X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week6.mdwn;h=25e52557481d221b2a73be6b45a1d26dfeb281af;hp=69881478360403e6f4ff7a2d8b4ca10517ff1656;hb=0d85c76d0d37b32bf99483b86828a7d2829db44e;hpb=e519121696a33c116b0942cb289e74d4d978b80c diff --git a/week6.mdwn b/week6.mdwn index 69881478..25e52557 100644 --- a/week6.mdwn +++ b/week6.mdwn @@ -1,9 +1,9 @@ [[!toc]] -Types, OCAML +Types, OCaml ------------ -OCAML has type inference: the system can often infer what the type of +OCaml has type inference: the system can often infer what the type of an expression must be, based on the type of other known expressions. For instance, if we type @@ -32,7 +32,7 @@ element: # (3) = 3;; - : bool = true -though OCAML, like many systems, refuses to try to prove whether two +though OCaml, like many systems, refuses to try to prove whether two functional objects may be identical: # (f) = f;; @@ -41,11 +41,11 @@ functional objects may be identical: Oh well. -Booleans in OCAML, and simple pattern matching +Booleans in OCaml, and simple pattern matching ---------------------------------------------- Where we would write `true 1 2` in our pure lambda calculus and expect -it to evaluate to `1`, in OCAML boolean types are not functions +it to evaluate to `1`, in OCaml boolean types are not functions (equivalently, are functions that take zero arguments). Selection is accomplished as follows: @@ -73,7 +73,7 @@ Compare with Unit and thunks --------------- -All functions in OCAML take exactly one argument. Even this one: +All functions in OCaml take exactly one argument. Even this one: # let f x y = x + y;; # f 2 3;; @@ -87,7 +87,7 @@ Here's how to tell that `f` has been curry'd: After we've given our `f` one argument, it returns a function that is still waiting for another argument. -There is a special type in OCAML called `unit`. There is exactly one +There is a special type in OCaml called `unit`. There is exactly one object in this type, written `()`. So # ();; @@ -145,7 +145,7 @@ So we can try our usual tricks: # (fun x -> true) omega;; - : bool = true -OCAML declined to try to evaluate the argument before applying the +OCaml declined to try to evaluate the argument before applying the functor. But remember that `omega` is a function too, so we can reverse the order of the arguments: @@ -176,14 +176,14 @@ Towards Monads So the integer division operation presupposes that its second argument (the divisor) is not zero, upon pain of presupposition failure. -Here's what my OCAML interpreter says: +Here's what my OCaml interpreter says: # 12/0;; Exception: Division_by_zero. So we want to explicitly allow for the possibility that division will return something other than a number. -We'll use OCAML's option type, which works like this: +We'll use OCaml's option type, which works like this: # type 'a option = None | Some of 'a;; # None;; @@ -200,7 +200,7 @@ let div (x:int) (y:int) = _ -> Some (x / y);; (* -val div : int -> int -> int option = +val div : int -> int -> int option = fun # div 12 3;; - : int option = Some 4 # div 12 0;; @@ -216,7 +216,7 @@ Error: This expression has type int option This starts off well: dividing 12 by 3, no problem; dividing 12 by 0, just the behavior we were hoping for. But we want to be able to use -the output of the safe division function as input for further division +the output of the safe-division function as input for further division operations. So we have to jack up the types of the inputs: 
@@ -240,7 +240,7 @@ val div : int option -> int option -> int option = 
 Beautiful, just what we need: now we can try to divide by anything we
 want, without fear that we're going to trigger any system errors.
 
-I prefer to line up the `match` alternatives by using OCAML's 
+I prefer to line up the `match` alternatives by using OCaml's 
 built-in tuple type:
 
 
@@ -271,17 +271,18 @@ val add : int option -> int option -> int option = 
 *)
 

 
-This works, but is somewhat disappointing: the `add` prediction
+This works, but is somewhat disappointing: the `add` operation
 doesn't trigger any presupposition of its own, so it is a shame that
 it needs to be adjusted because someone else might make trouble.
 
-But we can automate the adjustment.  The standard way in OCAML,
+But we can automate the adjustment.  The standard way in OCaml,
 Haskell, etc., is to define a `bind` operator (the name `bind` is not
 well chosen to resonate with linguists, but what can you do):
 
 
 let bind (x: int option) (f: int -> (int option)) = 
-  match x with None -> None | Some n -> f n;;
+  match x with None -> None | 
+               Some n -> f n;;
 
 let add (x: int option) (y: int option)  =
   bind x (fun x -> bind y (fun y -> Some (x + y)));;
@@ -301,7 +302,13 @@ let div (x: int option) (y: int option) =
 
 Compare the new definitions of `add` and `div` closely: the definition
 for `add` shows what it looks like to equip an ordinary operation to
-survive in a presupposition-filled world, and the definition of `div`
-shows exactly what extra needs to be added in order to trigger the
-no-division-by-zero presupposition.
+survive in dangerous presupposition-filled world.  Note that the new
+definition of `add` does not need to test whether its arguments are
+None objects or real numbers---those details are hidden inside of the
+`bind` function.
 
+The definition of `div` shows exactly what extra needs to be said in
+order to trigger the no-division-by-zero presupposition.
+
+For linguists: this is a complete theory of a particularly simply form
+of presupposition projection (every predicate is a hole).