X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=hints%2Fassignment_4_hint_3_alternate_1.mdwn;h=f75bad0efce8079169eac9a41b99cbd76c95e31f;hp=99a4202b59946a7da55a8cff6cef5837460dec3c;hb=f212354152a53c6a3ab018c7874570c600f463b9;hpb=8c28339b991f2d4f0940d295bf4d265a00751b05 diff --git a/hints/assignment_4_hint_3_alternate_1.mdwn b/hints/assignment_4_hint_3_alternate_1.mdwn index 99a4202b..f75bad0e 100644 --- a/hints/assignment_4_hint_3_alternate_1.mdwn +++ b/hints/assignment_4_hint_3_alternate_1.mdwn @@ -11,23 +11,48 @@ Alternate strategy for Y1, Y2 is implemented using regular, non-mutual recursion, like this (`u` is a variable not occurring free in `A`, `B`, or `C`): - let rec u g x = (let f = u g in A) - in let rec g y = (let f = u g in B) - in let f = u g in + let rec u g x = (let f = u g in A) in + let rec g y = (let f = u g in B) in + let f = u g in C or, expanded into the form we've been working with: - let u = Y (\u g x. (\f. A) (u g)) in - let g = Y ( \g y. (\f. B) (u g)) in - let f = u g in + let u = Y (\u g. (\f x. A) (u g)) in + let g = Y ( \g. (\f y. B) (u g)) in + let f = u g in C + We could abstract Y1 and Y2 combinators from this as follows: + + let Yu = \ff. Y (\u g. ff ( u g ) g) in + let Y2 = \ff gg. Y ( \g. gg (Yu ff g ) g) in + let Y1 = \ff gg. (Yu ff) (Y2 ff gg) in + let f = Y1 (\f g. A) (\f g. B) in + let g = Y2 (\f g. A) (\f g. B) in + C + + * Here's the same strategy extended to three mutually-recursive functions. `f`, `g` and `h`: - let u = Y (\u g h x. (\f. A) (u g h)) in - let w = Y ( \w h x. (\g. (\f. B) (u g h)) (w h)) in - let h = Y ( \h x. (\g. (\f. C) (u g h)) (w h)) in + let v = Y (\v g h. (\f x. A) (v g h)) in + let w = Y ( \w h. (\g. (\f y. B) (v g h)) (w h)) in + let h = Y ( \h. (\g. (\f z. C) (v g h)) (w h)) in let g = w h in - let f = u g h in + let f = v g h in D + + +