The three approches are:
-* Rethinking the list monad;
-* Montague's PTQ treatment of DPs as generalized quantifiers; and
-* Refunctionalizing zippers (Shan: zippers are defunctionalized continuations);
+[[!toc]]
Rethinking the list monad
-------------------------
then we can deduce the unit and the bind:
- runit x:'a -> 'a reader = fun (e:env) -> x
+ r_unit x:'a -> 'a reader = fun (e:env) -> x
Since the type of an `'a reader` is `fun e:env -> 'a` (by definition),
-the type of the `runit` function is `'a -> e:env -> 'a`, which is a
+the type of the `r_unit` function is `'a -> e: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.
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`, we'll have
+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 don't have an `e`, so we have to abstract over that
-variable:
+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) ...
This types to `env -> 'b reader`, but we want to end up with `env ->
-'b`. The easiest way to turn a 'b reader into a 'b is to apply it to
+'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
and a `z` that will turn our hand-crafted lists into standard Ocaml
lists, so that they will print out.
+<pre>
# let cons h t = h :: t;; (* Ocaml is stupid about :: *)
# l'_bind (fun f z -> f 1 (f 2 z))
(fun i -> fun f z -> f i (f (i+1) z)) cons [];;
- : int list = [1; 2; 2; 3]
+</pre>
Ta da!
elements of the course, has *no relevance whatsoever* to the topic of
continuations.
+Montague's PTQ treatment of DPs as generalized quantifiers
+----------------------------------------------------------
+
+We've hinted that Montague's treatment of DPs as generalized
+quantifiers embodies the spirit of continuations (see de Groote 2001,
+Barker 2002 for lengthy discussion). Let's see why.
+
+First, we'll need a type constructor. As you probably know,
+Montague replaced individual-denoting determiner phrases (with type `e`)
+with generalized quantifiers (with [extensional] type `(e -> t) -> t`.
+In particular, the denotation of a proper name like *John*, which
+might originally denote a object `j` of type `e`, came to denote a
+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
+
+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
+the bind follow naturally. We've done this enough times that I won't
+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_bind (u:('a -> 'b) -> 'b) (f: 'a -> ('c -> 'd) -> 'd): ('c -> 'd) -> 'd =
+ fun (k:'a -> 'b) -> u (fun (x:'a) -> f x 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)
+
+(I 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
+ let c_list_unit x = fun f -> f x;;
+ let c_list_bind u f = fun k -> u (fun x -> f x k);;
+
+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. 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
+-------------------------
+