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.
+> *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
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
+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.
understanding of the v2 lists will give you a helpful model.
In broad outline, a single stage of the search would look like before, except
-now f would receive two extra, "handler" arguments.
+now `f` would receive two extra, "handler" arguments.
f 3 <result of folding f and z over [2; 1]> <handler to continue folding leftwards> <handler to abort the traversal>
of the list multiplied to, because that would affect the answer you passed
along to the continue-leftwards handler.
-A **version 5** list encodes the kind of fold operation we're envisaging here, in
-the same way that v3 (and [v4](/advanced/#index1h1)) lists encoded the simpler fold operation.
-Roughly, the list `[5;4;3;2;1]` would look like this:
+A **version 5** list encodes the kind of fold operation we're envisaging here,
+in the same way that v3 (and [v4](/advanced_lambda/#index1h1)) lists encoded
+the simpler fold operation. Roughly, the list `[5;4;3;2;1]` would look like
+this:
\f z continue_leftwards_handler abort_handler.
discusses it in much more
detail](http://okmij.org/ftp/Streams.html#enumerator-stream).
-*Comments*:
-
-1. The technique deployed here, and in the v2 lists, and in our implementations
- of pairs and booleans, is known as **continuation-passing style** programming.
-
-2. We're still building the list as a right fold, so in a sense the
- application of `f` to the leftmost element `5` is "outermost". However,
- this "outermost" application is getting lifted, and passed as a *handler*
- to the next right application. Which is in turn getting lifted, and
- passed to its next right application, and so on. So if you
- trace the evaluation of the `extract_head` function to the list `[5;4;3;2;1]`,
- you'll see `1` gets passed as a "this is the head sofar" answer to its
- `continue_handler`; then that answer is discarded and `2` is
- passed as a "this is the head sofar" answer to *its* `continue_handler`,
- and so on. All those steps have to be evaluated to finally get the result
- that `5` is the outer/leftmost head of the list. That's not an efficient way
- to get the leftmost head.
-
- We could improve this by building lists as left folds when implementing them
- as continuation-passing style folds. We'd just replace above:
-
- let make_list = \h t. \f z continue_handler abort_handler.
- f h z (\z. t f z continue_handler abort_handler) abort_handler
-
- now `extract_head` should return the leftmost head directly, using its `abort_handler`:
-
- let extract_head = \lst larger_computation. lst
- (\hd sofar continue_handler abort_handler. abort_handler hd)
- junk
- larger_computation
- larger_computation
-
-3. To extract tails efficiently, too, it'd be nice to fuse the apparatus developed
- in these v5 lists with the ideas from [v4](/advanced/#index1h1) lists.
- But that also is left as an exercise.
+> *Comments*:
+
+> 1. The technique deployed here, and in the v2 lists, and in our
+> implementations of pairs and booleans, is known as
+> **continuation-passing style** programming.
+
+> 2. We're still building the list as a right fold, so in a sense the
+> application of `f` to the leftmost element `5` is "outermost". However,
+> this "outermost" application is getting lifted, and passed as a *handler*
+> to the next right application. Which is in turn getting lifted, and
+> passed to its next right application, and so on. So if you
+> trace the evaluation of the `extract_head` function to the list `[5;4;3;2;1]`,
+> you'll see `1` gets passed as a "this is the head sofar" answer to its
+> `continue_handler`; then that answer is discarded and `2` is
+> passed as a "this is the head sofar" answer to *its* `continue_handler`,
+> and so on. All those steps have to be evaluated to finally get the result
+> that `5` is the outer/leftmost head of the list. That's not an efficient way
+> to get the leftmost head.
+>
+> We could improve this by building lists as left folds when implementing them
+> as continuation-passing style folds. We'd just replace above:
+>
+> let make_list = \h t. \f z continue_handler abort_handler.
+> f h z (\z. t f z continue_handler abort_handler) abort_handler
+>
+> now `extract_head` should return the leftmost head directly, using its
+> `abort_handler`:
+>
+> let extract_head = \lst larger_computation. lst
+> (\hd sofar continue_handler abort_handler. abort_handler hd)
+> junk
+> larger_computation
+> larger_computation
+>
+> 3. To extract tails efficiently, too, it'd be nice to fuse the apparatus
+> developed in these v5 lists with the ideas from
+> [v4](/advanced_lambda/#index1h1) lists. But that is left as an exercise.
+