X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=zipper-lists-continuations.mdwn;h=08170a31eb8e9bac740ca5f35fba36a352811831;hp=2e84616ae05081cb6e7b9ccdced8487f641ef3e5;hb=cb47fc333272c7b1f1f77af11ae72643335c05dc;hpb=47bc0c16fec3c91dc1635ac6f42fe457e58fc41c
diff --git a/zipper-lists-continuations.mdwn b/zipper-lists-continuations.mdwn
index 2e84616a..08170a31 100644
--- a/zipper-lists-continuations.mdwn
+++ b/zipper-lists-continuations.mdwn
@@ -1,11 +1,14 @@
+
+[[!toc]]
+
Today we're going to encounter continuations. We're going to come at
them from three different directions, and each time we're going to end
up at the same place: a particular monad, which we'll call the
continuation monad.
-The three approches are:
+Much of this discussion benefited from detailed comments and
+suggestions from Ken Shan.
-[[!toc]]
Rethinking the list monad
-------------------------
@@ -73,7 +76,7 @@ The **State Monad** is similar. Once we've decided to use the following type co
Then our unit is naturally:
- let s_unit (x : 'a) : ('a state) = fun (s : store) -> (x, s)
+ let s_unit (a : 'a) : ('a state) = fun (s : store) -> (a, s)
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:
@@ -100,10 +103,13 @@ Our other familiar monad is the **List Monad**, which we were told
looks like this:
type 'a list = ['a];;
- l_unit (x : 'a) = [x];;
+ l_unit (a : 'a) = [a];;
l_bind u f = List.concat (List.map f u);;
-Recall that `List.map` take a function and a list and returns the
+Thinking through the list monad will take a little time, but doing so
+will provide a connection with continuations.
+
+Recall that `List.map` takes a function and a list and returns the
result to applying the function to the elements of the list:
List.map (fun i -> [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]]
@@ -117,17 +123,33 @@ And sure enough,
l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
-But where is the reasoning that led us to this unit and bind?
-And what is the type `['a]`? Magic.
-
-So let's indulge ourselves in a completely useless digression and see
-if we can gain some insight into the details of the List monad. Let's
-choose type constructor that we can peer into, using some of the
-technology we built up so laboriously during the first half of the
-course. We're going to use type 3 lists, partly because we know
-they'll give the result we want, but also because they're the coolest.
-These were the lists that made lists look like Church numerals with
-extra bits embdded in them:
+Now, why this unit, and why this bind? Well, ideally a unit should
+not throw away information, so we can rule out `fun x -> []` as an
+ideal unit. And units should not add more information than required,
+so there's no obvious reason to prefer `fun x -> [x,x]`. In other
+words, `fun x -> [x]` is a reasonable guess 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. 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
+have a collection of lists, one for each of the `'a`'s. One
+possibility is that we could gather them all up in a list, so that
+`bind' [1;2] (fun i -> [i;i]) ~~> [[1;1];[2;2]]`. But that restricts
+the object returned by the second argument of `bind` to always be of
+type `'b list list`. We can elimiate that restriction by flattening
+the list of lists into a single list. So there is some logic to the
+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 is
+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:
empty list: fun f z -> z
list with one element: fun f z -> f 1 z
@@ -167,7 +189,7 @@ general than an ordinary OCaml list, but we'll see how to map them
into OCaml lists soon. We don't need to fully grasp the role of the `'b`'s
in order to proceed to build a monad:
- l'_unit (x : 'a) : ('a, 'b) list = fun x -> fun f z -> f x z
+ l'_unit (a : 'a) : ('a, 'b) list = fun a -> fun f z -> f a z
No problem. Arriving at bind is a little more complicated, but
exactly the same principles apply, you just have to be careful and
@@ -279,9 +301,6 @@ lists, so that they will print out.
Ta da!
-To bad this digression, though it ties together various
-elements of the course, has *no relevance whatsoever* to the topic of
-continuations...
Montague's PTQ treatment of DPs as generalized quantifiers
----------------------------------------------------------
@@ -299,7 +318,7 @@ generalized quantifier `fun pred -> pred j` of type `(e -> t) -> t`.
