1 These notes may change in the next few days (today is 30 Nov 2010).
2 The material here benefited from many discussions with Ken Shan.
4 ##[[Tree and List Zippers]]##
6 ##[[Coroutines and Aborts]]##
8 ##[[From Lists to Continuations]]##
10 ##Same-fringe using a zipper-based coroutine##
12 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):
20 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.
23 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:
31 you won't move upwards at the same steps. Keep comparing "the next leafs" until they are different, or you exhaust the leafs of only one of the trees (then again the trees have different fringes), or you exhaust the leafs of both trees at the same time, without having found leafs with different labels. In this last case, the trees have the same fringe.
33 If your trees are very big---say, millions of leaves---you can imagine how this would be quicker and more memory-efficient than traversing each tree to construct a list of its fringe, and then comparing the two lists so built to see if they're equal. For one thing, the zipper method can abort early if the fringes diverge early, without needing to traverse or build a list containing the rest of each tree's fringe.
35 Let's sketch the implementation of this. We won't provide all the details for an implementation of the tree zipper, but we will sketch an interface for it.
37 First, we define a type for leaf-labeled, binary trees:
39 type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)
41 Next, the interface for our tree zippers. We'll help ourselves to OCaml's **record types**. These are nothing more than tuples with a pretty interface. Instead of saying:
43 # type blah = Blah of (int * int * (char -> bool));;
45 and then having to remember which element in the triple was which:
47 # let b1 = Blah (1, (fun c -> c = 'M'), 2);;
48 Error: This expression has type int * (char -> bool) * int
49 but an expression was expected of type int * int * (char -> bool)
51 # let b1 = Blah (1, 2, (fun c -> c = 'M'));;
52 val b1 : blah = Blah (1, 2, <fun>)
54 records let you attach descriptive labels to the components of the tuple:
56 # type blah_record = { height : int; weight : int; char_tester : char -> bool };;
57 # let b2 = { height = 1; weight = 2; char_tester = fun c -> c = 'M' };;
58 val b2 : blah_record = {height = 1; weight = 2; char_tester = <fun>}
59 # let b3 = { height = 1; char_tester = (fun c -> c = 'K'); weight = 3 };; (* also works *)
60 val b3 : blah_record = {height = 1; weight = 3; char_tester = <fun>}
62 These were the strategies to extract the components of an unlabeled tuple:
64 let h = fst some_pair;; (* accessor functions fst and snd are only predefined for pairs *)
66 let (h, w, test) = b1;; (* works for arbitrary tuples *)
69 | (h, w, test) -> ...;; (* same as preceding *)
71 Here is how you can extract the components of a labeled record:
73 let h = b2.height;; (* handy! *)
75 let {height = h; weight = w; char_tester = test} = b2
76 in (* go on to use h, w, and test ... *)
79 | {height = h; weight = w; char_tester = test} ->
80 (* go on to use h, w, and test ... *)
82 Anyway, using record types, we might define the tree zipper interface like so:
84 type 'a starred_level = Root | Starring_Left of 'a starred_nonroot | Starring_Right of 'a starred_nonroot
85 and 'a starred_nonroot = { parent : 'a starred_level; sibling: 'a tree };;
87 type 'a zipper = { level : 'a starred_level; filler: 'a tree };;
89 let rec move_botleft (z : 'a zipper) : 'a zipper =
90 (* returns z if the targetted node in z has no children *)
91 (* else returns move_botleft (zipper which results from moving down and left in z) *)
94 let {level; filler} = z
97 | Node(left, right) ->
98 let zdown = {level = Starring_Left {parent = level; sibling = right}; filler = left}
103 let rec move_right_or_up (z : 'a zipper) : 'a zipper option =
104 (* if it's possible to move right in z, returns Some (the result of doing so) *)
105 (* else if it's not possible to move any further up in z, returns None *)
106 (* else returns move_right_or_up (result of moving up in z) *)
109 let {level; filler} = z
111 | Starring_Left {parent; sibling = right} -> Some {level = Starring_Right {parent; sibling = filler}; filler = right}
113 | Starring_Right {parent; sibling = left} ->
114 let z' = {level = parent; filler = Node(left, filler)}
115 in move_right_or_up z'
119 The following function takes an 'a tree and returns an 'a zipper focused on its root:
121 let new_zipper (t : 'a tree) : 'a zipper =
122 {level = Root; filler = t}
125 Finally, we can use a mutable reference cell to define a function that enumerates a tree's fringe until it's exhausted:
127 let make_fringe_enumerator (t: 'a tree) =
128 (* create a zipper targetting the botleft of t *)
129 let zbotleft = move_botleft (new_zipper t)
130 (* create a refcell initially pointing to zbotleft *)
131 in let zcell = ref (Some zbotleft)
132 (* construct the next_leaf function *)
133 in let next_leaf () : 'a option =
136 (* extract label of currently-targetted leaf *)
137 let Leaf current = z.filler
138 (* update zcell to point to next leaf, if there is one *)
139 in let () = zcell := match move_right_or_up z with
141 | Some z' -> Some (move_botleft z')
142 (* return saved label *)
144 | None -> (* we've finished enumerating the fringe *)
147 (* return the next_leaf function *)
151 Here's an example of `make_fringe_enumerator` in action:
153 # let tree1 = Leaf 1;;
154 val tree1 : int tree = Leaf 1
155 # let next1 = make_fringe_enumerator tree1;;
156 val next1 : unit -> int option = <fun>
158 - : int option = Some 1
160 - : int option = None
162 - : int option = None
163 # let tree2 = Node (Node (Leaf 1, Leaf 2), Leaf 3);;
164 val tree2 : int tree = Node (Node (Leaf 1, Leaf 2), Leaf 3)
165 # let next2 = make_fringe_enumerator tree2;;
166 val next2 : unit -> int option = <fun>
168 - : int option = Some 1
170 - : int option = Some 2
172 - : int option = Some 3
174 - : int option = None
176 - : int option = None
178 You might think of it like this: `make_fringe_enumerator` returns a little subprogram that will keep returning the next leaf in a tree's fringe, in the form `Some ...`, until it gets to the end of the fringe. After that, it will keep returning `None`.
