There is a nice simple algorithm on wikipedia:
http://en.wikipedia.org/wiki/Combinatory_logic
(for both SKI and BCKW)

translated to haskell:

-- The anoying thing about the algorithm is that it is difficult to separate the SKI and LC expression types
-- it's easiest to just combine them.
data Expr = Apply Expr Expr
          | Lambda String Expr
          | Id String
          | S
          | K
          | I
      deriving (Show)

convert (Apply a b) = Apply (convert a) (convert b)
convert (Lambda x e) | not $ occursFree x e = Apply K (convert e)
convert (Lambda x (Id s)) | x == s = I
convert (Lambda x (Lambda y e)) | occursFree x e = convert (Lambda x (convert (Lambda y e)))
convert (Lambda x (Apply e1 e2)) = Apply (Apply S (convert $ Lambda x e1)) (convert $ Lambda x e2)
convert x = x

occursFree var (Apply a b) = (occursFree var a) || (occursFree var b)
occursFree var (Lambda a b) = if a == var then False else (occursFree var b)
occursFree var (Id a) = if a == var then True else False
occursFree var _ = False

testExpr = Lambda "x" $ Lambda "y" $ Apply (Id "y") (Id "x")

test = convert testExpr

Hope that helps,

- Job

2010/1/28 Du¹an Koláø <kolar@fit.vutbr.cz>

Dear cafe,

Could anyone provide a link to some paper/book (electronic version of both preferred, even if not free) that describes an algorithm of translation of untyped lambda calculus expression to a set of combinators? Preferably SKI or BCKW. I'm either feeding google with wrong question or there is no link available now...

Thanks,

Du¹an

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