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.
17 The closest we will come to metaphorical talk is to suggest that
18 monadic types place values inside of *boxes*, and that monads wrap
19 and unwrap boxes to expose or enclose the values 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 instantiated to,
48 we will be a "type box" of a certain sort that "contains" values of type `α`. For instance,
49 if `α list` is our box type, and `α` is the type `int`, then in this context, `int list`
50 is the type of a boxed integer.
52 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.
54 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
55 type variable instantiates to. So if our box type is `α list`, and `α` instantiates to the specific type `int`, we would write:
59 for the type of a boxed `int`. (We'll fool with the markup to make this show 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 values 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 <code><u>Q</u></code> is <code><u>bool</u></code>).
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>
84 We'll need a number of classes of functions to help us maneuver in the
85 presence of box types. We will want to define a different instance of
86 each of these for whichever box type we're dealing with. (This will
87 become clear shortly.)
89 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.
92 ## A family of functions for each box type ##
94 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.
95 >>>>>>> ecff6bbae7c00556584b51913b934bdade0cff40
97 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.)
99 <code>map (/mæp/): (P -> Q) -> <u>P</u> -> <u>Q</u></code>
101 <code>map2 (/mæptu/): (P -> Q -> R) -> <u>P</u> -> <u>Q</u> -> <u>R</u></code>
103 <code>mid (/εmaidεnt@tI/ aka unit, return, pure): P -> <u>P</u></code>
105 <code>m$ or mapply (/εm@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>
107 <code><=< or mcomp : (Q -> <u>R</u>) -> (P -> <u>Q</u>) -> (P -> <u>R</u>)</code>
109 <code>>=> or mpmoc (flip mcomp): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
111 <code>>>= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
113 <code>=<< or mdnib (flip mbind) (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
115 <code>join: <span style="border-bottom: 3px double;">P</span> -> <u>P</u></code>
118 Test1: <span style="border-bottom: 3px double"><d>P</d></span>
120 Test2: <span style="text-decoration: overline"><u>P</u></span>
122 Test3: <span style="font-style: italic;">XX</span>
124 Test4: <span class="double">YY</span>
127 The menagerie isn't quite as bewildering as you might suppose. Many of these will
128 be interdefinable. For example, here is how `mcomp` and `mbind` are related: <code>k <=< j ≡
129 \a. (j a >>= k)</code>.
131 In most cases of interest, instances of these systems of functions will provide
132 certain useful guarantees.
134 * ***Mappable*** (in Haskelese, "Functors") At the most general level, box types are *Mappable*
135 if there is a `map` function defined for that box type with the type given above. This
136 has to obey the following Map Laws:
140 * ***MapNable*** (in Haskelese, "Applicatives") A Mappable box type is *MapNable*
141 if there are in addition `map2`, `mid`, and `mapply`. (Given either
142 of `map2` and `mapply`, you can define the other, and also `map`.
143 Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
144 have to obey the following MapN Laws:
147 * ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
148 is in addition an associative `mcomp` having `mid` as its left and
149 right identity. That is, the following Monad Laws must hold:
151 mcomp (mcomp j k) l (that is, (j <=< k) <=< l) = mcomp j (mcomp k l)
152 mcomp mid k (that is, mid <=< k) = k
153 mcomp k mid (that is, k <=< mid) = k
155 If you have any of `mcomp`, `mpmoc`, `mbind`, or `join`, you can use them to define the others.
156 Also, with these functions you can define `m$` and `map2` from *MapNables*. So all you really need
157 are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `join`, on the other.
159 Here are some interdefinitions: TODO. Names in Haskell TODO.
163 To take a trivial (but, as we will see, still useful) example,
164 consider the identity box type Id: `α`. So if `α` is type `bool`,
165 then a boxed `α` is ... a `bool`. In terms of the box analogy, the
166 Identity box type is a completely invisible box. With the following
170 mcomp ≡ \f g x.f (g x)
172 Identity is a monad. Here is a demonstration that the laws hold:
174 mcomp mid k == (\fgx.f(gx)) (\p.p) k
178 mcomp k mid == (\fgx.f(gx)) k (\p.p)
182 mcomp (mcomp j k) l == mcomp ((\fgx.f(gx)) j k) l
183 ~~> mcomp (\x.j(kx)) l
184 == (\fgx.f(gx)) (\x.j(kx)) l
185 ~~> \x.(\x.j(kx))(lx)
187 mcomp j (mcomp k l) == mcomp j ((\fgx.f(gx)) k l)
188 ~~> mcomp j (\x.k(lx))
189 == (\fgx.f(gx)) j (\x.k(lx))
190 ~~> \x.j((\x.k(lx)) x)
193 Id is the favorite monad of mimes.
195 To take a slightly less trivial (and even more useful) example,
196 consider the box type `α list`, with the following operations:
201 mcomp: (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
202 mcomp f g a = concat (map f (g a))
203 = foldr (\b -> \gs -> (f b) ++ gs) [] (g a)
204 = [c | b <- g a, c <- f b]
206 These three definitions of `mcomp` are all equivalent, and it is easy to see that they obey the monad laws (see exercises).
208 In words, `mcomp f g a` feeds the `a` (which has type `α`) to `g`, which returns a list of `β`s;
209 each `β` in that list is fed to `f`, which returns a list of `γ`s. The
210 final result is the concatenation of those lists of `γ`s.
214 let f b = [b, b+1] in
215 let g a = [a*a, a+a] in
216 mcomp f g 7 ==> [49, 50, 14, 15]
218 `g 7` produced `[49, 14]`, which after being fed through `f` gave us `[49, 50, 14, 15]`.
