From: Chris Barker <barker@omega.(none)>
Date: Mon, 25 Oct 2010 02:10:28 +0000 (-0400)
Subject: added Curry-Howard
X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=f9bc566c3a2b3a401ac48de1c9c26dc68228c0ba;ds=sidebyside

added Curry-Howard
---

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 &sigma; and term `M` has type &tau;, then 
+the abstract `\xM` has type `&sigma; --> &tau;`.
+
+If a term `M` has type `&sigma; --> &tau`, and a term `N` has type
+&sigma;, then the application `MN` has type &tau;.
+
+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:
+
+Exchange: &Gamma;, A, B, &Delta; |- C
+          ---------------------------
+          $Gamma;, B, A, &Delta; |- C
+
+Contraction: &Gamma;, A, A |- B
+             -------------------
+             &Gamma;, A |- B
+
+Weakening: &Gamma; |- B
+           -----------------
+           &Gamma;, A |- B 
+
+Logical Rules:
+
+--> I:   &Gamma;, A |- B
+         -------------------
+         &Gamma; |- A --> B  
+
+--> E:   &Gamma; |- A --> B         &Gamma; |- A
+         -----------------------------------------
+         &Gamma; |- B
+</pre>
+
+`A`, `B`, etc. are variables over formulas.  
+&Gamma;, &Delta;, etc. are variables over (possibly empty) sequences
+of formulas.  `&Gamma; |- A` is a sequent, and is interpreted as
+claiming that if each of the formulas in &Gamma; 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:
+
+Exchange: &Gamma;, x:A, y:B, &Delta; |- R:C
+          --------------------------------------
+          &Gamma;, y:B, x:A, &Delta; |- R:C
+
+Contraction: &Gamma;, x:A, x:A |- R:B
+             --------------------------
+             &Gamma;, x:A |- R:B
+
+Weakening: &Gamma; |- R:B
+           --------------------- 
+           &Gamma;, x:A |- R:B     [x chosen fresh]
+
+Logical Rules:
+
+--> I:   &Gamma;, x:A |- R:B
+         -------------------------
+         &Gamma; |- \xM:A --> B  
+
+--> E:   &Gamma; |- f:(A --> B)      &Gamma; |- x:A
+         ---------------------------------------------
+         &Gamma; |- (fx):B
+</pre>
+
+In these labeling rules, if a sequence &Gamma; in a premise contains
+labeled formulas, those labels remain unchanged in the conclusion.
+
+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!
+
+[To do: add pairs and destructors; unit and negation...]
+
+Excercise: construct a proof whose labeling is the combinator S.