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4 # Applications of continuations to natural language
6 We've seen a number of applications of monads to natural language,
7 including presupposition projection, binding, intensionality, and the
8 dynamics of the GSV fragment.
10 In the past couple of weeks, we've introduced continuations, first as
11 a functional programming technique, then in terms of list and tree
12 zippers, then as a monad. In this lecture, we will generalize
13 continuations slightly beyond a monad, and then begin to outline some
14 of the applications of monads. In brief, the generalization can be
15 summarized in terms of types: instead of using a Kleisli arrow mapping
16 a type α to a continuized type (α -> ρ) -> ρ, we'll allow the result
17 types to differ, i.e., we'll map α to (α -> β) -> γ. This will be
18 crucial for some natural language applications.
20 Many (though not all) of the applications are discussed in detail in
21 Barker and Shan 2014, *Continuations in Natural Language*, OUP.
23 In terms of list zippers, the continuation of a focused element in
24 the list is the front part of the list.
26 list zipper for the list [a;b;c;d;e;f] with focus on d:
33 In terms of tree zippers, the continuation is the entire context of
34 the focused element--the entire rest of the tree.
36 [drawing of a broken tree]
38 Last week we had trouble computing the doubling task when there was more
39 than one shifty operator after moving from a list perspective to a
40 tree perspective. That is, it remained unclear why "aScSe" was
44 We'll burn through that conceptual fog today. The natural thing to
45 try would have been to defunctionalize the continuation-based solution
46 using a tree zipper. But that would not have been easy, since the
47 natural way to implement the doubling behavior of the shifty operator
48 would have been to simply copy the context provided by the zipper.
49 This would have produced two uncoordinated copies of the other shifty
50 operator, and we'd have been in the situation described in class of
51 having a reduction strategy that never reduced the number of shifty
52 operators below 2. (There are ways around this limitation of tree zippers,
53 but they are essentially equivalent to the technique given just below.)
55 Instead, we'll re-interpreting what the continuation monad was doing
56 in more or less defunctionalized terms by using Quantifier Raising, a technique
59 But first, motivating quantifier scope as a linguistic application.
61 # The primary application of continuations to natural language: scope-taking
63 We have seen that continuations allow a deeply-embedded element to
64 take control over (a portion of) the entire computation that contains
65 it. In natural language semantics, this is exactly what it means for
66 a scope-taking expression to take scope.
68 1. [Ann put a copy of [everyone]'s homeworks in her briefcase]
70 2. For every x, [Ann put a copy of x's homeworks in her briefcase]
72 The sentence in (1) can be paraphrased as in (2), in which the
73 quantificational DP *everyone* takes scope over the rest of the sentence.
74 Even if you suspect that there could be an analysis of (2) on which
75 "every student's term paper" could denote some kind of mereological
76 fusion of a set of papers, it is much more difficult to be satisfied
77 with a referential analysis when *every student* is replaced with
78 *no student*, or *fewer than three students*, and so on---see any
79 semantics text book for abundant discussion.
81 We can arrive at an analysis by expressing the meaning of
82 quantificational DP such as *everyone* using continuations:
84 3. everyone = shift (\k.∀x.kx)
86 Assuming there is an implicit reset at the top of the sentence (we'll
87 explicitly address determining where there is or isn't a reset), the
88 reduction rules for `shift` will apply the handler function (\k.∀x.kx)
89 to the remainder of the sentence after abstracting over the position
90 of the shift expression:
92 [Ann put a copy of [shift (\k.∀x.kx)]'s homeworks in her briefcase]
93 ~~> (\k.∀x.kx) (\v. Ann put a copy of v's homeworks in her briefcase)
94 ~~> ∀x. Ann put a copy of x's homeworks in her briefcase
96 (To be a bit pedantic, this reduction sequence is more suitable for
97 shift0 than for shift, but we're not being fussy here about subflavors
100 The standard technique for handling scope-taking in linguistics is
101 Quantifier Raising (QR). As you might suppose, the rule for Quantifier
102 Raising closely resembles the reduction rule for shift:
104 Quantifier Raising: given a sentence [... [QDP] ...], build a new
105 sentence [QDP (\x.[... [x] ...])].
107 Just to emphasize the similarity between QR and shift, we can use QR
108 to provide insight into the tree task that mystified us earlier.
110 \tree (. (a)((S)((d)((S)(e)))))
123 First we QR the lower shift operator
125 \tree (. (S) ((\\x) ((a)((S)((d)((x)(e)))))))
142 Next, we QR the upper shift operator
144 \tree (. (S) ((\\y) ((S) ((\\x) ((a)((y)((d)((x)(e)))))))))
165 We then evaluate, using the same value for the shift operator proposed before:
169 It will be easiest to begin evaluating this tree with the lower shift
170 operator (we get the same result if we start with the upper one).
171 The relevant value for k is (\x.a(y(d(x e)))). Then k "" is
172 a(y(d(""(e)))), and k(k "") is a(y(d((a(y(d(""(e)))))(e)))). In tree
175 \tree (. (S) ((\\y) ((a)((y)((d)(((a)((y)((d)(("")(e)))))(e)))))))
201 Repeating the process for the upper shift operator replaces each
202 occurrence of y with a copy of the whole tree.
