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8.4. SPACE-BOUNDED DISTINGUISHERS use .net 3 of 9 barcode implement togenerate code 39 on .net Microsoft Office Word Website Construction 8.17 pr .NET 3 of 9 ovides a black-box procedure for approximating the underlying predicate when given oracle access to a distinguisher (and this procedure is valid also in case the distinguisher is a non-deterministic machine).

Thus, under suitably strong (and yet plausible) assumptions, constant-round interactive proofs collapse to N P. We note that a stronger result, which deviates from the foregoing framework, has been subsequently obtained (cf. [167]).

Construction of randomness extractors. An even more radical instantiation of Construction 8.17 was used to obtain explicit constructions of randomness extractors (see Appendix D.

4). In this case, the predicate f is viewed as (an error correcting encoding of) a somewhat random function, and the construction makes sense because it refers to f in a black-box manner. In the analysis we rely on the fact that f can be approximated by combining relatively little information (regarding f ) with (black-box access to) a distinguisher for G f .

For further details, see D.4.2.

2.. 8.3.3.

2. Re ections Regarding Derandomization Part 1 of Theorem 8.19 is often summarized by saying that (under some reasonable assumptions) randomness is useless.

We believe that this interpretation is wrong even within the restricted context of traditional complexity classes, and is bluntly wrong if taken outside of the latter context. Let us elaborate. Taking a closer look at the proof of Theorem 8.

16 (which underlies Theorem 8.19), we note that a randomized algorithm A of time complexity t is emulated by a deterministic algorithm A of time complexity t = poly(t). Further noting that A = A G invokes A (as well as the canonical derandomizer G) for (t) times (because (k) = O(2k ) implies 2k = (t)), we infer that t = (t 2 ) must hold.

Thus, derandomization via (Part 1 of) Theorem 8.19 is not really for free. More importantly, we note that derandomization is not possible in various distributed settings, when both parties may protect their con icting interests by employing randomization.

Notable examples include most cryptographic primitives (e.g., encryption) as well as most types of probabilistic proof systems (e.

g., PCP). For further discussion, see 9 and Appendix C.

Additional settings where randomness makes a difference (either between impossibility and possibility or between formidable and affordable cost) include distributed computing (see [17]), communication complexity (see [148]), parallel architectures (see [151]), sampling (see Appendix D.3), and property testing (see Section 10.1.

2).. 8.4. Space-Bounded Distinguishers In the previous two sections we have considered generators that output sequences that look random to any ef cient procedures, where the latter were modeled by time-bounded computations. Speci cally, in Section 8.2 we considered indistinguishability by polynomialtime procedures.

A ner classi cation of time-bounded procedures is obtained by considering their space complexity, that is, restricting the space complexity of time-bounded computations. This restriction, which is the focus of 5, leads to the notion of pseudorandom generators that fool space-bounded distinguishers. Interestingly, in contrast to the notions of pseudorandom generators that were considered in Sections 8.

2 and 8.3, the existence of pseudorandom generators that fool space-bounded distinguishers can be established without relying on computational assumptions..

PSEUDORANDOM GENERATORS Prerequisites. Techn ically speaking, the current section is self-contained, but various definitional choices are justi ed by reference to 6.1.

5.1. Thus, we recommend Section 6.

1.5 as general background for the current section..

8.4.1. De nitional Issues Our main motivation for considering space-bounded distinguishers is to develop a notion of pseudorandomness that is adeqaute for space-bounded randomized algorithms. That is, such algorithms should essentially maintain their behavior when their source of internal coin tosses is replaced by a source of pseudorandom bits (which may be generated based on a much shorter random seed). We thus start by recalling and reviewing the natural notion of space-bounded randomized algorithms.

Unfortunately, natural notions of space-bounded computations are quite subtle, especially when non-determinism or randomization is concerned (see Sections 5.3 and 6.1.

5, respectively). Two major de nitional issues regarding randomized space-bounded computations are the need for imposing explicit time bounds and the type of access to the random-tape. 1.

Time bounds: The question is whether or not the space-bounded machines are restricted to time complexity that is at most exponential in their space complexity.30 Recall that such an upper bound follows automatically in the deterministic case (Theorem 5.3), and can be assumed without loss of generality in the non-deterministic case (see Section 5.

3.2), but it does not necessarily hold in the randomized case (see 6.1.

5.1). Furthermore, failing to restrict the time complexity of randomized spacebounded machines makes them unnatural and unintentionally too strong (see 6.

1.5.1 again).

As in Section 6.1.5, seeking a natural model of randomized space-bounded algorithms, we postulate that their time complexity must be at most exponential in their space complexity.

2. Access to the random-tape: Recall that randomized algorithms may be modeled as machines that are provided with the necessary randomness via a special randomtape. The question is whether the space-bounded machine has uni-directional or bi-directional (i.

e., unrestricted) access to its random-tape. (Allowing bi-directional access means that the randomness is recorded for free, that is, without being accounted for in the space bound (see discussions in Sections 5.

3 and 6.1.5).

) Recall that uni-directional access to the random-tape corresponds to the natural model of an on-line randomized machine, which determines its moves based on its internal coin tosses (and thus cannot store its past coin tosses for free ). Thus, as in Section 6.1.

5, we consider uni-directional access.31 Hence, we focus on randomized space-bounded computations that have time complexity that is at most exponential in their space complexity and access their random-tape in a uni-directional manner..

Alternatively, one c VS .NET 3 of 9 an ask whether these machines must always halt or only halt with probability approaching 1. It can be shown that the only way to ensure absolute halting is to have time complexity that is at most exponential in the space complexity.

(In the current discussion as well as throughout this section, we assume that the space complexity is at least logarithmic.) 31 We note that the fact that we restrict our attention to uni-directional access is instrumental in obtaining spacerobust generators without making intractability assumptions. Analogous generators for bi-directional space-bounded computations would imply hardness results of a breakthrough nature in the area.

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