-1. How would you define an operation to reverse a list? (Don't peek at the
+[[!toc]]
+
+#Reversing a list#
+
+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 <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.
+#Efficiently extracting tails and predecessors#
- 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.)
+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.
- 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 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.)
- 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.
+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.
- A clever approach would marry these two strategies.
+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.
- Version 3 makes the list `[a;b;c;d;e]` look like this:
+A clever approach would marry these two strategies.
- \f z. f a (f b (f c (f d (f e z))))
+Version 3 makes the list `[a;b;c;d;e]` look like this:
- or in other words:
+ \f z. f a (f b (f c (f d (f e z))))
- \f z. f a <the result of folding f and z over the tail>
+or in other words:
- Instead we could make it look like this:
+ \f z. f a <the result of folding f and z over the tail>
- \f z. f a <the tail itself> <the result of folding f and z over the tail>
+Instead we could make it look like this:
- 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`.
+ \f z. f a <the tail itself> <the result of folding f and z over the tail>
- Call this a **version 4** list. The empty list can be the same as in v3:
+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`.
- <pre><code>empty ≡ \f z. z</code></pre>
+Call this a **version 4** list. The empty list can be the same as in v3:
- The list constructor would be:
-
- <pre><code>make_list ≡ \h t. \f z. f h t (t f z)</code></pre>
-
- 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:
+<pre><code>empty ≡ \f z. z</code></pre>
- \s z. s <the result of applying s to z (pred 5)-many times>
-
- Instead we could make it look like this:
+The list constructor would be:
- \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](http://okmij.org/ftp/Computation/lambda-calc.html#p-numerals).
-
-
-
-3. **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.
+<pre><code>make_list ≡ \h t. \f z. f h t (t f z)</code></pre>
+
+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))))
- (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.)
+or in other words:
- 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)`.)
+ \s z. s <the result of applying s to z (pred 5)-many times>
- 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.
-
- 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** (see below).
-
-
-4. **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.
-
- 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.
+Instead we could make it look like this:
- 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.
+ \s z. s <pred 5> <the result of applying s to z (pred 5)-many times>
- 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.
+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.
- 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:
+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).
- pair (\x y. add x y)
- or:
- pair (\x y. x)
+#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.)
- to get the first element of the pair. Of course you can lift that if you want:
+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)`.)
- <pre><code>extract_fst ≡ \pair. pair (\x y. x)</code></pre>
+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.
+
+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** (see below).
+
+
+#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.
+
+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.
- 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.)
+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.
- > *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.
+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 v2 implementation of lists followed a similar strategy:
+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:
- v2list (\h t. do_something_with_h_and_t) result_if_empty
+ pair (\x y. add x y)
- 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)`.
+or:
- Now, what we've been imagining ourselves doing with the search through the v3
- list is something like this:
+ pair (\x y. x)
+to get the first element of the pair. Of course you can lift that if you want:
- larger_computation (search_through_the_list_for_3) other_arguments
+<pre><code>extract_fst ≡ \pair. pair (\x y. x)</code></pre>
- 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.
+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.)
- 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?
+> *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_search (\search_result. larger_computation search_result other_arguments)
+The v2 implementation of lists followed a similar strategy:
- What's the advantage of that, you say. Other than to show off how cleverly
- you can lift.
+ v2list (\h t. do_something_with_h_and_t) result_if_empty
- Well, think about it. Think about the difficulty we were having aborting the
- search. Does this switch-around offer us anything useful?
+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)`.
- It could.
+Now, what we've been imagining ourselves doing with the search through the v3
+list is something like this:
- 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.
+ larger_computation (search_through_the_list_for_3) other_arguments
- 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.
+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.
- Why would we do that, you say? Just more flamboyant lifting?
+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?
- 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)
+What's the advantage of that, you say. Other than to show off how cleverly
+you can lift.
- the_search (\search_result. larger_computation search_result other_arguments)
- ------------------------------------------------------------------
+Well, think about it. Think about the difficulty we were having aborting the
+search. Does this switch-around offer us anything useful?
