[[!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 = <fun>
# (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;;
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` 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
+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;;
# 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 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;;
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
# ();;
# f ();;
- : int = 3
-Let's have some fn: think of `rec` as our `Y` combinator. Then
+Now why would that be useful?
- # let rec f n = if (0 = n) then 1 else (n * (f (n - 1)));;
+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)));;
val f : int -> int = <fun>
# 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 id x = x;;
+ val id : 'a -> 'a = <fun>
+ # let unwrap (`Wrap a) = a;;
+ val unwrap : [< `Wrap of 'a ] -> 'a = <fun>
+ # let omega ((`Wrap x) as y) = x y;;
+ val omega : [< `Wrap of [> `Wrap of 'a ] -> 'b as 'a ] -> 'b = <fun>
+ # unwrap (omega (`Wrap id)) == id;;
+ - : bool = true
+ # unwrap (omega (`Wrap omega));;
+ <Infinite loop, need to control-c to interrupt>
+
+But we won't try to explain this now.]
- # let rec omega x = omega x;;
+
+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,
-the interpreter goes into an infinite loop, and you have to control-C
+ # blackhole 3;;
+
+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:
(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 = <fun>
- # (fun x -> true) omega;;
+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.
+
+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 variations in behavior:
- # let test = omega omega;;
- [Infinite loop, need to control c out]
+ # let test = blackhole blackhole;;
+ <Infinite loop, need to control-c to interrupt>
- # let test () = omega omega;;
+ # let test () = blackhole blackhole;;
val test : unit -> 'a = <fun>
# test;;
- : unit -> 'a = <fun>
# test ();;
- [Infinite loop, need to control c out]
+ <Infinite loop, need to control-c to interrupt>
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").
-Curry-Howard, take 1
---------------------
+Question: why do thunks work? We know that `blackhole ()` doesn't terminate, so why do expressions like:
-We will returnto the Curry-Howard correspondence a number of times
-during this course. It expresses a deep connection between logic,
-types, and computation. Today we'll discuss how the simply-typed
-lambda calculus corresponds to intuitionistic logic. This naturally
-give rise to the question of what sort of computation classical logic
-corresponds to---as we'll see later, the answer involves continuations.
+ let f = fun () -> blackhole ()
+ in true
-So at this point we have the simply-typed lambda calculus: a set of
-ground types, a set of functional types, and some typing rules, given
-roughly as follows:
+terminate?
-If a variable `x` has type σ and term `M` has type τ, then
-the abstract `\xM` has type `σ --> τ`.
+Bottom type, divergence
+-----------------------
-If a term `M` has type `σ --> &tau`, and a term `N` has type
-σ, then the application `MN` has type τ.
+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:
-These rules are clearly obverses of one another: the functional types
-that abstract builds up are taken apart by application.
+ type 'a option = None | Some of 'a;;
+ type 'a option = None | Some of 'a | bottom;;
-The next step in making sense out of the Curry-Howard corresponence is
-to present a logic. It will be a part of intuitionistic logic. We'll
-start with the implicational fragment (that is, the part of
-intuitionistic logic that only involves axioms and implications):
+Here are some exercises that may help better understand this. Figure out what is the type of each of the following:
-<pre>
-Axiom: ---------
- A |- A
+ fun x y -> y;;
-Structural Rules:
+ fun x (y:int) -> y;;
-Exchange: Γ, A, B, Δ |- C
- ---------------------------
- $Gamma;, B, A, Δ |- C
+ fun x y : int -> y;;
-Contraction: Γ, A, A |- B
- -------------------
- Γ, A |- B
+ let rec blackhole x = blackhole x in blackhole;;
-Weakening: Γ |- B
- -----------------
- Γ, A |- B
+ let rec blackhole x = blackhole x in blackhole 1;;
-Logical Rules:
+ let rec blackhole x = blackhole x in fun (y:int) -> blackhole y y y;;
---> I: Γ, A |- B
- -------------------
- Γ |- A --> B
+ let rec blackhole x = blackhole x in (blackhole 1) + 2;;
---> E: Γ |- A --> B Γ |- A
- -----------------------------------------
- Γ |- B
-</pre>
+ let rec blackhole x = blackhole x in (blackhole 1) || false;;
-`A`, `B`, etc. are variables over formulas.
-Γ, Δ, etc. are variables over (possibly empty) sequences
-of formulas. `Γ |- A` is a sequent, and is interpreted as
-claiming that if each of the formulas in Γ is true, then `A`
-must also be true.
+ let rec blackhole x = blackhole x in 2 :: (blackhole 1);;
-This logic allows derivations of theorems like the following:
+ let rec blackhole (x:'a) : 'a = blackhole x in blackhole
-<pre>
-------- Id
-A |- A
----------- Weak
-A, B |- A
-------------- --> I
-A |- B --> A
------------------ --> I
-|- A --> B --> A
-</pre>
-Should remind you of simple types. (What was `A --> B --> A` the type
-of again?)
+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.
-The easy way to grasp the Curry-Howard correspondence is to *label*
-the proofs. Since we wish to establish a correspondence between this
-logic and the lambda calculus, the labels will all be terms from the
-simply-typed lambda calculus. Here are the labeling rules:
+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.
<pre>
-Axiom: -----------
- x:A |- x:A
+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
+*)
+</pre>
-Structural Rules:
+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:
-Exchange: Γ, x:A, y:B, Δ |- R:C
- --------------------------------------
- Γ, y:B, x:A, Δ |- R:C
+<pre>
+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 = <fun>
+# 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
+*)
+</pre>
-Contraction: Γ, x:A, x:A |- R:B
- --------------------------
- Γ, x:A |- R:B
+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.
-Weakening: Γ |- R:B
- ---------------------
- Γ, x:A |- R:B [x chosen fresh]
+I prefer to line up the `match` alternatives by using OCaml's
+built-in tuple type:
-Logical Rules:
+<pre>
+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>
---> I: Γ, x:A |- R:B
- -------------------------
- Γ |- \xM:A --> B
+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:
---> E: Γ |- f:(A --> B) Γ |- x:A
- ---------------------------------------------
- Γ |- (fx):B
+<pre>
+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 = <fun>
+# add' (Some 12) (Some 4);;
+- : int option = Some 16
+# add' (div' (Some 12) (Some 0)) (Some 4);;
+- : int option = None
+*)
</pre>
-In these labeling rules, if a sequence Γ in a premise contains
-labeled formulas, those labels remain unchanged in the conclusion.
+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.
-Using these labeling rules, we can label the proof
-just given:
+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.
<pre>
------------- Id
-x:A |- x:A
----------------- Weak
-x:A, y:B |- x:A
-------------------------- --> I
-x:A |- (\y.x):(B --> A)
----------------------------- --> I
-|- (\x y. x):A --> B --> A
+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
+*)
</pre>
-We have derived the *K* combinator, and typed it at the same time!
+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.
-[To do: add pairs and destructors; unit and negation...]
+The definition of `div'` shows exactly what extra needs to be said in
+order to trigger the no-division-by-zero presupposition.
-Excercise: construct a proof whose labeling is the combinator S.
+For linguists: this is a complete theory of a particularly simply form
+of presupposition projection (every predicate is a hole).