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We continue the study of the trade-off between the length of PCP sand their query complexity, establishing the following main results(which refer to proofs of satisfiability of circuits of size <i>n</i>): 1 We present PCPs of length exp(&#213;(log log <i>n</i>)<sup>2</sup>)&#8226;<i>n</i> that can be verified by making <i>o</i>(log log<i>n</i>) Boolean(More)
The main result of this paper is a generic composition theorem for low error two-query probabilistically checkable proofs (PCPs). Prior to this work, composition of PCPs was well-understood only in the constant error regime. Existing composition methods in the low error regime were non-modular (i.e., very much tailored to the specific PCPs that were being(More)
We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka, the"short code" of Barak <i>et. al</i>. [FOCS 2012]) and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate this code for inapproximability results. In particular, we prove quasi-NP-hardness of the following problems on(More)
For a boolean formula &#966; on n variables, the associated property P<inf>&#966;</inf> is the collection of n-bit strings that satisfy &#966;. We prove that there are 3CNF properties that require a linear number of queries, even for adaptive tests. This contrasts with 2CNF properties that are testable with O(&#8730;n) queries[7]. Notice that for every bad(More)
Let <i>X</i> and <i>Y</i> be finite nonempty sets and <i>(X</i>,<i>Y</i>) a pair of random variables taking values in <i>X</i>?<i>Y</i>. We consider communication protocols between two parties, <b>Alice</b> and <b>Bob</b>, for generating <i>X</i> and <i>Y</i>. <b>Alice</b> is provided an <i>x</i> ? <i>X</i> generated according to the distribution of(More)
We show that every language in NP has a probabilistically checkable proof of proximity (i.e., proofs asserting that an instance is "close" to a member of the language), where the verifier's running time is polylogarithmic in the input size and the length of the probabilistically checkable proof is only polylogarithmically larger that the length of the(More)
Graphical models provide a convenient representation for a broad class of probability distributions. Due to their powerful and sophisticated modeling capabilities, such models have found numerous applications in machine learning and other areas. In this paper we consider the complexity of commonly encountered tasks involving graphical models such as the(More)
Most known constructions of probabilistically checkable proofs (PCPs) either blow up the proof size by a large polynomial, or have a high (though constant) query complexity. In this paper we give a transformation with slightly-super-cubic blowup in proof size, with a low query complexity. Specifically, the verifier probes the proof in 16 bits and rejects(More)
With distributed computing beginning to play a major role in modern Computer Science, the theory of grammar systems and distributed automata has been developed in order to model distributed computing. In this paper, we introduce the notion of distributed automata in the sequential sense. Distributed Automata are a group of automata working in unison to(More)
We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of degree-d polynomial threshold functions (PTFs). These bounds hold both for PTFs over the Boolean hypercube {-1,1}<sup>n</sup> and for PTFs over R<sup>n</sup> under the standard n-dimensional Gaussian distribution N(0,I<sub>n</sub>). Our bound on the Boolean(More)