These functions come together in several systems, and have to be defined in a way that coheres with the other functions in the system:
156 * ***Mappable*** (in Haskelese, "Functors") At the most general level, box types are *Mappable*
157 if there is a `map` function defined for that box type with the type given above. This
158 has to obey the following Map Laws:
160 <code>map (id : α -> α) == (id : <u>α</u> -> <u>α</u>)</code>
161 <code>map (g ○ f) == (map g) ○ (map f)</code>
163 Essentially these say that `map` is a homomorphism from the algebra of `(universe α -> β, operation ○, elsment id)` to that of <code>(<u>α</u> -> <u>β</u>, ○', id')</code>, where `○'` and `id'` are `○` and `id` restricted to arguments of type <code><u>_</u></code>. 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 <code><u>α</u></code>), 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.
165 > <small>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 <code><u>_</u></code> and of the `map` that goes together with it.</small>
168 * ***MapNable*** (in Haskelese, "Applicatives") A Mappable box type is *MapNable*
169 if there are in addition `map2`, `⇧`, and `mapply`. (Given either
170 of `map2` and `mapply`, you can define the other, and also `map`.
171 Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
172 have to obey the following MapN Laws:
174 1. <code>⇧(id : P->P) : <u>P</u> -> <u>P</u></code> is a left identity for `¢`, that is: `(⇧id) ¢ xs = xs`
175 2. `⇧(f a) = (⇧f) ¢ (⇧a)`
176 3. The `map2`ing 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)`.
177 4. 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),($2)] ¢ [succ,pred] == [2,0,3,1]`. This Law states a class of cases where the order is guaranteed not to matter.
178 5. A consequence of the laws already stated is that when the _left_-hand operand of `¢` is a `⇧`'d value, the order of `¢`ing doesn't matter either: `⇧f ¢ xs == map (flip ($)) xs ¢ ⇧f`.
180 <!-- Probably there's a shorter proof, but:
182 == ⇧T ¢ ((⇧id) ¢ xs) ¢ ⇧f, by 1
183 == ⇧(○) ¢ ⇧T ¢ ⇧id ¢ xs ¢ ⇧f, by 3
184 == ⇧($id) ¢ (⇧(○) ¢ ⇧T) ¢ xs ¢ ⇧f, by 4
185 == ⇧(○) ¢ ⇧($id) ¢ ⇧(○) ¢ ⇧T ¢ xs ¢ ⇧f, by 3
186 == ⇧((○) ($id)) ¢ ⇧(○) ¢ ⇧T ¢ xs ¢ ⇧f, by 2
187 == ⇧((○) ($id) (○)) ¢ ⇧T ¢ xs ¢ ⇧f, by 2
188 == ⇧id ¢ ⇧T ¢ xs ¢ ⇧f, by definitions of ○ and $
189 == ⇧T ¢ xs ¢ ⇧f, by 1
190 == ⇧($f) ¢ (⇧T ¢ xs), by 4
191 == ⇧(○) ¢ ⇧($f) ¢ ⇧T ¢ xs, by 3
192 == ⇧((○) ($f)) ¢ ⇧T ¢ xs, by 2
193 == ⇧((○) ($f) T) ¢ xs, by 2
194 == ⇧f ¢ xs, by definitions of ○ and $ and T == flip ($)
197 * ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
198 is in addition an associative `mcomp` having `⇧` as its left and
199 right identity. That is, the following Monad Laws must hold:
201 mcomp (mcomp j k) l (that is, (j <=< k) <=< l) == mcomp j (mcomp k l)
202 mcomp mid k (that is, ⇧ <=< k) == k
203 mcomp k mid (that is, k <=< ⇧) == k
205 You could just as well express the Monad laws using `>=>`:
207 l >=> (k >=> j) == (l >=> k) >=> j
211 If you studied algebra, you'll remember that a mon*oid* 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 <code>P -> <u>P</u></code>. 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.
213 In the Haskell manuals, they express the Monad Laws using `>>=` instead of the composition operators `>=>` or `<=<`. This looks similar, but doesn't have the same symmetry:
215 u >>= (\a -> k a >>= j) == (u >>= k) >>= j
219 (Also, Haskell calls `⇧` `return` or `pure`, but we've stuck to our terminology in this context.) Some authors try to make the first of those Laws look more symmetrical by writing it as:
221 (A >>= \a -> B) >>= \b -> C == A >>= (\a -> B >>= \b -> C)
223 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.
