1 We've talked about different implementations of lists in the Lambda Calculus; and we've also talked about using lists to implement other data structures, like sets. One thing we observed is that if you're going to implement a set using a list (this isn't an especially efficient implementation of a set, but let's do the best we can with it), it's helpful to make sure that the list is always sorted. That way, as you're searching through the list, you might come to a point before the end where you knew the element you're looking for wasn't going to be found anymore. So you wouldn't have to continue the search.
3 If you were implementing lists with `letrec` or a fixed point combinator, then at that point you could just have your traversal function return some appropriate result, instead of requesting that the recursion continue.
5 But what if you were using our right-fold or left-fold implementation of lists, rather than using `letrec` or a fixed point combinator? It's true that `letrec` and fixed point combinators are cool. But if you use them, the terms you construct won't be strongly normalizing. Whether their reduction stops at a normal form will depend on what evaluation order is operative. For efficiency, it can also be helpful to know how to get by without those powerful devices when they're not strictly necessary. Plus that knowledge could carry over to settings where `letrec` is no longer available, even in principle.
7 When dealing with a right-fold or left-fold implementation of lists, if your traversal function returns a result, that result automatically gets passed to the next stage of the traversal, as the updated seed value from the previous steps. If we make the seed value type complex, we could signal to the rest of the traversal that the job is done, they don't need to do any more work. For example, the seed value might be a pair of `false` and some default value until we reach a certain stage, and all the traversal steps until that stage have to do their normal work, but once we've gotten to the stage where we're ready to return a result, we make the seed value be a pair of `true` and the result we've computed. Then all the later traversal steps see that `true` and just pass the existing seed pair on to the rest of the traversal. At the end, we throw away the `true` and take the second member of the final seed value as our desired result.
9 That will work OK, but we still have to go through every step of the traversal. Let's think about whether we can modify the right-fold and/or left-fold implementation of lists to allow for a genuine early abort. We'd like to avoid any unnecessary steps of the traversal. If we've worked out our result mid-way through the traversal, we want to be able to deliver it _immediately_ to the larger computation in which our traversal was embedded. The problem with the fold-based implementations of lists we've got so far is that they are pre-programmed to traverse the whole list. We'd have to implement the folds differently to be able to achieve what we're now envisaging.
11 We worked out such an implementation in the homework session on Wed April 22. The scheme we used was that, whereas before our traversal functions would have an interface like this:
13 \current_list_element seed_value_so_far. do_something
15 Our traversal functions will instead now have an interface like this:
17 \current_list_element seed_value_so_far done_handler keep_going_handler.
18 if ... then done_handler (some_result) else keep_going_handler (another_result)
20 and we worked out that a left-fold implementation of the list `[10,20,30,40]` could look like this:
22 \f z done_handler. f 10 z done_handler (\z. [20,30,40] f z done_handler)
24 This is all reminiscent of the way in which our encodings of triples and pairs and such too their handler functions as arguments. Remember with triples, we say:
26 triple (\x y z. add x y)
32 to get the last element of the triple. Of course you can wrap this up in _another_ function, that takes the triple as _its_ argument, but at bottom, you are going to be supplying the handler to the triple. And then the handler gets the triple's _elements_, not the triple itself, as arguments. (Another of our implementations of lists followed a similar strategy.)
34 Something similar is happening here, where both the traversal function `f` and the fold function that implements the list both take such handler arguments. The only not-completely-straightforward thing going on in the fold function is that the `keep_going_handler` we pass to the traversal function `f` *encodes* how the fold should continue, using the updated seed value `z` from the current step, *without actually computing* the rest of the traversal. It's up to the body of `f` to decide whether to invoke the rest of the traversal, by supplying a value to that `keep_going_handler`, or to finish the traversal right now by supply a value to the `done_handler` instead.
36 A question came up in the session of why we need the `done_handler` in this scheme. We could just eliminate it and have the traversal function `f` choose between simply returning a value --- that'd abort the traversal, the way we did above by passing the value to `done_handler` --- or instead supplying a value to the `keep_going_handler`. And the answer is that yes, in this particular scheme, that is correct. (In a few weeks when we look at delimited continuations in terms of `reset`/`shift`, you'll see that what we just described is how the `abort` operation gets implemented in terms of `shift`.) However, sometimes it helps to express a basic case a bit more verbosely than seems immediately necessary, because then later generalizations will look more natural. That's true here. So let's just use the implementation as we've written it. If you prefer, you can just make the `done_handler` be the identity function.
