type 'a reader = env -> 'a
-then we can deduce the unit and the bind:
+then the choice of unit and bind is natural:
- let r_unit (x : 'a) : 'a reader = fun (e : env) -> x
+ let r_unit (a : 'a) : 'a reader = fun (e : env) -> 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
+specific case of the type of the *K* combinator. 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)
-We can deduce the correct `bind` function as follows:
+We can reason our way to the correct `bind` function as follows. We start by declaring the type:
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
+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
objects of type `'a`, the way we open a box in this monad is
by applying it to an environment:
- .... f (u e) ...
+ ... f (u e) ...
This subexpression types to `'b reader`, which is good. The only
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) ...
+ fun e -> f (u e) ...
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:
+'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.
+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.
-[This bind is a condensed version of the careful `let a = u e in ...`
+[The bind we cite here 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:
+The **State Monad** is similar. Once we've decided to use the following type constructor:
type 'a state = store -> ('a, store)
-So our unit is naturally
+Then our unit is naturally:
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:
+And we can reason our way to the bind function 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 =
+ ... f (...) ...
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,
however, we won't have an `'a`, we'll have a pair whose first element
is an `'a`. So we have to unpack the pair:
- ... let (a, s') = u s in ... (f a) ...
+ ... let (a, s') = u s in ... (f a) ...
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 : store) -> 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
+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.
principle types of these functions (rather than deducing what the
types should be):
-<pre>
-# fun f z -> z;;
-- : 'a -> 'b -> 'b = <fun>
-# fun f z -> f 1 z;;
-- : (int -> 'a -> 'b) -> 'a -> 'b = <fun>
-# fun f z -> f 2 (f 1 z);;
-- : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
-# fun f z -> f 3 (f 2 (f 1 z))
-- : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
-</pre>
-
-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, an int).
+ # fun f z -> z;;
+ - : 'a -> 'b -> 'b = <fun>
+ # fun f z -> f 1 z;;
+ - : (int -> 'a -> 'b) -> 'a -> 'b = <fun>
+ # fun f z -> f 2 (f 1 z);;
+ - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
+ # fun f z -> f 3 (f 2 (f 1 z))
+ - : (int -> 'a -> 'a) -> 'a -> 'a = <fun>
+
+We can see what the consistent, general principle types are at the end, 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, an int).
So here's our type constructor for our hand-rolled lists:
- type 'a list' = (int -> 'a -> 'a) -> 'a -> 'a
+ type 'b list' = (int -> 'b -> 'b) -> 'b -> 'b
Generalizing to lists that contain any kind of element (not just
ints), we have
l'_bind (u : ('a,'b) list') (f : 'a -> ('c, 'd) list') : ('c, 'd) list' = ...
-Unfortunately, we'll need to spell out the types:
+Unpacking the types gives:
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
+But it's a rookie mistake to quail before complicated types. You should
be no more intimiated by complex types than by a linguistic tree with
deeply embedded branches: complex structure created by repeated
application of simple rules.
As usual, we need to unpack the `u` box. Examine the type of `u`.
This time, `u` will only deliver up its contents if we give `u` as an
-argument a function expecting an `'a`. Once that argument is applied
-to an object of type `'a`, we'll have what we need. Thus:
+argument a function expecting an `'a` and a `'b`. `u` will fold that function over its type `'a` members, and that's how we'll get the `'a`s we need. Thus:
- .... u (fun (a : 'a) -> ... (f a) ... ) ...
+ ... u (fun (a : 'a) (b : 'b) -> ... 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
-order to deliver something of type `'b -> 'b`. The easiest way is to
-alias `'d` to `'b`, and provide `(f a)` with an argument of type `'c
--> 'b -> 'b`. Thus:
+In order for `u` to have the kind of argument it needs, the `... (f a) ...` has to evaluate to a result of type `'b`. The easiest way to do this is to collapse (or "unify") the types `'b` and `'d`, with the result that `f a` will have type `('c -> 'b -> 'b) -> 'b -> 'b`. Let's postulate an argument `k` of type `('c -> 'b -> 'b)` and supply it to `(f a)`:
- l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b)
- (f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b)
- : ('c -> 'b -> 'b) -> 'b -> 'b =
- .... u (fun (a : 'a) -> f a k) ...
+ ... u (fun (a : 'a) (b : 'b) -> ... f a k ... ) ...
+
+Now we have an argument `b` of type `'b`, so we can supply that to `(f a) k`, getting a result of type `'b`, as we need:
+
+ ... u (fun (a : 'a) (b : 'b) -> f a k b) ...
-[Exercise: can you arrive at a fully general bind for this type
-constructor, one that does not collapse `'d`'s with `'b`'s?]
+Now, we've used a `k` that we pulled out of nowhere, so we need to abstract over it:
-As usual, we have to abstract over `k`, but this time, no further
-adjustments are needed:
+ fun (k : 'c -> 'b -> 'b) -> u (fun (a : 'a) (b : 'b) -> f a k b)
+
+This whole expression has type `('c -> 'b -> 'b) -> 'b -> 'b`, which is exactly the type of a `('c, 'b) list'`. So we can hypothesize that we our bind is:
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 (a : 'a) -> f a k)
+ fun k -> u (fun a b -> f a k b)
+
+That is a function of the right type for our bind, but to check whether it works, we have to verify it (with the unit we chose) against the monad laws, and reason whether it will have the right behavior.
+
+Here's a way to persuade yourself that it will have the right behavior. First, it will be handy to eta-expand our `fun k -> u (fun a b -> f a k b)` to:
+
+ 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?
+
+... MORE...
-You should carefully check to make sure that this term is consistent
-with the typing.
Our theory is that this monad should be capable of exactly
replicating the behavior of the standard List monad. Let's test: