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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.
17 The closest we will come to metaphorical talk is to suggest that
18 monadic types place objects inside of *boxes*, and that monads wrap
19 and unwrap boxes to expose or enclose the objects inside of them. In
20 any case, our emphasis will be on starting with the abstract structure
21 of monads, followed by instances of monads from the philosophical and
22 linguistics literature.
24 ## Box types: type expressions with one free type variable
26 Recall that we've been using lower-case Greek letters
27 <code>α, β, γ, ...</code> as type variables. We'll
28 use `P`, `Q`, `R`, and `S` as schematic metavariables over type expressions, that may or may not contain unbound
29 type variables. For instance, we might have
38 A *box type* will be a type expression that contains exactly one free
39 type variable. (You could extend this to expressions with more free variables; then you'd have
40 to specify which one of them the box is capturing. But let's keep it simple.) Some examples (using OCaml's type conventions):
44 (α, R) tree (assuming R contains no free type variables)
47 The idea is that whatever type the free type variable α might be,
48 the boxed type will be a box that "contains" an object of type `α`.
49 For instance, if `α list` is our box type, and `α` is the type
50 `int`, then in this context, `int list` is the type of a boxed integer.
52 Warning: although our initial motivating examples are naturally thought of as "containers" (lists, trees, and so on, with `α`s as their "elments"), with later examples we discuss it will less intuitive to describe the box types that way. For example, where `R` is some fixed type, `R -> α` is a box type.
54 The *box type* is the type `α list` (or as we might just say, `list`); the *boxed type* is some specific instantiantion of the free type variable `α`. We'll often write boxed types as a box containing the instance of the free
55 type variable. So if our box type is `α list`, and `α` is instantiated with the specific type `int`, we would write:
59 for the type of a boxed `int`. (We'll fool with the markup to make this a genuine box later; for now it will just display as underlined.)
65 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:
69 That is, they are functions from objects of one type `P` to a boxed type `Q`, for some choice of type expressions `P` and `Q`.
70 For instance, the following are Kleisli arrows:
74 int list -> <u>int list</u>
76 In the first, `P` has become `int` and `Q` has become `bool`. (The boxed type <u>Q</u> is <u>bool</u>).
78 Note that the left-hand schema `P` is permitted to itself be a boxed type. That is, where
79 if `α list` is our box type, we can write the second arrow as
81 <u>int</u> -> <u>Q</u>
83 We'll need a number of classes of functions to help us maneuver in the
84 presence of box types. We will want to define a different instance of
85 each of these for whichever box type we're dealing with. (This will
86 become clear shortly.)
88 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.)
90 <code>map (/mæp/): (P -> Q) -> <u>P</u> -> <u>Q</u></code>
92 <code>map2 (/mæptu/): (P -> Q -> R) -> <u>P</u> -> <u>Q</u> -> <u>R</u></code>
94 <code>mid (/εmaidεnt@tI/ aka unit, return, pure): P -> <u>P</u></code>
96 <code>m$ or mapply (/εm@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>
98 <code><=< or mcomp : (Q -> <u>R</u>) -> (P -> <u>Q</u>) -> (P -> <u>R</u>)</code>
100 <code>>=> or mpmoc (m-flipcomp): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
102 <code>>>= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
104 <code>=<<mdnib (or m-flipbind) (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
106 <code>join: <u>2<u>P</u></u> -> <u>P</u></code>
108 The managerie isn't quite as bewildering as you might suppose. Many of these will
109 be interdefinable. For example, here is how `mcomp` and `mbind` are related: <code>k <=< j ≡
110 \a. (j a >>= k)</code>.
112 In most cases of interest, instances of these systems of functions will provide
113 certain useful guarantees.
115 * ***Mappable*** (in Haskelese, "Functors") At the most general level, box types are *Mappable*
116 if there is a `map` function defined for that box type with the type given above. This
117 has to obey the following Map Laws:
121 * ***MapNable*** (in Haskelese, "Applicatives") A Mappable box type is *MapNable*
122 if there are in addition `map2`, `mid`, and `mapply`. (Given either
123 of `map2` and `mapply`, you can define the other, and also `map`.
