## Testing Monotonicity (1999)

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Citations: | 55 - 12 self |

### BibTeX

@MISC{Goldreich99testingmonotonicity,

author = {Oded Goldreich and Shafi Goldwasser and Eric Lehman and Dana Ron and Alex Samorodnitsky},

title = {Testing Monotonicity},

year = {1999}

}

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### Abstract

We present a (randomized) test for monotonicity of Boolean functions. Namely, given the ability to query an unknown function f : f0; 1g 7! f0; 1g at arguments of its choice, the test always accepts a monotone f , and rejects f with high probability if it is ffl-far from being monotone (i.e., every monotone function differs from f on more than an ffl fraction of the domain).

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Citation Context ...ue of the function. Thus, knowing that a concept is monotone may be useful in various applications. In fact, this form of simplicity is exploited by Angluin's learning algorithm for monotone concepts =-=[4]-=-, which uses membership queries and has complexity that is linear in the number of terms in the DNF representation of the target concept. We note that an efficient tester for monotonicity is useful as... |

647 | Some optimal inapproximability results
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Citation Context ...rd tests (in this case of BCH codes), and that such tests can be defined and performed also for other error-correcting codes such as the Hadamard Code [5, 13, 14, 11, 12, 33, 37], and the "Long C=-=ode" [12, 29, 30, 37]-=-. For as much as error-correcting codes emerge naturally in the context of pcp, they do not seem to provide a natural representation of objects whose properties we may wish to investigate. That is, on... |

424 | Property testing and its connection to learning and approximation
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Citation Context ...bal measure we are interested in --- the minimum distance of the function to any monotone function. 1.1 Perspective Property Testing, as explicitly defined by Rubinfeld and Sudan [36] and extended in =-=[26]-=-, is best known by the special case of low degree testing 1 (see for example [17, 24, 36, 35, 7]), which plays a central role in the construction of probabilistically checkable proofs (pcp) [9, 8, 22,... |

403 | Non-deterministic exponential time has two-prover interactive protocols
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Citation Context ...ed in [26], is best known by the special case of low degree testing 1 (see for example [17, 24, 36, 35, 7]), which plays a central role in the construction of probabilistically checkable proofs (pcp) =-=[9, 8, 22, 6, 5, 35, 7]-=-. The recognition that property testing is a general notion has been implicit in the context of pcp: It is understood that low degree tests as used in this context are actually codeword tests (in this... |

345 | Self-testing/correcting with applications to numerical problems
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Citation Context ...monotone function. 1.1 Perspective Property Testing, as explicitly defined by Rubinfeld and Sudan [36] and extended in [26], is best known by the special case of low degree testing 1 (see for example =-=[17, 24, 36, 35, 7]-=-), which plays a central role in the construction of probabilistically checkable proofs (pcp) [9, 8, 22, 6, 5, 35, 7]. The recognition that property testing is a general notion has been implicit in th... |

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Citation Context ...al measure to the global measure we are interested in --- the minimum distance of the function to any monotone function. 1.1 Perspective Property Testing, as explicitly defined by Rubinfeld and Sudan =-=[36]-=- and extended in [26], is best known by the special case of low degree testing 1 (see for example [17, 24, 36, 35, 7]), which plays a central role in the construction of probabilistically checkable pr... |

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Citation Context ...in the class contains all instances having bn=2c+1 or more 1's, no instances having bn=2c \Gamma 1 or less 1's, and some subset of the instances having exactly bn=2c 1's. In contrast, "weak learn=-=ing" [32]-=- is possible in polynomial time. Specifically, the class of monotone concepts can be learned in polynomial time with error at most 1=2 \Gamma \Omega\Gamma1 = p n) [16] (though no polynomial-time learn... |

282 | Safra,A sub-constant error-probability low-degree test and sub-constant error-probability PCP characterization of NP
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Citation Context ...monotone function. 1.1 Perspective Property Testing, as explicitly defined by Rubinfeld and Sudan [36] and extended in [26], is best known by the special case of low degree testing 1 (see for example =-=[17, 24, 36, 35, 7]-=-), which plays a central role in the construction of probabilistically checkable proofs (pcp) [9, 8, 22, 6, 5, 35, 7]. The recognition that property testing is a general notion has been implicit in th... |

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Citation Context ...ed in [26], is best known by the special case of low degree testing 1 (see for example [17, 24, 36, 35, 7]), which plays a central role in the construction of probabilistically checkable proofs (pcp) =-=[9, 8, 22, 6, 5, 35, 7]-=-. The recognition that property testing is a general notion has been implicit in the context of pcp: It is understood that low degree tests as used in this context are actually codeword tests (in this... |

