## One-sided Versus Two-sided Randomness (1998)

Venue: | In Proceedings of the 16th Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science |

Citations: | 11 - 1 self |

### BibTeX

@INPROCEEDINGS{Buhrman98one-sidedversus,

author = {Harry Buhrman and Lance Fortnow},

title = {One-sided Versus Two-sided Randomness},

booktitle = {In Proceedings of the 16th Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science},

year = {1998},

pages = {100--109},

publisher = {Springer}

}

### OpenURL

### Abstract

We demonstrate how to use Lautemann's proof that BPP is in \Sigma p 2 to exhibit that BPP is in RP PromiseRP . Immediate consequences show that if PromiseRP is easy or if there exist quick hitting set generators then P = BPP. Our proof vastly simplifies the proofs of the later result due to Andreev, Clementi and Rolim and Andreev, Clementi, Rolim and Trevisan. Clementi, Rolimand Trevisan question whether the promise is necessary for the above results, i.e. whether BPP ` RP RP for instance. We give a relativized world where P = RP 6= BPP and thus the promise is indeed needed. 1 Introduction Andreev, Clementi and Rolim [ACR98] show how given access to a quick hitting set generator, one can approximate the size of easily describable sets. As an immediate consequence one gets that if quick hitting set generators exist then P = BPP. Andreev, Clementi, Rolim and Trevisan [ACRT97] simplify the proof and apply the result to simulating BPP with weak random sources. Much earlier, Lautema...

### Citations

145 | A complexity theoretic approach to randomness, in
- Sipser
- 1983
(Show Context)
Citation Context ...] simplify the proof and apply the result to simulating BPP with weak random sources. Much earlier, Lautemann [Lau83] gave a proof that BPP ` \Sigma p 2 = NP NP , simplifying work of G'acs and Sipser =-=[Sip83]-=-. Lautemann's proof uses two simple applications of the probabilistic method to get the existence results needed. As often with the case of the probabilistic method, the proof actually shows that the ... |

84 |
CREW PRAMs and decision trees
- Nisan
- 1991
(Show Context)
Citation Context ...how that if M is categorical then there is a polynomial time (relative to QBF) algorithm that computes for all x whether M(x) accepts or rejects. The core of this proof will be an argument from Nisan =-=[Nis91]-=-. The proof of Theorem 4.1 follows from Lemmas 4.5 and 4.7. Lemma 4.5 says that if we have a machine M(x) that is categorically R and we only consider oracles A such that at most 1=6 or at least 5=6 o... |

81 |
BPP and the polynomial hierarchy
- Lautemann
- 1983
(Show Context)
Citation Context ...k hitting set generators exist then P = BPP. Andreev, Clementi, Rolim and Trevisan [ACRT97] simplify the proof and apply the result to simulating BPP with weak random sources. Much earlier, Lautemann =-=[Lau83]-=- gave a proof that BPP ` \Sigma p 2 = NP NP , simplifying work of G'acs and Sipser [Sip83]. Lautemann's proof uses two simple applications of the probabilistic method to get the existence results need... |

75 |
AND RUSSELL IMPAGLIAZZO: Generic oracles and oracle classes
- BLUM
- 1987
(Show Context)
Citation Context ...ast possibility is that M QBF\PhiG happens to be an R machine but it is not categoric. This however can not happen since the genericity of G will diagonalize against such non-categoric machines. (See =-=[BI87]-=-) 2 Theorem 4.1 combined with Theorem 3.1 gives a relativized world where PromiseRP is not easy but P = RP. This corollary also follows from work of Impagliazzo and Naor [IN88]. Heller [Hel86] exhibit... |

47 | An oracle builder's toolkit
- Fenner, Fortnow, et al.
- 1993
(Show Context)
Citation Context ...tivized world where P = RP 6= BPP. Define the following function tower(0) = 2, tower(n+1) = 2 tower(n) , i.e. tower(n) is an exponential tower of n + 1 2's. We will use a special type of generic (see =-=[FFKL93]-=- for an overview) to prove the theorem. Definition 4.2 A BPP-generic oracle G is a type of generic oracle that is only defined at length n such that n = tower(m) for some m. Moreover at these lengths ... |

41 | Weak random sources, hitting sets, and BPP simulations
- Andreev, Clementi, et al.
- 1999
(Show Context)
Citation Context ...generator, one can approximate the size of easily describable sets. As an immediate consequence one gets that if quick hitting set generators exist then P = BPP. Andreev, Clementi, Rolim and Trevisan =-=[ACRT97]-=- simplify the proof and apply the result to simulating BPP with weak random sources. Much earlier, Lautemann [Lau83] gave a proof that BPP ` \Sigma p 2 = NP NP , simplifying work of G'acs and Sipser [... |

37 |
A new general derandomization method
- Andreev, Clementi, et al.
- 1998
(Show Context)
Citation Context ...cessary for the above results, i.e. whether BPP ` RP RP for instance. We give a relativized world where P = RP 6= BPP and thus the promise is indeed needed. 1 Introduction Andreev, Clementi and Rolim =-=[ACR98]-=- show how given access to a quick hitting set generator, one can approximate the size of easily describable sets. As an immediate consequence one gets that if quick hitting set generators exist then P... |

33 |
Decision trees downward closure
- Impagliazzo, Naor
- 1988
(Show Context)
Citation Context ...on to Definition 2.3. In particular we have PromiseRP is easy implies P = RP. The converse is not so simply provable, relativizable counterexamples easily follow from known results on generic oracles =-=[IN88]-=-. The oracle we develop in Section 4 also gives a relativizable counterexample. Definition 2.7 For any relativizable complexity class C, we say L is in C PromiseRP if there is a probabilistic polynomi... |

31 |
On relativized exponential and probabilistic complexity classes, Information and Control 71
- Heller
- 1986
(Show Context)
Citation Context ...es. (See [BI87]) 2 Theorem 4.1 combined with Theorem 3.1 gives a relativized world where PromiseRP is not easy but P = RP. This corollary also follows from work of Impagliazzo and Naor [IN88]. Heller =-=[Hel86]-=- exhibits a relativized world where BPP = NEXP. One might suspect that the techniques of Heller and those used in the proof of Theorem 4.1 may lead to an oracle A where P A = RP A and BPP A = NEXP A .... |

19 |
Probabilistic quantifiers and games
- Zachos
- 1988
(Show Context)
Citation Context ...proof of Theorem 4.1 may lead to an oracle A where P A = RP A and BPP A = NEXP A . We show this cannot happen. Theorem 4.8 In all relativized worlds, if P = RP and NP ` BPP then P = BPP. Proof Zachos =-=[Zac88]-=- shows that if NP ` BPP then NP = RP. We then have P = NP = \Sigma p 2 and thus P = BPP. These arguments all relativize. 2 Acknowledgments We thank Noam Nisan for suggesting this problem and Luca Trev... |

5 |
Recent advances towards proving P
- Clementi, Rolim, et al.
- 1998
(Show Context)
Citation Context ...rant CCR 92-53582. for all inputs. In PromiseRP we only need to solve instances where the machine rejects or accepts with probability at least one-half. A survey paper by Clementi, Rolim and Trevisan =-=[CRT98]-=- asks whether we can remove the promise in our result, i.e., whether BPP ` RP RP . We give a relativized counterexample to this conjecture by exhibiting and oracle A such that P A = RP A but P A 6= BP... |