A Probabilistic-Time Hierarchy Theorem for “Slightly Non-uniform” Algorithms
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Unlike other complexity measures such as deterministic and nondeterministic time and space, and non-uniform size, it is not known whether probabilistic time has a strict hierarchy. For example, as far as we know it may be that BPP is contained in the class BPtime(n). In fact, it may even be that the class BPtime(n logn ) is contained in the class BPtime(n).
We also discuss conditions under which a hierarchy theorem can be proven for fully uniform Turing machines. In particular we observe that such a theorem does hold if BPP has a complete problem.
KeywordsTuring Machine Complete Problem Universal Turing Machine Probabilistic Machine Promise Problem
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- 1.Karpinski, M., Verbeek, R.: Randomness, provability, and the separation of monte carlo time and space. In E. Börger, ed: Computation Theory and Logic. Volume 270 of LNCS. Springer (1987) 189-207.Google Scholar
- 3.Cai, J.-Y., Nerurkar, A., Sivakumar, D.: Hardness and hierarchy theorems for probabilistic quasi-polynomial time. In ACM, ed.: Proceedings of the thirty-first annual ACM Symposium on Theory of Computing: Atlanta, Georgia, May 1-4, 1999, New York, NY, USA, ACM Press (1999) 726–735CrossRefGoogle Scholar
- 4.Impagliazzo, R., Wigderson, A.: P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In: Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, El Paso, Texas (1997) 220–229Google Scholar
- 5.Sipser, M.: A complexity theoretic approach to randomness. In: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, Boston, Massachusetts (1983) 330–335Google Scholar
- 10.Levin, L.: Universal search problems (in russian). Problemy Peredachi Informatsii 9 (1973) 265–266 English translation in Trakhtenbrot, B. A.: A survey of Russian approaches to Perebor (brute-force search) algorithms. Annals of the History of Computing, 6 (1984), 384-400.Google Scholar
- 11.Schnorr, C. P.: Optimal algorithms for self-reducible problems. In Michaelson, S., Milner, R., eds.: Third International Colloquium on Automata, Languages and Programming, University of Edinburgh, Edinburgh University Press (1976) 322–337Google Scholar
- 12.Trevisan, L., Vadhan, S.: Pseudorandomness and average-case complexity via uniform reductions. In IEEE, ed.; Proceedings of 17th Conference on Computational Complexity, Montréal, Québec, May 21-24, IEEE (2002)Google Scholar