A way to test homework solutions for students

[lambda.git] / week2.mdwn

diff --git a/week2.mdwn b/week2.mdwn
index 1e3aa2b..a8e3ca4 100644 (file)
--- a/week2.mdwn
+++ b/week2.mdwn
@@ -318,17 +318,19 @@ will eta-reduce by n steps to:
 
        \x. M
 
+When we add eta-reduction to our proof system, we end up reconstruing the meaning of `~~>` and `<~~>` and "normal form", all in terms that permit eta-reduction as well. Sometimes these expressions will be annotated to indicate whether only beta-reduction is allowed (<code>~~><sub>&beta;</sub></code>) or whether both beta- and eta-reduction is allowed (<code>~~><sub>&beta;&eta;</sub></code>).
+
 The logical system you get when eta-reduction is added to the proof system has the following property:
 
 >      if `M`, `N` are normal forms with no free variables, then <code>M &equiv; N</code> iff `M` and `N` behave the same with respect to every possible sequence of arguments.
 
-That is, when `M` and `N` are (closed normal forms that are) syntactically distinct, there will always be some sequences of arguments <code>L<sub>1</sub>, ..., L<sub>n</sub></code> such that:
+This implies that, when `M` and `N` are (closed normal forms that are) syntactically distinct, there will always be some sequences of arguments <code>L<sub>1</sub>, ..., L<sub>n</sub></code> such that:
 
 <pre><code>M L<sub>1</sub> ... L<sub>n</sub> x y ~~> x
 N L<sub>1</sub> ... L<sub>n</sub> x y ~~> y
 </code></pre>
 
-That is, closed normal forms that are not just beta-reduced but also fully eta-reduced, will be syntactically different iff they yield different values for some arguments. That is, iff their extensions differ.
+So closed beta-plus-eta-normal forms will be syntactically different iff they yield different values for some arguments. That is, iff their extensions differ.
 
 So the proof theory with eta-reduction added is called "extensional," because its notion of normal form makes syntactic identity of closed normal forms coincide with extensional equivalence.
 
@@ -430,6 +432,7 @@ But is there any method for doing this in general---for telling, of any given co
 
 *      [Scooping the Loop Snooper](http://www.cl.cam.ac.uk/teaching/0910/CompTheory/scooping.pdf), a proof of the undecidability of the halting problem in the style of Dr Seuss by Geoffrey K. Pullum
 
+Interestingly, Church also set up an association between the lambda calculus and first-order predicate logic, such that, for arbitrary lambda formulas `M` and `N`, some formula would be provable in predicate logic iff `M` and `N` were convertible. So since the right-hand side is not decidable, questions of provability in first-order predicate logic must not be decidable either. This was the first proof of the undecidability of first-order predicate logic.
 
 
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