7 Conclusion and Open Problems

A cardinality version of Beigel's nonspeedup theorem. We have shown that the notion of enumeration complexity allows to characterize the query complexity, even in the fault tolerant case. Then we went on to determine for any nite function f : f0; 1g n ! S the extremal values of e(f; A) which can be obtained for some nonrecursive set A. It is open to determine the possible intermediate values. More formally, there is the following open problem. For any k 2 ! let V (f; k) = fA : e(f; A) kg. The \gen-eral inclusion problem" (GIP) is the question for which f : f0; 1g n ! S 1 , g : f0; 1g m ! S 2 and k 1 ; k 2 we have V (f; k 1) V (g; k 2). In particular, this would answer the question for which k 1 ; k 2 , the classes V (f; k 1) and V (f; k 2) coincide. By now we know that for s(f) > 1: V (f; 1) = V (f; k) ii 1 k < s(f). For the interesting special case f = id n , g = id m the inclusion problem has been completely solved in 8]. However, it is an open question whether GIP is decidable, since the solution of that special case does not generalize. This stands in contrast to the analogous inclusion problem in the context of polynomial time bounded computation (i.e. the version indicated in Remark (a) following Theorem 2.2) which has a positive answer: a combinatorial condition can be formulated similarly as for the special case treated in 8, Section 7].