-[[!toc]]
+These notes may change in the next few days (today is 30 Nov 2010).
+The material here benefited from many discussions with Ken Shan.
-##List Zippers##
+##[[Tree and List Zippers]]##
-Say you've got some moderately-complex function for searching through a list, for example:
+##[[Coroutines and Aborts]]##
- let find_nth (test : 'a -> bool) (n : int) (lst : 'a list) : (int * 'a) ->
- let rec helper (position : int) n lst =
- match lst with
- | [] -> failwith "not found"
- | x :: xs when test x -> (if n = 1
- then (position, x)
- else helper (position + 1) (n - 1) xs
- )
- | x :: xs -> helper (position + 1) n xs
- in helper 0 n lst;;
+##[[From Lists to Continuations]]##
-This searches for the `n`th element of a list that satisfies the predicate `test`, and returns a pair containing the position of that element, and the element itself. Good. But now what if you wanted to retrieve a different kind of information, such as the `n`th element matching `test`, together with its preceding and succeeding elements? In a real situation, you'd want to develop some good strategy for reporting when the target element doesn't have a predecessor and successor; but we'll just simplify here and report them as having some default value:
-
- let find_nth' (test : 'a -> bool) (n : int) (lst : 'a list) (default : 'a) : ('a * 'a * 'a) ->
- let rec helper (predecessor : 'a) n lst =
- match lst with
- | [] -> failwith "not found"
- | x :: xs when test x -> (if n = 1
- then (predecessor, x, match xs with [] -> default | y::ys -> y)
- else helper x (n - 1) xs
- )
- | x :: xs -> helper x n xs
- in helper default n lst;;
-
-This duplicates a lot of the structure of `find_nth`; it just has enough different code to retrieve different information when the matching element is found. But now what if you wanted to retrieve yet a different kind of information...?
-
-Ideally, there should be some way to factor out the code to find the target element---the `n`th element of the list satisfying the predicate `test`---from the code that retrieves the information you want once the target is found. We might build upon the initial `find_nth` function, since that returns the *position* of the matching element. We could hand that result off to some other function that's designed to retrieve information of a specific sort surrounding that position. But suppose our list has millions of elements, and the target element is at position 600512. The search function will already have traversed 600512 elements of the list looking for the target, then the retrieval function would have to *start again from the beginning* and traverse those same 600512 elements again. It could go a bit faster, since it doesn't have to check each element against `test` as it traverses. It already knows how far it has to travel. But still, this should seem a bit wasteful.
-
-Here's an idea. What if we had some way of representing a list as "broken" at a specific point. For example, if our base list is:
-
- [10; 20; 30; 40; 50; 60; 70; 80; 90]
-
-we might imagine the list "broken" at position 3 like this (positions are numbered starting from 0):
-
- 40;
- 30; 50;
- 20; 60;
- [10; 70;
- 80;
- 90]
-
-Then if we move one step forward in the list, it would be "broken" at position 4:
-
- 50;
- 40; 60;
- 30; 70;
- 20; 80;
- [10; 90]
-
-If we had some convenient representation of these "broken" lists, then our search function could hand *that* off to the retrieval function, and the retrieval function could start right at the position where the list was broken, without having to start at the beginning and traverse many elements to get there. The retrieval function would also be able to inspect elements both forwards and backwards from the position where the list was "broken".
-
-The kind of data structure we're looking for here is called a **list zipper**. To represent our first broken list, we'd use two lists: (1) containing the elements in the left branch, preceding the target element, *in the order reverse to their appearance in the base list*. (2) containing the target element and the rest of the list, in normal order. So:
-
- 40;
- 30; 50;
- 20; 60;
- [10; 70;
- 80;
- 90]
-
-would be represented as `([30; 20; 10], [40; 50; 60; 70; 80; 90])`. To move forward in the base list, we pop the head element `40` off of the head element of the second list in the zipper, and push it onto the first list, getting `([40; 30; 20; 10], [50; 60; 70; 80; 90])`. To move backwards again, we pop off of the first list, and push it onto the second. To reconstruct the base list, we just "move backwards" until the first list is empty. (This is supposed to evoke the image of zipping up a zipper; hence the data structure's name.)
