... 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. @@ -1115,7 +1125,7 @@ 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 @@ -1130,13 +1140,13 @@ 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 @@ -1220,7 +1230,7 @@ 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) @@ -1329,7 +1339,7 @@ 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 @@ -1360,7 +1370,7 @@ to monads that can be understood in terms of continuations? Manipulating trees with monads ------------------------------ -This thread develops an idea based on a detailed suggestion of Ken +This topic develops an idea based on a detailed suggestion of Ken Shan's. We'll build a series of functions that operate on trees, doing various things, including replacing leaves, counting nodes, and converting a tree to a list of leaves. The end result will be an @@ -1368,11 +1378,11 @@ application for continuations. From an engineering standpoint, we'll build a tree transformer that deals in monads. We can modify the behavior of the system by swapping -one monad for another. (We've already seen how adding a monad can add +one monad for another. We've already seen how adding a monad can add a layer of funtionality without disturbing the underlying system, for instance, in the way that the reader monad allowed us to add a layer of intensionality to an extensional grammar, but we have not yet seen -the utility of replacing one monad with other.) +the utility of replacing one monad with other. First, we'll be needing a lot of trees during the remainder of the course. Here's a type constructor for binary trees: