E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 807--818, 1994. 10  Home/Search   Document Details and Download   Summary   Related Articles   Check

....measure in E ( 11] in fact measure 1 in E ( 6] Finally, the question whether the last two inclusions between the second and the third column are proper is still open. It has been shown, however, that these questions cannot be resolved by relativizable techniques: namely, Allender and Strauss [1] have shown that, relative to some oracle, all n 2 random sets are p ttcomplete whereas Ambos Spies, Lempp, and Mainhardt [2] and, independently, Buhrman et al. 9] have given oracles relative to which no n 2 random set is p Tcomplete for E. This also shows that the measure in E of the ....

E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In: Proceedings of the 35th Annual IEEE Symposium an Foundations of Computer Science, p. 867818, IEEE Computer Society Press, 1994.

....measure in E ( 11] in fact measure 1 in E ( 6] Finally, the question whether the last two inclusions between the second and the third column are proper is still open. It has been shown, however, that these questions cannot be resolved by relativizable techniques: namely, Allender and Strauss [1] have shown that, relative to some oracle, all n 2 random sets are p ttcomplete whereas Ambos Spies, Lempp, and Mainhardt [2] and, independently, Buhrman et al. 9] have given oracles relative to which no n 2 random set is p Tcomplete for E. This also shows that the measure in E of the ....

E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In: Proceedings of the 35th Annual IEEE Symposium an Foundations of Computer Science, p. 867--818, IEEE Computer Society Press, 1994.

....p (X) 0) if there is a p computable supermartingale d such that X S 1 [d] In the context of resource bounded measure, it is interesting to ask for the measure of the class of all sets A for which E A is not CIR A (2 ffn ) hard. Building on initial results in [Lut93] it is shown in [AS94] that this class has p measure 0. Lemma 4 [AS94] For all 0 ff 1=3, p fA j E A is not CIR A (2 ffn ) hardg = 0. Lutz strengthened this to the following result that is more useful for some applications. Lemma 5 [Lut97] For all 0 ff 1=3 and all oracles B 2 E, p fA j E A is not ....

....d such that X S 1 [d] In the context of resource bounded measure, it is interesting to ask for the measure of the class of all sets A for which E A is not CIR A (2 ffn ) hard. Building on initial results in [Lut93] it is shown in [AS94] that this class has p measure 0. Lemma 4 [AS94] For all 0 ff 1=3, p fA j E A is not CIR A (2 ffn ) hardg = 0. Lutz strengthened this to the following result that is more useful for some applications. Lemma 5 [Lut97] For all 0 ff 1=3 and all oracles B 2 E, p fA j E A is not CIR A PhiB (2 ffn ) hardg = 0: As a consequence ....

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E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proc. 35th IEEE Symposium on the Foundations of Computer Science, 807--818. IEEE Computer Society Press, 1994.

....n Gamma1) not for DTIME(t(n) This fact is also responsible for problems in defining randomness for subexponential complexity classes. Here we will not discuss this problem, which has not been completely solved yet. For partial solutions we refer the reader to Allender and Strauss ([AS94], AS95] and Mayordomo [Ma94c] We conclude this section by introducing some basic notation. Let Sigma = f0; 1g denote the set of finite binary strings. For a string x, x(m) denotes the (m 1)th bit in x, i.e. x = x(0) x(n Gamma 1) where n = jxj is the length of x. is the empty ....

....ff ) hard set for NP is p selective (ff 2 (0; 1) see [Og95] Largeness Hypotheses for other polynomial complexity classes have been considered too. e.g. the result of Lautemann [La83] and Sipser and Gacs [Si83] that BPP Sigma p 2 Pi p 2 has been modified by Allender and Strauss [AS94] as follows: If p ( Delta p 2 ) 6= 0 then BPP Delta p 2 . For a survey of other recent results in this direction, see Lutz [Lu96] 10 Further Results Due to lack of space our survey on resource bounded measure and randomness and their applications in complexity theory cannot be ....

E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science, 807-818, IEEE Computer Society Press, 1994.

....0 ff 1 3 , the (nonrelativized) class H ff has pspace measure 1, so if E H ff = then E has measure 0 in ESPACE. Lutz [19] showed that, for 0 ff 1 3 , the set of all A satisfying E A H A ff = has pspace measure 0. This result was recently improved by Allender and Strauss [1], who proved that, for 0 ff 1 3 , the set of all A satisfying E A H A ff = has p measure 0. The following lemma is a small, but useful, extension of this fact. Lemma 3.2. For all 0 ff 1 3 and all S 2 E, p in A fi fi fi E A H A PhiS ff = oj = 0: Proof. For ....

....; has p measure 0. The following lemma is a small, but useful, extension of this fact. Lemma 3.2. For all 0 ff 1 3 and all S 2 E, p in A fi fi fi E A H A PhiS ff = oj = 0: Proof. For brevity, the notation and calculations of [19] are followed, while using the test language of [1]. Let 0 ff 1 3 and S 2 E. Without loss of generality, assume that ff is rational. For each A f0; 1g , define the test language C(A) n x fi fi fi pad(x) 2 A o ; where pad(x) x10 2 jxj : Let X = n A fi fi fi C(A) 62 H A PhiS ff o : Since C(A) 2 E A for all A ....

[Article contains additional citation context not shown here]

E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 807--818. IEEE Computer Society Press, 1994.

.... p (X) 0) if there is a p computable supermartingale d such that X S 1 [d] In the context of resource bounded measure, it is interesting to ask for the measure of the class of all sets A for which E A is not CIR A (2 ffn ) hard. Building on initial results in [13] it is shown in [1] that this class has p measure 0. Lemma 4 [1] For all 0 ff 1=3, p fA j E A is not CIR A (2 ffn ) hardg = 0. Lutz strengthened this to the following result that is more useful for some applications. Lemma 5 [14] For all 0 ff 1=3 and all oracles B 2 E, p fA j E A is not CIR ....

