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. It makes sense
+specific case 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 type:
+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) = ...
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
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.
+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. The only
+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
`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
+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 is
-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
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):
+principle types of these functions (rather than inferring what the
+types should be ourselves):
# fun f z -> z;;
- : 'a -> 'b -> 'b = <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 bit thrown in
+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:
(f : 'a -> ('c -> 'd -> 'd) -> 'd -> 'd)
: ('c -> 'd -> 'd) -> 'd -> 'd = ...
+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` 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:
+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) (b : 'b) -> ... f a ... ) ...
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 (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 (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:
+Note that `c_bind` 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
+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 a = fun f -> f a
Have we really discovered that lists are secretly continuations? Or
have we merely found a way of simulating lists using list
continuations? Well, strictly speaking, what we have done is shown
-that one particular implementation of lists---the left fold
+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
[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.
+
+
+Lists, zippers, 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)
+ | (sofar, 'S'::rest) -> t1 ((List.append sofar sofar), rest)
+ | (sofar, fst::rest) -> t1 (fst::sofar, rest);;
+
+# 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.
+
+
+
+
+
+
+