Fixed-Polynomial Size Circuit Bounds

Abstract

Abstract-InWe explore questions about fixed-polynomial size circuit lower bounds around and beyond the algebrization barrier. We find several connections, including• The following are equivalent:where ONP is the class of languages accepted obliviously by NP machines, with witnesses for "yes" instances depending only on the input length.• For a large number of natural classes C and all k 1, C is in SIZE(n k ) if and only if C/1 ∩ P/poly is in SIZE(n k ).• If there is a d such that MATIME(n) ⊆ NTIME(n d ), then P NP does not have O(n k ) size circuits for any k > 0.• One cannot show n 2 -size circuit lower bounds for ⊕P without new nonrelativizing techniques. In particular, the proof that PP ⊆ SIZE(n k ) for all k relies on the (relativizing) result that P PP ⊆ MA =⇒ PP ⊆ SIZE(n k ), and we give an oracle relative to which P ⊕P ⊆ MA and ⊕P ⊆ SIZE(n 2 ) both hold.

A full characterization of quantum advice

“…This notion has been studied before: Chakaravarthy and Roy [16] and Fortnow and Santhanam [17] defined the complexity class ONP ('Oblivious NP'), which is like NP except that the witness can depend only on the input length. Independently, Aaronson [5] defined the complexity class YP, 6 which is easily seen to equal ONP ∩ coONP.…”

Input-Oblivious Proof Systems and a Uniform Complexity Perspective on P/poly

“…But does every set in N P ∩ P/poly have such universal NP-witnesses? Denoting the class of sets having input-oblivious NP-proofs by ON P, Fortnow et al [6] showed that Theorem 1.1 (on the power of input-oblivious NP-proofs [6]):…”

“…While the first part of this note is devoted to surveying the input-oblivious aspect of the work of Fortnow et al [6], in the second part we define and study input-oblivious versions of various forms of probabilistic pproof systems. In particular, we consider input-oblivious versions of interactive proof systems (i.e., IP), zero-knowledge proof systems (i.e., ZK), and probabilistically checkable proof systems (i.e., PCP).…”

“…That is, we ask what is the power of input-oblivious provers in each of these proof systems. We start with the case of N P, studied in [6].…”

“…In this work, following the work of Fortnow, Santhanam, and Williams [6], we ask how is the expressive power of these proof system effected when the prescribed prover is only given the length of the claim to be proved. We stress that we restrict the power of the prover being referred to in the completeness condition, but maintain the original formulation of the soundness condition.…”