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Sub-Constant Error Probabilistically Checkable Proof of Almost-Linear Size

Dana Moshkovitz, Ran Raz
2009 Computational Complexity  
We show a construction of a P CP with both sub-constant error and almost-linear size.  ...  Its main ingredients are a recent low degree test with both sub-constant error and almost-linear size [12] and a new method for constructing a short list of balanced curves.  ...  Sub-Constant Error and Almost-Linear Size The question of whether there exist P CP s with both sub-constant error and almost-linear size was open.  ... 
doi:10.1007/s00037-009-0278-0 fatcat:vd5rlal3xfapjjmkahwtm5e46e

The tale of the PCP theorem

Dana Moshkovitz
2012 XRDS Crossroads The ACM Magazine for Students  
increase by more than a sub-linear factor to make the format probabilistically checkable.  ...  Overall, the reduction from "is there a proof of size n?" to the gap problem is essentially equivalent to a transformation of a proof to a probabilistically checkable format.  ... 
doi:10.1145/2090276.2090286 fatcat:aim67yzrxrhsnjfs72dkfomzae

Page 2646 of Mathematical Reviews Vol. , Issue 2001D [page]

2001 Mathematical Reviews  
(465-474 (electronic)); Ran Raz and Shmuel Safra, A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP (475- 484 (electronic)); Sanjeev Arora  ...  Yau, Eigenvalues, flows and separators of graphs (749 (electronic)); Lance Fortnow and Michael Sipser, Retraction of probabilistic computation and linear time (750 (electronic)).  ... 

Some Applications of Coding Theory in Computational Complexity [article]

Luca Trevisan
2004 arXiv   pre-print
Locally testable codes are error-correcting codes with sub-linear time error-detection algorithms, and they are the combinatorial core of probabilistically checkable proofs.  ...  Locally decodable codes are error-correcting codes with sub-linear time error-correcting algorithms.  ...  Acknowledgements Most of my understanding of coding theory comes from Madhu Sudan. Working with Salil Vadhan on various research project has clarified connections to complexity.  ... 
arXiv:cs/0409044v1 fatcat:46gkw3tgnfbz7gfkwyrzgbw4wm

Sub-Constant Error Low Degree Test of Almost-Linear Size

Dana Moshkovitz, Ran Raz
2008 SIAM journal on computing (Print)  
Previous constructions of almost-linear size testers obtained only constant error [13, 7] .  ...  We design and analyze new low degree testers that have both sub-constant error o(1) and almost-linear size n 1+o(1) (where n = |F| m ).  ...  However, at this point, these constructions achieve neither sub-constant error nor almost-linear size.  ... 
doi:10.1137/060656838 fatcat:ggfdacgvsvdolbec3vkzk2fqg4

Sub-constant error low degree test of almost-linear size

Dana Moshkovitz, Ran Raz
2006 Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06  
Previous constructions of almost-linear size testers obtained only constant error [13, 7] .  ...  We design and analyze new low degree testers that have both sub-constant error o(1) and almost-linear size n 1+o(1) (where n = |F| m ).  ...  However, at this point, these constructions achieve neither sub-constant error nor almost-linear size.  ... 
doi:10.1145/1132516.1132520 dblp:conf/stoc/MoshkovitzR06 fatcat:iew4s3or3fex7dkgigpjucztcy

Unconditional Lower Bounds against Advice [chapter]

Harry Buhrman, Lance Fortnow, Rahul Santhanam
2009 Lecture Notes in Computer Science  
We also consider the question of whether these lower bounds can be made to work on almost all input lengths rather than on infinitely many.  ...  For the probabilistic classes, no lower bounds for uniform exponential time against advice were known before.  ...  We also thank many other participants of Dagstuhl Seminar 08381 on the Computational Complexity of Discrete Problems, discussions with whom helped shape this paper.  ... 
doi:10.1007/978-3-642-02927-1_18 fatcat:t44e2glkrzd37dw4sviqs72gwy

Composition of Low-Error 2-Query PCPs Using Decodable PCPs

Irit Dinur, Prahladh Harsha
2009 2009 50th Annual IEEE Symposium on Foundations of Computer Science  
The main result of this paper is a generic composition theorem for low error two-query probabilistically checkable proofs (PCPs).  ...  In a recent breakthrough, Moshkovitz and Raz [In Proc. 49th IEEE Symp. on Foundations of Comp. Science (FOCS), 2008] constructed almost linear-sized low-error 2-query PCPs for every language in NP.  ...  Special thanks to Oded Goldreich for many helpful comments on an earlier version of this manuscript.  ... 
doi:10.1109/focs.2009.8 dblp:conf/focs/DinurH09 fatcat:4ymy3vyienguzmn4wgwobw3j4e

