. __|___ | | a __|___ | | S __|__ | | d _|__ | | S eFirst we QR the lower shift operator, replacing it with a variable and abstracting over that variable.

. ___|___ | | S ___|___ | | \x __|___ | | a __|___ | | S __|__ | | d _|__ | | x eNext, we QR the upper shift operator

. ___|___ | | S ___|____ | | \y ___|___ | | S ___|___ | | \x __|___ | | a __|___ | | y __|__ | | d _|__ | | x eWe then evaluate, using the same value for the shift operator proposed before: S = shift = \k.k(k "") It will be easiest to begin evaluating this tree with the lower shift operator (we get the same result if we start with the upper one). The relevant value for k is (\x.a(y(d(x e)))). Then k "" is a(y(d(""(e)))), and k(k "") is a(y(d((a(y(d(""(e)))))(e)))). In tree form:

. ___|___ | | S ___|____ | | \y ___|___ | | a ___|___ | | y ___|___ | | d ___|___ | | __|___ e | | a __|___ | | y __|___ | | d __|__ | | "" eRepeating the process for the upper shift operator replaces each occurrence of y with a copy of the whole tree.

. | ______|______ | | a _________|__________ | | | ___|___ ___|___ | | | | d ___|____ a ___|____ | | | | ___|____ e "" ___|___ | | | | a ____|_____ d ___|___ | | | | | __|___ ___|___ e ___|___ | | | | | | d __|__ a ___|___ a ___|____ | | | | | | "" e "" __|___ "" ___|___ | | | | d __|__ d ___|___ | | | | "" e ___|___ e | | a ___|___ | | "" __|___ | | d __|__ | | "" eThe yield of this tree (the sequence of leaf nodes) is aadadeedaadadeedee, which is the expected output of the double-shifted tree. Exercise: the result is different, by the way, if the QR occurs in the opposite order. Three lessons: * Generalizing from one-sided, list-based continuation operators to two-sided, tree-based continuation operators is a dramatic increase in power and complexity. * Operators that compose multiple copies of a context can be hard to understand (though keep this in mind when we see the continuations-based analysis of coordination, which involves context doubling). * When considering two-sided, tree-based continuation operators, quantifier raising is a good tool for visualizing (defunctionalizing) the computation. ## Tower notation At this point, we have three ways of representing computations involving control operators such as shift and reset: using a CPS 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. [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.] 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 monads, we've thought of monadic types as values inside of a box. The box will often contain information in addition to the core object. For instance, in the Reader monad, a boxed int contains an expression of type int as the payload, but also contains a function that manipulates a list of information. It is natural to imagine separating a box into two regions, the payload and the hidden scratch space:

_______________ _______________ _______________ | [x->2, y->3] | | [x->2, y->3] | | [x->2, y->3] | ------------------- ------------------ ------------------ | | ¢ | | = | | | +2 | | y | | 5 | |______________| |______________| |______________|For people who are familiar with Discourse Representation Theory (Kamp 1981, Kamp and Reyle 1993), this separation of boxes into payload and discourse scorekeeping will be familiar (although many details differ). The general pattern is that monadic treatments separate computation into an at-issue (pre-monadic) computation with a layer at which 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. Tower convention for types:

γ | β (α -> β) -> γ can be equivalently written ----- αTower convention for values:

g[] \k.g[k(x)] can be equivalently written --- xIf \k.g[k(x)] has type (α -> β) -> γ, then k has type (α -> β). Here "g[ ]" is a *context*, that is, an expression with (exactly) one hole in it. For instance, we might have g[x] = \forall x.P[x]. We'll use a simply-typed system with two atomic types, DP (the type of individuals) and S (the type of truth values). Then in the spirit of monadic thinking, we'll have a way of lifting an arbitrary value into the tower system: [] β|β LIFT (x:α) = \k.kx : (α -> β) -> β == -- : --- x α Obviously, LIFT is exactly the midentity (the unit) for the continuation monad. Notice that LIFT requires the result type of the continuation argument and the result type of the overall expression to match (here, both are β). The name LIFT comes from Partee's 1987 theory of type-shifters for determiner phrases. Importantly, LIFT applied to an individual-denoting expression yields the generalized quantifier proposed by Montague as the denotation for proper names: [] S|S LIFT (j:DP) = \k.kx : (DP -> S) -> S == -- : --- j DP So if the proper name *John* denotes the individual j, LIFT(j) is the generalized quantifier that maps each property k of type DP -> S to true just in case kj is true. Crucially for the discussion here, LIFT does not apply only to DPs, as 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 Once we have expressions of type (α -> β) -> γ, we'll need to combine them. We'll use the ¢ operator from the continuation monad: g[] γ | δ h[] δ | ρ g[h[]] γ | ρ --- : ------- ¢ --- : ----- == ------ : ----- f α -> β x α fx β 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: ¢ (\k.g[kf]) (\k.h[kx]) = (\MNk.M(\m.N(\n.k(mn)))) (\k.g[kf]) (\k.h[kx]) ~~> \k.(\k.g[kf])(\m.(\k.h[kx])(\n.k(mn)) ~~> \k.g[(\k.h[kx])(\n.k(fn)) ~~> \k.g[h[k(fx)]] g[h[]] == ------ fx Not a monad (Wadler); would be if the types were Neverthless, obeys the monad laws. This is (almost) all we need to get some significant linguistic work done.