1 <!-- λ Λ ∀ ≡ α β γ ρ ω Ω -->
2 <!-- Loved this one: http://www.stephendiehl.com/posts/monads.html -->
4 sample <u>underlined</u> text, and then <span class="box">box1</span>, and then <span class="box2">box2</span> and then <span class="ul">ul</span> end.
10 The [[tradition in the functional programming
11 literature|https://wiki.haskell.org/Monad_tutorials_timeline]] is to
12 introduce monads using a metaphor: monads are spacesuits, monads are
13 monsters, monads are burritos. These metaphors can be helpful, and they
14 can be unhelpful. There's a backlash about the metaphors that tells people
15 to instead just look at the formal definition. We'll give that to you below, but it's
16 sometimes sloganized as
17 [A monad is just a monoid in the category of endofunctors, what's the problem?](http://stackoverflow.com/questions/3870088).
18 Without some intuitive guidance, this can also be unhelpful. We'll try to find a good balance.
20 The closest we will come to metaphorical talk is to suggest that
21 monadic types place values inside of *boxes*, and that monads wrap
22 and unwrap boxes to expose or enclose the values inside of them. In
23 any case, our emphasis will be on starting with the abstract structure
24 of monads, followed by instances of monads from the philosophical and
25 linguistics literature.
27 ## Box types: type expressions with one free type variable ##
29 Recall that we've been using lower-case Greek letters
30 <code>α, β, γ, ...</code> as type variables. We'll
31 use `P`, `Q`, `R`, and `S` as schematic metavariables over type expressions, that may or may not contain unbound
32 type variables. For instance, we might have
41 A *box type* will be a type expression that contains exactly one free
42 type variable. (You could extend this to expressions with more free variables; then you'd have
43 to specify which one of them the box is capturing. But let's keep it simple.) Some examples (using OCaml's type conventions):
47 (α, R) tree (assuming R contains no free type variables)
50 The idea is that whatever type the free type variable `α` might be instantiated to,
51 we will be a "type box" of a certain sort that "contains" values of type `α`. For instance,
52 if `α list` is our box type, and `α` is the type `int`, then in this context, `int list`
53 is the type of a boxed integer.
55 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.
57 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
58 type variable instantiates to. So if our box type is `α list`, and `α` instantiates to the specific type `int`, we would write:
62 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.)
68 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:
72 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`.
73 For instance, the following are Kleisli arrows:
77 int list -> <u>int list</u>
79 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>).
81 Note that the left-hand schema `P` is permitted to itself be a boxed type. That is, where
82 if `α list` is our box type, we can write the second arrow as
84 <u>int</u> -> <u>Q</u>
86 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.
89 ## A family of functions for each box type ##
91 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.
93 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.)
95 <code>map (/mæp/): (P -> Q) -> <u>P</u> -> <u>Q</u></code>
97 <code>map2 (/mæptu/): (P -> Q -> R) -> <u>P</u> -> <u>Q</u> -> <u>R</u></code>
99 <code>mid (/εmaidεnt@tI/ aka unit, return, pure): P -> <u>P</u></code>
101 <code>m$ or mapply (/εm@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>
103 <code><=< or mcomp : (Q -> <u>R</u>) -> (P -> <u>Q</u>) -> (P -> <u>R</u>)</code>
105 <code>>=> or mpmoc (flip mcomp): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
107 <code>>>= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
109 <code>=<< or mdnib (flip mbind) (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
111 <code>join: <span class="double">P</span> -> <u>P</u></code>
114 The menagerie isn't quite as bewildering as you might suppose. Many of these will
115 be interdefinable. For example, here is how `mcomp` and `mbind` are related: <code>k <=< j ≡
116 \a. (j a >>= k)</code>.
118 In most cases of interest, instances of these systems of functions will provide
119 certain useful guarantees.
121 * ***Mappable*** (in Haskelese, "Functors") At the most general level, box types are *Mappable*
122 if there is a `map` function defined for that box type with the type given above. This
123 has to obey the following Map Laws:
127 * ***MapNable*** (in Haskelese, "Applicatives") A Mappable box type is *MapNable*
128 if there are in addition `map2`, `mid`, and `mapply`. (Given either
129 of `map2` and `mapply`, you can define the other, and also `map`.
