##Tree Zippers##
-Now how could we translate a zipper-like structure over to trees? What we're aiming for is a way to keep track of where we are in a tree, in the same way that the "broken" lists let us keep track of where we are in the base list. In a particular application, you may only need to implement this for binary trees---trees where internal nodes always have exactly two subtrees. But to get the guiding idea, it's helpful first to think about trees that permit nodes to have many subtrees. Suppose we have the following tree:
+Now how could we translate a zipper-like structure over to trees? What we're aiming for is a way to keep track of where we are in a tree, in the same way that the "broken" lists let us keep track of where we are in the base list.
+
+It's important to set some ground rules for what will follow. If you don't understand these ground rules you will get confused. First off, for many uses of trees one wants some of the nodes or leafs in the tree to be *labeled* with additional information. It's important not to conflate the label with the node itself. Numerically one and the same piece of information---for example, the same `int`---could label two nodes of the tree without those nodes thereby being identical, as here:
+
+ root
+ / \
+ / \
+ / \ label 1
+ / \
+ label 1 label 2
+
+The leftmost leaf and the rightmost leaf have the same label; but they are different leafs. The leftmost leaf has a sibling leaf with the label 2; the rightmost leaf has no siblings that are leafs. Sometimes when one is diagramming trees, one will annotate the nodes with the labels, as above. Other times, when one is diagramming trees, one will instead want to annotate the nodes with tags to make it easier to refer to particular parts of the tree. So for instance, I could diagram the same tree as above as:
+
+ 1
+ / \
+ 2 \
+ / \ 5
+ / \
+ 3 4
+
+Here I haven't drawn what the labels are. The leftmost leaf, the node tagged "3" in this diagram, doesn't have the label `3`. It has the label 1, as we said before. I just haven't put that into the diagram. The node tagged "2" doesn't have the label `2`. It doesn't have any label. The tree in this example only has information labeling its leafs, not any of its inner nodes. The identity of its inner nodes is exhausted by their position in the tree.
+
+That is a second thing to note. In what follows, we'll only be working with *leaf-labeled* trees. In some uses of trees, one also wants labels on inner nodes. But we won't be discussing any such trees now. Our trees only have labels on their leafs. The diagrams below will tag all of the nodes, as in the second diagram above, and won't display what the leafs' labels are.
+
+Final introductory comment: in particular applications, you may only need to work with binary trees---trees where internal nodes always have exactly two subtrees. That is what we'll work with in the homework, for example. But to get the guiding idea of how tree zippers work, it's helpful first to think about trees that permit nodes to have many subtrees. So that's how we'll start.
+
+Suppose we have the following tree:
9200
/ | \
20 50 80 91 92 93 94 95 96
1 2 3 4 5 6 7 8 9
-and we want to represent that we're at the node marked `50`. We might use the following metalanguage notation to specify this:
+This is a leaf-labeled tree whose labels aren't displayed. The `9200` and so on are tags to make it easier for us to refer to particular parts of the tree.
