cps tweak, another Seasoned cite

[lambda.git] / from_list_zippers_to_continuations.mdwn

diff --git a/from_list_zippers_to_continuations.mdwn b/from_list_zippers_to_continuations.mdwn
index d6eb685..dcd11ce 100644 (file)
--- a/from_list_zippers_to_continuations.mdwn
+++ b/from_list_zippers_to_continuations.mdwn
@@ -6,9 +6,9 @@ to continuations is to re-functionalize a zipper.  Then the
 concreteness and understandability of the zipper provides a way of
 understanding an equivalent treatment using continuations.
 
 concreteness and understandability of the zipper provides a way of
 understanding an equivalent treatment using continuations.
 

-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']`.
+Let's work with lists of `char`s for a change.  We'll sometimes write
+"abSd" as an abbreviation for  
+`['a'; 'b'; 'S'; 'd']`.

 
 We will set out to compute a deceptively simple-seeming **task: given a
 string, replace each occurrence of 'S' in that string with a copy of
 
 We will set out to compute a deceptively simple-seeming **task: given a
 string, replace each occurrence of 'S' in that string with a copy of


@@ -26,7 +26,7 @@ In linguistic terms, this is a kind of anaphora
 resolution, where `'S'` is functioning like an anaphoric element, and
 the preceding string portion is the antecedent.
 
 resolution, where `'S'` is functioning like an anaphoric element, and
 the preceding string portion is the antecedent.
 

-This deceptively simple task gives rise to some mind-bending complexity.
+This task can give rise to considerable 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'):
 


@@ -58,7 +58,7 @@ versus
                     *
        ~~> ...
 
                     *
        ~~> ...
 

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


@@ -70,15 +70,16 @@ This is a task well-suited to using a zipper.  We'll define a function
 `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
 move elements to the unzipped part by pulling the zipper down until 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
 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, at which point the zipped half of the
+zipper will be empty.

 
        type 'a list_zipper = ('a list) * ('a list);;
        
        let rec tz (z : char list_zipper) =
 
        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 *)
+          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'; 'b'; 'S'; 'd']);;
        - : char list = ['a'; 'b'; 'a'; 'b'; 'd']


@@ -86,14 +87,15 @@ entire list has been unzipped (and so the zipped half of the zipper is empty).
        # tz ([], ['a'; 'S'; 'b'; 'S']);;
        - : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b']
 
        # tz ([], ['a'; 'S'; 'b'; 'S']);;
        - : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b']
 

-Note that this implementation enforces the evaluate-leftmost rule.
-Task completed.
+Note that the direction in which the zipper unzips 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
 
 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

-arguments to `tz` each time it is (recurcively) called.  Note that the
-lines with left-facing arrows (`<--`) show (recursive) calls to `tz`,
+arguments to `tz` each time it is called, including when it is called
+recursively within one of the `match` clauses.  Note that the
+lines with left-facing arrows (`<--`) show (both initial and 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.
 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.


@@ -101,10 +103,10 @@ simple list.
        # #trace tz;;
        t1 is now traced.
        # tz ([], ['a'; 'b'; 'S'; 'd']);;
        # #trace tz;;
        t1 is now traced.
        # tz ([], ['a'; 'b'; 'S'; 'd']);;

-       tz <-- ([], ['a'; 'b'; 'S'; 'd'])
+       tz <-- ([], ['a'; 'b'; 'S'; 'd'])       (* Initial call *)

        tz <-- (['a'], ['b'; 'S'; 'd'])         (* Pull zipper *)
        tz <-- (['b'; 'a'], ['S'; 'd'])         (* Pull zipper *)
        tz <-- (['a'], ['b'; 'S'; 'd'])         (* Pull zipper *)
        tz <-- (['b'; 'a'], ['S'; 'd'])         (* Pull zipper *)

-       tz <-- (['b'; 'a'; 'b'; 'a'], ['d'])    (* Special step *)
+       tz <-- (['b'; 'a'; 'b'; 'a'], ['d'])    (* Special 'S' step *)

        tz <-- (['d'; 'b'; 'a'; 'b'; 'a'], [])  (* Pull zipper *)
        tz --> ['a'; 'b'; 'a'; 'b'; 'd']        (* Output reversed *)
        tz --> ['a'; 'b'; 'a'; 'b'; 'd']
        tz <-- (['d'; 'b'; 'a'; 'b'; 'a'], [])  (* Pull zipper *)
        tz --> ['a'; 'b'; 'a'; 'b'; 'd']        (* Output reversed *)
        tz --> ['a'; 'b'; 'a'; 'b'; 'd']


@@ -117,16 +119,16 @@ 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
 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.
+concrete zipper using procedures instead.

 
 

-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
+Think of a list as a procedural recipe: `['a'; 'b'; 'c'; 'd']` is the result of
+the computation `'a'::('b'::('c'::('d'::[])))` (or, in our old style,
+`make_list 'a' (make_list 'b' (make_list 'c' (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)  
 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)  
+>      (2)  make a new list whose first element is 'c' 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)
 >      -----------------------------------------  
 >      (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)


@@ -134,20 +136,15 @@ recipe for constructing the list goes like this:
 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

-(or group of steps) a **continuation** of the recipe.  So in this
+(or group of steps) a **continuation** of the previous steps.  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
 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)`.
+list) -> 'a'::('b'::tail)`. What is the continuation of the 4th step? That is, after we've built up `'a'::('b'::('c'::('d'::[])))`, what more has to happen to that for it to become the list `['a'; 'b'; 'c'; 'd']`? Nothing! Its continuation is the function that does nothing: `fun tail -> tail`.
+
+In what follows, we'll be thinking about the result list that we're building up in this procedural way. We'll treat our input list just as a plain old static list data structure, that we recurse through in the normal way we're accustomed to. We won't need a zipper data structure, because the continuation-based representation of our result list will take over the same role.

