lists-to-contin tweaks

[lambda.git] / from_lists_to_continuations.mdwn

diff --git a/from_lists_to_continuations.mdwn b/from_lists_to_continuations.mdwn
index 0b3f464..1d0ad42 100644 (file)
--- a/from_lists_to_continuations.mdwn
+++ b/from_lists_to_continuations.mdwn
@@ -1,13 +1,12 @@
-

 Refunctionalizing zippers: from lists to continuations
 ------------------------------------------------------
 
 If zippers are continuations reified (defuntionalized), then one route
 to continuations is to re-functionalize a zipper.  Then the
 concreteness and understandability of the zipper provides a way of
 Refunctionalizing zippers: from lists to continuations
 ------------------------------------------------------
 
 If zippers are continuations reified (defuntionalized), then one route
 to continuations is to re-functionalize a zipper.  Then the
 concreteness and understandability of the zipper provides a way of

-understanding and equivalent treatment using continuations.
+understanding an equivalent treatment using continuations.

 
 

-Let's work with lists of chars for a change.  To maximize readability, we'll
+Let's work with lists of `char`s for a change.  To maximize readability, we'll

 indulge in an abbreviatory convention that "abSd" abbreviates the
 list `['a'; 'b'; 'S'; 'd']`.
 
 indulge in an abbreviatory convention that "abSd" abbreviates the
 list `['a'; 'b'; 'S'; 'd']`.
 


@@ -20,9 +19,7 @@ updated version.
 
 Expected behavior:
 
 
 Expected behavior:
 

-<pre>
-t "abSd" ~~> "ababd"
-</pre>   
+       t "abSd" ~~> "ababd"

 
 
 In linguistic terms, this is a kind of anaphora
 
 
 In linguistic terms, this is a kind of anaphora


@@ -33,39 +30,33 @@ This deceptively simple task gives rise to some mind-bending complexity.
 Note that it matters which 'S' you target first (the position of the *
 indicates the targeted 'S'):
 
 Note that it matters which 'S' you target first (the position of the *
 indicates the targeted 'S'):
 

-<pre>
-    t "aSbS" 
-        *
-~~> t "aabS" 
-          *
-~~> "aabaab"
-</pre>
+           t "aSbS"
+               *
+       ~~> t "aabS"
+                 *
+       ~~> "aabaab"

 
 versus
 
 
 versus
 

-<pre>
-    t "aSbS"
-          *
-~~> t "aSbaSb" 
-        *
-~~> t "aabaSb"
-           *
-~~> "aabaaabab"
-</pre>   
+           t "aSbS"
+                 *
+       ~~> t "aSbaSb"
+               *
+       ~~> t "aabaSb"
+                  *
+       ~~> "aabaaabab"

 
 versus
 
 
 versus
 

-<pre>
-    t "aSbS"
-          *
-~~> t "aSbaSb"
-           *
-~~> t "aSbaaSbab"
-            *
-~~> t "aSbaaaSbaabab"
-             *
-~~> ...
-</pre>
+           t "aSbS"
+                 *
+       ~~> t "aSbaSb"
+                  *
+       ~~> t "aSbaaSbab"
+                   *
+       ~~> t "aSbaaaSbaabab"
+                    *
+       ~~> ...

 
 Aparently, this task, as simple as it is, is a form of computation,
 and the order in which the `'S'`s get evaluated can lead to divergent
 
 Aparently, this task, as simple as it is, is a form of computation,
 and the order in which the `'S'`s get evaluated can lead to divergent


@@ -76,109 +67,105 @@ guarantees termination, and a final string without any `'S'` in it.
 
 This is a task well-suited to using a zipper.  We'll define a function
 `tz` (for task with zippers), which accomplishes the task by mapping a
 
 This is a task well-suited to using a zipper.  We'll define a function
 `tz` (for task with zippers), which accomplishes the task by mapping a

-char list zipper to a char list.  We'll call the two parts of the
+`char list zipper` to a `char list`.  We'll call the two parts of the

 zipper `unzipped` and `zipped`; we start with a fully zipped list, and
 zipper `unzipped` and `zipped`; we start with a fully zipped list, and

-move elements to the zipped part by pulling the zipped down until the
+move elements to the unzipped part by pulling the zipper down until the

 entire list has been unzipped (and so the zipped half of the zipper is empty).
 
 entire list has been unzipped (and so the zipped half of the zipper is empty).
 