38 With that general scheme, here is the empty list:
40 \f z done_handler. done_handler z
42 and here is the `cons` operation:
44 \x xs. \f z done_handler. f x z done_handler (\z. xs f z done_handler)
46 (The latter can just be read off of our construction of `[10,20,30,40]`; I just substituted `x` for `10` and `xs` for `[20, 30, 40]`.)
48 Here's an example of how to get the head of such a list:
50 xs (\x z done keep_going. done x) err done_handler
52 `err` is what's returned if you ask for the `head` of the empty list.
54 Here's how to get the length of such a list:
56 xs (\x z done keep_going. keep_going (succ z)) 0 done_handler
58 Here there is no opportunity to abort early with a correct value, so our traversal function always delivers its output to the `keep_going` handler.
60 Here's how to get the last element of such a list:
62 xs (\x z done keep_going. keep_going x) err done_handler
64 This is similar to getting the first element, except that each step delivers its output to the `keep_going` handler rather than to the `done` handler. That ensures that we will only get the output of the last step, when the traversal function is applied to the last member of the list. If the list is empty, then we'll get the `err` value, just as with the function that tries to extract the list's head.
66 One thing to note is that there are limits to how much you can immunize yourself against doing unnecessary work. A demon evaluator who got to custom-pick the evaluation order (including doing reductions underneath lambdas when he wanted to) could ensure that lots of unnecessary computations got performed, despite your best efforts. We don't yet have any way to prevent that. (Later we will see some ways to *computationally force* the evaluation order we prefer. Of course, in any real computing environment you'll mostly know what evaluation order you're dealing with, and you can mostly program efficiently around that.) The current scheme at least makes our result not *computationally depend on* what happens further on in the traversal, once we've passed a result to the `done_handler`. We don't even depend on the later steps in the traversal cooperating to pass our result through.
68 All of that gave us a left-fold implementation of lists. (Perhaps if you were _aiming_ for a left-fold implementation of lists, you would make the traversal function `f` take its `current_list_element` and `seed_value` arguments in the flipped order, but let's not worry about that.)
70 Now, let's think about how to get a right-fold implementation. It's not profoundly different, but it does require us to change our interface a little. Our left-fold implementation of `[10,20,30,40]`, above, looked like this (now we abbreviate some of the variables):
72 \f z d. f 10 z d (\z. [20,30,40] f z d)
74 Expanding the definition of `[20,30,40]`, and all the successive tails, this comes to:
76 \f z d. f 10 z d (\z. f 20 z d (\z. f 30 z d (\z. f 40 z d d)))
78 For a right-fold implementation, that should instead look like roughly like this:
80 \f z d. f 40 z d (\z. f 30 z d (\z. f 20 z d (\z. f 10 z d d)))
82 Now suppose we had just the implementation of the tail of the list, `[20,30,40]`, that is:
84 \f z d. f 40 z d (\z. f 30 z d (\z. f 20 z d d))
86 How should we take that value and transform it into the preceding value, which represents `10` consed onto that tail? I can't see how to do it in a general way, and I expect it's just not possible. Essentially what we want is to take that second `d` in the innermost function `\z. f 20 z d d`, we want to replace that second `d` with something like `(\z. f 10 z d d)`. But how can we replace just the second `d` without also replacing the first `d`, and indeed all the other bound occurrences of `d` in the expansion of `[20,30,40]`.
