This "handler" encodes the search's having finished, and delivering a final
answer to whatever else you wanted your program to do with the result of the
search. If you like, at any stage in the search you might just give an argument
- to this handler, instead of giving an argument to the handler that continues
+ to *this* handler, instead of giving an argument to the handler that continues
the list traversal leftwards. Semantically, this would amount to *aborting* the
list traversal! (As we've said before, whether the rest of the list traversal
really gets evaluated will depend on what evaluation order is in place. But
f 3 <result of folding f and z over [2; 1]> <handler to continue folding leftwards> <handler to abort the traversal>
- `f`'s job would be to check whether 3 matches the element we're searching for
- (here also 3), and if it does, just evaluate to the result of passing `true` to
+ `f`'s job would be to check whether `3` matches the element we're searching for
+ (here also `3`), and if it does, just evaluate to the result of passing `true` to
the abort handler. If it doesn't, then evaluate to the result of passing
`false` to the continue-leftwards handler.
of the list multiplied to, because that would affect the answer you passed
along to the continue-leftwards handler.
- A **version 5** list would encode this kind of fold operation over the list, in
+ A **version 5** list encodes the kind of fold operation we're envisaging here, in
the same way that v3 (and v4) lists encoded the simpler fold operation.
Roughly, the list `[5;4;3;2;1]` would look like this:
\f z continue_leftwards_handler abort_handler.
- <fold f and z over [4; 3; 2; 1]>
+ <fold f and z over [4;3;2;1]>
(\result_of_fold_over_4321. f 5 result_of_fold_over_4321 continue_leftwards_handler abort_handler)
abort_handler
+ ; or, expanding the fold over [4;3;2;1]:
\f z continue_leftwards_handler abort_handler.
(\continue_leftwards_handler abort_handler.
- <fold f and z over [3; 2; 1]>
+ <fold f and z over [3;2;1]>
(\result_of_fold_over_321. f 4 result_of_fold_over_321 continue_leftwards_handler abort_handler)
abort_handler
)
(\result_of_fold_over_4321. f 5 result_of_fold_over_4321 continue_leftwards_handler abort_handler)
abort_handler
- and so on
+ ; and so on
- Remarks: the `larger_computation_handler` should be supplied as both the
+ Remarks: the `larger_computation` handler should be supplied as both the
`continue_leftwards_handler` and the `abort_handler` for the leftmost
- application, where the head `5` is supplied to `f`. Because the result of this
+ application, where the head `5` is supplied to `f`; because the result of this
application should be passed to the larger computation, whether it's a "fall
off the left end of the list" result or it's a "I'm finished, possibly early"
- result. The `larger_computation_handler` also then gets passed to the next
+ result. The `larger_computation` handler also then gets passed to the next
rightmost stage, where the head `4` is supplied to `f`, as the `abort_handler` to
use if that stage decides it has an early answer.
Finally, notice that we don't have the result of applying `f` to `4` etc given as
an argument to the application of `f` to `5` etc. Instead, we pass
- (\result_of_fold_over_4321. f 5 result_of_fold_over_4321 one_handler another_handler)
+ (\result_of_fold_over_4321. f 5 result_of_fold_over_4321 <one_handler> <another_handler>)
*to* the application of `f` to `4` as its "continue" handler. The application of `f`
to `4` can decide whether this handler, or the other, "abort" handler, should be
of the complex expression semantically depending only on this, not on that. A
demon evaluator who custom-picked the evaluation order to make things maximally
bad for you could ensure that all the semantically unnecessary computations got
- evaluated anyway. At this stage, we don't have any way to prevent that. Later,
- we'll see ways to semantically guarantee one evaluation order rather than
+ evaluated anyway. We don't have any way to prevent that. Later,
+ we'll see ways to *semantically guarantee* one evaluation order rather than
another. Though even then the demonic evaluation-order-chooser could make it
take unnecessarily long to compute the semantically guaranteed result. Of
course, in any real computing environment you'll know you're dealing with a
; here's the abort_handler
larger_computation in
let extract_tail = ; left as exercise
- ;; for real efficiency, it'd be nice to fuse the apparatus developed
- ;; in these v5 lists with the ideas from the v4 lists, above
- ;; but that also is left as an exercise
These functions are used like this:
your reach. And once you have followed it, you'll be well on your way to
appreciating the full terrible power of continuations.
-<!-- (Silly [cultural reference](http://www.newgrounds.com/portal/view/33440).) -->
+ <!-- (Silly [cultural reference](http://www.newgrounds.com/portal/view/33440).) -->
Of course, like everything elegant and exciting in this seminar, [Oleg
discusses it in much more
detail](http://okmij.org/ftp/Streams.html#enumerator-stream).
+ *Comments*:
+
+ 1. The technique deployed here, and in the v2 lists, and in our implementations
+ of pairs and booleans, is known as **continuation-passing style** programming.
+
+ 2. We're still building the list as a right fold, so in a sense the
+ application of `f` to the leftmost element `5` is "outermost". However,
+ this "outermost" application is getting lifted, and passed as a *handler*
+ to the next right application. Which is in turn getting lifted, and
+ passed to its next right application, and so on. So if you
+ trace the evaluation of the `extract_head` function to the list `[5;4;3;2;1]`,
+ you'll see `1` gets passed as a "this is the head sofar" answer to its
+ `continue_handler`; then that answer is discarded and `2` is
+ passed as a "this is the head sofar" answer to *its* `continue_handler`,
+ and so on. All those steps have to be evaluated to finally get the result
+ that `5` is the outer/leftmost head of the list. That's not an efficient way
+ to get the leftmost head.
+
+ We could improve this by building lists as left folds when implementing them
+ as continuation-passing style folds. We'd just replace above:
+
+ let make_list = \h t. \f z continue_handler abort_handler.
+ f h z (\z. t f z continue_handler abort_handler) abort_handler
+
+ now `extract_head` can return the leftmost head directly, using its `abort_handler`:
+
+ let extract_head = \lst larger_computation. lst
+ ; here's our f
+ (\hd sofar continue_handler abort_handler. abort_handler hd)
+ ; here's our z
+ junk
+ ; here's the continue_handler for the leftmost application of f
+ larger_computation
+ ; here's the abort_handler
+ larger_computation in
+
+ 3. To extract tails efficiently, too, it'd be nice to fuse the apparatus developed
+ in these v5 lists with the ideas from the v4 lists, above.
+ But that also is left as an exercise.
5. Implementing (self-balancing) trees