130 Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
131 have to obey the following MapN Laws:
136 * ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
137 is in addition an associative `mcomp` having `mid` as its left and
138 right identity. That is, the following Monad Laws must hold:
140 mcomp (mcomp j k) l (that is, (j <=< k) <=< l) = mcomp j (mcomp k l)
141 mcomp mid k (that is, mid <=< k) = k
142 mcomp k mid (that is, k <=< mid) = k
144 If you have any of `mcomp`, `mpmoc`, `mbind`, or `join`, you can use them to define the others.
145 Also, with these functions you can define `m$` and `map2` from *MapNables*. So all you really need
146 are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `join`, on the other.
148 Here are some interdefinitions: TODO
150 Names in Haskell: TODO
152 The name "bind" is not well chosen from our perspective, but this is too deeply entrenched by now.
156 To take a trivial (but, as we will see, still useful) example,
157 consider the Identity box type: `α`. So if `α` is type `bool`,
158 then a boxed `α` is ... a `bool`. In terms of the box analogy, the
159 Identity box type is a completely invisible box. With the following
163 mcomp ≡ \f g x.f (g x)
165 Identity is a monad. Here is a demonstration that the laws hold:
167 mcomp mid k == (\fgx.f(gx)) (\p.p) k
171 mcomp k mid == (\fgx.f(gx)) k (\p.p)
175 mcomp (mcomp j k) l == mcomp ((\fgx.f(gx)) j k) l
176 ~~> mcomp (\x.j(kx)) l
177 == (\fgx.f(gx)) (\x.j(kx)) l
178 ~~> \x.(\x.j(kx))(lx)
180 mcomp j (mcomp k l) == mcomp j ((\fgx.f(gx)) k l)
181 ~~> mcomp j (\x.k(lx))
182 == (\fgx.f(gx)) j (\x.k(lx))
183 ~~> \x.j((\x.k(lx)) x)
186 The Identity Monad is favored by mimes.
188 To take a slightly less trivial (and even more useful) example,
189 consider the box type `α list`, with the following operations:
194 mcomp: (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
195 mcomp f g a = concat (map f (g a))
196 = foldr (\b -> \gs -> (f b) ++ gs) [] (g a)
197 = [c | b <- g a, c <- f b]
199 The last three definitions of `mcomp` are all equivalent, and it is easy to see that they obey the monad laws (see exercises TODO).
201 In words, `mcomp f g a` feeds the `a` (which has type `α`) to `g`, which returns a list of `β`s;
202 each `β` in that list is fed to `f`, which returns a list of `γ`s. The
203 final result is the concatenation of those lists of `γ`s.
207 let f b = [b, b+1] in
208 let g a = [a*a, a+a] in
209 mcomp f g 7 ==> [49, 50, 14, 15]
211 `g 7` produced `[49, 14]`, which after being fed through `f` gave us `[49, 50, 14, 15]`.
213 Contrast that to `m$` (`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:
215 let gs = [(\a->a*a),(\a->a+a)] in
217 mapply gs xs ==> [49, 25, 14, 10]
220 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.
225 Integer division presupposes that its second argument
226 (the divisor) is not zero, upon pain of presupposition failure.
227 Here's what my OCaml interpreter says:
230 Exception: Division_by_zero.
232 Say we want to explicitly allow for the possibility that
233 division will return something other than a number.
234 To do that, we'll use OCaml's `option` type, which works like this:
236 # type 'a option = None | Some of 'a;;
240 - : int option = Some 3
242 So if a division is normal, we return some number, but if the divisor is
243 zero, we return `None`. As a mnemonic aid, we'll prepend a `safe_` to the start of our new divide function.
246 let safe_div (x:int) (y:int) =
249 | _ -> Some (x / y);;
252 val safe_div : int -> int -> int option = fun
254 - : int option = Some 6
256 - : int option = None
257 # safe_div (safe_div 12 2) 3;;
259 Error: This expression has type int option
260 but an expression was expected of type int
264 This starts off well: dividing `12` by `2`, no problem; dividing `12` by `0`,
265 just the behavior we were hoping for. But we want to be able to use
266 the output of the safe-division function as input for further division
267 operations. So we have to jack up the types of the inputs:
270 let safe_div2 (u:int option) (v:int option) =
276 | Some y -> Some (x / y));;
279 val safe_div2 : int option -> int option -> int option = <fun>
280 # safe_div2 (Some 12) (Some 2);;
281 - : int option = Some 6
282 # safe_div2 (Some 12) (Some 0);;
283 - : int option = None
284 # safe_div2 (safe_div2 (Some 12) (Some 0)) (Some 3);;
285 - : int option = None
289 Calling the function now involves some extra verbosity, but it gives us what we need: now we can try to divide by anything we
290 want, without fear that we're going to trigger system errors.
