1 <!-- λ Λ ∀ ≡ α β γ ρ ω Ω -->
2 <!-- Loved this one: http://www.stephendiehl.com/posts/monads.html -->
7 The [[tradition in the functional programming
8 literature|https://wiki.haskell.org/Monad_tutorials_timeline]] is to
9 introduce monads using a metaphor: monads are spacesuits, monads are
10 monsters, monads are burritos. These metaphors can be helpful, and they
11 can be unhelpful. There's a backlash about the metaphors that tells people
12 to instead just look at the formal definition. We'll give that to you below, but it's
13 sometimes sloganized as
14 [A monad is just a monoid in the category of endofunctors, what's the problem?](http://stackoverflow.com/questions/3870088).
15 Without some intuitive guidance, this can also be unhelpful. We'll try to find a good balance.
18 The closest we will come to metaphorical talk is to suggest that
19 monadic types place values inside of *boxes*, and that monads wrap
20 and unwrap boxes to expose or enclose the values inside of them. In
21 any case, our emphasis will be on starting with the abstract structure
22 of monads, followed by instances of monads from the philosophical and
23 linguistics literature.
25 > <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 maps. Their "monoidal" character is captured in the Monad Laws, where a "monoid"---don't confuse with a mon*ad*---is a simpler algebraic notion, meaning a universe with some associative operation that has an identity. 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:
26 [1](http://en.wikipedia.org/wiki/Outline_of_category_theory)
27 [2](http://lambda1.jimpryor.net/advanced_topics/monads_in_category_theory/)
28 [3](http://en.wikibooks.org/wiki/Haskell/Category_theory)
29 [4](https://wiki.haskell.org/Category_theory), where you should follow the further links discussing Functors, Natural Transformations, and Monads.</small>
32 ## Box types: type expressions with one free type variable ##
34 Recall that we've been using lower-case Greek letters
35 <code>α, β, γ, ...</code> as type variables. We'll
36 use `P`, `Q`, `R`, and `S` as schematic metavariables over type expressions, that may or may not contain unbound
37 type variables. For instance, we might have
46 A *box type* will be a type expression that contains exactly one free
47 type variable. (You could extend this to expressions with more free variables; then you'd have
48 to specify which one of them the box is capturing. But let's keep it simple.) Some examples (using OCaml's type conventions):
52 (α, R) tree (assuming R contains no free type variables)
55 The idea is that whatever type the free type variable `α` might be instantiated to,
56 we will have a "type box" of a certain sort that "contains" values of type `α`. For instance,
57 if `α list` is our box type, and `α` is the type `int`, then in this context, `int list`
58 is the type of a boxed integer.
60 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` is a box type.
62 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
63 type variable instantiates to. So if our box type is `α list`, and `α` instantiates to the specific type `int`, we would write:
65 <code><u>int</u></code>
67 for the type of a boxed `int`.
73 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:
75 <code>P -> <u>Q</u></code>
77 That is, they are functions from values of one type `P` to a boxed type `Q`, for some choice of type expressions `P` and `Q`.
78 For instance, the following are Kleisli arrows:
80 <code>int -> <u>bool</u></code>
82 <code>int list -> <u>int list</u></code>
84 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>).
86 Note that the left-hand schema `P` is permitted to itself be a boxed type. That is, where
87 if `α list` is our box type, we can write the second type as:
89 <code><u>int</u> -> <u>int list</u></code>
91 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 arrows.
94 ## A family of functions for each box type ##
96 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.
98 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.)
100 <code>map (/mæp/): (P -> Q) -> <u>P</u> -> <u>Q</u></code>
102 <code>map2 (/mæptu/): (P -> Q -> R) -> <u>P</u> -> <u>Q</u> -> <u>R</u></code>
104 <code>mid (/εmaidεnt@tI/ aka unit, return, pure): P -> <u>P</u></code>
106 <code>m$ or mapply (/εm@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>
108 <code><=< or mcomp : (Q -> <u>R</u>) -> (P -> <u>Q</u>) -> (P -> <u>R</u>)</code>
110 <code>>=> (flip mcomp, should we call it mpmoc?): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
112 <code>>>= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
114 <code>=<< (flip mbind, should we call it mdnib?) (Q -> <u>R</u>) -> (<u>Q</u>) -> (<u>R</u>)</code>
116 <code>join: <span class="box2">P</span> -> <u>P</u></code>
119 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.
