Weizmann Logo
Electronic Colloquium on Computational Complexity

Under the auspices of the Computational Complexity Foundation (CCF)

Login | Register | Classic Style

Reports tagged with Probabilistically Checkable Proofs:
TR98-007 | 12th January 1998
Luca Trevisan

Recycling Queries in PCPs and in Linearity Tests

We study query-efficient Probabilistically Checkable
Proofs (PCPs) and linearity tests. We focus on the number
of amortized query bits. A testing algorithm uses $q$ amortized
query bits if, for some constant $k$, it reads $qk$ bits and has
error probability at most $2^{-k}$. The best known ... more >>>

TR98-078 | 1st December 1998
Vikraman Arvind, K.V. Subrahmanyam, Vinodchandran Variyam

The Query Complexity of Program Checking by Constant-Depth Circuits

In this paper we study program checking (in the
sense of Blum and Kannan) using constant-depth circuits as
checkers. Our focus is on the number of queries made by the
checker to the program being checked and we term this as the
query complexity of the checker for the given ... more >>>

TR99-025 | 2nd July 1999
Yonatan Aumann, Johan Hastad, Michael O. Rabin, Madhu Sudan

Linear Consistency Testing

We extend the notion of linearity testing to the task of checking
linear-consistency of multiple functions. Informally, functions
are ``linear'' if their graphs form straight lines on the plane.
Two such functions are ``consistent'' if the lines have the same
slope. We propose a variant of a test of ... more >>>

TR00-062 | 25th August 2000
Venkatesan Guruswami, Johan Hastad, Madhu Sudan

Hardness of approximate hypergraph coloring

We introduce the notion of covering complexity of a probabilistic
verifier. The covering complexity of a verifier on a given input is
the minimum number of proofs needed to ``satisfy'' the verifier on
every random string, i.e., on every random string, at least one of the
given proofs must be ... more >>>

TR01-027 | 23rd March 2001
Marius Zimand

Probabilistically Checkable Proofs The Easy Way

We present a weaker variant of the PCP Theorem that admits a
significantly easier proof. In this
variant the prover only has $n^t$ time to compute each
bit of his answer, for an arbitray but fixed constant
$t$, in contrast to
being all powerful. We show that
3SAT ... more >>>

TR02-030 | 3rd June 2002
Lars Engebretsen, Jonas Holmerin, Alexander Russell

Inapproximability Results for Equations over Finite Groups

Revisions: 1

An equation over a finite group G is an expression of form
w_1 w_2...w_k = 1_G, where each w_i is a variable, an inverted
variable, or a constant from G; such an equation is satisfiable
if there is a setting of the variables to values in G ... more >>>

TR02-040 | 20th June 2002
Lars Engebretsen, Jonas Holmerin

Three-Query PCPs with Perfect Completeness over non-Boolean Domains

We study non-Boolean PCPs that have perfect completeness and read
three positions from the proof. For the case when the proof consists
of values from a domain of size d for some integer constant d
>= 2, we construct a non-adaptive PCP with perfect completeness
more >>>

TR04-021 | 23rd March 2004
Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, Salil Vadhan

Robust PCPs of Proximity, Shorter PCPs and Applications to Coding

We continue the study of the trade-off between the length of PCPs
and their query complexity, establishing the following main results
(which refer to proofs of satisfiability of circuits of size $n$):
We present PCPs of length $\exp(\tildeO(\log\log n)^2)\cdot n$
that can be verified by making $o(\log\log n)$ Boolean queries.
more >>>

TR04-060 | 22nd July 2004
Eli Ben-Sasson, Madhu Sudan

Simple PCPs with Poly-log Rate and Query Complexity

We give constructions of PCPs of length n*polylog(n) (with respect
to circuits of size n) that can be verified by making polylog(n)
queries to bits of the proof. These PCPs are not only shorter than
previous ones, but also simpler. Our (only) building blocks are
Reed-Solomon codes and the bivariate ... more >>>

TR04-065 | 28th July 2004
Luca Trevisan

Inapproximability of Combinatorial Optimization Problems

Revisions: 1

We survey results on the hardness of approximating combinatorial
optimization problems.

more >>>

TR05-019 | 9th February 2005
Venkatesan Guruswami, Atri Rudra

Tolerant Locally Testable Codes

An error-correcting code is said to be {\em locally testable} if it has an
efficient spot-checking procedure that can distinguish codewords
from strings that are far from every codeword, looking at very few
locations of the input in doing so. Locally testable codes (LTCs) have
generated ... more >>>

