U. Vazirani, "Randomness, Adversaries and Computation," Ph.D. Thesis, University of California, Berkeley, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....known. They proved that it is impossible to extract even a single almost random bit from one such source (so Vazirani [Va1, Va3] used two independent sources) In light of this result, one might give up hope for simulating randomized algorithms with one semi random source. Nevertheless, VV] and [Va2] showed how to efficiently simulate all algorithms in RP and BPP with one semi random source, for any constant ff 0. Note that these simulations use R = poly(n) bits from the semi random source, where n denotes the length of the input to the RP or BPP machine; we shall let n and R denote ....

....We then pick a family F that satisfies Lemma 3. 2 with parameters k = c 2) log n and ffl = n Gamma(c 1) Now we use the following modification of a lemma from the final version of [Zu2] which, using the Leftover Hash Lemma in the manner of [IZ] essentially strengthened related lemmas in [Va2] and [CG] Lemma 4.1. Let F be a function family mapping l bits to b Gamma 2k bits, satisfying Lemma 3.2 with parameters k = c 2) log n and ffl = n Gamma(c 1) Let D be a block wise ffi source on f0; 1g ml . If Y = Y 1 : Ym is chosen according to D and f is chosen uniformly at ....

U. Vazirani, "Randomness, Adversaries and Computation," Ph.D. Thesis, University of California, Berkeley, 1986.

....Because there are at most 2 7k such sequences s, Pr[9s with 7k 2 incorrect answers] 2 7k 5 Gamma7k=2 2 Gammak ; which is what we wanted to show. Remark. The above directly implies a version of Vazirani s result that quasi random sources can be used to perform BPP computations[V2]. Specifically, there is a ffi Gamma1 such that if each r bit string occurs with probability at most 2 ffir , the above construction works. To see this, say a prg needs Cr bits to reduce the probability of error to 2 Gammar . Then there are 2 (C Gamma1)r bad strings of length Cr. ....

U. Vazirani, "Randomness, Adversaries and Computation, " PhD Thesis, University of California, Berkeley, 1986.

....to extract even a single almost random bit from one such source (so Vazirani [Va2, Va3] showed how to extract almost random bits from two independent sources) In light of this result, one might give up hope for simulating randomized algorithms with one semi random source. Nevertheless, VV] and [Va1] showed how to simulate RP and BPP with one semi random source. Chor and Goldreich [CG1] generalized this model by assuming no sequence of l bits has too high a probability of being output. More precisely, 1 Definition 2 [CG1] An (l; ffi) PRB source outputs R bits as R=l blocks Y 1 ; Y ....

....this previous work, it is natural to ask: what is the most general model of a weak source for which we can simulate randomized algorithms Do we need the randomness in some particular form, or will any form suffice Of course, if BPP = P, then we don t need randomness at all. Yet we follow [VV] [Va1], and [CG1] and deal with a more abstract BPP problem: let an adversary label strings in f0; 1g r either yes or no, provided that at least 3=4 of the strings have the same label. We wish to find out whether the majority say yes or no, with high probability. It is clear that randomness ....

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U. Vazirani, "Randomness, Adversaries and Computation," Ph.D. Thesis, University of California, Berkeley, 1986.

....the bits output by an unknown Markov chain into a sequence of perfectly random bits. However, for more general faulty sources, it can be shown (see e.g. SV86] that to construct such a direct conversion f is impossible. On the other hand, the following broader idea of conversion (introduced in [VV85, Vaz86]) works for simulating randomized algorithms: Suppose the randomized computation C we wish to simulate needs m random bits. The simulation first requests R = R(m) bits from the faulty source. Then it maps this sequence of R(m) bits to a set of t(m) strings each of m bits, called test strings, ....

....as a black box simulation. Applying this framework, efficient simulations of RP and BPP under various models of weak sources have been extensively studied. For example, Santha and Vazirani [SV86] studied the class of weak sources called slightly random sources , which was further examined in [VV85, Vaz86, Vaz87a, Vaz87b]. A more general model PRB sources is considered later by Chor and Goldreich [CG88] In [Zuc90, Zuc91] Zuckerman introduced the model of ffi sources which generalizes all the previous models. Let D be a probability distribution on a set X. The min entropy of D is defined to be the maximum real ....

