The Two Queries Assumption and Arthur-Merlin Classes

Abstract

We explore the implications of the two queries assumption, $P^{SAT[1]}=P^{SAT[2]}_{||}$, with respect to the polynomial hierarchy (PH) and Arthur-Merlin classes. We prove the following results under the assumption $P^{SAT[1]}=P^{SAT[2]}_{||}$:

• 1 AM = MA

• 2 There exists no relativizable proof for PH ⊆ AM

• 3 Every problem in PH can be solved by a non-uniform variant of an Arthur-Merlin(AM) protocol where Arthur(the verifier) has access to one bit of advice.

• 4 $PH = P^{SAT[1],MA[1]}_{||}$

Under the two queries assumption, Chakaravarthy and Roy showed that PH collapses to $NO^p_2$ [5]. Since $NP \subseteq MA \subseteq NO^p_2$ unconditionally, our result on relativizability improves upon the result by Buhrman and Fortnow that we cannot show that PH ⊆ NP using relativizable proof techniques [3]. However, we show a containment of PH in a non-uniform variant of AM where Arthur has one bit of advice. This also improves upon the result by Kadin that PH ⊂ NP _/poly [11]. Our fourth result shows that simulating MA in a P ^SAT[1] machine is as hard as collapsing PH to P ^SAT[1].