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
-------------------------
To construct a monad, the key element is to settle on a type
-constructor, and the monad naturally follows from that. I'll remind
+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
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 we're done.
+[This bind is a simplified version of the careful `let a = u e in ...`
+constructions we provided in earlier lectures. We use the simplified
+versions here in order to emphasize similarities of structure across
+monads; the official bind is still the one with the plethora of `let`'s.]
+
The **State Monad** is similar. We somehow intuit that we want to use
the following type constructor:
But where is the reasoning that led us to this unit and bind?
And what is the type `['a]`? Magic.
-So let's take a *completely useless digressing* 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. I'm
-going to use type 3 lists, partly because I know they'll give the
-result I want, but also because they're my favorite. These were the
-lists that made lists look like Church numerals with extra bits
-embdded in them:
+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 I know
+they'll give the result I 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:
empty list: fun f z -> z
list with one element: fun f z -> f 1 z
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, and int).
+thrown in (in this case, an int).
So here's our type constructor for our hand-rolled lists:
So an `('a, 'b) list'` is a list containing elements of type `'a`,
where `'b` is the type of some part of the plumbing. This is more
general than an ordinary Ocaml list, but we'll see how to map them
-into Ocaml lists soon. We don't need to grasp the role of the `'b`'s
+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
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!
-Just for mnemonic purposes (sneaking in an instance of eta reduction
-to the definition of unit), we can summarize the result as follows:
+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
+----------------------------------------------------------
+
+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 we won't
+belabor the construction of the bind function, the derivation is
+similar to the List monad just given:
+
+<pre>
+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)
+</pre>
+
+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'_unit x = fun f -> f x
l'_bind u f = fun k -> u (fun x -> f x k)
-To bad this digression, though it ties together various
-elements of the course, has *no relevance whatsoever* to the topic of
-continuations.
+(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
+ 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
+-------------------------