180 Using these fringe enumerators, we can write our `same_fringe` function like this:
182 let same_fringe (t1 : 'a tree) (t2 : 'a tree) : bool =
183 let next1 = make_fringe_enumerator t1
184 in let next2 = make_fringe_enumerator t2
185 in let rec loop () : bool =
186 match next1 (), next2 () with
187 | Some a, Some b when a = b -> loop ()
193 The auxiliary `loop` function will keep calling itself recursively until a difference in the fringes has manifested itself---either because one fringe is exhausted before the other, or because the next leaves in the two fringes have different labels. If we get to the end of both fringes at the same time (`next1 (), next2 ()` matches the pattern `None, None`) then we've established that the trees do have the same fringe.
195 The technique illustrated here with our fringe enumerators is a powerful and important one. It's an example of what's sometimes called **cooperative threading**. A "thread" is a subprogram that the main computation spawns off. Threads are called "cooperative" when the code of the main computation and the thread fixes when control passes back and forth between them. (When the code doesn't control this---for example, it's determined by the operating system or the hardware in ways that the programmer can't predict---that's called "preemptive threading.") Cooperative threads are also sometimes called *coroutines* or *generators*.
197 With cooperative threads, one typically yields control to the thread, and then back again to the main program, multiple times. Here's the pattern in which that happens in our `same_fringe` function:
199 main program next1 thread next2 thread
200 ------------ ------------ ------------
203 (paused) calculate first leaf
204 (paused) <--- return it
205 start next2 (paused) starting
206 (paused) (paused) calculate first leaf
207 (paused) (paused) <-- return it
208 compare leaves (paused) (paused)
209 call loop again (paused) (paused)
210 call next1 again (paused) (paused)
211 (paused) calculate next leaf (paused)
212 (paused) <-- return it (paused)
215 If you want to read more about these kinds of threads, here are some links:
217 <!-- * [[!wikipedia Computer_multitasking]]
218 * [[!wikipedia Thread_(computer_science)]] -->
220 * [[!wikipedia Coroutine]]
221 * [[!wikipedia Iterator]]
222 * [[!wikipedia Generator_(computer_science)]]
223 * [[!wikipedia Fiber_(computer_science)]]
224 <!-- * [[!wikipedia Green_threads]]
225 * [[!wikipedia Protothreads]] -->
227 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.
229 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:
231 > function fringe_enumerator (tree)
233 coroutine.yield (tree.leaf)
235 fringe_enumerator (tree.left)
236 fringe_enumerator (tree.right)
240 > function same_fringe (tree1, tree2)
241 local next1 = coroutine.wrap (fringe_enumerator)
242 local next2 = coroutine.wrap (fringe_enumerator)
243 local function loop (leaf1, leaf2)
244 if leaf1 or leaf2 then
245 return leaf1 == leaf2 and loop( next1(), next2() )
246 elseif not leaf1 and not leaf2 then
252 return loop (next1(tree1), next2(tree2))
255 > return same_fringe ( {leaf=1}, {leaf=2})
258 > return same_fringe ( {leaf=1}, {leaf=1})
261 > return same_fringe ( {left = {leaf=1}, right = {left = {leaf=2}, right = {leaf=3}}},
262 {left = {left = {leaf=1}, right = {leaf=2}}, right = {leaf=3}} )
265 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.
268 ##Exceptions and Aborts##
270 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.
272 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:
274 # let lst = [1; 2] in
276 Error: This expression has type int list
277 but an expression was expected of type string list
279 But you may also have encountered other kinds of error, that arise while your program is running. For example:
282 Exception: Division_by_zero.
283 # List.nth [1;2] 10;;
284 Exception: Failure "nth".
286 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`:
289 if n < 0 then invalid_arg "List.nth" else
290 let rec nth_aux l n =
292 | [] -> failwith "nth"
293 | a::l -> if n = 0 then a else nth_aux l (n-1)
296 Notice the two clauses `invalid_arg "List.nth"` and `failwith "nth"`. These are two helper functions which are shorthand for:
298 raise (Invalid_argument "List.nth");;
299 raise (Failure "nth");;
301 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:
305 the effect is for the program to immediately stop without evaluating any further code:
307 # let xcell = ref 0;;
308 val xcell : int ref = {contents = 0}
309 # let ex = Failure "test"
312 Exception: Failure "test".
316 Notice that the line `xcell := 1` was never evaluated, so the contents of `xcell` are still `0`.
318 I said when you evaluate the expression:
322 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:
327 else if x = 2 then raise (Failure "two")
328 else raise (Failure "three")
329 with Failure "two" -> 20
331 val foo : int -> int = <fun>
337 Exception: Failure "three".
339 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.
