It differs from the version 3 `make_list` only in adding the extra argument
`t` to the new, outer application of `f`.
- Similarly, 5 as a v3 or Church numeral looks like this:
+ Similarly, `five` as a v3 or Church numeral looks like this:
\s z. s (s (s (s (s z))))
of a new list with the added member prepended to the old list. That is:
let empty_set = empty in
- ; see the library for definition of any
+ ; see the library for definitions of any and eq
let make_set = \new_member old_set. any (eq new_member) old_set
; if any element in old_set was eq new_member
old_set
d)`.)
So, if we were searching the list that implements some set to see if the number
- 5 belonged to it, once we get to elements in the list that are larger than 5,
- we can stop. If we haven't found 5 already, we know it's not in the rest of the
+ `5` belonged to it, once we get to elements in the list that are larger than `5`,
+ we can stop. If we haven't found `5` already, we know it's not in the rest of the
list either.
This is an improvement, but it's still a "linear" search through the list.
parts of the list that have head `4` and head `5`, too.
We *can* avoid *some* unneccessary computation. The search function can detect
- that the result we've accumulated so far during the fold is now true, so we
+ that the result we've accumulated so far during the fold is now `true`, so we
don't need to bother comparing `4` or `5` to `3` for equality. That will simplify the
computation to some degree, since as we said, numerical comparison in the
system we're working in is moderately expensive.
It would be better if there were some way to "abort" the list traversal. If,
having found the element we're looking for (or having determined that the
element isn't going to be found), we could just immediately stop traversing the
- list with our answer. Continuations will turn out to let us do that.
+ list with our answer. **Continuations** will turn out to let us do that.
We won't try yet to fully exploit the terrible power of continuations. But
there's a way that we can gain their benefits here locally, without yet having
to get the first element of the pair. Of course you can lift that if you want:
- extract_1st === \pair. pair (\x y. x)
+ <pre><code>extract_fst ≡ \pair. pair (\x y. x)</code></pre>
but at a lower level, the pair is still accepting its handler as an argument,
rather than the handler taking the pair as an argument. (The handler gets *the
pair's elements*, not the pair itself, as arguments.)
+ > *Terminology*: we'll try to use names of the form `get_foo` for handlers, and
+ names of the form `extract_foo` for lifted versions of them, that accept the
+ lists (or whatever data structure we're working with) as arguments. But we may
+ sometimes forget.
+
The v2 implementation of lists followed a similar strategy:
v2list (\h t. do_something_with_h_and_t) result_if_empty
- If the v2list here is not empty, then this will reduce to the result of
+ If the `v2list` here is not empty, then this will reduce to the result of
supplying the list's head and tail to the handler `(\h t.
do_something_with_h_and_t)`.
What if the way we implemented the search procedure looked something like this?
- At a given stage in the search, we wouldn't just apply some function f to the
- head at this stage and the result accumulated so far, from folding the same
- function (and a base value) to the tail at this stage. And then pass the result
- of doing so leftward along the rest of the list.
+ At a given stage in the search, we wouldn't just apply some function `f` to the
+ head at this stage and the result accumulated so far (from folding the same
+ function, and a base value, to the tail at this stage)...and then pass the result
+ of that application to the embedding, more leftward computation.
- We'd also give that function a "handler" that expected the result of the
- current stage as an argument, and evaluated to passing that result leftwards
- along the rest of the list.
+ We'd *instead* give `f` a "handler" that expects the result of the current
+ stage *as an argument*, and then evaluates to what you'd get by passing that
+ result leftwards up the list, as before.
Why would we do that, you say? Just more flamboyant lifting?
Well, no, there's a real point here. If we give the function a "handler" that
- encodes the normal continuation of the fold leftwards through the list. We can
- give it another "handler" as well. We can also give it the underlined handler:
+ encodes the normal continuation of the fold leftwards through the list, we can
+ also give it other "handlers" too. For example, we can also give it the underlined handler:
the_search (\search_result. larger_computation search_result other_arguments)
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` should return the leftmost head directly, using its `abort_handler`:
+
+ let extract_head = \lst larger_computation. lst
+ (\hd sofar continue_handler abort_handler. abort_handler hd)
+ junk
+ larger_computation
+ larger_computation
+
+ 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