* 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
+ 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
-is implemented using regular, non-mutual recursion, like this (`u` is a variable not occurring free in `A`, `B`, or `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 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
+ 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
-or, expanded into the form we've been working with:
+ <!--
+ Or in Y1of3, Y2of3, Y3of3 form:
- 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
+ let Yv = \ff. Y (\v g h. ff ( v g h) g h) in
+ let Yw = \ff gg. Y ( \w h. (\g. gg (Yv ff g h) g h) ( w h)) in
+ let Y3of3 = \ff gg hh. Y ( \h. (\g. hh (Yv ff g h) g h) (Yw ff gg h)) in
+ let Y2of3 = \ff gg hh. Yw ff gg (Y3of3 ff gg hh) in
+ let Y1of3 = \ff gg hh. Yv ff (Y2of3 ff gg hh) (Y3of3 ff gg hh) in
+ let f = Y1of3 (\f g h. A) (\f g h. B) (\f g h. C) in
+ let g = Y2of3 (\f g h. A) (\f g h. B) (\f g h. C) in
+ let h = Y3of3 (\f g h. A) (\f g h. B) (\f g h. C) in
+ D
+ -->