341 So what I should have said is that when you evaluate the expression:
345 *and that exception is never caught*, then the effect is for the program to immediately stop.
347 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.
349 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:
353 raise (Failure "blah")
354 with Failure "fooey" -> 10
355 with Failure "blah" -> 20;;
358 The matching `try ... with ...` block need not *lexically surround* the site where the error was raised:
363 with Failure "blah" -> 20
365 raise (Failure "blah")
369 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`.
371 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:
380 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.
382 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:
400 Okay, so that's our second execution pattern.
402 ##What do these have in common?##
404 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.
406 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:
415 we can imagine a box:
418 +---------------------------+
420 | (if x = 1 then 10 |
421 | else abort 20) + 1 |
423 +---------------------------+
426 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.
429 # #require "delimcc";;
431 # let reset body = let p = new_prompt () in push_prompt p (body p);;
432 val reset : ('a Delimcc.prompt -> unit -> 'a) -> 'a = <fun>
433 # let foo x = reset(fun p () -> (shift p (fun k -> if x = 1 then k 10 else 20)) + 1) in (foo 1) + 100;;
435 # let foo x = reset(fun p () -> (shift p (fun k -> if x = 1 then k 10 else 20)) + 1) in (foo 2) + 100;;
442 --------------------------------------
444 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.
447 ##Introducing Continuations##
449 A continuation is "the rest of the program." Or better: an **delimited continuation** is "the rest of the program, up to a certain boundary." An **undelimited continuation** is "the rest of the program, period."
451 Even if you haven't read specifically about this notion (for example, even if you haven't read Chris and Ken's work on using continuations in natural language semantics), you'll have brushed shoulders with it already several times in this course.
453 A naive semantics for atomic sentences will say the subject term is of type `e`, and the predicate of type `e -> t`, and that the subject provides an argument to the function expressed by the predicate.
455 Monatague proposed we instead take subject terms to be of type `(e -> t) -> t`, and that now it'd be the predicate (still of type `e -> t`) that provides an argument to the function expressed by the subject.
457 If all the subject did then was supply an `e` to the `e -> t` it receives as an argument, we wouldn't have gained anything we weren't already able to do. But of course, there are other things the subject can do with the `e -> t` it receives as an argument. For instance, it can check whether anything in the domain satisfies that `e -> t`; or whether most things do; and so on.
459 This inversion of who is the argument and who is the function receiving the argument is paradigmatic of working with continuations. We did the same thing ourselves back in the early days of the seminar, for example in our implementation of pairs. In the untyped lambda calculus, we identified the pair `(x, y)` with a function:
461 \handler. handler x y
463 A pair-handling function would accept the two elements of a pair as arguments, and then do something with one or both of them. The important point here is that the handler was supplied as an argument to the pair. Eventually, the handler would itself be supplied with arguments. But only after it was supplied as an argument to the pair. This inverts the order you'd expect about what is the data or argument, and what is the function that operates on it.
465 Consider a complex computation, such as:
467 1 + 2 * (1 - g (3 + 4))
469 Part of this computation---`3 + 4`---leads up to supplying `g` with an argument. The rest of the computation---`1 + 2 * (1 - ___)`---waits for the result of applying `g` to that argument and will go on to do something with it (inserting the result into the `___` slot). That "rest of the computation" can be regarded as a function:
471 \result. 1 + 2 * (1 - result)
473 This function will be applied to whatever is the result of `g (3 + 4)`. So this function can be called the *continuation* of that application of `g`. For some purposes, it's useful to be able to invert the function/argument order here, and rather than supplying the result of applying `g` to the continuation, we instead supply the continuation to `g`. Well, not to `g` itself, since `g` only wants a single `int` argument. But we might build some `g`-like function which accepts not just an `int` argument like `g` does, but also a continuation argument.
475 Go back and read the material on "Aborting a Search Through a List" in [[Week4]] for an example of doing this.
477 In very general terms, the strategy is to work with functions like this:
481 ... if you want to abort early, supply an argument to k ...
482 ... do more stuff ...
484 in let gcon = fun result -> 1 + 2 * (1 - result)
485 in gcon (g' gcon (3 + 4))
487 It's a convention to use variables like `k` for continuation arguments. If the function `g'` never supplies an argument to its contination argument `k`, but instead just finishes evaluating to a normal result, that normal result will be delivered to `g'`'s continuation `gcon`, just as happens when we don't pass around any explicit continuation variables.
489 The above snippet of OCaml code doesn't really capture what happens when we pass explicit continuation variables. For suppose that inside `g'`, we do supply an argument to `k`. That would go into the `result` parameter in `gcon`. But then what happens once we've finished evaluating the application of `gcon` to that `result`? In the OCaml snippet above, the final value would then bubble up through the context in the body of `g'` where `k` was applied, and eventually out to the final line of the snippet, where it once again supplied an argument to `gcon`. That's not what happens with a real continuation. A real continuation works more like this:
493 ... if you want to abort early, supply an argument to k ...
494 ... do more stuff ...
496 in let gcon = fun result ->
497 let final_value = 1 + 2 * (1 - result)
498 in end_program_with final_value
499 in gcon (g' gcon (3 + 4))
501 So once we've finished evaluating the application of `gcon` to a `result`, the program is finished. (This is how undelimited continuations behave. We'll discuss delimited continuations later.)
