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

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 ⊆ σ_mTIME(n^k) ∩ Π_mTIME(n^k), then for every r there exists k_r such that P^NP[nr] ⊆ σ_mTIME(n^k_r) ∩ Π_mTIME(n^k_r);

(B) If there exists a superpolynomial time-constructible function f such that NTIME(f) ⊆ Σ^p_m ∪ Π^p_m, then additionally P^NP[nr] ⊈ Σ^p_m ∪ Π^p_m.

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