```+ ta            tb          tc
+ .             .           .
+_|__          _|__        _|__
+|  |          |  |        |  |
+1  .          .  3        1  .
+  _|__       _|__           _|__
+  |  |       |  |           |  |
+  2  3       1  2           3  2
+
+let ta = Node (Leaf 1, Node (Leaf 2, Leaf 3));;
+let tb = Node (Node (Leaf 1, Leaf 2), Leaf 3);;
+let tc = Node (Leaf 1, Node (Leaf 3, Leaf 2));;
+```
+ +So `ta` and `tb` are different trees that have the same fringe, but +`ta` and `tc` are not. + +The simplest solution is to map each tree to a list of its leaves, +then compare the lists. But because we will have computed the entire +fringe before starting the comparison, if the fringes differ in an +early position, we've wasted our time examining the rest of the trees. The second solution was to use tree zippers and mutable state to -simulate coroutines. We would unzip the first tree until we found the -next leaf, then store the zipper structure in the mutable variable -while we turned our attention to the other tree. Because we stop as -soon as we find the first mismatched leaf, this solution does not have -the flaw just mentioned of the solution that maps both trees to a list -of leaves before beginning comparison. +simulate coroutines (see [[coroutines and aborts]]). In that +solution, we pulled the zipper on the first tree until we found the +next leaf, then stored the zipper structure in the mutable variable +while we turned our attention to the other tree. Because we stopped +as soon as we find the first mismatched leaf, this solution does not +have the flaw just mentioned of the solution that maps both trees to a +list of leaves before beginning comparison. Since zippers are just continuations reified, we expect that the solution in terms of zippers can be reworked using continuations, and -this is indeed the case. To make this work in the most convenient -way, we need to use the fully general type for continuations just mentioned. - -tree_monadize (fun a k -> a, k a) t1 (fun t -> 0);; +this is indeed the case. Before we can arrive at a solution, however, +we must define a data structure called a stream: + + type 'a stream = End | Next of 'a * (unit -> 'a stream);; + +A stream is like a list in that it contains a series of objects (all +of the same type, here, type `'a`). The first object in the stream +corresponds to the head of a list, which we pair with a stream +representing the rest of a the list. There is a special stream called +`End` that represents a stream that contains no (more) elements, +analogous to the empty list `[]`. + +Actually, we pair each element not with a stream, but with a thunked +stream, that is, a function from the unit type to streams. The idea +is that the next element in the stream is not computed until we forced +the thunk by applying it to the unit: + +
```+# let rec make_int_stream i = Next (i, fun () -> make_int_stream (i + 1));;
+val make_int_stream : int -> int stream =
+# let int_stream = make_int_stream 1;;
+val int_stream : int stream = Next (1, )         (* First element: 1 *)
+# match int_stream with Next (i, rest) -> rest;;
+- : unit -> int stream =                         (* Rest: a thunk *)
+
+(* Force the thunk to compute the second element *)
+# (match int_stream with Next (i, rest) -> rest) ();;
+- : int stream = Next (2, )
+```
+ +You can think of `int_stream` as a functional object that provides +access to an infinite sequence of integers, one at a time. It's as if +we had written `[1;2;...]` where `...` meant "continue indefinitely". + +So, with streams in hand, we need only rewrite our continuation tree +monadizer so that instead of mapping trees to lists, it maps them to +streams. Instead of + + # tree_monadize (fun a k -> a :: k a) t1 (fun t -> []);; + - : int list = [2; 3; 5; 7; 11] +as above, we have + + # tree_monadize (fun i k -> Next (i, fun () -> k ())) t1 (fun _ -> End);; + - : int stream = Next (2, ) + +We can see the first element in the stream, the first leaf (namely, +2), but in order to see the next, we'll have to force a thunk. + +Then to complete the same-fringe function, we simply convert both +trees into leaf-streams, then compare the streams element by element. +The code is enitrely routine, but for the sake of completeness, here it is: + +
```+let rec compare_streams stream1 stream2 =
+    match stream1, stream2 with
+    | End, End -> true (* Done!  Fringes match. *)