+
+Suppose we want to represent that we're *at* the node marked `50`. We might use the following metalanguage notation to specify this:
- {parent = ...; siblings = [node 20; *; node 80]}, * filled by node 50
+ {parent = ...; siblings = [subtree 20; *; subtree 80]}, * filled by subtree 50
-This is modeled on the notation suggested above for list zippers. Here `node 20` refers not to a `int` label associated with that node, but rather to the whole subtree rooted at that node:
+This is modeled on the notation suggested above for list zippers. Here `subtree 20` refers to the whole subtree rooted at node `20`:
20
/ | \
1 2 3
-Similarly for `node 50` and `node 80`. We haven't said yet what goes in the `parent = ...` slot. Well, the parent of a subtree targetted on `node 50` should intuitively be a tree targetted on `node 500`:
+Similarly for `subtree 50` and `subtree 80`. We haven't said yet what goes in the `parent = ...` slot. Well, the parent of a subtree targetted on `node 50` should intuitively be a tree targetted on `node 500`:
- {parent = ...; siblings = [*; node 920; node 950]}, * filled by node 500
+ {parent = ...; siblings = [*; subtree 920; subtree 950]}, * filled by subtree 500
And the parent of that targetted subtree should intuitively be a tree targetted on `node 9200`:
- {parent = None; siblings = [*]}, * filled by node 9200
+ {parent = None; siblings = [*]}, * filled by tree 9200
This tree has no parents because it's the root of the base tree. Fully spelled out, then, our tree targetted on `node 50` would be:
parent = {
parent = None;
siblings = [*]
- }, * filled by node 9200;
- siblings = [*; node 920; node 950]
- }, * filled by node 500;
- siblings = [node 20; *; node 80]
- }, * filled by node 50
+ }, * filled by tree 9200;
+ siblings = [*; subtree 920; subtree 950]
+ }, * filled by subtree 500;
+ siblings = [subtree 20; *; subtree 80]
+ }, * filled by subtree 50
-In fact, there's some redundancy in this structure, at the points where we have `* filled by node 9200` and `* filled by node 500`. Most of `node 9200`---with the exception of any label attached to node `9200` itself---is determined by the rest of this structure; and so too with `node 500`. So we could really work with:
-
- {
- parent = {
- parent = {
- parent = None;
- siblings = [*]
- }, label for * position (at node 9200);
- siblings = [*; node 920; node 950]
- }, label for * position (at node 500);
- siblings = [node 20; *; node 80]
- }, * filled by node 50
-
-Or, if we only had labels on the leafs of our tree:
+In fact, there's some redundancy in this structure, at the points where we have `* filled by tree 9200` and `* filled by subtree 500`. Since node 9200 doesn't have any label attached to it, the subtree rooted in it is determined by the rest of this structure; and so too with `subtree 500`. So we could really work with:
{
parent = {
parent = None;
siblings = [*]
},
- siblings = [*; node 920; node 950]
+ siblings = [*; subtree 920; subtree 950]
},
- siblings = [node 20; *; node 80]
- }, * filled by node 50
+ siblings = [subtree 20; *; subtree 80]
+ }, * filled by subtree 50
-We're understanding the `20` here in `node 20` to just be a metalanguage marker to help us theorists keep track of which node we're referring to. We're supposing the tree structure itself doesn't associate any informative labelling information with those nodes. It only associates informative labels with the tree leafs. (We haven't represented any such labels in our diagrams.)
-We still do need to keep track of what fills the outermost targetted position---`* filled by node 50`---because that contain a subtree of arbitrary complexity, that is not determined by the rest of this data structure.
+We still do need to keep track of what fills the outermost targetted position---`* filled by subtree 50`---because that contain a subtree of arbitrary complexity, that is not determined by the rest of this data structure.
For simplicity, I'll continue to use the abbreviated form:
- {parent = ...; siblings = [node 20; *; node 80]}, * filled by node 50
+ {parent = ...; siblings = [subtree 20; *; subtree 80]}, * filled by subtree 50
-But that should be understood as standing for the more fully-spelled-out structure. Structures of this sort are called **tree zippers**, for a reason that will emerge. They should already seem intuitively similar to list zippers, though, at least in what we're using them to represent. I think it may initially be more helpful to call these **targetted trees**, though, and so will be switching back and forth between this different terms.
+But that should be understood as standing for the more fully-spelled-out structure. Structures of this sort are called **tree zippers**. They should already seem intuitively similar to list zippers, at least in what we're using them to represent. I think it may also be helpful to call them **targetted trees**, though, and so will be switching back and forth between these different terms.
Moving left in our targetted tree that's targetted on `node 50` would be a matter of shifting the `*` leftwards:
- {parent = ...; siblings = [*; node 50; node 80]}, * filled by node 20
+ {parent = ...; siblings = [*; subtree 50; subtree 80]}, * filled by subtree 20
and similarly for moving right. If the sibling list is implemented as a list zipper, you should already know how to do that. If one were designing a tree zipper for a more restricted kind of tree, however, such as a binary tree, one would probably not represent siblings with a list zipper, but with something more special-purpose and economical.