- This "handler" encodes the search's having finished, and delivering a final
- answer to whatever else you wanted your program to do with the result of the
- search. If you like, at any stage in the search you might just give an argument
- to *this* handler, instead of giving an argument to the handler that continues
- the list traversal leftwards. Semantically, this would amount to *aborting* the
- list traversal! (As we've said before, whether the rest of the list traversal
- really gets evaluated will depend on what evaluation order is in place. But
- semantically we'll have avoided it. Our larger computation won't depend on the
- rest of the list traversal having been computed.)
+It could.
- Do you have the basic idea? Think about how you'd implement it. A good
- understanding of the v2 lists will give you a helpful model.
+What if the way we implemented the search procedure looked something like this?
- In broad outline, a single stage of the search would look like before, except
- now f would receive two extra, "handler" arguments.
+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.
- f 3 <result of folding f and z over [2; 1]> <handler to continue folding leftwards> <handler to abort the traversal>
+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.
- `f`'s job would be to check whether `3` matches the element we're searching for
- (here also `3`), and if it does, just evaluate to the result of passing `true` to
- the abort handler. If it doesn't, then evaluate to the result of passing
- `false` to the continue-leftwards handler.
+Why would we do that, you say? Just more flamboyant lifting?
- In this case, `f` wouldn't need to consult the result of folding `f` and `z` over `[2;
- 1]`, since if we had found the element `3` in more rightward positions of the
- list, we'd have called the abort handler and this application of `f` to `3` etc
- would never be needed. However, in other applications the result of folding `f`
- and `z` over the more rightward parts of the list would be needed. Consider if
- you were trying to multiply all the elements of the list, and were going to
- abort (with the result `0`) if you came across any element in the list that was
- zero. If you didn't abort, you'd need to know what the more rightward elements
- of the list multiplied to, because that would affect the answer you passed
- along to the continue-leftwards handler.
+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:
- A **version 5** list encodes the kind of fold operation we're envisaging here, in
- the same way that v3 (and v4) lists encoded the simpler fold operation.
- Roughly, the list `[5;4;3;2;1]` would look like this:
+ the_search (\search_result. larger_computation search_result other_arguments)
+ ------------------------------------------------------------------
- \f z continue_leftwards_handler abort_handler.
- <fold f and z over [4;3;2;1]>
- (\result_of_fold_over_4321. f 5 result_of_fold_over_4321 continue_leftwards_handler abort_handler)
- abort_handler
+This "handler" encodes the search's having finished, and delivering a final
+answer to whatever else you wanted your program to do with the result of the
+search. If you like, at any stage in the search you might just give an argument
+to *this* handler, instead of giving an argument to the handler that continues
+the list traversal leftwards. Semantically, this would amount to *aborting* the
+list traversal! (As we've said before, whether the rest of the list traversal
+really gets evaluated will depend on what evaluation order is in place. But
+semantically we'll have avoided it. Our larger computation won't depend on the
+rest of the list traversal having been computed.)
- ; or, expanding the fold over [4;3;2;1]:
+Do you have the basic idea? Think about how you'd implement it. A good
+understanding of the v2 lists will give you a helpful model.
- \f z continue_leftwards_handler abort_handler.
- (\continue_leftwards_handler abort_handler.
- <fold f and z over [3;2;1]>
- (\result_of_fold_over_321. f 4 result_of_fold_over_321 continue_leftwards_handler abort_handler)
- abort_handler
- )
- (\result_of_fold_over_4321. f 5 result_of_fold_over_4321 continue_leftwards_handler abort_handler)
- abort_handler
+In broad outline, a single stage of the search would look like before, except
+now f would receive two extra, "handler" arguments.
- ; and so on
-
- Remarks: the `larger_computation` handler should be supplied as both the
- `continue_leftwards_handler` and the `abort_handler` for the leftmost
- application, where the head `5` is supplied to `f`; because the result of this
- application should be passed to the larger computation, whether it's a "fall
- off the left end of the list" result or it's a "I'm finished, possibly early"
- result. The `larger_computation` handler also then gets passed to the next
- rightmost stage, where the head `4` is supplied to `f`, as the `abort_handler` to
- use if that stage decides it has an early answer.