....d such that X S 1 [d] In the context of resource bounded measure, it is interesting to ask for the measure of the class of all sets A for which E A is not CIR A (2 ffn ) hard. Building on initial results in [13] it is shown in [1] that this class has p measure 0. Lemma 4 [1] For all 0 ff 1=3, p fA j E A is not CIR A (2 ffn ) hardg = 0. Lutz strengthened this to the following result that is more useful for some applications. Lemma 5 [14] For all 0 ff 1=3 and all oracles B 2 E, p fA j E A is not CIR A PhiB (2 ffn ) hardg = 0: As a consequence ....

[Article contains additional citation context not shown here]

E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proc. 35th IEEE Symposium on the Foundations of Computer Science, 807--818. IEEE Computer Society Press, 1994.

....time class ( 2 1) not for ( This fact is also responsible for problems in defining randomness for subexponential complexity classes. Here we will not discuss this problem, which has not been completely solved yet. For partial solutions we refer the reader to Allender and Strauss ([AS94], AS95] and Mayordomo [Ma94c] We conclude this section by introducing some basic notation. Let 6 = 0 1 denote the set of finite binary strings. For a string , denotes the ( 1)th bit in , i.e. 0) 1) where = is the length of . is the empty string. denotes that the string is extended ....

E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In , 807-818, IEEE Computer Society Press, 1994.

....2 ) 6= 0 follows from, and is thus at least as plausible as, the hypothesis p (NP) 6= 0. The first consequence of p ( Delta P 2 ) 6= 0 is a tightening of the result, due to Lautemann [Lau83] and Sipser and G acs [Sip83] that BPP Sigma P 2 Pi P 2 . Theorem 12.11 (Allender and Strauss [AS94]) If p ( Delta P 2 ) 6= 0, then BPP Delta P 2 . A slight strengthening of the proof of Theorem 12.11 yields the following. Theorem 12.12 (Lutz [Luta] If p ( Delta P 2 ) 6= 0, then for all k 1, BPP( Sigma P k ) Delta P k 1 . Theorem 12.12 has consequences for lowness and ....

....between genericity and measure, but more investigation is needed to fully understand the relative power of these two methods. 8. One of the most challenging tasks remaining is the development of measure in subexponential complexity classes. Mayordomo [May94c, May94b] and Allender and Strauss [AS94] have proposed (inequivalent) definitions of measure in PSPACE, and Allender and Strauss [AS94, AS95] have investigated various formulations of measure in P and other subexponential classes, but at the time of this writing, many issues are unresolved. Resource bounded measure is a powerful ....

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E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 807-- 818. IEEE Computer Society Press, 1994.

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E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 807--818. IEEE Computer Society, 1994.

....First and foremost, many recent investigations in complexity theory focus on the resource bounded measure of the upper P r span P Gamma1 r (A) fBjA P r Bg of a language A. Such investigations include work on small span theorems [9, 14, 4, 11, 7] and work on the BPP versus E question [1, 7, 8]. In general, the upper P r span of a language is closed upwards, but not downwards, under P r reductions. Our second reason for interest in upward closure conditions is that the above mentioned results of Breutzmann and Lutz [5] do not fully establish the invariance of measures of ....

....an induced probability measure (dual to the martingale dilation technique introduced in [5] and a new, improved positive bias reduction of one bias sequence to another. We also note three easy consequences of our Bias Invariance Theorem. First, in combination with work of Allender and Strauss [1] and Buhrman, van Melkebeek, Regan, Sivakumar, and Strauss [8] it implies that, if there is any strongly positive P sequence of biases fi such 3 that the complete P T degree for E 2 does not have fi measure 1 in E 2 , then E 6 BPP. Second, in combination with the work of Regan, ....

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E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 807--818, Piscataway, NJ, 1994. IEEE Computer Society Press.

.... (improving the result by Hartmanis and Yesha [35] that P #= BPP implies E #= ESPACE) and that the # P T hard problems for BPP form a set of pspace measure 1 [57] improving the result by Bennett and Gill [14] that this set has classical measure 1) More significantly, Allender and Strauss [1] improved this latter result by showing that the # P T hard problems for BPP form a set of p measure 1. Thus almost every problem in E is # P T hard for BPP (and similarly for E 2 ) It follows easily that, if the # P T complete degree does not have measure 1 in E (or does not have ....

....for E. 3. There is a decision problem that is weakly # P T complete, but not # P T complete, for E 2 . 4. The # P T complete degree does not have measure 1 in E 2 . 5. BPP #= E 2 . Using results of Juedes and Lutz [42] Ambos Spies, Terwijn, and Zheng [5] and Allender and Strauss [1], it is easy to see that 1 # 2 # 3 # 4 # 5 What else can be proven about the relative strengths of these five conditions Recent, more sophisticated pseudorandom generator constructions appear to be verging on a complete derandomization of BPP. Impagliazzo and Wigderson [38] improving ....

[Article contains additional citation context not shown here]

E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 807--818, Piscataway, NJ, 1994. IEEE Computer Society Press.

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E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 807--818, 1994. 10

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E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 807--818. IEEE Computer Society Press, 1994.

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E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proc. 35th IEEE Symposium on the Foundations of Computer Science, 807-818. IEEE Computer Society Press, 1994.

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E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 807--818, 1994.

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E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 807-818, 1994.

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E. Allender and M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Annual IEEE Foundations of Computer Science Conference, pages 807--818, 1994.

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