+++ /dev/null
-[[!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 = <fun>
-
-Since `+` is only defined on integers, it has type
-
- # (+);;
- - : int -> int -> int = <fun>
-
-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 <bool expression> with true -> <expression1> | false -> <expression2>
-
-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 = <fun>
-
-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 = <fun>
-
-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 = <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:
-
- # 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>
- # (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 = <fun>
-
- # test;;
- - : unit -> 'a = <fun>
-
- # 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").
-
-Curry-Howard, take 1
---------------------
-
-We will return to 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.
-
-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:
-
-If a variable `x` has type σ and term `M` has type τ, then
-the abstract `\xM` has type σ `-->` τ.
-
-If a term `M` has type σ `-->` τ, and a term `N` has type
-σ, then the application `MN` has type τ.
-
-These rules are clearly obverses of one another: the functional types
-that abstract builds up are taken apart by application.
-
-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):
-
-<pre>
-Axiom: ---------
- A |- A
-
-Structural Rules:
-
- Γ, A, B, Δ |- C
-Exchange: ---------------------------
- Γ, B, A, Δ |- C
-
- Γ, A, A |- B
-Contraction: -------------------
- Γ, A |- B
-
- Γ |- B
-Weakening: -----------------
- Γ, A |- B
-
-Logical Rules:
-
- Γ, A |- B
---> I: -------------------
- Γ |- A --> B
-
- Γ |- A --> B Γ |- A
---> E: -----------------------------------
- Γ |- B
-</pre>
-
-`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.
-
-This logic allows derivations of theorems like the following:
-
-<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?)
-
-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:
-
-<pre>
-Axiom: -----------
- x:A |- x:A
-
-Structural Rules:
-
- Γ, x:A, y:B, Δ |- R:C
-Exchange: -------------------------------
- Γ, y:B, x:A, Δ |- R:C
-
- Γ, x:A, x:A |- R:B
-Contraction: --------------------------
- Γ, x:A |- R:B
-
- Γ |- R:B
-Weakening: ---------------------
- Γ, x:A |- R:B [x chosen fresh]
-
-Logical Rules:
-
- Γ, x:A |- R:B
---> I: -------------------------
- Γ |- \xM:A --> B
-
- Γ |- f:(A --> B) Γ |- x:A
---> E: -------------------------------------
- Γ |- (fx):B
-</pre>
-
-In these labeling rules, if a sequence Γ in a premise contains
-labeled formulas, those labels remain unchanged in the conclusion.
-
-What is means for a variable `x` to be chosen *fresh* is that
-`x` must be distinct from any other variable in any of the labels
-used in the proof.
-
-Using these labeling rules, we can label the proof
-just given:
-
-<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
-</pre>
-
-We have derived the *K* combinator, and typed it at the same time!
-
-Need a proof that involves application, and a proof with cut that will
-show beta reduction, so "normal" proof.
-
-[To do: add pairs and destructors; unit and negation...]
-
-Excercise: construct a proof whose labeling is the combinator S.