Version 4 lists: Efficiently extracting tails
Version 3 lists and Church numerals are lovely, because they have their recursive capacity built into their very bones. 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 <the result of folding f and z over the tail>
Instead we could make it look like this:
\f z. f a <the tail itself> <the result of folding f and z over the tail>
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 can be the same as in v3:
empty ≡ \f z. zThe 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, five as a v3 or Church numeral looks like this:
\s z. s (s (s (s (s z))))
or in other words:
\s z. s <the result of applying s to z (pred 5)-many times>
Instead we could make it look like this:
\s z. s <pred 5> <the result of applying s to z (pred 5)-many times>
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.