X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week11.mdwn;h=3daf56c5398e8fb01f508d9cb3177197e741e79c;hp=357711a0c19765910ff7b2089b57da25666dc8cd;hb=9c7ca26e6dd0c1ce1b6cd653e27a083b7379a5dd;hpb=fd7ca2e46f24f795d9bd8d14b49ba25d4e2b277f
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+++ b/week11.mdwn
@@ -1,258 +1,11 @@
These notes may change in the next few days (today is 30 Nov 2010).
The material here benefited from many discussions with Ken Shan.
-[[!toc]]
+##[[Tree and List Zippers]]##
-##List Zippers##
-
-Say you've got some moderately-complex function for searching through a list, for example:
-
- 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;;
-
-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.
+##[[Coroutines and Aborts]]##
+##[[From Lists to Continuations]]##
##Same-fringe using a zipper-based coroutine##
@@ -499,10 +252,10 @@ It's possible to build cooperative threads without using those tools, however. S
return loop (next1(tree1), next2(tree2))
end
- > return same_fringe ( {leaf=1}, {leaf=2} )
+ > return same_fringe ( {leaf=1}, {leaf=2})
false
- > return same_fringe ( {leaf=1}, {leaf=1} )
+ > return same_fringe ( {leaf=1}, {leaf=1})
true
> return same_fringe ( {left = {leaf=1}, right = {left = {leaf=2}, right = {leaf=3}}},
@@ -758,253 +511,17 @@ So now, guess what would be the result of doing the following:
-
-Refunctionalizing zippers: from lists to continuations
-------------------------------------------------------
-
-If zippers are continuations reified (defuntionalized), then one route
-to continuations is to re-functionalize a zipper. Then the
-concreteness and understandability of the zipper provides a way of
-understanding and equivalent treatment using continuations.
-
-Let's work with lists of chars for a change. To maximize readability, we'll
-indulge in an abbreviatory convention that "abSd" abbreviates the
-list `['a'; 'b'; 'S'; 'd']`.
-
-We will set out to compute a deceptively simple-seeming **task: given a
-string, replace each occurrence of 'S' in that string with a copy of
-the string up to that point.**
-
-We'll define a function `t` (for "task") that maps strings to their
-updated version.
-
-Expected behavior:
-
-
-t "abSd" ~~> "ababd"
-
-
-
-In linguistic terms, this is a kind of anaphora
-resolution, where `'S'` is functioning like an anaphoric element, and
-the preceding string portion is the antecedent.
-
-This deceptively simple task gives rise to some mind-bending complexity.
-Note that it matters which 'S' you target first (the position of the *
-indicates the targeted 'S'):
-
-
- t "aSbS"
- *
-~~> t "aabS"
- *
-~~> "aabaab"
-
-
-versus
-
-
- t "aSbS"
- *
-~~> t "aSbaSb"
- *
-~~> t "aabaSb"
- *
-~~> "aabaaabab"
-
-
-versus
-
-
- t "aSbS"
- *
-~~> t "aSbaSb"
- *
-~~> t "aSbaaSbab"
- *
-~~> t "aSbaaaSbaabab"
- *
-~~> ...
-
-
-Aparently, this task, as simple as it is, is a form of computation,
-and the order in which the `'S'`s get evaluated can lead to divergent
-behavior.
-
-For now, we'll agree to always evaluate the leftmost `'S'`, which
-guarantees termination, and a final string without any `'S'` in it.
-
-This is a task well-suited to using a zipper. We'll define a function
-`tz` (for task with zippers), which accomplishes the task by mapping a
-char list zipper to a char list. We'll call the two parts of the
-zipper `unzipped` and `zipped`; we start with a fully zipped list, and
-move elements to the zipped part by pulling the zipped down until the
-entire list has been unzipped (and so the zipped half of the zipper is empty).
-
-
-type 'a list_zipper = ('a list) * ('a list);;
-
-let rec tz (z:char list_zipper) =
- match z with (unzipped, []) -> List.rev(unzipped) (* Done! *)
- | (unzipped, 'S'::zipped) -> tz ((List.append unzipped unzipped), zipped)
- | (unzipped, target::zipped) -> tz (target::unzipped, zipped);; (* Pull zipper *)
-
-# tz ([], ['a'; 'b'; 'S'; 'd']);;
-- : char list = ['a'; 'b'; 'a'; 'b'; 'd']
-
-# tz ([], ['a'; 'S'; 'b'; 'S']);;
-- : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b']
-
-
-Note that this implementation enforces the evaluate-leftmost rule.
