+
+[[!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
-------------------------
type 'a reader = env -> 'a
-then we can deduce the unit and the bind:
+then the choice of unit and bind is natural:
- let r_unit (x : 'a) : 'a reader = fun (e : env) -> x
+ 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
r_bind : ('a reader) -> ('a -> 'b reader) -> ('b reader)
-We can deduce the correct `bind` function as follows:
+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:
- let r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) =
+ let r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = ...
-We have to open up the `u` box and get out the `'a` object in order to
+Now we have to open up the `u` box and get out the `'a` object in order to
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) ...
+ ... 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:
- fun e -> f (u e) ...
+ fun e -> f (u e) ...
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:
+'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.
+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.
-[This bind is a condensed version of the careful `let a = u e in ...`
+[The bind we cite here is a condensed version of the careful `let a = u e in ...`
constructions we provided in earlier lectures. We use the condensed
version here in order to emphasize similarities of structure across
monads.]
-The **State Monad** is similar. We somehow intuit that we want to use
-the following type constructor:
+The **State Monad** is similar. Once we've decided to use the following type constructor:
type 'a state = store -> ('a, store)
-So our unit is naturally
+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 deduce the bind in a way similar to the reasoning given above.
-First, we need to apply `f` to the contents of the `u` box:
+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:
let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state =
+ ... f (...) ...
But unlocking the `u` box is a little more complicated. As before, we
need to posit a state `s` that we can apply `u` to. Once we do so,
however, we won't have an `'a`, we'll have a pair whose first element
is an `'a`. So we have to unpack the pair:
- ... let (a, s') = u s in ... (f a) ...
+ ... let (a, s') = u s in ... (f a) ...
Abstracting over the `s` and adjusting the types gives the result:
- let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state =
- fun (s : store) -> let (a, s') = u s in f a s'
+ let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state =
+ fun (s : store) -> let (a, s') = u s in f a s'
-The **Option Monad** doesn't follow the same pattern so closely, so we
+The **Option/Maybe Monad** doesn't follow the same pattern so closely, so we
won't pause to explore it here, though conceptually its unit and bind
follow just as naturally from its type constructor.
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]]
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 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
+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
+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: this is
+just List.concat applied to the output of List.map. 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
+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:
empty list: fun f z -> z
list with one element: fun f z -> f 1 z
list with three elements: fun f z -> f 3 (f 2 (f 1 z))
and so on. To save time, we'll let the OCaml interpreter infer the
-principle types of these functions (rather than deducing what the
-types should be):
-
-<pre>
-# fun f z -> z;;
-- : 'a -> 'b -> 'b = <fun>
-# fun f z -> f 1 z;;
-- : (int -> 'a -> 'b) -> 'a -> 'b = <fun>
-# fun f z -> f 2 (f 1 z);;
-- : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
-# fun f z -> f 3 (f 2 (f 1 z))
-- : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
-</pre>
-
-Finally, we're getting consistent principle types, so we can stop.
-These types should remind you of the simply-typed lambda calculus
-types for Church numerals (`(o -> o) -> o -> o`) with one extra bit
-thrown in (in this case, an int).
+principle types of these functions (rather than inferring what the
+types should be ourselves):
+
+ # fun f z -> z;;
+ - : 'a -> 'b -> 'b = <fun>
+ # fun f z -> f 1 z;;
+ - : (int -> 'a -> 'b) -> 'a -> 'b = <fun>
+ # fun f z -> f 2 (f 1 z);;
+ - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
+ # fun f z -> f 3 (f 2 (f 1 z))
+ - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
+
+We can see what the consistent, general principle types are at the end, so we
+can stop. These types should remind you of the simply-typed lambda calculus
+types for Church numerals (`(o -> o) -> o -> o`) with one extra type
+thrown in, the type of the element a the head of the list
+(in this case, an int).
So here's our type constructor for our hand-rolled lists:
- type 'a list' = (int -> 'a -> 'a) -> 'a -> 'a
+ type 'b list' = (int -> 'b -> 'b) -> 'b -> 'b
Generalizing to lists that contain any kind of element (not just
ints), we have
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
l'_bind (u : ('a,'b) list') (f : 'a -> ('c, 'd) list') : ('c, 'd) list' = ...
-Unfortunately, we'll need to spell out the types:
+Unpacking the types gives:
l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
(f : 'a -> ('c -> 'd -> 'd) -> 'd -> 'd)
: ('c -> 'd -> 'd) -> 'd -> 'd = ...
-It's a rookie mistake to quail before complicated types. You should
+Perhaps a bit intimiating.
+But it's a rookie mistake to quail before complicated types. You should
be no more intimiated by complex types than by a linguistic tree with
deeply embedded branches: complex structure created by repeated
application of simple rules.
+[This would be a good time to try to build your own term for the types
+just given. Doing so (or attempting to do so) will make the next
+paragraph much easier to follow.]
+
As usual, we need to unpack the `u` box. Examine the type of `u`.
