From: Chris Barker Date: Mon, 25 Oct 2010 13:32:42 +0000 (-0400) Subject: moved curry-howard, added Wadler paper X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=37309e8fca0040224958bf089c024d301e39de09 moved curry-howard, added Wadler paper --- diff --git a/curry-howard b/curry-howard new file mode 100644 index 00000000..7840e377 --- /dev/null +++ b/curry-howard @@ -0,0 +1,142 @@ +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): + +
+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
+
+ +`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:
+
+          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
+
+ +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: + +
+------------  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! + +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. diff --git a/wadler-monads.pdf b/wadler-monads.pdf new file mode 100644 index 00000000..ad23fcd4 Binary files /dev/null and b/wadler-monads.pdf differ