## Probabilistic Techniques In Structural Complexity Theory (1996)

Venue: | 19 |

Citations: | 1 - 0 self |

### BibTeX

@TECHREPORT{Sivakumar96probabilistictechniques,

author = {D. Sivakumar},

title = {Probabilistic Techniques In Structural Complexity Theory},

institution = {19},

year = {1996}

}

### OpenURL

### Abstract

viii 1 Introduction 1 1.1 This Dissertation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.2 Preliminaries : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.3 A Brief Tour of some Gems in Complexity Theory : : : : : : : : : : : : : : : : 7 2 Sparse Hard Sets 10 2.1 Background and Motivation : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.1.1 The Sparse Set Connection : : : : : : : : : : : : : : : : : : : : : : : : : 11 2.1.2 The Isomorphism Conjecture for NP : : : : : : : : : : : : : : : : : : : : 12 2.1.3 The Conjectures of Hartmanis for P and NL : : : : : : : : : : : : : : : : 13 2.1.4 The Connection to Parallel Complexity : : : : : : : : : : : : : : : : : : 14 2.1.5 The Circuit-Value Problem : : : : : : : : : : : : : : : : : : : : : : : : : 15 2.1.6 Overview of the Proof Technique : : : : : : : : : : : : : : : : : : : : : 15 2.2 The First Stage: an RNC 2 Simulation : : : : : : : : : : : : : : : : : : : : : : : 16 2.2....

### Citations

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- 2000
(Show Context)
Citation Context ...t in applications of resource-bounded measure theory, for it says that if we provesD(D) (C ) = 0, then we would have minimally proved that C 6= D. This is also an instance of the probabilistic method =-=[AS92a]-=- where, in order to prove that there is a language in D that is not in C , we demonstrate that almost every language in D (i.e. a measure one subset of D) is not in C . To substantiate Condition (1) a... |

267 | Simple constructions of almost k-wise independent random variables
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Citation Context ...ard for P under logspace or NC 2 many-one reductions, then P ` RNC 2 . 2.3 Stage Two: An NC 2 Solution via Derandomization In this section, we use a small sample space construction due to Alon et al. =-=[AGHP92]-=-, and generalize their result concerning the construction. We apply the generalization to derandomize the probabilistic simulation of Section 2.2. Under the hypothesis about sparse hard sets, this yie... |

220 |
A fast and simple randomized parallel algorithm for the maximal independent set problem
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- 1986
(Show Context)
Citation Context ...s that of pairwise independence. Returning to our large-cut example, simple and elegant constructions of sample spaces of size O(n) are known for generating n pairwise independent0-1 random variables =-=[ABI86]-=-, in stark contrast to the sample space f0;1g n of size 2 n . Historically, research on pseudorandom generators has primarily been from a cryptographic viewpoint, and their usefulness is fairly well-u... |

100 |
Deterministic simulation in LOGSPACE
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Citation Context ...considerable savings from the obvious O(r 2 ) random bits. Expander graphs have been studied extensively, and have proved to be a very valuable tool in computation theory. Ajtai, Komlos and Szemeredi =-=[AKS87]-=- showed how to (partially) derandomize space-bounded randomized algorithms by using expanders; Cohen and Wigderson [CW89] and Impagliazzo and Zuckerman [IZ89] showed how to amplify the success probabi... |

47 | Measure on small complexity classes with applications for BPP
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(Show Context)
Citation Context ...guages in NP that require deterministic time more than 2 n c ; there would be NP-complete sets under Turing (Cook) reductions that are not complete under many-one (Karp) reductions [LM94]; BPP ` P NP =-=[AS94a]-=-; the NP-complete satisfiability problem would, in fact, be hard for a measure one subset of EXP [JL93]. Thus, Lutz [Lut93] argues, we should study the hypothesis (NPjEXP) 6= 0 for its explanatory pow... |

40 | Probabilistic checking of proofs - Arora, Safra - 1992 |

23 |
Some consequences of the existence of pseudorandom generators
- Allender
- 1989
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Citation Context ...." Hypotheses about the existence of pseudorandom generators of considerable hardness have been considered before and some exciting results have been shown. For example, Yao [Yao82] (see also [BH=-=89], [All89]-=-) has shown that if generators of hardness 2 n e exist, then randomized polynomial-time algorithms can be simulated in deterministic quasipolynomial time. The equivalence of our hypothesis about stron... |

14 | Measure on P: Robustness of the notion
- Allender, Strauss
- 1995
(Show Context)
Citation Context ...ieve anything new. However, given the restriction on dependency sets, it appears that this relaxation might be useful in proving more measure zero results, and this is indeed so. Allender and Strauss =-=[AS95]-=- show that one can construct martingales that take advantage of this relaxation, if, in addition, we redefine success as achieving a lim sup of + rather than achieving a limit of infinity. This is ver... |

6 |
codes, PCP’s and inapproximability
- Reductions
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Citation Context ... with high probability, if indeed j(~a) = TRUE. This result has found numerous applications in proving the limitations on finding approximate solutions to hard combinatorial problems. Recently, Arora =-=[Aro95]-=- has shown the limitations of this approach in proving inapproximability; not coincidentally, his work is based on results about quasirandom constructions! Chapter 2 Sparse Hard Sets 2.1 Background an... |

1 | Toward a measure for P
- Allender, Strauss
- 1994
(Show Context)
Citation Context ...d. Then the length of the recursive chain of calls to compute k(w) will be jwjs2 jxj , and cannot be done in time polynomial in jxj, which is needed to place L in P. We note that Allender and Strauss =-=[AS94b]-=- were the first to consider the idea of defining a measure on P. Allender and Strauss [AS94a] realized that measure conservation was the stumbling block, and found a way to circumvent the problem outl... |