edits

[lambda.git] / zipper-lists-continuations.mdwn

diff --git a/zipper-lists-continuations.mdwn b/zipper-lists-continuations.mdwn
index 637f54d..0ef9436 100644 (file)
--- a/zipper-lists-continuations.mdwn
+++ b/zipper-lists-continuations.mdwn
@@ -153,7 +153,7 @@ types should be):
 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
 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).
+thrown in (in this case, an int).

 
 So here's our type constructor for our hand-rolled lists:
 
 
 So here's our type constructor for our hand-rolled lists:
 


@@ -167,7 +167,7 @@ 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
 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
+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
 in order to proceed to build a monad:
 
     l'_unit (x:'a):(('a, 'b) list) = fun x -> fun f z -> f x z


@@ -243,16 +243,9 @@ lists, so that they will print out.
 
 Ta da!
 
 
 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
 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
 ----------------------------------------------------------
 
 Montague's PTQ treatment of DPs as generalized quantifiers
 ----------------------------------------------------------


@@ -278,10 +271,12 @@ 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:
 
 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:
 
 How similar is it to the List monad?  Let's examine the type
 constructor and the terms from the list monad derived above:


@@ -299,16 +294,15 @@ 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
 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
 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
 -------------------------
 
 Refunctionalizing zippers
 -------------------------