Proving SAT does not have small circuits with an application to the two queries problem

Abstract

We show that if SAT does not have small circuits, then there must exist a small number of satisfiable formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if P NP[1] = P NP [2] , then the polynomial-time hierarchy collapses to S p 2 ⊆ Σ p 2 ∩ Π p 2 . Even showing that the hierarchy collapsed to Σ p 2 remained open prior to this paper.

Amplifying ZPP^SAT[1] and the Two Queries Problem

“…He extended the result of Fortnow, Pavan and Sengupta [11] and showed that ZPP SAT [1] = ZPP SAT [2] =⇒ PH ⊆ S P 2 .…”

“…Buhrman and Fortnow [2] improved this technique and made it work for queries to a Σ P 2 oracle. Fortnow, Pavan and Sengupta [11] then showed that P SAT [1] = P SAT [2] =⇒ PH ⊆ S P 2 † Address: Department of Computer Science and Electrical Engineering, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA. Email: chang@umbc.edu, suresh1@umbc.edu.…”

Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds

“…So how does the algorithm of Bshouty, Cleve, Gavaldà, Kannan, and Tamon (1996) behave when SAT is not even in the hypothesis class SIZE(n k+3 )? Fortnow, Pavan, and Sengupta (2008) observed that in this case the algorithm outputs a poly(n)-long list of SAT instances such that every circuit of size n k fails to compute correctly the SAT-value of at least one of them.…”

“…Overview. Our starting point is a result by Fortnow, Pavan, and Sengupta (2008), which essentially states that if SAT does not have size-n k+3 circuits, then there is a short list of counterexamples for every size-n k circuit. More precisely, their result is that if SAT is hard for size-n k+3 circuits, then there is a list of satisfiable formulae L = (φ 1 , .…”

“…In particular, if the algorithm runs in deterministic polynomial-time with access to an NP oracle, then there is a function in E NP that requires exponentially large circuits. Next, in section Section 4.3 we show (in Theorem 4.5), based on ideas from Bshouty, Cleve, Gavaldà, Kannan, and Tamon (1996); Chakaravarthy and Roy (2008) ;Fortnow, Pavan, and Sengupta (2008), that there is a deterministic algorithm that uses an oracle to prAM for learning counterexamples (recall that the result of Fortnow, Pavan, and Sengupta (2008) only gives a randomized algorithm, so it is not good for us). By the hypothesis of Theorem 4.1, we can replace the prAM oracle with an NP oracle and then the lower bound follows in Section 4.4.…”