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

Venue: | Theoretical Computer Science |

Citations: | 54 - 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

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19 |
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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... |

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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... |

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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... |