a functional programming technique, then in terms of list and tree
zippers, then as a monad. In this lecture, we will generalize
continuations slightly beyond a monad, and then begin to outline some
-of the applications of monads. In brief, the generalization can be
-summarized in terms of types: instead of using a Kleisli arrow mapping
-a type α to a continuized type (α -> ρ) -> ρ, we'll allow the result
-types to differ, i.e., we'll map α to (α -> β) -> γ. This will be
-crucial for some natural language applications.
+of the applications of the generalized continuations.
Many (though not all) of the applications are discussed in detail in
Barker and Shan 2014, *Continuations in Natural Language*, OUP.
-In terms of list zippers, the continuation of a focused element in
-the list is the front part of the list.
+To review, in terms of list zippers, the continuation of a focused
+element in the list is the front part of the list.
list zipper for the list [a;b;c;d;e;f] with focus on d:
In terms of tree zippers, the continuation is the entire context of
the focused element--the entire rest of the tree.
-[drawing of a broken tree]
+[drawing of a tree zipper]
-Last week we had trouble computing the doubling task when there was more
-than one shifty operator after moving from a list perspective to a
-tree perspective. That is, it remained unclear why "aScSe" was
+We explored continuations first in a list setting, then in a tree
+setting, using the doubling task as an example.
- "aacaceecaacaceecee"
+ "abSd" ~~> "ababd"
+ "ab#deSfg" ~~> "abdedefg"
-We'll burn through that conceptual fog today. The natural thing to
-try would have been to defunctionalize the continuation-based solution
-using a tree zipper. But that would not have been easy, since the
-natural way to implement the doubling behavior of the shifty operator
-would have been to simply copy the context provided by the zipper.
-This would have produced two uncoordinated copies of the other shifty
-operator, and we'd have been in the situation described in class of
-having a reduction strategy that never reduced the number of shifty
-operators below 2. (There are ways around this limitation of tree zippers,
-but they are essentially equivalent to the technique given just below.)
+The "S" functions like a shifty operator, and "#" functions like a reset.
+
+Although the list version of the doubling task was easy to understand
+thoroughly, the tree version was significantly more challenging. In
+particular, it remained unclear why
+
+ "aScSe" ~~> "aacaceecaacaceecee"
+
+We'll burn through that conceptual fog today by learning more about
+how to work with continuations.
+
+The natural thing to try would have been to defunctionalize the
+continuation-based solution using a tree zipper. But that would not
+have been easy, since the natural way to implement the doubling
+behavior of the shifty operator would have been to simply copy the
+context provided by the zipper. This would have produced two
+uncoordinated copies of the other shifty operator, and we'd have been
+in the situation described in class of having a reduction strategy
+that never reduced the number of shifty operators below 2. The
+limitation is that zippers by themselves don't provide a natural way
+to establish a dependency between two distant elements of a data
+structure. (There are ways around this limitation of tree zippers,
+but they are essentially equivalent to the technique given just
+below.)
Instead, we'll re-interpreting what the continuation monad was doing
-in more or less defunctionalized terms by using Quantifier Raising, a technique
-from linguistics.
+in more or less defunctionalized terms by using Quantifier Raising, a
+technique from linguistics.
But first, motivating quantifier scope as a linguistic application.
Quantifier Raising (QR). As you might suppose, the rule for Quantifier
Raising closely resembles the reduction rule for shift:
- Quantifier Raising: given a sentence [... [QDP] ...], build a new
- sentence [QDP (\x.[... [x] ...])].
+ Quantifier Raising: given a sentence of the form
+
+ [... [QDP] ...],
+
+ build a new sentence of the form
+
+ [QDP (\x.[... [x] ...])].
Here, QDP is a scope-taking quantificational DP.
Just to emphasize the similarity between QR and shift, we can use QR
-to provide insight into the tree task that mystified us earlier.
+to provide insight into the tree version of the doubling task that
+mystified us earlier. Here's the starting point:
<!--
\tree (. (a)((S)((d)((S)(e)))))
* Generalizing from one-sided, list-based continuation
operators to two-sided, tree-based continuation operators is a
- dramatic increase in power and complexity.
+ dramatic increase in power and complexity. (De Groote's dynamic
+ montague semantics continuations are the one-sided, list-based variety.)
* Operators that
compose multiple copies of a context can be hard to understand
transform, lifting into a continuation monad, and by using QR.
QR is the traditional system in linguistics, but it will not be
-adequate for us in general. The reason has to do with order. As
-we've discussed, especially with respect to the CPS transform,
-continuations allow fine-grained control over the order of evaluation.
