1 Today we're going to encounter continuations. We're going to come at
2 them from three different directions, and each time we're going to end
3 up at the same place: a particular monad, which we'll call the
6 The three approches are:
10 Rethinking the list monad
11 -------------------------
13 To construct a monad, the key element is to settle on a type
14 constructor, and the monad naturally follows from that. I'll remind
15 you of some examples of how monads follow from the type constructor in
16 a moment. This will involve some review of familair material, but
17 it's worth doing for two reasons: it will set up a pattern for the new
18 discussion further below, and it will tie together some previously
19 unconnected elements of the course (more specifically, version 3 lists
22 For instance, take the **Reader Monad**. Once we decide that the type
25 type 'a reader = fun e:env -> 'a
27 then we can deduce the unit and the bind:
29 r_unit x:'a -> 'a reader = fun (e:env) -> x
31 Since the type of an `'a reader` is `fun e:env -> 'a` (by definition),
32 the type of the `r_unit` function is `'a -> e:env -> 'a`, which is a
33 specific case of the type of the *K* combinator. So it makes sense
34 that *K* is the unit for the reader monad.
36 Since the type of the `bind` operator is required to be
38 r_bind:('a reader) -> ('a -> 'b reader) -> ('b reader)
40 We can deduce the correct `bind` function as follows:
42 r_bind (u:'a reader) (f:'a -> 'b reader):('b reader) =
44 We have to open up the `u` box and get out the `'a` object in order to
45 feed it to `f`. Since `u` is a function from environments to
46 objects of type `'a`, the way we open a box in this monad is
47 by applying it to an environment:
51 This subexpression types to `'b reader`, which is good. The only
52 problem is that we invented an environment `e` that we didn't already have ,
53 so we have to abstract over that variable to balance the books:
57 This types to `env -> 'b reader`, but we want to end up with `env ->
58 'b`. Once again, the easiest way to turn a `'b reader` into a `'b` is to apply it to
59 an environment. So we end up as follows:
61 r_bind (u:'a reader) (f:'a -> 'b reader):('b reader) = f (u e) e
65 The **State Monad** is similar. We somehow intuit that we want to use
66 the following type constructor:
68 type 'a state = 'store -> ('a, 'store)
70 So our unit is naturally
72 let s_unit (x:'a):('a state) = fun (s:'store) -> (x, s)
74 And we deduce the bind in a way similar to the reasoning given above.
75 First, we need to apply `f` to the contents of the `u` box:
77 let s_bind (u:'a state) (f:'a -> ('b state)):('b state) =
79 But unlocking the `u` box is a little more complicated. As before, we
80 need to posit a state `s` that we can apply `u` to. Once we do so,
81 however, we won't have an `'a`, we'll have a pair whose first element
82 is an `'a`. So we have to unpack the pair:
84 ... let (a, s') = u s in ... (f a) ...
86 Abstracting over the `s` and adjusting the types gives the result:
88 let s_bind (u:'a state) (f:'a -> ('b state)):('b state) =
89 fun (s:state) -> let (a, s') = u s in f a s'
91 The **Option Monad** doesn't follow the same pattern so closely, so we
92 won't pause to explore it here, though conceptually its unit and bind
93 follow just as naturally from its type constructor.
95 Our other familiar monad is the **List Monad**, which we were told
100 l_bind u f = List.concat (List.map f u);;
102 Recall that `List.map` take a function and a list and returns the
103 result to applying the function to the elements of the list:
105 List.map (fun i -> [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]]
107 and List.concat takes a list of lists and erases the embdded list
110 List.concat [[1; 2]; [2; 3]] ~~> [1; 2; 2; 3]
114 l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
116 But where is the reasoning that led us to this unit and bind?
117 And what is the type `['a]`? Magic.
119 So let's take a *completely useless digressing* and see if we can
120 gain some insight into the details of the List monad. Let's choose
121 type constructor that we can peer into, using some of the technology
122 we built up so laboriously during the first half of the course. I'm
123 going to use type 3 lists, partly because I know they'll give the
124 result I want, but also because they're my favorite. These were the
125 lists that made lists look like Church numerals with extra bits
128 empty list: fun f z -> z
129 list with one element: fun f z -> f 1 z
130 list with two elements: fun f z -> f 2 (f 1 z)
131 list with three elements: fun f z -> f 3 (f 2 (f 1 z))
133 and so on. To save time, we'll let the Ocaml interpreter infer the
134 principle types of these functions (rather than deducing what the
139 - : 'a -> 'b -> 'b = <fun>
141 - : (int -> 'a -> 'b) -> 'a -> 'b = <fun>
142 # fun f z -> f 2 (f 1 z);;
143 - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
144 # fun f z -> f 3 (f 2 (f 1 z))
145 - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
148 Finally, we're getting consistent principle types, so we can stop.
149 These types should remind you of the simply-typed lambda calculus
150 types for Church numerals (`(o -> o) -> o -> o`) with one extra bit
151 thrown in (in this case, and int).
153 So here's our type constructor for our hand-rolled lists:
155 type 'a list' = (int -> 'a -> 'a) -> 'a -> 'a
157 Generalizing to lists that contain any kind of element (not just
160 type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
162 So an `('a, 'b) list'` is a list containing elements of type `'a`,
163 where `'b` is the type of some part of the plumbing. This is more
164 general than an ordinary Ocaml list, but we'll see how to map them
165 into Ocaml lists soon. We don't need to grasp the role of the `'b`'s
166 in order to proceed to build a monad:
168 l'_unit (x:'a):(('a, 'b) list) = fun x -> fun f z -> f x z
170 No problem. Arriving at bind is a little more complicated, but
171 exactly the same principles apply, you just have to be careful and
174 l'_bind (u:('a,'b) list') (f:'a -> ('c, 'd) list'): ('c, 'd) list' = ...