88 The difficulty here is that our traversal function `f` expects two handlers, but we are only giving the fold function we implement the list as a single handler. That single handler gets fed twice to the traversal function. One time it may be transformed, but at the end of the traversal, as with `\z. f 20 z d d`, there's nothing left to do to "keep going", so here it's just the single handler `d` fed to `f` twice. But we can see that in order to implement `cons` for a right-folding traversal, we don't want it to be the single handler `d` fed to `f` twice. It'd work better if we implemented `[20,30,40]` like this:
90 \f z d g. f 40 z d (\z. f 30 z d (\z. f 20 z d g))
92 Notice that now the fold function we implement the list as takes *two* handlers, `d` (for "done") and `g` (for "keep going"). Generally we'll *invoke* the fold function *by supplying the same handler function to both of these*. However, it's still useful to have the list be defined so that they're separate arguments. For now we can `cons` `10` onto the list by just substituting `(\z. f 10 z d g)` in for the bound `g`. That is:
94 [10,20,30,40] ≡ \f z d g. [20,30,40] f z d (\z. f 10 z d g)
96 Spelling this out, here are the implementations of the functions we defined before, only now for the right-fold lists:
99 cons x xs = \f z d g. xs f z d (\z. f x z d g)
100 head xs = xs (\x z d g. g x) err done_handler done_handler
101 length xs = xs (\x z d g. g (succ z)) 0 done_handler done_handler
102 last xs = xs (\x z d g. d x) err done_handler done_handler
104 *Exercise*: when considering just the implementation of `null`, both `\f z d g. g z` and `\f z d g. d z` may seem like reasonable candidates. What would go wrong with the rest of our scheme is we had instead used the latter?
106 To extract tails efficiently, too, it'd be nice to merge the apparatus developed above with the ideas from [[another of our implementations of lists|week3_lists#v3-lists]], where we passed the traversal function `f` not merely the current element and *result of traversing the list's tail* (or in the present case, a modified `keep_going` handler which encodes how to do that), but also *the list's tail itself*. That would make some computations more efficient. But we leave this as an exercise.
108 Of course, like everything elegant and exciting in this seminar, [Oleg
109 discusses it in much more
110 detail](http://okmij.org/ftp/Streams.html#enumerator-stream).
118 We said that the sorted-list implementation of a set was more efficient than
119 the unsorted-list implementation, because as you were searching through the
120 list, you could come to a point where you knew the element wasn't going to be
121 found. So you wouldn't have to continue the search.
123 If your implementation of lists was, say v1 lists plus the Y-combinator, then
124 this is exactly right. When you get to a point where you know the answer, you
125 can just deliver that answer, and not branch into any further recursion. If
126 you've got the right evaluation strategy in place, everything will work out
129 But what if we wanted to use v3 lists instead?
131 > Why would we want to do that? The advantage of the v3 lists and v3 (aka
132 "Church") numerals is that they have their recursive capacity built into their
133 very bones. So for many natural operations on them, you won't need to use a fixed
136 > Why is that an advantage? Well, if you use a fixed point combinator, then
137 the terms you get won't be strongly normalizing: whether their reduction stops
138 at a normal form will depend on what evaluation order you use. Our online
139 [[lambda evaluator]] uses normal-order reduction, so it finds a normal form if
140 there's one to be had. But if you want to build lambda terms in, say, Scheme,
141 and you wanted to roll your own recursion as we've been doing, rather than
142 relying on Scheme's native `let rec` or `define`, then you can't use the
143 fixed-point combinators `Y` or <code>Θ</code>. Expressions using them
144 will have non-terminating reductions, with Scheme's eager/call-by-value
145 strategy. There are other fixed-point combinators you can use with Scheme (in
146 the [week 3 notes](/week3/#index7h2) they were <code>Y′</code> and
147 <code>Θ′</code>. But even with them, evaluation order still
148 matters: for some (admittedly unusual) evaluation strategies, expressions using
149 them will also be non-terminating.
151 > The fixed-point combinators may be the conceptual stars. They are cool and
152 mathematically elegant. But for efficiency and implementation elegance, it's
153 best to know how to do as much as you can without them. (Also, that knowledge
154 could carry over to settings where the fixed point combinators are in principle
158 So again, what if we're using v3 lists? What options would we have then for
159 aborting a search or list traversal before it runs to completion?
161 Suppose we're searching through the list `[5;4;3;2;1]` to see if it
162 contains the number `3`. The expression which represents this search would have
163 something like the following form:
165 ..................<eq? 1 3> ~~>
166 .................. false ~~>
167 .............<eq? 2 3> ~~>
168 ............. false ~~>
169 .........<eq? 3 3> ~~>
173 Of course, whether those reductions actually followed in that order would
174 depend on what reduction strategy was in place. But the result of folding the
175 search function over the part of the list whose head is `3` and whose tail is `[2;
176 1]` will *semantically* depend on the result of applying that function to the
177 more rightmost pieces of the list, too, regardless of what order the reduction
178 is computed by. Conceptually, it will be easiest if we think of the reduction
179 happening in the order displayed above.