Moving downward in the tree would be a matter of constructing a tree targetted on some child of `node 20`, with the first part of the targetted tree above as its parent:
{
- parent = {parent = ...; siblings = [*; node 50; node 80]};
+ parent = {parent = ...; siblings = [*; subtree 50; subtree 80]};
siblings = [*; leaf 2; leaf 3]
}, * filled by leaf 1
-How would we move upward in a tree? Well, we'd build a regular, untargetted tree with a root node---let's call it `20`---and whose children are given by the outermost sibling list in the targetted tree above, after inserting the targetted subtree into the `*` position:
+How would we move upward in a tree? Well, we'd build a regular, untargetted tree with a root node---let's call it `20'`---and whose children are given by the outermost sibling list in the targetted tree above, after inserting the targetted subtree into the `*` position:
- node 20
+ node 20'
/ | \
/ | \
leaf 1 leaf 2 leaf 3
-We'll call this new untargetted tree `node 20`. The result of moving upward from our previous targetted tree, targetted on `leaf 1`, would be the outermost `parent` element of that targetted tree, with `node 20` being the subtree that fills that parent's target position `*`:
+We'll call this new untargetted tree `subtree 20'`. The result of moving upward from our previous targetted tree, targetted on `leaf 1`, would be the outermost `parent` element of that targetted tree, with `subtree 20'` being the subtree that fills that parent's target position `*`:
{
parent = ...;
- siblings = [*; node 50; node 80]
- }, * filled by node 20
+ siblings = [*; subtree 50; subtree 80]
+ }, * filled by subtree 20'
Or, spelling that structure out fully:
parent = None;
siblings = [*]
},
- siblings = [*; node 920; node 950]
+ siblings = [*; subtree 920; subtree 950]
},
- siblings = [*; node 50; node 80]
- }, * filled by node 20
+ siblings = [*; subtree 50; subtree 80]
+ }, * filled by subtree 20'
Moving upwards yet again would get us:
parent = None;
siblings = [*]
},
- siblings = [*; node 920; node 950]
- }, * filled by node 500
+ siblings = [*; subtree 920; subtree 950]
+ }, * filled by subtree 500'
-where `node 500` refers to a tree built from a root node whose children are given by the list `[*; node 50; node 80]`, with `node 20` inserted into the `*` position. Moving upwards yet again would get us:
+where `subtree 500'` refers to a tree built from a root node whose children are given by the list `[*; subtree 50; subtree 80]`, with `subtree 20'` inserted into the `*` position. Moving upwards yet again would get us:
{
parent = None;
siblings = [*]
- }, * filled by node 9200
+ }, * filled by tree 9200'
where the targetted element is the root of our base tree. Like the "moving backward" operation for the list zipper, this "moving upward" operation is supposed to be reminiscent of closing a zipper, and that's why these data structures are called zippers.
* Huet, Gerard. ["Functional Pearl: The Zipper"](http://www.st.cs.uni-sb.de/edu/seminare/2005/advanced-fp/docs/huet-zipper.pdf) Journal of Functional Programming 7 (5): 549-554, September 1997.
* As always, [Oleg](http://okmij.org/ftp/continuations/Continuations.html#zipper) takes this a few steps deeper.
-##Same-fringe using a tree zipper##
-Recall back in [[Assignment4]], we asked you to enumerate the "fringe" of a leaf-labeled tree. Both of these trees:
+##Same-fringe using a zipper-based coroutine##
+
+Recall back in [[Assignment4]], we asked you to enumerate the "fringe" of a leaf-labeled tree. Both of these trees (here I *am* drawing the labels in the diagram):
. .
/ \ / \
/ \ / \
1 2 2 3
-have the same fringe: `[1;2;3]`. We also asked you to write a function that determined when two trees have the same fringe. The way you approached that back then was to enumerate each tree's fringe, and then compare the two lists for equality. Today, and then again in a later class, we'll encounter two new ways to approach the problem of determining when two trees have the same fringe.