-
-We had some discussio in seminar of the right way to understand the "zipper" metaphor. I think it's best to think of the tab of the zipper being here:
-
- t
- a
- b
- 40;
- 30; 50;
- 20; 60;
- [10; 70;
- 80;
- 90]
-
-And imagine that you're just seeing the left half of a real-zipper, rotated 60 degrees counter-clockwise. When the list is all "zipped up", we've "move backwards" to the state where the first element is targetted:
-
- ([], [10; 20; 30; 40; 50; 60; 70; 80; 90])
-
-However you understand the "zipper" metaphor, this is a very handy datastructure, and it will become even more handy when we translate it over to more complicated base structures, like trees. To help get a good conceptual grip on how to do that, it's useful to introduce a kind of symbolism for talking about zippers. This is just a metalanguage notation, for us theorists; we don't need our programs to interpret the notation. We'll use a specification like this:
-
- [10; 20; 30; *; 50; 60; 70; 80; 90], * filled by 40
-
-to represent a list zipper where the break is at position 3, and the element occupying that position is 40. For a list zipper, this is implemented using the pairs-of-lists structure described above.
-
-
-##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.
-
-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
- / | \
- / | \
- / | \
- / | \
- / | \
- 500 920 950
- / | \ / | \ / | \
- 20 50 80 91 92 93 94 95 96
- 1 2 3 4 5 6 7 8 9
-
-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 = [subtree 20; *; subtree 80]}, * filled by subtree 50
-
-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 `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 = [*; 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 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 = {
- parent = None;
- siblings = [*]
- }, * 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 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 = {
- parent = None;
- siblings = [*]
- },
- siblings = [*; subtree 920; subtree 950]
- },
- siblings = [subtree 20; *; subtree 80]
- }, * filled by subtree 50
-
-
-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 = [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**. 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 = [*; 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 = [*; 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:
-
- node 20'
- / | \
- / | \
- leaf 1 leaf 2 leaf 3
-
-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 = [*; subtree 50; subtree 80]
- }, * filled by subtree 20'
-
-Or, spelling that structure out fully:
-
- {
- parent = {
- parent = {
- parent = None;
- siblings = [*]
- },
- siblings = [*; subtree 920; subtree 950]
- },
- siblings = [*; subtree 50; subtree 80]
- }, * filled by subtree 20'
-
-Moving upwards yet again would get us:
-
- {
- parent = {
- parent = None;
- siblings = [*]
- },
- siblings = [*; subtree 920; subtree 950]
- }, * filled by subtree 500'
-
-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 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.
-
-We haven't given you a real implementation of the tree zipper, but only a suggestive notation. We have however told you enough that you should be able to implement it yourself. Or if you're lazy, you can read:
-
-* [[!wikipedia Zipper (data structure)]]
-* 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##
+##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):
<!-- * [[!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 both of these patterns, we need to have some way to take a snapshot of where we are in the evaluation of a complex piece of code, so that we might later resume execution at that point. In the coroutine example, the two threads need to have a snapshot of where they were in the enumeration of their tree's leaves. In the abort example, we need to have a snapshot of where to pick up again if some embedded piece of code aborts. Sometimes we might distill that snapshot into a datastructure like a zipper. But we might not always know how to do so; and learning how to think about these snapshots without the help of zippers will help us see patterns and similarities we might otherwise miss.
+
+A more general way to think about these snapshots is to think of the code we're taking a snapshot of as a *function.* For example, in this code:
+
+ let foo x =
+ try
+ (if x = 1 then 10
+ else abort 20) + 1
+ end
+ in (foo 2) + 1;;
+
+we can imagine a box:
+
+ let foo x =
+ +---------------------------+
+ | try |
+ | (if x = 1 then 10 |
+ | else abort 20) + 1 |
+ | end |
+ +---------------------------+
+ in (foo 2) + 1;;
+
+and as we're about to enter the box, we want to take a snapshot of the code *outside* the box. If we decide to abort, we'd be aborting to that snapshotted code.
+
+<!--
+# #require "delimcc";;
+# open Delimcc;;
+# let reset body = let p = new_prompt () in push_prompt p (body p);;
+val reset : ('a Delimcc.prompt -> unit -> 'a) -> 'a = <fun>
+# let foo x = reset(fun p () -> (shift p (fun k -> if x = 1 then k 10 else 20)) + 1) in (foo 1) + 100;;
+- : int = 111
+# let foo x = reset(fun p () -> (shift p (fun k -> if x = 1 then k 10 else 20)) + 1) in (foo 2) + 100;;
+- : int = 120
+-->
+
+
+
+
+--------------------------------------
+
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.
<!-- (1, 7) ... explain why not (1, 8) -->
+##[[List Monad as Continuation Monad]]##
+
+##[[Manipulating Trees with Monads]]##