226 > <small>In Category Theory discussion, the Monad Laws are instead expressed in terms of `join` (which they call `μ`) and `⇧` (which they call `η`). These are assumed to be "natural transformations" for their box type, which means that they satisfy these equations with that box type's `map`:
227 > <pre>map f ○ ⇧ == ⇧ ○ f<br>map f ○ join == join ○ map (map f)</pre>
228 > The Monad Laws then take the form:
229 > <pre>join ○ (map join) == join ○ join<br>join ○ ⇧ == id == join ○ map ⇧</pre>
230 > 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.<p>
231 > The Category Theorist would state these Laws like this, where `M` is the endofunctor that takes us from type `α` to type <code><u>α</u></code>:
232 > <pre>μ ○ M(μ) == μ ○ μ<br>μ ○ η == id == μ ○ M(η)</pre>
233 > 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.
234 > We link to further reading about the Category Theory origins of Monads below.</small>
236 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.
238 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.
240 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.
242 With `map`, you can supply an `f` such that `map f [3,2,0,1] == [[3,3,3],[2,2],[],[1]]`. 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.
244 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`).
247 Some global transformations that we work with in semantics, like Veltman's test functions, can't directly be expressed in terms of the general Monad operations. For example, there's no `j` such that `xs >>= j == mzero` if `xs` anywhere contains the value `1`. That would instead be defined as a special-purpose operation, specific to some restricted class of Monads.
251 ## Interdefinitions and Subsidiary notions##
253 We said above that various of these box type operations can be defined in terms of others. Here is a list of various ways in which they're related. We try to stick to the consistent typing conventions that:
256 f : α -> β; g and h have types of the same form
257 also sometimes these will have types of the form α -> β -> γ
258 note that α and β are permitted to be, but needn't be, boxed types
259 j : α -> <u>β</u>; k and l have types of the same form
260 u : <u>α</u>; v and xs and ys have types of the same form
262 w : <span class="box2">α</span>
265 But we may sometimes slip.
267 Here are some ways the different notions are related:
270 j >=> k ≡= \a. (j a >>= k)
271 u >>= k == (id >=> k) u; or ((\(). u) >=> k) ()
272 u >>= k == join (map k u)
274 map2 f xs ys == xs >>= (\x. ys >>= (\y. ⇧(f x y)))
275 map2 f xs ys == (map f xs) ¢ ys, using ¢ as an infix operator
276 fs ¢ xs == fs >>= (\f. map f xs)
279 map f u == u >>= ⇧ ○ f
283 Here are some other monadic notion that you may sometimes encounter:
285 * <code>mzero</code> is a value of type <code><u>α</u></code> 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`.
287 * 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.*>`.
289 * 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]`. <!-- You might expect Haskell to call `<*` `<<`, but they don't. I thought they used to use `<<` for `flip (>>)` instead, but now I can't confirm that this was ever the case. -->
291 * <code>mapconst</code> 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 ()`.
297 To take a trivial (but, as we will see, still useful) example,
298 consider the Identity box type: `α`. So if `α` is type `bool`,
299 then a boxed `α` is ... a `bool`. That is, <code><u>α</u> == α</code>.
300 In terms of the box analogy, the Identity box type is a completely invisible box. With the following
303 mid (* or ⇧ *) ≡ \p. p, that is, our familiar combinator I
304 mcomp (* or <=< *) ≡ \f g x. f (g x), that is, ordinary function composition (○) (aka the B combinator)
306 Identity is a monad. Here is a demonstration that the laws hold:
308 mcomp mid k ≡ (\fgx.f(gx)) (\p.p) k
312 mcomp k mid ≡ (\fgx.f(gx)) k (\p.p)
316 mcomp (mcomp j k) l ≡ mcomp ((\fgx.f(gx)) j k) l
317 ~~> mcomp (\x.j(kx)) l
318 ≡ (\fgx.f(gx)) (\x.j(kx)) l
319 ~~> \x.(\x.j(kx))(lx)
321 mcomp j (mcomp k l) ≡ mcomp j ((\fgx.f(gx)) k l)
322 ~~> mcomp j (\x.k(lx))
323 ≡ (\fgx.f(gx)) j (\x.k(lx))
324 ~~> \x.j((\x.k(lx)) x)
327 The Identity monad is favored by mimes.
333 To take a slightly less trivial (and even more useful) example,
334 consider the box type `α list`, with the following operations:
339 mcomp : (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
340 mcomp k j a = concat (map k (j a)) = List.flatten (List.map k (j a))
341 = foldr (\b ks -> (k b) ++ ks) [] (j a) = List.fold_right (fun b ks -> List.append (k b) ks) [] (j a)
342 = [c | b <- j a, c <- k b]
344 In the first two definitions of `mcomp`, we give the definition first in Haskell and then in the equivalent OCaml. The three different definitions of `mcomp` (one for each line) are all equivalent, and it is easy to show that they obey the Monad Laws. (You will do this in the homework.)
