We study how the nondeterminism versus determinism problem and the time versus space problem are related to the problem of derandomization. In particular, we show two ways of derandomizing the complexity class AM under uniform assumptions, which was only known previously under non-uniform assumptions [13, 14]. First, we prove that either AM = NP or it appears to any nondeterministic polynomial time adversary that NP is contained in deterministic subexponential time infinitely often. This implies that to any nondeterministic polynomial time adversary, the graph non-isomorphism problem appears to have subexponential-size proofs infinitely often, the first nontrivial derandomization of this problem without any assumption. Next, we show that either all BPP = P, AM = NP, and PH P hold, or for any t(n) = 2 n) , DTIME(t(n)) DSPACE(t (n)) infinitely often for any constant > 0. Similar tradeoffs also hold for a whole range of parameters. This improves previous results [17, 5] ...

We prove (mostly tight) space lower bounds for "streaming " (or "on-line") computations of four fundamental combinatorial objects: error-correcting codes, universal hash functions, extractors, and dispersers. Streaming computations for these objects are motivated algorithmically by massive data set applications and complexity-theoretically by pseudorandomness and derandomization for spacebounded probabilistic algorithms.

We use derandomization to show that sequences of positive pspace-dimension -- in fact, even positive # k -dimension for suitable k -- have, for many purposes, the full power of random oracles. For example, we show that, if S is any binary sequence whose # 3 -dimension is positive, then and, moreover, every BPP promise problem is P -separable. We prove analogous results at higher levels of the polynomial-time hierarchy. The dimension-almost-class of a complexity class denoted by dimalmost-C, is the class consisting of all problems A such that A for all but a Hausdor# dimension 0 set of oracles S. Our results yield several characterizations of complexity classes, such as BPP = dimalmost-P and AM = dimalmost-NP, that refine previously known results on almost-classes. They also yield results, such as Promise-BPP = almost-P-Sep = dimalmost-P-Sep, in which even the almost-class appears to be a new characterization.

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A hitting-set generator is a deterministic algorithm which generates a set of strings that intersects every dense set recognizable by a small circuit. A polynomial time hitting-set generator readily implies RP = P . Andreev et. al. (ICALP'96, and JACM 1998) showed that if polynomial-time hitting-set generator in fact implies the much stronger conclusion BPP = P . We simplify and improve their (and later) constructions. Keywords: Derandomization, RP, BPP , one-sided error versus two-sided error A preliminary version of this work has appeared in the proceedings of Random99. y Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel. oded@wisdom.weizmann.ac.il. z MIT Laboratory for Computer Science, 545 Technology Square, Cambridge, MA 02139. salil@theory.lcs.mit.edu. Supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. x Institute of Computer Science, The Hebrew University of Jerusalem, Givat-Ram, Jerusalem, Israel. avi@cs.huji.ac....

Most of the hypotheses of full derandomization fall into two sets of equivalent statements: Those equivalent to the existence of ecient pseudorandom generators and those equivalent to approximating the accepting probability of a circuit. We give the rst relativized world where these sets of equivalent statements are not equivalent to each other.