121 Haskell's name "bind" for `>>=` is not well chosen from our perspective, but this is too deeply entrenched by now. We've at least preprended an `m` to the front of it.
123 Haskell's names "return" and "pure" for `mid` are even less well chosen, and we think it will be clearer in our discussion to use a different name. (Also, in other theoretical contexts this notion goes by other names, anyway, like `unit` or `η` --- having nothing to do with `η`-reduction in the Lambda Calculus.) In the handout we called `mid` `𝟭`. But now we've decided that `mid` is better. (Think of it as "m" plus "identity", not as the start of "midway".)
125 The menagerie isn't quite as bewildering as you might suppose. Many of these will
126 be interdefinable. For example, here is how `mcomp` and `mbind` are related: <code>k <=< j ≡
127 \a. (j a >>= k)</code>.
129 We will move freely back and forth between using `>=>` and using `<=<` (aka `mcomp`), which
130 is just `>=>` with its arguments flipped. `<=<` has the virtue that it corresponds more
131 closely to the ordinary mathematical symbol `○`. But `>=>` has the virtue
132 that its types flow more naturally from left to right.
134 These functions come together in several systems, and have to be defined in a way that coheres with the other functions in the system:
136 * ***Mappable*** (in Haskelese, "Functors") At the most general level, box types are *Mappable*
137 if there is a `map` function defined for that box type with the type given above. This
138 has to obey the following Map Laws:
140 <code>map (id : α -> α) == (id : <u>α</u> -> <u>α</u>)</code>
141 <code>map (g ○ f) == (map g) ○ (map f)</code>
143 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.
145 > <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>
148 * ***MapNable*** (in Haskelese, "Applicatives") A Mappable box type is *MapNable*
149 if there are in addition `map2`, `mid`, and `mapply`. (Given either
150 of `map2` and `mapply`, you can define the other, and also `map`.
151 Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
152 have to obey the following MapN Laws:
157 * ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
158 is in addition an associative `mcomp` having `mid` as its left and
159 right identity. That is, the following Monad Laws must hold:
161 mcomp (mcomp j k) l (that is, (j <=< k) <=< l) == mcomp j (mcomp k l)
162 mcomp mid k (that is, mid <=< k) == k
163 mcomp k mid (that is, k <=< mid) == k
165 You could just as well express the Monad laws using `>=>`:
167 l >=> (k >=> j) == (l >=> k) >-> j
171 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 `m$` and `map2` from *MapNables*. So with Monads, all you really need to get the whole system of functions are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `join`, on the other.
173 In practice, you will often work with `>>=`. In the Haskell manuals, they express the Monad Laws using `>>=` instead of the composition operators. This looks similar, but doesn't have the same symmetry:
175 u >>= (\a -> k a >>= j) == (u >>= k) >>= j
179 Also, Haskell calls `mid` `return` or `pure`, but we've stuck to our terminology in this context.
181 > <small>In Category Theory discussion, the Monad Laws are instead expressed in terms of `join` (which they call `μ`) and `mid` (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`:
182 > <pre>map f ○ mid == mid ○ f<br>map f ○ join == join ○ map (map f)</pre>
183 > The Monad Laws then take the form:
184 > <pre>join ○ (map join) == join ○ join<br>join ○ mid == id == join ○ map mid</pre>
185 > Or, as the Category Theorist would state it, where `M` is the endofunctor that takes us from type `α` to type <code><u>α</u></code>:
186 > <pre>μ ○ M(μ) == μ ○ μ<br>μ ○ η = id == μ ○ M(η)</pre></small>
189 Here are some interdefinitions: TODO
191 Names in Haskell: TODO
195 To take a trivial (but, as we will see, still useful) example,
196 consider the Identity box type: `α`. So if `α` is type `bool`,
197 then a boxed `α` is ... a `bool`. That is, <code><u>α</u> = α</code>.
