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A finite model-theoretical proof of a property of bounded query classes within PH

Published online by Cambridge University Press:  12 March 2014

Leszek Aleksander Kołodziejczyk
Leszek Aleksander Kołodziejczyk*
Affiliation:
Institute of Philosophy, Warsaw University, Krakowskie Przedmieście 3, 00-047 Warsaw, Poland, E-mail: 1.kolodziejczyk@zodiac.mimuw.edu.pl
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Abstract.

We use finite model theory (in particular, the method of FM-truth definitions, introduced in [MM01] and developed in [K04], and a normal form result akin to those of [Ste93] and [G97]) to prove:

Let m ≥ 2. Then:

(A) If there exists k such that NP⊆ Σm TIME(nk)∩ Πm TIME(nk), then for every r there exists kr such that :

(B) If there exists a superpolynomial time-constructible function f such that NTIME(f), then additionally .

This strengthens a result by Mocas [M96] that for any r, .

In addition, we use FM-truth definitions to give a simple sufficient condition for the arity hierarchy to be strict over finite models.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 4 , December 2004 , pp. 1105 - 1116
Copyright
Copyright © Association for Symbolic Logic 2004

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References

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