+`prefact` if we are forced to.
+
+
+#Q. You claimed that the Ackerman function couldn't be implemented using our primitive recursion techniques (such as the techniques that allow us to define addition and multiplication). But you haven't shown that it is possible to define the Ackerman function using full recursion.#
+
+
+A. OK:
+
+ A(m,n) =
+ | when m == 0 -> n + 1
+ | else when n == 0 -> A(m-1,1)
+ | else -> A(m-1, A(m,n-1))
+
+ let A = Y (\A m n. iszero m (succ n) (iszero n (A (pred m) 1) (A (pred m) (A m (pred n)))))
+
+So for instance:
+
+ A 1 2
+ ~~> A 0 (A 1 1)
+ ~~> A 0 (A 0 (A 1 0))
+ ~~> A 0 (A 0 (A 0 1))
+ ~~> A 0 (A 0 2)
+ ~~> A 0 3
+ ~~> 4
+
+`A 1 x` is to `A 0 x` as addition is to the successor function;
+`A 2 x` is to `A 1 x` as multiplication is to addition;
+`A 3 x` is to `A 2 x` as exponentiation is to multiplication---
+so `A 4 x` is to `A 3 x` as hyper-exponentiation is to exponentiation...
+
+#Q. What other questions should I be asking?#
+
+* What is it about the variant fixed-point combinators that makes
+ them compatible with a call-by-value evaluation strategy?
+
+* How do you know that the Ackerman function can't be computed
+ using primitive recursion techniques?
+
+* What *exactly* is primitive recursion?
+
+* I hear that `Y` delivers the *least* fixed point. Least
+ according to what ordering? How do you know it's least?
+ Is leastness important?
+
+
+
+#Sets#
+
+You're now already in a position to implement sets: that is, collections with
+no intrinsic order where elements can occur at most once. Like lists, we'll
+understand the basic set structures to be *type-homogenous*. So you might have
+a set of integers, or you might have a set of pairs of integers, but you
+wouldn't have a set that mixed both types of elements. Something *like* the
+last option is also achievable, but it's more difficult, and we won't pursue it
+now. In fact, we won't talk about sets of pairs, either. We'll just talk about
+sets of integers. The same techniques we discuss here could also be applied to
+sets of pairs of integers, or sets of triples of booleans, or sets of pairs
+whose first elements are booleans, and whose second elements are triples of
+integers. And so on.
+
+(You're also now in a position to implement *multi*sets: that is, collections
+with no intrinsic order where elements can occur multiple times: the multiset
+{a,a} is distinct from the multiset {a}. But we'll leave these as an exercise.)
+
+The easiest way to implement sets of integers would just be to use lists. When
+you "add" a member to a set, you'd get back a list that was either identical to
+the original list, if the added member already was present in it, or consisted
+of a new list with the added member prepended to the old list. That is:
+
+ let empty_set = empty in
+ ; 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
+ ; else
+ make_list new_member old_set
+
+Think about how you'd implement operations like `set_union`,
+`set_intersection`, and `set_difference` with this implementation of sets.
+
+The implementation just described works, and it's the simplest to code.
+However, it's pretty inefficient. If you had a 100-member set, and you wanted
+to create a set which had all those 100-members and some possibly new element
+`e`, you might need to check all 100 members to see if they're equal to `e`
+before concluding they're not, and returning the new list. And comparing for
+numeric equality is a moderately expensive operation, in the first place.
+
+(You might say, well, what's the harm in just prepending `e` to the list even
+if it already occurs later in the list. The answer is, if you don't keep track
+of things like this, it will likely mess up your implementations of
+`set_difference` and so on. You'll have to do the book-keeping for duplicates
+at some point in your code. It goes much more smoothly if you plan this from
+the very beginning.)
+
+How might we make the implementation more efficient? Well, the *semantics* of
+sets says that they have no intrinsic order. That means, there's no difference
+between the set {a,b} and the set {b,a}; whereas there is a difference between
+the *list* `[a;b]` and the list `[b;a]`. But this semantic point can be respected
+even if we *implement* sets with something ordered, like list---as we're
+already doing. And we might *exploit* the intrinsic order of lists to make our
+implementation of sets more efficient.
+
+What we could do is arrange it so that a list that implements a set always
+keeps in elements in some specified order. To do this, there'd have *to be*
+some way to order its elements. Since we're talking now about sets of numbers,
+that's easy. (If we were talking about sets of pairs of numbers, we'd use
+"lexicographic" ordering, where `(a,b) < (c,d)` iff `a < c or (a == c and b <
+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
+list either.
+
+*Comment*: This is an improvement, but it's still a "linear" search through the list.
+There are even more efficient methods, which employ "binary" searching. They'd
+represent the set in such a way that you could quickly determine whether some
+element fell in one half, call it the left half, of the structure that
+implements the set, if it belonged to the set at all. Or that it fell in the
+right half, it it belonged to the set at all. And then the same sort of
+determination could be made for whichever half you were directed to. And then
+for whichever quarter you were directed to next. And so on. Until you either
+found the element or exhausted the structure and could then conclude that the
+element in question was not part of the set. These sorts of structures are done
+using [binary trees](/implementing_trees).
