X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week6.mdwn;h=ac94b79c36373bf7377477a83206231c8ec97f2c;hp=1839455a6542b7df4b9339222295d6f871b14358;hb=afc4f7a71546e785d23f59b46fd9d7f035748a60;hpb=80ad862c64373ac07b6e33236a47a50e98583d62

diff --git a/week6.mdwn b/week6.mdwn
index 1839455a..ac94b79c 100644
--- a/week6.mdwn
+++ b/week6.mdwn
@@ -44,9 +44,10 @@ Oh well.
 Booleans in OCAML, and simple pattern matching
 ----------------------------------------------
 
-Where we would write `true 1 2` and expect it to evaluate to `1`, in
-OCAML boolean types are not functions (equivalently, are functions
-that take zero arguments).  Choices are made as follows:
+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
+(equivalently, are functions that take zero arguments).  Selection is
+accomplished as follows:
 
     # if true then 1 else 2;;
     - : int = 1
@@ -69,8 +70,8 @@ Compare with
     # match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;;
     - : int = 9
 
-Unit
-----
+Unit and thunks
+---------------
 
 All functions in OCAML take exactly one argument.  Even this one:
 
@@ -135,7 +136,7 @@ Oh, one more thing: lambda expressions look like this:
     # (fun x -> x);;
     - : 'a -> 'a = <fun>
     # (fun x -> x) true;;
-    - : book = true
+    - : bool = true
 
 (But `(fun x -> x x)` still won't work.)
 
@@ -152,7 +153,7 @@ reverse the order of the arguments:
 
 Infinite loop.
 
-Now consider the following differences:
+Now consider the following variations in behavior:
 
     # let test = omega omega;;
     [Infinite loop, need to control c out]
@@ -170,3 +171,137 @@ We can use functions that take arguments of type unit to control
 execution.  In Scheme parlance, functions on the unit type are called
 *thunks* (which I've always assumed was a blend of "think" and "chunk").
 
+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:
+
+    # 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:
+
+    # type 'a option = None | Some of 'a;;
+    # None;;
+    - : 'a option = None
+    # Some 3;;
+    - : int option = Some 3
+
+So if a division is normal, we return some number, but if the divisor is
+zero, we return None:
+
+<pre>
+let div (x:int) (y:int) = 
+  match y with 0 -> None |
+               _ -> Some (x / y);;
+
+(*
+val div : int -> int -> int option = fun
+# div 12 3;;
+- : int option = Some 4
+# div 12 0;;
+- : int option = None
+# div (div 12 3) 2;;
+Characters 4-14:
+  div (div 12 3) 2;;
+      ^^^^^^^^^^
+Error: This expression has type int option
+       but an expression was expected of type int
+*)
+</pre>
+
+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
+operations.  So we have to jack up the types of the inputs:
+
+<pre>
+let div (x:int option) (y:int option) = 
+  match y with None -> None |
+               Some 0 -> None |
+               Some n -> (match x with None -> None |
+                                       Some m -> Some (m / n));;
+
+(*
+val div : int option -> int option -> int option = <fun>
+# div (Some 12) (Some 4);;
+- : int option = Some 3
+# div (Some 12) (Some 0);;
+- : int option = None
+# div (div (Some 12) (Some 0)) (Some 4);;
+- : int option = None
+*)
+</pre>
+
+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 
+built-in tuple type:
+
+<pre>
+let div (x:int option) (y:int option) = 
+  match (x, y) with (None, _) -> None |
+                    (_, None) -> None |
+                    (_, Some 0) -> None |
+                    (Some m, Some n) -> Some (m / n);;
+</pre>
+
+So far so good.  But what if we want to combine division with 
+other arithmetic operations?  We need to make those other operations 
+aware of the possibility that one of their arguments will trigger a
+presupposition failure:
+
+<pre>
+let add (x:int option) (y:int option) = 
+  match (x, y) with (None, _) -> None |
+                    (_, None) -> None |
+                    (Some m, Some n) -> Some (m + n);;
+
+(*
+val add : int option -> int option -> int option = <fun>
+# add (Some 12) (Some 4);;
+- : int option = Some 16
+# add (div (Some 12) (Some 0)) (Some 4);;
+- : int option = None
+*)
+</pre>
+
+This works, but is somewhat disappointing: the `add` prediction
+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,
+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;;
+
+let add (x: int option) (y: int option)  =
+  bind x (fun x -> bind y (fun y -> Some (x + y)));;
+
+let div (x: int option) (y: int option) =
+  bind x (fun x -> bind y (fun y -> if (0 = y) then None else Some (x / y)));;
+
+(*
+#  div (div (Some 12) (Some 2)) (Some 4);;
+- : int option = Some 1
+#  div (div (Some 12) (Some 0)) (Some 4);;
+- : int option = None
+# add (div (Some 12) (Some 0)) (Some 4);;
+- : int option = None
+*)
+</pre>
+
+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.
+