+have the same fringe: `[1;2;3]`. We also asked you to write a function that determined when two trees have the same fringe. The way you approached that back then was to enumerate each tree's fringe, and then compare the two lists for equality. Today, and then again in a later class, we'll encounter new ways to approach the problem of determining when two trees have the same fringe.
Supposing you did work out an implementation of the tree zipper, then one way to determine whether two trees have the same fringe would be: go downwards (and leftwards) in each tree as far as possible. Compare the targetted leaves. If they're different, stop because the trees have different fringes. If they're the same, then for each tree, move rightward if possible; if it's not (because you're at the rightmost position in a sibling list), more upwards then try again to move rightwards. Repeat until you are able to move rightwards. Once you do move rightwards, go downwards (and leftwards) as far as possible. Then you'll be targetted on the next leaf in the tree's fringe. The operations it takes to get to "the next leaf" may be different for the two trees. For example, in these trees:
Anyway, using record types, we might define the tree zipper interface like so:
- type 'a starred_tree = Root | Starring_Left of 'a starred_pair | Starring_Right of 'a starred_pair
- and 'a starred_pair = { parent : 'a starred_tree; sibling: 'a tree }
- and 'a zipper = { tree : 'a starred_tree; filler: 'a tree };;
+ type 'a starred_level = Root | Starring_Left of 'a starred_nonroot | Starring_Right of 'a starred_nonroot
+ and 'a starred_nonroot = { parent : 'a starred_level; sibling: 'a tree };;
+
+ type 'a zipper = { level : 'a starred_level; filler: 'a tree };;
let rec move_botleft (z : 'a zipper) : 'a zipper =
(* returns z if the targetted node in z has no children *)
(* else returns move_botleft (zipper which results from moving down and left in z) *)
<!--
- let {tree; filler} = z
+ let {level; filler} = z
in match filler with
| Leaf _ -> z
| Node(left, right) ->
- let zdown = {tree = Starring_Left {parent = tree; sibling = right}; filler = left}
+ let zdown = {level = Starring_Left {parent = level; sibling = right}; filler = left}
in move_botleft zdown
;;
-->
(* else returns move_right_or_up (result of moving up in z) *)
<!--
- let {tree; filler} = z
- in match tree with
- | Starring_Left {parent; sibling = right} -> Some {tree = Starring_Right {parent; sibling = filler}; filler = right}
+ let {level; filler} = z
+ in match level with
+ | Starring_Left {parent; sibling = right} -> Some {level = Starring_Right {parent; sibling = filler}; filler = right}
| Root -> None
| Starring_Right {parent; sibling = left} ->
- let z' = {tree = parent; filler = Node(left, filler)}
+ let z' = {level = parent; filler = Node(left, filler)}
in move_right_or_up z'
;;
-->
The following function takes an 'a tree and returns an 'a zipper focused on its root:
let new_zipper (t : 'a tree) : 'a zipper =
- {tree = Root; filler = t}
+ {level = Root; filler = t}
;;
Finally, we can use a mutable reference cell to define a function that enumerates a tree's fringe until it's exhausted:
let make_fringe_enumerator (t: 'a tree) =
- (* create a zipper targetting the root of t *)
- let zstart = new_zipper t
- in let zbotleft = move_botleft zstart
+ (* create a zipper targetting the botleft of t *)
+ let zbotleft = move_botleft (new_zipper t)
(* create a refcell initially pointing to zbotleft *)
in let zcell = ref (Some zbotleft)
(* construct the next_leaf function *)
in let next_leaf () : 'a option =
match !zcell with
- | None -> (* we've finished enumerating the fringe *)
- None
| Some z -> (
(* extract label of currently-targetted leaf *)
let Leaf current = z.filler
| Some z' -> Some (move_botleft z')
(* return saved label *)
in Some current
+ | None -> (* we've finished enumerating the fringe *)
+ None
)
(* return the next_leaf function *)
in next_leaf
<!-- * [[!wikipedia Computer_multitasking]]
* [[!wikipedia Thread_(computer_science)]] -->
+
* [[!wikipedia Coroutine]]
* [[!wikipedia Iterator]]
* [[!wikipedia Generator_(computer_science)]]
The way we built cooperative threads here crucially relied on two heavyweight tools. First, it relied on our having a data structure (the tree zipper) capable of being a static snapshot of where we left off in the tree whose fringe we're enumerating. Second, it relied on our using mutable reference cells so that we could update what the current snapshot (that is, tree zipper) was, so that the next invocation of the `next_leaf` function could start up again where the previous invocation left off.