198 In terms of the box analogy, the Identity box type is a completely invisible box. With the following
202 mcomp ≡ \f g x.f (g x)
204 Identity is a monad. Here is a demonstration that the laws hold:
206 mcomp mid k ≡ (\fgx.f(gx)) (\p.p) k
210 mcomp k mid ≡ (\fgx.f(gx)) k (\p.p)
214 mcomp (mcomp j k) l ≡ mcomp ((\fgx.f(gx)) j k) l
215 ~~> mcomp (\x.j(kx)) l
216 ≡ (\fgx.f(gx)) (\x.j(kx)) l
217 ~~> \x.(\x.j(kx))(lx)
219 mcomp j (mcomp k l) ≡ mcomp j ((\fgx.f(gx)) k l)
220 ~~> mcomp j (\x.k(lx))
221 ≡ (\fgx.f(gx)) j (\x.k(lx))
222 ~~> \x.j((\x.k(lx)) x)
225 The Identity monad is favored by mimes.
227 To take a slightly less trivial (and even more useful) example,
228 consider the box type `α list`, with the following operations:
233 mcomp : (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
234 mcomp k j a = concat (map k (j a)) = List.flatten (List.map k (j a))
235 = foldr (\b ks -> (k b) ++ ks) [] (j a) = List.fold_right (fun b ks -> List.append (k b) ks) [] (j a)
236 = [c | b <- j a, c <- k b]
238 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.)
240 In words, `mcomp k j a` feeds the `a` (which has type `α`) to `j`, which returns a list of `β`s;
241 each `β` in that list is fed to `k`, which returns a list of `γ`s. The
242 final result is the concatenation of those lists of `γ`s.
246 let j a = [a*a, a+a] in
247 let k b = [b, b+1] in
248 mcomp k j 7 ==> [49, 50, 14, 15]
250 `j 7` produced `[49, 14]`, which after being fed through `k` gave us `[49, 50, 14, 15]`.
252 Contrast that to `m$` (`mapply`, which operates not on two *box-producing functions*, but instead on two *values of a boxed type*, one containing functions to be applied to the values in the other box, via some predefined scheme. Thus:
254 let js = [(\a->a*a),(\a->a+a)] in
256 mapply js xs ==> [49, 25, 14, 10]
259 As we illustrated in class, there are clear patterns shared between lists and option types and trees, so perhaps you can see why people want to figure out the general structures. But it probably isn't obvious yet why it would be useful to do so. To a large extent, this will only emerge over the next few classes. But we'll begin to demonstrate the usefulness of these patterns by talking through a simple example, that uses the monadic functions of the Option/Maybe box type.
264 Integer division presupposes that its second argument
265 (the divisor) is not zero, upon pain of presupposition failure.
266 Here's what my OCaml interpreter says:
269 Exception: Division_by_zero.
271 Say we want to explicitly allow for the possibility that
272 division will return something other than a number.
273 To do that, we'll use OCaml's `option` type, which works like this:
275 # type 'a option = None | Some of 'a;;
279 - : int option = Some 3
281 So if a division is normal, we return some number, but if the divisor is
282 zero, we return `None`. As a mnemonic aid, we'll prepend a `safe_` to the start of our new divide function.
285 let safe_div (x:int) (y:int) =
288 | _ -> Some (x / y);;
291 val safe_div : int -> int -> int option = fun
293 - : int option = Some 6
295 - : int option = None
296 # safe_div (safe_div 12 2) 3;;
298 Error: This expression has type int option
299 but an expression was expected of type int
303 This starts off well: dividing `12` by `2`, no problem; dividing `12` by `0`,
304 just the behavior we were hoping for. But we want to be able to use
305 the output of the safe-division function as input for further division
306 operations. So we have to jack up the types of the inputs:
309 let safe_div2 (u:int option) (v:int option) =
315 | Some y -> Some (x / y));;
318 val safe_div2 : int option -> int option -> int option = <fun>
319 # safe_div2 (Some 12) (Some 2);;
320 - : int option = Some 6
321 # safe_div2 (Some 12) (Some 0);;
322 - : int option = None
323 # safe_div2 (safe_div2 (Some 12) (Some 0)) (Some 3);;
324 - : int option = None
328 Calling the function now involves some extra verbosity, but it gives us what we need: now we can try to divide by anything we
329 want, without fear that we're going to trigger system errors.