U. Vazirani, "Randomness, Adversaries and Computation," Ph.D. Thesis, University of California, Berkeley, 1986.

.... with a relatively small extra cost of random bits [KPS85, CG86] The problems of extracting fully random bits from the outputs of imperfect random sources and simulating probabilistic algorithms with such outputs have been studied for a variety of imperfect random sources [VN51, Blu84, SV84, VV85, Vaz86, CG85, CGH 85, BOL85, KKL88, LLS87] Bit extraction has been proved possible in some cases [VN51, Blu84, CGH 85] yielding immediately simulations of probabilistic algorithms with these sources. In other cases bit extraction was proved to be impossible [SV84, CG85, CGH 85, BOL85, ....

....yielding immediately simulations of probabilistic algorithms with these sources. In other cases bit extraction was proved to be impossible [SV84, CG85, CGH 85, BOL85, KKL88, LLS87] Simulations of probabilistic algorithms when bit extraction is impossible, have been given in some cases [VV85, Vaz86, CG85] 1.2 The Approach of This Paper. The approach of this paper is to translate the simulation of probabilistic algorithms to the following statistical problem: Given a finite domain X estimate with high confidence the size of a sufficiently large subset S ae X , using a small number of ....

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U. Vazirani. Randomness adversaries and computation. PhD thesis, University of California Berkeley, 1986.

....07974, USA. A portion of this work was done while the author was in the Computer Science Department at Stanford and while visiting DEC Systems Research Center. Partially supported by NSF Grant CCR 9010517 and an OTL grant. E mail: phillips research.att.com of the computation. A number of authors [17, 19, 4, 14] consider the same problem from the source s viewpoint. There are many ways for the source to deviate from perfect randomness, but for most models the source is an adversary who can select a strategy from a given repertoire, and whose goal is to derail the randomized algorithm. While the ....

.... can be obtained using Markov decision theory, but we are mainly interested in quantitative questions: how much can the controller s bias affect the stochastic process, and which stochastic processes are the hardest to influence The only instances of this problem for which answers are available [17, 19, 4, 14, 6, 2, 3, 10] are (either originally thus stated, or easily translated into these terms) random walks on a finite tree, which is directed from the root to the leaves. The decision which branch to take is determined locally and the controller can bias this decision with a certain probability. The present paper ....

U. Vazirani, "Randomness, Adversaries and Computation," PhD Thesis, University of California, Berkeley, 1986.

....uniform distribution over all n bit strings is unbiased (0biased) with respect to all linear tests. A linear test can be interpreted as trying to refute the randomness of a probability space by taking a fixed linear combination of the bits in the sample. The following lemma, attributed to Vazirani [29] (see also [30] 11] links the ability to pass linear tests with almost independence. Lemma 1 (Vazirani) Let S n ae f0; 1g n be a sample space that is ffl biased with respect to linear tests of size at most k. Then the sample space S n is ( 1 Gamma 2 Gammak )ffl; k) independent (in ....

U.V. Vazirani, "Randomness, Adversaries and Computation", Ph.D. Thesis, EECS, UC Berkeley, 1986.

....= X x j (x) Gamma (x)j ffl The maximum bias of the exclusive or of certain bit positions in strings chosen according to the distribution ; namely, maxbias( def = max S 6= j (fx : Phi i2S x i = 0g) Gamma (fx : Phi i2S x i = 1g)j The XOR Lemma, commonly attributed to U.V. Vazirani [16] 1 , states that stat( N Delta maxbias( The proof is based on viewing distributions as elements in an N dimensional 1 The special case where the maxbias is zero appears in Chor et al. 5] vector space and observing that the two measures considered by the lemma are merely two norms ....

....to different orthonormal bases. It turns out that stat( p N Delta maxbias( It seems that the previously inferior bound was due to a less careful way of using the same underlying ideas. As motivation to the XOR Lemma, we point out that it has been used in numerous works (e.g. Vazirani [16], Naor and Naor [12] In a typical application, first a upper bound is proved on the maxbias of the constructed distribution and then the XOR Lemma is applied to derive a bound on the statistical difference from the uniform distribution. 1.2 Preliminaries: the XOR Lemma and vector spaces ....

U.V. Vazirani, "Randomness, Adversaries and Computation", Ph.D. Thesis, EECS, UC Berkeley, 1986.

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Vazirani, U., "Randomness, adversaries and computation", Ph.D. thesis, UC Berkeley, 1986.

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