1 #Miscellaneous challenges and advanced topics, for untyped lambda calculus#
3 1. How would you define an operation to reverse a list? (Don't peek at the
4 [[lambda_library]]! Try to figure it out on your own.) Choose whichever
5 implementation of list you like. Even then, there are various strategies you
9 2. An advantage of the v3 lists and v3 (aka "Church") numerals is that they
10 have a recursive capacity built into their skeleton. So for many natural
11 operations on them, you won't need to use a fixed point combinator. Why is
12 that an advantage? Well, if you use a fixed point combinator, then the terms
14 won't be strongly normalizing: whether their reduction stops at a normal form
15 will depend on what evaluation order you use. Our online [[lambda evaluator]]
16 uses normal-order reduction, so it finds a normal form if there's one to be
17 had. But if you want to build lambda terms in, say, Scheme, and you wanted to
18 roll your own recursion as we've been doing, rather than relying on Scheme's
19 native `let rec` or `define`, then you can't use the fixed-point combinators
20 `Y` or <code>Θ</code>. Expressions using them will have non-terminating
21 reductions, with Scheme's eager/call-by-value strategy. There are other
22 fixed-point combinators you can use with Scheme (in the [[week 3]] notes they
23 were <code>Y′</code> and <code>Θ′</code>. But even with
24 those evaluation order still matters: for some (admittedly unusual)
25 evaluation strategies, expressions using them will also be non-terminating.
27 The fixed-point combinators may be the conceptual heros. They are cool and
28 mathematically elegant. But for efficiency and implementation elegance, it's
29 best to know how to do as much as you can without them. (Also, that knowledge
30 could carry over to settings where the fixed point combinators are in
31 principle unavailable.)
33 This is why the v3 lists and numbers are so lovely. However, one disadvantage
34 to them is that it's relatively inefficient to extract a list's tail, or get a
35 number's predecessor. To get the tail of the list [a;b;c;d;e], one will
36 basically be performing some operation that builds up the tail afresh: at
37 different stages, one will have built up [e], then [d;e], then [c;d;e], and
38 finally [b;c;d;e]. With short lists, this is no problem, but with longer lists
39 it takes longer and longer. And it may waste more of your computer's memory
40 than you'd like. Similarly for obtaining a number's predecessor.
42 The v1 lists and numbers on the other hand, had the tail and the predecessor
43 right there as an element, easy for the taking. The problem was just that the
44 v1 lists and numbers didn't have recursive capacity built into them, in the
45 way the v3 implementations do.
47 A clever approach would marry these two strategies.
49 Version 3 makes the list [a; b; c; d; e] look like this:
51 \f z. f a (f b (f c (f d (f e z))))
55 \f z. f a <the result of folding f and z over the tail>
57 Instead we could make it look like this:
59 \f z. f a <the tail itself> <the result of folding f and z over the tail>
61 That is, now f is a function expecting *three* arguments: the head of the
62 current list, the tail of the current list, and the result of continuing to
63 fold f over the tail, with a given base value z.
65 Call this a **version 4** list. The empty list could be the same:
69 The list constructor would be:
71 make_list === \h t. \f z. f h t (t f z)
73 It differs from the version 3 `make_list` only in adding the extra argument
74 `t` to the new, outer application of `f`.
76 Similarly, 5 as a v3 or Church numeral looks like this:
78 \s z. s (s (s (s (s z))))
82 \s z. s <the result of applying s to z (pred 5)-many times>
84 Instead we could make it look like this:
86 \s z. s <pred 5> <the result of applying s to z (pred 5)-many times>
88 That is, now s is a function expecting *two* arguments: the predecessor of the
89 current number, and the result of continuing to apply s to the base value z
90 predecessor-many times.
92 Jim had the pleasure of "inventing" these implementations himself. However,
93 unsurprisingly, he wasn't the first to do so. See for example [Oleg's report
94 on P-numerals](http://okmij.org/ftp/Computation/lambda-calc.html#p-numerals).
100 You're now already in a position to implement sets: that is, collections with
101 no intrinsic order where elements can occur at most once. Like lists, we'll
102 understand the basic set structures to be *type-homogenous*. So you might have
103 a set of integers, or you might have a set of pairs of integers, but you
104 wouldn't have a set that mixed both types of elements. Something *like* the
105 last option is also achievable, but it's more difficult, and we won't pursue it
106 now. In fact, we won't talk about sets of pairs, either. We'll just talk about
107 sets of integers. The same techniques we discuss here could also be applied to
108 sets of pairs of integers, or sets of triples of booleans, or sets of pairs
109 whose first elements are booleans, and whose second elements are triples of
112 (You're also now in a position to implement *multi*sets: that is, collections
113 with no intrinsic order where elements can occur multiple times: the multiset
114 {a,a} is distinct from the multiset {a}. But we'll leave these as an exercise.)
