X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment4.mdwn;h=9f56675892ddaabde44640fa0f227c124ca6dcc5;hp=cd89a134af191c0c53d2fc37e0001bb394d65548;hb=4cea4f69242f3a229292186c8fa652942f31f8b8;hpb=48b6ae024e8268541a5b6be4d87f4894e357e4a1
diff --git a/assignment4.mdwn b/assignment4.mdwn
index cd89a134..9f566758 100644
--- a/assignment4.mdwn
+++ b/assignment4.mdwn
@@ -12,46 +12,78 @@ can use.
#Comparing lists for equality#
+
-- blah
+
- Suppose you have two lists of integers, `left` and `right`. You want to determine whether those lists are equal: that is, whether they have all the same members in the same order. (Equality for the lists we're working with is *extensional*, or parasitic on the equality of their members, and the list structure. Later in the course we'll see lists which aren't extensional in this way.)
+
+How would you implement such a list comparison?
+
+(See [[hints/Assignment 4 hint 2]] if you need some hints.)
-
-#Mutually-recursive functions#
+
+#Enumerating the fringe of a leaf-labeled tree#
+
+First, read this: [[Implementing trees]]
- blah
+
+(See [[hints/Assignment 4 hint 3]] if you need some hints.)
-#Enumerating the fringe of a leaf-labeled tree#
+#Mutually-recursive functions#
-[[Implementing trees]]
+
+- (Challenging.) One way to define the function `even` is to have it hand off part of the work to another function `odd`:
+ let even = \x. iszero x
+ ; if x == 0 then result is
+ true
+ ; else result turns on whether x's pred is odd
+ (odd (pred x))
+
+At the same tme, though, it's natural to define `odd` in such a way that it hands off part of the work to `even`:
+
+ let odd = \x. iszero x
+ ; if x == 0 then result is
+ false
+ ; else result turns on whether x's pred is even
+ (even (pred x))
+
+Such a definition of `even` and `odd` is called **mutually recursive**. If you trace through the evaluation of some sample numerical arguments, you can see that eventually we'll always reach a base step. So the recursion should be perfectly well-grounded:
+
+ even 3
+ ~~> iszero 3 true (odd (pred 3))
+ ~~> odd 2
+ ~~> iszero 2 false (even (pred 2))
+ ~~> even 1
+ ~~> iszero 1 true (odd (pred 1))
+ ~~> odd 0
+ ~~> iszero 0 false (even (pred 0))
+ ~~> false
+
+But we don't yet know how to implement this kind of recursion in the lambda calculus.
+
+The fixed point operators we've been working with so far worked like this:
+
+ let X = Y T in
+ X <~~> T X
+
+Suppose we had a pair of fixed point operators, `Y1` and `Y2`, that operated on a *pair* of functions `T1` and `T2`, as follows:
+
+ let X1 = Y1 T1 T2 in
+ let X2 = Y2 T1 T2 in
+ X1 <~~> T1 X1 X2 and
+ X2 <~~> T2 X1 X2
+
+If we gave you such a `Y1` and `Y2`, how would you implement the above definitions of `even` and `odd`?
+
+
+
- (More challenging.) Using our derivation of Y from the [[Week2]] notes as a model, construct a pair `Y1` and `Y2` that behave in the way described.
+
+(See [[hints/Assignment 4 hint 4]] if you need some hints.)
-
-- blah