+It's possible to build cooperative threads without using those tools, however. Some languages have a native syntax for them. Here's how we'd write the same-fringe solution above using native coroutines in the language Lua:
+
+ > function fringe_enumerator (tree)
+ if tree.leaf then
+ coroutine.yield (tree.leaf)
+ else
+ fringe_enumerator (tree.left)
+ fringe_enumerator (tree.right)
+ end
+ end
+
+ > function same_fringe (tree1, tree2)
+ local next1 = coroutine.wrap (fringe_enumerator)
+ local next2 = coroutine.wrap (fringe_enumerator)
+ local function loop (leaf1, leaf2)
+ if leaf1 or leaf2 then
+ return leaf1 == leaf2 and loop( next1(), next2() )
+ elseif not leaf1 and not leaf2 then
+ return true
+ else
+ return false
+ end
+ end
+ return loop (next1(tree1), next2(tree2))
+ end
+
+ > return same_fringe ( {leaf=1}, {leaf=2})
+ false
+
+ > return same_fringe ( {leaf=1}, {leaf=1})
+ true
+
+ > return same_fringe ( {left = {leaf=1}, right = {left = {leaf=2}, right = {leaf=3}}},
+ {left = {left = {leaf=1}, right = {leaf=2}}, right = {leaf=3}} )
+ true
+
+We're going to think about the underlying principles to this execution pattern, and instead learn how to implement it from scratch---without necessarily having zippers to rely on.
+
+
+##Exceptions and Aborts##
+
+To get a better understanding of how that execution patter works, we'll add yet a second execution pattern to our plate, and then think about what they have in common.
+
+While writing OCaml code, you've probably come across errors. In fact, you've probably come across errors of two sorts. One sort of error comes about when you've got syntax errors or type errors and the OCaml interpreter isn't even able to understand your code:
+
+ # let lst = [1; 2] in
+ "a" :: lst;;
+ Error: This expression has type int list
+ but an expression was expected of type string list
+
+But you may also have encountered other kinds of error, that arise while your program is running. For example:
+
+ # 1/0;;
+ Exception: Division_by_zero.
+ # List.nth [1;2] 10;;
+ Exception: Failure "nth".
+
+These "Exceptions" are **run-time errors**. OCaml will automatically detect some of them, like when you attempt to divide by zero. Other exceptions are *raised* by code. For instance, here is the implementation of `List.nth`:
+
+ let nth l n =
+ if n < 0 then invalid_arg "List.nth" else
+ let rec nth_aux l n =
+ match l with
+ | [] -> failwith "nth"
+ | a::l -> if n = 0 then a else nth_aux l (n-1)
+ in nth_aux l n
+
+Notice the two clauses `invalid_arg "List.nth"` and `failwith "nth"`. These are two helper functions which are shorthand for:
+
+ raise (Invalid_argument "List.nth");;
+ raise (Failure "nth");;
+
+where `Invalid_argument "List.nth"` is a value of type `exn`, and so too `Failure "nth"`. When you have some value `ex` of type `exn` and evaluate the expression:
+
+ raise ex
+
+the effect is for the program to immediately stop without evaluating any further code:
+
+ # let xcell = ref 0;;
+ val xcell : int ref = {contents = 0}
+ # let ex = Failure "test"
+ in let _ = raise ex
+ in xcell := 1;;
+ Exception: Failure "test".
+ # !xcell;;
+ - : int = 0
+
+Notice that the line `xcell := 1` was never evaluated, so the contents of `xcell` are still `0`.