116 The easiest way to implement sets of integers would just be to use lists. When
117 you "add" a member to a set, you'd get back a list that was either identical to
118 the original list, if the added member already was present in it, or consisted
119 of a new list with the added member prepended to the old list. That is:
121 let empty_set = empty in
122 ; see the library for definition of any
123 let make_set = \new_member old_set. any (eq new_member) old_set
124 ; if any element in old_set was eq new_member
127 make_list new_member old_set
129 Think about how you'd implement operations like `set_union`,
130 `set_intersection`, and `set_difference` with this implementation of sets.
132 The implementation just described works, and it's the simplest to code.
133 However, it's pretty inefficient. If you had a 100-member set, and you wanted
134 to create a set which had all those 100-members and some possibly new element
135 `e`, you might need to check all 100 members to see if they're equal to `e`
136 before concluding they're not, and returning the new list. And comparing for
137 numeric equality is a moderately expensive operation, in the first place.
139 (You might say, well, what's the harm in just prepending `e` to the list even
140 if it already occurs later in the list. The answer is, if you don't keep track
141 of things like this, it will likely mess up your implementations of
142 `set_difference` and so on. You'll have to do the book-keeping for duplicates
143 at some point in your code. It goes much more smoothly if you plan this from
146 How might we make the implementation more efficient? Well, the *semantics* of
147 sets says that they have no intrinsic order. That means, there's no difference
148 between the set {a,b} and the set {b,a}; whereas there is a difference between
149 the *list* [a;b] and the list [b;a]. But this semantic point can be respected
150 even if we *implement* sets with something ordered, like list---as we're
151 already doing. And we might *exploit* the intrinsic order of lists to make our
152 implementation of sets more efficient.
154 What we could do is arrange it so that a list that implements a set always
155 keeps in elements in some specified order. To do this, there'd have *to be*
156 some way to order its elements. Since we're talking now about sets of numbers,
157 that's easy. (If we were talking about sets of pairs of numbers, we'd use
158 "lexicographic" ordering, where `(a,b) < (c,d)` iff `a < c or (a == c and b <
161 So, if we were searching the list that implements some set to see if the number
162 5 belonged to it, once we get to elements in the list that are larger than 5,
163 we can stop. If we haven't found 5 already, we know it's not in the rest of the
166 This is an improvement, but it's still a "linear" search through the list.
167 There are even more efficient methods, which employ "binary" searching. They'd
168 represent the set in such a way that you could quickly determine whether some
169 element fell in one half, call it the left half, of the structure that
170 implements the set, if it belonged to the set at all. Or that it fell in the
171 right half, it it belonged to the set at all. And then the same sort of
172 determination could be made for whichever half you were directed to. And then
173 for whichever quarter you were directed to next. And so on. Until you either
174 found the element or exhausted the structure and could then conclude that the
175 element in question was not part of the set. These sorts of structures are done
176 using **binary trees** (see below).
179 4. **Aborting a search through a list**
181 We said that the sorted-list implementation of a set was more efficient than
182 the unsorted-list implementation, because as you were searching through the
183 list, you could come to a point where you knew the element wasn't going to be
184 found. So you wouldn't have to continue the search.
186 If your implementation of lists was, say v1 lists plus the Y-combinator, then
187 this is exactly right. When you get to a point where you know the answer, you
188 can just deliver that answer, and not branch into any further recursion. If
189 you've got the right evaluation strategy in place, everything will work out
192 But what if you're using v3 lists? What options would you have then for
195 Well, suppose we're searching through the list [5; 4; 3; 2; 1] to see if it
196 contains the number 3. The expression which represents this search would have
197 something like the following form:
199 ..................<eq? 1 3> ~~>
200 .................. false ~~>
201 .............<eq? 2 3> ~~>
202 ............. false ~~>
203 .........<eq? 3 3> ~~>
207 Of course, whether those reductions actually followed in that order would
208 depend on what reduction strategy was in place. But the result of folding the
209 search function over the part of the list whose head is 3 and whose tail is [2;
210 1] will *semantically* depend on the result of applying that function to the
211 more rightmost pieces of the list, too, regardless of what order the reduction
212 is computed by. Conceptually, it will be easiest if we think of the reduction
213 happening in the order displayed above.
215 Well, once we've found a match between our sought number 3 and some member of
216 the list, we'd like to avoid any further unnecessary computations and just
217 deliver the answer `true` as "quickly" or directly as possible to the larger
218 computation in which the search was embedded.
220 With a Y-combinator based search, as we said, we could do this by just not
221 following a recursion branch.