-
- Finally, notice that we don't have the result of applying `f` to `4` etc given as
- an argument to the application of `f` to `5` etc. Instead, we pass
-
- (\result_of_fold_over_4321. f 5 result_of_fold_over_4321 <one_handler> <another_handler>)
-
- *to* the application of `f` to `4` as its "continue" handler. The application of `f`
- to `4` can decide whether this handler, or the other, "abort" handler, should be
- given an argument and constitute its result.
-
-
- I'll say once again: we're using temporally-loaded vocabulary throughout this,
- but really all we're in a position to mean by that are claims about the result
- of the complex expression semantically depending only on this, not on that. A
- demon evaluator who custom-picked the evaluation order to make things maximally
- bad for you could ensure that all the semantically unnecessary computations got
- evaluated anyway. We don't have any way to prevent that. Later,
- we'll see ways to *semantically guarantee* one evaluation order rather than
- another. Though even then the demonic evaluation-order-chooser could make it
- take unnecessarily long to compute the semantically guaranteed result. Of
- course, in any real computing environment you'll know you're dealing with a
- fixed evaluation order and you'll be able to program efficiently around that.
-
- In detail, then, here's what our v5 lists will look like:
-
- let empty = \f z continue_handler abort_handler. continue_handler z in
- let make_list = \h t. \f z continue_handler abort_handler.
- t f z (\sofar. f h sofar continue_handler abort_handler) abort_handler in
- let isempty = \lst larger_computation. lst
- ; here's our f
- (\hd sofar continue_handler abort_handler. abort_handler false)
- ; here's our z
- true
- ; here's the continue_handler for the leftmost application of f
- larger_computation
- ; here's the abort_handler
- larger_computation in
- let extract_head = \lst larger_computation. lst
- ; here's our f
- (\hd sofar continue_handler abort_handler. continue_handler hd)
- ; here's our z
- junk
- ; here's the continue_handler for the leftmost application of f
- larger_computation
- ; here's the abort_handler
- larger_computation in
- let extract_tail = ; left as exercise
-
- These functions are used like this:
+ f 3 <result of folding f and z over [2; 1]> <handler to continue folding leftwards> <handler to abort the traversal>
- let my_list = make_list a (make_list b (make_list c empty) in
- extract_head my_list larger_computation
+`f`'s job would be to check whether `3` matches the element we're searching for
+(here also `3`), and if it does, just evaluate to the result of passing `true` to
+the abort handler. If it doesn't, then evaluate to the result of passing
+`false` to the continue-leftwards handler.
- If you just want to see `my_list`'s head, the use `I` as the
- `larger_computation`.
+In this case, `f` wouldn't need to consult the result of folding `f` and `z` over `[2;
+1]`, since if we had found the element `3` in more rightward positions of the
+list, we'd have called the abort handler and this application of `f` to `3` etc
+would never be needed. However, in other applications the result of folding `f`
+and `z` over the more rightward parts of the list would be needed. Consider if
+you were trying to multiply all the elements of the list, and were going to
+abort (with the result `0`) if you came across any element in the list that was
+zero. If you didn't abort, you'd need to know what the more rightward elements
+of the list multiplied to, because that would affect the answer you passed
+along to the continue-leftwards handler.
- What we've done here does take some work to follow. But it should be within
- your reach. And once you have followed it, you'll be well on your way to
- appreciating the full terrible power of continuations.
+A **version 5** list encodes the kind of fold operation we're envisaging here, in
+the same way that v3 (and v4) lists encoded the simpler fold operation.
+Roughly, the list `[5;4;3;2;1]` would look like this:
- <!-- (Silly [cultural reference](http://www.newgrounds.com/portal/view/33440).) -->
- Of course, like everything elegant and exciting in this seminar, [Oleg
- discusses it in much more
- detail](http://okmij.org/ftp/Streams.html#enumerator-stream).
+ \f z continue_leftwards_handler abort_handler.
+ <fold f and z over [4;3;2;1]>
+ (\result_of_fold_over_4321. f 5 result_of_fold_over_4321 continue_leftwards_handler abort_handler)
+ abort_handler
- *Comments*:
+ ; or, expanding the fold over [4;3;2;1]:
- 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.
+ \f z continue_leftwards_handler abort_handler.
+ (\continue_leftwards_handler abort_handler.
