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An analysis of the W*-hierarchy

Published online by Cambridge University Press:  12 March 2014

Yijia Chen ,
Jörg Flum  and
Martin Grohe
Yijia Chen
Affiliation:
Basics, Department of Computer Science, Shanghai Jiaotong University, Shanghai 200030, China, E-mail: [email protected]
Jörg Flum
Affiliation:
Abteilung für Mathematische Logik, Universität Freiburg, Eckerstr. 1, 79104 Freiburg, Germany, E-mail: [email protected]
Martin Grohe
Affiliation:
Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany, E-mail: [email protected]
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Abstract

We observe that the W*-hierarchy, a variant (introduced by Downey, Fellows, and Taylor [7]) of the better known W-hierarchy, coincides with the W-hierarchy, though not level wise, but just as a whole hierarchy. More precisely, we prove that W[t] ⊆ W* [t] ⊆ W[2t − 2] for each t ≥ 2. It was known before that W[1] = W*[1] and W[2] = W*[2].

Our second main result is a new logical characterization of the W*-hierarchy in terms of “Fagin-definable problems.” As a by-product, we also obtain an improvement of our earlier characterization of the hierarchy in terms of model-checking problems. Furthermore, we obtain new complete problems for the classes W[3] and W*[3].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 2 , June 2007 , pp. 513 - 534
Copyright
Copyright © Association for Symbolic Logic 2007

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References

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