X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week6.mdwn;h=2a4586a01c9b36c546e3aeacbb599a14c3e581d8;hp=25e52557481d221b2a73be6b45a1d26dfeb281af;hb=HEAD;hpb=472e280120834cc613fe8b7ff045a99e956636d3 diff --git a/week6.mdwn b/week6.mdwn deleted file mode 100644 index 25e52557..00000000 --- a/week6.mdwn +++ /dev/null @@ -1,314 +0,0 @@ -[[!toc]] - -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. - -For instance, if we type - - # let f x = x + 3;; - -The system replies with - - val f : int -> int = - -Since `+` is only defined on integers, it has type - - # (+);; - - : int -> int -> int = - -The parentheses are there to turn off the trick that allows the two -arguments of `+` to surround it in infix (for linguists, SOV) argument -order. That is, - - # 3 + 4 = (+) 3 4;; - - : bool = true - -In general, tuples with one element are identical to their one -element: - - # (3) = 3;; - - : bool = true - -though OCaml, like many systems, refuses to try to prove whether two -functional objects may be identical: - - # (f) = f;; - Exception: Invalid_argument "equal: functional value". - -Oh well. - - -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 -(equivalently, are functions that take zero arguments). Selection is -accomplished as follows: - - # if true then 1 else 2;; - - : int = 1 - -The types of the `then` clause and of the `else` clause must be the -same. - -The `if` construction can be re-expressed by means of the following -pattern-matching expression: - - match with true -> | false -> - -That is, - - # match true with true -> 1 | false -> 2;; - - : int = 1 - -Compare with - - # match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;; - - : int = 9 - -Unit and thunks ---------------- - -All functions in OCaml take exactly one argument. Even this one: - - # let f x y = x + y;; - # f 2 3;; - - : int = 5 - -Here's how to tell that `f` has been curry'd: - - # f 2;; - - : int -> int = - -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 -object in this type, written `()`. So - - # ();; - - : unit = () - -Just as you can define functions that take constants for arguments - - # let f 2 = 3;; - # f 2;; - - : int = 3;; - -you can also define functions that take the unit as its argument, thus - - # let f () = 3;; - val f : unit -> int = - -Then the only argument you can possibly apply `f` to that is of the -correct type is the unit: - - # f ();; - - : int = 3 - -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)));; - val f : int -> int = - # f 5;; - - : int = 120 - -We can't define a function that is exactly analogous to our ω. -We could try `let rec omega x = x x;;` what happens? However, we can -do this: - - # let rec omega x = omega x;; - -By the way, what's the type of this function? -If you then apply this omega to an argument, - - # omega 3;; - -the interpreter goes into an infinite loop, and you have to control-C -to break the loop. - -Oh, one more thing: lambda expressions look like this: - - # (fun x -> x);; - - : 'a -> 'a = - # (fun x -> x) true;; - - : bool = true - -(But `(fun x -> x x)` still won't work.) - -So we can try our usual tricks: - - # (fun x -> true) omega;; - - : bool = true - -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: - - # omega (fun x -> true);; - -Infinite loop. - -Now consider the following variations in behavior: - - # let test = omega omega;; - [Infinite loop, need to control c out] - - # let test () = omega omega;; - val test : unit -> 'a = - - # test;; - - : unit -> 'a = - - # test ();; - [Infinite loop, need to control c out] - -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: - -
-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
-*)
-
- -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: - -
-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);;
-- : int option = Some 3
-# div (Some 12) (Some 0);;
-- : int option = None
-# div (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-
- -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: - -
-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 -aware of the possibility that one of their arguments will trigger a -presupposition failure: - -
-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);;
-- : int option = Some 16
-# add (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-
- -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 -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;;
-
-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
-*)
-
- -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).