X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week15_continuation_applications.mdwn;h=cc5c9e2918ea22fa4153e612b0130f1a85aa6b7c;hp=ec0cb5d60183ff0f06635a1bd3351794607a2b5e;hb=88b8a69e01f705bf5000ce75cd3f70caea1a929d;hpb=3763823a34161836b06f956662ad4e42372b4739 diff --git a/topics/_week15_continuation_applications.mdwn b/topics/_week15_continuation_applications.mdwn index ec0cb5d6..cc5c9e29 100644 --- a/topics/_week15_continuation_applications.mdwn +++ b/topics/_week15_continuation_applications.mdwn @@ -104,11 +104,14 @@ Raising closely resembles the reduction rule for shift: Quantifier Raising: given a sentence [... [QDP] ...], build a new sentence [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. \tree (. (a)((S)((d)((S)(e))))) +
. __|___ | | @@ -119,11 +122,13 @@ a __|___ d _|__ | | S e +First we QR the lower shift operator \tree (. (S) ((\\x) ((a)((S)((d)((x)(e))))))) +
. ___|___ | | @@ -138,11 +143,13 @@ S ___|___ d _|__ | | x e +Next, we QR the upper shift operator \tree (. (S) ((\\y) ((S) ((\\x) ((a)((y)((d)((x)(e))))))))) +
. ___|___ | | @@ -161,6 +168,7 @@ S ___|____ d _|__ | | x e +We then evaluate, using the same value for the shift operator proposed before: @@ -174,6 +182,7 @@ form: \tree (. (S) ((\\y) ((a)((y)((d)(((a)((y)((d)(("")(e)))))(e))))))) +
. ___|___ | | @@ -196,6 +205,7 @@ S ___|____ d __|__ | | "" e +Repeating the process for the upper shift operator replaces each @@ -203,6 +213,7 @@ occurrence of y with a copy of the whole tree. \tree (. ((a)((((a)(("")((d)(((a)(("")((d)(("")(e)))))(e))))))((d)(((a)((((a)(("")((d)(((a)(("")((d)(("")(e)))))(e))))))((d)(("")(e)))))(e)))))) +
. | ______|______ @@ -235,8 +246,10 @@ a ___|____ | | d __|__ | | "" e +-The yield of this tree (the sequence of leaf nodes) is aadadeedaadadeedee. +The 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 a different order. @@ -285,15 +298,14 @@ 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 | |______________| |______________| |______________| - - -(Imagine the + operation has been lifted into the Reader monad too.) +For people who are familiar with Discourse Representation Theory (Kamp 1981, Kamp and Reyle 1993), this separation of boxes into payload and