```-# fun f z -> z;;
-- : 'a -> 'b -> 'b =
-# 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 =
-```
- -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). + # fun f z -> z;; + - : 'a -> 'b -> 'b = + # 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 = + +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 @@ -161,63 +163,104 @@ ints), we have 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 +general than an ordinary OCaml list, but we'll see how to map them +into OCaml lists soon. We don't need to fully 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 + l'_unit (a : 'a) : ('a, 'b) list = fun a -> fun f z -> f a 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' = ... + 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) + 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 (x:'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 (x:'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: -[Excercise: can you arrive at a fully general bind for this type -constructor, one that does not collapse `'d`'s with `'b`'s?] + ... u (fun (a : 'a) (b : 'b) -> f a k b) ... -As usual, we have to abstract over `k`, but this time, no further -adjustments are needed: +Now, we've used a `k` that we pulled out of nowhere, so we need to abstract over it: - l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b) - (f: 'a -> ('c -> 'b -> 'b) -> 'b -> 'b) + 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 (x:'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? + +Suppose we have a list' whose contents are `[1; 2; 4; 8]`---that is, our list' will be `fun f z -> f 1 (f 2 (f 4 (f 8 z)))`. We call that list' `u`. Suppose we also have a function `f` that for each `int` we give it, gives back a list of the divisors of that `int` that are greater than 1. Intuitively, then, binding `u` to `f` should give us: + + concat (map f u) = + concat [[]; [2]; [2; 4]; [2; 4; 8]] = + [2; 2; 4; 2; 4; 8] + +Or rather, it should give us a list' version of that, which takes a function `k` and value `z` as arguments, and returns the right fold of `k` and `z` over those elements. What does our formula + + fun k z -> u (fun a b -> f a k b) z + +do? Well, for each element `a` in `u`, it applies `f` to that `a`, getting one of the lists: -You should carefully check to make sure that this term is consistent -with the typing. + [] + [2] + [2; 4] + [2; 4; 8] -Our theory is that this monad should be capable of exactly -replicating the behavior of the standard List monad. Let's test: +(or rather, their list' versions). Then it takes the accumulated result `b` of previous steps in the fold, and it folds `k` and `b` over the list generated by `f a`. The result of doing so is passed on to the next step as the accumulated result so far. + +So if, for example, we let `k` be `+` and `z` be `0`, then the computation would proceed: + + 0 ==> + right-fold + and 0 over [2; 4; 8] = 2+4+8+0 ==> + right-fold + and 2+4+8+0 over [2; 4] = 2+4+2+4+8+0 ==> + right-fold + and 2+4+2+4+8+0 over [2] = 2+2+4+2+4+8+0 ==> + right-fold + and 2+2+4+2+4+8+0 over [] = 2+2+4+2+4+8+0 + +which indeed is the result of right-folding + and 0 over `[2; 2; 4; 2; 4; 8]`. If you trace through how this works, you should be able to persuade yourself that our formula: + + fun k z -> u (fun a b -> f a k b) z + +will deliver just the same folds, for arbitrary choices of `k` and `z` (with the right types), and arbitrary list's `u` and appropriately-typed `f`s, as + + fun k z -> List.fold_right k (concat (map f u)) z + +would. + +For future reference, we might make two eta-reductions to our formula, so that we have instead: + + let l'_bind = fun k -> u (fun a -> f a k);; + +Let's make some more tests: l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3] @@ -225,29 +268,20 @@ replicating the behavior of the standard List monad. Let's test: l'_bind (fun f z -> f 1 (f 2 z)) (fun i -> fun f z -> f i (f (i+1) z)) ~~> -Sigh. Ocaml won't show us our own list. So we have to choose an `f` -and a `z` that will turn our hand-crafted lists into standard Ocaml +Sigh. OCaml won't show us our own list. So we have to choose an `f` +and a `z` that will turn our hand-crafted lists into standard OCaml lists, so that they will print out. -
```-# 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]
-```
+ # 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. +continuations... Montague's PTQ treatment of DPs as generalized quantifiers ---------------------------------------------------------- @@ -265,45 +299,44 @@ 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 + gqize (a : e) = fun (p : e -> t) -> p a 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 +the bind follow naturally. We've done this enough times that we 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) + 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) 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) + l'_unit a = fun f -> f a + l'_bind u f = fun k -> u (fun a -> f a k) -(I performed a sneaky but valid eta reduction in the unit term.) +(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. To emphasize the parallel, we can -instantiate the type of the list' monad using the Ocaml list type: +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 +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. +versa (not to mention our intuitions about primitive recursion in +Church numerals too). 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 -------------------------