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 = env -> 'a
27 then the choice of unit and bind is natural:
29 let r_unit (a : 'a) : 'a reader = fun (e : env) -> a
31 Since the type of an `'a reader` is `env -> 'a` (by definition),
32 the type of the `r_unit` function is `'a -> env -> 'a`, which is a
33 specific case of the type of the *K* combinator. 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 reason our way to the correct `bind` function as follows. We start by declaring the type:
42 let r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) =
44 Now 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 an environment. So we end up as follows:
60 r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) =
63 And we're done. This gives us a bind function of the right type. We can then check whether, in combination with the unit function we chose, it satisfies the monad laws, and behaves in the way we intend. And it does.
65 [The bind we cite here is a condensed version of the careful `let a = u e in ...`
66 constructions we provided in earlier lectures. We use the condensed
67 version here in order to emphasize similarities of structure across
70 The **State Monad** is similar. Once we've decided to use the following type constructor:
72 type 'a state = store -> ('a, store)
74 Then our unit is naturally:
76 let s_unit (a : 'a) : ('a state) = fun (s : store) -> (a, s)
78 And we can reason our way to the bind function in a way similar to the reasoning given above. First, we need to apply `f` to the contents of the `u` box:
80 let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state =
83 But unlocking the `u` box is a little more complicated. As before, we
84 need to posit a state `s` that we can apply `u` to. Once we do so,
85 however, we won't have an `'a`, we'll have a pair whose first element
86 is an `'a`. So we have to unpack the pair:
88 ... let (a, s') = u s in ... (f a) ...
90 Abstracting over the `s` and adjusting the types gives the result:
92 let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state =
93 fun (s : store) -> let (a, s') = u s in f a s'
95 The **Option/Maybe Monad** doesn't follow the same pattern so closely, so we
96 won't pause to explore it here, though conceptually its unit and bind
97 follow just as naturally from its type constructor.
99 Our other familiar monad is the **List Monad**, which we were told
102 type 'a list = ['a];;
103 l_unit (a : 'a) = [a];;
104 l_bind u f = List.concat (List.map f u);;
106 Recall that `List.map` take a function and a list and returns the
107 result to applying the function to the elements of the list:
109 List.map (fun i -> [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]]
111 and List.concat takes a list of lists and erases the embdded list
114 List.concat [[1; 2]; [2; 3]] ~~> [1; 2; 2; 3]
118 l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
120 But where is the reasoning that led us to this unit and bind?
121 And what is the type `['a]`? Magic.
123 So let's indulge ourselves in a completely useless digression and see
124 if we can gain some insight into the details of the List monad. Let's
125 choose type constructor that we can peer into, using some of the
126 technology we built up so laboriously during the first half of the
127 course. We're going to use type 3 lists, partly because we know
128 they'll give the result we want, but also because they're the coolest.
129 These were the lists that made lists look like Church numerals with
130 extra bits embdded in them:
132 empty list: fun f z -> z
133 list with one element: fun f z -> f 1 z
134 list with two elements: fun f z -> f 2 (f 1 z)
135 list with three elements: fun f z -> f 3 (f 2 (f 1 z))
137 and so on. To save time, we'll let the OCaml interpreter infer the
138 principle types of these functions (rather than deducing what the
142 - : 'a -> 'b -> 'b = <fun>
144 - : (int -> 'a -> 'b) -> 'a -> 'b = <fun>
145 # fun f z -> f 2 (f 1 z);;
146 - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
147 # fun f z -> f 3 (f 2 (f 1 z))
148 - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
150 We can see what the consistent, general principle types are at the end, so we
151 can stop. These types should remind you of the simply-typed lambda calculus
152 types for Church numerals (`(o -> o) -> o -> o`) with one extra bit thrown in
153 (in this case, an int).
155 So here's our type constructor for our hand-rolled lists:
157 type 'b list' = (int -> 'b -> 'b) -> 'b -> 'b
159 Generalizing to lists that contain any kind of element (not just
162 type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
164 So an `('a, 'b) list'` is a list containing elements of type `'a`,
165 where `'b` is the type of some part of the plumbing. This is more
166 general than an ordinary OCaml list, but we'll see how to map them
167 into OCaml lists soon. We don't need to fully grasp the role of the `'b`'s
168 in order to proceed to build a monad:
170 l'_unit (a : 'a) : ('a, 'b) list = fun a -> fun f z -> f a z
172 No problem. Arriving at bind is a little more complicated, but
173 exactly the same principles apply, you just have to be careful and
176 l'_bind (u : ('a,'b) list') (f : 'a -> ('c, 'd) list') : ('c, 'd) list' = ...