-<pre>
-type 'a list_zipper = ('a list) * ('a list);;
-
-let rec tz (z:char list_zipper) = 
-    match z with (unzipped, []) -> List.rev(unzipped) (* Done! *)
-               | (unzipped, 'S'::zipped) -> tz ((List.append unzipped unzipped), zipped) 
-               | (unzipped, target::zipped) -> tz (target::unzipped, zipped);; (* Pull zipper *)
-
-# tz ([], ['a'; 'b'; 'S'; 'd']);;
-- : char list = ['a'; 'b'; 'a'; 'b'; 'd']
-
-# tz ([], ['a'; 'S'; 'b'; 'S']);;
-- : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b']
-</pre>
+       type 'a list_zipper = ('a list) * ('a list);;
+       
+       let rec tz (z : char list_zipper) =
+           match z with
+           | (unzipped, []) -> List.rev(unzipped) (* Done! *)
+           | (unzipped, 'S'::zipped) -> tz ((List.append unzipped unzipped), zipped)
+           | (unzipped, target::zipped) -> tz (target::unzipped, zipped);; (* Pull zipper *)
+       
+       # tz ([], ['a'; 'b'; 'S'; 'd']);;
+       - : char list = ['a'; 'b'; 'a'; 'b'; 'd']
+       
+       # tz ([], ['a'; 'S'; 'b'; 'S']);;
+       - : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b']

 
 Note that this implementation enforces the evaluate-leftmost rule.
 Task completed.
 
 One way to see exactly what is going on is to watch the zipper in
 action by tracing the execution of `tz`.  By using the `#trace`
 
 Note that this implementation enforces the evaluate-leftmost rule.
 Task completed.
 
 One way to see exactly what is going on is to watch the zipper in
 action by tracing the execution of `tz`.  By using the `#trace`

-directive in the Ocaml interpreter, the system will print out the
+directive in the OCaml interpreter, the system will print out the

 arguments to `tz` each time it is (recurcively) called.  Note that the
 lines with left-facing arrows (`<--`) show (recursive) calls to `tz`,
 giving the value of its argument (a zipper), and the lines with
 right-facing arrows (`-->`) show the output of each recursive call, a
 arguments to `tz` each time it is (recurcively) called.  Note that the
 lines with left-facing arrows (`<--`) show (recursive) calls to `tz`,
 giving the value of its argument (a zipper), and the lines with
 right-facing arrows (`-->`) show the output of each recursive call, a

-simple list.  
-
-<pre>
-# #trace tz;;
-t1 is now traced.
-# tz ([], ['a'; 'b'; 'S'; 'd']);;
-tz <-- ([], ['a'; 'b'; 'S'; 'd'])
-tz <-- (['a'], ['b'; 'S'; 'd'])         (* Pull zipper *)
-tz <-- (['b'; 'a'], ['S'; 'd'])         (* Pull zipper *)
-tz <-- (['b'; 'a'; 'b'; 'a'], ['d'])    (* Special step *)
-tz <-- (['d'; 'b'; 'a'; 'b'; 'a'], [])  (* Pull zipper *)
-tz --> ['a'; 'b'; 'a'; 'b'; 'd']        (* Output reversed *)
-tz --> ['a'; 'b'; 'a'; 'b'; 'd']
-tz --> ['a'; 'b'; 'a'; 'b'; 'd']
-tz --> ['a'; 'b'; 'a'; 'b'; 'd']
-tz --> ['a'; 'b'; 'a'; 'b'; 'd']
-- : char list = ['a'; 'b'; 'a'; 'b'; 'd'] 
-</pre>
+simple list.
+
+       # #trace tz;;
+       t1 is now traced.
+       # tz ([], ['a'; 'b'; 'S'; 'd']);;
+       tz <-- ([], ['a'; 'b'; 'S'; 'd'])
+       tz <-- (['a'], ['b'; 'S'; 'd'])         (* Pull zipper *)
+       tz <-- (['b'; 'a'], ['S'; 'd'])         (* Pull zipper *)
+       tz <-- (['b'; 'a'; 'b'; 'a'], ['d'])    (* Special step *)
+       tz <-- (['d'; 'b'; 'a'; 'b'; 'a'], [])  (* Pull zipper *)
+       tz --> ['a'; 'b'; 'a'; 'b'; 'd']        (* Output reversed *)
+       tz --> ['a'; 'b'; 'a'; 'b'; 'd']
+       tz --> ['a'; 'b'; 'a'; 'b'; 'd']
+       tz --> ['a'; 'b'; 'a'; 'b'; 'd']
+       tz --> ['a'; 'b'; 'a'; 'b'; 'd']
+       - : char list = ['a'; 'b'; 'a'; 'b'; 'd']

 
 The nice thing about computations involving lists is that it's so easy
 to visualize them as a data structure.  Eventually, we want to get to
 a place where we can talk about more abstract computations.  In order
 to get there, we'll first do the exact same thing we just did with
 
 The nice thing about computations involving lists is that it's so easy
 to visualize them as a data structure.  Eventually, we want to get to
 a place where we can talk about more abstract computations.  In order
 to get there, we'll first do the exact same thing we just did with

