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
----------------------------------------------------------
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)
+<pre>
+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)
+</pre>
How similar is it to the List monad? Let's examine the type
constructor and the terms from the list monad derived above:
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