503 So now, guess what would be the result of doing the following:
507 in let gcon = fun result ->
508 let final_value = (1, result)
509 in end_program_with final_value
510 in gcon (g' gcon (3 + 4))
512 <!-- (1, 7) ... explain why not (1, 8) -->
514 Rethinking the list monad
515 -------------------------
517 To construct a monad, the key element is to settle on a type
518 constructor, and the monad more or less naturally follows from that.
519 We'll remind you of some examples of how monads follow from the type
520 constructor in a moment. This will involve some review of familair
521 material, but it's worth doing for two reasons: it will set up a
522 pattern for the new discussion further below, and it will tie together
523 some previously unconnected elements of the course (more specifically,
524 version 3 lists and monads).
526 For instance, take the **Reader Monad**. Once we decide that the type
529 type 'a reader = env -> 'a
531 then the choice of unit and bind is natural:
533 let r_unit (a : 'a) : 'a reader = fun (e : env) -> a
535 The reason this is a fairly natural choice is that because the type of
536 an `'a reader` is `env -> 'a` (by definition), the type of the
537 `r_unit` function is `'a -> env -> 'a`, which is an instance of the
538 type of the *K* combinator. So it makes sense that *K* is the unit
539 for the reader monad.
541 Since the type of the `bind` operator is required to be
543 r_bind : ('a reader) -> ('a -> 'b reader) -> ('b reader)
545 We can reason our way to the traditional reader `bind` function as
546 follows. We start by declaring the types determined by the definition
549 let r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = ...
551 Now we have to open up the `u` box and get out the `'a` object in order to
552 feed it to `f`. Since `u` is a function from environments to
553 objects of type `'a`, the way we open a box in this monad is
554 by applying it to an environment:
560 This subexpression types to `'b reader`, which is good. The only
561 problem is that we made use of an environment `e` that we didn't already have,
562 so we must abstract over that variable to balance the books:
566 [To preview the discussion of the Curry-Howard correspondence, what
567 we're doing here is constructing an intuitionistic proof of the type,
568 and using the Curry-Howard labeling of the proof as our bind term.]
570 This types to `env -> 'b reader`, but we want to end up with `env ->
571 'b`. Once again, the easiest way to turn a `'b reader` into a `'b` is to apply it to an environment. So we end up as follows:
574 r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = f (u e) e
577 And we're done. This gives us a bind function of the right type. We can then check whether, in combination with the unit function we chose, it satisfies the monad laws, and behaves in the way we intend. And it does.
579 [The bind we cite here is a condensed version of the careful `let a = u e in ...`
580 constructions we provided in earlier lectures. We use the condensed
581 version here in order to emphasize similarities of structure across
584 The **State Monad** is similar. Once we've decided to use the following type constructor:
586 type 'a state = store -> ('a, store)
588 Then our unit is naturally:
590 let s_unit (a : 'a) : ('a state) = fun (s : store) -> (a, s)
592 And we can reason our way to the bind function in a way similar to the reasoning given above. First, we need to apply `f` to the contents of the `u` box:
594 let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state =
597 But unlocking the `u` box is a little more complicated. As before, we
598 need to posit a state `s` that we can apply `u` to. Once we do so,
599 however, we won't have an `'a`, we'll have a pair whose first element
600 is an `'a`. So we have to unpack the pair:
602 ... let (a, s') = u s in ... (f a) ...
604 Abstracting over the `s` and adjusting the types gives the result:
606 let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state =
607 fun (s : store) -> let (a, s') = u s in f a s'
609 The **Option/Maybe Monad** doesn't follow the same pattern so closely, so we
610 won't pause to explore it here, though conceptually its unit and bind
611 follow just as naturally from its type constructor.
613 Our other familiar monad is the **List Monad**, which we were told
616 type 'a list = ['a];;
617 l_unit (a : 'a) = [a];;
618 l_bind u f = List.concat (List.map f u);;
620 Thinking through the list monad will take a little time, but doing so
621 will provide a connection with continuations.
623 Recall that `List.map` takes a function and a list and returns the
624 result to applying the function to the elements of the list:
626 List.map (fun i -> [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]]
628 and List.concat takes a list of lists and erases the embdded list
631 List.concat [[1; 2]; [2; 3]] ~~> [1; 2; 2; 3]
635 l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
637 Now, why this unit, and why this bind? Well, ideally a unit should
638 not throw away information, so we can rule out `fun x -> []` as an
639 ideal unit. And units should not add more information than required,
640 so there's no obvious reason to prefer `fun x -> [x,x]`. In other
641 words, `fun x -> [x]` is a reasonable choice for a unit.
643 As for bind, an `'a list` monadic object contains a lot of objects of
644 type `'a`, and we want to make use of each of them (rather than
645 arbitrarily throwing some of them away). The only
646 thing we know for sure we can do with an object of type `'a` is apply
647 the function of type `'a -> 'a list` to them. Once we've done so, we
648 have a collection of lists, one for each of the `'a`'s. One
649 possibility is that we could gather them all up in a list, so that
650 `bind' [1;2] (fun i -> [i;i]) ~~> [[1;1];[2;2]]`. But that restricts
651 the object returned by the second argument of `bind` to always be of
652 type `'b list list`. We can elimiate that restriction by flattening
653 the list of lists into a single list: this is
654 just List.concat applied to the output of List.map. So there is some logic to the
655 choice of unit and bind for the list monad.
