ass5: omega->blackhole

[lambda.git] / week6.mdwn

diff --git a/week6.mdwn b/week6.mdwn
index 6988147..25e5255 100644 (file)
--- a/week6.mdwn
+++ b/week6.mdwn
@@ -1,9 +1,9 @@
 [[!toc]]
 
 [[!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 
 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
 
     # (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;;
 functional objects may be identical:
 
     # (f) = f;;


@@ -41,11 +41,11 @@ functional objects may be identical:
 Oh well.
 
 
 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
 ----------------------------------------------
 
 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:
 
 (equivalently, are functions that take zero arguments).  Selection is
 accomplished as follows:
 


@@ -73,7 +73,7 @@ Compare with
 Unit and thunks
 ---------------
 
 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;;
 
     # 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.
 
 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
 
     # ();;
 object in this type, written `()`.  So
 
     # ();;


@@ -145,7 +145,7 @@ So we can try our usual tricks:
     # (fun x -> true) omega;;
     - : bool = true
 
     # (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:
 
 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.
 
 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.
 
     # 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;;
 
     # type 'a option = None | Some of 'a;;
     # None;;


@@ -200,7 +200,7 @@ let div (x:int) (y:int) =
                _ -> Some (x / y);;
 
 (*
                _ -> Some (x / y);;
 
 (*

-val div : int -> int -> int option = <fun>
+val div : int -> int -> int option = fun

 # div 12 3;;
 - : int option = Some 4
 # div 12 0;;
 # 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
 
 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:
 
 <pre>
 operations.  So we have to jack up the types of the inputs:
 
 <pre>


@@ -240,7 +240,7 @@ val div : int option -> int option -> int option = <fun>
 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.
 
 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:
 
 <pre>
 built-in tuple type:
 
 <pre>


@@ -271,17 +271,18 @@ val add : int option -> int option -> int option = <fun>
 *)
 </pre>
 
 *)
 </pre>
 

-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.
 
 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):
 
 <pre>
 let bind (x: int option) (f: int -> (int option)) = 
 Haskell, etc., is to define a `bind` operator (the name `bind` is not
 well chosen to resonate with linguists, but what can you do):
 
 <pre>
 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)));;
 
 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
 
 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).