346 In words, `mcomp k j a` feeds the `a` (which has type `α`) to `j`, which returns a list of `β`s;
347 each `β` in that list is fed to `k`, which returns a list of `γ`s. The
348 final result is the concatenation of those lists of `γ`s.
352 let j a = [a*a, a+a] in
353 let k b = [b, b+1] in
354 mcomp k j 7 ==> [49, 50, 14, 15]
356 `j 7` produced `[49, 14]`, which after being fed through `k` gave us `[49, 50, 14, 15]`.
358 Contrast that to `¢` (`mapply`), which operates not on two *box-producing functions*, but instead on two *boxed type values*, one containing functions to be applied to the values in the other box, via some predefined scheme. Thus:
360 let js = [(\a->a*a),(\a->a+a)] in
362 mapply js xs ==> [49, 25, 14, 10]
364 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.
366 Sticking with the "crossing" strategy, here's how to motivate our implementation of `<=<`. Recall that we have on the one hand, an operation `filter` for lists, that applies a predicate to each element of the list, and returns a list containing only those elements which satisfied the predicate. But the elements it does retain, it retains unaltered. On the other hand, we have the operation `map` for lists, that is capable of _changing_ the list elements in the result. But it doesn't have the power to throw list elements away; elements in the source map one-to-one to elements in the result. In many cases, we want something in between `filter` and `map`. We want to be able to ignore or discard some list elements, and possibly also to change the list elements we keep. One way of doing this is to have a function `optmap`, defined like this:
368 let rec optmap (f : α -> β option) (xs : α list) : β list =
373 | None -> optmap f xs'
374 | Some b -> b :: optmap f xs')
376 Then we retain only those `α`s for which `f` returns `Some b`; when `f` returns `None`, we just leave out any corresponding element in the result.
378 That can be helpful, but it only enables us to have _zero or one_ elements in the result corresponding to a given element in the source list. What if we sometimes want more? Well, here is a more general approach:
380 let rec catmap (k : α -> β list) (xs : α list) : β list =
383 | x' :: xs' -> List.append (k x') (catmap k xs')
385 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.
387 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.
390 [3, 2, 0, 1] >>=<sub>α list</sub> (\a -> dup a a) ==> [3, 3, 3, 2, 2, 1]
392 Some a >>=<sub>α option</sub> (\a -> Some 0) ==> Some 0
393 None >>=<sub>α option</sub> (\a -> Some 0) ==> None
394 Some a >>=<sub>α option</sub> (\a -> None ) ==> None
400 . 3 >>=<sub>(α,unit) tree</sub> (\a -> / \ ) ==> / \ .
408 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.
411 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**.
413 For the first failure, we noted that it's easy to define a `map` operation for the box type `R -> α`, for a fixed type `R`. You `map` a function of type `P -> Q` over a value of the boxed type <code><u>P</u></code>, that is a value of type `R -> P`, by just returning a function that takes some `R` as input, first supplies it to your `R -> P` value, and then supplies the result to your `map`ped function of type `P -> Q`. (We will be working with this Mappable extensively; in fact it's not just a Mappable but more specifically a Monad.)
415 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 <code><u>P</u></code>, that is `P -> R`, and returns values of the boxed type <code><u>Q</u></code>.