-This time, `u` will only deliver up its contents if we give `u` as an
-argument a function expecting an `'a`. Once that argument is applied
-to an object of type `'a`, we'll have what we need. Thus:
+This time, `u` will only deliver up its contents if we give `u` an
+argument that is a function expecting an `'a` and a `'b`. `u` will
+fold that function over its type `'a` members, and that's how we'll get the `'a`s we need. Thus:
- .... u (fun (a : 'a) -> ... (f a) ... ) ...
+ ... u (fun (a : 'a) (b : 'b) -> ... f a ... ) ...
-In order for `u` to have the kind of argument it needs, we have to
-adjust `(f a)` (which has type `('c -> 'd -> 'd) -> 'd -> 'd`) in
-order to deliver something of type `'b -> 'b`. The easiest way is to
-alias `'d` to `'b`, and provide `(f a)` with an argument of type `'c
--> 'b -> 'b`. Thus:
+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)`:
- l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
- (f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
- : ('c -> 'b -> 'b) -> 'b -> 'b =
- .... u (fun (a : 'a) -> f a k) ...
+ ... u (fun (a : 'a) (b : 'b) -> ... f a k ... ) ...
+
+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:
+
+ ... u (fun (a : 'a) (b : 'b) -> f a k b) ...
+
+Now, we've used a `k` that we pulled out of nowhere, so we need to abstract over it:
-[Exercise: can you arrive at a fully general bind for this type
-constructor, one that does not collapse `'d`'s with `'b`'s?]
+ fun (k : 'c -> 'b -> 'b) -> u (fun (a : 'a) (b : 'b) -> f a k b)
-As usual, we have to abstract over `k`, but this time, no further
-adjustments are needed:
+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:
l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
(f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
: ('c -> 'b -> 'b) -> 'b -> 'b =
- fun (k : 'c -> 'b -> 'b) -> u (fun (a : 'a) -> f a k)
+ fun k -> u (fun a b -> f a k b)
+
+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.
+
+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:
+
+ fun k z -> u (fun a b -> f a k b) z
+
+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?
+
+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:
+
+ concat (map f u) =
+ concat [[]; [2]; [2; 4]; [2; 4; 8]] =
+ [2; 2; 4; 2; 4; 8]
+
+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
+
+ fun k z -> u (fun a b -> f a k b) z
+
+do? Well, for each element `a` in `u`, it applies `f` to that `a`, getting one of the lists:
+
+ []
+ [2]
+ [2; 4]
+ [2; 4; 8]
-You should carefully check to make sure that this term is consistent
-with the typing.
+(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.
-Our theory is that this monad should be capable of exactly
-replicating the behavior of the standard List monad. Let's test:
+So if, for example, we let `k` be `+` and `z` be `0`, then the computation would proceed:
+
+ 0 ==>
+ right-fold + and 0 over [2; 4; 8] = 2+4+8+0 ==>
+ right-fold + and 2+4+8+0 over [2; 4] = 2+4+2+4+8+0 ==>
+ right-fold + and 2+4+2+4+8+0 over [2] = 2+2+4+2+4+8+0 ==>
+ right-fold + and 2+2+4+2+4+8+0 over [] = 2+2+4+2+4+8+0
+
+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:
+
+ fun k z -> u (fun a b -> f a k b) z
+
+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
+
+ fun k z -> List.fold_right k (concat (map f u)) z
+
+would.
+
+For future reference, we might make two eta-reductions to our formula, so that we have instead:
+
+ let l'_bind = fun k -> u (fun a -> f a k);;
+
+Let's make some more tests:
l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
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
----------------------------------------------------------
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,
+This function is what Partee 1987 calls LIFT, and it would be
+reasonable to use it here, but we will avoid that name, given that we
+use that word to refer to other functions.
+
+This function wraps up an individual in a box. That is to say,
we are in the presence of a monad. The type constructor, the unit and
the bind follow naturally. We've done this enough times that we won't
belabor the construction of the bind function, the derivation is
-similar to the List monad just given:
+highly 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)
+
+Note that `c_bind` is exactly the `gqize` function that Montague used
+to lift individuals into the continuation monad.
-How similar is it to the List monad? Let's examine the type
-constructor and the terms from the list monad derived above:
+That last bit in `c_bind` looks familiar---we just saw something like
+it in the List monad. 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.
-
-Refunctionalizing zippers
--------------------------
+continuations? Well, strictly speaking, what we have done is shown
+that one particular implementation of lists---the right 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?
Manipulating trees with monads
------------------------------
</pre>
Notice that we've counted each internal node twice---it's a good
-excerice to adjust the code to count each node once.
+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
<pre>
-# treemonadizer (fun x -> [[x; square x]]) t1;;
+# treemonadizer (fun x -> [ [x; square x] ]) t1;;
- : int list tree list =
[Node
(Node (Leaf [2; 4], Leaf [3; 9]),
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:
+
+<pre>
+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));;
+</pre>
+
+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)))
+<pre>
+ . .
+ __|__ __|__
+ | | | |
+ a1 . fa1 .