178 Unpacking the types gives:
180 l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
181 (f : 'a -> ('c -> 'd -> 'd) -> 'd -> 'd)
182 : ('c -> 'd -> 'd) -> 'd -> 'd = ...
184 But it's a rookie mistake to quail before complicated types. You should
185 be no more intimiated by complex types than by a linguistic tree with
186 deeply embedded branches: complex structure created by repeated
187 application of simple rules.
189 As usual, we need to unpack the `u` box. Examine the type of `u`.
190 This time, `u` will only deliver up its contents if we give `u` as an
191 argument a function expecting an `'a` and a `'b`. `u` will fold that function over its type `'a` members, and that's how we'll get the `'a`s we need. Thus:
193 ... u (fun (a : 'a) (b : 'b) -> ... f a ... ) ...
195 In order for `u` to have the kind of argument it needs, the `... (f a) ...` has to evaluate to a result of type `'b`. The easiest way to do this is to collapse (or "unify") the types `'b` and `'d`, with the result that `f a` will have type `('c -> 'b -> 'b) -> 'b -> 'b`. Let's postulate an argument `k` of type `('c -> 'b -> 'b)` and supply it to `(f a)`:
197 ... u (fun (a : 'a) (b : 'b) -> ... f a k ... ) ...
199 Now we have an argument `b` of type `'b`, so we can supply that to `(f a) k`, getting a result of type `'b`, as we need:
201 ... u (fun (a : 'a) (b : 'b) -> f a k b) ...
203 Now, we've used a `k` that we pulled out of nowhere, so we need to abstract over it:
205 fun (k : 'c -> 'b -> 'b) -> u (fun (a : 'a) (b : 'b) -> f a k b)
207 This whole expression has type `('c -> 'b -> 'b) -> 'b -> 'b`, which is exactly the type of a `('c, 'b) list'`. So we can hypothesize that we our bind is:
209 l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
210 (f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
211 : ('c -> 'b -> 'b) -> 'b -> 'b =
212 fun k -> u (fun a b -> f a k b)
214 That is a function of the right type for our bind, but to check whether it works, we have to verify it (with the unit we chose) against the monad laws, and reason whether it will have the right behavior.
216 Here's a way to persuade yourself that it will have the right behavior. First, it will be handy to eta-expand our `fun k -> u (fun a b -> f a k b)` to:
218 fun k z -> u (fun a b -> f a k b) z
220 Now let's think about what this does. It's a wrapper around `u`. In order to behave as the list which is the result of mapping `f` over each element of `u`, and then joining (`concat`ing) the results, this wrapper would have to accept arguments `k` and `z` and fold them in just the same way that the list which is the result of mapping `f` and then joining the results would fold them. Will it?
222 Suppose we have a list' whose contents are `[1; 2; 4; 8]`---that is, our list' will be `fun f z -> f 1 (f 2 (f 4 (f 8 z)))`. We call that list' `u`. Suppose we also have a function `f` that for each `int` we give it, gives back a list of the divisors of that `int` that are greater than 1. Intuitively, then, binding `u` to `f` should give us:
225 concat [[]; [2]; [2; 4]; [2; 4; 8]] =
228 Or rather, it should give us a list' version of that, which takes a function `k` and value `z` as arguments, and returns the right fold of `k` and `z` over those elements. What does our formula
230 fun k z -> u (fun a b -> f a k b) z
232 do? Well, for each element `a` in `u`, it applies `f` to that `a`, getting one of the lists:
239 (or rather, their list' versions). Then it takes the accumulated result `b` of previous steps in the fold, and it folds `k` and `b` over the list generated by `f a`. The result of doing so is passed on to the next step as the accumulated result so far.