+ <fold f and z over [3;2;1]>
+ (\result_of_fold_over_321. f 4 result_of_fold_over_321 continue_leftwards_handler abort_handler)
+ abort_handler
+ )
+ (\result_of_fold_over_4321. f 5 result_of_fold_over_4321 continue_leftwards_handler abort_handler)
+ abort_handler
- We could improve this by building lists as left folds when implementing them
- as continuation-passing style folds. We'd just replace above:
+ ; and so on
- let make_list = \h t. \f z continue_handler abort_handler.
- f h z (\z. t f z continue_handler abort_handler) abort_handler
+Remarks: the `larger_computation` handler should be supplied as both the
+`continue_leftwards_handler` and the `abort_handler` for the leftmost
+application, where the head `5` is supplied to `f`; because the result of this
+application should be passed to the larger computation, whether it's a "fall
+off the left end of the list" result or it's a "I'm finished, possibly early"
+result. The `larger_computation` handler also then gets passed to the next
+rightmost stage, where the head `4` is supplied to `f`, as the `abort_handler` to
+use if that stage decides it has an early answer.
+
+Finally, notice that we don't have the result of applying `f` to `4` etc given as
+an argument to the application of `f` to `5` etc. Instead, we pass
+
+ (\result_of_fold_over_4321. f 5 result_of_fold_over_4321 <one_handler> <another_handler>)
+
+*to* the application of `f` to `4` as its "continue" handler. The application of `f`
+to `4` can decide whether this handler, or the other, "abort" handler, should be
+given an argument and constitute its result.
+
+
+I'll say once again: we're using temporally-loaded vocabulary throughout this,
+but really all we're in a position to mean by that are claims about the result
+of the complex expression semantically depending only on this, not on that. A
+demon evaluator who custom-picked the evaluation order to make things maximally
+bad for you could ensure that all the semantically unnecessary computations got
+evaluated anyway. We don't have any way to prevent that. Later,
+we'll see ways to *semantically guarantee* one evaluation order rather than
+another. Though even then the demonic evaluation-order-chooser could make it
+take unnecessarily long to compute the semantically guaranteed result. Of
+course, in any real computing environment you'll know you're dealing with a
+fixed evaluation order and you'll be able to program efficiently around that.
+
+In detail, then, here's what our v5 lists will look like:
+
+ let empty = \f z continue_handler abort_handler. continue_handler z in
+ let make_list = \h t. \f z continue_handler abort_handler.
+ t f z (\sofar. f h sofar continue_handler abort_handler) abort_handler in
+ let isempty = \lst larger_computation. lst
+ ; here's our f
+ (\hd sofar continue_handler abort_handler. abort_handler false)
+ ; here's our z
+ true
+ ; here's the continue_handler for the leftmost application of f
+ larger_computation
+ ; here's the abort_handler
+ larger_computation in
+ let extract_head = \lst larger_computation. lst
+ ; here's our f
+ (\hd sofar continue_handler abort_handler. continue_handler hd)
+ ; here's our z
+ junk
+ ; here's the continue_handler for the leftmost application of f
+ larger_computation
+ ; here's the abort_handler
+ larger_computation in
+ let extract_tail = ; left as exercise
+
+These functions are used like this:
+
+ let my_list = make_list a (make_list b (make_list c empty) in
+ extract_head my_list larger_computation
+
+If you just want to see `my_list`'s head, the use `I` as the
+`larger_computation`.
+
+What we've done here does take some work to follow. But it should be within
+your reach. And once you have followed it, you'll be well on your way to
+appreciating the full terrible power of continuations.
+
+<!-- (Silly [cultural reference](http://www.newgrounds.com/portal/view/33440).) -->
+
+Of course, like everything elegant and exciting in this seminar, [Oleg
+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:
- now `extract_head` should return the leftmost head directly, using its `abort_handler`:
+ 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
+ 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 the v4 lists, above.
- But that also is left as an exercise.
+3. To extract tails efficiently, too, it'd be nice to fuse the apparatus developed
+ in these v5 lists with the ideas from the v4 lists, above.
+ But that also is left as an exercise.
-5. **Implementing (self-balancing) trees**
+#Implementing trees#
- In [[Assignment3]] we proposed a very ad-hoc-ish implementation of trees.
+In [[Assignment3]] we proposed a very ad-hoc-ish implementation of trees.
