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. We'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 [This bind is a simplified version of the careful `let a = u e in ...`
66 constructions we provided in earlier lectures. We use the simplified
67 versions here in order to emphasize similarities of structure across
68 monads; the official bind is still the one with the plethora of `let`'s.]
70 The **State Monad** is similar. We somehow intuit that we want to use
71 the following type constructor:
73 type 'a state = 'store -> ('a, 'store)
75 So our unit is naturally
77 let s_unit (x:'a):('a state) = fun (s:'store) -> (x, s)
79 And we deduce the bind in a way similar to the reasoning given above.
80 First, we need to apply `f` to the contents of the `u` box:
82 let s_bind (u:'a state) (f:'a -> ('b state)):('b state) =
84 But unlocking the `u` box is a little more complicated. As before, we
85 need to posit a state `s` that we can apply `u` to. Once we do so,
86 however, we won't have an `'a`, we'll have a pair whose first element
87 is an `'a`. So we have to unpack the pair:
89 ... let (a, s') = u s in ... (f a) ...
91 Abstracting over the `s` and adjusting the types gives the result:
93 let s_bind (u:'a state) (f:'a -> ('b state)):('b state) =
94 fun (s:state) -> let (a, s') = u s in f a s'
96 The **Option Monad** doesn't follow the same pattern so closely, so we
97 won't pause to explore it here, though conceptually its unit and bind
98 follow just as naturally from its type constructor.
100 Our other familiar monad is the **List Monad**, which we were told
103 type 'a list = ['a];;
104 l_unit (x:'a) = [x];;
105 l_bind u f = List.concat (List.map f u);;
107 Recall that `List.map` take a function and a list and returns the
108 result to applying the function to the elements of the list:
110 List.map (fun i -> [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]]
112 and List.concat takes a list of lists and erases the embdded list
115 List.concat [[1; 2]; [2; 3]] ~~> [1; 2; 2; 3]
119 l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
121 But where is the reasoning that led us to this unit and bind?
122 And what is the type `['a]`? Magic.
124 So let's indulge ourselves in a completely useless digression and see
125 if we can gain some insight into the details of the List monad. Let's
126 choose type constructor that we can peer into, using some of the
127 technology we built up so laboriously during the first half of the
128 course. We're going to use type 3 lists, partly because I know
129 they'll give the result I want, but also because they're the coolest.
130 These were the lists that made lists look like Church numerals with
131 extra bits embdded in them:
133 empty list: fun f z -> z
134 list with one element: fun f z -> f 1 z
135 list with two elements: fun f z -> f 2 (f 1 z)
136 list with three elements: fun f z -> f 3 (f 2 (f 1 z))
138 and so on. To save time, we'll let the Ocaml interpreter infer the
139 principle types of these functions (rather than deducing what the
144 - : 'a -> 'b -> 'b = <fun>
146 - : (int -> 'a -> 'b) -> 'a -> 'b = <fun>
147 # fun f z -> f 2 (f 1 z);;
148 - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
149 # fun f z -> f 3 (f 2 (f 1 z))
150 - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
153 Finally, we're getting consistent principle types, so we can stop.
154 These types should remind you of the simply-typed lambda calculus
155 types for Church numerals (`(o -> o) -> o -> o`) with one extra bit
156 thrown in (in this case, an int).
158 So here's our type constructor for our hand-rolled lists:
160 type 'a list' = (int -> 'a -> 'a) -> 'a -> 'a
162 Generalizing to lists that contain any kind of element (not just
165 type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
167 So an `('a, 'b) list'` is a list containing elements of type `'a`,
168 where `'b` is the type of some part of the plumbing. This is more
169 general than an ordinary Ocaml list, but we'll see how to map them
170 into Ocaml lists soon. We don't need to fully grasp the role of the `'b`'s
171 in order to proceed to build a monad:
173 l'_unit (x:'a):(('a, 'b) list) = fun x -> fun f z -> f x z
175 No problem. Arriving at bind is a little more complicated, but
176 exactly the same principles apply, you just have to be careful and
179 l'_bind (u:('a,'b) list') (f:'a -> ('c, 'd) list'): ('c, 'd) list' = ...