-Task completed.
-
-One way to see exactly what is going on is to watch the zipper in
-action by tracing the execution of `tz`. By using the `#trace`
-directive in the Ocaml interpreter, the system will print out the
-arguments to `tz` each time it is (recurcively) called. Note that the
-lines with left-facing arrows (`<--`) show (recursive) calls to `tz`,
-giving the value of its argument (a zipper), and the lines with
-right-facing arrows (`-->`) show the output of each recursive call, a
-simple list.
-
-
-# #trace tz;;
-t1 is now traced.
-# tz ([], ['a'; 'b'; 'S'; 'd']);;
-tz <-- ([], ['a'; 'b'; 'S'; 'd'])
-tz <-- (['a'], ['b'; 'S'; 'd']) (* Pull zipper *)
-tz <-- (['b'; 'a'], ['S'; 'd']) (* Pull zipper *)
-tz <-- (['b'; 'a'; 'b'; 'a'], ['d']) (* Special step *)
-tz <-- (['d'; 'b'; 'a'; 'b'; 'a'], []) (* Pull zipper *)
-tz --> ['a'; 'b'; 'a'; 'b'; 'd'] (* Output reversed *)
-tz --> ['a'; 'b'; 'a'; 'b'; 'd']
-tz --> ['a'; 'b'; 'a'; 'b'; 'd']
-tz --> ['a'; 'b'; 'a'; 'b'; 'd']
-tz --> ['a'; 'b'; 'a'; 'b'; 'd']
-- : char list = ['a'; 'b'; 'a'; 'b'; 'd']
-
-
-The nice thing about computations involving lists is that it's so easy
-to visualize them as a data structure. Eventually, we want to get to
-a place where we can talk about more abstract computations. In order
-to get there, we'll first do the exact same thing we just did with
-concrete zipper using procedures.
-
-Think of a list as a procedural recipe: `['a'; 'b'; 'S'; 'd']`
-is the result of the computation `a::(b::(S::(d::[])))` (or, in our old
-style, `makelist a (makelist b (makelist S (makelist c empty)))`).
-The recipe for constructing the list goes like this:
-
-
-(0) Start with the empty list []
-(1) make a new list whose first element is 'd' and whose tail is the list constructed in step (0)
-(2) make a new list whose first element is 'S' and whose tail is the list constructed in step (1)
------------------------------------------
-(3) make a new list whose first element is 'b' and whose tail is the list constructed in step (2)
-(4) make a new list whose first element is 'a' and whose tail is the list constructed in step (3)
-
-
-What is the type of each of these steps? Well, it will be a function
-from the result of the previous step (a list) to a new list: it will
-be a function of type `char list -> char list`. We'll call each step
-a **continuation** of the recipe. So in this context, a continuation
-is a function of type `char list -> char list`. For instance, the
-continuation corresponding to the portion of the recipe below the
-horizontal line is the function `fun (tail:char list) -> a::(b::tail)`.
-
-This means that we can now represent the unzipped part of our
-zipper--the part we've already unzipped--as a continuation: a function
-describing how to finish building the list. We'll write a new
-function, `tc` (for task with continuations), that will take an input
-list (not a zipper!) and a continuation and return a processed list.
-The structure and the behavior will follow that of `tz` above, with
-some small but interesting differences. We've included the orginal
-`tz` to facilitate detailed comparison:
-
-
-let rec tz (z:char list_zipper) =
- match z with (unzipped, []) -> List.rev(unzipped) (* Done! *)
- | (unzipped, 'S'::zipped) -> tz ((List.append unzipped unzipped), zipped)
- | (unzipped, target::zipped) -> tz (target::unzipped, zipped);; (* Pull zipper *)
-
-let rec tc (l: char list) (c: (char list) -> (char list)) =
- match l with [] -> List.rev (c [])
- | 'S'::zipped -> tc zipped (fun x -> c (c x))
- | target::zipped -> tc zipped (fun x -> target::(c x));;
-
-# tc ['a'; 'b'; 'S'; 'd'] (fun x -> x);;
-- : char list = ['a'; 'b'; 'a'; 'b']
-
-# tc ['a'; 'S'; 'b'; 'S'] (fun x -> x);;
-- : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b']
-
-
-To emphasize the parallel, I've re-used the names `zipped` and
-`target`. The trace of the procedure will show that these variables
-take on the same values in the same series of steps as they did during
-the execution of `tz` above. There will once again be one initial and
-four recursive calls to `tc`, and `zipped` will take on the values
-`"bSd"`, `"Sd"`, `"d"`, and `""` (and, once again, on the final call,
-the first `match` clause will fire, so the the variable `zipper` will
-not be instantiated).
