... 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: +problem is that we made use of an environment `e` that we didn't already have, +so we must abstract over that variable to balance the books: fun e -> f (u e) ... +[To preview the discussion of the Curry-Howard correspondence, what +we're doing here is constructing an intuitionistic proof of the type, +and using the Curry-Howard labeling of the proof as our bind term.] + 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) = 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. @@ -122,7 +130,7 @@ so there's no obvious reason to prefer `fun x -> [x,x]`. In other words, `fun x -> [x]` is a reasonable choice for a unit. As for bind, an `'a list` monadic object contains a lot of objects of -type `'a`, and we want to make some use of each of them (rather than +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 @@ -137,13 +145,13 @@ 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 -considered monads. But we can't do that if we leave the list type -as a primitive Ocaml type. However, we know several ways of implementing +considered monads. But we can't do that if we leave the list type as +a primitive Ocaml type. However, we know several ways of implementing lists using just functions. In what follows, we're going to use type -3 lists (the right fold implementation), though it's important to -wonder how things would change if we used some other strategy for -implementating lists. These were the lists that made lists look like -Church numerals with extra bits embdded in them: +3 lists, the right fold implementation (though it's important and +intriguing to wonder how things would change if we used some other +strategy for implementating lists). 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 @@ -227,7 +235,7 @@ Now, we've used a `k` that we pulled out of nowhere, so we need to abstract over 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: +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 our bind is: l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b) (f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b) @@ -336,7 +344,7 @@ highly similar to the List monad just given: 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_bind` is exactly the `gqize` function that Montague used +Note that `c_unit` is exactly the `gqize` function that Montague used to lift individuals into the continuation monad. That last bit in `c_bind` looks familiar---we just saw something like diff --git a/week11.mdwn b/week11.mdwn index 3daf56c5..d0ecb704 100644 --- a/week11.mdwn +++ b/week11.mdwn @@ -511,378 +511,6 @@ So now, guess what would be the result of doing the following: -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 familair -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 - - type 'a reader = env -> 'a - -then the choice of unit and bind is natural: - - let r_unit (a : 'a) : 'a reader = fun (e : env) -> a - -The reason this is a fairly natural choice is that because the type of -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. - -Since the type of the `bind` operator is required to be - - 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) = ... - -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) ... -- -This subexpression types to `'b reader`, which is good. The only -problem is that we made use of an environment `e` that we didn't already have, -so we must abstract over that variable to balance the books: - - fun e -> f (u e) ... - -[To preview the discussion of the Curry-Howard correspondence, what -we're doing here is constructing an intuitionistic proof of the type, -and using the Curry-Howard labeling of the proof as our bind term.] - -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 -- -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. - -[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. Once we've decided to use the following type constructor: - - type 'a state = store -> ('a, store) - -Then our unit is naturally: - - let s_unit (a : 'a) : ('a state) = fun (s : store) -> (a, s) - -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) ... - -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' - -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 -looks like this: - - 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 -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: - - List.map (fun i -> [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]] - -and List.concat takes a list of lists and erases the embdded list -boundaries: - - List.concat [[1; 2]; [2; 3]] ~~> [1; 2; 2; 3] - -And sure enough, - - l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3] - -Now, why this unit, and why this bind? Well, ideally a unit should -not throw away information, so we can rule out `fun x -> []` as an -ideal unit. And units should not add more information than required, -so there's no obvious reason to prefer `fun x -> [x,x]`. In other -words, `fun x -> [x]` is a reasonable choice for a unit. - -As for bind, an `'a list` monadic object contains a lot of objects of -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 -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 elimiate 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. - -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 -considered monads. But we can't do that if we leave the list type as -a primitive Ocaml type. However, we know several ways of implementing -lists using just functions. In what follows, we're going to use type -3 lists, the right fold implementation (though it's important and -intriguing to wonder how things would change if we used some other -strategy for implementating lists). 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 -principle types of these functions (rather than inferring what the -types should be ourselves): - - # fun f z -> z;; - - : 'a -> 'b -> 'b =