From: Chris Barker Date: Mon, 25 Oct 2010 18:10:15 +0000 (-0400) Subject: added proto-monad X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=e519121696a33c116b0942cb289e74d4d978b80c added proto-monad --- diff --git a/assignment5.mdwn b/assignment5.mdwn index ccf402ab..43c3ef55 100644 --- a/assignment5.mdwn +++ b/assignment5.mdwn @@ -1,4 +1,17 @@ +Assignment 5 + Types and OCAML +--------------- + +0. Recall that the S combinator is given by \x y z. x z (y z). + Give two different typings for this function in OCAML. + To get you started, here's one typing for K: + + # let k (y:'a) (n:'b) = y;; + val k : 'a -> 'b -> 'a = + # k 1 true;; + - : int = 1 + 1. Which of the following expressions is well-typed in OCAML? For those that are, give the type of the expression as a whole. @@ -108,3 +121,20 @@ or of `match`. That is, you must keep the `let` statements, though you're allowed to adjust what `b`, `y`, and `n` get assigned to. [[Hint assignment 5 problem 3]] + +4. Baby monads. Read the lecture notes for week 6, then write a + function `lift` that generalized the correspondence between + and + `add`: that is, `lift` takes any two-place operation on integers + and returns a version that takes arguments of type `int option` + instead, returning a result of `int option`. In other words, + `lift` will have type + + (int -> int -> int) -> (int option) -> (int option) -> (int option) + + so that `lift (+) (Some 3) (Some 4)` will evalute to `Some 7`. + Don't worry about why you need to put `+` inside of parentheses. + You should make use of `bind` in your definition of `lift`: + + let bind (x: int option) (f: int -> (int option)) = + match x with None -> None | Some n -> f n;; + diff --git a/curry-howard b/curry-howard index 7840e377..f3c1be75 100644 --- a/curry-howard +++ b/curry-howard @@ -139,4 +139,28 @@ 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. +Excercise: construct a proof whose labeling is the combinator S, +something like this: + + --------- Ax --------- Ax ------- Ax + !a --> !a !b --> !b c --> c + ----------------------- L-> -------- L! + !a,!a->!b --> !b !c --> c +--------- Ax ---------------------------------- L-> +!a --> !a !a,!b->!c,!a->!b --> c +------------------------------------------ L-> + !a,!a,!a->!b->!c,!a->!b --> c + ----------------------------- C! + !a,!a->!b->!c,!a->!b --> c + ------------------------------ L! + !a,!a->!b->!c,! (!a->!b) --> c + ---------------------------------- L! + !a,! (!a->!b->!c),! (!a->!b) --> c + ----------------------------------- R! + !a,! (!a->!b->!c),! (!a->!b) --> !c + ------------------------------------ R-> + ! (!a->!b->!c),! (!a->!b) --> !a->!c + ------------------------------------- R-> + ! (!a->!b) --> ! (!a->!b->!c)->!a->!c + --------------------------------------- R-> + --> ! (!a->!b)->! (!a->!b->!c)->!a->!c diff --git a/week6.mdwn b/week6.mdwn index f26285f3..69881478 100644 --- a/week6.mdwn +++ b/week6.mdwn @@ -171,145 +171,137 @@ 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 --------------------- +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:
```-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 =
+# 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
+*)
```
-`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:
```--------  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 =
+# 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
+*)
```
-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:
```-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);;
+```
- 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 +
```+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
+*)
```
-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):
```-------------  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
+*)
```
-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.