## Resource Bounded Randomness and Weakly Complete Problems (1994)

### Cached

### Download Links

- [ftp.fwi.uva.nl]
- [www.mathematik.uni-muenchen.de]
- DBLP

### Other Repositories/Bibliography

Venue: | Theoretical Computer Science |

Citations: | 37 - 6 self |

### BibTeX

@ARTICLE{Ambos-spies94resourcebounded,

author = {Klaus Ambos-spies and Sebastiaan A. Terwijn and Xizhong Zheng},

title = {Resource Bounded Randomness and Weakly Complete Problems},

journal = {Theoretical Computer Science},

year = {1994},

volume = {172},

pages = {369--377}

}

### Years of Citing Articles

### OpenURL

### Abstract

We introduce and study resource bounded random sets based on Lutz's concept of resource bounded measure ([7, 8]). We concentrate on n c - randomness (c 2) which corresponds to the polynomial time bounded (p-) measure of Lutz, and which is adequate for studying the internal and quantitative structure of E = DTIME(2 lin ). However we will also comment on E2 = DTIME(2 pol ) and its corresponding (p2 -) measure. First we show that the class of n c -random sets has p-measure 1. This provides a new, simplified approach to p-measure 1-results. Next we compare randomness with genericity (in the sense of [2, 3]) and we show that n c+1 - random sets are n c -generic, whereas the converse fails. From the former we conclude that n c -random sets are not p-btt-complete for E. Our technical main results describe the distribution of the n c -random sets under p-m-reducibility. We show that every n c -random set in E has n k -random predecessors in E for any k 1, whereas the amou...

### Citations

334 |
The definition of random sequences
- Martin-Lof
- 1966
(Show Context)
Citation Context ...etters c, k, n always denote elements of N . 2 Resource Bounded Measure and Randomness Lutz's resource bounded measure theory is inspired by earlier effectivizations of Lebesgue measure by Martin-Lof =-=[12]-=- and Schnorr [14]. It is based on the concept of a computable martingale. For technical convenience our definition of a martingale slightly differs from the one of Lutz (our martingales are called den... |

173 | Almost everywhere high nonuniform complexity
- Lutz
- 1992
(Show Context)
Citation Context ...zhong Zheng 3 Nanjing University Department of Mathematics Nanjing 210008, P.R. China Abstract We introduce and study resource bounded random sets based on Lutz's concept of resource bounded measure (=-=[7, 8]-=-). We concentrate on n c - randomness (cs2) which corresponds to the polynomial time bounded (p-) measure of Lutz, and which is adequate for studying the internal and quantitative structure of E = DTI... |

152 |
Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische Begründung der Wahrscheinlichkeitstheorie
- Schnorr
(Show Context)
Citation Context ...erties of the random sets not required for the study of the weakly complete sets. In particular we relate randomness to genericity. In Section 2 we introduce the randomness concept. Following Schnorr =-=[14]-=- and Lutz [9] we say that a set A is t(n)-random if A does not belong to any class of t(n)-measure 0. So a t(n)-random set has all properties which occur with t(n)-measure 1. It is easy to show that, ... |

94 | The quantitative structure of exponential time
- Lutz
- 1997
(Show Context)
Citation Context ...olynomial time bounded (p-) measure for the study of the class E = DTIME(2 linear ), of exponential time computable sets, and he and others already obtained interesting results along these lines (see =-=[9]-=- for a survey). Juedes and Lutz [5] used this new measure approach to prove new and reprove old results on the strong intractability of p-m-complete sets for E, like the result of Orponen and Schoning... |

70 | Equivalence of measures of complexity classes
- Breutzmann, Lutz
- 1999
(Show Context)
Citation Context ...zhong Zheng 3 Nanjing University Department of Mathematics Nanjing 210008, P.R. China Abstract We introduce and study resource bounded random sets based on Lutz's concept of resource bounded measure (=-=[7, 8]-=-). We concentrate on n c - randomness (cs2) which corresponds to the polynomial time bounded (p-) measure of Lutz, and which is adequate for studying the internal and quantitative structure of E = DTI... |

47 | The complexity and distribution of hard problems - Juedes, Lutz - 1995 |

43 | Measure, stochasticity, and the density of hard languages
- Lutz, Mayordomo
- 1994
(Show Context)
Citation Context ...n ) = 1: 2 Note that many more much stronger properties than the above can be proven (such as the various stochastic properties from probability theory, or such as the Weak Stochasticity Theorem from =-=[11]-=-), but we will not need these in the sequel. 3 Resource Bounded Genericity and Randomness Ambos-Spies, Fleischhack, and Huwig [2, 3] introduced different types of resource bounded genericity. Here we ... |

24 |
Diagonalizations over polynomial time computable sets
- Ambos-Spies, Fleischhack, et al.
- 1987
(Show Context)
Citation Context ...measure. First we show that the class of n c -random sets has p-measure 1. This provides a new, simplified approach to p-measure 1-results. Next we compare randomness with genericity (in the sense of =-=[2, 3]-=-) and we show that n c+1 - random sets are n c -generic, whereas the converse fails. From the former we conclude that n c -random sets are not p-btt-complete for E. Our technical main results describe... |

20 |
The density and complexity of polynomial cores for intractable sets
- Orponen, Schoning
- 1986
(Show Context)
Citation Context ...for a survey). Juedes and Lutz [5] used this new measure approach to prove new and reprove old results on the strong intractability of p-m-complete sets for E, like the result of Orponen and Schoning =-=[13]-=- that any p-m-complete set A for E has a dense polynomial complexity core. As Lutz observed, the measure approach does not require p-m-completeness (or hardness) but only a weaker property of the comp... |

19 |
Diagonalizing over deterministic polynomial time
- Ambos-Spies, Fleischhack, et al.
- 1988
(Show Context)
Citation Context ...measure. First we show that the class of n c -random sets has p-measure 1. This provides a new, simplified approach to p-measure 1-results. Next we compare randomness with genericity (in the sense of =-=[2, 3]-=-) and we show that n c+1 - random sets are n c -generic, whereas the converse fails. From the former we conclude that n c -random sets are not p-btt-complete for E. Our technical main results describe... |

14 | Weakly hard problems
- Lutz
- 1993
(Show Context)
Citation Context ... that every n c -random set in E has n k -random predecessors in E for any ks1, whereas the amount of randomness of the successors is bounded. We apply this result to answer a question raised by Lutz =-=[10]-=-: We show that the class of weakly complete sets has measure 1 in E and that there are weakly complete problems which are not p-btt-complete for E. 1 This research was done while the second and third ... |

11 |
Terwijn, Genericity and measure for exponential time
- Ambos-Spies, Neis, et al.
- 1994
(Show Context)
Citation Context ...late randomness to the resource bounded genericity concepts introduced by Ambos-Spies, Fleischhack, and Huwig in [2, 3]. These genericity concepts were recently used by Ambos-Spies, Neis, and Terwijn =-=[4]-=- 2 to investigate the p-measure on E. In particular they observed that the class of n c -generic sets has p-measure 1, so that the properties shared by all generic sets occur with p-measure 1. By stud... |

2 |
Resource-bounded genericity, to appear in Computability, Enumerability and Unsolvability
- Ambos-Spies
(Show Context)
Citation Context ...y t(n) (measured in the length n of the previously built part A x of A) so that these subrequirements may be described by t(n)-bounded conditions. For a more detailed discussion of these concepts see =-=[1, 3]-=-. The proof of the next theorem is essentially the same as the proof in [4] showing that the n c -generic sets have p-measure 1. Theorem 3.1 Let A be n c+1 -random. Then A is n c -generic. Hence any p... |