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;fp=hints%2Fassignment_4_hint_3_alternate_1.mdwn;h=0000000000000000000000000000000000000000;hp=f75bad0efce8079169eac9a41b99cbd76c95e31f;hb=fd698b815e417dec463cb0f0e9ed056ab83daed6;hpb=573a8b36ce653c84c2aecb2b81ef99128cb41d13 diff --git a/hints/assignment_4_hint_3_alternate_1.mdwn b/hints/assignment_4_hint_3_alternate_1.mdwn deleted file mode 100644 index f75bad0e..00000000 --- a/hints/assignment_4_hint_3_alternate_1.mdwn +++ /dev/null @@ -1,58 +0,0 @@ -Alternate strategy for Y1, Y2 - -* This is (in effect) the strategy used by OCaml. The mutually recursive: - - let rec - f x = A ; A may refer to f or g - and - g y = B ; B may refer to f or g - in - C - - 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 - C - - or, expanded into the form we've been working with: - - 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 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 = v g h in - D - - -