657 Yet we can still desire to go deeper, and see if the appropriate bind
658 behavior emerges from the types, as it did for the previously
659 considered monads. But we can't do that if we leave the list type as
660 a primitive Ocaml type. However, we know several ways of implementing
661 lists using just functions. In what follows, we're going to use type
662 3 lists, the right fold implementation (though it's important and
663 intriguing to wonder how things would change if we used some other
664 strategy for implementating lists). These were the lists that made
665 lists look like Church numerals with extra bits embdded in them:
667 empty list: fun f z -> z
668 list with one element: fun f z -> f 1 z
669 list with two elements: fun f z -> f 2 (f 1 z)
670 list with three elements: fun f z -> f 3 (f 2 (f 1 z))
672 and so on. To save time, we'll let the OCaml interpreter infer the
673 principle types of these functions (rather than inferring what the
674 types should be ourselves):
677 - : 'a -> 'b -> 'b = <fun>
679 - : (int -> 'a -> 'b) -> 'a -> 'b = <fun>
680 # fun f z -> f 2 (f 1 z);;
681 - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
682 # fun f z -> f 3 (f 2 (f 1 z))
683 - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
685 We can see what the consistent, general principle types are at the end, so we
686 can stop. These types should remind you of the simply-typed lambda calculus
687 types for Church numerals (`(o -> o) -> o -> o`) with one extra type
688 thrown in, the type of the element a the head of the list
689 (in this case, an int).
691 So here's our type constructor for our hand-rolled lists:
693 type 'b list' = (int -> 'b -> 'b) -> 'b -> 'b
695 Generalizing to lists that contain any kind of element (not just
698 type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
700 So an `('a, 'b) list'` is a list containing elements of type `'a`,
701 where `'b` is the type of some part of the plumbing. This is more
702 general than an ordinary OCaml list, but we'll see how to map them
703 into OCaml lists soon. We don't need to fully grasp the role of the `'b`'s
704 in order to proceed to build a monad:
706 l'_unit (a : 'a) : ('a, 'b) list = fun a -> fun f z -> f a z
708 No problem. Arriving at bind is a little more complicated, but
709 exactly the same principles apply, you just have to be careful and
712 l'_bind (u : ('a,'b) list') (f : 'a -> ('c, 'd) list') : ('c, 'd) list' = ...
714 Unpacking the types gives:
716 l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
717 (f : 'a -> ('c -> 'd -> 'd) -> 'd -> 'd)
718 : ('c -> 'd -> 'd) -> 'd -> 'd = ...
720 Perhaps a bit intimiating.
721 But it's a rookie mistake to quail before complicated types. You should
722 be no more intimiated by complex types than by a linguistic tree with
723 deeply embedded branches: complex structure created by repeated
724 application of simple rules.
726 [This would be a good time to try to build your own term for the types
727 just given. Doing so (or attempting to do so) will make the next
728 paragraph much easier to follow.]
730 As usual, we need to unpack the `u` box. Examine the type of `u`.
731 This time, `u` will only deliver up its contents if we give `u` an
732 argument that is a function expecting an `'a` and a `'b`. `u` will
733 fold that function over its type `'a` members, and that's how we'll get the `'a`s we need. Thus:
735 ... u (fun (a : 'a) (b : 'b) -> ... f a ... ) ...
737 In order for `u` to have the kind of argument it needs, the `... (f a) ...` has to evaluate to a result of type `'b`. The easiest way to do this is to collapse (or "unify") the types `'b` and `'d`, with the result that `f a` will have type `('c -> 'b -> 'b) -> 'b -> 'b`. Let's postulate an argument `k` of type `('c -> 'b -> 'b)` and supply it to `(f a)`:
739 ... u (fun (a : 'a) (b : 'b) -> ... f a k ... ) ...
741 Now we have an argument `b` of type `'b`, so we can supply that to `(f a) k`, getting a result of type `'b`, as we need:
743 ... u (fun (a : 'a) (b : 'b) -> f a k b) ...
745 Now, we've used a `k` that we pulled out of nowhere, so we need to abstract over it:
747 fun (k : 'c -> 'b -> 'b) -> u (fun (a : 'a) (b : 'b) -> f a k b)
749 This whole expression has type `('c -> 'b -> 'b) -> 'b -> 'b`, which is exactly the type of a `('c, 'b) list'`. So we can hypothesize that our bind is:
751 l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
752 (f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
753 : ('c -> 'b -> 'b) -> 'b -> 'b =
754 fun k -> u (fun a b -> f a k b)
756 That is a function of the right type for our bind, but to check whether it works, we have to verify it (with the unit we chose) against the monad laws, and reason whether it will have the right behavior.
758 Here's a way to persuade yourself that it will have the right behavior. First, it will be handy to eta-expand our `fun k -> u (fun a b -> f a k b)` to:
760 fun k z -> u (fun a b -> f a k b) z
762 Now let's think about what this does. It's a wrapper around `u`. In order to behave as the list which is the result of mapping `f` over each element of `u`, and then joining (`concat`ing) the results, this wrapper would have to accept arguments `k` and `z` and fold them in just the same way that the list which is the result of mapping `f` and then joining the results would fold them. Will it?
