Let the N-Cuil Problem be as follows: Given two situations x and x', and some constant N >=0, determine whether x' is a member of k‽(x), for any k such that 0 <= k <= N.
It seems to me that k‽(x) is an infinite set (when k > 0), in which case the N-Cuil Problem is not decidable, but is recursively enumerable. This is trivially true for N = 0, and is easy to see for N = 1. I suspect this also to be true for all N >= 0, but I am not positive. Thoughts?