expand name 'towards monads'

[lambda.git] / week6.mdwn

diff --git a/week6.mdwn b/week6.mdwn
index 25e5255..81e7a83 100644 (file)
--- a/week6.mdwn
+++ b/week6.mdwn
@@ -6,11 +6,11 @@ Types, OCaml
 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.
 
 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 
+For instance, if we type

 
     # let f x = x + 3;;
 
 
     # let f x = x + 3;;
 

-The system replies with 
+The system replies with

 
     val f : int -> int = <fun>
 
 
     val f : int -> int = <fun>
 


@@ -65,7 +65,7 @@ That is,
     # match true with true -> 1 | false -> 2;;
     - : int = 1
 
     # match true with true -> 1 | false -> 2;;
     - : int = 1
 

-Compare with 
+Compare with

 
     # match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;;
     - : int = 9
 
     # match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;;
     - : int = 9


@@ -112,7 +112,7 @@ correct type is the unit:
 
 Let's have some fn: think of `rec` as our `Y` combinator.  Then
 
 
 Let's have some fn: think of `rec` as our `Y` combinator.  Then
 

-    # let rec f n = if (0 = n) then 1 else (n * (f (n - 1)));; 
+    # let rec f n = if (0 = n) then 1 else (n * (f (n - 1)));;

     val f : int -> int = <fun>
     # f 5;;
     - : int = 120
     val f : int -> int = <fun>
     # f 5;;
     - : int = 120


@@ -171,8 +171,8 @@ 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").
 
 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
---------------
+Dividing by zero: 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.


@@ -181,7 +181,7 @@ Here's what my OCaml interpreter says:
     # 12/0;;
     Exception: Division_by_zero.
 
     # 12/0;;
     Exception: Division_by_zero.
 

-So we want to explicitly allow for the possibility that 
+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:
 
 division will return something other than a number.
 We'll use OCaml's option type, which works like this:
 


@@ -192,22 +192,22 @@ We'll use OCaml's option type, which works like this:
     - : int option = Some 3
 
 So if a division is normal, we return some number, but if the divisor is
     - : int option = Some 3
 
 So if a division is normal, we return some number, but if the divisor is

-zero, we return None:
+zero, we return None. As a mnemonic aid, we'll append a `'` to the end of our new divide function.

 
 <pre>
 
 <pre>

-let div (x:int) (y:int) = 
+let div' (x:int) (y:int) =

   match y with 0 -> None |
                _ -> Some (x / y);;
 
 (*
   match y with 0 -> None |
                _ -> Some (x / y);;
 
 (*

-val div : int -> int -> int option = fun
-# div 12 3;;
+val div' : int -> int -> int option = fun
+# div' 12 3;;

 - : int option = Some 4
 - : int option = Some 4

-# div 12 0;;
+# div' 12 0;;

 - : int option = None
 - : int option = None

-# div (div 12 3) 2;;
+# div' (div' 12 3) 2;;

 Characters 4-14:
 Characters 4-14:

-  div (div 12 3) 2;;
+  div' (div' 12 3) 2;;

       ^^^^^^^^^^
 Error: This expression has type int option
        but an expression was expected of type int
       ^^^^^^^^^^
 Error: This expression has type int option
        but an expression was expected of type int


@@ -220,19 +220,19 @@ 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>

-let div (x:int option) (y:int option) = 
+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));;
 
 (*
   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);;
+val div' : int option -> int option -> int option = <fun>
+# div' (Some 12) (Some 4);;

 - : int option = Some 3
 - : int option = Some 3

-# div (Some 12) (Some 0);;
+# div' (Some 12) (Some 0);;

 - : int option = None
 - : int option = None

-# div (div (Some 12) (Some 0)) (Some 4);;
+# div' (div' (Some 12) (Some 0)) (Some 4);;

 - : int option = None
 *)
 </pre>
 - : int option = None
 *)
 </pre>


@@ -240,74 +240,74 @@ 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>

-let div (x:int option) (y:int option) = 
+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>
 
   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 
+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>
 aware of the possibility that one of their arguments will trigger a
 presupposition failure:
 
 <pre>

-let add (x:int option) (y:int option) = 
+let add' (x:int option) (y:int option) =

   match (x, y) with (None, _) -> None |
                     (_, None) -> None |
                     (Some m, Some n) -> Some (m + n);;
 
 (*
   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);;
+val add' : int option -> int option -> int option = <fun>
+# add' (Some 12) (Some 4);;

 - : int option = Some 16
 - : int option = Some 16

-# add (div (Some 12) (Some 0)) (Some 4);;
+# add' (div' (Some 12) (Some 0)) (Some 4);;

 - : int option = None
 *)
 </pre>
 
 - : int option = None
 *)
 </pre>
 

-This works, but is somewhat disappointing: the `add` operation
+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,
 Haskell, etc., is to define a `bind` operator (the name `bind` is not
 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):
+well chosen to resonate with linguists, but what can you do). To continue our mnemonic association, we'll put a `'` after the name "bind" as well.

 
 <pre>
 
 <pre>

-let bind (x: int option) (f: int -> (int option)) = 
-  match x with None -> None | 
+let bind' (x: int option) (f: int -> (int option)) =
+  match x with None -> None |

                Some n -> f n;;
 
                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)));;

 
 

-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)));;
+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);;
+#  div' (div' (Some 12) (Some 2)) (Some 4);;

 - : int option = Some 1
 - : int option = Some 1

-#  div (div (Some 12) (Some 0)) (Some 4);;
+#  div' (div' (Some 12) (Some 0)) (Some 4);;

 - : int option = None
 - : int option = None

-# add (div (Some 12) (Some 0)) (Some 4);;
+# add' (div' (Some 12) (Some 0)) (Some 4);;

 - : int option = None
 *)
 </pre>
 
 - : 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
+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 dangerous presupposition-filled world.  Note that the new
 survive in dangerous presupposition-filled world.  Note that the new

-definition of `add` does not need to test whether its arguments are
+definition of `add'` does not need to test whether its arguments are

 None objects or real numbers---those details are hidden inside of the
 None objects or real numbers---those details are hidden inside of the

-`bind` function.
+`bind'` function.

 
 

-The definition of `div` shows exactly what extra needs to be said in
+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
 order to trigger the no-division-by-zero presupposition.
 
 For linguists: this is a complete theory of a particularly simply form