- Think about how you'd implement them in a more principled way. You could
- use any of the version 1 -- version 5 implementation of lists as a model.
+Think about how you'd implement them in a more principled way. You could
+use any of the version 1 -- version 5 implementation of lists as a model.
- To keep things simple, I recommend starting with the version 3 pattern. And
- stick to binary trees.
+To keep things simple, we'll stick to binary trees. A node will either be a
+*leaf* of the tree, or it will have exactly two children.
- There are two kinds of trees to think about. In one sort of tree, it's only
- the tree's *leaves* that are labeled:
+There are two kinds of trees to think about. In one sort of tree, it's only
+the tree's leaves that are labeled:
- +
- / \
- + 3
- / \
- 1 2
+ .
+ / \
+ . 3
+ / \
+ 1 2
- Linguists often use trees of this sort. The inner, non-leaf nodes of the
-tree have associated values. But what values they are can be determined from
+Linguists often use trees of this sort. The inner, non-leaf nodes of the
+tree do have associated values. But what values they are can be determined from
the structure of the tree and the values of the node's left and right children.
So the inner node doesn't need its own independent label.
- In another sort of tree, the tree's inner nodes are also labeled:
+In another sort of tree, the tree's inner nodes are also labeled:
- 4
- / \
- 2 5
- / \
- 1 3
+ 4
+ / \
+ 2 5
+ / \
+ 1 3
- When you want to efficiently arrange an ordered collection, so that it's
- easy to do a binary search through it, this is the way you usually structure
- your data.
+When you want to efficiently arrange an ordered collection, so that it's
+easy to do a binary search through it, this is the way you usually structure
+your data.
- These latter sorts of trees can helpfully be thought of as ones where
- *only* the inner nodes are labeled. Leaves can be thought of as special,
- dead-end branches with no label:
-
- .4.
- / \
- 2 5
- / \ / \
- 1 3 x x
- / \ / \
- x x x x
+These latter sorts of trees can helpfully be thought of as ones where
+*only* the inner nodes are labeled. Leaves can be thought of as special,
+dead-end branches with no label:
- In our earlier discussion of lists, we said they could be thought of as
- data structures of the form:
+ .4.
+ / \
+ 2 5
+ / \ / \
+ 1 3 x x
+ / \ / \
+ x x x x
- Empty_list | Non_empty_list (its_head, its_tail)
+In our earlier discussion of lists, we said they could be thought of as
+data structures of the form:
- And that could in turn be implemented in v2 form as:
+ Empty_list | Non_empty_list (its_head, its_tail)
- the_list (\head tail. non_empty_handler) empty_handler
+And that could in turn be implemented in v2 form as:
- Similarly, the leaf-labeled tree:
+ the_list (\head tail. non_empty_handler) empty_handler
- +
- / \
- + 3
- / \
- 1 2
+Similarly, the leaf-labeled tree:
- can be thought of as a data structure of the form:
+ .
+ / \
+ . 3
+ / \
+ 1 2
- Node (its_left_subtree, its_right_subtree) | Leaf (its_label)
+can be thought of as a data structure of the form:
- and that could be implemented in v2 form as:
+ Leaf (its_label) | Non_leaf (its_left_subtree, its_right_subtree)
- the_tree (\left right. node_handler) (\label. lead_handler)
+and that could be implemented in v2 form as:
- And the node-labeled tree:
+ the_tree (\left right. non_leaf_handler) (\label. leaf_handler)
- .4.
- / \
- 2 5
- / \ / \
- 1 3 x x
- / \ / \
- x x x x
+And the node-labeled tree:
- can be thought of as a data structure of the form:
+ .4.
+ / \
+ 2 5
+ / \ / \
+ 1 3 x x
+ / \ / \
+ x x x x
- Node (its_left_subtree, its_label, its_right_subtree) | Leaf
+can be thought of as a data structure of the form:
- and that could be implemented in v2 form as:
+ Leaf | Non_leaf (its_left_subtree, its_label, its_right_subtree)
- the_tree (\left label right. node_handler) leaf_result
+and that could be implemented in v2 form as:
+ the_tree (\left label right. non_leaf_handler) leaf_result
- What would correspond to "folding" a function `f` and base value `z` over a
- tree? Well, if it's an empty tree:
- x
+What would correspond to "folding" a function `f` and base value `z` over a
+tree? Well, if it's an empty tree:
- we should presumably get back `z`. And if it's a simple, non-empty tree:
+ x
- 1
- / \
- x x
+we should presumably get back `z`. And if it's a simple, non-empty tree:
- we should expect something like `f z 1 z`, or `f <result of folding f and z
- over left subtree> label_of_this_node <result of folding f and z over right
- subtree>`. (It's not important what order we say `f` has to take its arguments
- in.)