181 Once we've found a match between our sought number `3` and some member of
182 the list, we'd like to avoid any further unnecessary computations and just
183 deliver the answer `true` as "quickly" or directly as possible to the larger
184 computation in which the search was embedded.
186 With a Y-combinator based search, as we said, we could do this by just not
187 following a recursion branch.
189 But with the v3 lists, the fold is "pre-programmed" to continue over the whole
190 list. There is no way for us to bail out of applying the search function to the
191 parts of the list that have head `4` and head `5`, too.
193 We *can* avoid *some* unneccessary computation. The search function can detect
194 that the result we've accumulated so far during the fold is now `true`, so we
195 don't need to bother comparing `4` or `5` to `3` for equality. That will simplify the
196 computation to some degree, since as we said, numerical comparison in the
197 system we're working in is moderately expensive.
199 However, we're still going to have to traverse the remainder of the list. That
200 `true` result will have to be passed along all the way to the leftmost head of
201 the list. Only then can we deliver it to the larger computation in which the
204 It would be better if there were some way to "abort" the list traversal. If,
205 having found the element we're looking for (or having determined that the
206 element isn't going to be found), we could just immediately stop traversing the
207 list with our answer. **Continuations** will turn out to let us do that.
209 We won't try yet to fully exploit the terrible power of continuations. But
210 there's a way that we can gain their benefits here locally, without yet having
211 a fully general machinery or understanding of what's going on.
213 The key is to recall how our implementations of booleans and pairs worked.
214 Remember that with pairs, we supply the pair "handler" to the pair as *an
215 argument*, rather than the other way around:
223 to get the first element of the pair. Of course you can lift that if you want:
225 <pre><code>extract_fst ≡ \pair. pair (\x y. x)</code></pre>
227 but at a lower level, the pair is still accepting its handler as an argument,
228 rather than the handler taking the pair as an argument. (The handler gets *the
229 pair's elements*, not the pair itself, as arguments.)
231 > *Terminology*: we'll try to use names of the form `get_foo` for handlers, and
232 names of the form `extract_foo` for lifted versions of them, that accept the
233 lists (or whatever data structure we're working with) as arguments. But we may
236 The v2 implementation of lists followed a similar strategy:
238 v2list (\h t. do_something_with_h_and_t) result_if_empty
240 If the `v2list` here is not empty, then this will reduce to the result of
241 supplying the list's head and tail to the handler `(\h t.
242 do_something_with_h_and_t)`.
244 Now, what we've been imagining ourselves doing with the search through the v3
245 list is something like this:
248 larger_computation (search_through_the_list_for_3) other_arguments
250 That is, the result of our search is supplied as an argument (perhaps together
251 with other arguments) to the "larger computation". Without knowing the
252 evaluation order/reduction strategy, we can't say whether the search is
253 evaluated before or after it's substituted into the larger computation. But
254 semantically, the search is the argument and the larger computation is the
255 function to which it's supplied.
257 What if, instead, we did the same kind of thing we did with pairs and v2
258 lists? That is, what if we made the larger computation a "handler" that we
259 passed as an argument to the search?
261 the_search (\search_result. larger_computation search_result other_arguments)
263 What's the advantage of that, you say. Other than to show off how cleverly
266 Well, think about it. Think about the difficulty we were having aborting the
267 search. Does this switch-around offer us anything useful?
271 What if the way we implemented the search procedure looked something like this?
273 At a given stage in the search, we wouldn't just apply some function `f` to the
274 head at this stage and the result accumulated so far (from folding the same
275 function, and a base value, to the tail at this stage)...and then pass the result
276 of that application to the embedding, more leftward computation.
278 We'd *instead* give `f` a "handler" that expects the result of the current
279 stage *as an argument*, and then evaluates to what you'd get by passing that
280 result leftwards up the list, as before.
