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 m TIME( n k )∩ Π m TIME( n k ), then for every r there exists k r 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.