We're going to come at continuations from three different directions, and each
time we're going to end up at the same place: a particular monad, which we'll
-call the continuation monad.
+call the Continuation monad.
-Rethinking the list monad
+Rethinking the List monad
-------------------------
-To construct a monad, the key element is to settle on a type
-constructor, and the monad more or less 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 familiar
+To construct a monad, the key element is to settle on how to implement its type, and
+the monad more or less naturally follows from that.
+We'll remind you of some examples of how monads follow from their types
+in a moment. This will involve some review of familiar
material, but it's worth doing for two reasons: it will set up a
pattern for the new discussion further below, and it will tie together
some previously unconnected elements of the course (more specifically,
version 3 lists and monads).
-For instance, take the **Reader Monad**. Once we decide that the type
-constructor is
+For instance, take the **Reader monad**. Once we decide to define its type as:
type 'a reader = env -> 'a
an `'a reader` is `env -> 'a` (by definition), the type of the
`r_unit` function is `'a -> env -> 'a`, which is an instance of the
type of the **K** combinator. So it makes sense that **K** is the unit
-for the reader monad.
+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 reason our way to the traditional reader `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
This types to `env -> 'b reader`, but we want to end up with `env ->
'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 =
+ fun e -> f (u e) e
And we're done. This gives us a bind function of the right type. We can then check whether, in combination with the unit function we chose, it satisfies the monad laws, and behaves in the way we intend. And it does.
version here in order to emphasize similarities of structure across
monads.]
-The **State Monad** is similar. Once we've decided to use the following type constructor:
+The **State monad** is similar. Once we've decided to use the following type constructor:
type 'a state = store -> ('a, store)
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/Maybe Monad** doesn't follow the same pattern so closely, so we
+The **Option/Maybe monad** doesn't follow the same pattern so closely, so we
won't pause to explore it here, though conceptually its unit and bind
follow just as naturally from its type constructor.
-Our other familiar monad is the **List Monad**, which we were told
+Our other familiar monad is the **List monad**, which we were told
looks like this:
- type 'a list = ['a];;
+ (* type 'a list = ['a];; *)
l_unit (a : 'a) = [a];;
l_bind u f = List.concat (List.map f u);;
-Thinking through the list monad will take a little time, but doing so
+Thinking through the List monad will take a little time, but doing so
will provide a connection with continuations.
Recall that `List.map` takes a function and a list and returns the
-result to applying the function to the elements of the list:
+result of applying the function to the elements of the list:
List.map (fun i -> [i; i+1]) [1; 2] ~~> [[1; 2]; [2; 3]]
-and `List.concat` takes a list of lists and erases the embedded list
+and `List.concat` takes a list of lists and erases one level of embedded list
boundaries:
List.concat [[1; 2]; [2; 3]] ~~> [1; 2; 2; 3]
type `'a`, and we want to make 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
+the function of type `'a -> 'b list` to them. Once we've done so, we
have a collection of lists, one for each of the `'a`'s. One
possibility is that we could gather them all up in a list, so that
`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 eliminate that restriction by flattening
+type `('something list) list`. We can eliminate that restriction by flattening
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.
+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
can stop. These types should remind you of the simply-typed lambda calculus
types for Church numerals (`(o -> o) -> o -> o`) with one extra type
thrown in, the type of the element at the head of the list
-(in this case, an int).
+(in this case, an `int`).
So here's our type constructor for our hand-rolled lists:
fun k z -> u (fun a b -> f a k b) z
-Now let's think about what this does. It's a wrapper around `u`. In order to behave as the list which is the result of mapping `f` over each element of `u`, and then joining (`concat`ing) the results, this wrapper would have to accept arguments `k` and `z` and fold them in just the same way that the list which is the result of mapping `f` and then joining the results would fold them. Will it?
+Now let's think about what this does. It's a wrapper around `u`. In order to behave as the list `v` which is the result of mapping `f` over each element of `u`, and then joining (`concat`ing) the results, this wrapper would have to accept arguments `k` and `z` and fold them in just the same way that `v` would.
+Will it?