417 For the second failure, that is cases of Mappables that are not MapNables, we cited box types like `(R, α)`, for arbitrary fixed types `R`. The `map` operation for these is defined by `map f (r,a) = (r, f a)`. For certain choices of `R` these can be MapNables too. The easiest case is when `R` is the type of `()`. But when we look at the MapNable Laws, we'll see that they impose constraints we cannot satisfy for *every* choice of the fixed type `R`. Here's why. We'll need to define `⇧a = (r0, a)` for some specific `r0` of type `R`. The choice can't depend on the value of `a`, because `⇧` needs to work for `a`s of _any_ type. Then the MapNable Laws will entail:
419 1. (r0,id) ¢ (r,x) == (r,x)
420 2. (r0,f x) == (r0,f) ¢ (r0,x)
421 3. (r0,(○)) ¢ (r'',f) ¢ (r',g) ¢ (r,x) == (r'',f) ¢ ((r',g) ¢ (r,x))
422 4. (r'',f) ¢ (r0,x) == (r0,($x)) ¢ (r'',f)
423 5. (r0,f) ¢ (r,x) == (r,($x)) ¢ (r0,f)
425 Now we are not going to be able to write a `¢` function that inspects the second element of its left-hand operand to check if it's the `id` function; the identity of functions is not decidable. So the only way to satisfy Law 1 will be to have the first element of the result (`r`) be taken from the first element of the right-hand operand in _all_ the cases when the first element of the left-hand operand is `r0`. But then that means that the result of the lhs of Law 5 will also have a first element of `r`; so, turning now to the rhs of Law 5, we see that `¢` must use the first element of its _left_-hand operand (here again `r`) at least in those cases when the first element of its right-hand operand is `r0`. If our `R` type has a natural *monoid* structure, we could just let `r0` be the monoid's identity, and have `¢` combine other `R`s using the monoid's operation. Alternatively, if the `R` type is one that we can safely apply the predicate `(r0==)` to, then we could define `¢` something like this:
427 let (¢) (r1,f) (r2,x) = ((if r0==r1 then r2 else if r0==r2 then r1 else ...), ...)
429 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 `int`s to `bool`s, for example, we won't have any way to write a `¢` that satisfies the constraints stated above.
431 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 `ZipList`s), these are claimed to exemplify that failure. But we aren't now in a position to demonstrate that to you.
434 ## Further Reading ##
436 As we mentioned above, the notions of Monads have their origin in Category Theory, where they are mostly specified in terms of (what we call) `⇧` and `join`. For advanced study, here are some further links on the relation between monads as we're working with them and monads as they appear in Category Theory:
437 [1](http://en.wikipedia.org/wiki/Outline_of_category_theory)
438 [2](http://lambda1.jimpryor.net/advanced_topics/monads_in_category_theory/)
439 [3](http://en.wikibooks.org/wiki/Haskell/Category_theory)
440 [4](https://wiki.haskell.org/Category_theory) <small>(where you should follow the further links discussing Functors, Natural Transformations, and Monads)</small>
441 [5](http://www.stephendiehl.com/posts/monads.html)
444 Here are some papers that introduced Monads into functional programming:
446 * 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. <!-- http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf --> However, the next two papers should be accessible.
448 * [Philip Wadler. The essence of functional programming](http://homepages.inf.ed.ac.uk/wadler/papers/essence/essence.ps):
449 invited talk, *19'th Symposium on Principles of Programming Languages*, ACM Press, Albuquerque, January 1992.
450 <!-- This paper explores the use monads to structure functional programs. No prior knowledge of monads or category theory is required.
451 Monads increase the ease with which programs may be modified. They can mimic the effect of impure features such as exceptions, state, and continuations; and also provide effects not easily achieved with such features. The types of a program reflect which effects occur.
452 The first section is an extended example of the use of monads. A simple interpreter is modified to support various extra features: error messages, state, output, and non-deterministic choice. The second section describes the relation between monads and continuation-passing style. The third section sketches how monads are used in a compiler for Haskell that is written in Haskell. -->
454 * [Philip Wadler. Monads for Functional Programming](http://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf):
455 in M. Broy, editor, *Marktoberdorf Summer School on Program Design
456 Calculi*, Springer Verlag, NATO ASI Series F: Computer and systems
457 sciences, Volume 118, August 1992. Also in J. Jeuring and E. Meijer,
458 editors, *Advanced Functional Programming*, Springer Verlag,
459 LNCS 925, 1995. Some errata fixed August 2001.
460 <!-- The use of monads to structure functional programs is described. Monads provide a convenient framework for simulating effects found in other languages, such as global state, exception handling, output, or non-determinism. Three case studies are looked at in detail: how monads ease the modification of a simple evaluator; how monads act as the basis of a datatype of arrays subject to in-place update; and how monads can be used to build parsers. -->
462 Here is some other reading:
464 * [Yet Another Haskell Tutorial on Monad Laws](http://en.wikibooks.org/wiki/Haskell/YAHT/Monads#Definition)
465 * [Haskell wikibook on Understanding Monads](http://en.wikibooks.org/wiki/Haskell/Understanding_monads)
466 * [Haskell wikibook on Advanced Monads](http://en.wikibooks.org/wiki/Haskell/Advanced_monads)
467 * [Haskell wiki on Monad Laws](http://www.haskell.org/haskellwiki/Monad_laws)
469 There's a long list of monad tutorials linked at the [[Haskell wiki|https://wiki.haskell.org/Monad_tutorials_timeline]] (we linked to this at the top of the page), and on our own [[Offsite Reading|/readings]] 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.