From: Chris Barker Date: Tue, 18 Jan 2011 04:11:50 +0000 (-0500) Subject: edits X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=675ab516c47d0bbf6fe8501bd15b0e044996e327 edits --- diff --git a/hints/assignment_4_hint_3_alternate_2.mdwn b/hints/assignment_4_hint_3_alternate_2.mdwn index 1a86fa66..b2ddef82 100644 --- a/hints/assignment_4_hint_3_alternate_2.mdwn +++ b/hints/assignment_4_hint_3_alternate_2.mdwn @@ -7,8 +7,9 @@ rest on your own. Let's remind ourselves of how the ordinary Y combinator works, since we're going to attack mutual recursion in the same way. -First, we define a recursive proto-function that takes the true -function (the fixed point) as an argument, then calls that argument. +First, we define a recursive proto-function called `protomonoeven` +that takes the true even function (the fixed point) as an argument, +then calls that argument: 
 let true = \y n . y in
@@ -22,8 +23,8 @@ Y protomonoeven 3
So in the equation `X = Y T`, `X`, the function that computes whether -a number is even or not, is the fixed point of the protoeven function, -i.e., `X = Y protoeven`. +a number is even or not, is the fixed point of the `protomonoeven` function, +i.e., `X = Y protomonoeven`. Obviously, we don't need mutual recursion to define the "even" function (since we just defined it without mutual recursion). @@ -35,7 +36,7 @@ But we *can* define it using mutual recursion, as shown in assignment Since we have two functions, we'll need two fixed points (`X1` and `X2`), and therefore two fixed point combinators to compute them (`Y1` -and `Y2`), and two protofunction (`protoeven` and `protoodd`) to feed +and `Y2`), and two protofunctions (`protoeven` and `protoodd`) to feed to the combinators. Furthermore, since the fixed points depend on both the even function and the odd function, the combinators will need to take two arguments, namely, both of the protofunctions: @@ -44,13 +45,13 @@ to take two arguments, namely, both of the protofunctions: odd = X2 = Y2 protoeven protoodd Writing protoeven and protoodd is easy. Analogously to the ordinary -case, they will take arguments that we want to find fixed points for: +case, they will take as arguments the very fixed points we're looking for: protoeven = \e o n . iszero n true (o (pred n)) protoodd = \e o n . iszero n false (e (pred n)) -Notice that protoeven doesn't make use of its first argument (the one -that will eventually correspond to the even function itself). This is +Notice that `protoeven` doesn't make use of its first argument. +Likewise, `protoodd` doesn't make use of its second argument. This is an expository weakness in the choice of the example, which is particularly simple.