-concrete zipper using procedures.  
-
-Think of a list as a procedural recipe: `['a'; 'b'; 'S'; 'd']` 
-is the result of the computation `a::(b::(S::(d::[])))` (or, in our old
-style, `makelist a (makelist b (makelist S (makelist c empty)))`).
-The recipe for constructing the list goes like this:
-
-<pre>
-(0)  Start with the empty list []
-(1)  make a new list whose first element is 'd' and whose tail is the list constructed in step (0)
-(2)  make a new list whose first element is 'S' and whose tail is the list constructed in step (1)
------------------------------------------
-(3)  make a new list whose first element is 'b' and whose tail is the list constructed in step (2)
-(4)  make a new list whose first element is 'a' and whose tail is the list constructed in step (3)
-</pre>
+concrete zipper using procedures.
+
+Think of a list as a procedural recipe: `['a'; 'b'; 'S'; 'd']` is the result of
+the computation `'a'::('b'::('S'::('d'::[])))` (or, in our old style,
+`make_list 'a' (make_list 'b' (make_list 'S' (make_list 'd' empty)))`). The
+recipe for constructing the list goes like this:
+
+>      (0)  Start with the empty list []  
+>      (1)  make a new list whose first element is 'd' and whose tail is the list constructed in step (0)  
+>      (2)  make a new list whose first element is 'S' and whose tail is the list constructed in step (1)  
+>      -----------------------------------------  
+>      (3)  make a new list whose first element is 'b' and whose tail is the list constructed in step (2)  
+>      (4)  make a new list whose first element is 'a' and whose tail is the list constructed in step (3)

 
 What is the type of each of these steps?  Well, it will be a function
 from the result of the previous step (a list) to a new list: it will
 be a function of type `char list -> char list`.  We'll call each step
 
 What is the type of each of these steps?  Well, it will be a function
 from the result of the previous step (a list) to a new list: it will
 be a function of type `char list -> char list`.  We'll call each step

-a **continuation** of the recipe.  So in this context, a continuation
-is a function of type `char list -> char list`.  For instance, the
-continuation corresponding to the portion of the recipe below the
-horizontal line is the function `fun (tail:char list) -> a::(b::tail)`.
+(or group of steps) a **continuation** of the recipe.  So in this
+context, a continuation is a function of type `char list -> char
+list`.  For instance, the continuation corresponding to the portion of
+the recipe below the horizontal line is the function `fun (tail : char
+list) -> 'a'::('b'::tail)`.

 
 This means that we can now represent the unzipped part of our
 
 This means that we can now represent the unzipped part of our

-zipper--the part we've already unzipped--as a continuation: a function
-describing how to finish building the list.  We'll write a new
+zipper---the part we've already unzipped---as a continuation: a function
+describing how to finish building a list.  We'll write a new

 function, `tc` (for task with continuations), that will take an input
 list (not a zipper!) and a continuation and return a processed list.
 The structure and the behavior will follow that of `tz` above, with
 some small but interesting differences.  We've included the orginal
 `tz` to facilitate detailed comparison:
 
 function, `tc` (for task with continuations), that will take an input
 list (not a zipper!) and a continuation and return a processed list.
 The structure and the behavior will follow that of `tz` above, with
 some small but interesting differences.  We've included the orginal
 `tz` to facilitate detailed comparison:
 

-<pre>
-let rec tz (z:char list_zipper) = 
-    match z with (unzipped, []) -> List.rev(unzipped) (* Done! *)
-               | (unzipped, 'S'::zipped) -> tz ((List.append unzipped unzipped), zipped) 
-               | (unzipped, target::zipped) -> tz (target::unzipped, zipped);; (* Pull zipper *)
-
-let rec tc (l: char list) (c: (char list) -> (char list)) =
-  match l with [] -> List.rev (c [])
-             | 'S'::zipped -> tc zipped (fun x -> c (c x))
-             | target::zipped -> tc zipped (fun x -> target::(c x));;
-
-# tc ['a'; 'b'; 'S'; 'd'] (fun x -> x);;
-- : char list = ['a'; 'b'; 'a'; 'b']
-
-# tc ['a'; 'S'; 'b'; 'S'] (fun x -> x);;
-- : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b']
-</pre>
+       let rec tz (z : char list_zipper) =
+           match z with
+           | (unzipped, []) -> List.rev(unzipped) (* Done! *)
+           | (unzipped, 'S'::zipped) -> tz ((List.append unzipped unzipped), zipped)
+           | (unzipped, target::zipped) -> tz (target::unzipped, zipped);; (* Pull zipper *)
+       
+       let rec tc (l: char list) (c: (char list) -> (char list)) =
+           match l with
+           | [] -> List.rev (c [])
+           | 'S'::zipped -> tc zipped (fun tail -> c (c tail))
+           | target::zipped -> tc zipped (fun tail -> target::(c tail));;
+       
+       # tc ['a'; 'b'; 'S'; 'd'] (fun tail -> tail);;
+       - : char list = ['a'; 'b'; 'a'; 'b']
+       
+       # tc ['a'; 'S'; 'b'; 'S'] (fun tail -> tail);;
+       - : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b']