-One of the main empirical claims of Barker and Shan 2014 is that
-natural language is sensitive to evaluation order. Unlike other
-presentations of continuations, QR does not lend itself to reasoning
-about evaluation order, so we will need to use a different strategy.
+adequate for us in general. The reason has to do with evaluation
+order. As we've discussed, especially with respect to the CPS
+transform, continuations allow fine-grained control over the order of
+evaluation. One of the main empirical claims of Barker and Shan 2014
+is that natural language is sensitive to evaluation order. Unlike
+other presentations of continuations, QR does not lend itself to
+reasoning about evaluation order, so we will need to use a different
+strategy.
[Note to self: it is interesting to consider what it would take to
reproduce the analyses giving in Barker and Shan in purely QR terms.
Simple quantificational binding using parasitic scope should be easy,
but how reconstruction would work is not so clear.]
+## Introducting the tower notation
+
We'll present tower notation, then comment and motivate several of its
features as we consider various applications. For now, we'll motivate
the tower notation by thinking about box types. In the discussion of
space:
<pre>
- _______________ _______________ _______________
+ _______________ _______________ _______________
| [x->2, y->3] | | [x->2, y->3] | | [x->2, y->3] |
------------------- ------------------ ------------------
| | ¢ | | = | |
- | +2 | | y | | 5 |
- |______________| |______________| |______________|
+ | +2 | | y | | 5 |
+ |______________| |______________| |______________|
</pre>
For people who are familiar with Discourse Representation Theory (Kamp
side-effects occur.
The tower notation is a precise way of articulating continuation-based
-computations into a payload and (potentially multiple) layers of side-effects.
-We won't keep the outer box, but we will keep the horizontal line
-dividing main effects from side-effects.
+computations into a payload and (potentially multiple) layers of
+side-effects. Visually, we won't keep the outer box, but we will keep
+the horizontal line dividing main effects from side-effects.
Tower convention for types:
<pre>
α
</pre>
+Read these types counter-clockwise starting at the bottom.
+
Tower convention for values:
<pre>
g[]
We'll use a simply-typed system with two atomic types, DP (the type of
individuals) and S (the type of truth values).
+## LIFT
+
Then in the spirit of monadic thinking, we'll have a way of lifting an
arbitrary value into the tower system:
proposed by Montague as the denotation for proper names:
[] S|S
- LIFT (j:DP) = \k.kx : (DP -> S) -> S == -- : ---
+ LIFT (j:DP) = \k.kj : (DP -> S) -> S == -- : ---
j DP
So if the proper name *John* denotes the individual j, LIFT(j) is the
in Montague and Partee, but to any expression whatsoever. For
instance, here is LIFT applied to a lexical verb phrase:
- [] S|S
- LIFT (left:DP\S) = \k.kx : (DP\S -> S) -> S == ---- : ---
- left DP
+ [] S|S
+ LIFT (left:DP->S) = \k.kx : ((DP->S) -> S) -> S == ---- : ---
+ left DP
Once we have expressions of type (α -> β) -> γ, we'll need to combine
them. We'll use the ¢ operator from the continuation monad:
Note that the types below the horizontal line combine just like
functional application (i.e, f:(α->β) (x:α) = fx:β).
-To demonstrate that this is indeed the continuation monad's ¢
-operator:
+## Not quite a monad
+
+This discussion is based on Wadler's paper `Composable continuations'.
+
+The unit and the combination rule work very much like we are used to
+from the various monads we've studied.
+
+In particular, we can easily see that the ¢ operator defined just
+above is exactly the same as the ¢ operator from the continuation monad:
¢ (\k.g[kf]) (\k.h[kx])
= (\MNk.M(\m.N(\n.k(mn)))) (\k.g[kf]) (\k.h[kx])
== ------
fx
-Not a monad (Wadler); would be if the types were
-Neverthless, obeys the monad laws.
+However, these continuations do not form an official monad.
+One way to see this is to consider the type of the bind operator.
+The type of the bind operator in a genuine monad is
+
+ mbind : Mα -> (α -> Mβ) -> Mβ
+
+In particular, the result type of the second argument (Mβ) must be
+identical to the result type of the bind operator overall. For the
+continuation monad, this means that mbind has the following type:
+
+ ((α -> γ) -> ρ)
+ -> α -> ((β -> δ) -> ρ)
+ -> ((β -> δ) -> ρ)
+
+But the type of the bind operator in our generalized continuation
+system (call it "kbind") is
+
+ kbind :
+ ((α -> γ) -> ρ)
+ -> α -> ((β -> δ) -> γ)
+ -> ((β -> δ) -> ρ)
+
+Note that `(β -> δ) -> γ` is not the same as `(β -> δ) -> ρ`.