+ 1
+ / \
+ x x
- A v3-style implementation of node-labeled trees, then, might be:
+we should expect something like `f z 1 z`, or `f <result of folding f and z
+over left subtree> label_of_this_node <result of folding f and z over right
+subtree>`. (It's not important what order we say `f` has to take its arguments
+in.)
- let empty_tree = \f z. z in
- let make_tree = \left label right. \f z. f (left f z) label (right f z) in
- ...
+A v3-style implementation of node-labeled trees, then, might be:
- Think about how you might implement other tree operations, such as getting
+ let empty_tree = \f z. z in
+ let make_tree = \left label right. \f z. f (left f z) label (right f z) in
+ ...
+
+Think about how you might implement other tree operations, such as getting
the label of the root (topmost node) of a tree; extracting the left subtree of
a node; and so on.
- Think about different ways you might implement v3-style trees.
+Think about different ways you might implement leaf-labeled trees.
- If you had one tree and wanted to make a larger tree out of it, adding in a
+If you had one tree and wanted to make a larger tree out of it, adding in a
new element, how would you do that?
- When using trees to represent linguistic structures, one doesn't have
+When using trees to represent linguistic structures, one doesn't have
latitude about *how* to build a larger tree. The linguistic structure you're
trying to represent will determine where the new element should be placed, and
where the previous tree should be placed.
- However, when using trees as a computational tool, one usually does have
+However, when using trees as a computational tool, one usually does have
latitude about how to structure a larger tree---in the same way that we had the
freedom to implement our sets with lists whose members were just appended in
the order we built the set up, or instead with lists whose members were ordered
numerically.
- When building a new tree, one strategy for where to put the new element and
+When building a new tree, one strategy for where to put the new element and
where to put the existing tree would be to always lean towards a certain side.
For instance, to add the element `2` to the tree:
- 1
- / \
- x x
+ 1
+ / \
+ x x
- we might construct the following tree:
+we might construct the following tree:
- 1
- / \
- x 2
- / \
- x x
+ 1
+ / \
+ x 2
+ / \
+ x x
- or perhaps we'd do it like this instead:
+or perhaps we'd do it like this instead:
- 2
- / \
- x 1
- / \
- x x
+ 2
+ / \
+ x 1
+ / \
+ x x
- However, if we always leaned to the right side in this way, then the tree
+However, if we always leaned to the right side in this way, then the tree
would get deeper and deeper on that side, but never on the left:
- 1
+ 1
+ / \
+ x 2
+ / \
+ x 3
/ \
- x 2
+ x 4
/ \
- x 3
- / \
- x 4
- / \
- x 5
- / \
- x x
-
- and that wouldn't be so useful if you were using the tree as an arrangement
+ x 5
+ / \
+ x x
+
+and that wouldn't be so useful if you were using the tree as an arrangement
to enable *binary searches* over the elements it holds. For that, you'd prefer
the tree to be relatively "balanced", like this:
- .4.
- / \
- 2 5
- / \ / \
- 1 3 x x
- / \ / \
- x x x x
+ .4.
+ / \
+ 2 5
+ / \ / \
+ 1 3 x x
+ / \ / \
+ x x x x
- Do you have any ideas about how you might efficiently keep the new trees
+Do you have any ideas about how you might efficiently keep the new trees
you're building pretty "balanced" in this way?
- This is a large topic in computer science. There's no need for you to learn
-any of the various strategies that they've developed for doing this. But
+This is a large topic in computer science. There's no need for you to learn
+the various strategies that they've developed for doing this. But
thinking in broad brush-strokes about what strategies might be promising will
help strengthen your understanding of trees, and useful ways to implement them
in a purely functional setting like the lambda calculus.