220 Contrast that to `m$` (`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:
222 let gs = [(\a->a*a),(\a->a+a)] in
224 mapply gs xs ==> [49, 25, 14, 10]
227 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.
232 Integer division presupposes that its second argument
233 (the divisor) is not zero, upon pain of presupposition failure.
234 Here's what my OCaml interpreter says:
237 Exception: Division_by_zero.
239 Say we want to explicitly allow for the possibility that
240 division will return something other than a number.
241 To do that, we'll use OCaml's `option` type, which works like this:
243 # type 'a option = None | Some of 'a;;
247 - : int option = Some 3
249 So if a division is normal, we return some number, but if the divisor is
250 zero, we return `None`. As a mnemonic aid, we'll prepend a `safe_` to the start of our new divide function.
253 let safe_div (x:int) (y:int) =
256 | _ -> Some (x / y);;
259 val safe_div : int -> int -> int option = fun
261 - : int option = Some 6
263 - : int option = None
264 # safe_div (safe_div 12 2) 3;;
265 # safe_div (safe_div 12 2) 3;;
267 Error: This expression has type int option
268 but an expression was expected of type int
272 This starts off well: dividing 12 by 2, no problem; dividing 12 by 0,
273 just the behavior we were hoping for. But we want to be able to use
274 the output of the safe-division function as input for further division
275 operations. So we have to jack up the types of the inputs:
278 let safe_div2 (u:int option) (v:int option) =
281 | Some x -> (match v with
283 | Some y -> Some (x / y));;
286 val safe_div2 : int option -> int option -> int option = <fun>
287 # safe_div2 (Some 12) (Some 2);;
288 - : int option = Some 6
289 # safe_div2 (Some 12) (Some 0);;
290 - : int option = None
291 # safe_div2 (safe_div2 (Some 12) (Some 0)) (Some 3);;
292 - : int option = None
296 Beautiful, just what we need: now we can try to divide by anything we
297 want, without fear that we're going to trigger any system errors.
299 I prefer to line up the `match` alternatives by using OCaml's
303 let safe_div2 (u:int option) (v:int option) =
307 | (_, Some 0) -> None
308 | (Some x, Some y) -> Some (x / y);;
311 So far so good. But what if we want to combine division with
312 other arithmetic operations? We need to make those other operations
313 aware of the possibility that one of their arguments has triggered a
314 presupposition failure:
317 let safe_add (u:int option) (v:int option) =
321 | (Some x, Some y) -> Some (x + y);;
324 val safe_add : int option -> int option -> int option = <fun>
325 # safe_add (Some 12) (Some 4);;
326 - : int option = Some 16
327 # safe_add (safe_div (Some 12) (Some 0)) (Some 4);;
328 - : int option = None
332 This works, but is somewhat disappointing: the `safe_add` operation
333 doesn't trigger any presupposition of its own, so it is a shame that
334 it needs to be adjusted because someone else might make trouble.
336 But we can automate the adjustment. The standard way in OCaml,
337 Haskell, and other functional programming languages, is to use the monadic
338 `bind` operator, `>>=`. (The name "bind" is not well chosen from our
339 perspective, but this is too deeply entrenched by now.) As mentioned above,
340 there needs to be a different `>>=` operator for each Monad or box type you'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 `>>=` or `bind` operator is 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 `bind` operation by hand:
348 let bind (u: int option) (f: int -> (int option)) =
353 let safe_add3 (u: int option) (v: int option) =
354 bind u (fun x -> bind v (fun y -> Some (x + y)));;
356 (* This is really just `map2 (+)`, using the `map2` operation that corresponds to
357 definition of `bind`. *)
359 let safe_div3 (u: int option) (v: int option) =
360 bind u (fun x -> bind v (fun y -> if 0 = y then None else Some (x / y)));;
362 (* This goes back to some of the simplicity of the original safe_div, without the complexity
363 introduced by safe_div2. *)
366 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:
368 safe_div3 :: Maybe Int -> Maybe Int -> Maybe Int
369 safe_div3 u v = do {x <- u;
371 if 0 == y then Nothing else return (x `div` y)}
373 Let's see our new functions in action:
377 # safe_div3 (safe_div3 (Some 12) (Some 2)) (Some 3);;
378 - : int option = Some 2
379 # safe_div3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
380 - : int option = None
381 # safe_add3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
382 - : int option = None
386 Compare the new definitions of `safe_add3` and `safe_div3` closely: the definition
387 for `safe_add3` shows what it looks like to equip an ordinary operation to
388 survive in dangerous presupposition-filled world. Note that the new
389 definition of `safe_add3` does not need to test whether its arguments are
390 None values or real numbers---those details are hidden inside of the
393 The definition of `safe_div3` shows exactly what extra needs to be said in
394 order to trigger the no-division-by-zero presupposition. Here, too, we don't
395 need to keep track of what presuppositions may have already failed
396 for whatever reason on our inputs.
398 (Linguistics note: Dividing by zero is supposed to feel like a kind of
399 presupposition failure. If we wanted to adapt this approach to
400 building a simple account of presupposition projection, we would have
401 to do several things. First, we would have to make use of the
402 polymorphism of the `option` type. In the arithmetic example, we only
403 made use of `int option`s, but when we're composing natural language
404 expression meanings, we'll need to use types like `N option`, `Det option`,
405 `VP option`, and so on. But that works automatically, because we can use
406 any type for the `'a` in `'a option`. Ultimately, we'd want to have a
407 theory of accommodation, and a theory of the situations in which
408 material within the sentence can satisfy presuppositions for other
409 material that otherwise would trigger a presupposition violation; but,
410 not surprisingly, these refinements will require some more
411 sophisticated techniques than the super-simple Option monad.)