TR05-086 | 14th August 2005
Dana Moshkovitz, Ran Raz

Sub-Constant Error Low Degree Test of Almost Linear Size

Revisions: 1

Given a function f:F^m \rightarrow F over a finite
field F, a low degree tester tests its proximity to
an m-variate polynomial of total degree at most d
over F. The tester is usually given access to an oracle
A providing the supposed restrictions of f to
affine subspaces of ... more >>>

TR05-116 | 12th October 2005
Alex Samorodnitsky, Luca Trevisan

Gowers Uniformity, Influence of Variables, and PCPs

Gowers introduced, for d\geq 1, the notion of dimension-d uniformity U^d(f)
of a function f: G -> \C, where G is a finite abelian group and \C are the
complex numbers. Roughly speaking, if a function has small Gowers uniformity
of dimension d, then it ``looks random'' on ... more >>>

TR07-026 | 21st November 2006
Dana Moshkovitz, Ran Raz

Sub-Constant Error Probabilistically Checkable Proof of Almost Linear Size

We show a construction of a PCP with both sub-constant error and
almost-linear size. Specifically, for some constant alpha in (0,1),
we construct a PCP verifier for checking satisfiability of
Boolean formulas that on input of size n uses log n + O((log
n)^{1-alpha}) random bits to query a constant ... more >>>

TR07-115 | 19th November 2007
Or Meir

Combinatorial Construction of Locally Testable Codes

An error correcting code is said to be locally testable if there is a test that checks whether a given string is a codeword, or rather far from the code, by reading only a constant number of symbols of the string. Locally Testable Codes (LTCs) were first systematically studied by ... more >>>

TR08-071 | 6th August 2008
Dana Moshkovitz, Ran Raz

Two Query PCP with Sub-Constant Error

We show that the NP-Complete language 3Sat has a PCP
verifier that makes two queries to a proof of almost-linear size
and achieves sub-constant probability of error $o(1)$. The
verifier performs only projection tests, meaning that the answer
to the first query determines at most one accepting answer to the
more >>>

TR10-017 | 10th February 2010
Jonathan Ullman, Salil Vadhan

PCPs and the Hardness of Generating Synthetic Data

Revisions: 4

Assuming the existence of one-way functions, we show that there is no
polynomial-time, differentially private algorithm $A$ that takes a database
$D\in (\{0,1\}^d)^n$ and outputs a ``synthetic database'' $\hat{D}$ all of whose two-way
marginals are approximately equal to those of $D$. (A two-way marginal is the fraction
of database rows ... more >>>

TR10-038 | 10th March 2010
Dieter van Melkebeek, Holger Dell

Satisfiability Allows No Nontrivial Sparsification Unless The Polynomial-Time Hierarchy Collapses

Revisions: 1

Consider the following two-player communication process to decide a language $L$: The first player holds the entire input $x$ but is polynomially bounded; the second player is computationally unbounded but does not know any part of $x$; their goal is to cooperatively decide whether $x$ belongs to $L$ at small ... more >>>

TR12-052 | 27th April 2012
Mohammad Mahmoody, David Xiao

Languages with Efficient Zero-Knowledge PCPs are in SZK

A Zero-Knowledge PCP (ZK-PCP) is a randomized PCP such that the view of any (perhaps cheating) efficient verifier can be efficiently simulated up to small statistical distance. Kilian, Petrank, and Tardos (STOC '97) constructed ZK-PCPs for all languages in $NEXP$. Ishai, Mahmoody, and Sahai (TCC '12), motivated by cryptographic applications, ... more >>>

TR13-085 | 13th June 2013
Eli Ben-Sasson, Yohay Kaplan, Swastik Kopparty, Or Meir, Henning Stichtenoth

Constant rate PCPs for circuit-SAT with sublinear query complexity

The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up ... more >>>

TR13-122 | 5th September 2013
Irit Dinur, Venkatesan Guruswami

PCPs via low-degree long code and hardness for constrained hypergraph coloring

Revisions: 1

We develop new techniques to incorporate the recently proposed ``short code" (a low-degree version of the long code) into the construction and analysis of PCPs in the classical ``Label Cover + Fourier Analysis'' framework. As a result, we obtain more size-efficient PCPs that yield improved hardness results for approximating CSPs ... more >>>

TR16-046 | 23rd March 2016
Eli Ben-Sasson, Alessandro Chiesa, Ariel Gabizon, Michael Riabzev, Nicholas Spooner

Short Interactive Oracle Proofs with Constant Query Complexity, via Composition and Sumcheck

Revisions: 2

We study *interactive oracle proofs* (IOPs) (Ben-Sasson, Chiesa, Spooner '16), which combine aspects of probabilistically checkable proofs (PCPs) and interactive proofs (IPs). We present IOP constructions and general techniques that enable us to obtain tradeoffs in proof length versus query complexity that are not known to be achievable via PCPs ... more >>>

ISSN 1433-8092 | Imprint