208 |
Proof verification and intractability of approximation problems
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Citation Context ...ed in [26], is best known by the special case of low degree testing 1 (see for example [17, 24, 36, 35, 7]), which plays a central role in the construction of probabilistically checkable proofs (pcp) =-=[9, 8, 22, 6, 5, 35, 7]-=-. The recognition that property testing is a general notion has been implicit in the context of pcp: It is understood that low degree tests as used in this context are actually codeword tests (in this... |

206 | Free bits, PCPs and non-approximability — towards tight results
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- 1998
(Show Context)
Citation Context ...tests as used in this context are actually codeword tests (in this case of BCH codes), and that such tests can be defined and performed also for other error-correcting codes such as the Hadamard Code =-=[5, 13, 14, 11, 12, 33, 37], and the -=-"Long Code" [12, 29, 30, 37]. For as much as error-correcting codes emerge naturally in the context of pcp, they do not seem to provide a natural representation of objects whose properties w... |

185 |
Approximating clique is almost NPcomplete
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- 1991
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Citation Context |

171 | Efficient probabilistically checkable proofs and applications to approximations
- Bellare, Goldwasser, et al.
- 1993
(Show Context)
Citation Context ...tests as used in this context are actually codeword tests (in this case of BCH codes), and that such tests can be defined and performed also for other error-correcting codes such as the Hadamard Code =-=[5, 13, 14, 11, 12, 33, 37], and the -=-"Long Code" [12, 29, 30, 37]. For as much as error-correcting codes emerge naturally in the context of pcp, they do not seem to provide a natural representation of objects whose properties w... |

168 | On the learnability of Boolean formulae
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(Show Context)
Citation Context ...gorithm for the class of all monotone concepts when the allowed error is O(1= p n): Learning (under the uniform distribution) requires \Omega\Gammau n = p n) examples (and at least that many queries) =-=[31]-=-. 3 1.5 Extensions 1.5.1 Other Domain Alphabets Let \Sigma be a finite alphabet, and ! \Sigma a (total) order on \Sigma. Then we can extend the notion of monotonicity to Boolean functions over \Sigma ... |

158 | Efficient testing of large graphs - Alon, Fischer, et al. - 2000 |

146 | Improved low-degree testing and its applications
- Arora, Sudan
(Show Context)
Citation Context ...monotone function. 1.1 Perspective Property Testing, as explicitly defined by Rubinfeld and Sudan [36] and extended in [26], is best known by the special case of low degree testing 1 (see for example =-=[17, 24, 36, 35, 7]-=-), which plays a central role in the construction of probabilistically checkable proofs (pcp) [9, 8, 22, 6, 5, 35, 7]. The recognition that property testing is a general notion has been implicit in th... |

120 | Property testing in bounded degree graphs
- Goldreich, Ron
(Show Context)
Citation Context ... the graph. The study of Property Testing as applied to natural representation of non-algebraic objects was initiated in [26]. In particular, Property Testing as applied to graphs has been studied in =-=[26, 27, 28, 2, 3, 34, 15]-=-, where graphs are either represented by their adjacency matrix (most adequate for dense graphs), or by their incidence lists (adequate for sparse graphs). In this work we consider property testing as... |

117 | Improved non-approximability results
- Bellare, Sudan
- 1994
(Show Context)
Citation Context ...tests as used in this context are actually codeword tests (in this case of BCH codes), and that such tests can be defined and performed also for other error-correcting codes such as the Hadamard Code =-=[5, 13, 14, 11, 12, 33, 37], and the -=-"Long Code" [12, 29, 30, 37]. For as much as error-correcting codes emerge naturally in the context of pcp, they do not seem to provide a natural representation of objects whose properties w... |

83 | sublinear bipartite tester for bounded degree graphs
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Citation Context ... the graph. The study of Property Testing as applied to natural representation of non-algebraic objects was initiated in [26]. In particular, Property Testing as applied to graphs has been studied in =-=[26, 27, 28, 2, 3, 34, 15]-=-, where graphs are either represented by their adjacency matrix (most adequate for dense graphs), or by their incidence lists (adequate for sparse graphs). In this work we consider property testing as... |

83 | Graph Algorithms. Computer Science - Even - 1979 |

80 | Selftesting/correcting for polynomials and for approximate functions
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Citation Context |

57 |
Probabilistic checkable proofs: A new characterization of np
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56 | Linearity testing in characteristic two
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Citation Context |

50 | Testing k-colorability
- Alon, Krivelevich
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Citation Context ... the graph. The study of Property Testing as applied to natural representation of non-algebraic objects was initiated in [26]. In particular, Property Testing as applied to graphs has been studied in =-=[26, 27, 28, 2, 3, 34, 15]-=-, where graphs are either represented by their adjacency matrix (most adequate for dense graphs), or by their incidence lists (adequate for sparse graphs). In this work we consider property testing as... |