+
+These more general types work out fine when plugged into the
+continuation monad's bind operator:
+
+ kbind u f = \k. u (\x. f x k )
+ β->δ (α->γ)->ρ α α->(β->δ)->γ α β->δ
+ -----------------
+ (β->δ)->γ
+ ------------------------
+ γ
+ --------------------
+ α->γ
+ ---------------------
+ ρ
+ ---------------
+ (β->δ)->ρ
+
+Neverthless, it's easy to see that the generalized continuation system
+obeys the monad laws. We haven't spent much time proving monad laws,
+so this seems like a worthy occasion on which to give some details.
+Since we're working with bind here, we'll use the version of the monad
+laws that are expressed in terms of bind:
+
+ Prove u >>= ⇧ == u:
+
+ u >>= (\ak.ka)
+ == (\ufk.u(\x.fxk)) u (\ak.ka)
+ ~~> \k.u(\x.(\ak.ka)xk)
+ ~~> \k.u(\x.kx)
+ ~~> \k.uk
+ ~~> u
+
+The last two steps are eta reductions.
+
+ Prove ⇧a >>= f == f a:
+
+ ⇧a >>= f
+ == (\ak.ka)a >>= f
+ ~~> (\k.ka) >>= f
+ == (\ufk.u(\x.fxk)) (\k.ka) f
+ ~~> \k.(\k.ka)(\x.fxk)
+ ~~> \k.fak
+ ~~> fa
+
+ Prove u >>= (\a.fa >>= g) == (u >>= f) >>= g:
+
+ u >>= (\a.fa >>= g)
+ == u >>= (\a.(\k.fa(\x.gxk)))
+ == (\ufk.u(\x.fxk)) u (\a.(\k.fa(\x.gxk)))
+ == \k.u(\x.(\a.(\k.fa(\x.gxk)))xk)
+ ~~> \k.u(\x.fx(\y.gyk))
+
+ (u >>= f) >>= g
+ == (\ufk.u(\x.fxk)) u f >>= g
+ ~~> (\k.u(\x.fxk)) >>= g
+ == (\ufk.u(\x.fxk)) (\k.u(\x.fxk)) g
+ ~~> \k.(\k.u(\x.fxk))(\y.gyk)
+ ~~> \k.u(\x.fx(\y.gyk))
+
+The fact that the monad laws hold means that we can rely on any
+reasoning that depends on the monad laws.
+
+## Syntactic refinements: subtypes of implication
+
+Because natural language allows the functor to be on the left or on
+the right, we replace the type arrow -> with a left-leaning version \
+and a right-leaning version, as follows:
+
+ α/β β = α
+ β β\α = α
+
+This means (without adding some fancy footwork, as in Charlow 2014) we
+need two versions of ¢ too, one for each direction for the unmonadized types.
+
+Just to be explicit, here are the two versions:
+
+ g[] γ | δ h[] δ | ρ g[h[]] γ | ρ
+ --- : ------- ¢ --- : ----- == ------ : -----
+ f α/β x α fx β
+
+ h[] δ | ρ g[] γ | δ g[h[]] γ | ρ
+ --- : ----- ¢ --- : ------- == ------ : -----
+ x α f β\α fx β
+
+With respect to types, they differ only in replacing α -> β with α/β
+(top version) and with β\α (bottom version). With respect to
+syntactic order, they differ in a parallel way with respect to whether
+the function f is on the left of its argument x (top version) or on
+the right (bottom version).
+
+Logically, separating -> into \\ and / corresponds to rejecting the
+structural rule of exchange.
+
+Note that the \\ and / only govern types at the bottom of the tower.
+That is, we currently still have arrow at the higher-order levels of
+the types, if we undo the tower notation:
+
+ γ|δ
+ --- == ((α/β) -> δ) -> γ
+ α/β
-Oh, one more thing: because natural language allows the functor to be
-on the left or on the right, we replace the type arrow -> with a
-left-leaning version \ and a right-leaning version, as follows:
+We'll change these arrows into left-leaning and right-leaning versions
+too, according to the following scheme:
- α/β β = α
- β β\α = α
+ γ|δ
+ --- == γ//((α/β) \\\\ δ)
+ α/β
-This means we need two versions of ¢ too (see Barker and Shan 2014
-chapter 1 for full details).
+As we'll see in a minute, each of these for implications (\\, /, \\\\,
+//) will have a syntactic interpretation:
-This is (almost) all we need to get some significant linguistic work
-done.
+ \ argument is adjacent on the left of the functor
+ / argument is adjacent on the right of the functor
+ \\ argument is surrounded by the functor
+ // argument surrounds the functor