# fun f z -> z;; - : 'a -> 'b -> 'b =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, and int). So here's our type constructor for our hand-rolled lists: type 'a list' = (int -> 'a -> 'a) -> 'a -> 'a Generalizing to lists that contain any kind of element (not just ints), we have type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b So an `('a, 'b) list'` is a list containing elements of type `'a`, where `'b` is the type of some part of the plumbing. This is more general than an ordinary Ocaml list, but we'll see how to map them into Ocaml lists soon. We don't need to grasp the role of the `'b`'s in order to proceed to build a monad: l'_unit (x:'a):(('a, 'b) list) = fun x -> fun f z -> f x z No problem. Arriving at bind is a little more complicated, but exactly the same principles apply, you just have to be careful and systematic about it. l'_bind (u:('a,'b) list') (f:'a -> ('c, 'd) list'): ('c, 'd) list' = ... Unfortunately, we'll need to spell out the types: 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 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: .... u (fun (x:'a) -> ... (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: l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b) (f: 'a -> ('c -> 'b -> 'b) -> 'b -> 'b) : ('c -> 'b -> 'b) -> 'b -> 'b = .... u (fun (x:'a) -> f a k) ... [Excercise: can you arrive at a fully general bind for this type constructor, one that does not collapse `'d`'s with `'b`'s?] As usual, we have to abstract over `k`, but this time, no further adjustments are needed: 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 (x:'a) -> f a k) 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: l_bind [1;2] (fun i -> [i, i+1]) ~~> [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 f z -> f 1 z;; - : (int -> 'a -> 'b) -> 'a -> 'b = # fun f z -> f 2 (f 1 z);; - : (int -> 'a -> 'a) -> 'a -> 'a = # fun f z -> f 3 (f 2 (f 1 z)) - : (int -> 'a -> 'a) -> 'a -> 'a =

# 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 [];; - : int list = [1; 2; 2; 3]Ta da! Just for mnemonic purposes (sneaking in an instance of eta reduction to the definition of unit), we can summarize the result as follows: type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b l'_unit x = fun f -> f x l'_bind u f = fun k -> u (fun x -> f x k) To bad this digression, though it ties together various elements of the course, has *no relevance whatsoever* to the topic of continuations. 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 (x:e) = fun (p:e->t) -> p x This function wraps up an individual in a fancy 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 I won't belabor the construction of the bind function, the derivation is similar to the List monad just given: type 'a continuation = ('a -> 'b) -> 'b c_unit (x:'a) = fun (p:'a -> 'b) -> p x c_bind (u:('a -> 'b) -> 'b) (f: 'a -> ('c -> 'd) -> 'd): ('c -> 'd) -> 'd = fun (k:'a -> 'b) -> u (fun (x:'a) -> f x k) 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 x = fun f -> f x l'_bind u f = fun k -> u (fun x -> f x k) (I 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. To emphasize the parallel, we can instantiate the type of the list' monad using the Ocaml list type: type 'a c_list = ('a -> 'a list) -> 'a list let c_list_unit x = fun f -> f x;; let c_list_bind u f = fun k -> u (fun x -> f x k);; Have we really discovered that lists are secretly continuations? Or have we merely found a way of simulating lists using list continuations? Both perspectives are valid, and we can use our intuitions about the list monad to understand continuations, and vice versa. The connections will be expecially relevant when we consider indefinites and Hamblin semantics on the linguistic side, and non-determinism on the list monad side. Refunctionalizing zippers -------------------------