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 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
+So let's make 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. I'm
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!
indefinites and Hamblin semantics on the linguistic side, and
non-determinism on the list monad side.
+Refunctionalizing zippers
+-------------------------
+