241 So if, for example, we let `k` be `+` and `z` be `0`, then the computation would proceed:
244 right-fold + and 0 over [2; 4; 8] = 2+4+8+0 ==>
245 right-fold + and 2+4+8+0 over [2; 4] = 2+4+2+4+8+0 ==>
246 right-fold + and 2+4+2+4+8+0 over [2] = 2+2+4+2+4+8+0 ==>
247 right-fold + and 2+2+4+2+4+8+0 over [] = 2+2+4+2+4+8+0
249 which indeed is the result of right-folding + and 0 over `[2; 2; 4; 2; 4; 8]`. If you trace through how this works, you should be able to persuade yourself that our formula:
251 fun k z -> u (fun a b -> f a k b) z
253 will deliver just the same folds, for arbitrary choices of `k` and `z` (with the right types), and arbitrary list's `u` and appropriately-typed `f`s, as
255 fun k z -> List.fold_right k (concat (map f u)) z
259 For future reference, we might make two eta-reductions to our formula, so that we have instead:
261 let l'_bind = fun k -> u (fun a -> f a k);;
263 Let's make some more tests:
266 l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
268 l'_bind (fun f z -> f 1 (f 2 z))
269 (fun i -> fun f z -> f i (f (i+1) z)) ~~> <fun>
271 Sigh. OCaml won't show us our own list. So we have to choose an `f`
272 and a `z` that will turn our hand-crafted lists into standard OCaml
273 lists, so that they will print out.
275 # let cons h t = h :: t;; (* OCaml is stupid about :: *)
276 # l'_bind (fun f z -> f 1 (f 2 z))
277 (fun i -> fun f z -> f i (f (i+1) z)) cons [];;
278 - : int list = [1; 2; 2; 3]
282 To bad this digression, though it ties together various
283 elements of the course, has *no relevance whatsoever* to the topic of
286 Montague's PTQ treatment of DPs as generalized quantifiers
287 ----------------------------------------------------------
289 We've hinted that Montague's treatment of DPs as generalized
290 quantifiers embodies the spirit of continuations (see de Groote 2001,
291 Barker 2002 for lengthy discussion). Let's see why.
293 First, we'll need a type constructor. As you probably know,
294 Montague replaced individual-denoting determiner phrases (with type `e`)
295 with generalized quantifiers (with [extensional] type `(e -> t) -> t`.
296 In particular, the denotation of a proper name like *John*, which
297 might originally denote a object `j` of type `e`, came to denote a
298 generalized quantifier `fun pred -> pred j` of type `(e -> t) -> t`.
299 Let's write a general function that will map individuals into their
300 corresponding generalized quantifier:
302 gqize (a : e) = fun (p : e -> t) -> p a
304 This function wraps up an individual in a fancy box. That is to say,
305 we are in the presence of a monad. The type constructor, the unit and
306 the bind follow naturally. We've done this enough times that we won't
307 belabor the construction of the bind function, the derivation is
308 similar to the List monad just given:
310 type 'a continuation = ('a -> 'b) -> 'b
311 c_unit (a : 'a) = fun (p : 'a -> 'b) -> p a
312 c_bind (u : ('a -> 'b) -> 'b) (f : 'a -> ('c -> 'd) -> 'd) : ('c -> 'd) -> 'd =
313 fun (k : 'a -> 'b) -> u (fun (a : 'a) -> f a k)
315 How similar is it to the List monad? Let's examine the type
316 constructor and the terms from the list monad derived above:
318 type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
319 l'_unit a = fun f -> f a
320 l'_bind u f = fun k -> u (fun a -> f a k)
322 (We performed a sneaky but valid eta reduction in the unit term.)
324 The unit and the bind for the Montague continuation monad and the
325 homemade List monad are the same terms! In other words, the behavior
326 of the List monad and the behavior of the continuations monad are
327 parallel in a deep sense. To emphasize the parallel, we can
328 instantiate the type of the list' monad using the OCaml list type:
330 type 'a c_list = ('a -> 'a list) -> 'a list
332 Have we really discovered that lists are secretly continuations? Or
333 have we merely found a way of simulating lists using list
334 continuations? Both perspectives are valid, and we can use our
335 intuitions about the list monad to understand continuations, and vice
336 versa (not to mention our intuitions about primitive recursion in
337 Church numerals too). The connections will be expecially relevant
338 when we consider indefinites and Hamblin semantics on the linguistic
339 side, and non-determinism on the list monad side.
341 Refunctionalizing zippers
342 -------------------------