X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week6.mdwn;h=7a5ae2c2af56aa9e84d0c19dc144f2a2962a77a4;hp=5898788187b1c1cf4fb5e53631f024aaee09103e;hb=966f3179a2866846d6b0e347b32ebe56da8cdd5e;hpb=177c81322f3487ac795fbcb58e8fb765bbc1f86d diff --git a/week6.mdwn b/week6.mdwn index 58987881..7a5ae2c2 100644 --- a/week6.mdwn +++ b/week6.mdwn @@ -1,16 +1,16 @@ [[!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 +For instance, if we type # let f x = x + 3;; -The system replies with +The system replies with val f : int -> int = @@ -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: @@ -65,7 +65,7 @@ That is, # match true with true -> 1 | false -> 2;; - : int = 1 -Compare with +Compare with # match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;; - : int = 9 @@ -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 # ();; @@ -112,7 +112,7 @@ correct type is the unit: 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 = # f 5;; - : int = 120 @@ -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 +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;; @@ -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 -zero, we return None: +zero, we return None. As a mnemonic aid, we'll append a `'` to the end of our new divide function.
```-let div (x:int) (y:int) =
+let div' (x:int) (y:int) =
match y with 0 -> None |
_ -> Some (x / y);;

(*
-val div : int -> int -> int option = \
-# div 12 3;;
+val div' : int -> int -> int option = fun
+# div' 12 3;;
- : int option = Some 4
-# div 12 0;;
+# div' 12 0;;
- : int option = None
-# div (div 12 3) 2;;
+# div' (div' 12 3) 2;;
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
@@ -216,23 +216,23 @@ 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:

-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));;

(*
-val div : int option -> int option -> int option =
-# div (Some 12) (Some 4);;
+val div' : int option -> int option -> int option =
+# div' (Some 12) (Some 4);;
- : int option = Some 3
-# div (Some 12) (Some 0);;
+# div' (Some 12) (Some 0);;
- : int option = None
-# div (div (Some 12) (Some 0)) (Some 4);;
+# div' (div' (Some 12) (Some 0)) (Some 4);;
- : int option = None
*)

@@ -240,68 +240,75 @@ 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:

-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);;

-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:

-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);;

(*
-val add : int option -> int option -> int option =
-# add (Some 12) (Some 4);;
+val add' : int option -> int option -> int option =
+# add' (Some 12) (Some 4);;
- : int option = Some 16
-# add (div (Some 12) (Some 0)) (Some 4);;
+# add' (div' (Some 12) (Some 0)) (Some 4);;
- : int option = None
*)

-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):
+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.

-let bind (x: int option) (f: int -> (int option)) =
-  match x with None -> None | Some n -> f n;;
+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 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
-#  div (div (Some 12) (Some 0)) (Some 4);;
+#  div' (div' (Some 12) (Some 0)) (Some 4);;
- : int option = None
-# add (div (Some 12) (Some 0)) (Some 4);;
+# add' (div' (Some 12) (Some 0)) (Some 4);;
- : int option = None
*)

-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.
+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
+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).
```