331 I prefer to line up the `match` alternatives by using OCaml's
335 let safe_div2 (u:int option) (v:int option) =
339 | (_, Some 0) -> None
340 | (Some x, Some y) -> Some (x / y);;
343 So far so good. But what if we want to combine division with
344 other arithmetic operations? We need to make those other operations
345 aware of the possibility that one of their arguments has already triggered a
346 presupposition failure:
349 let safe_add (u:int option) (v:int option) =
353 | (Some x, Some y) -> Some (x + y);;
356 val safe_add : int option -> int option -> int option = <fun>
357 # safe_add (Some 12) (Some 4);;
358 - : int option = Some 16
359 # safe_add (safe_div (Some 12) (Some 0)) (Some 4);;
360 - : int option = None
364 This works, but is somewhat disappointing: the `safe_add` operation
365 doesn't trigger any presupposition of its own, so it is a shame that
366 it needs to be adjusted because someone else might make trouble.
368 But we can automate the adjustment, using the monadic machinery we introduced above.
369 As we said, there needs to be different `>>=`, `map2` and so on operations for each
370 monad or box type we're working with.
371 Haskell finesses this by "overloading" the single symbol `>>=`; you can just input that
372 symbol and it will calculate from the context of the surrounding type constraints what
373 monad you must have meant. In OCaml, the monadic operators are not pre-defined, but we will
374 give you a library that has definitions for all the standard monads, as in Haskell.
375 For now, though, we will define our `>>=` and `map2` operations by hand:
378 let (>>=) (u : 'a option) (j : 'a -> 'b option) : 'b option =
383 let map2 (f : 'a -> 'b -> 'c) (u : 'a option) (v : 'b option) : 'c option =
384 u >>= (fun x -> v >>= (fun y -> Some (f x y)));;
386 let safe_add3 = map2 (+);; (* that was easy *)
388 let safe_div3 (u: int option) (v: int option) =
389 u >>= (fun x -> v >>= (fun y -> if 0 = y then None else Some (x / y)));;
392 Haskell has an even more user-friendly notation for defining `safe_div3`, namely:
394 safe_div3 :: Maybe Int -> Maybe Int -> Maybe Int
395 safe_div3 u v = do {x <- u;
397 if 0 == y then Nothing else Just (x `div` y)}
399 Let's see our new functions in action:
403 # safe_div3 (safe_div3 (Some 12) (Some 2)) (Some 3);;
404 - : int option = Some 2
405 # safe_div3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
406 - : int option = None
407 # safe_add3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
408 - : int option = None
412 Compare the new definitions of `safe_add3` and `safe_div3` closely: the definition
413 for `safe_add3` shows what it looks like to equip an ordinary operation to
414 survive in dangerous presupposition-filled world. Note that the new
415 definition of `safe_add3` does not need to test whether its arguments are
416 `None` values or real numbers---those details are hidden inside of the
419 Note also that our definition of `safe_div3` recovers some of the simplicity of
420 the original `safe_div`, without the complexity introduced by `safe_div2`. We now
421 add exactly what extra is needed to track the no-division-by-zero presupposition. Here, too, we don't
422 need to keep track of what other presuppositions may have already failed
423 for whatever reason on our inputs.
425 (Linguistics note: Dividing by zero is supposed to feel like a kind of
426 presupposition failure. If we wanted to adapt this approach to
427 building a simple account of presupposition projection, we would have
428 to do several things. First, we would have to make use of the
429 polymorphism of the `option` type. In the arithmetic example, we only
430 made use of `int option`s, but when we're composing natural language
431 expression meanings, we'll need to use types like `N option`, `Det option`,
432 `VP option`, and so on. But that works automatically, because we can use
433 any type for the `'a` in `'a option`. Ultimately, we'd want to have a
434 theory of accommodation, and a theory of the situations in which
435 material within the sentence can satisfy presuppositions for other
436 material that otherwise would trigger a presupposition violation; but,
437 not surprisingly, these refinements will require some more
438 sophisticated techniques than the super-simple Option/Maybe monad.)