## Some 3CNF properties are hard to test (2003)

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Venue: | In Proc. 35th ACM Symp. on Theory of Computing |

Citations: | 56 - 11 self |

### BibTeX

@INPROCEEDINGS{Ben-sasson03some3cnf,

author = {Eli Ben-sasson and Prahladh Harsha and Sofya Raskhodnikova},

title = {Some 3CNF properties are hard to test},

booktitle = {In Proc. 35th ACM Symp. on Theory of Computing},

year = {2003},

pages = {345--354}

}

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### Abstract

Abstract. For a Boolean formula ϕ on n variables, the associated property Pϕ is the collection of n-bit strings that satisfy ϕ. We study the query complexity of tests that distinguish (with high probability) between strings in Pϕ and strings that are far from Pϕ in Hamming distance. We prove that there are 3CNF formulae (with O(n) clauses) such that testing for the associated property requires Ω(n) queries, even with adaptive tests. This contrasts with 2CNF formulae, whose associated properties are always testable with O ( √ n) queries [E. Fischer et al., Monotonicity testing over general poset domains, in Proceedings of the 34th Annual ACM Symposium on Theory of Computing, ACM, New York, 2002, pp. 474–483]. Notice that for every negative instance (i.e., an assignment that does not satisfy ϕ) there are three bit queries that witness this fact. Nevertheless, finding such a short witness requires reading a constant fraction of the input, even when the input is very far from satisfying the formula that is associated with the property. A property is linear if its elements form a linear space. We provide sufficient conditions for linear properties to be hard to test, and in the course of the proof include the following observations which are of independent interest: 1. In the context of testing for linear properties, adaptive two-sided error tests have no more power than nonadaptive one-sided error tests. Moreover, without loss of generality, any test for a linear property is a linear test. A linear test verifies that a portion of the input satisfies a set of linear constraints, which define the property, and rejects if and only if it finds a falsified constraint. A linear test is by definition nonadaptive and, when applied to linear properties, has a one-sided error. 2. Random low density parity check codes (which are known to have linear distance and constant rate) are not locally testable. In fact, testing such a code of length n requires Ω(n) queries.

### Citations

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(Show Context)
Citation Context ...ts μ>0,δ <μ c , and any integer d> 2μc2 (μ c −δ) 2 , a random (c, d)-regular graph is with high probability a (δ, μ)-right odd expander. Proof. In the proof, we make use of the following theorem (see =-=[MR95]-=-). Theorem 6.7 (Azuma’s inequality). If X0,...,Xt is a martingale sequence such that |Xi − Xi+1| ≤1 for all i, then Pr[|Xt − X0| ≥λ √ t] ≤ 2e −λ2 /2 . Fix T ⊆ R |T | = t ≥ μm. Let X = |N odd (T )|. We... |

467 |
Low Density Parity Check Codes
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(Show Context)
Citation Context ...elected variables, and each variable appears in a constant number of constraints. Such linear spaces are called random low density parity check (LDPC) codes. 5 These codes were introduced by Gallager =-=[Gal63]-=-, who showed that they have constant rate and (with high probability) large minimal distance. It is possible to show that with high probability the random constraints are linearly independent. Our bad... |

431 | Property testing and its connection to learning and approximation
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- 1998
(Show Context)
Citation Context ..., Massachusetts Institute of Technology, Cambridge, MA 02139. 1s2 E. BEN-SASSON, P. HARSHA, AND S. RASKHODNIKOVA applied to combinatorial objects, especially graphs, by Goldreich, Goldwasser, and Ron =-=[GGR98]-=-. Property testing has recently become quite an active research area; see [Ron01, Fis01] for surveys on the topic. One of the important problems in property testing is characterizing properties that c... |

349 | Self-testing/correcting with applications to numerical problems
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- 1993
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Citation Context ...n nonadaptive and have only a one-sided error (members of V are always accepted). Since the inception of property testing, linear properties have been invariably tested by linear tests (starting with =-=[BLR93]-=-). The following theorem shows this is not a coincidence. Theorem 3.3 (linear properties have linear tests). If a linear property V ⊆ F n over a finite field F has a two-sided error adaptive (ε, η+,η−... |

332 | Robust characterization of polynomials with applications to program testing
- Rubinfeld, Sudan
- 1996
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Citation Context ... a property, or is far from it. “Far” usually means that many characters of the input have to be modified to obtain an element in the set. Property testing was first formulated by Rubinfeld and Sudan =-=[RS96]-=- in the context of linear functions and was ∗Received by the editors July 30, 2004; accepted for publication (in revised form) March 17, 2005; published electronically September 8, 2005. A preliminary... |