Let's write a general function that will map individuals into their
corresponding generalized quantifier:
- gqize (x : e) = fun (p : e -> t) -> p x
+ gqize (a : e) = fun (p : e -> t) -> p a
This function wraps up an individual in a fancy box. That is to say,
we are in the presence of a monad. The type constructor, the unit and
@@ -308,36 +327,425 @@ belabor the construction of the bind function, the derivation is
similar to the List monad just given:
type 'a continuation = ('a -> 'b) -> 'b
- c_unit (x : 'a) = fun (p : 'a -> 'b) -> p x
+ c_unit (a : 'a) = fun (p : 'a -> 'b) -> p a
c_bind (u : ('a -> 'b) -> 'b) (f : 'a -> ('c -> 'd) -> 'd) : ('c -> 'd) -> 'd =
- fun (k : 'a -> 'b) -> u (fun (x : 'a) -> f x k)
+ fun (k : 'a -> 'b) -> u (fun (a : 'a) -> f a k)
How similar is it to the List monad? Let's examine the type
constructor and the terms from the list monad derived above:
type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b
- l'_unit x = fun f -> f x
- l'_bind u f = fun k -> u (fun x -> f x k)
+ l'_unit a = fun f -> f a
+ l'_bind u f = fun k -> u (fun a -> f a k)
(We performed a sneaky but valid eta reduction in the unit term.)
The unit and the bind for the Montague continuation monad and the
homemade List monad are the same terms! In other words, the behavior
of the List monad and the behavior of the continuations monad are
-parallel in a deep sense. To emphasize the parallel, we can
-instantiate the type of the list' monad using the OCaml list type:
-
- type 'a c_list = ('a -> 'a list) -> 'a list
+parallel in a deep sense.
Have we really discovered that lists are secretly continuations? Or
have we merely found a way of simulating lists using list
-continuations? Both perspectives are valid, and we can use our
-intuitions about the list monad to understand continuations, and vice
-versa (not to mention our intuitions about primitive recursion in
-Church numerals too). The connections will be expecially relevant
-when we consider indefinites and Hamblin semantics on the linguistic
-side, and non-determinism on the list monad side.
+continuations? Well, strictly speaking, what we have done is shown
+that one particular implementation of lists---the left fold
+implementation---gives rise to a continuation monad fairly naturally,
+and that this monad can reproduce the behavior of the standard list
+monad. But what about other list implementations? Do they give rise
+to monads that can be understood in terms of continuations?
Refunctionalizing zippers
-------------------------
+Manipulating trees with monads
+------------------------------
+
+This thread 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
+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
+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.)
+
+First, we'll be needing a lot of trees during the remainder of the
+course. Here's a type constructor for binary trees:
+
+ type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)
+
+These are trees in which the internal nodes do not have labels. [How
+would you adjust the type constructor to allow for labels on the
+internal nodes?]
+
+We'll be using trees where the nodes are integers, e.g.,
+
+
+
+let t1 = Node ((Node ((Leaf 2), (Leaf 3))),
+ (Node ((Leaf 5),(Node ((Leaf 7),
+ (Leaf 11))))))
+
+ .
+ ___|___
+ | |
+ . .
+_|__ _|__
+| | | |
+2 3 5 .
+ _|__
+ | |
+ 7 11
+
+
+Our first task will be to replace each leaf with its double:
+
+
+let rec treemap (newleaf:'a -> 'b) (t:'a tree):('b tree) =
+ match t with Leaf x -> Leaf (newleaf x)
+ | Node (l, r) -> Node ((treemap newleaf l),
+ (treemap newleaf r));;
+
+`treemap` takes a function that transforms old leaves into new leaves,
+and maps that function over all the leaves in the tree, leaving the
+structure of the tree unchanged. For instance:
+
+
+let double i = i + i;;
+treemap double t1;;
+- : int tree =
+Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
+
+ .
+ ___|____
+ | |
+ . .
+_|__ __|__
+| | | |
+4 6 10 .
+ __|___
+ | |
+ 14 22
+
+
+We could have built the doubling operation right into the `treemap`
+code. However, because what to do to each leaf is a parameter, we can
+decide to do something else to the leaves without needing to rewrite
+`treemap`. For instance, we can easily square each leaf instead by
+supplying the appropriate `int -> int` operation in place of `double`:
+
+
+let square x = x * x;;
+treemap square t1;;
+- : int tree =ppp
+Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
+
+
+Note that what `treemap` does is take some global, contextual
+information---what to do to each leaf---and supplies that information
+to each subpart of the computation. In other words, `treemap` has the
+behavior of a reader monad. Let's make that explicit.
+
+In general, we're on a journey of making our treemap function more and
+more flexible. So the next step---combining the tree transducer with
+a reader monad---is to have the treemap function return a (monadized)
+tree that is ready to accept any `int->int` function and produce the
+updated tree.
+
+\tree (. (. (f2) (f3))(. (f5) (.(f7)(f11))))
+
+\f .
+ ____|____
+ | |
+ . .
+__|__ __|__
+| | | |
+f2 f3 f5 .