-
-I have not called the functional argument `unzipped`, although that is
-what the parallel would suggest. The reason is that `unzipped` is a
-list, but `c` is a function. That's the most crucial difference, the
-point of the excercise, and it should be emphasized. For instance,
-you can see this difference in the fact that in `tz`, we have to glue
-together the two instances of `unzipped` with an explicit `List.append`.
-In the `tc` version of the task, we simply compose `c` with itself:
-`c o c = fun x -> c (c x)`.
-
-Why use the identity function as the initial continuation? Well, if
-you have already constructed the list "abSd", what's the next step in
-the recipe to produce the desired result (which is the same list,
-"abSd")? Clearly, the identity continuation.
-
-A good way to test your understanding is to figure out what the
-continuation function `c` must be at the point in the computation when
-`tc` is called with the first argument `"Sd"`. Two choices: is it
-`fun x -> a::b::x`, or it is `fun x -> b::a::x`?
-The way to see if you're right is to execute the following
-command and see what happens:
-
- tc ['S'; 'd'] (fun x -> 'a'::'b'::x);;
-
-There are a number of interesting directions we can go with this task.
-The task was chosen because the computation can be viewed as a
-simplified picture of a computation using continuations, where `'S'`
-plays the role of a control operator with some similarities to what is
-often called `shift`. In the analogy, the list portrays a string of
-functional applications, where `[f1; f2; f3; x]` represents `f1(f2(f3
-x))`. The limitation of the analogy is that it is only possible to
-represent computations in which the applications are always
-right-branching, i.e., the computation `((f1 f2) f3) x` cannot be
-directly represented.
-
-One possibile development is that we could add a special symbol `'#'`,
-and then the task would be to copy from the target `'S'` only back to
-the closest `'#'`. This would allow the task to simulate delimited
-continuations (for right-branching computations).
-
-The task is well-suited to the list zipper because the list monad has
-an intimate connection with continuations. The following section
-makes this connection. We'll return to the list task after talking
-about generalized quantifiers below.
-
-
Rethinking the list monad
-------------------------
To construct a monad, the key element is to settle on a type
-constructor, and the monad naturally follows from that. We'll remind
-you of some examples of how monads follow from the type constructor in
-a moment. This will involve some review of familair material, but
-it's worth doing for two reasons: it will set up a pattern for the new
-discussion further below, and it will tie together some previously
-unconnected elements of the course (more specifically, version 3 lists
-and monads).
+constructor, and the monad more or less naturally follows from that.
+We'll remind you of some examples of how monads follow from the type
+constructor in a moment. This will involve some review of familair
+material, but it's worth doing for two reasons: it will set up a
+pattern for the new discussion further below, and it will tie together
+some previously unconnected elements of the course (more specifically,
+version 3 lists and monads).
For instance, take the **Reader Monad**. Once we decide that the type
constructor is
@@ -1015,17 +532,19 @@ then the choice of unit and bind is natural:
let r_unit (a : 'a) : 'a reader = fun (e : env) -> a
-Since the type of an `'a reader` is `env -> 'a` (by definition),
-the type of the `r_unit` function is `'a -> env -> 'a`, which is a
-specific case of the type of the *K* combinator. So it makes sense
-that *K* is the unit for the reader monad.
+The reason this is a fairly natural choice is that because the type of
+an `'a reader` is `env -> 'a` (by definition), the type of the
+`r_unit` function is `'a -> env -> 'a`, which is an instance of the
+type of the *K* combinator. So it makes sense that *K* is the unit
+for the reader monad.
Since the type of the `bind` operator is required to be
r_bind : ('a reader) -> ('a -> 'b reader) -> ('b reader)
-We can reason our way to the correct `bind` function as follows. We
-start by declaring the types determined by the definition of a bind operation:
+We can reason our way to the traditional reader `bind` function as
+follows. We start by declaring the types determined by the definition
+of a bind operation:
let r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = ...