181 Unfortunately, we'll need to spell out the types:
183 l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b)
184 (f: 'a -> ('c -> 'd -> 'd) -> 'd -> 'd)
185 : ('c -> 'd -> 'd) -> 'd -> 'd = ...
187 It's a rookie mistake to quail before complicated types. You should
188 be no more intimiated by complex types than by a linguistic tree with
189 deeply embedded branches: complex structure created by repeated
190 application of simple rules.
192 As usual, we need to unpack the `u` box. Examine the type of `u`.
193 This time, `u` will only deliver up its contents if we give `u` as an
194 argument a function expecting an `'a`. Once that argument is applied
195 to an object of type `'a`, we'll have what we need. Thus:
197 .... u (fun (x:'a) -> ... (f a) ... ) ...
199 In order for `u` to have the kind of argument it needs, we have to
200 adjust `(f a)` (which has type `('c -> 'd -> 'd) -> 'd -> 'd`) in
201 order to deliver something of type `'b -> 'b`. The easiest way is to
202 alias `'d` to `'b`, and provide `(f a)` with an argument of type `'c
205 l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b)
206 (f: 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
207 : ('c -> 'b -> 'b) -> 'b -> 'b =
208 .... u (fun (x:'a) -> f a k) ...
210 [Excercise: can you arrive at a fully general bind for this type
211 constructor, one that does not collapse `'d`'s with `'b`'s?]
213 As usual, we have to abstract over `k`, but this time, no further
214 adjustments are needed:
216 l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b)
217 (f: 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
218 : ('c -> 'b -> 'b) -> 'b -> 'b =
219 fun (k:'c -> 'b -> 'b) -> u (fun (x:'a) -> f a k)
221 You should carefully check to make sure that this term is consistent
224 Our theory is that this monad should be capable of exactly
225 replicating the behavior of the standard List monad. Let's test:
228 l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
230 l'_bind (fun f z -> f 1 (f 2 z))
231 (fun i -> fun f z -> f i (f (i+1) z)) ~~> <fun>
233 Sigh. Ocaml won't show us our own list. So we have to choose an `f`
234 and a `z` that will turn our hand-crafted lists into standard Ocaml
235 lists, so that they will print out.
238 # let cons h t = h :: t;; (* Ocaml is stupid about :: *)
239 # l'_bind (fun f z -> f 1 (f 2 z))
240 (fun i -> fun f z -> f i (f (i+1) z)) cons [];;
241 - : int list = [1; 2; 2; 3]
246 To bad this digression, though it ties together various
247 elements of the course, has *no relevance whatsoever* to the topic of
250 Montague's PTQ treatment of DPs as generalized quantifiers
251 ----------------------------------------------------------
253 We've hinted that Montague's treatment of DPs as generalized
254 quantifiers embodies the spirit of continuations (see de Groote 2001,
255 Barker 2002 for lengthy discussion). Let's see why.
257 First, we'll need a type constructor. As you probably know,
258 Montague replaced individual-denoting determiner phrases (with type `e`)
259 with generalized quantifiers (with [extensional] type `(e -> t) -> t`.
260 In particular, the denotation of a proper name like *John*, which
261 might originally denote a object `j` of type `e`, came to denote a
262 generalized quantifier `fun pred -> pred j` of type `(e -> t) -> t`.
263 Let's write a general function that will map individuals into their
264 corresponding generalized quantifier:
266 gqize (x:e) = fun (p:e->t) -> p x
268 This function wraps up an individual in a fancy box. That is to say,
269 we are in the presence of a monad. The type constructor, the unit and
270 the bind follow naturally. We've done this enough times that we won't
271 belabor the construction of the bind function, the derivation is
272 similar to the List monad just given:
275 type 'a continuation = ('a -> 'b) -> 'b
276 c_unit (x:'a) = fun (p:'a -> 'b) -> p x
277 c_bind (u:('a -> 'b) -> 'b) (f: 'a -> ('c -> 'd) -> 'd): ('c -> 'd) -> 'd =
278 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 (We 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 -------------------------