124 Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
125 have to obey the following MapN Laws:
128 * ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
129 is in addition an associative `mcomp` having `mid` as its left and
130 right identity. That is, the following Monad Laws must hold:
132 mcomp (mcomp j k) l (that is, (j <=< k) <=< l) = mcomp j (mcomp k l)
133 mcomp mid k (that is, mid <=< k) = k
134 mcomp k mid (that is, k <=< mid) = k
136 If you have any of `mcomp`, `mpmoc`, `mbind`, or `join`, you can use them to define the others.
137 Also, with these functions you can define `m$` and `map2` from *MapNables*. So all you really need
138 are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `join`, on the other.
140 Here are some interdefinitions: TODO. Names in Haskell TODO.
144 To take a trivial (but, as we will see, still useful) example,
145 consider the identity box type Id: `α`. So if `α` is type `bool`,
146 then a boxed `α` is ... a `bool`. In terms of the box analogy, the
147 Identity box type is a completely invisible box. With the following
151 mcomp ≡ \f g x.f (g x)
153 Identity is a monad. Here is a demonstration that the laws hold:
155 mcomp mid k == (\fgx.f(gx)) (\p.p) k
159 mcomp k mid == (\fgx.f(gx)) k (\p.p)
163 mcomp (mcomp j k) l == mcomp ((\fgx.f(gx)) j k) l
164 ~~> mcomp (\x.j(kx)) l
165 == (\fgx.f(gx)) (\x.j(kx)) l
166 ~~> \x.(\x.j(kx))(lx)
168 mcomp j (mcomp k l) == mcomp j ((\fgx.f(gx)) k l)
169 ~~> mcomp j (\x.k(lx))
170 == (\fgx.f(gx)) j (\x.k(lx))
171 ~~> \x.j((\x.k(lx)) x)
174 Id is the favorite monad of mimes.
176 To take a slightly less trivial (and even more useful) example,
177 consider the box type `α list`, with the following operations:
182 mcomp: (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
183 mcomp f g a = concat (map f (g a))
184 = foldr (\b -> \gs -> (f b) ++ gs) [] (g a)
185 = [c | b <- g a, c <- f b]
187 These three definitions of `mcomp` are all equivalent, and it is easy to see that they obey the monad laws (see exercises).
189 In words, `mcomp f g a` feeds the `a` (which has type `α`) to `g`, which returns a list of `β`s;
190 each `β` in that list is fed to `f`, which returns a list of `γ`s. The
191 final result is the concatenation of those lists of `γ`s.
195 let f b = [b, b+1] in
196 let g a = [a*a, a+a] in
197 mcomp f g 7 ==> [49, 50, 14, 15]
199 `g 7` produced `[49, 14]`, which after being fed through `f` gave us `[49, 50, 14, 15]`.
201 Contrast that to `m$` (`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:
203 let gs = [(\a->a*a),(\a->a+a)] in
205 mapply gs xs ==> [49, 25, 14, 10]
208 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.
214 Integer division presupposes that its second argument
215 (the divisor) is not zero, upon pain of presupposition failure.
216 Here's what my OCaml interpreter says:
219 Exception: Division_by_zero.
221 Say we want to explicitly allow for the possibility that
222 division will return something other than a number.
223 To do that, we'll use OCaml's `option` type, which works like this:
225 # type 'a option = None | Some of 'a;;
229 - : int option = Some 3
231 So if a division is normal, we return some number, but if the divisor is
232 zero, we return `None`. As a mnemonic aid, we'll prepend a `safe_` to the start of our new divide function.
235 let safe_div (x:int) (y:int) =
238 | _ -> Some (x / y);;
241 val safe_div : int -> int -> int option = fun
243 - : int option = Some 6
245 - : int option = None
246 # safe_div (safe_div 12 2) 3;;
247 # safe_div (safe_div 12 2) 3;;
249 Error: This expression has type int option
250 but an expression was expected of type int
254 This starts off well: dividing 12 by 2, no problem; dividing 12 by 0,
255 just the behavior we were hoping for. But we want to be able to use
256 the output of the safe-division function as input for further division
257 operations. So we have to jack up the types of the inputs:
260 let safe_div2 (u:int option) (v:int option) =
263 | Some x -> (match v with
265 | Some y -> Some (x / y));;
268 val safe_div2 : int option -> int option -> int option = <fun>
269 # safe_div2 (Some 12) (Some 2);;
270 - : int option = Some 6
271 # safe_div2 (Some 12) (Some 0);;
272 - : int option = None
273 # safe_div2 (safe_div2 (Some 12) (Some 0)) (Some 3);;
274 - : int option = None
278 Beautiful, just what we need: now we can try to divide by anything we
279 want, without fear that we're going to trigger any system errors.
