On Randomized Versus Deterministic Computation

In contrast to deterministic or nondeterministic computation, it is a fundamental open problem in randomized computation how to separate different randomized time classes (at this point we do not even know how to separate linear randomized time from O(n^(log n)) randomized time) or how to compare them relative to corresponding deterministic time classes. In another words we are far from understanding the power of "random coin tosses" in the computation, and the possible ways of simulating them deterministically.In this paper we study the relative power of linear and polynomial randomized time compared with exponential deterministic time. Surprisingly, we are able to construct an oracle A such that exponential time (with or without the oracle A) is simulated by linear time Las Vegas algorithms using the oracle A. We are also able to prove, for the first time, that in some situations the randomized reductions are exponentially more powerful than deterministic ones (cf. [Adleman, Manders, 1977]).Furthermore, a set B is constructed such that Monte Carlo polynomial time (BPP) under the oracle B is exponentially more powerful than deterministic time with nondeterministic oracles. This strengthens considerably a result of Stockmeyer [St85] about the polynomial time hierarchy that for some decidable oracle B, (BPP^B) (not subseteq)(Delta^2)(P^B). Under our oracle BPP^B is exponentially more powerful than (Delta subscript 2)(P^B), and B does not add any power to (Delta subscript 2)(EXPTIME).

ICSI Technical Report TR-92-078

Technical Report