204 \tree (. ((a)((((a)(("")((d)(((a)(("")((d)(("")(e)))))(e))))))((d)(((a)((((a)(("")((d)(((a)(("")((d)(("")(e)))))(e))))))((d)(("")(e)))))(e))))))
210 a _________|__________
221 ___|___ e ___|___ | |
223 a ___|___ a ___|____ | |
239 The yield of this tree (the sequence of leaf nodes) is aadadeedaadadeedee.
241 Exercise: the result is different, by the way, if the QR occurs in a
246 * Generalizing from one-sided, list-based continuation
247 operators to two-sided, tree-based continuation operators is a
248 dramatic increase in power and complexity.
251 compose multiple copies of a context can be hard to understand.
253 * When considering two-sided, tree-based continuation operators,
254 quantifier raising is a good tool for visualizing (defunctionalizing)
259 At this point, we have three ways of representing computations
260 involving control operators such as shift and reset: using a CPS
261 transform, lifting into a continuation monad, and by using QR.
263 QR is the traditional system in linguistics, but it will not be
264 adequate for us in general. The reason has to do with order. As
265 we've discussed, especially with respect to the CPS transform,
266 continuations allow fine-grained control over the order of evaluation.
267 One of the main empirical claims of Barker and Shan 2014 is that
268 natural language is sensitive to evaluation order. Unlike other
269 presentations of continuations, QR does not lend itself to reasoning
270 about evaluation order, so we will need to use a different strategy.
272 [Note to self: it is interesting to consider what it would take to
273 reproduce the analyses giving in Barker and Shan in purely QR terms.
274 Simple quantificational binding using parasitic scope should be easy,
275 but how reconstruction would work is not so clear.]
277 We'll present tower notation, then comment and motivate several of its
278 features as we consider various applications. For now, we'll motivate
279 the tower notation by thinking about box types. In the discussion of
280 monads, we've thought of monadic types as values inside of a box. The
281 box will often contain information in addition to the core object.
282 For instance, in the Reader monad, a boxed int contains an expression
283 of type int as the payload, but also contains a function that
284 manipulates a list of information. It is natural to imagine
285 separating a box into two regions, the payload and the hidden scratch
288 _______________ _______________ _______________
289 | [x->2, y->3] | | [x->2, y->3] | | [x->2, y->3] |
290 ------------------- ------------------ ------------------
293 |______________| |______________| |______________|
296 (Imagine the + operation has been lifted into the Reader monad too.)
298 For people who are familiar with Discourse Representation Theory (Kamp
299 1981, Kamp and Reyle 1993), this separation of boxes into payload and
300 discourse scorekeeping will be familiar (although many details differ).
302 The general pattern is that monadic treatments separate computation
303 into an at-issue (pre-monadic) computation with a layer at which
306 The tower notation is a precise way of articulating continuation-based
307 computations into a payload and (potentially multiple) layers of side-effects.
308 We won't keep the outer box, but we will keep the horizontal line
309 dividing main effects from side-effects.
311 Tower convention for types:
313 (α -> β) -> γ can be equivalently written -----
316 Tower convention for values:
318 \k.g[k(x)] can be equivalently written ---
321 If \k.g[k(x)] has type (α -> β) -> γ, then k has type (α -> β).
323 Here "g[ ]" is a *context*, that is, an expression with (exactly) one
324 hole in it. For instance, we might have g[x] = \forall x.P[x].
326 We'll use a simply-typed system with two atomic types, DP (the type of
327 individuals) and S (the type of truth values).
329 Then in the spirit of monadic thinking, we'll have a way of lifting an
330 arbitrary value into the tower system:
333 LIFT (x:α) = \k.kx : (α -> β) -> γ == --- : ---
336 Obviously, LIFT is exactly the midentity (the unit) for the continuation monad.
337 The name comes from Partee's 1987 theory of type-shifters for
338 determiner phrases. Importantly, LIFT applied to an
339 individual-denoting expression yields the generalized quantifier
340 proposed by Montague as the denotation for proper names:
343 LIFT (j:DP) = \k.kx : (DP -> S) -> S == -- : ---
346 So if the proper name *John* denotes the individual j, LIFT(j) is the
347 generalized quantifier that maps each property k of type DP -> S to true
348 just in case kj is true.
350 Once we have expressions of type (α -> β) -> γ, we'll need to combine
351 them. We'll use the ¢ operator from the continuation monad:
353 g[] γ | δ h[] δ | ρ g[h[]] γ | ρ
354 --- : ------- ¢ --- : ----- == ------ : -----
357 Note that the types below the horizontal line combine just like
358 functional application (i.e, f:(α->β) (x:α) = fx:β).
360 To demonstrate that this is indeed the continuation monad's ¢
363 ¢ (\k.g[kf]) (\k.h[kx])
364 = (\MNk.M(\m.N(\n.k(mn)))) (\k.g[kf]) (\k.h[kx])
365 ~~> \k.(\k.g[kf])(\m.(\k.h[kx])(\n.k(mn))
366 ~~> \k.g[(\k.h[kx])(\n.k(fn))
373 Not a monad (Wadler); would be if the types were
374 Neverthless, obeys the monad laws.
376 This is (almost) all we need to get some significant linguistic work