## On Pseudorandomness and Resource-Bounded Measure (1997)

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Venue: | Theoretical Computer Science |

Citations: | 46 - 3 self |

### BibTeX

@ARTICLE{Arvind97onpseudorandomness,

author = {V. Arvind and Johannes Köbler},

title = {On Pseudorandomness and Resource-Bounded Measure},

journal = {Theoretical Computer Science},

year = {1997},

volume = {255},

pages = {235--249}

}

### Years of Citing Articles

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### Abstract

In this paper we extend a key result of Nisan and Wigderson [17] to the nondeterministic setting: for all ff ? 0 we show that if there is a language in E = DTIME(2 O(n) ) that is hard to approximate by nondeterministic circuits of size 2 ffn , then there is a pseudorandom generator that can be used to derandomize BP \Delta NP (in symbols, BP \Delta NP = NP). By applying this extension we are able to answer some open questions in [14] regarding the derandomization of the classes BP \Delta \Sigma P k and BP \Delta \Theta P k under plausible measure theoretic assumptions. As a consequence, if \Theta P 2 does not have p-measure 0, then AM " coAM is low for \Theta P 2 . Thus, in this case, the graph isomorphism problem is low for \Theta P 2 . By using the NisanWigderson design of a pseudorandom generator we unconditionally show the inclusion MA ` ZPP NP and that MA " coMA is low for ZPP NP . 1 Introduction In recent years, following the development of resource-bounded meas...

### Citations

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Relativized circuit complexity
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Citation Context ...Here, the subscript r 2R f0; 1g p(n) means that the probability is taken by choosing r uniformly at random from f0; 1g p(n) . We recall the definition of oracle circuits first introduced by Wilson in =-=[25]-=-. The definition below is essentially from Lutz and Schmidt [16]. An oracle circuit is a directed acyclic graph fl = (V; E), with vertex set V consisting of inputs, standard gates (that compute AND, O... |

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A complexity theory based on boolean algebra
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Citation Context ...(2 ffn )-hard. Hence, by Theorem 20 it follows that L 2 BP \Delta fsBg ` BPP A\PhisB jj = P A\PhisB jj ` C: For example, using the fact that PP is closed under polynomial-time truthtable reducibility =-=[8]-=-, it follows that ifsp (PP) 6= 0, then BP \Delta PP = PP. 14 6 MA is Contained in ZPP NP In this section we apply the Nisan-Wigderson generator to show that MA is contained in ZPP NP and, as a consequ... |

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Citation Context ...ed to derandomize the Arthur-Merlin class AM = BP \Delta NP, provided it is built from a set in E that is hard to approximate by nondeterministic circuits of size 2 ffn for some ff ? 0. Very recently =-=[9]-=-, the result of Nisan and Wigderson has been improved by weakening the assumption that there exists a set A in E that is hard to approximate: it actually suffices that A has worst-case circuit complex... |

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Citation Context ...from P 6= NP. Two prominent examples of such results are: there are Turing-complete sets for NP that are not many-one complete [15], there are NP problems for which search does not reduce to decision =-=[15, 7]-=-. Recently, Lutz [14] has shown that the hypothesissp (NP) 6= 0 (in fact, the possibly weaker hypothesissp (\Delta P k ) 6= 0, ks2) implies that BP \Delta \Delta P k = \Delta P k 1 (in other words, BP... |

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Hitting Sets Derandomize BPP
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Citation Context ...rst-case circuit complexity 2 \Omega\Gamma n) . We leave it as an open question whether a similar improvement is possible for the non-deterministic case. (For related results on derandomizing BPP see =-=[2, 3]-=-.) In Section 4 we apply our extension of the Nisan and Wigderson result to the non-deterministic case to answer some questions left open by Lutz in [14]. We show that for all ks2,sp (\Delta P k ) 6= ... |

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Citation Context ...a quantifier simulation technique is used to show that NP BPP (a subclass of MA) is contained in ZPP NP . We notice that the result MA ` ZPP NP has been shown independently using different techniques =-=[GZ97]-=-. The proof of the next theorem makes use of the fact that there are many n-ary boolean functions that are CIR(2 ffn )-hard (Lemma 12). Theorem 25 MA is contained in ZPP NP . Proof. Let L be a set in ... |

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Citation Context ...rst-case circuit complexity 2 \Omega\Gamma n) . We leave it as an open question whether a similar improvement is possible for the non-deterministic case. (For related results on derandomizing BPP see =-=[2, 3]-=-.) In Section 4 we apply our extension of the Nisan and Wigderson result to the non-deterministic case to answer some questions left open by Lutz in [14]. We show that for all ks2,sp (\Delta P k ) 6= ... |

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Citation Context ...ontained in ZPP NP In this section we apply the Nisan-Wigderson generator to show that MA is contained in ZPP NP and, as a consequence, that MA " coMA is low for ZPP NP . This improves on a resul=-=t of [27]-=- where a quantifier simulation technique is used to show that NP BPP (a subclass of MA) is contained in ZPP NP . The proof of the next theorem also makes use of the fact that there are many n-ary bool... |