282 Why would we do that, you say? Just more flamboyant lifting?
284 Well, no, there's a real point here. If we give the function a "handler" that
285 encodes the normal continuation of the fold leftwards through the list, we can
286 also give it other "handlers" too. For example, we can also give it the underlined handler:
289 the_search (\search_result. larger_computation search_result other_arguments)
290 ------------------------------------------------------------------
292 This "handler" encodes the search's having finished, and delivering a final
293 answer to whatever else you wanted your program to do with the result of the
294 search. If you like, at any stage in the search you might just give an argument
295 to *this* handler, instead of giving an argument to the handler that continues
296 the list traversal leftwards. Semantically, this would amount to *aborting* the
297 list traversal! (As we've said before, whether the rest of the list traversal
298 really gets evaluated will depend on what evaluation order is in place. But
299 semantically we'll have avoided it. Our larger computation won't depend on the
300 rest of the list traversal having been computed.)
302 Do you have the basic idea? Think about how you'd implement it. A good
303 understanding of the v2 lists will give you a helpful model.
305 In broad outline, a single stage of the search would look like before, except
306 now `f` would receive two extra, "handler" arguments. We'll reserve the name `f` for the original fold function, and use `f2` for the function that accepts two additional handler arguments. To get the general idea, you can regard these as interchangeable. If the extra precision might help, then you can pay attention to when we're talking about the handler-taking `f2` or the original `f`. You'll only be *supplying* the `f2` function; the idea will be that the behavior of the original `f` will be implicitly encoded in `f2`'s behavior.
308 f2 3 <sofar value that would have resulted from folding f and z over [2; 1]> <handler to continue folding leftwards> <handler to abort the traversal>
310 `f2`'s job would be to check whether `3` matches the element we're searching for
311 (here also `3`), and if it does, just evaluate to the result of passing `true` to
312 the abort handler. If it doesn't, then evaluate to the result of passing
313 `false` to the continue-leftwards handler.
315 In this case, `f2` wouldn't need to consult the result of folding `f` and `z`
316 over `[2; 1]`, since if we had found the element `3` in more rightward
317 positions of the list, we'd have called the abort handler and this application
318 of `f2` to `3` etc would never be needed. However, in other applications the
319 result of folding `f` and `z` over the more rightward parts of the list would
320 be needed. Consider if you were trying to multiply all the elements of the
321 list, and were going to abort (with the result `0`) if you came across any
322 element in the list that was zero. If you didn't abort, you'd need to know what
323 the more rightward elements of the list multiplied to, because that would
324 affect the answer you passed along to the continue-leftwards handler.
326 A **version 5** list encodes the kind of fold operation we're envisaging here,
327 in the same way that v3 (and [v4](/advanced_lambda/#index1h1)) lists encoded
328 the simpler fold operation. Roughly, the list `[5;4;3;2;1]` would look like
332 \f2 z continue_leftwards_handler abort_handler.
333 <fold f2 and z over [4;3;2;1]>
334 (\result_of_folding_over_4321. f2 5 result_of_folding_over_4321 continue_leftwards_handler abort_handler)
337 ; or, expanding the fold over [4;3;2;1]:
339 \f2 z continue_leftwards_handler abort_handler.
340 (\continue_leftwards_handler abort_handler.
341 <fold f2 and z over [3;2;1]>
342 (\result_of_folding_over_321. f2 4 result_of_folding_over_321 continue_leftwards_handler abort_handler)
345 (\result_of_folding_over_4321. f2 5 result_of_folding_over_4321 continue_leftwards_handler abort_handler)
350 Remarks: the `larger_computation` handler should be supplied as both the
351 `continue_leftwards_handler` and the `abort_handler` for the leftmost
352 application, where the head `5` is supplied to `f2`; because the result of this
353 application should be passed to the larger computation, whether it's a "fall
354 off the left end of the list" result or it's a "I'm finished, possibly early"
355 result. The `larger_computation` handler also then gets passed to the next
356 rightmost stage, where the head `4` is supplied to `f2`, as the `abort_handler` to
357 use if that stage decides it has an early answer.
359 Finally, notice that we're not supplying the application of `f2` to `4` etc as an argument to the application of `f2` to `5` etc---at least, not directly. Instead, we pass
361 (\result_of_folding_over_4321. f2 5 result_of_folding_over_4321 <one_handler> <another_handler>)
363 *to* the application of `f2` to `4` as its "continue" handler. The application of `f2`
364 to `4` can decide whether this handler, or the other, "abort" handler, should be
365 given an argument and constitute its result.