764 Suppose we have a list' whose contents are `[1; 2; 4; 8]`---that is, our list' will be `fun f z -> f 1 (f 2 (f 4 (f 8 z)))`. We call that list' `u`. Suppose we also have a function `f` that for each `int` we give it, gives back a list of the divisors of that `int` that are greater than 1. Intuitively, then, binding `u` to `f` should give us:
767 concat [[]; [2]; [2; 4]; [2; 4; 8]] =
770 Or rather, it should give us a list' version of that, which takes a function `k` and value `z` as arguments, and returns the right fold of `k` and `z` over those elements. What does our formula
772 fun k z -> u (fun a b -> f a k b) z
774 do? Well, for each element `a` in `u`, it applies `f` to that `a`, getting one of the lists:
781 (or rather, their list' versions). Then it takes the accumulated result `b` of previous steps in the fold, and it folds `k` and `b` over the list generated by `f a`. The result of doing so is passed on to the next step as the accumulated result so far.
783 So if, for example, we let `k` be `+` and `z` be `0`, then the computation would proceed:
786 right-fold + and 0 over [2; 4; 8] = 2+4+8+0 ==>
787 right-fold + and 2+4+8+0 over [2; 4] = 2+4+2+4+8+0 ==>
788 right-fold + and 2+4+2+4+8+0 over [2] = 2+2+4+2+4+8+0 ==>
789 right-fold + and 2+2+4+2+4+8+0 over [] = 2+2+4+2+4+8+0
791 which indeed is the result of right-folding + and 0 over `[2; 2; 4; 2; 4; 8]`. If you trace through how this works, you should be able to persuade yourself that our formula:
793 fun k z -> u (fun a b -> f a k b) z
795 will deliver just the same folds, for arbitrary choices of `k` and `z` (with the right types), and arbitrary list's `u` and appropriately-typed `f`s, as
797 fun k z -> List.fold_right k (concat (map f u)) z
801 For future reference, we might make two eta-reductions to our formula, so that we have instead:
803 let l'_bind = fun k -> u (fun a -> f a k);;
805 Let's make some more tests:
808 l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
810 l'_bind (fun f z -> f 1 (f 2 z))
811 (fun i -> fun f z -> f i (f (i+1) z)) ~~> <fun>
813 Sigh. OCaml won't show us our own list. So we have to choose an `f`
814 and a `z` that will turn our hand-crafted lists into standard OCaml
815 lists, so that they will print out.
817 # let cons h t = h :: t;; (* OCaml is stupid about :: *)
818 # l'_bind (fun f z -> f 1 (f 2 z))
819 (fun i -> fun f z -> f i (f (i+1) z)) cons [];;
820 - : int list = [1; 2; 2; 3]
825 Montague's PTQ treatment of DPs as generalized quantifiers
826 ----------------------------------------------------------
828 We've hinted that Montague's treatment of DPs as generalized
829 quantifiers embodies the spirit of continuations (see de Groote 2001,
830 Barker 2002 for lengthy discussion). Let's see why.
832 First, we'll need a type constructor. As you probably know,
833 Montague replaced individual-denoting determiner phrases (with type `e`)
834 with generalized quantifiers (with [extensional] type `(e -> t) -> t`.
835 In particular, the denotation of a proper name like *John*, which
836 might originally denote a object `j` of type `e`, came to denote a
837 generalized quantifier `fun pred -> pred j` of type `(e -> t) -> t`.
838 Let's write a general function that will map individuals into their
839 corresponding generalized quantifier:
841 gqize (a : e) = fun (p : e -> t) -> p a
843 This function is what Partee 1987 calls LIFT, and it would be
844 reasonable to use it here, but we will avoid that name, given that we
845 use that word to refer to other functions.
847 This function wraps up an individual in a box. That is to say,
848 we are in the presence of a monad. The type constructor, the unit and
849 the bind follow naturally. We've done this enough times that we won't
850 belabor the construction of the bind function, the derivation is
851 highly similar to the List monad just given:
853 type 'a continuation = ('a -> 'b) -> 'b
854 c_unit (a : 'a) = fun (p : 'a -> 'b) -> p a
855 c_bind (u : ('a -> 'b) -> 'b) (f : 'a -> ('c -> 'd) -> 'd) : ('c -> 'd) -> 'd =
856 fun (k : 'a -> 'b) -> u (fun (a : 'a) -> f a k)
858 Note that `c_unit` is exactly the `gqize` function that Montague used
859 to lift individuals into the continuation monad.
861 That last bit in `c_bind` looks familiar---we just saw something like
862 it in the List monad. How similar is it to the List monad? Let's
863 examine the type constructor and the terms from the list monad derived
866 type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
867 l'_unit a = fun f -> f a
868 l'_bind u f = fun k -> u (fun a -> f a k)
870 (We performed a sneaky but valid eta reduction in the unit term.)
872 The unit and the bind for the Montague continuation monad and the
873 homemade List monad are the same terms! In other words, the behavior
874 of the List monad and the behavior of the continuations monad are
875 parallel in a deep sense.
877 Have we really discovered that lists are secretly continuations? Or
878 have we merely found a way of simulating lists using list
879 continuations? Well, strictly speaking, what we have done is shown
880 that one particular implementation of lists---the right fold
881 implementation---gives rise to a continuation monad fairly naturally,
882 and that this monad can reproduce the behavior of the standard list
883 monad. But what about other list implementations? Do they give rise
884 to monads that can be understood in terms of continuations?
886 Manipulating trees with monads
887 ------------------------------
889 This topic develops an idea based on a detailed suggestion of Ken
890 Shan's. We'll build a series of functions that operate on trees,
891 doing various things, including replacing leaves, counting nodes, and
892 converting a tree to a list of leaves. The end result will be an
893 application for continuations.