 
 To emphasize the parallel, I've re-used the names `zipped` and
 `target`.  The trace of the procedure will show that these variables
 
 To emphasize the parallel, I've re-used the names `zipped` and
 `target`.  The trace of the procedure will show that these variables


@@ -194,42 +181,40 @@ what the parallel would suggest.  The reason is that `unzipped` is a
 list, but `c` is a function.  That's the most crucial difference, the
 point of the excercise, and it should be emphasized.  For instance,
 you can see this difference in the fact that in `tz`, we have to glue
 list, but `c` is a function.  That's the most crucial difference, the
 point of the excercise, and it should be emphasized.  For instance,
 you can see this difference in the fact that in `tz`, we have to glue

-together the two instances of `unzipped` with an explicit `List.append`.
-In the `tc` version of the task, we simply compose `c` with itself: 
-`c o c = fun x -> c (c x)`.
+together the two instances of `unzipped` with an explicit (and
+relatively inefficient) `List.append`.
+In the `tc` version of the task, we simply compose `c` with itself:
+`c o c = fun tail -> c (c tail)`.

 
 

-Why use the identity function as the initial continuation?  Well, if
-you have already constructed the list "abSd", what's the next step in
-the recipe to produce the desired result (which is the same list,
-"abSd")?  Clearly, the identity continuation.
+A call `tc ['a'; 'b'; 'S'; 'd']` yields a partially-applied function; it still waits for another argument, a continuation of type `char list -> char list`. We have to give it an "initial continuation" to get started. Here we supply *the identity function* as the initial continuation. Why did we choose that? Well, if
+you have already constructed the initial list `"abSd"`, what's the desired continuation? What's the next step in the recipe to produce the desired result, i.e, the very same list, `"abSd"`?  Clearly, the identity function.

 
 A good way to test your understanding is to figure out what the
 continuation function `c` must be at the point in the computation when
 `tc` is called with the first argument `"Sd"`.  Two choices: is it
 
 A good way to test your understanding is to figure out what the
 continuation function `c` must be at the point in the computation when
 `tc` is called with the first argument `"Sd"`.  Two choices: is it

-`fun x -> a::b::x`, or it is `fun x -> b::a::x`?  
-The way to see if you're right is to execute the following 
-command and see what happens:
+`fun tail -> 'a'::'b'::tail`, or it is `fun tail -> 'b'::'a'::tail`?  The way to see if
+you're right is to execute the following command and see what happens:

 
 

-    tc ['S'; 'd'] (fun x -> 'a'::'b'::x);;
+    tc ['S'; 'd'] (fun tail -> 'a'::'b'::tail);;

 
 There are a number of interesting directions we can go with this task.
 
 There are a number of interesting directions we can go with this task.

-The task was chosen because the computation can be viewed as a
+The reason this task was chosen is because it can be viewed as a

 simplified picture of a computation using continuations, where `'S'`
 simplified picture of a computation using continuations, where `'S'`

-plays the role of a control operator with some similarities to what is
-often called `shift`.  In the analogy, the list portrays a string of
-functional applications, where `[f1; f2; f3; x]` represents `f1(f2(f3
-x))`.  The limitation of the analogy is that it is only possible to
-represent computations in which the applications are always
-right-branching, i.e., the computation `((f1 f2) f3) x` cannot be
-directly represented.
-
-One possibile development is that we could add a special symbol `'#'`,
+plays the role of a continuation operator. (It works like the Scheme operators `shift` or `control`; the differences between them don't manifest themselves in this example.) In the analogy, the input list portrays a
+sequence of functional applications, where `[f1; f2; f3; x]` represents
+`f1(f2(f3 x))`.  The limitation of the analogy is that it is only
+possible to represent computations in which the applications are
+always right-branching, i.e., the computation `((f1 f2) f3) x` cannot
+be directly represented.
+
+One way to extend this exercise would be to add a special symbol `'#'`,

 and then the task would be to copy from the target `'S'` only back to
 and then the task would be to copy from the target `'S'` only back to

-the closest `'#'`.  This would allow the task to simulate delimited
-continuations (for right-branching computations).
+the closest `'#'`.  This would allow our task to simulate delimited
+continuations with embedded prompts.
+
+The reason the task is well-suited to the list zipper is in part
+because the list monad has an intimate connection with continuations.
+The following section explores this connection.  We'll return to the
+list task after talking about generalized quantifiers below.

 
 

-The task is well-suited to the list zipper because the list monad has
-an intimate connection with continuations.  The following section
-makes this connection.  We'll return to the list task after talking
-about generalized quantifiers below.