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`.
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>
83 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.
86 ## A family of functions for each box type ##
88 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.
90 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.)
92 <code>map (/mæp/): (P -> Q) -> <u>P</u> -> <u>Q</u></code>
94 <code>map2 (/mæptu/): (P -> Q -> R) -> <u>P</u> -> <u>Q</u> -> <u>R</u></code>
96 <code>mid (/εmaidεnt@tI/ aka unit, return, pure): P -> <u>P</u></code>
98 <code>m$ or mapply (/εm@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>
100 <code><=< or mcomp : (Q -> <u>R</u>) -> (P -> <u>Q</u>) -> (P -> <u>R</u>)</code>
102 <code>>=> or mpmoc (flip mcomp): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
104 <code>>>= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
106 <code>=<< or mdnib (flip mbind) (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
108 <code>join: <span class="box2">P</span> -> <u>P</u></code>
111 The menagerie isn't quite as bewildering as you might suppose. Many of these will
112 be interdefinable. For example, here is how `mcomp` and `mbind` are related: <code>k <=< j ≡
113 \a. (j a >>= k)</code>.
115 In most cases of interest, instances of these systems of functions will provide
116 certain useful guarantees.
118 * ***Mappable*** (in Haskelese, "Functors") At the most general level, box types are *Mappable*
119 if there is a `map` function defined for that box type with the type given above. This
120 has to obey the following Map Laws:
124 * ***MapNable*** (in Haskelese, "Applicatives") A Mappable box type is *MapNable*
125 if there are in addition `map2`, `mid`, and `mapply`. (Given either
126 of `map2` and `mapply`, you can define the other, and also `map`.
127 Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
128 have to obey the following MapN Laws:
133 * ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
134 is in addition an associative `mcomp` having `mid` as its left and
135 right identity. That is, the following Monad Laws must hold:
137 mcomp (mcomp j k) l (that is, (j <=< k) <=< l) = mcomp j (mcomp k l)
138 mcomp mid k (that is, mid <=< k) = k
139 mcomp k mid (that is, k <=< mid) = k
141 If you have any of `mcomp`, `mpmoc`, `mbind`, or `join`, you can use them to define the others.
142 Also, with these functions you can define `m$` and `map2` from *MapNables*. So all you really need
143 are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `join`, on the other.
145 Here are some interdefinitions: TODO
147 Names in Haskell: TODO
149 The name "bind" is not well chosen from our perspective, but this is too deeply entrenched by now.
153 To take a trivial (but, as we will see, still useful) example,
154 consider the Identity box type: `α`. So if `α` is type `bool`,
155 then a boxed `α` is ... a `bool`. In terms of the box analogy, the
156 Identity box type is a completely invisible box. With the following
160 mcomp ≡ \f g x.f (g x)
162 Identity is a monad. Here is a demonstration that the laws hold:
164 mcomp mid k == (\fgx.f(gx)) (\p.p) k
168 mcomp k mid == (\fgx.f(gx)) k (\p.p)
172 mcomp (mcomp j k) l == mcomp ((\fgx.f(gx)) j k) l
173 ~~> mcomp (\x.j(kx)) l
174 == (\fgx.f(gx)) (\x.j(kx)) l
175 ~~> \x.(\x.j(kx))(lx)
177 mcomp j (mcomp k l) == mcomp j ((\fgx.f(gx)) k l)
178 ~~> mcomp j (\x.k(lx))
179 == (\fgx.f(gx)) j (\x.k(lx))
180 ~~> \x.j((\x.k(lx)) x)
183 The Identity Monad is favored by mimes.
185 To take a slightly less trivial (and even more useful) example,
186 consider the box type `α list`, with the following operations:
191 mcomp: (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
192 mcomp f g a = concat (map f (g a))
193 = foldr (\b -> \gs -> (f b) ++ gs) [] (g a)
194 = [c | b <- g a, c <- f b]
196 The last three definitions of `mcomp` are all equivalent, and it is easy to see that they obey the monad laws (see exercises TODO).
198 In words, `mcomp f g a` feeds the `a` (which has type `α`) to `g`, which returns a list of `β`s;
199 each `β` in that list is fed to `f`, which returns a list of `γ`s. The
200 final result is the concatenation of those lists of `γ`s.
204 let f b = [b, b+1] in
205 let g a = [a*a, a+a] in
206 mcomp f g 7 ==> [49, 50, 14, 15]
208 `g 7` produced `[49, 14]`, which after being fed through `f` gave us `[49, 50, 14, 15]`.
210 Contrast that to `m$` (`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:
212 let gs = [(\a->a*a),(\a->a+a)] in
214 mapply gs xs ==> [49, 25, 14, 10]
217 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.
222 Integer division presupposes that its second argument
223 (the divisor) is not zero, upon pain of presupposition failure.
224 Here's what my OCaml interpreter says:
227 Exception: Division_by_zero.
229 Say we want to explicitly allow for the possibility that
230 division will return something other than a number.
231 To do that, we'll use OCaml's `option` type, which works like this:
233 # type 'a option = None | Some of 'a;;
237 - : int option = Some 3
239 So if a division is normal, we return some number, but if the divisor is
240 zero, we return `None`. As a mnemonic aid, we'll prepend a `safe_` to the start of our new divide function.