19 |
On resource-bounded measure and pseudorandomness
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Citation Context ...nary versions of this paper appeared as Ulmer Informatik-Bericht Nr. 97-05 [AK97a] and in the Proceedings of the 17th Conference on Foundations of Software Technology and Theoretical Computer Science =-=[AK97b]-=-. 1 \Delta P 2 machines). It also follows fromsp (\Delta P 2 ) 6= 0 that if NP ` P/poly then PH = \Delta P 2 . Thus the results of Lutz's paper [Lut97] have opened up a study of derandomization of ran... |

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Citation Context ...now that the test language C(A) = fx j x10 2 jxj 2 Ag is in E A and is NCIR(2 ffn )-hard. Hence, we can assume that A is sparse and therefore we get BP \Delta NP ` NP= log, by using a census argument =-=[10]. Corollary 18 -=-Ifsp (NP " coNP) 6= 0, then BP \Delta NP = NP. Proof. Assuming thatsp (NP " coNP) 6= 0, similar to the proof of Theorem 16 it follows that there is a set A 2 NP " coNP such that NP A = ... |

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Citation Context ...is taken by choosing r uniformly at random from f0; 1g p(n) . We recall the definition of oracle circuits first introduced by Wilson in [25]. The definition below is essentially from Lutz and Schmidt =-=[16]-=-. An oracle circuit is a directed acyclic graph fl = (V; E), with vertex set V consisting of inputs, standard gates (that compute AND, OR, and NOT), a special output gate, and oracle gates. The inputs... |

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Structural Complexity I
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Citation Context ...ean function g defines a finite language fx 2 \Sigma n j g(x) = 1g that we denote by L g . The definitions of complexity classes we consider like P, NP, AM, E, EXP etc. can be found in standard books =-=[6, 5, 18]-=-. By log we denote the function log x = maxf1; dlog 2 xeg and h\Delta; \Deltai denotes a standard pairing function. For a class C of sets and a class F of functions from 1 to \Sigma , let C=F [11] be ... |

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Citation Context ...e unconditionally show the inclusion MA ` ZPP NP and that MA " coMA is low for ZPP NP . 1 Introduction In recent years, following the development of resource-bounded measure theory, pioneered by =-=Lutz [12, 13]-=-, plausible complexity-theoretic assumptions like P 6= NP have been replaced by the possibly stronger, but arguably plausible measuretheoretic assumptionsp (NP) 6= 0. With this assumption as hypothesi... |

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Citation Context ...a \Sigma P k = BP \Delta NP B ` NP B =FP A ` \Sigma P k =FP \Sigma P k "Tally ` \Theta P k+1 " \Sigma P k = log; where the inclusion in \Sigma P k = log follows by a census argument [Kad89] =-=(see also [KT94]-=-). Also, by combining Theorem 16 with Theorem 10 we easily get the following result. Corollary 19 Let D be a complexity class. Thensp (D) 6= 0 implies that for every oracle B 2 E there is a set A in D... |

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Citation Context ...ept else reject else output ? We note that Theorem 24 cannot be further improved to AM ` ZPP NP by relativizing techniques since there is an oracle relative to which AM is not contained in \Sigma P 2 =-=[20]. From the-=- closure properties of MA (namely that MA is closed under conjunctive truth-table reductions) it easily follows that NP MA"coMA ` MA. From Theorem 24 we have MA ` ZPP NP . Hence, NP MA"coMA ... |

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Citation Context ...ircuits). The following lemma shows that g D is also secure against small nondeterministic circuits provided g is hard to approximate by nondeterministic circuits of a certain size. As pointed out in =-=[19]-=-, this appears somewhat counter-intuitive since a nondeterministic circuit c might guess the seed given to the pseudorandom generator g D and then verify that the guess is correct. But note that in ou... |

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oning, Probabilistic complexity classes and lowness
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Citation Context ...e the class of sets A such that there is a set B 2 C and a function h 2 F such that for all x 2 \Sigma , x 2 A , hx; h(1 jxj )i 2 B: The function h is called an advice function for A. The BP-operator =-=[21]-=- assigns to each complexity class C a randomized version 3 BP \Delta C as follows. A set L belongs to BP \Delta C if there exist a polynomial p and a set D 2 C such that for all x, jxj = n x 2 L ) Pro... |

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Citation Context ...y-theoretic conclusions have been derived, which are not known to follow from P 6= NP. Two prominent examples of such results are: there are Turing-complete sets for NP that are not many-one complete =-=[15]-=-, there are NP problems for which search does not reduce to decision [15, 7]. Recently, Lutz [14] has shown that the hypothesissp (NP) 6= 0 (in fact, the possibly weaker hypothesissp (\Delta P k ) 6= ... |

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