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 Also, in the handout we called `mid` `𝟭`. But now we've decided that `mid`
122 is better. (Think of it as "m" plus "identity", not as the start of "midway".)
124 The menagerie isn't quite as bewildering as you might suppose. Many of these will
125 be interdefinable. For example, here is how `mcomp` and `mbind` are related: <code>k <=< j ≡
126 \a. (j a >>= k)</code>.
128 We will move freely back and forth between using `>=>` and using `<=<` (aka `mcomp`), which
129 is just `>=>` with its arguments flipped. `<=<` has the virtue that it corresponds more
130 closely to the ordinary mathematical symbol `○`. But `>=>` has the virtue
131 that its types flow more naturally from left to right.
133 These functions come together in several systems, and have to be defined in a way that coheres with the other functions in the system:
135 * ***Mappable*** (in Haskelese, "Functors") At the most general level, box types are *Mappable*
136 if there is a `map` function defined for that box type with the type given above. This
137 has to obey the following Map Laws:
139 <code>map (id : α -> α) = (id : <u>α</u> -> <u>α</u>)</code>
140 <code>map (g ○ f) = (map g) ○ (map f)</code>
142 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.
144 > <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>
147 * ***MapNable*** (in Haskelese, "Applicatives") A Mappable box type is *MapNable*
148 if there are in addition `map2`, `mid`, and `mapply`. (Given either
149 of `map2` and `mapply`, you can define the other, and also `map`.
150 Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
151 have to obey the following MapN Laws:
156 * ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
157 is in addition an associative `mcomp` having `mid` as its left and
158 right identity. That is, the following Monad Laws must hold:
160 mcomp (mcomp j k) l (that is, (j <=< k) <=< l) = mcomp j (mcomp k l)
161 mcomp mid k (that is, mid <=< k) = k
162 mcomp k mid (that is, k <=< mid) = k
164 If you have any of `mcomp`, `mpmoc`, `mbind`, or `join`, you can use them to define the others.
165 Also, with these functions you can define `m$` and `map2` from *MapNables*. So all you really need
166 are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `join`, on the other.
168 Here are some interdefinitions: TODO
170 Names in Haskell: TODO
172 The name "bind" is not well chosen from our perspective, but this is too deeply entrenched by now.
176 To take a trivial (but, as we will see, still useful) example,
177 consider the Identity box type: `α`. So if `α` is type `bool`,
178 then a boxed `α` is ... a `bool`. That is, <code><u>α</u> = α</code>.
179 In terms of the box analogy, the Identity box type is a completely invisible box. With the following
183 mcomp ≡ \f g x.f (g x)
185 Identity is a monad. Here is a demonstration that the laws hold:
187 mcomp mid k ≡ (\fgx.f(gx)) (\p.p) k
191 mcomp k mid ≡ (\fgx.f(gx)) k (\p.p)
195 mcomp (mcomp j k) l ≡ mcomp ((\fgx.f(gx)) j k) l
196 ~~> mcomp (\x.j(kx)) l
197 ≡ (\fgx.f(gx)) (\x.j(kx)) l
198 ~~> \x.(\x.j(kx))(lx)
200 mcomp j (mcomp k l) ≡ mcomp j ((\fgx.f(gx)) k l)
201 ~~> mcomp j (\x.k(lx))
202 ≡ (\fgx.f(gx)) j (\x.k(lx))
203 ~~> \x.j((\x.k(lx)) x)
206 The Identity monad is favored by mimes.
208 To take a slightly less trivial (and even more useful) example,
209 consider the box type `α list`, with the following operations:
214 mcomp : (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
215 mcomp k j a = concat (map k (j a)) = List.flatten (List.map k (j a))
216 = foldr (\b ks -> (k b) ++ ks) [] (j a) = List.fold_right (fun b ks -> List.append (k b) ks) [] (j a)
217 = [c | b <- j a, c <- k b]
219 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.)
221 In words, `mcomp k j a` feeds the `a` (which has type `α`) to `j`, which returns a list of `β`s;
222 each `β` in that list is fed to `k`, which returns a list of `γ`s. The
223 final result is the concatenation of those lists of `γ`s.
227 let j a = [a*a, a+a] in
228 let k b = [b, b+1] in
229 mcomp k j 7 ==> [49, 50, 14, 15]
231 `j 7` produced `[49, 14]`, which after being fed through `k` gave us `[49, 50, 14, 15]`.
233 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:
235 let js = [(\a->a*a),(\a->a+a)] in
237 mapply js xs ==> [49, 25, 14, 10]
240 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.
245 Integer division presupposes that its second argument
246 (the divisor) is not zero, upon pain of presupposition failure.
247 Here's what my OCaml interpreter says:
250 Exception: Division_by_zero.
252 Say we want to explicitly allow for the possibility that
253 division will return something other than a number.
254 To do that, we'll use OCaml's `option` type, which works like this:
256 # type 'a option = None | Some of 'a;;
260 - : int option = Some 3
262 So if a division is normal, we return some number, but if the divisor is
263 zero, we return `None`. As a mnemonic aid, we'll prepend a `safe_` to the start of our new divide function.