+ __|___
+ | |
+ f7 f11
+
+
+That is, we want to transform the ordinary tree `t1` (of type `int
+tree`) into a reader object of type `(int->int)-> int tree`: something
+that, when you apply it to an `int->int` function returns an `int
+tree` in which each leaf `x` has been replaced with `(f x)`.
+
+With previous readers, we always knew which kind of environment to
+expect: either an assignment function (the original calculator
+simulation), a world (the intensionality monad), an integer (the
+Jacobson-inspired link monad), etc. In this situation, it will be
+enough for now to expect that our reader will expect a function of
+type `int->int`.
+
+
+type 'a reader = (int->int) -> 'a;; (* mnemonic: e for environment *)
+let reader_unit (x:'a): 'a reader = fun _ -> x;;
+let reader_bind (u: 'a reader) (f:'a -> 'c reader):'c reader = fun e -> f (u e) e;;
+
+
+It's easy to figure out how to turn an `int` into an `int reader`:
+
+
+let int2int_reader (x:'a): 'b reader = fun (op:'a -> 'b) -> op x;;
+int2int_reader 2 (fun i -> i + i);;
+- : int = 4
+
+
+But what do we do when the integers are scattered over the leaves of a
+tree? A binary tree is not the kind of thing that we can apply a
+function of type `int->int` to.
+
+
+let rec treemonadizer (f:'a -> 'b reader) (t:'a tree):('b tree) reader =
+ match t with Leaf x -> reader_bind (f x) (fun x' -> reader_unit (Leaf x'))
+ | Node (l, r) -> reader_bind (treemonadizer f l) (fun x ->
+ reader_bind (treemonadizer f r) (fun y ->
+ reader_unit (Node (x, y))));;
+
+
+This function says: give me a function `f` that knows how to turn
+something of type `'a` into an `'b reader`, and I'll show you how to
+turn an `'a tree` into an `'a tree reader`. In more fanciful terms,
+the `treemonadizer` function builds plumbing that connects all of the
+leaves of a tree into one connected monadic network; it threads the
+monad through the leaves.
+
+
+# treemonadizer int2int_reader t1 (fun i -> i + i);;
+- : int tree =
+Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
+
+
+Here, our environment is the doubling function (`fun i -> i + i`). If
+we apply the very same `int tree reader` (namely, `treemonadizer
+int2int_reader t1`) to a different `int->int` function---say, the
+squaring function, `fun i -> i * i`---we get an entirely different
+result:
+
+
+# treemonadizer int2int_reader t1 (fun i -> i * i);;
+- : int tree =
+Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
+
+
+Now that we have a tree transducer that accepts a monad as a
+parameter, we can see what it would take to swap in a different monad.
+For instance, we can use a state monad to count the number of nodes in
+the tree.
+
+
+type 'a state = int -> 'a * int;;
+let state_unit x i = (x, i+.5);;
+let state_bind u f i = let (a, i') = u i in f a (i'+.5);;
+
+
+Gratifyingly, we can use the `treemonadizer` function without any
+modification whatsoever, except for replacing the (parametric) type
+`reader` with `state`:
+
+
+let rec treemonadizer (f:'a -> 'b state) (t:'a tree):('b tree) state =
+ match t with Leaf x -> state_bind (f x) (fun x' -> state_unit (Leaf x'))
+ | Node (l, r) -> state_bind (treemonadizer f l) (fun x ->
+ state_bind (treemonadizer f r) (fun y ->
+ state_unit (Node (x, y))));;
+
+
+Then we can count the number of nodes in the tree:
+
+
+# treemonadizer state_unit t1 0;;
+- : int tree * int =
+(Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 13)
+
+ .
+ ___|___
+ | |
+ . .
+_|__ _|__
+| | | |
+2 3 5 .
+ _|__
+ | |
+ 7 11
+
+
+Notice that we've counted each internal node twice---it's a good
+exercise to adjust the code to count each node once.
+
+One more revealing example before getting down to business: replacing
+`state` everywhere in `treemonadizer` with `list` gives us
+
+
+# treemonadizer (fun x -> [ [x; square x] ]) t1;;
+- : int list tree list =
+[Node
+ (Node (Leaf [2; 4], Leaf [3; 9]),
+ Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))]
+
+
+Unlike the previous cases, instead of turning a tree into a function
+from some input to a result, this transformer replaces each `int` with
+a list of `int`'s.
+
+Now for the main point. What if we wanted to convert a tree to a list
+of leaves?