223 But with the v3 lists, the fold is "pre-programmed" to continue over the whole
224 list. There is no way for us to bail out of applying the search function to the
225 parts of the list that have head 4 and head 5, too.
227 We *can* avoid some unneccessary computation. The search function can detect
228 that the result we've accumulated so far during the fold is now true, so we
229 don't need to bother comparing 4 or 5 to 3 for equality. That will simplify the
230 computation to some degree, since as we said, numerical comparison in the
231 system we're working in is moderately expensive.
233 However, we're still going to have to traverse the remainder of the list. That
234 `true` result will have to be passed along all the way to the leftmost head of
235 the list. Only then can we deliver it to the larger computation in which the
238 It would be better if there were some way to "abort" the list traversal. If,
239 having found the element we're looking for (or having determined that the
240 element isn't going to be found), we could just immediately stop traversing the
241 list with our answer. Continuations will turn out to let us do that.
243 We won't try yet to fully exploit the terrible power of continuations. But
244 there's a way that we can gain their benefits here locally, without yet having
245 a fully general machinery or understanding of what's going on.
247 The key is to recall how our implementations of booleans and pairs worked.
248 Remember that with pairs, we supply the pair "handler" to the pair as *an
249 argument*, rather than the other way around:
257 to get the first element of the pair. Of course you can lift that if you want:
259 extract_1st === \pair. pair (\x y. x)
261 but at a lower level, the pair is still accepting its handler as an argument,
262 rather than the handler taking the pair as an argument. (The handler gets *the
263 pair's elements*, not the pair itself, as arguments.)
265 The v2 implementation of lists followed a similar strategy:
267 v2list (\h t. do_something_with_h_and_t) result_if_empty
269 If the v2list here is not empty, then this will reduce to the result of
270 supplying the list's head and tail to the handler `(\h t.
271 do_something_with_h_and_t)`.
273 Now, what we've been imagining ourselves doing with the search through the v3
274 list is something like this:
277 larger_computation (search_through_the_list_for_3) other_arguments
279 That is, the result of our search is supplied as an argument (perhaps together
280 with other arguments) to the "larger computation". Without knowing the
281 evaluation order/reduction strategy, we can't say whether the search is
282 evaluated before or after it's substituted into the larger computation. But
283 semantically, the search is the argument and the larger computation is the
284 function to which it's supplied.
286 What if, instead, we did the same kind of thing we did with pairs and v2 lists? That is, what if we made the larger computation a "handler" that we passed as an argument to the search?
288 the_search (\search_result. larger_computation search_result other_arguments)
290 What's the advantage of that, you say. Other than to show off how cleverly you can lift.
292 Well, think about it. Think about the difficulty we were having aborting the
293 search. Does this switch-around offer us anything useful?
297 What if the way we implemented the search procedure looked something like this?
299 At a given stage in the search, we wouldn't just apply some function f to the
300 head at this stage and the result accumulated so far, from folding the same
301 function (and a base value) to the tail at this stage. And then pass the result
302 of doing so leftward along the rest of the list.
304 We'd also give that function a "handler" that expected the result of the
305 current stage as an argument, and evaluated to passing that result leftwards
306 along the rest of the list.
308 Why would we do that, you say? Just more flamboyant lifting?
310 Well, no, there's a real point here. If we give the function a "handler" that
311 encodes the normal continuation of the fold leftwards through the list. We can
312 give it another "handler" as well. We can also give it the underlined handler:
315 the_search (\search_result. larger_computation search_result other_arguments)
316 ------------------------------------------------------------------
318 This "handler" encodes the search's having finished, and delivering a final
319 answer to whatever else you wanted your program to do with the result of the
320 search. If you like, at any stage in the search you might just give an argument
321 to this handler, instead of giving an argument to the handler that continues
322 the list traversal leftwards. Semantically, this would amount to *aborting* the
323 list traversal! (As we've said before, whether the rest of the list traversal
324 really gets evaluated will depend on what evaluation order is in place. But
325 semantically we'll have avoided it. Our larger computation won't depend on the
326 rest of the list traversal having been computed.)
328 Do you have the basic idea? Think about how you'd implement it. A good
329 understanding of the v2 lists will give you a helpful model.
331 In broad outline, a single stage of the search would look like before, except
332 now f would receive two extra, "handler" arguments.
334 f 3 <result of folding f and z over [2; 1]> <handler to continue folding leftwards> <handler to abort the traversal>
336 f's job would be to check whether 3 matches the element we're searching for
337 (here also 3), and if it does, just evaluate to the result of passing `true` to
338 the abort handler. If it doesn't, then evaluate to the result of passing
339 `false` to the continue-leftwards handler.
