The three approches are:
-[[toc!]]
+[[!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
For instance, take the **Reader Monad**. Once we decide that the type
constructor is
- type 'a reader = fun e:env -> 'a
+ type 'a reader = env -> 'a
then we can deduce the unit and the bind:
- runit x:'a -> 'a reader = fun (e:env) -> x
+ let 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
+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. 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)
+ r_bind : ('a reader) -> ('a -> 'b reader) -> ('b reader)
We can deduce the correct `bind` function as follows:
- 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
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
+ r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = f (u e) e
And we're done.
+[This bind 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:
- type 'a state = 'store -> ('a, 'store)
+ type 'a state = store -> ('a, store)
So our unit is naturally
- let s_unit (x:'a):('a state) = fun (s:'store) -> (x, s)
+ let s_unit (x : 'a) : ('a state) = fun (s : store) -> (x, 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:
- let s_bind (u:'a state) (f:'a -> ('b state)):('b state) =
+ let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state =
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,
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:state) -> 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
won't pause to explore it here, though conceptually its unit and bind
looks like this:
type 'a list = ['a];;
- l_unit (x:'a) = [x];;
+ l_unit (x : 'a) = [x];;
l_bind u f = List.concat (List.map f u);;
Recall that `List.map` take a function and a list and returns the
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 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:
empty list: fun f z -> z
list with one element: fun f z -> f 1 z
list with two elements: fun f z -> f 2 (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
+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):
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
+general than an ordinary OCaml list, but we'll see how to map them
+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 (x : 'a) : ('a, 'b) list = fun x -> fun f z -> f x z
No problem. Arriving at bind is a little more complicated, but
exactly the same principles apply, you just have to be careful and
systematic about it.
- l'_bind (u:('a,'b) list') (f:'a -> ('c, 'd) list'): ('c, 'd) list' = ...
+ l'_bind (u : ('a,'b) list') (f : 'a -> ('c, 'd) list') : ('c, 'd) list' = ...
Unfortunately, we'll need to spell out the types:
- l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b)
- (f: 'a -> ('c -> 'd -> 'd) -> 'd -> 'd)
+ 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
argument a function expecting an `'a`. Once that argument is applied
to an object of type `'a`, we'll have what we need. Thus:
- .... u (fun (x:'a) -> ... (f a) ... ) ...
+ .... u (fun (a : 'a) -> ... (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
alias `'d` to `'b`, and provide `(f a)` with an argument of type `'c
-> 'b -> 'b`. Thus:
- l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b)
- (f: 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
+ l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
+ (f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
: ('c -> 'b -> 'b) -> 'b -> 'b =
- .... u (fun (x:'a) -> f a k) ...
+ .... u (fun (a : 'a) -> f a k) ...
-[Excercise: can you arrive at a fully general bind for this type
+[Exercise: can you arrive at a fully general bind for this type
constructor, one that does not collapse `'d`'s with `'b`'s?]
As usual, we have to abstract over `k`, but this time, no further
adjustments are needed:
- l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b)
- (f: 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
+ 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 (x:'a) -> f a k)
+ fun (k : 'c -> 'b -> 'b) -> u (fun (a : 'a) -> f a k)
You should carefully check to make sure that this term is consistent
with the typing.
l'_bind (fun f z -> f 1 (f 2 z))
(fun i -> fun f z -> f i (f (i+1) z)) ~~> <fun>
-Sigh. Ocaml won't show us our own list. So we have to choose an `f`
-and a `z` that will turn our hand-crafted lists into standard Ocaml
+Sigh. OCaml won't show us our own list. So we have to choose an `f`
+and a `z` that will turn our hand-crafted lists into standard OCaml
lists, so that they will print out.
-# 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]
+ # 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]
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:
-
- 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)
-
To bad this digression, though it ties together various
elements of the course, has *no relevance whatsoever* to the topic of
-continuations.
+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 (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
+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:
- 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)
+ 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:
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.)
+(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:
+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
+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.
+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
+-------------------------