+
+
+#Aborting a search through a list#
+
+We said that the sorted-list implementation of a set was more efficient than
+the unsorted-list implementation, because as you were searching through the
+list, you could come to a point where you knew the element wasn't going to be
+found. So you wouldn't have to continue the search.
+
+If your implementation of lists was, say v1 lists plus the Y-combinator, then
+this is exactly right. When you get to a point where you know the answer, you
+can just deliver that answer, and not branch into any further recursion. If
+you've got the right evaluation strategy in place, everything will work out
+fine.
+
+--
+An advantage of the v3 lists and v3 (aka "Church") numerals is that they
+have a recursive capacity built into their skeleton. So for many natural
+operations on them, you won't need to use a fixed point combinator. Why is
+that an advantage? Well, if you use a fixed point combinator, then the terms
+you get
+won't be strongly normalizing: whether their reduction stops at a normal form
+will depend on what evaluation order you use. Our online [[lambda evaluator]]
+uses normal-order reduction, so it finds a normal form if there's one to be
+had. But if you want to build lambda terms in, say, Scheme, and you wanted to
+roll your own recursion as we've been doing, rather than relying on Scheme's
+native `let rec` or `define`, then you can't use the fixed-point combinators
+`Y` or <code>Θ</code>. Expressions using them will have non-terminating
+reductions, with Scheme's eager/call-by-value strategy. There are other
+fixed-point combinators you can use with Scheme (in the [week 3 notes](/week3/#index7h2) they
+were <code>Y′</code> and <code>Θ′</code>. But even with
+them, evaluation order still matters: for some (admittedly unusual)
+evaluation strategies, expressions using them will also be non-terminating.
+
+The fixed-point combinators may be the conceptual stars. They are cool and
+mathematically elegant. But for efficiency and implementation elegance, it's
+best to know how to do as much as you can without them. (Also, that knowledge
+could carry over to settings where the fixed point combinators are in
+principle unavailable.)
+
+This is why the v3 lists and numbers are so lovely..
+
+--
+
+But what if you're using v3 lists? What options would you have then for
+aborting a search?
+
+Well, suppose we're searching through the list `[5;4;3;2;1]` to see if it
+contains the number `3`. The expression which represents this search would have
+something like the following form:
+
+ ..................<eq? 1 3> ~~>
+ .................. false ~~>
+ .............<eq? 2 3> ~~>
+ ............. false ~~>
+ .........<eq? 3 3> ~~>
+ ......... true ~~>
+ ?
+
+Of course, whether those reductions actually followed in that order would
+depend on what reduction strategy was in place. But the result of folding the
+search function over the part of the list whose head is `3` and whose tail is `[2;
+1]` will *semantically* depend on the result of applying that function to the
+more rightmost pieces of the list, too, regardless of what order the reduction
+is computed by. Conceptually, it will be easiest if we think of the reduction
+happening in the order displayed above.
+
+Well, once we've found a match between our sought number `3` and some member of
+the list, we'd like to avoid any further unnecessary computations and just
+deliver the answer `true` as "quickly" or directly as possible to the larger
+computation in which the search was embedded.
+
+With a Y-combinator based search, as we said, we could do this by just not
+following a recursion branch.
+
+But with the v3 lists, the fold is "pre-programmed" to continue over the whole
+list. There is no way for us to bail out of applying the search function to the
+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
+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.
+
+However, we're still going to have to traverse the remainder of the list. That
+`true` result will have to be passed along all the way to the leftmost head of
+the list. Only then can we deliver it to the larger computation in which the
+search was embedded.
+
+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.
+
+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
+a fully general machinery or understanding of what's going on.
+
+The key is to recall how our implementations of booleans and pairs worked.
+Remember that with pairs, we supply the pair "handler" to the pair as *an
+argument*, rather than the other way around:
+
+ pair (\x y. add x y)
+
+or:
+
+ pair (\x y. x)
+
+to get the first element of the pair. Of course you can lift that if you want:
+
+<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
+supplying the list's head and tail to the handler `(\h t.
+do_something_with_h_and_t)`.
+
+Now, what we've been imagining ourselves doing with the search through the v3
+list is something like this:
+
+
+ larger_computation (search_through_the_list_for_3) other_arguments
+
+That is, the result of our search is supplied as an argument (perhaps together
+with other arguments) to the "larger computation". Without knowing the
+evaluation order/reduction strategy, we can't say whether the search is
+evaluated before or after it's substituted into the larger computation. But
+semantically, the search is the argument and the larger computation is the
+function to which it's supplied.
+
+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?
+
+ the_search (\search_result. larger_computation search_result other_arguments)
+
+What's the advantage of that, you say. Other than to show off how cleverly
+you can lift.
+
+Well, think about it. Think about the difficulty we were having aborting the
+search. Does this switch-around offer us anything useful?
+
+It could.
+
+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 that application to the embedding, more leftward computation.
+
+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
+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)
+ ------------------------------------------------------------------