657 research outputs found
On the Power of Many One-Bit Provers
We study the class of languages, denoted by \MIP[k, 1-\epsilon, s], which
have -prover games where each prover just sends a \emph{single} bit, with
completeness and soundness error . For the case that
(i.e., for the case of interactive proofs), Goldreich, Vadhan and Wigderson
({\em Computational Complexity'02}) demonstrate that \SZK exactly
characterizes languages having 1-bit proof systems with"non-trivial" soundness
(i.e., ). We demonstrate that for the case that
, 1-bit -prover games exhibit a significantly richer structure:
+ (Folklore) When , \MIP[k, 1-\epsilon, s]
= \BPP;
+ When , \MIP[k,
1-\epsilon, s] = \SZK;
+ When , \AM \subseteq \MIP[k, 1-\epsilon,
s];
+ For and sufficiently large , \MIP[k, 1-\epsilon, s]
\subseteq \EXP;
+ For , \MIP[k, 1, 1-\epsilon, s] = \NEXP.
As such, 1-bit -prover games yield a natural "quantitative" approach to
relating complexity classes such as \BPP,\SZK,\AM, \EXP, and \NEXP.
We leave open the question of whether a more fine-grained hierarchy (between
\AM and \NEXP) can be established for the case when
Property Testing via Set-Theoretic Operations
Given two testable properties and , under
what conditions are the union, intersection or set-difference of these two
properties also testable? We initiate a systematic study of these basic
set-theoretic operations in the context of property testing. As an application,
we give a conceptually different proof that linearity is testable, albeit with
much worse query complexity. Furthermore, for the problem of testing
disjunction of linear functions, which was previously known to be one-sided
testable with a super-polynomial query complexity, we give an improved analysis
and show it has query complexity O(1/\eps^2), where \eps is the distance
parameter.Comment: Appears in ICS 201
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