... 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. @@ -121,7 +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 @@ -136,13 +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 @@ -226,7 +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) @@ -301,3 +310,66 @@ lists, so that they will print out. Ta da! + +Montague's PTQ treatment of DPs as generalized quantifiers +---------------------------------------------------------- + +We've hinted that Montague's treatment of DPs as generalized +quantifiers embodies the spirit of continuations (see de Groote 2001, +Barker 2002 for lengthy discussion). Let's see why. + +First, we'll need a type constructor. As you probably know, +Montague replaced individual-denoting determiner phrases (with type `e`) +with generalized quantifiers (with [extensional] type `(e -> t) -> t`. +In particular, the denotation of a proper name like *John*, which +might originally denote a object `j` of type `e`, came to denote a +generalized quantifier `fun pred -> pred j` of type `(e -> t) -> t`. +Let's write a general function that will map individuals into their +corresponding generalized quantifier: + + 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 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 +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 = + 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. + +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 + l'_unit a = fun f -> f a + l'_bind u f = fun k -> u (fun a -> f a k) + +(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. + +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, +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? + +