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 The menagerie isn't quite as bewildering as you might suppose. Many of these will
120 be interdefinable. For example, here is how `mcomp` and `mbind` are related: <code>k <=< j ≡
121 \a. (j a >>= k)</code>.
123 In most cases of interest, instances of these systems of functions will provide
124 certain useful guarantees.
126 * ***Mappable*** (in Haskelese, "Functors") At the most general level, box types are *Mappable*
127 if there is a `map` function defined for that box type with the type given above. This
128 has to obey the following Map Laws:
130 <code>map (id : α -> α) = (id : <u>α</u> -> <u>α</u>)</code>
131 <code>map (g ○ f) = (map g) ○ (map f)</code>
133 Essentially these say that `map` is a homomorphism from `(α -> β, ○, id)` to <code>(<u>α</u> -> <u>β</u>, ○', id')</code>, where `○'` and `id'` are `○` and `id` restricted to arguments of type <code><u>_</u></code>.
136 * ***MapNable*** (in Haskelese, "Applicatives") A Mappable box type is *MapNable*
137 if there are in addition `map2`, `mid`, and `mapply`. (Given either
138 of `map2` and `mapply`, you can define the other, and also `map`.
139 Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
140 have to obey the following MapN Laws:
145 * ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
146 is in addition an associative `mcomp` having `mid` as its left and
147 right identity. That is, the following Monad Laws must hold:
149 mcomp (mcomp j k) l (that is, (j <=< k) <=< l) = mcomp j (mcomp k l)
150 mcomp mid k (that is, mid <=< k) = k
151 mcomp k mid (that is, k <=< mid) = k
153 If you have any of `mcomp`, `mpmoc`, `mbind`, or `join`, you can use them to define the others.
154 Also, with these functions you can define `m$` and `map2` from *MapNables*. So all you really need
155 are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `join`, on the other.
157 Here are some interdefinitions: TODO
159 Names in Haskell: TODO
161 The name "bind" is not well chosen from our perspective, but this is too deeply entrenched by now.
165 To take a trivial (but, as we will see, still useful) example,
166 consider the Identity box type: `α`. So if `α` is type `bool`,
167 then a boxed `α` is ... a `bool`. That is, <code><u>α</u> = α</code>.
168 In terms of the box analogy, the Identity box type is a completely invisible box. With the following
172 mcomp ≡ \f g x.f (g x)
174 Identity is a monad. Here is a demonstration that the laws hold:
176 mcomp mid k ≡ (\fgx.f(gx)) (\p.p) k
180 mcomp k mid ≡ (\fgx.f(gx)) k (\p.p)
184 mcomp (mcomp j k) l ≡ mcomp ((\fgx.f(gx)) j k) l
185 ~~> mcomp (\x.j(kx)) l
186 ≡ (\fgx.f(gx)) (\x.j(kx)) l
187 ~~> \x.(\x.j(kx))(lx)
189 mcomp j (mcomp k l) ≡ mcomp j ((\fgx.f(gx)) k l)
190 ~~> mcomp j (\x.k(lx))
191 ≡ (\fgx.f(gx)) j (\x.k(lx))
192 ~~> \x.j((\x.k(lx)) x)
195 The Identity monad is favored by mimes.
197 To take a slightly less trivial (and even more useful) example,
198 consider the box type `α list`, with the following operations:
203 mcomp: (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
204 mcomp f g a = concat (map f (g a))
205 = foldr (\b -> \gs -> (f b) ++ gs) [] (g a)
206 = [c | b <- g a, c <- f b]
208 The last three definitions of `mcomp` are all equivalent, and it is easy to see that they obey the monad laws (see exercises TODO).
210 In words, `mcomp f g a` feeds the `a` (which has type `α`) to `g`, which returns a list of `β`s;
211 each `β` in that list is fed to `f`, which returns a list of `γ`s. The
212 final result is the concatenation of those lists of `γ`s.
216 let f b = [b, b+1] in
217 let g a = [a*a, a+a] in
218 mcomp f g 7 ==> [49, 50, 14, 15]
220 `g 7` produced `[49, 14]`, which after being fed through `f` gave us `[49, 50, 14, 15]`.
222 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:
224 let gs = [(\a->a*a),(\a->a+a)] in
226 mapply gs xs ==> [49, 25, 14, 10]
229 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 identify 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.
234 Integer division presupposes that its second argument
235 (the divisor) is not zero, upon pain of presupposition failure.
236 Here's what my OCaml interpreter says:
239 Exception: Division_by_zero.
241 Say we want to explicitly allow for the possibility that
242 division will return something other than a number.
243 To do that, we'll use OCaml's `option` type, which works like this:
245 # type 'a option = None | Some of 'a;;
249 - : int option = Some 3
251 So if a division is normal, we return some number, but if the divisor is
252 zero, we return `None`. As a mnemonic aid, we'll prepend a `safe_` to the start of our new divide function.
255 let safe_div (x:int) (y:int) =
258 | _ -> Some (x / y);;
261 val safe_div : int -> int -> int option = fun
263 - : int option = Some 6
265 - : int option = None
266 # safe_div (safe_div 12 2) 3;;
268 Error: This expression has type int option
269 but an expression was expected of type int
273 This starts off well: dividing `12` by `2`, no problem; dividing `12` by `0`,
274 just the behavior we were hoping for. But we want to be able to use
275 the output of the safe-division function as input for further division
276 operations. So we have to jack up the types of the inputs:
279 let safe_div2 (u:int option) (v:int option) =
285 | Some y -> Some (x / y));;
288 val safe_div2 : int option -> int option -> int option = <fun>
289 # safe_div2 (Some 12) (Some 2);;
290 - : int option = Some 6
291 # safe_div2 (Some 12) (Some 0);;
292 - : int option = None
293 # safe_div2 (safe_div2 (Some 12) (Some 0)) (Some 3);;
294 - : int option = None
298 Calling the function now involves some extra verbosity, but it gives us what we need: now we can try to divide by anything we
299 want, without fear that we're going to trigger system errors.
301 I prefer to line up the `match` alternatives by using OCaml's
305 let safe_div2 (u:int option) (v:int option) =
309 | (_, Some 0) -> None
310 | (Some x, Some y) -> Some (x / y);;
313 So far so good. But what if we want to combine division with
314 other arithmetic operations? We need to make those other operations
315 aware of the possibility that one of their arguments has already triggered a
316 presupposition failure:
319 let safe_add (u:int option) (v:int option) =
323 | (Some x, Some y) -> Some (x + y);;
326 val safe_add : int option -> int option -> int option = <fun>
327 # safe_add (Some 12) (Some 4);;
328 - : int option = Some 16
329 # safe_add (safe_div (Some 12) (Some 0)) (Some 4);;
330 - : int option = None
334 This works, but is somewhat disappointing: the `safe_add` operation
335 doesn't trigger any presupposition of its own, so it is a shame that
336 it needs to be adjusted because someone else might make trouble.
338 But we can automate the adjustment, using the monadic machinery we introduced above.
339 As we said, there needs to be different `>>=`, `map2` and so on operations for each
340 monad or box type we're working with.
341 Haskell finesses this by "overloading" the single symbol `>>=`; you can just input that
342 symbol and it will calculate from the context of the surrounding type constraints what
343 monad you must have meant. In OCaml, the monadic operators are not pre-defined, but we will
344 give you a library that has definitions for all the standard monads, as in Haskell.
345 For now, though, we will define our `>>=` and `map2` operations by hand:
348 let (>>=) (u : 'a option) (j : 'a -> 'b option) : 'b option =
353 let map2 (f : 'a -> 'b -> 'c) (u : 'a option) (v : 'b option) : 'c option =
354 u >>= (fun x -> v >>= (fun y -> Some (f x y)));;
356 let safe_add3 = map2 (+);; (* that was easy *)
358 let safe_div3 (u: int option) (v: int option) =
359 u >>= (fun x -> v >>= (fun y -> if 0 = y then None else Some (x / y)));;
362 Haskell has an even more user-friendly notation for defining `safe_div3`, namely:
364 safe_div3 :: Maybe Int -> Maybe Int -> Maybe Int
365 safe_div3 u v = do {x <- u;
367 if 0 == y then Nothing else Just (x `div` y)}
369 Let's see our new functions in action:
373 # safe_div3 (safe_div3 (Some 12) (Some 2)) (Some 3);;
374 - : int option = Some 2
375 # safe_div3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
376 - : int option = None
377 # safe_add3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
378 - : int option = None
382 Compare the new definitions of `safe_add3` and `safe_div3` closely: the definition
383 for `safe_add3` shows what it looks like to equip an ordinary operation to
384 survive in dangerous presupposition-filled world. Note that the new
385 definition of `safe_add3` does not need to test whether its arguments are
386 `None` values or real numbers---those details are hidden inside of the
389 Note also that our definition of `safe_div3` recovers some of the simplicity of
390 the original `safe_div`, without the complexity introduced by `safe_div2`. We now
391 add exactly what extra is needed to track the no-division-by-zero presupposition. Here, too, we don't
392 need to keep track of what other presuppositions may have already failed
393 for whatever reason on our inputs.
395 (Linguistics note: Dividing by zero is supposed to feel like a kind of
396 presupposition failure. If we wanted to adapt this approach to
397 building a simple account of presupposition projection, we would have
398 to do several things. First, we would have to make use of the
399 polymorphism of the `option` type. In the arithmetic example, we only
400 made use of `int option`s, but when we're composing natural language
401 expression meanings, we'll need to use types like `N option`, `Det option`,
402 `VP option`, and so on. But that works automatically, because we can use
403 any type for the `'a` in `'a option`. Ultimately, we'd want to have a
404 theory of accommodation, and a theory of the situations in which
405 material within the sentence can satisfy presuppositions for other
406 material that otherwise would trigger a presupposition violation; but,
407 not surprisingly, these refinements will require some more
408 sophisticated techniques than the super-simple Option/Maybe monad.)