X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week6.mdwn;h=2a4586a01c9b36c546e3aeacbb599a14c3e581d8;hp=16149724e49ddc67d3a0f278ab2b6acb0a26e8da;hb=67ed83b1ad44c7590cf7d7c1ec3a079bc5140a61;hpb=7ead05816d92955744e53ea78d54efc3c08176dd diff --git a/week6.mdwn b/week6.mdwn index 16149724..2a4586a0 100644 --- a/week6.mdwn +++ b/week6.mdwn @@ -1,7 +1,36 @@ [[!toc]] -Types, OCaml ------------- +Polymorphic Types and System F +------------------------------ + +[Notes still to be added. Hope you paid attention during seminar.] + + + + +Types in 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. @@ -44,9 +73,9 @@ Oh well. `==` operator instead of the `=` operator. Later when we discuss mutation, we'll discuss the difference between these two equality operations. Scheme has a similar pair, which they name `eq?` and `equal?`. In Python, -these are `is` and `==` respectively. It's unfortunate that OCaml uses `==` for the opposite operation that Python and many other languages use it for. In any case, OCaml will understand `(f) == f` even though it doesn't understand +these are `is` and `==` respectively. It's unfortunate that OCaml uses `==` for the opposite operation that Python and many other languages use it for. In any case, OCaml will accept `(f) == f` even though it doesn't accept `(f) = f`. However, don't expect it to figure out in general when two functions -are identical. (That question is not Turing computable.) +are equivalent. (That question is not Turing computable.) # (f) == (fun x -> x + 3);; - : bool = false @@ -210,8 +239,8 @@ Now consider the following variations in behavior: # test ();; -We can use functions that take arguments of type unit to control -execution. In Scheme parlance, functions on the unit type are called +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"). Question: why do thunks work? We know that `blackhole ()` doesn't terminate, so why do expressions like: @@ -249,6 +278,8 @@ Here are some exercises that may help better understand this. Figure out what is let rec blackhole x = blackhole x in 2 :: (blackhole 1);; +By the way, what's the type of this: + let rec blackhole (x:'a) : 'a = blackhole x in blackhole @@ -265,145 +296,8 @@ whether an expression that would diverge if we tried to fully evaluate it does diverge. As we consider richer languages, thunks will become more useful. +Towards Monads +-------------- + +This has now been moved to the start of [[week7]]. -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. -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. As a mnemonic aid, we'll append a `'` to the end of our new divide function. - -
-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). 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 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).