X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week6.mdwn;h=16149724e49ddc67d3a0f278ab2b6acb0a26e8da;hp=1839455a6542b7df4b9339222295d6f871b14358;hb=7ead05816d92955744e53ea78d54efc3c08176dd;hpb=80ad862c64373ac07b6e33236a47a50e98583d62 diff --git a/week6.mdwn b/week6.mdwn index 1839455a..16149724 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;; @@ -40,13 +40,28 @@ functional objects may be identical: Oh well. +[Note: There is a limited way you can compare functions, using the +`==` 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 +`(f) = f`. However, don't expect it to figure out in general when two functions +are identical. (That question is not Turing computable.) -Booleans in OCAML, and simple pattern matching + # (f) == (fun x -> x + 3);; + - : bool = false + +Here OCaml says (correctly) that the two functions don't stand in the `==` relation, which basically means they're not represented in the same chunk of memory. However as the programmer can see, the functions are extensionally equivalent. The meaning of `==` is rather weird.] + + + +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, they're functions that take zero arguments). Instead, selection is +accomplished as follows: # if true then 1 else 2;; - : int = 1 @@ -64,15 +79,15 @@ 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 -Unit ----- +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;; @@ -86,7 +101,7 @@ 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 # ();; @@ -109,25 +124,45 @@ correct type is the unit: # f ();; - : int = 3 -Let's have some fn: think of `rec` as our `Y` combinator. Then +Now why would that be useful? + +Let's have some fun: 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 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: +We could try `let rec omega x = x x;;` what happens? + +[Note: if you want to learn more OCaml, you might come back here someday and try: - # let rec omega x = omega x;; + # let id x = x;; + val id : 'a -> 'a = + # let unwrap (`Wrap a) = a;; + val unwrap : [< `Wrap of 'a ] -> 'a = + # let omega ((`Wrap x) as y) = x y;; + val omega : [< `Wrap of [> `Wrap of 'a ] -> 'b as 'a ] -> 'b = + # unwrap (omega (`Wrap id)) == id;; + - : bool = true + # unwrap (omega (`Wrap omega));; + + +But we won't try to explain this now.] + + +Even if we can't (easily) express omega in OCaml, we can do this: + + # let rec blackhole x = blackhole x;; By the way, what's the type of this function? -If you then apply this omega to an argument, - # omega 3;; +If you then apply this `blackhole` function to an argument, + + # blackhole 3;; -the interpreter goes into an infinite loop, and you have to control-C +the interpreter goes into an infinite loop, and you have to type control-c to break the loop. Oh, one more thing: lambda expressions look like this: @@ -135,38 +170,240 @@ Oh, one more thing: lambda expressions look like this: # (fun x -> x);; - : 'a -> 'a = # (fun x -> x) true;; - - : book = true + - : bool = true (But `(fun x -> x x)` still won't work.) -So we can try our usual tricks: +You may also see this: + + # (function x -> x);; + - : 'a -> 'a = + +This works the same as `fun` in simple cases like this, and slightly differently in more complex cases. If you learn more OCaml, you'll read about the difference between them. - # (fun x -> true) omega;; +We can try our usual tricks: + + # (fun x -> true) blackhole;; - : 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 +OCaml declined to try to fully reduce the argument before applying the +lambda function. Question: Why is that? Didn't we say that OCaml is a call-by-value/eager language? + +Remember that `blackhole` is a function too, so we can reverse the order of the arguments: - # omega (fun x -> true);; + # blackhole (fun x -> true);; 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] + # let test = blackhole blackhole;; + - # let test () = omega omega;; + # let test () = blackhole blackhole;; 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"). +Question: why do thunks work? We know that `blackhole ()` doesn't terminate, so why do expressions like: + + let f = fun () -> blackhole () + in true + +terminate? + +Bottom type, divergence +----------------------- + +Expressions that don't terminate all belong to the **bottom type**. This is a subtype of every other type. That is, anything of bottom type belongs to every other type as well. More advanced type systems have more examples of subtyping: for example, they might make `int` a subtype of `real`. But the core type system of OCaml doesn't have any general subtyping relations. (Neither does System F.) Just this one: that expressions of the bottom type also belong to every other type. It's as if every type definition in OCaml, even the built in ones, had an implicit extra clause: + + type 'a option = None | Some of 'a;; + type 'a option = None | Some of 'a | bottom;; + +Here are some exercises that may help better understand this. Figure out what is the type of each of the following: + + fun x y -> y;; + + fun x (y:int) -> y;; + + fun x y : int -> y;; + + let rec blackhole x = blackhole x in blackhole;; + + let rec blackhole x = blackhole x in blackhole 1;; + + let rec blackhole x = blackhole x in fun (y:int) -> blackhole y y y;; + + let rec blackhole x = blackhole x in (blackhole 1) + 2;; + + let rec blackhole x = blackhole x in (blackhole 1) || false;; + + let rec blackhole x = blackhole x in 2 :: (blackhole 1);; + + let rec blackhole (x:'a) : 'a = blackhole x in blackhole + + +Back to thunks: the reason you'd want to control evaluation with thunks is to +manipulate when "effects" happen. In a strongly normalizing system, like the +simply-typed lambda calculus or System F, there are no "effects." In Scheme and +OCaml, on the other hand, we can write programs that have effects. One sort of +effect is printing (think of the [[damn]] example at the start of term). +Another sort of effect is mutation, which we'll be looking at soon. +Continuations are yet another sort of effect. None of these are yet on the +table though. The only sort of effect we've got so far is *divergence* or +non-termination. So the only thing thunks are useful for yet is controlling +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. + + + +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).