895 From an engineering standpoint, we'll build a tree transformer that
896 deals in monads. We can modify the behavior of the system by swapping
897 one monad for another. We've already seen how adding a monad can add
898 a layer of funtionality without disturbing the underlying system, for
899 instance, in the way that the reader monad allowed us to add a layer
900 of intensionality to an extensional grammar, but we have not yet seen
901 the utility of replacing one monad with other.
903 First, we'll be needing a lot of trees during the remainder of the
904 course. Here's a type constructor for binary trees:
906 type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)
908 These are trees in which the internal nodes do not have labels. [How
909 would you adjust the type constructor to allow for labels on the
912 We'll be using trees where the nodes are integers, e.g.,
916 let t1 = Node ((Node ((Leaf 2), (Leaf 3))),
917 (Node ((Leaf 5),(Node ((Leaf 7),
932 Our first task will be to replace each leaf with its double:
935 let rec treemap (newleaf:'a -> 'b) (t:'a tree):('b tree) =
936 match t with Leaf x -> Leaf (newleaf x)
937 | Node (l, r) -> Node ((treemap newleaf l),
938 (treemap newleaf r));;
940 `treemap` takes a function that transforms old leaves into new leaves,
941 and maps that function over all the leaves in the tree, leaving the
942 structure of the tree unchanged. For instance:
945 let double i = i + i;;
948 Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
962 We could have built the doubling operation right into the `treemap`
963 code. However, because what to do to each leaf is a parameter, we can
964 decide to do something else to the leaves without needing to rewrite
965 `treemap`. For instance, we can easily square each leaf instead by
966 supplying the appropriate `int -> int` operation in place of `double`:
969 let square x = x * x;;
972 Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
975 Note that what `treemap` does is take some global, contextual
976 information---what to do to each leaf---and supplies that information
977 to each subpart of the computation. In other words, `treemap` has the
978 behavior of a reader monad. Let's make that explicit.
980 In general, we're on a journey of making our treemap function more and
981 more flexible. So the next step---combining the tree transducer with
982 a reader monad---is to have the treemap function return a (monadized)
983 tree that is ready to accept any `int->int` function and produce the
986 \tree (. (. (f2) (f3))(. (f5) (.(f7)(f11))))
1000 That is, we want to transform the ordinary tree `t1` (of type `int
1001 tree`) into a reader object of type `(int->int)-> int tree`: something
1002 that, when you apply it to an `int->int` function returns an `int
1003 tree` in which each leaf `x` has been replaced with `(f x)`.
1005 With previous readers, we always knew which kind of environment to
1006 expect: either an assignment function (the original calculator
1007 simulation), a world (the intensionality monad), an integer (the
1008 Jacobson-inspired link monad), etc. In this situation, it will be
1009 enough for now to expect that our reader will expect a function of
1013 type 'a reader = (int->int) -> 'a;; (* mnemonic: e for environment *)
1014 let reader_unit (x:'a): 'a reader = fun _ -> x;;
1015 let reader_bind (u: 'a reader) (f:'a -> 'c reader):'c reader = fun e -> f (u e) e;;
1018 It's easy to figure out how to turn an `int` into an `int reader`:
1021 let int2int_reader (x:'a): 'b reader = fun (op:'a -> 'b) -> op x;;
1022 int2int_reader 2 (fun i -> i + i);;
1026 But what do we do when the integers are scattered over the leaves of a
1027 tree? A binary tree is not the kind of thing that we can apply a
1028 function of type `int->int` to.
1031 let rec treemonadizer (f:'a -> 'b reader) (t:'a tree):('b tree) reader =
1032 match t with Leaf x -> reader_bind (f x) (fun x' -> reader_unit (Leaf x'))
1033 | Node (l, r) -> reader_bind (treemonadizer f l) (fun x ->
1034 reader_bind (treemonadizer f r) (fun y ->
1035 reader_unit (Node (x, y))));;
1038 This function says: give me a function `f` that knows how to turn
1039 something of type `'a` into an `'b reader`, and I'll show you how to
1040 turn an `'a tree` into an `'a tree reader`. In more fanciful terms,
1041 the `treemonadizer` function builds plumbing that connects all of the
1042 leaves of a tree into one connected monadic network; it threads the
1043 monad through the leaves.
1046 # treemonadizer int2int_reader t1 (fun i -> i + i);;
1048 Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
1051 Here, our environment is the doubling function (`fun i -> i + i`). If
1052 we apply the very same `int tree reader` (namely, `treemonadizer
1053 int2int_reader t1`) to a different `int->int` function---say, the
1054 squaring function, `fun i -> i * i`---we get an entirely different
1058 # treemonadizer int2int_reader t1 (fun i -> i * i);;
1060 Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
1063 Now that we have a tree transducer that accepts a monad as a
1064 parameter, we can see what it would take to swap in a different monad.
1065 For instance, we can use a state monad to count the number of nodes in
1069 type 'a state = int -> 'a * int;;
1070 let state_unit x i = (x, i+.5);;
1071 let state_bind u f i = let (a, i') = u i in f a (i'+.5);;
1074 Gratifyingly, we can use the `treemonadizer` function without any
1075 modification whatsoever, except for replacing the (parametric) type
1076 `reader` with `state`:
1079 let rec treemonadizer (f:'a -> 'b state) (t:'a tree):('b tree) state =
1080 match t with Leaf x -> state_bind (f x) (fun x' -> state_unit (Leaf x'))
1081 | Node (l, r) -> state_bind (treemonadizer f l) (fun x ->
1082 state_bind (treemonadizer f r) (fun y ->
1083 state_unit (Node (x, y))));;
1086 Then we can count the number of nodes in the tree:
1089 # treemonadizer state_unit t1 0;;
1090 - : int tree * int =
1091 (Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 13)
1105 Notice that we've counted each internal node twice---it's a good
1106 exercise to adjust the code to count each node once.