50 |
The influence of variables in product spaces
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- 1992
(Show Context)
Citation Context ...Our main results are proved using shifting of Boolean functions (associated with subsets of f0; 1g n ). Various shifting techniques play an important role in extremal set theory (cf., [23] as well as =-=[1, 19]-=-). Shifting a Boolean function means modifying the set of inputs on which the value of the function is 1. The modification is chosen accordingly to the desired application. A typical application is fo... |

35 | On the density of sets of vectors
- Alon
(Show Context)
Citation Context ...Our main results are proved using shifting of Boolean functions (associated with subsets of f0; 1g n ). Various shifting techniques play an important role in extremal set theory (cf., [23] as well as =-=[1, 19]-=-). Shifting a Boolean function means modifying the set of inputs on which the value of the function is 1. The modification is chosen accordingly to the desired application. A typical application is fo... |

35 |
Testing of the long code and hardness for clique
- H˚astad
- 1996
(Show Context)
Citation Context ...rd tests (in this case of BCH codes), and that such tests can be defined and performed also for other error-correcting codes such as the Hadamard Code [5, 13, 14, 11, 12, 33, 37], and the "Long C=-=ode" [12, 29, 30, 37]-=-. For as much as error-correcting codes emerge naturally in the context of pcp, they do not seem to provide a natural representation of objects whose properties we may wish to investigate. That is, on... |

30 |
The shifting technique in extremal set theory
- Frankl
- 1987
(Show Context)
Citation Context ... 1.6 Techniques Our main results are proved using shifting of Boolean functions (associated with subsets of f0; 1g n ). Various shifting techniques play an important role in extremal set theory (cf., =-=[23]-=- as well as [1, 19]). Shifting a Boolean function means modifying the set of inputs on which the value of the function is 1. The modification is chosen accordingly to the desired application. A typica... |

27 | Testing the diameter of graphs
- Parnas, Ron
- 2002
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25 |
Recycling Queries in PCPs and in Linearity Tests
- Trevisan
- 1998
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Citation Context |

19 | On learning monotone boolean functions
- Blum, Burch, et al.
- 1998
(Show Context)
Citation Context ...'s. In contrast, "weak learning" [32] is possible in polynomial time. Specifically, the class of monotone concepts can be learned in polynomial time with error at most 1=2 \Gamma \Omega\Gamm=-=a1 = p n) [16]-=- (though no polynomial-time learning algorithm can achieve an error of 1=2 \Gamma !(log(n)= p n)) [16]). 4 and Raskhodnikova have devised a transformation whose dependency on the size of the range is ... |

15 | Probabilistically Checkable Proofs and the Testing of Hadamard-like Codes
- Kiwi
- 1996
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13 | Testing acyclicity of directed graphs in sublinear time
- Bender, Ron
- 2000
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9 |
Improved Testing Algorithms for Monotonocity
- Dodis, Goldreich, et al.
(Show Context)
Citation Context ...omial-time learning algorithm can achieve an error of 1=2 \Gamma !(log(n)= p n)) [16]). 4 and Raskhodnikova have devised a transformation whose dependency on the size of the range is only logarithmic =-=[20]-=-. 1.5.3 Testing Unateness A function f : f0; 1g n ! f0; 1g is said to be unate if for every i 2 f1; : : : ; ng exactly one of the following holds: whenever the i th bit is flipped from 0 to 1 then the... |

2 |
An extension to testing monotonicity
- Batu
- 1998
(Show Context)
Citation Context ...: \Sigma n ! \Xi while losing a factor of j\Xij. We note that an alternative proof for the case \Sigma = f0; 1g, which is based on a previous analysis of our testing algorithm [25], was given by Batu =-=[10]-=-. Without loss of generality, let \Xi = f0; : : : ; bg. The definition of ffl M extends in the natural way to functions f : \Sigma n ! f0; 1; :::; bg. Given a function f : \Sigma n ! f0; 1; :::; bg, w... |

1 | Private communications - Doddis, Lehman - 1998 |

1 |
Testing monotinicity. Available from http://theory.lcs.mit.edu/˜danar
- Goldreich, Goldwasser, et al.
- 1998
(Show Context)
Citation Context ...ese dependencies can be removed (or reduced). In particular, we believe that the dependence on the size of the range can be removed (for more details, see discussion in the full version of this paper =-=[18]-=-). REMOVING THE DEPENDENCE ON n. Our algorithm (even for the base case), has a polynomial dependence on the dimension of the input, n. As shown in Proposition 4, some dependence of the query complexit... |