 
 

-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 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:
+So our new function `tc` (for task with continuations) takes an input list (not a zipper) and a also takes a continuation `k` (it's conventional to use `k` for continuation variables). `k` is a function that represents how the result list is going to continue being built up after this invocation of `tc` delivers up a value. When we invoke `tc` for the first time, we expect it to deliver as a value the very de-S'd list we're seeking, so the way for the list to continue being built up is for nothing to happen to it. That is, our initial invocation of `tc` will supply `fun tail -> tail` as the value for `k`. Here is the whole `tc` function. Its structure and behavior follows `tz` from above, which we've repeated here to facilitate detailed comparison:

 
        let rec tz (z : char list_zipper) =
            match z with
 
        let rec tz (z : char list_zipper) =
            match z with


@@ -155,11 +152,11 @@ some small but interesting differences.  We've included the orginal
            | (unzipped, 'S'::zipped) -> tz ((List.append unzipped unzipped), zipped)
            | (unzipped, target::zipped) -> tz (target::unzipped, zipped);; (* Pull zipper *)
        
            | (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)) =
+       let rec tc (l: char list) (k: (char list) -> (char list)) =

            match l with
            match l with

-           | [] -> List.rev (c [])
-           | 'S'::zipped -> tc zipped (fun tail -> c (c tail))
-           | target::zipped -> tc zipped (fun tail -> target::(c tail));;
+           | [] -> List.rev (k [])
+           | 'S'::zipped -> tc zipped (fun tail -> k (k tail))
+           | target::zipped -> tc zipped (fun tail -> target::(k tail));;

        
        # tc ['a'; 'b'; 'S'; 'd'] (fun tail -> tail);;
        - : char list = ['a'; 'b'; 'a'; 'b']
        
        # tc ['a'; 'b'; 'S'; 'd'] (fun tail -> tail);;
        - : char list = ['a'; 'b'; 'a'; 'b']


@@ -167,40 +164,46 @@ some small but interesting differences.  We've included the orginal
        # tc ['a'; 'S'; 'b'; 'S'] (fun tail -> tail);;
        - : char list = ['a'; 'a'; 'b'; 'a'; '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
+To emphasize the parallel, we've re-used the names `zipped` and

 `target`.  The trace of the procedure will show that these variables
 take on the same values in the same series of steps as they did during
 `target`.  The trace of the procedure will show that these variables
 take on the same values in the same series of steps as they did during

-the execution of `tz` above.  There will once again be one initial and
+the execution of `tz` above: there will once again be one initial and

 four recursive calls to `tc`, and `zipped` will take on the values
 `"bSd"`, `"Sd"`, `"d"`, and `""` (and, once again, on the final call,
 four recursive calls to `tc`, and `zipped` will take on the values
 `"bSd"`, `"Sd"`, `"d"`, and `""` (and, once again, on the final call,

-the first `match` clause will fire, so the the variable `zipper` will
+the first `match` clause will fire, so the the variable `zipped` will

 not be instantiated).
 
 not be instantiated).
 

-I have not called the functional argument `unzipped`, although that is
-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
+We have not named the continuation argument `unzipped`, although that is
+what the parallel would suggest.  The reason is that `unzipped` (in
+`tz`) is a list, but `k` (in `tc`) is a function.  That's the most crucial 
+difference between the solutions---it's 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
 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 (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)`.
+together the two instances of `unzipped` with an explicit (and,
+computationally speaking, relatively inefficient) `List.append`.
+In the `tc` version of the task, we simply compose `k` with itself:
+`k o k = fun tail -> k (k tail)`.

 
 

-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 call `tc ['a'; 'b'; 'S'; 'd']` would yield a partially-applied function; it would still wait for another argument, a continuation of type `char list -> char list`. So we have to give it an "initial continuation" to get started. As mentioned above, we supply *the identity function* as the initial continuation. Why did we choose that? Again, if
+you have already constructed the result list `"ababd"`, what's the desired continuation? What's the next step in the recipe to produce the desired result, i.e, the very same list, `"ababd"`?  Clearly, the identity function.

 
 A good way to test your understanding is to figure out what the
 
 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 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:
+continuation function `k` must be at the point in the computation when
+`tc` is applied to the argument `"Sd"`.  Two choices: is it
+`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 tail -> 'a'::'b'::tail);;
 
 There are a number of interesting directions we can go with this task.
 
     tc ['S'; 'd'] (fun tail -> 'a'::'b'::tail);;
 
 There are a number of interesting directions we can go with this task.

-The reason this task was chosen is because it can be viewed as a
+The reason this task was chosen is because the task itself (as opposed
+to the functions used to implement the task) 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 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
+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.
+See Ken Shan's paper [Shift to control](http://www.cs.rutgers.edu/~ccshan/recur/recur.pdf),
+which inspired some of the discussion in this topic.)
+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
 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


@@ -213,8 +216,7 @@ the closest `'#'`.  This would allow our task to simulate delimited
 continuations with embedded `prompt`s (also called `reset`s).
 
 The reason the task is well-suited to the list zipper is in part
 continuations with embedded `prompt`s (also called `reset`s).
 
 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.
+because the List monad has an intimate connection with continuations.
+We'll explore this next.