+ _|__ __|__
+ | | | |
+ . a5 . fa5
+ bind _|__ f = __|__
+ | | | |
+ . a4 . fa4
+ __|__ __|___
+ | | | |
+ a2 a3 fa2 fa3
+</pre>
+
+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))) ))
+<pre>
+ .
+ ____|____
+ . . | |
+bind __|__ f = __|_ = . .
+ | | | | __|__ __|__
+ a1 a2 fa1 fa2 | | | |
+ a1 a1 a1 a1
+</pre>
+
+Now when we bind this tree to `g`, we get
+
+<pre>
+ .
+ ____|____
+ | |
+ . .
+ __|__ __|__
+ | | | |
+ ga1 ga1 ga1 ga1
+</pre>
+
+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.
+
+
+Refunctionalizing zippers: from lists to continuations
+------------------------------------------------------
+
+Let's work with lists of chars for a change. To maximize readability, we'll
+indulge in an abbreviatory convention that "abc" abbreviates the
+list `['a'; 'b'; 'c']`.
+
+Task 1: replace each occurrence of 'S' with a copy of the string up to
+that point.
+
+Expected behavior:
+
+<pre>
+t1 "abSe" ~~> "ababe"
+</pre>
+
+
+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'):
+
+<pre>
+ t1 "aSbS"
+ *
+~~> t1 "aabS"
+ *
+~~> "aabaab"
+</pre>
+
+versus
+
+<pre>
+ t1 "aSbS"
+ *
+~~> t1 "aSbaSb"
+ *
+~~> t1 "aabaSb"
+ *
+~~> "aabaaabab"
+</pre>
+
+versus
+
+<pre>
+ t1 "aSbS"
+ *
+~~> t1 "aSbaSb"
+ *
+~~> t1 "aSbaaSbab"
+ *
+~~> t1 "aSbaaaSbaabab"
+ *
+~~> ...
+</pre>
+
+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, as usual, we'll agree to always evaluate the leftmost `'S'`.
+
+This is a task well-suited to using a zipper.
+
+<pre>
+type 'a list_zipper = ('a list) * ('a list);;
+
+let rec t1 (z:char list_zipper) =
+ match z with (sofar, []) -> List.rev(sofar) (* Done! *)
+ | (sofar, 'S'::rest) -> t1 ((List.append sofar sofar), rest)
+ | (sofar, fst::rest) -> t1 (fst::sofar, rest);; (* Move zipper *)
+
+# t1 ([], ['a'; 'b'; 'S'; 'e']);;
+- : char list = ['a'; 'b'; 'a'; 'b'; 'e']
+
+# t1 ([], ['a'; 'S'; 'b'; 'S']);;
+- : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b']
+</pre>
+
+Note that this implementation enforces the evaluate-leftmost rule.
+Task 1 completed.
+
+One way to see exactly what is going on is to watch the zipper in
+action by tracing the execution of `t1`. By using the `#trace`
+directive in the Ocaml interpreter, the system will print out the
+arguments to `t1` each time it is (recurcively) called:
+
+<pre>
+# #trace t1;;
+t1 is now traced.
+# t1 ([], ['a'; 'b'; 'S'; 'e']);;
+t1 <-- ([], ['a'; 'b'; 'S'; 'e'])
+t1 <-- (['a'], ['b'; 'S'; 'e'])
+t1 <-- (['b'; 'a'], ['S'; 'e'])
+t1 <-- (['b'; 'a'; 'b'; 'a'], ['e'])
+t1 <-- (['e'; 'b'; 'a'; 'b'; 'a'], [])
+t1 --> ['a'; 'b'; 'a'; 'b'; 'e']
+t1 --> ['a'; 'b'; 'a'; 'b'; 'e']
+t1 --> ['a'; 'b'; 'a'; 'b'; 'e']
+t1 --> ['a'; 'b'; 'a'; 'b'; 'e']
+t1 --> ['a'; 'b'; 'a'; 'b'; 'e']
+- : char list = ['a'; 'b'; 'a'; 'b'; 'e']
+</pre>
+
+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'; 'c']` means (1)
+start with the empty list `[]`; (2) make a new list whose first
+element is 'c' and whose tail is the list construted in the previous
+step; (3) make a new list whose first element is 'b' and whose tail is
+the list constructed in the previous step; and (4) make a new list
+whose first element is 'a' and whose tail is the list constructed in
+the previous step.
+
+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`.
+
+This means that we can now represent the sofar part of our zipper--the
+part we've already unzipped--as a continuation, a function describing
+how to finish building the list:
+
+<pre>
+let rec t1c (l: char list) (c: (char list) -> (char list)) =
+ match l with [] -> c []
+ | 'S'::rest -> t1c rest (fun x -> c (c x))
+ | a::rest -> t1c rest (fun x -> List.append (c x) [a]);;
+
+# t1c ['a'; 'b'; 'S'] (fun x -> x);;
+- : char list = ['a'; 'b'; 'a'; 'b']
+
+# t1c ['a'; 'S'; 'b'; 'S'] (fun x -> x);;
+- : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b']
+</pre>
+
+Note that we don't need to do any reversing.
+