215 | Many hard examples for resolution - Chvátal, Szemerédi - 1988 |

161 | Efficient testing of large graphs - Alon, Fischer, et al. |

133 | The art of uninformed decisions: A primer to property testing. The Computational Complexity Column of The Bulletin of the European Association for Theoretical Computer - Fischer |

122 | Property testing in bounded degree graphs
- Goldreich, Ron
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Citation Context ...a constant fraction of the input, even if the assignment is far from any satisfying one. A similar phenomenon is exhibited by Goldreich and Ron for testing bipartiteness in 3-regular, n-vertex graphs =-=[GR02]-=-. They showed a lower bound of Ω( √ n) on the query complexity; despite this, short witnesses of nonbipartiteness do exist in the form of odd cycles of length poly(log n). Our result strengthens this ... |

80 | Regular languages are testable with a constant number of queries
- Alon, Krivelevich, et al.
(Show Context)
Citation Context ...ber of queries to the input. A series of works identified classes of properties testable with constant query complexity. Goldreich, Goldwasser, and Ron [GGR98] found many such properties. Alon et al. =-=[AKNS01]-=- proved that all regular languages are testable with constant complexity. Newman [New02] extended their result to properties that can be computed by oblivious read-once constant-width branching progra... |

79 | Robust PCPs of proximity, shorter PCPs, and applications to coding - Ben-Sasson, Goldreich, et al. |

74 | Three theorems regarding testing graph properties. Random Struct
- Goldreich, Trevisan
(Show Context)
Citation Context ...neralize to read-twice branching programs. Several papers [AFKS00, Fis05] worked on the logical characterization of graph properties testable with a constant number of queries. Goldreich and Trevisan =-=[GT03]-=- provide a characterization of properties testable with a constant number of queries and one-sided error in the framework of graph partition properties. Linear lower bounds. The published linear lower... |

68 | Locally testable codes and PCPs of almost linear length
- Goldreich, Sudan
- 2002
(Show Context)
Citation Context ...ically checkable proof (PCP) constructions and are of fundamental importance in theoretical computer science. Recently, Ben-Sasson et al. [BSVW03, BGH + 04], following the work of Goldreich and Sudan =-=[GS02]-=-, proved the existence of such codes, which achieve linear distance and near constant rate, resulting in better PCP constructions. As mentioned earlier, the vector spaces we use (which are hard to tes... |

51 | Monotonicity testing over general poset domains - Fischer, Lehman, et al. |

43 | Randomness-efficient low degree tests and short PCPs via epsilon-biased sets - Ben-Sasson, Sudan, et al. - 2003 |

43 | Property testing (a tutorial). In: Handbook of Randomized Computing - Ron - 2001 |

29 | Computationally efficient error-correcting codes and holographic proofs - Spielman - 1995 |

28 | A lower bound for testing 3-colorability in bounded-degree graphs
- Bogdanov, Obata, et al.
(Show Context)
Citation Context ...and Ron [GGR98], later extended by Goldreich and Trevisan [GT03] to monotone graph properties in NP, and the bound for testing 3-coloring in bounded degree graphs due to Bogdanov, Obata, and Trevisan =-=[BOT02]-=-. In addition, there is a simple and elegant unpublished linear lower bound, observed by Madhu Sudan in a personal communication to the authors. His property consists of polynomials of degree at most ... |

26 | Functions that have readtwice constant width branching programs are not necessarily testable. Random Structures and Algorithms
- Fischer, Newman, et al.
(Show Context)
Citation Context ...r languages are testable with constant complexity. Newman [New02] extended their result to properties that can be computed by oblivious read-once constant-width branching programs. Fischer and Newman =-=[FN04]-=- demonstrated a property that requires superconstant query complexity and is computable by a read-twice constant-width branching program, thus showing that Newman’s result does not generalize to read-... |

21 | Testing membership in languages that have small width branching programs
- Newman
(Show Context)
Citation Context ...with constant query complexity. Goldreich, Goldwasser, and Ron [GGR98] found many such properties. Alon et al. [AKNS01] proved that all regular languages are testable with constant complexity. Newman =-=[New02]-=- extended their result to properties that can be computed by oblivious read-once constant-width branching programs. Fischer and Newman [FN04] demonstrated a property that requires superconstant query ... |

15 | Testing satisfiability
- Alon, Shapira
(Show Context)
Citation Context ... in our setting the input is an assignment to a fixed kCNF formula.) The exact version of this problem is a classical NP-complete problem. The property testing version was studied by Alon and Shapira =-=[AS03]-=-. They showed that satisfiability of kCNF formulae is testable with complexity independent of the input size. 8 In contrast, our problem is very easy in its exact version but hard in its property test... |

5 | Testing graphs for colorability properties. Random Struct. Algorithms 26(3 - Fischer - 2005 |