368 I'll say once again: we're using temporally-loaded vocabulary throughout this,
369 but really all we're in a position to mean by that are claims about the result
370 of the complex expression semantically depending only on this, not on that. A
371 demon evaluator who custom-picked the evaluation order to make things maximally
372 bad for you could ensure that all the semantically unnecessary computations got
373 evaluated anyway. We don't yet know any way to prevent that. Later, we'll see
374 ways to *guarantee* one evaluation order rather than another. Of
375 course, in any real computing environment you'll know in advance that you're
376 dealing with a fixed evaluation order and you'll be able to program efficiently
379 In detail, then, here's what our v5 lists will look like:
381 let empty = \f2 z continue_handler abort_handler. continue_handler z in
382 let make_list = \h t. \f2 z continue_handler abort_handler.
383 t f2 z (\sofar. f2 h sofar continue_handler abort_handler) abort_handler in
384 let isempty = \lst larger_computation. lst
386 (\hd sofar continue_handler abort_handler. abort_handler false)
389 ; here's the continue_handler for the leftmost application of f2
391 ; here's the abort_handler
392 larger_computation in
393 let extract_head = \lst larger_computation. lst
395 (\hd sofar continue_handler abort_handler. continue_handler hd)
398 ; here's the continue_handler for the leftmost application of f2
400 ; here's the abort_handler
401 larger_computation in
402 let extract_tail = ; left as exercise
404 These functions are used like this:
406 let my_list = make_list a (make_list b (make_list c empty) in
407 extract_head my_list larger_computation
409 If you just want to see `my_list`'s head, the use `I` as the
410 `larger_computation`.
412 What we've done here does take some work to follow. But it should be within
413 your reach. And once you have followed it, you'll be well on your way to
414 appreciating the full terrible power of continuations.
416 (Silly [cultural reference](http://www.newgrounds.com/portal/view/33440).)
421 > 1. The technique deployed here, and in the v2 lists, and in our
422 > implementations of pairs and booleans, is known as
423 > **continuation-passing style** programming.
425 > 2. We're still building the list as a right fold, so in a sense the
426 > application of `f2` to the leftmost element `5` is "outermost". However,
427 > this "outermost" application is getting lifted, and passed as a *handler*
428 > to the next right application. Which is in turn getting lifted, and
429 > passed to its next right application, and so on. So if you
430 > trace the evaluation of the `extract_head` function to the list `[5;4;3;2;1]`,
431 > you'll see `1` gets passed as a "this is the head sofar" answer to its
432 > `continue_handler`; then that answer is discarded and `2` is
433 > passed as a "this is the head sofar" answer to *its* `continue_handler`,
434 > and so on. All those steps have to be evaluated to finally get the result
435 > that `5` is the outer/leftmost head of the list. That's not an efficient way
436 > to get the leftmost head.
438 > We could improve this by building lists as **left folds**. What's that?
440 > Well, the right fold of `f` over a list `[a;b;c;d;e]`, using starting value z, is:
442 > f a (f b (f c (f d (f e z))))
444 > The left fold on the other hand starts combining `z` with elements from the left. `f z a` is then combined with `b`, and so on:
446 > f (f (f (f (f z a) b) c) d) e
448 > or, if we preferred the arguments to each `f` flipped:
450 > f e (f d (f c (f b (f a z))))
452 > Recall we implemented v3 lists as their own right-fold functions. We could
453 > instead implement lists as their own left-fold functions. To do that with our
454 > v5 lists, we'd replace above:
456 > let make_list = \h t. \f2 z continue_handler abort_handler.
457 > f2 h z (\z. t f2 z continue_handler abort_handler) abort_handler
459 > Having done that, now `extract_head` can return the leftmost head
460 > directly, using its `abort_handler`:
462 > let extract_head = \lst larger_computation. lst
463 > (\hd sofar continue_handler abort_handler. abort_handler hd)
468 > 3. To extract tails efficiently, too, it'd be nice to fuse the apparatus
469 > developed in these v5 lists with the ideas from
470 > [v4](/advanced_lambda/#index1h1) lists. But that is left as an exercise.