#Miscellaneous challenges and advanced topics, for untyped lambda calculus#
1. How would you define an operation to reverse a list? (Don't peek at the
[[lambda_library]]! Try to figure it out on your own.) Choose whichever
implementation of list you like. Even then, there are various strategies you
can use.
2. 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 `Θ`

. 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 they
were `Y′`

and `Θ′`

. But even with
those 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 heros. 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. However, one disadvantage
to them is that it's relatively inefficient to extract a list's tail, or get a
number's predecessor. To get the tail of the list [a;b;c;d;e], one will
basically be performing some operation that builds up the tail afresh: at
different stages, one will have built up [e], then [d;e], then [c;d;e], and
finally [b;c;d;e]. With short lists, this is no problem, but with longer lists
it takes longer and longer. And it may waste more of your computer's memory
than you'd like. Similarly for obtaining a number's predecessor.
The v1 lists and numbers on the other hand, had the tail and the predecessor
right there as an element, easy for the taking. The problem was just that the
v1 lists and numbers didn't have recursive capacity built into them, in the
way the v3 implementations do.
A clever approach would marry these two strategies.
Version 3 makes the list [a; b; c; d; e] look like this:
\f z. f a (f b (f c (f d (f e z))))
or in other words:
\f z. f a
Instead we could make it look like this:
\f z. f a
That is, now f is a function expecting *three* arguments: the head of the
current list, the tail of the current list, and the result of continuing to
fold f over the tail, with a given base value z.
Call this a **version 4** list. The empty list could be the same:
empty === \f z. z
The list constructor would be:
make_list === \h t. \f z. f h t (t f z)
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:
\s z. s (s (s (s (s z))))
or in other words:
\s z. s
Instead we could make it look like this:
\s z. s
That is, now s is a function expecting *two* arguments: the predecessor of the
current number, and the result of continuing to apply s to the base value z
predecessor-many times.
Jim had the pleasure of "inventing" these implementations himself. However,
unsurprisingly, he wasn't the first to do so. See for example [Oleg's report
on P-numerals](http://okmij.org/ftp/Computation/lambda-calc.html#p-numerals).
3. Implementing (self-balancing) trees