243 let safe_div (x:int) (y:int) =
246 | _ -> Some (x / y);;
249 val safe_div : int -> int -> int option = fun
251 - : int option = Some 6
253 - : int option = None
254 # safe_div (safe_div 12 2) 3;;
256 Error: This expression has type int option
257 but an expression was expected of type int
261 This starts off well: dividing `12` by `2`, no problem; dividing `12` by `0`,
262 just the behavior we were hoping for. But we want to be able to use
263 the output of the safe-division function as input for further division
264 operations. So we have to jack up the types of the inputs:
267 let safe_div2 (u:int option) (v:int option) =
273 | Some y -> Some (x / y));;
276 val safe_div2 : int option -> int option -> int option = <fun>
277 # safe_div2 (Some 12) (Some 2);;
278 - : int option = Some 6
279 # safe_div2 (Some 12) (Some 0);;
280 - : int option = None
281 # safe_div2 (safe_div2 (Some 12) (Some 0)) (Some 3);;
282 - : int option = None
286 Calling the function now involves some extra verbosity, but it gives us what we need: now we can try to divide by anything we
287 want, without fear that we're going to trigger system errors.
289 I prefer to line up the `match` alternatives by using OCaml's
293 let safe_div2 (u:int option) (v:int option) =
297 | (_, Some 0) -> None
298 | (Some x, Some y) -> Some (x / y);;
301 So far so good. But what if we want to combine division with
302 other arithmetic operations? We need to make those other operations
303 aware of the possibility that one of their arguments has already triggered a
304 presupposition failure:
307 let safe_add (u:int option) (v:int option) =
311 | (Some x, Some y) -> Some (x + y);;
314 val safe_add : int option -> int option -> int option = <fun>
315 # safe_add (Some 12) (Some 4);;
316 - : int option = Some 16
317 # safe_add (safe_div (Some 12) (Some 0)) (Some 4);;
318 - : int option = None
322 This works, but is somewhat disappointing: the `safe_add` operation
323 doesn't trigger any presupposition of its own, so it is a shame that
324 it needs to be adjusted because someone else might make trouble.
326 But we can automate the adjustment, using the monadic machinery we introduced above.
327 As we said, there needs to be different `>>=`, `map2` and so on operations for each
328 Monad or box type we're working with.
329 Haskell finesses this by "overloading" the single symbol `>>=`; you can just input that
330 symbol and it will calculate from the context of the surrounding type constraints what
331 monad you must have meant. In OCaml, the monadic operators are not pre-defined, but we will
332 give you a library that has definitions for all the standard monads, as in Haskell.
333 For now, though, we will define our `>>=` and `map2` operations by hand:
336 let (>>=) (u : 'a option) (j : 'a -> 'b option) : 'b option =
341 let map2 (f : 'a -> 'b -> 'c) (u : 'a option) (v : 'b option) : 'c option =
342 u >>= (fun x -> v >>= (fun y -> Some (f x y)));;
344 let safe_add3 = map2 (+);; (* that was easy *)
346 let safe_div3 (u: int option) (v: int option) =
347 u >>= (fun x -> v >>= (fun y -> if 0 = y then None else Some (x / y)));;
350 Haskell has an even more user-friendly notation for defining `safe_div3`, namely:
352 safe_div3 :: Maybe Int -> Maybe Int -> Maybe Int
353 safe_div3 u v = do {x <- u;
355 if 0 == y then Nothing else Just (x `div` y)}
357 Let's see our new functions in action:
361 # safe_div3 (safe_div3 (Some 12) (Some 2)) (Some 3);;
362 - : int option = Some 2
363 # safe_div3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
364 - : int option = None
365 # safe_add3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
366 - : int option = None
370 Compare the new definitions of `safe_add3` and `safe_div3` closely: the definition
371 for `safe_add3` shows what it looks like to equip an ordinary operation to
372 survive in dangerous presupposition-filled world. Note that the new
373 definition of `safe_add3` does not need to test whether its arguments are
374 `None` values or real numbers---those details are hidden inside of the
377 Note also that our definition of `safe_div3` recovers some of the simplicity of
378 the original `safe_div`, without the complexity introduced by `safe_div2`. We now
379 add exactly what extra is needed to track the no-division-by-zero presupposition. Here, too, we don't
380 need to keep track of what other presuppositions may have already failed
381 for whatever reason on our inputs.
383 (Linguistics note: Dividing by zero is supposed to feel like a kind of
384 presupposition failure. If we wanted to adapt this approach to
385 building a simple account of presupposition projection, we would have
386 to do several things. First, we would have to make use of the
387 polymorphism of the `option` type. In the arithmetic example, we only
388 made use of `int option`s, but when we're composing natural language
389 expression meanings, we'll need to use types like `N option`, `Det option`,
390 `VP option`, and so on. But that works automatically, because we can use
391 any type for the `'a` in `'a option`. Ultimately, we'd want to have a
392 theory of accommodation, and a theory of the situations in which
393 material within the sentence can satisfy presuppositions for other
394 material that otherwise would trigger a presupposition violation; but,
395 not surprisingly, these refinements will require some more
396 sophisticated techniques than the super-simple Option monad.)