281 I prefer to line up the `match` alternatives by using OCaml's
285 let safe_div2 (u:int option) (v:int option) =
289 | (_, Some 0) -> None
290 | (Some x, Some y) -> Some (x / y);;
293 So far so good. But what if we want to combine division with
294 other arithmetic operations? We need to make those other operations
295 aware of the possibility that one of their arguments has triggered a
296 presupposition failure:
299 let safe_add (u:int option) (v:int option) =
303 | (Some x, Some y) -> Some (x + y);;
306 val safe_add : int option -> int option -> int option = <fun>
307 # safe_add (Some 12) (Some 4);;
308 - : int option = Some 16
309 # safe_add (safe_div (Some 12) (Some 0)) (Some 4);;
310 - : int option = None
314 This works, but is somewhat disappointing: the `safe_add` operation
315 doesn't trigger any presupposition of its own, so it is a shame that
316 it needs to be adjusted because someone else might make trouble.
318 But we can automate the adjustment. The standard way in OCaml,
319 Haskell, and other functional programming languages, is to use the monadic
320 `bind` operator, `>>=`. (The name "bind" is not well chosen from our
321 perspective, but this is too deeply entrenched by now.) As mentioned above,
322 there needs to be a different `>>=` operator for each Monad or box type you're working with.
323 Haskell finesses this by "overloading" the single symbol `>>=`; you can just input that
324 symbol and it will calculate from the context of the surrounding type constraints what
325 monad you must have meant. In OCaml, the `>>=` or `bind` operator is not pre-defined, but we will
326 give you a library that has definitions for all the standard monads, as in Haskell.
327 For now, though, we will define our `bind` operation by hand:
330 let bind (u: int option) (f: int -> (int option)) =
335 let safe_add3 (u: int option) (v: int option) =
336 bind u (fun x -> bind v (fun y -> Some (x + y)));;
338 (* This is really just `map2 (+)`, using the `map2` operation that corresponds to
339 definition of `bind`. *)
341 let safe_div3 (u: int option) (v: int option) =
342 bind u (fun x -> bind v (fun y -> if 0 = y then None else Some (x / y)));;
344 (* This goes back to some of the simplicity of the original safe_div, without the complexity
345 introduced by safe_div2. *)
348 The above definitions look even simpler if you focus on the fact that `safe_add3` can be written as simply `map2 (+)`, and that `safe_div3` could be written as `u >>= fun x -> v >>= fun y -> if 0 = y then None else Some (x / y)`. Haskell has an even more user-friendly notation for this, namely:
350 safe_div3 :: Maybe Int -> Maybe Int -> Maybe Int
351 safe_div3 u v = do {x <- u;
353 if 0 == y then Nothing else return (x `div` y)}
355 Let's see our new functions in action:
359 # safe_div3 (safe_div3 (Some 12) (Some 2)) (Some 3);;
360 - : int option = Some 2
361 # safe_div3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
362 - : int option = None
363 # safe_add3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
364 - : int option = None
368 Compare the new definitions of `safe_add3` and `safe_div3` closely: the definition
369 for `safe_add3` shows what it looks like to equip an ordinary operation to
370 survive in dangerous presupposition-filled world. Note that the new
371 definition of `safe_add3` does not need to test whether its arguments are
372 None objects or real numbers---those details are hidden inside of the
375 The definition of `safe_div3` shows exactly what extra needs to be said in
376 order to trigger the no-division-by-zero presupposition. Here, too, we don't
377 need to keep track of what presuppositions may have already failed
378 for whatever reason on our inputs.
380 (Linguistics note: Dividing by zero is supposed to feel like a kind of
381 presupposition failure. If we wanted to adapt this approach to
382 building a simple account of presupposition projection, we would have
383 to do several things. First, we would have to make use of the
384 polymorphism of the `option` type. In the arithmetic example, we only
385 made use of `int option`s, but when we're composing natural language
386 expression meanings, we'll need to use types like `N option`, `Det option`,
387 `VP option`, and so on. But that works automatically, because we can use
388 any type for the `'a` in `'a option`. Ultimately, we'd want to have a
389 theory of accommodation, and a theory of the situations in which
390 material within the sentence can satisfy presuppositions for other
391 material that otherwise would trigger a presupposition violation; but,
392 not surprisingly, these refinements will require some more
393 sophisticated techniques than the super-simple Option monad.)