The approximability of NP-hard problems

Sanjeev Arora
1998 Proceedings of the thirtieth annual ACM symposium on Theory of computing - STOC '98  
in size-such that a probabilistic checker can verify their correctness by examining only a constant number of bits in them.  ...  ] and PH [92] , probabilistically checkable codes, zero-knowledge proofs, checkable VLSI computations, etc.  ... 
doi:10.1145/276698.276784 dblp:conf/stoc/Arora98 fatcat:yu5xnmrhsbd2xk7m6aanrb5ule

Probabilistic Proof Systems: A Primer

Oded Goldreich
2007 Foundations and Trends® in Theoretical Computer Science  
Probabilistically Checkable Proof Systems Probabilistically checkable proof systems can be viewed as standard (deterministic) proof systems that are augmented with a probabilistic procedure capable of  ...  In this primer, we shall survey three types of probabilistic proof systems, called interactive proofs, zero-knowledge proofs, and probabilistic checkable proofs.  ... 
doi:10.1561/0400000023 fatcat:pcglbvm2xjfxnjm3w2p5ljzdem

Probabilistic Checking Against Non-Signaling Strategies from Linearity Testing

Alessandro Chiesa, Peter Manohar, Igor Shinkar, Michael Wagner
2018 Innovations in Theoretical Computer Science  
A central tool for these applications is probabilistically checkable proofs (PCPs) that are sound against non-signaling strategies.  ...  Our result offers a new length-vs-query tradeoff when compared to the non-signaling PCP of Kalai, Raz, and Rothblum (STOC 2013 and 2014) and, moreover, may serve as an intermediate step to a proof of a  ...  sub-constant soundness error of 2 −O(log S) .  ... 
doi:10.4230/lipics.itcs.2019.25 dblp:conf/innovations/ChiesaMS19 fatcat:7io7utcuhbb4pd6awocpevulqy

Composition of Low-Error 2-Query PCPs Using Decodable PCPs

Irit Dinur, Prahladh Harsha
2013 SIAM journal on computing (Print)  
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 wellunderstood only in the constant error regime.  ...  Special thanks to Oded Goldreich for many helpful comments on an earlier version of this manuscript.  ... 
doi:10.1137/100788161 fatcat:3k2qhgnkvrahri42ynnykctyle

Composition of Low-Error 2-Query PCPs Using Decodable PCPs [chapter]

Irit Dinur, Prahladh Harsha
2010 Lecture Notes in Computer Science  
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 wellunderstood only in the constant error regime.  ...  Special thanks to Oded Goldreich for many helpful comments on an earlier version of this manuscript.  ... 
doi:10.1007/978-3-642-16367-8_21 fatcat:57tg7catbndm7dyeaelrfgee54

Hard properties with (very) short PCPPs and their applications [article]

Omri Ben-Eliezer, Eldar Fischer, Amit Levi, Ron D. Rothblum
2019 arXiv   pre-print
We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear.  ...  proof size is O(n·log^(ℓ)n), where log^(ℓ) denotes the ℓ times iterated log function (e.g., log^(2)n = loglog n).  ...  Introduction Probabilistically checkable proofs (PCPs) are one of the landmark achievements in theoretical computer science.  ... 
arXiv:1909.03255v2 fatcat:hfkyuaaeu5bjdhleu2w6bf2r7a

Improved 3LIN Hardness via Linear Label Cover

Prahladh Harsha, Subhash Khot, Euiwoong Lee, Devanathan Thiruvenkatachari, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization  
We prove that for every constant c and ε = (log n) −c , there is no polynomial time algorithm that when given an instance of 3-LIN with n variables where an (1 − ε)-fraction of the clauses are satisfiable  ...  , finds an assignment that satisfies atleast ( 1 2 + ε)-fraction of clauses unless NP ⊆ BPP.  ...  Later in a celebrated paper, Moshkovitz and Raz [12] gave an improved hardness of Label Cover that achieves sub-constant error under polynomial time reductions.  ... 
doi:10.4230/lipics.approx-random.2019.9 dblp:conf/approx/HarshaKLT19 fatcat:vjy5cuwaqbambgcza7ev3gjlu4
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