341 In this case, f wouldn't need to consult the result of folding f and z over [2;
342 1], since if we had found the element 3 in more rightward positions of the
343 list, we'd have called the abort handler and this application of f to 3 etc
344 would never be needed. However, in other applications the result of folding f
345 and z over the more rightward parts of the list would be needed. Consider if
346 you were trying to multiply all the elements of the list, and were going to
347 abort (with the result 0) if you came across any element in the list that was
348 zero. If you didn't abort, you'd need to know what the more rightward elements
349 of the list multiplied to, because that would affect the answer you passed
350 along to the continue-leftwards handler.
352 A **version 5** list would encode this kind of fold operation over the list, in
353 the same way that v3 (and v4) lists encoded the simpler fold operation.
354 Roughly, the list [5; 4; 3; 2; 1] would look like this:
357 \f z continue_leftwards_handler abort_handler.
358 <fold f and z over [4; 3; 2; 1]>
359 (\result_of_fold_over_4321. f 5 result_of_fold_over_4321 continue_leftwards_handler abort_handler)
363 \f z continue_leftwards_handler abort_handler.
364 (\continue_leftwards_handler abort_handler.
365 <fold f and z over [3; 2; 1]>
366 (\result_of_fold_over_321. f 4 result_of_fold_over_321 continue_leftwards_handler abort_handler)
369 (\result_of_fold_over_4321. f 5 result_of_fold_over_4321 continue_leftwards_handler abort_handler)
374 Remarks: the `larger_computation_handler` should be supplied as both the
375 `continue_leftwards_handler` and the `abort_handler` for the leftmost
376 application, where the head 5 is supplied to f. Because the result of this
377 application should be passed to the larger computation, whether it's a "fall
378 off the left end of the list" result or it's a "I'm finished, possibly early"
379 result. The `larger_computation_handler` also then gets passed to the next
380 rightmost stage, where the head 4 is supplied to f, as the `abort_handler` to
381 use if that stage decides it has an early answer.
383 Finally, notice that we don't have the result of applying f to 4 etc given as
384 an argument to the application of f to 5 etc. Instead, we pass
386 (\result_of_fold_over_4321. f 5 result_of_fold_over_4321 one_handler another_handler)
388 *to* the application of f to 4 as its "continue" handler. The application of f
389 to 4 can decide whether this handler, or the other, "abort" handler, should be
390 given an argument and constitute its result.
393 I'll say once again: we're using temporally-loaded vocabulary throughout this,
394 but really all we're in a position to mean by that are claims about the result
395 of the complex expression semantically depending only on this, not on that. A
396 demon evaluator who custom-picked the evaluation order to make things maximally
397 bad for you could ensure that all the semantically unnecessary computations got
398 evaluated anyway. At this stage, we don't have any way to prevent that. Later,
399 we'll see ways to semantically guarantee one evaluation order rather than
400 another. Though even then the demonic evaluation-order-chooser could make it
401 take unnecessarily long to compute the semantically guaranteed result. Of
402 course, in any real computing environment you'll know you're dealing with a
403 fixed evaluation order and you'll be able to program efficiently around that.
405 In detail, then, here's what our v5 lists will look like:
407 let empty = \f z continue_handler abort_handler. continue_handler z in
408 let make_list = \h t. \f z continue_handler abort_handler.
409 t f z (\sofar. f h sofar continue_handler abort_handler) abort_handler in
410 let isempty = \lst larger_computation. lst
412 (\hd sofar continue_handler abort_handler. abort_handler false)
415 ; here's the continue_handler for the leftmost application of f
417 ; here's the abort_handler
418 larger_computation in
419 let extract_head = \lst larger_computation. lst
421 (\hd sofar continue_handler abort_handler. continue_handler hd)
424 ; here's the continue_handler for the leftmost application of f
426 ; here's the abort_handler
427 larger_computation in
428 let extract_tail = ; left as exercise
429 ;; for real efficiency, it'd be nice to fuse the apparatus developed
430 ;; in these v5 lists with the ideas from the v4 lists, above
431 ;; but that also is left as an exercise
433 These functions are used like this:
435 let my_list = make_list a (make_list b (make_list c empty) in
436 extract_head my_list larger_computation
438 If you just want to see `my_list`'s head, the use `I` as the
439 `larger_computation`.
441 What we've done here does take some work to follow. But it should be within
442 your reach. And once you have followed it, you'll be well on your way to
443 appreciating the full terrible power of continuations. (Silly [cultural
444 reference](http://www.newgrounds.com/portal/view/33440).)
446 Of course, like everything elegant and exciting in this seminar, [Oleg
447 discusses it in much more
448 detail](http://okmij.org/ftp/Streams.html#enumerator-stream).
452 5. Implementing (self-balancing) trees