Even with a fold-based representation of numbers, and pred/equal/subtraction, some computable functions are going to be out of our reach. Fibonacci: doable without Y, but takes some ingenuity And so on... Need a general method, where f(n) doesn't just depend on f(n-1) or (f(n-1),f(n-2),..). Looks like Ackermann function is simplest example that MUST be done with Y, Everything simpler could be done using only fixed iteration limits. A(y,x) = | when x == 0 -> y + 1 | when y == 0 -> A(x-1,1) | _ -> A(x-1, A(x,y-1)) A(0,y) = y+1 A(1,y) = y+2 A(2,y) = 2y + 3 A(3,y) = 2^(y+3) -3 A(4,y) = 2^(2^(2^...2)) [y+3 2s] - 3 Some algorithms can also be done more efficiently / intelligibly with general mechanism for recursion. How to do recursion with omega. fixed point combinators