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Boolean operations, joins, and the extended low hierarchy

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Abstract

We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy than either of the sets. In particular, there exist sets that are not in the second level of the extended low hierarchy, EL2, yet their join is in EL2. That is, in terms of extended lowness, the join operator can lower complexity. Since in a strong intuitive sense the join does not lower complexity, our result suggests that the extended low hierarchy is unnatural as a complexity measure. We also study the closure properties of EL2 and prove that EL2 is not closed under certain Boolean operations. To this end, we establish the first known (and optimal) EL2 lower bounds for certain notions generalizing P-selectivity, which may be regarded as an interesting result in its own right.

Keywords

Closure properties
Computational complexity
Extended low hierarchy
Information extraction algorithms
Joins

Cited by (0)

1

Supported in part by grants NSF-INT-9513368/DAAD-315-PRO-fo-ab, NSF-CCR-8957604, NSF-INT-9116781/JSPS-ENG-207, and NSF-CCR-9322513. Work done in part while visiting Friedrich-Schiller-Universität Jena and the Tokyo Institute of Technology.

2

Supported in part by a postdoctoral fellowship from the Chinese Academy of Sciences, and by grant NSF-CCR-8957604. Work done in part while visiting the University of Rochester and while at McMaster University.

3

Supported in part by a DAAD research visit grant, a NATO Postdoctoral Science Fellowship from the Deutscher Akademischer Austauschdienst (“Gemeinsames Hochschulsonderprogramm III von Bund und Ländern”), and by grants NSF-INT-9513368/DAAD-315-PRO-fo-ab, NSF-CCR-9322513, and NSF-CCR-8957604. Work done in part while visiting the University of Rochester. Current address: Department of Computer Science, University of Rochester, Rochester, NY 14627, USA.

4

Supported in part by grant NSF-INT-9116781/JSPS-ENG-207. Work done in part while visiting the University of Rochester.