On relativized exponential and probabilistic complexity classes

An oracle X is constructed such that the exponential complexity class Δ^EP,X₂ equals the probabilistic class R(R(X)). This shows that it will be difficult to prove that Δ^EP₂ is different from R(R), although it seems very unlikely that these two classes are equal. The result subsumes several known results about relativized computations:

• (i)

  the existence of relativized polynomial hierarchies extending two levels (Long, T., 1978, Dissertation, Purdue Univ., Lafayette, Ind.; Heller, H., 1984(a), SIAM J. Comput. 13, 717–725; Heller, H., 1984(b), Math. Systems Theory 17, 71–84);

• (ii)

  the existence of an oracle X such that BPP(X) ⊄ Δ^P,X₂ (Stockmeyer, L., 1983, “Proc. 15th STOC” pp. 118–126),

• (iii)

  the existence of an oracle X such that NP(X) is polynomially Turing reducible to a sparse set (Wilson, C., 1983, “Proc. 24th FOCS,”, pp. 329–334; Immerman, N., and Mahaney, S., 1983, “Conference on Computational Complexity Theory,” Santa Barbara, March 21–25).

The result shows possible inclusion relations for nonrelativized complexity classes and points out that certain results about probabilistic complexity classes and about polynomial size circuits cannot be improved unless methods are applied which do not relativize.