+
+
+type ('a, 'r) continuation = ('a -> 'r) -> 'r;;
+let continuation_unit x c = c x;;
+let continuation_bind u f c = u (fun a -> f a c);;
+
+let rec treemonadizer (f:'a -> ('b, 'r) continuation) (t:'a tree):(('b tree), 'r) continuation =
+ match t with Leaf x -> continuation_bind (f x) (fun x' -> continuation_unit (Leaf x'))
+ | Node (l, r) -> continuation_bind (treemonadizer f l) (fun x ->
+ continuation_bind (treemonadizer f r) (fun y ->
+ continuation_unit (Node (x, y))));;
+
+
+We use the continuation monad described above, and insert the
+`continuation` type in the appropriate place in the `treemonadizer` code.
+We then compute:
+
+
+# treemonadizer (fun a c -> a :: (c a)) t1 (fun t -> []);;
+- : int list = [2; 3; 5; 7; 11]
+
+
+We have found a way of collapsing a tree into a list of its leaves.
+
+The continuation monad is amazingly flexible; we can use it to
+simulate some of the computations performed above. To see how, first
+note that an interestingly uninteresting thing happens if we use the
+continuation unit as our first argument to `treemonadizer`, and then
+apply the result to the identity function:
+
+
+# treemonadizer continuation_unit t1 (fun x -> x);;
+- : int tree =
+Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11)))
+
+
+That is, nothing happens. But we can begin to substitute more
+interesting functions for the first argument of `treemonadizer`:
+
+
+(* Simulating the tree reader: distributing a operation over the leaves *)
+# treemonadizer (fun a c -> c (square a)) t1 (fun x -> x);;
+- : int tree =
+Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
+
+(* Simulating the int list tree list *)
+# treemonadizer (fun a c -> c [a; square a]) t1 (fun x -> x);;
+- : int list tree =
+Node
+ (Node (Leaf [2; 4], Leaf [3; 9]),
+ Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))
+
+(* Counting leaves *)
+# treemonadizer (fun a c -> 1 + c a) t1 (fun x -> 0);;
+- : int = 5
+
+
+We could simulate the tree state example too, but it would require
+generalizing the type of the continuation monad to
+
+ type ('a -> 'b -> 'c) continuation = ('a -> 'b) -> 'c;;
+
+The binary tree monad
+---------------------
+
+Of course, by now you may have realized that we have discovered a new
+monad, the binary tree monad:
+
+
+type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);;
+let tree_unit (x:'a) = Leaf x;;
+let rec tree_bind (u:'a tree) (f:'a -> 'b tree):'b tree =
+ match u with Leaf x -> f x
+ | Node (l, r) -> Node ((tree_bind l f), (tree_bind r f));;
+
+
+For once, let's check the Monad laws. The left identity law is easy:
+
+ Left identity: bind (unit a) f = bind (Leaf a) f = fa
+
+To check the other two laws, we need to make the following
+observation: it is easy to prove based on `tree_bind` by a simple
+induction on the structure of the first argument that the tree
+resulting from `bind u f` is a tree with the same strucure as `u`,
+except that each leaf `a` has been replaced with `fa`:
+
+\tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5)))
+
+ . .
+ __|__ __|__
+ | | | |
+ a1 . fa1 .
+ _|__ __|__
+ | | | |
+ . a5 . fa5
+ bind _|__ f = __|__
+ | | | |
+ . a4 . fa4
+ __|__ __|___
+ | | | |
+ a2 a3 fa2 fa3
+
+
+Given this equivalence, the right identity law
+
+ Right identity: bind u unit = u
+
+falls out once we realize that
+
+ bind (Leaf a) unit = unit a = Leaf a
+
+As for the associative law,
+
+ Associativity: bind (bind u f) g = bind u (\a. bind (fa) g)
+
+we'll give an example that will show how an inductive proof would
+proceed. Let `f a = Node (Leaf a, Leaf a)`. Then
+
+\tree (. (. (. (. (a1)(a2)))))
+\tree (. (. (. (. (a1) (a1)) (. (a1) (a1))) ))
+
+ .
+ ____|____
+ . . | |
+bind __|__ f = __|_ = . .
+ | | | | __|__ __|__
+ a1 a2 fa1 fa2 | | | |
+ a1 a1 a1 a1
+
+
+Now when we bind this tree to `g`, we get
+
+
+ .
+ ____|____
+ | |
+ . .
+ __|__ __|__
+ | | | |
+ ga1 ga1 ga1 ga1
+
+
+At this point, it should be easy to convince yourself that
+using the recipe on the right hand side of the associative law will
+built the exact same final tree.
+
+So binary trees are a monad.
+
+Haskell combines this monad with the Option monad to provide a monad
+called a
+[SearchTree](http://hackage.haskell.org/packages/archive/tree-monad/0.2.1/doc/html/src/Control-Monad-SearchTree.html#SearchTree)
+that is intended to
+represent non-deterministic computations as a tree.