+
+I said when you evaluate the expression:
+
+ raise ex
+
+the effect is for the program to immediately stop. That's not exactly true. You can also programmatically arrange to *catch* errors, without the program necessarily stopping. In OCaml we do that with a `try ... with PATTERN -> ...` construct, analogous to the `match ... with PATTERN -> ...` construct:
+
+ # let foo x =
+ try
+ if x = 1 then 10
+ else if x = 2 then raise (Failure "two")
+ else raise (Failure "three")
+ with Failure "two" -> 20
+ ;;
+ val foo : int -> int = <fun>
+ # foo 1;;
+ - : int = 10
+ # foo 2;;
+ - : int = 20
+ # foo 3;;
+ Exception: Failure "three".
+
+Notice what happens here. If we call `foo 1`, then the code between `try` and `with` evaluates to `10`, with no exceptions being raised. That then is what the entire `try ... with ...` block evaluates to; and so too what `foo 1` evaluates to. If we call `foo 2`, then the code between `try` and `with` raises an exception `Failure "two"`. The pattern in the `with` clause matches that exception, so we get instead `20`. If we call `foo 3`, we again raise an exception. This exception isn't matched by the `with` block, so it percolates up to the top of the program, and then the program immediately stops.
+
+So what I should have said is that when you evaluate the expression:
+
+ raise ex
+
+*and that exception is never caught*, then the effect is for the program to immediately stop.
+
+Of course, it's possible to handle errors in other ways too. There's no reason why the implementation of `List.nth` *had* to do things this way. They might instead have returned `Some a` when the list had an nth member `a`, and `None` when it does not. But it's pedagogically useful for us to think about this pattern now.
+
+When an exception is raised, it percolates up through the code that called it, until it finds a surrounding `try ... with ...` that matches it. That might not be the first `try ... with ...` that it encounters. For example:
+
+ # try
+ try
+ raise (Failure "blah")
+ with Failure "fooey" -> 10
+ with Failure "blah" -> 20;;
+ - : int = 20
+
+The matching `try ... with ...` block need not *lexically surround* the site where the error was raised:
+
+ # let foo b x =
+ try
+ b x
+ with Failure "blah" -> 20
+ in let bar x =
+ raise (Failure "blah")
+ in foo bar 0;;
+ - : int = 20
+
+Here we call `foo bar 0`, and `foo` in turn calls `bar 0`, and `bar` raises the exception. Since there's no matching `try ... with ...` block in `bar`, we percolate back up the history of *who called this function?* and find a matching `try ... with ...` block in `foo`. This catches the error and so then the `try ... with ...` block in `foo` that called `bar` in the first place will evaluate to `20`.
+
+OK, now this exception-handling apparatus does exemplify the second execution pattern we want to focus on. But it may bring it into clearer focus if we simplify the pattern even more. Imagine we could write code like this instead:
+
+ # let foo x =
+ try
+ (if x = 1 then 10
+ else abort 20) + 1
+ end
+ ;;
+
+then if we called `foo 1`, we'd get the result `11`. If we called `foo 2`, on the other hand, we'd get `20` (note, not `21`). This exemplifies the same interesting "jump out of this part of the code" behavior that the `try ... raise ... with ...` code does, but without the details of matching which exception was raised, and handling the exception to produce a new result.
+
+Many programming languages have this simplified exceution pattern, either instead of or alongside a `try ... with ...`-like pattern. In Lua and many other languages, `abort` is instead called `return`. The preceding example would be written:
+
+ > function foo(x)
+ local value
+ if (x == 1) then
+ value = 10
+ else
+ return 20
+ end
+ return value + 1
+ end
+
+ > return foo(1)
+ 11
+
+ > return foo(2)
+ 20
+
+Okay, so that's our second execution pattern.
+
+##What do these have in common?##
+
+
+
+
+
+
+
+--------------------------------------
+
In coming weeks, we'll learn about a different way to create threads, that relies on **continuations** rather than on those two tools. All of these tools are inter-related. As Oleg says, "Zipper can be viewed as a delimited continuation reified as a data structure." These different tools are also inter-related with monads. Many of these tools can be used to define the others. We'll explore some of the connections between them in the remaining weeks, but we encourage you to explore more.