+    | Next (next1, rest1), Next (next2, rest2) when next1 = next2 -> compare_streams (rest1 ()) (rest2 ())
+    | _ -> false;;
+
+let same_fringe t1 t2 =
+  let stream1 = tree_monadize (fun i k -> Next (i, fun () -> k ())) t1 (fun _ -> End) in
+  let stream2 = tree_monadize (fun i k -> Next (i, fun () -> k ())) t2 (fun _ -> End) in
+  compare_streams stream1 stream2;;
+```
+ +Notice the forcing of the thunks in the recursive call to +`compare_streams`. So indeed: + +
```+# same_fringe ta tb;;
+- : bool = true
+# same_fringe ta tc;;
+- : bool = false
+```
+ +Now, this implementation is a bit silly, since in order to convert the +trees to leaf streams, our tree_monadizer function has to visit every +node in the tree. But if we needed to compare each tree to a large +set of other trees, we could arrange to monadize each tree only once, +and then run compare_streams on the monadized trees. + +By the way, what if you have reason to believe that the fringes of +your trees are more likely to differ near the right edge than the left +edge? If we reverse evaluation order in the tree_monadizer function, +as shown above when we replaced leaves with their ordinal position, +then the resulting streams would produce leaves from the right to the +left. + +The idea of using continuations to characterize natural language meaning +------------------------------------------------------------------------ + +We might a philosopher or a linguist be interested in continuations, +especially if efficiency of computation is usually not an issue? +Well, the application of continuations to the same-fringe problem +shows that continuations can manage order of evaluation in a +well-controlled manner. In a series of papers, one of us (Barker) and +Ken Shan have argued that a number of phenomena in natural langauge +semantics are sensitive to the order of evaluation. We can't +reproduce all of the intricate arguments here, but we can give a sense +of how the analyses use continuations to achieve an analysis of +natural language meaning. + +**Quantification and default quantifier scope construal**. + +We saw in the copy-string example and in the same-fringe example that +local properties of a tree (whether a character is `S` or not, which +integer occurs at some leaf position) can control global properties of +the computation (whether the preceeding string is copied or not, +whether the computation halts or proceeds). Local control of +surrounding context is a reasonable description of in-situ +quantification. + + (1) John saw everyone yesterday. + +This sentence means (roughly) + + &Forall; x . yesterday(saw x) john + +That is, the quantifier *everyone* contributes a variable in the +direct object position, and a universal quantifier that takes scope +over the whole sentence. If we have a lexical meaning function like +the following: + +
```+let lex (s:string) k = match s with
+  | "everyone" -> Node (Leaf "forall x", k "x")
+  | "someone" -> Node (Leaf "exists y", k "y")
+  | _ -> k s;;
+
+let sentence1 = Node (Leaf "John",
+                      Node (Node (Leaf "saw",
+                                  Leaf "everyone"),
+                            Leaf "yesterday"));;
+```
+ +Then we can crudely approximate quantification as follows: + +
```+# tree_monadize lex sentence1 (fun x -> x);;
+- : string tree =
+Node
+ (Leaf "forall x",
+  Node (Leaf "John", Node (Node (Leaf "saw", Leaf "x"), Leaf "yesterday")))
+```
+ +In order to see the effects of evaluation order, +observe what happens when we combine two quantifiers in the same +sentence: + +
```+# let sentence2 = Node (Leaf "everyone", Node (Leaf "saw", Leaf "someone"));;
+# tree_monadize lex sentence2 (fun x -> x);;
+- : string tree =
+Node
+ (Leaf "forall x",
+  Node (Leaf "exists y", Node (Leaf "x", Node (Leaf "saw", Leaf "y"))))
+```
+ +The universal takes scope over the existential. If, however, we +replace the usual tree_monadizer with tree_monadizer_rev, we get +inverse scope: + +
```+# tree_monadize_rev lex sentence2 (fun x -> x);;
+- : string tree =
+Node
+ (Leaf "exists y",
+  Node (Leaf "forall x", Node (Leaf "x", Node (Leaf "saw", Leaf "y"))))
+```
+ +There are many crucially important details about quantification that +are being simplified here, and the continuation treatment here is not +scalable for a number of reasons. Nevertheless, it will serve to give +an idea of how continuations can provide insight into the behavior of +quantifiers. The Binary Tree monad -- 2.11.0