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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 focussed 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 focussed 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
54 Instead, we'll re-interpreting what the continuation monad was doing
55 in defunctionalized terms by using Quantifier Raising (a technique
58 But first, motivating quantifier scope as a linguistic application.
60 # The primary application of continuations to natural language: scope-taking
62 We have seen that continuations allow a deeply-embedded element to
63 take control over (a portion of) the entire computation that contains
64 it. In natural language semantics, this is exactly what it means for
65 a scope-taking expression to take scope.
67 1. [Ann put a copy of [everyone]'s homeworks in her briefcase]
69 2. For every x, [Ann put a copy of x's homeworks in her briefcase]
71 The sentence in (1) can be paraphrased as in (2), in which the
72 quantificational DP *every student* takes scope over the rest of the sentence.
73 Even if you suspect that there could be an analysis of (2) on which
74 "every student's term paper" could denote some kind of mereological
75 fusion of a set of papers, it is much more difficult to be satisfied
76 with a referential analysis when *every student* is replaced with
77 *no student*, or *fewer than three students*, and so on---see any
78 semantics text book for abundant discussion.
80 We can arrive at an analysis by expressing the meaning of
81 quantificational DP such as *everyone* using continuations:
83 3. everyone = shift (\k.∀x.kx)
85 Assuming there is an implicit reset at the top of the sentence (we'll
86 explicitly address determining where there is or isn't a reset), the
87 reduction rules for `shift` will apply the handler function (\k.∀x.kx)
88 to the remainder of the sentence after abstracting over the position
89 of the shift expression:
91 [Ann put a copy of [shift (\k.∀x.kx)]'s homeworks in her briefcase]
92 ~~> (\k.∀x.kx) (\v. Ann put a copy of v's homeworks in her briefcase)
93 ~~> ∀x. Ann put a copy of x's homeworks in her briefcase
95 (To be a bit pedantic, this reduction sequence is more suitable for
96 shift0 than for shift, but we're not being fussy here about subflavors
99 The standard technique for handling scope-taking in linguistics is
100 Quantifier Raising (QR). As you might suppose, the rule for Quantifier
101 Raising closely resembles the reduction rule for shift:
103 Quantifier Raising: given a sentence [... [QDP] ...], build a new
104 sentence [QDP (\x.[... [x] ...])].
106 Just to emphasize the similarity between QR and shift, we can use QR
107 to provide insight into the tree task that mystified us earlier.
109 \tree (. (a)((S)((d)((S)(e)))))
122 First we QR the lower shift operator
124 \tree (. (S) ((\\x) ((a)((S)((d)((x)(e)))))))
141 Next, we QR the upper shift operator
143 \tree (. (S) ((\\y) ((S) ((\\x) ((a)((y)((d)((x)(e)))))))))
164 We then evaluate, using the same value for the shift operator proposed before:
168 It will be easiest to begin evaluating this tree with the lower shift
169 operator (we get the same result if we start with the upper one).
170 The relevant value for k is (\x.a(y(d(x e)))). Then k "" is
171 a(y(d(""(e)))), and k(k "") is a(y(d((a(y(d(""(e)))))(e)))). In tree
174 \tree (. (S) ((\\y) ((a)((y)((d)(((a)((y)((d)(("")(e)))))(e)))))))
200 Repeating the process for the upper shift operator replaces each
201 occurrence of y with a copy of the whole tree.
203 \tree (. ((a)((((a)(("")((d)(((a)(("")((d)(("")(e)))))(e))))))((d)(((a)((((a)(("")((d)(((a)(("")((d)(("")(e)))))(e))))))((d)(("")(e)))))(e))))))
209 a _________|__________
220 ___|___ e ___|___ | |
222 a ___|___ a ___|____ | |
238 The yield of this tree (the sequence of leaf nodes) is aadadeedaadadeedee.
240 Exercise: the result is different, by the way, if the QR occurs in a
245 * Generalizing from one-sided, list-based continuation
246 operators to two-sided, tree-based continuation operators is a
247 dramatic increase in power and complexity.
250 compose multiple copies of a context can be hard to understand.
252 * When considering two-sided, tree-based continuation operators,
253 quantifier raising is a good tool for visualizing (defunctionalizing)
258 At this point, we have three ways of representing computations
259 involving control operators such as shift and reset: using a CPS
260 transform, lifting into a continuation monad, and by using QR.