176 Unfortunately, we'll need to spell out the types:
178 l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b)
179 (f: 'a -> ('c -> 'd -> 'd) -> 'd -> 'd)
180 : ('c -> 'd -> 'd) -> 'd -> 'd = ...
182 It's a rookie mistake to quail before complicated types. You should
183 be no more intimiated by complex types than by a linguistic tree with
184 deeply embedded branches: complex structure created by repeated
185 application of simple rules.
187 As usual, we need to unpack the `u` box. Examine the type of `u`.
188 This time, `u` will only deliver up its contents if we give `u` as an
189 argument a function expecting an `'a`. Once that argument is applied
190 to an object of type `'a`, we'll have what we need. Thus:
192 .... u (fun (x:'a) -> ... (f a) ... ) ...
194 In order for `u` to have the kind of argument it needs, we have to
195 adjust `(f a)` (which has type `('c -> 'd -> 'd) -> 'd -> 'd`) in
196 order to deliver something of type `'b -> 'b`. The easiest way is to
197 alias `'d` to `'b`, and provide `(f a)` with an argument of type `'c
200 l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b)
201 (f: 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
202 : ('c -> 'b -> 'b) -> 'b -> 'b =
203 .... u (fun (x:'a) -> f a k) ...
205 [Excercise: can you arrive at a fully general bind for this type
206 constructor, one that does not collapse `'d`'s with `'b`'s?]
208 As usual, we have to abstract over `k`, but this time, no further
209 adjustments are needed:
211 l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b)
212 (f: 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
213 : ('c -> 'b -> 'b) -> 'b -> 'b =
214 fun (k:'c -> 'b -> 'b) -> u (fun (x:'a) -> f a k)
216 You should carefully check to make sure that this term is consistent
219 Our theory is that this monad should be capable of exactly
220 replicating the behavior of the standard List monad. Let's test:
223 l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
225 l'_bind (fun f z -> f 1 (f 2 z))
226 (fun i -> fun f z -> f i (f (i+1) z)) ~~> <fun>
228 Sigh. Ocaml won't show us our own list. So we have to choose an `f`
229 and a `z` that will turn our hand-crafted lists into standard Ocaml
230 lists, so that they will print out.
233 # let cons h t = h :: t;; (* Ocaml is stupid about :: *)
234 # l'_bind (fun f z -> f 1 (f 2 z))
235 (fun i -> fun f z -> f i (f (i+1) z)) cons [];;
236 - : int list = [1; 2; 2; 3]
241 Just for mnemonic purposes (sneaking in an instance of eta reduction
242 to the definition of unit), we can summarize the result as follows:
244 type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
245 l'_unit x = fun f -> f x
246 l'_bind u f = fun k -> u (fun x -> f x k)
248 To bad this digression, though it ties together various
249 elements of the course, has *no relevance whatsoever* to the topic of
252 Montague's PTQ treatment of DPs as generalized quantifiers
253 ----------------------------------------------------------
255 We've hinted that Montague's treatment of DPs as generalized
256 quantifiers embodies the spirit of continuations (see de Groote 2001,
257 Barker 2002 for lengthy discussion). Let's see why.
259 First, we'll need a type constructor. As you probably know,
260 Montague replaced individual-denoting determiner phrases (with type `e`)
261 with generalized quantifiers (with [extensional] type `(e -> t) -> t`.
262 In particular, the denotation of a proper name like *John*, which
263 might originally denote a object `j` of type `e`, came to denote a
264 generalized quantifier `fun pred -> pred j` of type `(e -> t) -> t`.
265 Let's write a general function that will map individuals into their
266 corresponding generalized quantifier:
268 gqize (x:e) = fun (p:e->t) -> p x
270 This function wraps up an individual in a fancy box. That is to say,
271 we are in the presence of a monad. The type constructor, the unit and
272 the bind follow naturally. We've done this enough times that I won't
273 belabor the construction of the bind function, the derivation is
274 similar to the List monad just given:
276 type 'a continuation = ('a -> 'b) -> 'b
277 c_unit (x:'a) = fun (p:'a -> 'b) -> p x
278 c_bind (u:('a -> 'b) -> 'b) (f: 'a -> ('c -> 'd) -> 'd): ('c -> 'd) -> 'd =
279 fun (k:'a -> 'b) -> u (fun (x:'a) -> f x k)
281 How similar is it to the List monad? Let's examine the type
282 constructor and the terms from the list monad derived above:
284 type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
285 l'_unit x = fun f -> f x
286 l'_bind u f = fun k -> u (fun x -> f x k)
288 (I performed a sneaky but valid eta reduction in the unit term.)
290 The unit and the bind for the Montague continuation monad and the
291 homemade List monad are the same terms! In other words, the behavior
292 of the List monad and the behavior of the continuations monad are
293 parallel in a deep sense. To emphasize the parallel, we can
294 instantiate the type of the list' monad using the Ocaml list type:
296 type 'a c_list = ('a -> 'a list) -> 'a list
297 let c_list_unit x = fun f -> f x;;
298 let c_list_bind u f = fun k -> u (fun x -> f x k);;
300 Have we really discovered that lists are secretly continuations?
301 Or have we merely found a way of simulating lists using list
302 continuations? Both perspectives are valid, and we can use our
303 intuitions about the list monad to understand continuations, and vice
304 versa. The connections will be expecially relevant when we consider
305 indefinites and Hamblin semantics on the linguistic side, and
306 non-determinism on the list monad side.
308 Refunctionalizing zippers
309 -------------------------