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
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
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