266 let safe_div (x:int) (y:int) =
269 | _ -> Some (x / y);;
272 val safe_div : int -> int -> int option = fun
274 - : int option = Some 6
276 - : int option = None
277 # safe_div (safe_div 12 2) 3;;
279 Error: This expression has type int option
280 but an expression was expected of type int
284 This starts off well: dividing `12` by `2`, no problem; dividing `12` by `0`,
285 just the behavior we were hoping for. But we want to be able to use
286 the output of the safe-division function as input for further division
287 operations. So we have to jack up the types of the inputs:
290 let safe_div2 (u:int option) (v:int option) =
296 | Some y -> Some (x / y));;
299 val safe_div2 : int option -> int option -> int option = <fun>
300 # safe_div2 (Some 12) (Some 2);;
301 - : int option = Some 6
302 # safe_div2 (Some 12) (Some 0);;
303 - : int option = None
304 # safe_div2 (safe_div2 (Some 12) (Some 0)) (Some 3);;
305 - : int option = None
309 Calling the function now involves some extra verbosity, but it gives us what we need: now we can try to divide by anything we
310 want, without fear that we're going to trigger system errors.
312 I prefer to line up the `match` alternatives by using OCaml's
316 let safe_div2 (u:int option) (v:int option) =
320 | (_, Some 0) -> None
321 | (Some x, Some y) -> Some (x / y);;
324 So far so good. But what if we want to combine division with
325 other arithmetic operations? We need to make those other operations
326 aware of the possibility that one of their arguments has already triggered a
327 presupposition failure:
330 let safe_add (u:int option) (v:int option) =
334 | (Some x, Some y) -> Some (x + y);;
337 val safe_add : int option -> int option -> int option = <fun>
338 # safe_add (Some 12) (Some 4);;
339 - : int option = Some 16
340 # safe_add (safe_div (Some 12) (Some 0)) (Some 4);;
341 - : int option = None
345 This works, but is somewhat disappointing: the `safe_add` operation
346 doesn't trigger any presupposition of its own, so it is a shame that
347 it needs to be adjusted because someone else might make trouble.
349 But we can automate the adjustment, using the monadic machinery we introduced above.
350 As we said, there needs to be different `>>=`, `map2` and so on operations for each
351 monad or box type we're working with.
352 Haskell finesses this by "overloading" the single symbol `>>=`; you can just input that
353 symbol and it will calculate from the context of the surrounding type constraints what
354 monad you must have meant. In OCaml, the monadic operators are not pre-defined, but we will
355 give you a library that has definitions for all the standard monads, as in Haskell.
356 For now, though, we will define our `>>=` and `map2` operations by hand:
359 let (>>=) (u : 'a option) (j : 'a -> 'b option) : 'b option =
364 let map2 (f : 'a -> 'b -> 'c) (u : 'a option) (v : 'b option) : 'c option =
365 u >>= (fun x -> v >>= (fun y -> Some (f x y)));;
367 let safe_add3 = map2 (+);; (* that was easy *)
369 let safe_div3 (u: int option) (v: int option) =
370 u >>= (fun x -> v >>= (fun y -> if 0 = y then None else Some (x / y)));;
373 Haskell has an even more user-friendly notation for defining `safe_div3`, namely:
375 safe_div3 :: Maybe Int -> Maybe Int -> Maybe Int
376 safe_div3 u v = do {x <- u;
378 if 0 == y then Nothing else Just (x `div` y)}
380 Let's see our new functions in action:
384 # safe_div3 (safe_div3 (Some 12) (Some 2)) (Some 3);;
385 - : int option = Some 2
386 # safe_div3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
387 - : int option = None
388 # safe_add3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
389 - : int option = None
393 Compare the new definitions of `safe_add3` and `safe_div3` closely: the definition
394 for `safe_add3` shows what it looks like to equip an ordinary operation to
395 survive in dangerous presupposition-filled world. Note that the new
396 definition of `safe_add3` does not need to test whether its arguments are
397 `None` values or real numbers---those details are hidden inside of the
400 Note also that our definition of `safe_div3` recovers some of the simplicity of
401 the original `safe_div`, without the complexity introduced by `safe_div2`. We now
402 add exactly what extra is needed to track the no-division-by-zero presupposition. Here, too, we don't
403 need to keep track of what other presuppositions may have already failed
404 for whatever reason on our inputs.
406 (Linguistics note: Dividing by zero is supposed to feel like a kind of
407 presupposition failure. If we wanted to adapt this approach to
408 building a simple account of presupposition projection, we would have
409 to do several things. First, we would have to make use of the
410 polymorphism of the `option` type. In the arithmetic example, we only
411 made use of `int option`s, but when we're composing natural language
412 expression meanings, we'll need to use types like `N option`, `Det option`,
413 `VP option`, and so on. But that works automatically, because we can use
414 any type for the `'a` in `'a option`. Ultimately, we'd want to have a
415 theory of accommodation, and a theory of the situations in which
416 material within the sentence can satisfy presuppositions for other
417 material that otherwise would trigger a presupposition violation; but,
418 not surprisingly, these refinements will require some more
419 sophisticated techniques than the super-simple Option/Maybe monad.)