1108 One more revealing example before getting down to business: replacing
1109 `state` everywhere in `treemonadizer` with `list` gives us
1112 # treemonadizer (fun x -> [ [x; square x] ]) t1;;
1113 - : int list tree list =
1115 (Node (Leaf [2; 4], Leaf [3; 9]),
1116 Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))]
1119 Unlike the previous cases, instead of turning a tree into a function
1120 from some input to a result, this transformer replaces each `int` with
1123 Now for the main point. What if we wanted to convert a tree to a list
1127 type ('a, 'r) continuation = ('a -> 'r) -> 'r;;
1128 let continuation_unit x c = c x;;
1129 let continuation_bind u f c = u (fun a -> f a c);;
1131 let rec treemonadizer (f:'a -> ('b, 'r) continuation) (t:'a tree):(('b tree), 'r) continuation =
1132 match t with Leaf x -> continuation_bind (f x) (fun x' -> continuation_unit (Leaf x'))
1133 | Node (l, r) -> continuation_bind (treemonadizer f l) (fun x ->
1134 continuation_bind (treemonadizer f r) (fun y ->
1135 continuation_unit (Node (x, y))));;
1138 We use the continuation monad described above, and insert the
1139 `continuation` type in the appropriate place in the `treemonadizer` code.
1143 # treemonadizer (fun a c -> a :: (c a)) t1 (fun t -> []);;
1144 - : int list = [2; 3; 5; 7; 11]
1147 We have found a way of collapsing a tree into a list of its leaves.
1149 The continuation monad is amazingly flexible; we can use it to
1150 simulate some of the computations performed above. To see how, first
1151 note that an interestingly uninteresting thing happens if we use the
1152 continuation unit as our first argument to `treemonadizer`, and then
1153 apply the result to the identity function:
1156 # treemonadizer continuation_unit t1 (fun x -> x);;
1158 Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11)))
1161 That is, nothing happens. But we can begin to substitute more
1162 interesting functions for the first argument of `treemonadizer`:
1165 (* Simulating the tree reader: distributing a operation over the leaves *)
1166 # treemonadizer (fun a c -> c (square a)) t1 (fun x -> x);;
1168 Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
1170 (* Simulating the int list tree list *)
1171 # treemonadizer (fun a c -> c [a; square a]) t1 (fun x -> x);;
1174 (Node (Leaf [2; 4], Leaf [3; 9]),
1175 Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))
1177 (* Counting leaves *)
1178 # treemonadizer (fun a c -> 1 + c a) t1 (fun x -> 0);;
1182 We could simulate the tree state example too, but it would require
1183 generalizing the type of the continuation monad to
1185 type ('a -> 'b -> 'c) continuation = ('a -> 'b) -> 'c;;
1187 The binary tree monad
1188 ---------------------
1190 Of course, by now you may have realized that we have discovered a new
1191 monad, the binary tree monad:
1194 type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);;
1195 let tree_unit (x:'a) = Leaf x;;
1196 let rec tree_bind (u:'a tree) (f:'a -> 'b tree):'b tree =
1197 match u with Leaf x -> f x
1198 | Node (l, r) -> Node ((tree_bind l f), (tree_bind r f));;
1201 For once, let's check the Monad laws. The left identity law is easy:
1203 Left identity: bind (unit a) f = bind (Leaf a) f = fa
1205 To check the other two laws, we need to make the following
1206 observation: it is easy to prove based on `tree_bind` by a simple
1207 induction on the structure of the first argument that the tree
1208 resulting from `bind u f` is a tree with the same strucure as `u`,
1209 except that each leaf `a` has been replaced with `fa`:
1211 \tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5)))
1228 Given this equivalence, the right identity law
1230 Right identity: bind u unit = u
1232 falls out once we realize that
1234 bind (Leaf a) unit = unit a = Leaf a
1236 As for the associative law,
1238 Associativity: bind (bind u f) g = bind u (\a. bind (fa) g)
1240 we'll give an example that will show how an inductive proof would
1241 proceed. Let `f a = Node (Leaf a, Leaf a)`. Then
1243 \tree (. (. (. (. (a1)(a2)))))
1244 \tree (. (. (. (. (a1) (a1)) (. (a1) (a1))) ))
1249 bind __|__ f = __|_ = . .
1251 a1 a2 fa1 fa2 | | | |
1255 Now when we bind this tree to `g`, we get
1267 At this point, it should be easy to convince yourself that
1268 using the recipe on the right hand side of the associative law will
1269 built the exact same final tree.
1271 So binary trees are a monad.
1273 Haskell combines this monad with the Option monad to provide a monad
1275 [SearchTree](http://hackage.haskell.org/packages/archive/tree-monad/0.2.1/doc/html/src/Control-Monad-SearchTree.html#SearchTree)
1277 represent non-deterministic computations as a tree.
1279 ##[[List Monad as Continuation Monad]]##
1281 ##[[Manipulating Trees with Monads]]##