-Suppose we have a list' whose contents are `[1; 2; 4; 8]`---that is, our list' will be `fun f z -> f 1 (f 2 (f 4 (f 8 z)))`. We call that list' `u`. Suppose we also have a function `f` that for each `int` we give it, gives back a list of the divisors of that `int` that are greater than 1. Intuitively, then, binding `u` to `f` should give us:
+Suppose we have a list' whose contents are `[1; 2; 4; 8]`---that is, our list' `u` will be `fun f z -> f 1 (f 2 (f 4 (f 8 z)))`. Suppose we also have a function `f` that for each `int` we give it, gives back a list of the divisors of that `int` that are greater than 1. Intuitively, then, binding `u` to `f` should give us:
+ v =
List.concat (List.map f u) =
List.concat [[]; [2]; [2; 4]; [2; 4; 8]] =
[2; 2; 4; 2; 4; 8]
-Or rather, it should give us a list' version of that, which takes a function `k` and value `z` as arguments, and returns the right fold of `k` and `z` over those elements. What does our formula
+Or rather, it should give us a list' version of that, which takes a function `k` and value `z` as arguments, and returns the right fold of `k` and `z` over those elements. Does our formula
fun k z -> u (fun a b -> f a k b) z
-do? Well, for each element `a` in `u`, it applies `f` to that `a`, getting one of the lists:
+do that? Well, for each element `a` in `u`, it applies `f` to that `a`, getting one of the lists:
[] ; result of applying f to leftmost a
[2]
[2; 4]
[2; 4; 8] ; result of applying f to rightmost a
-(or rather, their list' versions). Then it takes the accumulated result `b` of previous steps in the fold, and it folds `k` and `b` over the list generated by `f a`. The result of doing so is passed on to the next step as the accumulated result so far.
+(or rather, their list' versions). Then it takes the accumulated result `b` of previous steps in the `k`,`z`-fold, and it folds `k` and `b` over the list generated by `f a`. The result of doing so is passed on to the next step of the `k`,`z`-fold as the new accumulated result `b`.
So if, for example, we let `k` be `+` and `z` be `0`, then the computation would proceed:
will deliver just the same folds, for arbitrary choices of `k` and `z` (with the right types), and arbitrary `list'`s `u` and appropriately-typed `f`s, as
+ fun k z -> List.fold_right k v z =
fun k z -> List.fold_right k (List.concat (List.map f u)) z
would.
Let's make some more tests:
- l_bind [1; 2] (fun i -> [i; i+1]) ~~> [1; 2; 2; 3]
+ # l_bind [1; 2] (fun i -> [i; i+1]);;
+ - : int list = [1; 2; 2; 3]
- l'_bind (fun f z -> f 1 (f 2 z))
- (fun i -> fun f z -> f i (f (i+1) z)) ~~> <fun>
+ # l'_bind (fun f z -> f 1 (f 2 z)) (fun i -> fun f z -> f i (f (i+1) z));;
+ - : (int -> '_a -> '_a) -> '_a -> '_a = <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
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 [];;
+ # 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!
Let's write a general function that will map individuals into their
corresponding generalized quantifier:
- gqize (a : e) = fun (p : e -> t) -> p a
+ gqize (a : e) = fun (p : e -> t) -> p a
-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 is what Partee 1987 calls LIFT, which is not an unreasonable name. But we will avoid her term here, since that word has been used to refer to other functions in our discussion.
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
+belabor the construction of the bind function. The derivation is
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 =
+ let c_unit (a : 'a) = fun (p : 'a -> 'b) -> p a
+ let 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)
Note that `c_unit` is exactly the `gqize` function that Montague used
-to lift individuals into the continuation monad.
+to lift individuals into generalized quantifiers.
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
- let l'_unit a = fun k z -> k a z
+ type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b;;
+ (* that is of the form ('a -> 'r) -> 'r, where 'r = 'b -> 'b *)
+ let l'_unit a = fun k z -> k a z;;
This can be eta-reduced to:
let l'_unit a = fun k -> k a
+and:
+
let l'_bind u f =
- (* we mentioned three versions of this, the fully eta-expanded: *)
+ (* we mentioned three versions of this, the eta-expanded: *)
fun k z -> u (fun a b -> f a k b) z
(* an intermediate version, and the fully eta-reduced: *)
fun k -> u (fun a -> f a k)
-Consider the most eta-reduced versions of `l'_unit` and `l'_bind`. They're the same as the unit and bind for the Montague continuation monad! In other words, the behavior of our v3-list monad and the behavior of the continuations monad are
+Consider the most eta-reduced versions of `l'_unit` and `l'_bind`. They're the same as the unit and bind for the Montague Continuation monad! In other words, the behavior of our v3-List monad and the behavior of the continuations monad are
parallel in a deep sense.
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 right fold
-implementation---gives rise to a continuation monad fairly naturally,
+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
to monads that can be understood in terms of continuations?