@@ -1034,19 +553,26 @@ feed it to `f`. Since `u` is a function from environments to
objects of type `'a`, the way we open a box in this monad is
by applying it to an environment:
+
... f (u e) ...
+
This subexpression types to `'b reader`, which is good. The only
-problem is that we invented an environment `e` that we didn't already have ,
-so we have to abstract over that variable to balance the books:
+problem is that we made use of an environment `e` that we didn't already have,
+so we must abstract over that variable to balance the books:
fun e -> f (u e) ...
+[To preview the discussion of the Curry-Howard correspondence, what
+we're doing here is constructing an intuitionistic proof of the type,
+and using the Curry-Howard labeling of the proof as our bind term.]
+
This types to `env -> 'b reader`, but we want to end up with `env ->
'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:
- r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) =
- f (u e) e
+
+r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = f (u e) e
+
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.
@@ -1115,7 +641,7 @@ so there's no obvious reason to prefer `fun x -> [x,x]`. In other
words, `fun x -> [x]` is a reasonable choice for a unit.
As for bind, an `'a list` monadic object contains a lot of objects of
-type `'a`, and we want to make some use of each of them (rather than
+type `'a`, and we want to make use of each of them (rather than
arbitrarily throwing some of them away). The only
thing we know for sure we can do with an object of type `'a` is apply
the function of type `'a -> 'a list` to them. Once we've done so, we
@@ -1130,13 +656,13 @@ choice of unit and bind for the list monad.
Yet we can still desire to go deeper, and see if the appropriate bind
behavior emerges from the types, as it did for the previously
-considered monads. But we can't do that if we leave the list type
-as a primitive Ocaml type. However, we know several ways of implementing
+considered monads. But we can't do that if we leave the list type as
+a primitive Ocaml type. However, we know several ways of implementing
lists using just functions. In what follows, we're going to use type
-3 lists (the right fold implementation), though it's important to
-wonder how things would change if we used some other strategy for
-implementating lists. These were the lists that made lists look like
-Church numerals with extra bits embdded in them:
+3 lists, the right fold implementation (though it's important and
+intriguing to wonder how things would change if we used some other
+strategy for implementating lists). These were the lists that made
+lists look like Church numerals with extra bits embdded in them:
empty list: fun f z -> z
list with one element: fun f z -> f 1 z
@@ -1220,7 +746,7 @@ Now, we've used a `k` that we pulled out of nowhere, so we need to abstract over
fun (k : 'c -> 'b -> 'b) -> u (fun (a : 'a) (b : 'b) -> f a k b)
-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 we our bind is:
+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:
l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
(f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
@@ -1329,7 +855,7 @@ highly similar to the List monad just given:
c_bind (u : ('a -> 'b) -> 'b) (f : 'a -> ('c -> 'd) -> 'd) : ('c -> 'd) -> 'd =
fun (k : 'a -> 'b) -> u (fun (a : 'a) -> f a k)
-Note that `c_bind` is exactly the `gqize` function that Montague used
+Note that `c_unit` is exactly the `gqize` function that Montague used
to lift individuals into the continuation monad.
That last bit in `c_bind` looks familiar---we just saw something like
@@ -1360,7 +886,7 @@ to monads that can be understood in terms of continuations?
Manipulating trees with monads
------------------------------
-This thread develops an idea based on a detailed suggestion of Ken
+This topic develops an idea based on a detailed suggestion of Ken
Shan's. We'll build a series of functions that operate on trees,
doing various things, including replacing leaves, counting nodes, and
converting a tree to a list of leaves. The end result will be an
@@ -1368,11 +894,11 @@ application for continuations.
From an engineering standpoint, we'll build a tree transformer that
deals in monads. We can modify the behavior of the system by swapping
-one monad for another. (We've already seen how adding a monad can add
+one monad for another. We've already seen how adding a monad can add
a layer of funtionality without disturbing the underlying system, for
instance, in the way that the reader monad allowed us to add a layer
of intensionality to an extensional grammar, but we have not yet seen
-the utility of replacing one monad with other.)
+the utility of replacing one monad with other.
First, we'll be needing a lot of trees during the remainder of the
course. Here's a type constructor for binary trees:
@@ -1750,4 +1276,7 @@ called a
that is intended to
represent non-deterministic computations as a tree.
+##[[List Monad as Continuation Monad]]##
+
+##[[Manipulating Trees with Monads]]##