292 I prefer to line up the `match` alternatives by using OCaml's
296 let safe_div2 (u:int option) (v:int option) =
300 | (_, Some 0) -> None
301 | (Some x, Some y) -> Some (x / y);;
304 So far so good. But what if we want to combine division with
305 other arithmetic operations? We need to make those other operations
306 aware of the possibility that one of their arguments has already triggered a
307 presupposition failure:
310 let safe_add (u:int option) (v:int option) =
314 | (Some x, Some y) -> Some (x + y);;
317 val safe_add : int option -> int option -> int option = <fun>
318 # safe_add (Some 12) (Some 4);;
319 - : int option = Some 16
320 # safe_add (safe_div (Some 12) (Some 0)) (Some 4);;
321 - : int option = None
325 This works, but is somewhat disappointing: the `safe_add` operation
326 doesn't trigger any presupposition of its own, so it is a shame that
327 it needs to be adjusted because someone else might make trouble.
329 But we can automate the adjustment, using the monadic machinery we introduced above.
330 As we said, there needs to be different `>>=`, `map2` and so on operations for each
331 Monad or box type we're working with.
332 Haskell finesses this by "overloading" the single symbol `>>=`; you can just input that
333 symbol and it will calculate from the context of the surrounding type constraints what
334 monad you must have meant. In OCaml, the monadic operators are not pre-defined, but we will
335 give you a library that has definitions for all the standard monads, as in Haskell.
336 For now, though, we will define our `>>=` and `map2` operations by hand:
339 let (>>=) (u : 'a option) (j : 'a -> 'b option) : 'b option =
344 let map2 (f : 'a -> 'b -> 'c) (u : 'a option) (v : 'b option) : 'c option =
345 u >>= (fun x -> v >>= (fun y -> Some (f x y)));;
347 let safe_add3 = map2 (+);; (* that was easy *)
349 let safe_div3 (u: int option) (v: int option) =
350 u >>= (fun x -> v >>= (fun y -> if 0 = y then None else Some (x / y)));;
353 Haskell has an even more user-friendly notation for defining `safe_div3`, namely:
355 safe_div3 :: Maybe Int -> Maybe Int -> Maybe Int
356 safe_div3 u v = do {x <- u;
358 if 0 == y then Nothing else Just (x `div` y)}
360 Let's see our new functions in action:
364 # safe_div3 (safe_div3 (Some 12) (Some 2)) (Some 3);;
365 - : int option = Some 2
366 # safe_div3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
367 - : int option = None
368 # safe_add3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
369 - : int option = None
373 Compare the new definitions of `safe_add3` and `safe_div3` closely: the definition
374 for `safe_add3` shows what it looks like to equip an ordinary operation to
375 survive in dangerous presupposition-filled world. Note that the new
376 definition of `safe_add3` does not need to test whether its arguments are
377 None values or real numbers---those details are hidden inside of the
380 Note also that our definition of `safe_div3` recovers some of the simplicity of
381 the original `safe_div`, without the complexity introduced by `safe_div2`. We now
382 add exactly what extra is needed to track the no-division-by-zero presupposition. Here, too, we don't
383 need to keep track of what other presuppositions may have already failed
384 for whatever reason on our inputs.
386 (Linguistics note: Dividing by zero is supposed to feel like a kind of
387 presupposition failure. If we wanted to adapt this approach to
388 building a simple account of presupposition projection, we would have
389 to do several things. First, we would have to make use of the
390 polymorphism of the `option` type. In the arithmetic example, we only
391 made use of `int option`s, but when we're composing natural language
392 expression meanings, we'll need to use types like `N option`, `Det option`,
393 `VP option`, and so on. But that works automatically, because we can use
394 any type for the `'a` in `'a option`. Ultimately, we'd want to have a
395 theory of accommodation, and a theory of the situations in which
396 material within the sentence can satisfy presuppositions for other
397 material that otherwise would trigger a presupposition violation; but,
398 not surprisingly, these refinements will require some more
399 sophisticated techniques than the super-simple Option monad.)