Abstract

We use finite model theory to prove:Let m ≥ 2. Then: If there exists k such that NP ⊆ σmTIME ∩ ΠmTIME, then for every r there exists kr such that PNP[nr] ⊆ σmTIME ∩ ΠmTIME; If there exists a superpolynomial time-constructible function f such that NTIME ⊆ 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.