A separation theorem in property testing

Alon, Noga; Shapira, A.

doi:10.1007/s00493-008-2321-1

Cited by 24 publications

(33 citation statements)

References 29 publications

(40 reference statements)

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“…This results in uniform (where the ε-testers are computable given ε, or equivalently, where there is a single tester that also takes ε as input) and nonuniform versions of testability. Although testable properties in the literature are usually uniformly testable, see Alon and Shapira [7] for a property that is testable only with uncomputable c(ε) and therefore only non-uniformly. Our results hold in both cases 8 and so we will not distinguish between them.…”

Section: Property Testing Definitionsmentioning

confidence: 99%

Testable and untestable classes of first-order formulae

Jordan

Zeugmann

2012

Journal of Computer and System Sciences

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In property testing, the goal is to distinguish structures that have some desired property from those that are far from having the property, based on only a small, random sample of the structure. We focus on the classification of first-order sentences according to their testability. This classification was initiated by Alon et al. [2], who showed that graph properties expressible with prefix ∃ * ∀ * are testable but that there is an untestable graph property expressible with quantifier prefix ∀ * ∃ * . The main results of the present paper are as follows. We prove that all (relational) properties expressible with quantifier prefix ∃ * ∀∃ * (Ackermann's class with equality) are testable and also extend the positive result of Alon et al.[2] to relational structures using a recent result by Austin and Tao [8]. Finally, we simplify the untestable property of Alon et al. [2] and show that prefixes ∀ 3 ∃, ∀ 2 ∃∀, ∀∃∀ 2 and ∀∃∀∃ can express untestable graph properties when equality is allowed.

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Section: Property Testing Definitionsmentioning

confidence: 99%

Testable and untestable classes of first-order formulae

Jordan

Zeugmann

2012

Journal of Computer and System Sciences

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“…6 As pointed out in [10], the statement of [41, Thm 2] should be corrected such that the auxiliary property Π ′ may depend on N and not only on Π. Thus, on input N and ǫ (and oracle access to an N -vertex graph G), the canonical tester checks whether a random induced subgraph of size s = O(q(N, ǫ)) has the property Π ′ , where Π ′ itself (or rather its intersection with the set of s-vertex graphs) may depend on N .…”

Section: Theorem 22 (Canonical Testers [41 Thm 2])mentioning

confidence: 91%

“…147 result shown in [10]). In contrast, negative results typically refer to a fixed value of the distance parameter.…”

Section: Organizationmentioning

confidence: 94%

“…4 That is, we refer to the standard (uniform) model of computation (cf., e.g., [31, Sec. 1.2.3]), which does not allow for hard-wiring some parameters (e.g., input length) into the computing device (as done in the case of non-uniform circuit families).147 result shown in [10]). In contrast, negative results typically refer to a fixed value of the distance parameter.…”

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confidence: 99%

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Introduction to Testing Graph Properties

Goldreich

2010

Property Testing

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Abstract. The aim of this article is to introduce the reader to the study of testing graph properties, while focusing on the main models and issues involved. No attempt is made to provide a comprehensive survey of this study, and specific results are often mentioned merely as illustrations of general themes.Keywords: Graph Properties, randomized algorithms, approximation problems.This survey was originally written for inclusion in [32]. The General ContextIn general, property testing is concerned with super-fast (probabilistic) algorithms for deciding whether a given object has a predetermined property or is far from any object having this property. Such algorithms, called testers, obtain local views of the object by making adequate queries; that is, the object is seen as a function and the tester gets oracle access to this function, and thus may be expected to work in time that is sub-linear in the size of the object.Looking at the foregoing formulation, we first note that property testing is concerned with promise problems (cf. [26,30]), rather than with standard decision problems. Specifically, objects that neither have the property nor are far from having the property are discarded. The exact formulation of these promise problems refers to a distance measure defined on the set of all relevant objects (i.e., this distance measure coupled with a distance parameter determine the set of objects that are far from the property). Thus, the choice of natural distance measures is crucial to the study of property testing. Secondly, we note that the requirement that the algorithms operate in sub-linear time (i.e., without reading their entire input) calls for a specification of the type of queries that these algorithms can make to their input. Thus, the choice of natural query types is also crucial to the study of property testing. These two general considerations will become concrete once we delve into the actual subject matter (i.e., testing graph properties). Why Graphs?Let us start with an empirical observation, taken from Shimon Even's book Graph Algorithms [25] (published in 1979): 142Graph theory has long become recognized as one of the more useful mathematical subjects for the computer science student to master. The approach which is natural in computer science is the algorithmic one; our interest is not so much in existence proofs or enumeration techniques, as it is in finding efficient algorithms for solving relevant problems, or alternatively showing evidence that no such algorithms exist. Although algorithmic graph theory was started by Euler, if not earlier, its development in the last ten years has been dramatic and revolutionary.Meditating on these facts, one may ask what is the source of this ubiquitous use of graphs in computer science. The most common answer is that graphs arise naturally as a model (or an abstraction) of numerous natural and artificial objects. Another answer is that graphs help visualize binary relations over finite sets. These two different answers correspond to two types of models...

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“…The uniform case is strictly more difficult (see, e.g., Alon and Shapira [3]). We are interested in proving untestability, and our results hold even in the non-uniform case.…”

Section: Preliminariesmentioning

confidence: 99%

Untestable Properties Expressible with Four First-Order Quantifiers

Jordan

Zeugmann

2010

Language and Automata Theory and Applications

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Abstract. In property testing, the goal is to distinguish between structures that have some desired property and those that are far from having the property, after examining only a small, random sample of the structure. We focus on the classification of first-order sentences based on their quantifier prefixes and vocabulary into testable and untestable classes. This classification was initiated by Alon et al.[1], who showed that graph properties expressible with quantifier patterns ∃ * ∀ * are testable but that there is an untestable graph property expressible with quantifier pattern ∀ * ∃ * . In the present paper, their untestable example is simplified. In particular, it is shown that there is an untestable graph property expressible with each of the following quantifier patterns: ∀∃∀∃, ∀∃∀ 2 , ∀ 2 ∃∀ and ∀ 3 ∃.

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A separation theorem in property testing

Cited by 24 publications

References 29 publications

Testable and untestable classes of first-order formulae

Testable and untestable classes of first-order formulae

Introduction to Testing Graph Properties

Untestable Properties Expressible with Four First-Order Quantifiers

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