262 QR is the traditional system in linguistics, but it will not be
263 adequate for us in general. The reason has to do with order. As
264 we've discussed, especially with respect to the CPS transform,
265 continuations allow fine-grained control over the order of evaluation.
266 One of the main empirical claims of Barker and Shan 2014 is that
267 natural language is sensitive to evaluation order. Unlike other
268 presentations of continuations, QR does not lend itself to reasoning
269 about evaluation order, so we will need to use a different strategy.
271 [Note to self: it is interesting to consider what it would take to
272 reproduce the analyses giving in Barker and Shan in purely QR terms.
273 Simple quantificational binding using parasitic scope should be easy,
274 but how reconstruction would work is not so clear.]
276 We'll present tower notation, then comment and motivate several of its
277 features as we consider various applications. For now, we'll motivate
278 the tower notation by thinking about box types. In the discussion of
279 monads, we've thought of monadic types as values inside of a box. The
280 box will often contain information in addition to the core object.
281 For instance, in the Reader monad, a boxed int contains an expression
282 of type int as the payload, but also contains a function that
283 manipulates a list of information. It is natural to imagine
284 separating a box into two regions, the payload and the hidden scratch
287 _______________ _______________ _______________
288 | [x->2, y->3] | | [x->2, y->3] | | [x->2, y->3] |
289 ------------------- ------------------ ------------------
292 |______________| |______________| |______________|
295 (Imagine the + operation has been lifted into the Reader monad too.)
297 For people who are familiar with Discourse Representation Theory (Kamp
298 1981, Kamp and Reyle 1993), this separation of boxes into payload and
299 discourse scorekeeping will be familiar (although many details differ).
301 The general pattern is that monadic treatments separate computation
302 into an at-issue (pre-monadic) computation with a layer at which
305 The tower notation is a precise way of articulating continuation-based
306 computations into a payload and (potentially multiple) layers of side-effects.
307 We won't keep the outer box, but we will keep the horizontal line
308 dividing main effects from side-effects.
310 Tower convention for types:
312 (α -> β) -> γ can be equivalently written -----
315 Tower convention for values:
317 \k.g[k(x)] can be equivalently written ---
320 If \k.g[k(x)] has type (α -> β) -> γ, then k has type (α -> β).
322 Here "g[ ]" is a *context*, that is, an expression with (exactly) one
323 hole in it. For instance, we might have g[x] = \forall x.P[x].
325 We'll use a simply-typed system with two atomic types, DP (the type of
326 individuals) and S (the type of truth values).
328 Then in the spirit of monadic thinking, we'll have a way of lifting an
329 arbitrary value into the tower system:
332 LIFT (x:α) = \k.kx : (α -> β) -> γ == --- : ---
335 Obviously, LIFT is exactly the midentity (the unit) for the continuation monad.
336 The name comes from Partee's 1987 theory of type-shifters for
337 determiner phrases. Importantly, LIFT applied to an
338 individual-denoting expression yields the generalized quantifier
339 proposed by Montague as the denotation for proper names:
342 LIFT (j:DP) = \k.kx : (DP -> S) -> S == -- : ---
345 So if the proper name *John* denotes the individual j, LIFT(j) is the
346 generalized quantifier that maps each property k of type DP -> S to true
347 just in case kj is true.
349 Once we have expressions of type (α -> β) -> γ, we'll need to combine
350 them. We'll use the ¢ operator from the continuation monad:
352 g[] γ | δ h[] δ | ρ g[h[]] γ | ρ
353 --- : ------- ¢ --- : ----- == ------ : -----
356 Note that the types below the horizontal line combine just like
357 functional application (i.e, f:(α->β) (x:α) = fx:β).
359 To demonstrate that this is indeed the continuation monad's ¢
362 ¢ (\k.g[kf]) (\k.h[kx])
363 = (\MNk.M(\m.N(\n.k(mn)))) (\k.g[kf]) (\k.h[kx])
364 ~~> \k.(\k.g[kf])(\m.(\k.h[kx])(\n.k(mn))
365 ~~> \k.g[(\k.h[kx])(\n.k(fn))
372 Not a monad (Wadler); would be if the types were
373 Neverthless, obeys the monad laws.
375 This is (almost) all we need to get some significant linguistic work