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December 2004 A finite model-theoretical proof of a property of bounded query classes within PH
Leszek Aleksander Kołodziejczyk
J. Symbolic Logic 69(4): 1105-1116 (December 2004). DOI: 10.2178/jsl/1102022213

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 ⊆ σmTIME(nk) ∩ ΠmTIME(nk), then for every r there exists kr such that PNP[nr] ⊆ σmTIME(nkr) ∩ ΠmTIME(nkr);

(B) If there exists a superpolynomial time-constructible function f such that NTIME(f) ⊆ Σpm ∪ Πpm, then additionally PNP[nr] ⊈ Σpm ∪ Πpm.

This strengthens a result by Mocas [M96] that for any r, PNP[nr] ⊈ NEXP.

In addition, we use FM-truth definitions to give a simple sufficient condition for the σ11 arity hierarchy to be strict over finite models.

Citation

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Leszek Aleksander Kołodziejczyk. "A finite model-theoretical proof of a property of bounded query classes within PH." J. Symbolic Logic 69 (4) 1105 - 1116, December 2004. https://doi.org/10.2178/jsl/1102022213

Information

Published: December 2004
First available in Project Euclid: 2 December 2004

zbMATH: 1081.03027
MathSciNet: MR2135657
Digital Object Identifier: 10.2178/jsl/1102022213

Rights: Copyright © 2004 Association for Symbolic Logic

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Vol.69 • No. 4 • December 2004
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