From f9bc566c3a2b3a401ac48de1c9c26dc68228c0ba Mon Sep 17 00:00:00 2001 From: Chris Barker Date: Sun, 24 Oct 2010 22:10:28 -0400 Subject: [PATCH] added Curry-Howard --- week6.mdwn | 135 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 135 insertions(+) diff --git a/week6.mdwn b/week6.mdwn index 221e0207..79b12f71 100644 --- a/week6.mdwn +++ b/week6.mdwn @@ -171,3 +171,138 @@ 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 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. + +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 --> &tau`, 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): + +
```+Axiom: ---------
+        A |- A
+
+Structural Rules:
+
+Exchange: Γ, A, B, Δ |- C
+          ---------------------------
+          \$Gamma;, B, A, Δ |- C
+
+Contraction: Γ, A, A |- B
+             -------------------
+             Γ, A |- B
+
+Weakening: Γ |- B
+           -----------------
+           Γ, A |- B
+
+Logical Rules:
+
+--> I:   Γ, A |- B
+         -------------------
+         Γ |- A --> B
+
+--> E:   Γ |- A --> B         Γ |- A
+         -----------------------------------------
+         Γ |- B
+```
+ +`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: + +
```+-------  Id
+A |- A
+---------- Weak
+A, B |- A
+------------- --> I
+A |- B --> A
+----------------- --> I
+|- A --> B --> A
+```
+ +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: + +
```+Axiom: -----------
+       x:A |- x:A
+
+Structural Rules:
+
+Exchange: Γ, x:A, y:B, Δ |- R:C
+          --------------------------------------
+          Γ, y:B, x:A, Δ |- R:C
+
+Contraction: Γ, x:A, x:A |- R:B
+             --------------------------
+             Γ, x:A |- R:B
+
+Weakening: Γ |- R:B
+           ---------------------
+           Γ, x:A |- R:B     [x chosen fresh]
+
+Logical Rules:
+
+--> I:   Γ, x:A |- R:B
+         -------------------------
+         Γ |- \xM:A --> B
+
+--> E:   Γ |- f:(A --> B)      Γ |- x:A
+         ---------------------------------------------
+         Γ |- (fx):B
+```
+ +In these labeling rules, if a sequence Γ in a premise contains +labeled formulas, those labels remain unchanged in the conclusion. + +Using these labeling rules, we can label the proof +just given: + +
```+------------  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
+```
+ +We have derived the *K* combinator, and typed it at the same time! + +[To do: add pairs and destructors; unit and negation...] + +Excercise: construct a proof whose labeling is the combinator S. -- 2.11.0