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FixedPolynomial Size Circuit Bounds
"... Abstract—In 1982, Kannan showed that Σ P 2 does not have n ksized circuits for any k. Do smaller classes also admit such circuit lower bounds? Despite several improvements of Kannan’s result, we still cannot prove that P NP does not have linear size circuits. Work of Aaronson and Wigderson provides ..."
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Abstract—In 1982, Kannan showed that Σ P 2 does not have n ksized circuits for any k. Do smaller classes also admit such circuit lower bounds? Despite several improvements of Kannan’s result, we still cannot prove that P NP does not have linear size circuits. Work of Aaronson and Wigderson provides strong evidence – the “algebrization ” barrier – that current techniques have inherent limitations in this respect. We explore questions about fixedpolynomial size circuit lower bounds around and beyond the algebrization barrier. We find several connections, including The following are equivalent: – NP is in SIZE(n k) (has O(n k)size circuit families) for some k – For each c, P NP[nc] k is in SIZE(n) for some k – ONP/1 is in SIZE(n k) for some k, 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 2size 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. I.
Electronic Colloquium on Computational Complexity, 2006. 51 Ehud Friedgut. Necessary and sufficient conditions for sharp thresholds of graph properties and the kSAT problem
 J. of the AMS
, 1999
"... We initiate a study of inputoblivious proof systems, and present a few preliminary results regarding such systems. Our results offer a perspective on the intersection of the nonuniform complexity class P/poly with uniform complexity classes such as N P and IP. In particular, we provide a uniform c ..."
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We initiate a study of inputoblivious proof systems, and present a few preliminary results regarding such systems. Our results offer a perspective on the intersection of the nonuniform complexity class P/poly with uniform complexity classes such as N P and IP. In particular, we provide a uniform complexity formulation of the conjecture N P ⊂ P/poly and a uniform complexity characterization of the class IP ∩P/poly. These (and similar) results offer a perspective on the attempt to prove circuit lower bounds for complexity classes such as N P, PSPACE,
The 1versus2 queries problem revisited
"... The 1versus2 queries problem, which has been extensively studied in computational complexity theory, asks in its generality whether every efficient algorithm that makes at most 2 queries to a Σ p k complete set Lk has an efficient simulation that makes at most 1 query to Lk. We obtain solutions t ..."
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The 1versus2 queries problem, which has been extensively studied in computational complexity theory, asks in its generality whether every efficient algorithm that makes at most 2 queries to a Σ p k complete set Lk has an efficient simulation that makes at most 1 query to Lk. We obtain solutions to this problem under hypotheses weaker than previously considered. We prove that:
A Full Characterization of Quantum Advice
"... We prove the following surprising result: given any quantum state ρ on n qubits, there exists a local Hamiltonian H on poly (n) qubits (e.g., a sum of twoqubit interactions), such that any ground state of H can be used to simulate ρ on all quantum circuits of fixed polynomial size. In terms of comp ..."
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We prove the following surprising result: given any quantum state ρ on n qubits, there exists a local Hamiltonian H on poly (n) qubits (e.g., a sum of twoqubit interactions), such that any ground state of H can be used to simulate ρ on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly ⊆ QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly ⊆ PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in power to untrusted quantum advice combined with trusted classical advice. Proving our main result requires combining a large number of previous tools—including a result of Alon et al. on learning of realvalued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on ‘QMA+ superverifiers’—and also creating some new ones. The main new tool is a socalled majoritycertificates lemma, which is related to boosting in machine learning, and which seems likely to find independent applications. In its simplest version, this lemma says the following. Given any set S of Boolean functions on n variables, any function f ∈ S can be expressed as the pointwise majority of m = O (n) functions f1,..., fm ∈ S, such that each fi is the unique function in S compatible with O (log S) input/output constraints.
A Note on the KarpLipton Collapse for the Exponential Hierarchy
, 2007
"... We extend previous collapsing results involving the exponential hierarchy by using recent hardnessrandomness tradeoff results. Specifically, we show that if the second level of the exponential hierarchy has polynomialsized circuits, then it collapses all the way down to MA. ..."
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We extend previous collapsing results involving the exponential hierarchy by using recent hardnessrandomness tradeoff results. Specifically, we show that if the second level of the exponential hierarchy has polynomialsized circuits, then it collapses all the way down to MA.
InputOblivious Proof Systems and a Uniform Complexity Perspective on P/poly
, 2011
"... An inputoblivious proof system is a proof system in which the proof does not depend on the claim being proved. Inputoblivious versions of N P and MA were introduced in passing by Fortnow, Santhanam, and Williams (CCC 2009), who also showed that those classes are related to questions on circuit com ..."
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An inputoblivious proof system is a proof system in which the proof does not depend on the claim being proved. Inputoblivious versions of N P and MA were introduced in passing by Fortnow, Santhanam, and Williams (CCC 2009), who also showed that those classes are related to questions on circuit complexity. In this note we wish to highlight the notion of inputoblivious proof systems, and initiate a more systematic study of them. We begin by describing in detail the results of Fortnow et al., and discussing their connection to circuit complexity. We then extend the study to inputoblivious versions of IP, PCP, and ZK, and present few preliminary results regarding those versions.
FINDING IRREFUTABLE CERTIFICATES FOR S p 2 via . . .
, 2008
"... We show that S p 2 ⊆ PprAM, where S p 2 is the symmetric alternation class and prAM refers to the promise version of the ArthurMerlin class AM. This is derived as a consequence of our main result that presents an FP prAM algorithm for finding a small set of “collectively irrefutable certificates ” ..."
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We show that S p 2 ⊆ PprAM, where S p 2 is the symmetric alternation class and prAM refers to the promise version of the ArthurMerlin class AM. This is derived as a consequence of our main result that presents an FP prAM algorithm for finding a small set of “collectively irrefutable certificates ” of a given S2type matrix. The main result also yields some new consequences of the hypothesis that NP has polynomial size circuits. It is known that the above hypothesis implies a collapse of the polynomial time hierarchy (PH) to S p 2 ⊆ ZPPNP [5, 14]. Under the same hypothesis, we show that PH collapses to P prMA. We also describe an FP prMA algorithm for learning polynomial size circuits for SAT, assuming such circuits exist. For the same problem, the previously best known result was a ZPP NP algorithm [4].