## Almost Every Set in Exponential Time is P-Bi-Immune (1994)

Venue: | Theoretical Computer Science |

Citations: | 53 - 5 self |

### BibTeX

@ARTICLE{Mayordomo94almostevery,

author = {Elvira Mayordomo},

title = {Almost Every Set in Exponential Time is P-Bi-Immune},

journal = {Theoretical Computer Science},

year = {1994},

volume = {136},

pages = {392--400}

}

### Years of Citing Articles

### OpenURL

### Abstract

. A set A is P-bi-immune if neither A nor its complement has an infinite subset in P. We investigate here the abundance of P-bi-immune languages in linearexponential time (E). We prove that the class of P-bi-immune sets has measure 1 in E. This implies that `almost' every language in E is P-bi-immune, that is to say, almost every set recognizable in linear exponential time has no algorithm that recognizes it and works in polynomial time on an infinite number of instances. A bit further, we show that every p-random (pseudo-random) language is E-bi-immune. Regarding the existence of P-bi-immune sets in NP, we show that if NP does not have measure 0 in E, then NP contains a P-bi-immune set. Another consequence is that the class of p m -complete languages for E has measure 0 in E. In contrast, it is shown that in E, and even in REC, the class of P-bi-immune languages lacks the property of Baire (the Baire category analogue of Lebesgue measurability). * This work was supported by a Spani...

### Citations

173 | Almost everywhere high nonuniform complexity
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- 1992
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162 |
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Citation Context ...sive function theory. Flajolet and Steyaert transformed it into the complexity theoretic setting in [6] and [7]. Hartmanis and Berman show that E (linear exponential time) contains a P-bi-immune set (=-=[10]-=-, observed in [12]), and an application of [9] yields that for all c ? 0 there exists a DTIME(2 cn )-bi-immune set in E. P-bi-immunity is also studied in detail by Balc'azar and Schoning in [2], where... |

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50 |
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- 1979
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24 |
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- 1981
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23 | On reducibility to complex or sparse sets
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- 1975
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20 |
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- 1976
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Citation Context ...nsformed it into the complexity theoretic setting in [6] and [7]. Hartmanis and Berman show that E (linear exponential time) contains a P-bi-immune set ([10], observed in [12]), and an application of =-=[9]-=- yields that for all c ? 0 there exists a DTIME(2 cn )-bi-immune set in E. P-bi-immunity is also studied in detail by Balc'azar and Schoning in [2], where several characterizations are presented; for ... |

19 |
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Citation Context ...certain two-person combinatorial game is intractable because it issp m -complete for E. We want to know whether completeness is a typical property in E. We studysp m -completeness, that from [11] and =-=[4]-=- is exactly the same assp 1\Gammatt -completeness. Corollary 23. The class ofsp m -complete languages for E has measure 0 in E. The class ofsp m -complete languages for NE has measure 0 in E 2 . Proof... |

16 |
On the structure of complete sets: Almost everywhere complexity and infinitely often speedup
- Berman
- 1976
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Citation Context ...same assp 1\Gammatt -completeness. Corollary 23. The class ofsp m -complete languages for E has measure 0 in E. The class ofsp m -complete languages for NE has measure 0 in E 2 . Proof . As proven in =-=[3]-=-, nosp m -complete set for E is P-bi-immune, so the class ofsp m - complete sets is included in a measure 0 in E class by Theorem 19, and from Lemma 18 it has measure 0 in E. The second part is analog... |

7 |
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Citation Context ...et of A or its complement is in C. The notion of immunity was first introduced by Post [18] in recursive function theory. Flajolet and Steyaert transformed it into the complexity theoretic setting in =-=[6]-=- and [7]. Hartmanis and Berman show that E (linear exponential time) contains a P-bi-immune set ([10], observed in [12]), and an application of [9] yields that for all c ? 0 there exists a DTIME(2 cn ... |

7 |
Measure and Category, Graduate Texts
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- 1980
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3 |
A Note on 1-Truth-Table Hard Languages
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- 1993
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Citation Context ... using a certain two-person combinatorial game is intractable because it issp m -complete for E. We want to know whether completeness is a typical property in E. We studysp m -completeness, that from =-=[11]-=- and [4] is exactly the same assp 1\Gammatt -completeness. Corollary 23. The class ofsp m -complete languages for E has measure 0 in E. The class ofsp m -complete languages for NE has measure 0 in E 2... |

2 |
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Citation Context ...or its complement is in C. The notion of immunity was first introduced by Post [18] in recursive function theory. Flajolet and Steyaert transformed it into the complexity theoretic setting in [6] and =-=[7]-=-. Hartmanis and Berman show that E (linear exponential time) contains a P-bi-immune set ([10], observed in [12]), and an application of [9] yields that for all c ? 0 there exists a DTIME(2 cn )-bi-imm... |

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Relativizations comparing NP and exponential time
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- 1983
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Citation Context ...o almost every language in E 2 is E-bi-immune. The existence of P-bi-immune sets inside NP has been proven in certain relativizations. (See for instance the oracle constructed by Gasarch and Homer in =-=[8]-=-.) We obtain here a sufficient condition for the existence of P-bi-immune sets in NP: if NP does not have measure 0 in E, then NP contains a P-bi-immune set. Lutz has proposed investigation of the hyp... |