From: Jim Pryor <profjim@jimpryor.net>
Date: Sun, 3 Oct 2010 02:41:19 +0000 (-0400)
Subject: week 4: added html comment
X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=97ac6ce93d149beb471ad64da9e47e0416efbe82

week 4: added html comment

Signed-off-by: Jim Pryor <profjim@jimpryor.net>
---

diff --git a/week4.mdwn b/week4.mdwn
index d46981e0..5870a6a3 100644
--- a/week4.mdwn
+++ b/week4.mdwn
@@ -299,3 +299,33 @@ Fortunately, the Church numberals are well behaved with respect to
 types.  They can all be given the type (&sigma; --> &sigma;) -->
 &sigma; --> &sigma;.
 
+
+
+
+
+<!--
+
+Mau integrate some mention of this at some point.
+
+http://okmij.org/ftp/Computation/lambda-calc.html#predecessor
+
+
+Predecessor and lists are not representable in simply typed lambda-calculus
+
+    The predecessor of a Church-encoded numeral, or, generally, the encoding of a list with the car and cdr operations are both impossible in the simply typed lambda-calculus. Henk Barendregt's ``The impact of the lambda-calculus in logic and computer science'' (The Bulletin of Symbolic Logic, v3, N2, June 1997) has the following phrase, on p. 186:
+
+        Even for a function as simple as the predecessor lambda definability remained an open problem for a while. From our present knowledge it is tempting to explain this as follows. Although the lambda calculus was conceived as an untyped theory, typeable terms are more intuitive. Now the functions addition and multiplication are defineable by typeable terms, while [101] and [108] have characterized the lambda-defineable functions in the (simply) typed lambda calculus and the predecessor is not among them [the story of the removal of Kleene's four wisdom teeth is skipped...]
+        Ref 108 is R.Statman: The typed lambda calculus is not elementary recursive. Theoretical Comp. Sci., vol 9 (1979), pp. 73-81.
+
+    Since list is a generalization of numeral -- with cons being a successor, append being the addition, tail (aka cdr) being the predecessor -- it follows then the list cannot be encoded in the simply typed lambda-calculus.
+
+    To encode both operations, we need either inductive (generally, recursive) types, or System F with its polymorphism. The first approach is the most common. Indeed, the familiar definition of a list
+
+         data List a = Nil | Cons a (List a)
+
+    gives an (iso-) recursive data type (in Haskell. In ML, it is an inductive data type).
+
+    Lists can also be represented in System F. As a matter of fact, we do not need the full System F (where the type reconstruction is not decidable). We merely need the extension of the Hindley-Milner system with higher-ranked types, which requires a modicum of type annotations and yet is able to infer the types of all other terms. This extension is supported in Haskell and OCaml. With such an extension, we can represent a list by its fold, as shown in the code below. It is less known that this representation is faithful: we can implement all list operations, including tail, drop, and even zip.
+
+
+-->