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
---------------------
+Towards Monads
+--------------
-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 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:
-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:
+ # 12/0;;
+ Exception: Division_by_zero.
-If a variable `x` has type σ and term `M` has type τ, then
-the abstract `\xM` has type σ `-->` τ.
+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:
-If a term `M` has type σ `-->` τ, and a term `N` has type
-σ, then the application `MN` has type τ.
+ # type 'a option = None | Some of 'a;;
+ # None;;
+ - : 'a option = None
+ # Some 3;;
+ - : int option = Some 3
-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):
+So if a division is normal, we return some number, but if the divisor is
+zero, we return None:
<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
+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>
-`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:
+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:
<pre>
-------- Id
-A |- A
----------- Weak
-A, B |- A
-------------- --> I
-A |- B --> A
------------------ --> I
-|- A --> B --> A
+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>
-Should remind you of simple types. (What was `A --> B --> A` the type
-of again?)
+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.
-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:
+I prefer to line up the `match` alternatives by using OCAML's
+built-in tuple type:
<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:
+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>
- Γ, x:A |- R:B
---> I: -------------------------
- 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:
- Γ |- f:(A --> B) Γ |- x:A
---> E: -------------------------------------
- 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.
-
-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.
+This works, but is somewhat disappointing: the `add` prediction
+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):
<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!
-
-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...]
+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 a presupposition-filled world, and the definition of `div`
+shows exactly what extra needs to be added in order to trigger the
+no-division-by-zero presupposition.
-Excercise: construct a proof whose labeling is the combinator S.